question bank with sample papers - don …...2 ch-1: rational numbers rational number-any number...

DON BOSCO SCHOOL , ALAKNANDA

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QUESTION BANK WITH

SAMPLE PAPERS CLASS 8

“The study of mathematics, like the Nile, begins in

minuteness but ends in magnificence”. Charles Caleb

Colton

1

PREFACE

“Mathematics, rightly viewed, possesses not only truth, but supreme

beauty, sublimely pure, and capable of a stern perfection such as only

the greatest art can show.”― Bertrand Russell,

This Mathematics booklet has been prepared with a vision that

.Children learn to enjoy mathematics rather than fear it.

• Children learn important mathematics • Children see mathematics as something to talk about, to communicate through, to discuss among themselves, to work together on. • Children pose and solve meaningful problems. • Children use abstractions to perceive relationships, to see structures, to reason out things, to argue the truth or falsity of statements. • Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and the basic content areas of school Mathematics. • Engage every child in class with the conviction that everyone can learn mathematics. This booklet has a simplified introduction to every chapter and the

questions are arranged in progression of difficulty starting with MCQ followed by Basics and Application.

Sample papers have been added in the end so that the student is

aware of the pattern of the question papers and can prepare accordingly

We sincerely hope that this will benefit all the students in learning, practicing and testing themselves enabling them to score well in the Exam.

2

Ch-1: Rational Numbers

Rational Number- Any Number that can be expressed in the form p/q , where p and q are

integers and q ≠ 0, is known as rational number. The collection or group of rational numbers

is denoted by Q.

Properties of a Rational Number

Closure- Rational numbers are closed under addition, subtraction and multiplication. For

eg.- If p and q are any two rational numbers, then and the sum, difference and product of

these rational numbers is also a rational number. This is known as the closure law

Commutativity- Rational numbers are commutative under addition and multiplication. If p

and q are two rational numbers, then:

Commutative law under addition says- p + q = q + p.

Commutative law under multiplication says p x q = q x p.

Note- Rational numbers, integers and whole numbers are commutative under addition and

multiplication. Rational numbers, integers and whole numbers are non commutative under

subtraction and division.

Associativity- Rational numbers are associative under addition and multiplication. If a, b, c

are rational numbers, then:

Associative property under addition: p + (q + r) = (p + q) + r

Associative property under multiplication: p(qr) = (pq)r

Role of zero and one- 0 is the additive identity for rational numbers. 1 is the multiplicative

identity for rational numbers.

Multiplicative inverse- If the product of two rational numbers is 1, then they are called

multiplicative inverse of each other. Eg. 4/9 * 9/4 = 1

MCQ

1. What should be added to -5/4 to get -1?

I. -1/4

II. 1/4

III. 1

IV. -3/4

2. What should be subtracted from -5/4 to get -1?

I. -1/4

II. 1/4

III. 1

IV. -3/4

3. Which of the following is the identity element?

I. 1

II. -1

III. 0

IV. None of these

4. Which of the following is the Multiplicative identity for rational numbers?

I. 1

II. -1

III. 0

3

IV. None of these

5. Which of the following is neither appositive nor a negative rational number?

I. 1

II. 0

III. Such a rational number doesn‟t exist

IV. None of these

6. Which of the following lies between 0 and -1?

I. 0

II. -3

III. -2/3

IV. 4/3

7. Which of the following is the reciprocal of a

I. 1

II. a

III. 1/a

IV. -1/a

8. Which of the following is the product of 7/8 and -4/21?

I. -1/6

II. 1/12

III. -16/63

IV. -147/16

9. Which of the following is the product of (-7/8) and 4/21?

I. -1/6

II. 12

III. -63/16

IV. -16/147

10. Which of the following is the reciprocal of the reciprocal of a rational number?

I. -1

II. 1

III. 0

IV. The number itself

SET-1

1. The multiplicative inverse of ¾ is ______________

2. ____ is the identity for the addition of rational numbers.

3. What is the multiplicative identity for rational numbers.

4. What is the additive inverse of 3/5?

5. Which rational number does not have a reciprocal?

6. Write.

(i) The rational numbers those are equal to their reciprocals.

(ii) The rational number that is equal to its negative.

7. Give a rational number which when added to it gives the same number.

8. By what rational number should 22/7 be divided, to get the number - 11/24?

4

9. Represent the following rational numbers on the number line. (i) 3/10 (ii) 8/7 (iii) 1.345

(iv)21/7

10. If you subtract 1/8 from a number and multiply the result by 1/4, you get 1/16. What is

the number?

11. Which of the following can be expressed as terminating or non - terminating?

(a) 1/3 (b) -14/15 (c) 1/2

12. Find two rational numbers between:

(i) -3 and 3.

(ii) 0 and 1.

13. Insert six rational numbers between:

(i) -1/4 and -2/5

(ii) 21/12 and 12/21

14. Find the sum of 1/8 and 2/9.

SET-2

1. Fill in the blanks:

(i) The product of a number and its reciprocal is _________.

(ii) The rational number _________ has no reciprocal.

(iii) The reciprocal of the reciprocal of a number is _________.

(iv) The rational number _________ is neither positive nor negative.

(v) _________ is the only rational number which is equal its additive inverse.

2. Write:

(i) A rational number which has no reciprocal.

(ii) A rational number whose product with a given rational number is equal to the given

rational number.

(iii) A rational number which is equal to its reciprocal.

3. Verify that:

(i) and are there same

(ii)

4. Find:

5. Find:

6. Find three rational numbers between and

7. Find five rational numbers between and

8. Find 10 rational numbers between and

9. Write the rational number represented by the points A, B, and C on the following number

line:

10. The product of two rational numbers is If one of them is then find the other

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MISCELLANEOUS

1. Express each of the rational numbers in the standard form:

(i) 30

−72 (ii)

95

105 (iii)

− 54

− 81

2. Express −2

3 as a rational number with numerator 4.

3. Express −2

3 as a rational number with denominator 9.

4. What is a rational number? ( write the definition)

5. Represent −4

5 on the number line.

6. Arrange the following rational numbers in ascending order:

−7

4 ,

5

6 ,

7

−12

7. Represent 1

2 𝑎𝑛𝑑

2

3 on the same number line.

8. What should be added to −7

8 to get

−3

2 ?

9. What should be subtracted from 5

7 to get

1

21 ?

10. The sum of two numbers is −6

11 . if one of the numbers is

−64

77, find the other.

11 .Find: 3

5+

−2

3 +

−11

5 +

4

3

12 .Subtract the sum of −5

6 𝑎𝑛𝑑

4

5 from the sum of

−3

5 𝑎𝑛𝑑

7

15

13. Simplify: 7

2 𝑋

5

3+

1

6 𝑋

3

2 −

12

8 𝑋

4

3

14. By what number −5

14 be multiplied so that the product should be

−1

6 ?

15. By what number −63

15 be divided so as to get – 3 ?

16. The cost of 23

5 m of cloth is Rs.65. Find the cost of cloth per metre.

17. If a = 2

5 𝑎𝑛𝑑 𝑏 =

1

3, 𝑡ℎ𝑒𝑛 find ( a+b) ÷ ( a – b )

MCQ ANSWERS

1. (II)

2. (I)

3. (III)

4. (I)

5. (II)

6. (III)

2. 7. (III)

8. (I)

9. (I)

10. (IV)

6

SET 1 ANSWERS

1) ( 4/3)

2) 0

3) 1

4) ( -3/5)

5) 0

6) (i) 1 and – 1 (ii) 0

7) 0

8) ( -48/7)

10) ( 3/8)

11) (a) non – terminating (b) non – terminating (c) terminating

12) (i) (1,2 or any other such numbers) (ii) ( 1/2, 1/3 or any other such numbers)

13) (i) (51/200,-52/200,-53/200,-54/200,-55/200,-56/200 or any other such numbers)

(ii) (145/252,146/252,147/252,148/252,149/252,150/252or any other such numbers)

14) (25/72)

SET-2 ANSWERS

1) (i) 1 (ii) 0 (iii) original number (iv) 0 (v) 0

2) (i) 0 (ii) 1 (iii) 1 or -1

3) (i) LHS = RHS = -59/42 (ii) LHS = 4/105 , RHS = 16/35, LHS ≠ RHS

4) 125/462)

5) ( ½)

6) (10/21,11/21,12/21or any such numbers)

7) (11/40,12/40,13/40,14/40,15/40 or any such numbers)

8) ( 0/15,1/15,2/15,3/15,4/15,5/15,6/15,7/15,8/15,9/15 or any such numbers)

9) ( A = -3/13, B = - 7 /13, C = -10/13 ) 10) (14/27)

MISCELLANEOUS ANSWERS

1) (i) ( −5

12 ) (ii) :

19

21, (iii)

2

3 2) -

16

12 6) (

−7

4 <

7

−12 <

5

6 )

8) (−5

8 ) 9) 2

3 10) 2

7 11) (−14

15 )

12) ( −1

10 ) 13) 49

12 14) 7

15 15) 7

5

16) Rs.25 17) 11

7

Ch-2 : Linear Equation in One Variable

1. Rules for solving a linear equation in one variable:-

Add the same number on both the sides of the equation.

Subtract the same number from both the sides the equation.

Multiply both the sides of the equation by the same number.

Divide both the sides of the equation by the same non zero number.

Transpose a term from LHS to RHS or vice-versa.

2. Transposing means taking a term from one side to the other side of an equation while

changing its sign.

3. The highest power of a linear equation is always one.

4. An equation may have linear expressions on both sides of the equation.

5. Such equations help us to solve various problems based on numbers, age, perimeters,

combination of coins etc.

MCQ

1. What do we get when we transpose 5/2 to RHS in the equation x/4 + 5/2 = -3/4?

I. x/4 = -3/4 + 5/2

II. x/4 = -5/2 + ¾

III. x/4 = -3/4 + (-5/2)

IV. none of these

2. In the equation 3x = 4-x, transposing -x to LHS we get

I. 3x - x = 4

II. 3x + x = 4

III. -3x + x = 4

IV. -3x - x = 4

3. If x/3 + 1 = 7/15, then which of the following is correct?

I. x/3 = 7/15 - 1

II. x/3 = -7/15 + 1

III. x/3 = -7/15 - 1

IV. none of these

4. If 7x+15 = 50, then which of the following is the solution of the equation?

I. -5

II. 65/7

III. 5

IV. 1/5

5. If 2x/5 = 4, the value of x is

I. 10

II. -10

III. -8/5

IV. 8/5

6. If the sum of two consecutive numbers is 71 and one number is x. Then the linear equation

to represent this situation is:

I. x + (x+1) = 71

II. x + (x+2) = 71

III. x + x = 71

IV. none of these

7. Two years ago my age was x years, then what was my age 5 years ago?

I. X + 7

8

II. X - 2 - 5

III. X - 5

IV. X – 3

8. How old will I be after 10 years, if my age before 10 years was „x‟ years?

I. X + 20

II. X – 20

III. X + 10

IV. X ‐ 10

9. If the sum of two consecutive numbers is 15 and greater of them is x then the smaller

number is:

I. 16

II. 14

III. 8

IV. 7

10. If x is an even number, which is the next odd number?

I. X + 1

II. X + 2

III. X - 1

IV. X – 2

SET 1

1. Solve the following Equations

a) (2x - 5)/(3x - 1) = (2x - 1)/(3x + 2)

b) (3 - 7x)/(15 + 2x) = 0

c) (0.4y - 3) / (1.5y + 9) = -7/5

d) 2/(3x - 1) + 3/(3x + 1) = 5/2x

e) 2/(x - 3) + 5/(x - 1) = 5/(x - 1) - 2/(x - 2)

f) 15(x - 5) - 3(x - 9) + 5(x + 6) = 0

g) y/2 - 1/2 = y/3 + ¼ h) (0.5y - 9) / 0.25 = 4y – 3

i) [17(2 - y) - 5(y + 12)] / (1 - 7y) = 8

2. Sunita is as twice as old as Ashima. If six years is subtracted from Ashima's age and 4

years added to Sunitas age, then Sunita will be four times Ashima‟s age. How old were

they two years ago?

3. The sum of two twin prime numbers is 60. Find the prime nos.

4. Of the three angles of a triangle, the second one is one third of the first and the third angle

is 26 degrees more than the first angle. Find all the three angles of the triangle.

SET-2

1. Solve x/3 + 1/5 = x/2 – 1/4 2. Show that x = 4 is a solution of the equation x + 7 – 8x/3 = 17/6 – 5x/8 3. Find x for the equation: (2 + x)(7 – x)/(5 – x)(4 + x) = 1 4. A number is such that it is as much greater than 45 as it is less than 75. Find the number. 5. Divide 40 into two parts such that 1/4th of one part is 3/8th of the other. 6. x + 3x/2 = 35. Find x.

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7. A is twice old as B. Five years ago A was 3 times as old as B. Find their present ages. 8. Solve : (x + 3)/6 + 1 = (6x – 1)/3 9. The digits of a 2-digit number differ by 5. If the digits are interchanged and the resulting number is added to the original number, we get99. Find the original number. 10. Solve : 5x – 3 = 3x + 7 11. Find the solution of 3x-4 = 12 12. Solve: 5x-9 = 8 13. What should be subtracted from thrice the rational number -8/3 to get 5/2? 14. The sum of three consecutive multiples of 7 is 63. Find these multiples. 15. Solve 3x/4 – 7/4 = 5x + 12 16. Perimeter of a rectangle is 13cm. if its width is 11/4 cm, find its length. 17. The present of Sita’s father is three times the present age of Sita. After six years sum of their ages will be 68 years. Find their present ages. 18. The digits of a two-digit number differ by 3. If digits are interchanged and the resulting number is added to the original number, we get 121. Find the original number. 19. (x-2)/(x+1) = ½. Find x 20. Sanjay will be 3 times as old as he was 4 years ago after 18 years. Find his present age. 21. If the sum of two numbers is 30 and their ratio is 2:3 then find the numbers. 22. The numerator of a fraction is 2 less than the denominator. If one is added to its denominator, it becomes 1/2 find the fraction.

MISCELLANEOUS

1.Solve : 3

5 + x =

13

5

2. Solve: 10x + 7 = 3x – 21

3. Solve: 𝑥 −3

5=

𝑥−5

3

4. Solve: 6𝑥+1

2=

7𝑥−3

3− 1

5. Solve: 𝑥 − 𝑥−2

7+ 36 =

9𝑥+7

2

6. Solve: 2

3𝑥−

1

12=

3

2𝑥

7. Divide 120 into two parts such that 2

3 of one part is equal to

2

5 of the other part.

8. A number is such that it is as much greater than 17 as it is less than 63. Find the number. 9. A number consists of two digits whose sum is 6. If 18 is added to the number, its digits are reversed. Find the number. 10. The numerator of a fraction is 4 less than its denominator. If 2 is added to the

numerator, then the fraction becomes 5

7 . Find the fraction.

11. Sumit is twice as old as Saksham. If four years are added to Sumit’s age and six years are subtracted from Saksham’s age, then Sumit’s age will be four times Saksham’s age. What are their present ages?

10

12. Five years ago, a man was 7 times as old as his son. Five years hence, he will be three times as old as his son. Find their present ages. 13.The purse of Sunita contains fifty paise and one rupee coins in the ratio 3:2. The total amount in the purse is Rs.70. Find the number of 50 paise coins and Rs.1 coins. 14. A steamer goes down stream and covers the distance between two ports in 6 hours. While it covers the same distance upstream in 8 hours. If the speed of the stream is 2km/hr, find the speed of the steamer in still water. 15. The length of a rectangle exceeds its breadth by 4cm. If length and breadth are increased by 3cm each, the area of the new rectangle will be 81sq.cm more than that of the given rectangle. Find the length and breadth of the given rectangle.

MCQ ANSWERS

1. (II)

2. (II)

3. (I)

4. (III)

5. (I)

6. (I)

7. (IV)

8. (I)

9. (IV)

10. (I)

SET 1 ANSWERS

1). (a) -11/6 (b) 3/7 (c) - 96/25 (d) 5/3 (e) 7/3 (f) -1/6

(g) 9/2 (h) -16.5 (i) 1

2) Ashima 12Years, Sunita 26 Years

3) Prime no‟s = 29, 31

4) 66, 22, 92

SET-2 ANSWERS

1) 27/10 3) LHS = RHS = 1/3 4) 60 5) 24. 16 6) 14 7) 20,10 8) 11/10 9) 72 10) 5 11) 16/3 12) 17/5 13) -21/2) 14) 14,21, 28 15) -55/7 16) 15/4 17) 14, 42 18) 47 19) 5 20) 15 21) 12, 18 22) 3/5


1) 2 2) - 4 3) -15/4 4) 60 5) 9 6) -10 7) 45 and 75 8) 40 9) 24 10) 3/7 11) 28years and 14years 12) 40yeras and 10 years 13) 60 and 40 14) 14km/hr 15) 14cm, 10cm

11

Ch-3 : Understanding Quadrilaterals

Introduction

Polygon – Polygon is a combination of two Greek words Polus + Gonia, in which Polus

means many and Gonia means Corner or angle.

Classification of Polygons

Polygons are classified as per their sides or vertices they have.

(a) Triangle – A triangle has three sides and three vertices. A triangle is of three types:

Equilateral, Isosceles and Scalene.

b) Quadrilateral – A quadrilateral has four sides and consecutively four vertices.

(c) Pentagon – (Penta stands for five) A pentagon has five sides and five vertices.

(d) Hexagon – (Hexa stands for six) A hexagon has six sides and six vertices.

(e) Heptagon – (Hepta stands for seven) A heptagon has seven sides and seven vertices.

(f) Octagon – (Octa stands for eight) an octagon has eight sides and eight vertices.

(g) Nonagon – (Nona stands for nine) A nonagon has nine sides and nine vertices.

(h) Decagon – (Deca stands for ten) A decagon has ten sides and ten vertices.

Diagonals

A line segments which connects two non-consecutive vertices of a polygon is called diagonal.

Quadrilateral

This is the combination of two Latin words; Quardi + Latus. Quadri . means four and Latus

means side.

So, a polygon that has four sides is known as a quadrilateral. In a quadrilateral, sides are

straight lines and are two dimensional. Square, rectangle, rhombus, parallelogram, etc. are the

examples of quadrilateral.

Formula for angle sum of a polygon = (n – 2) × 180°.

Where „n‟ is the number of sides

Example:

A triangle has three sides,

Thus, Angle sum of a triangle = (3 – 2) × 180° = 1 × 180° = 180°

MCQ

1. Which of the following quadrilaterals has two pairs of adjacent sides equal and diagonals

intersecting at right angles?

(i) square

(ii) rhombus

(iii) kite

(iv) rectangle.

2. Which of the following quadrilaterals has a pair of opposite sides parallel?

(i) rhombus

(ii) trapezium

(iii) kite

(iv) rectangle.

12

3. Which of the following quadrilaterals is a regular quadrilateral?

(i) rectangle

(ii) square

(iii) rhombus

(iv) kite.

4. Which of the quadrilaterals has all angles as right angles, opposite sides equal and

diagonals bisect-each other?

(i) rectangle

(ii) rhombus

(iii) square

(iv) none of these.

5. Which of the parallelograms has all sides equal and diagonals bisect each other at right

angle?

(i) square

(ii) rectangle

(iii) rhombus

(iv) trapezium.

6. In an isosceles parallelogram, we have:

(i) pair of parallel sides as equal

(ii) pair of non-parallel sides as equal

(iii) pair of non-parallel sides as perpendicular

(iv) none of these.

7. Which of the following is true for the adjacent angles of a parallelogram?

(i) they are equal to each other

(ii) they are complementary angles

(iii) they are supplementary angles

(iv) none of these.

8. The sides of a pentagon are produced in order. Which of the following is the sum of its

exterior angles?

(i) 540°

(ii) 180°

(iii) 720°

(iv) 360°

9. Which of the following is a formula to find the sum of interior angles of quadrilaterals of

n-sides?

(i)

(ii)

(iii)

(iv) (n – 2) × 180°

10. Diagonals of which of the following quadrilaterals do not bisect it into two congruent

triangles?

(i) rhombus

(ii) trapezium

(iii) square

(iv) rectangle.

13

SET 1

1. One angle of a parallelogram is of measure 70°. Find the measures of the remaining angles

of the parallelogram.

2. In the given figure PR is a diagonal of the parallelogram PQRS.

(i) Is PS = RQ? Why?

(ii) Is SR = PQ? Why?

(iii) Is PR = RP? Why?

(iv) Is ΔPSR ≌ ΔRQP? Why?

3. The perimeter of a parallelogram is 150 cm. One of its side is greater than the other by 25

cm. Find length of all sides of the parallelogram.

4. Lengths of adjacent sides of a parallelogram is 3 cm and 4 cm. Find its perimeter.

5. In a parallelogram, the ratio of the adjacent sides is 4 : 5 and its perimeter is 72 cm then,

find the sides of the parallelogram.

6. The adjacent figure HOPE is a parallelogram. Find the angle measure x, y and z.

State the properties you use to find them.

7. Find value of x and y in the following figures.

(i) where ABCD is a parallelogram.

(ii) where PQRS is a rhombus.

8. Find x, y, z in the given parallelogram ABCD.

14

SET 2

1. State whether True or False.

(a) All rectangles are squares.

(b) All rhombuses are parallelograms.

(c) All square are rhombuses and also rectangles.

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

2. PQRS is a parallelogram such that m∠R = 110°, then find m∠P and ∠S.

3. Two opposite angles of a parallelogram are (5x – 8)° and (2x + 82)°. Find the measures of

each angle of the parallelogram.

4. JKLM is a parallelogram. If m∠J = 70°, then find all other angles.

5. 5. The exterior angle of a regular polygon is one-fifth of its interior angle. How many

sides the polygon has?

6. In a quadrilateral ABCD, DO and CO are the bisectors of ∠D and ∠C respectively. Prove

that

MISCELLANEOUS

1.Thefour angles of a quadrilateral are in the ratio 2:3:6:9. Find the angles. 2. Find the number of sides of a regular polygon whose each exterior angle has a measure of 450. 3. How many sides does a regular polygon have if each of its interior angle is 165? 4. How many sides does a regular polygon have if each of its exterior angle is 240. 5. What is the measure of each interior angle of a regular hexagon? 6. The measure of the angles of a pentagon are ( x – 5 )0, ( x – 6 )0, (2x – 7 )0, x0 and ( 2x – 2 )0. Find the value of x. 7. Find the measure of x in each of the following:

15

8. Determine the number of sides of a regular polygon whose exterior and interior angles are in the ratio 1:5 9. The ratio of the adjacent sides of a parallelogram is 3:5 and its perimeter is 96cm. Find the sides of the parallelogram. 10. Two adjacent angles of a parallelogram are ( 3x – 4 )0 and ( 3x + 10 )0. Find the angles of the parallelogram. 11. Find x and y from the parallelograms given below:

12. The diagonals of a rhombus are 8cm and 6cm. Find the length of each side of the rhombus. 13. Two opposite angles of a parallelogram are (5x – 2 )0 and ( 40 – x )0. Find x. 14. The diagonals of a rectangle ABCD intersect at O. If ∠ BOC = 520, find ∠ ODA. (Ans: 640) 15. Two diagonals of a rectangle are of length ( 4x + 15 )cm and ( 2x + 25), then find x.

MCQ ANSWERS

1. (iii) 2. (ii) 3. (ii) 4. (i) 5. (iii)

6. (ii) 7. (iii) 8. (iv) 9. (iv) 10. (ii)

16

SET 1 ANSWERS

1) 1100, 70

0, 110

0

2) (i) yes, opposite sides of a parallelogram are equal

(ii) yes, opposite sides of a parallelogram are equal

(iii) yes, common side

(iv) yes, by SSS congruence rule

3) 50cm,25cm,50cm,25cm

4) 14cm

5) 16cm, 20cm

6) x = 1100, y = 40

0, z = 30

0

7) (i) x = 3, y = 2 (ii) x = 500, y = 40

0

8) x = 250 , 45

0, z = 110

0

SET 2 ANSWERS

1) (a) False (b) True (c) True (d) False (e) False (f) True (g) True (h) True

2) m∠P = 110° and and m∠S = 70° 3) 142°, 38°, 142°, 38°

4) m∠M = 110°, m∠L = 70° and m∠K = 110° 5) n = 12


1) 360, 540, 1080 and 1620 2) 8 3) 24 4) 15 5) 1200 6) 800 7) 600 , 1300 8) 12

9) 18cm, 30cm 10) 830, 970, 830, 970 11) (i) x = 6, y = 7, (ii) x = 12, y = 2 12) 5cm 13) 7 14) 640 15) 5

17

Ch-4 : Practical Geometry

WE KNOW THAT A triangle has six elements – 3 sides and 3 angles. To construct a unique triangle, 3 elements

out of six elements are required under a certain combination. A quadrilateral has 8 elements –

4 sides and 4 angles. In addition to these elements a quadrilateral has 2 diagonals which play

an important role in determining the size and shape of a quadrilateral. Thus a quadrilateral

has 10 elements (4 sides, 4 angles and 2 diagonals) or measurements.

CONSTRUCTING A QUADRILATERAL To construct a unique quadrilateral we need to know 5 measurements (elements).

Note: To construct a unique quadrilateral simply the knowledge of any five elements is not

sufficient. We will need to know a combination of specific 5 elements.

VARIOUS COMBINATIONS OF ELEMENTS FOR CONSTRUCTING A UNIQUE

QUADRILATERAL

With the help of the following measurement we can construct quadrilaterals.

(i) Four sides and a diagonal.

(ii) Three sides and two diagonals.

(iii) Four sides and an angle.

(iv) Three sides and two included angles.

(v) Two adjacent sides and three angles.

(vi) Using special properties of a square or a rhombus, etc.

SET 1

1. Construct a rhombus whose diagonals are 4.8 cm and 6.3 cm.

2. Draw a parallelogram whose adjacent sides are 2.8 cm and 3.8 cm.

3. Draw a rectangle whose adjacent sides are 4.5 cm and 2.3 cm.

4. Construct a quadrilateral ABCD, where AB = 4.3 cm, BC = 5.2 cm, CD = 6.5 cm, ∠B =

105° and ∠C = 60°.

5. Construct a quadrilateral PQRS where PQ = 5.4 cm, ∠P = 60°. ∠Q = 105°, ∠R = 75° and

∠S = 120°.

6. Construct a rhombus whose diagonals are 4.5cm and 6.2 cm.

7. Draw a parallelogram whose adjacent sides are 2.8 cm and 4.8 cm.

8. Draw a rectangle whose adjacent sides are 3 cm and 5 cm.

9. Construct a quadrilateral ABCD, where AB= 4.3 cm, BC= 5.2 cm, CD= 6.5 cm, ∟B=

105° and ∟C= 60°.

10. Construct a quadrilateral PQRS where, PQ= 5.4 cm, ∟P= 6°, ∟Q= 105°, ∟R=75° and

∟S= 120°

11. Construct a quadrilateral ABCD in which AB= 5 cm, BC= 6.5 cm, angle A= 75°, angle

B= 105° and angle C= 120°.

12. Draw a line segment of length 10 cm and divide it into 4 equal parts.

13. Construct a quadrilateral WXYZ when WX= 3.3 cm, XY= 4cm, YZ= 4.1 cm, WZ= 3.6

cm and XZ= 5.5 cm.

14. Construct a rhombus whose diagonals are 6.2 cm and 8.4 cm.

15. Construct a quadrilateral BEST, given ES= 4.5cm, BT= 5.5 cm, St= 5 cm, the diagonal

BS= 5.5 cm and diagonal ET= 7 cm. Find Angle E, Angle T and RE.

16. Construct a parallelogram BEAT, BE=5 cm, EA= 6cm and Angle R= 85°.

18

Ch-5 :Data Handling

Representation of data

For data to be useful, it is very important to collect complete, accurate & relevant data.

After collection of data, it is necessary to represent data in a precise manner so that it is easily

understood. Representation of data in a visual manner is known as its Graphical Presentation

or simply, Graphs. Data can also be presented in the form of a table; however a graphical

form is easier to understand. Consider the family of Mr. Khanna. They are very fond of

eating ice-creams. Some like butterscotch, some like vanilla flavor, some like strawberry and

the rest of them likes chocolate. Some demand cones & some of them demand cups. A child

of VIII class decides to construct the graphs according to their amount of ice creams

consumed. There are 7 ways of representing the data.

Pictograph Pictorial representation of numerical data, using picture symbol is known as a pictograph.

Most business and industrial organizations use this method to represent their data.

denotes cones and

represents cups.

If we draw the pictograph, it will be in the above way where represents cone &

represents cups.

FACTS THAT MATTER

• The numerical information is called data.

• Data can be arranged and presented by grouped frequency distribution.

• Frequency is the number of times a particular entry occurs.

• Histogram is a special type of bar graph in which the class intervals are shown on the

horizontal axis and heights of the bars correspond to the frequency of the class.

• In a histogram there is no gap between bars.

• A circle graph or a pie chart shows the relation ship between a whole and its parts.

• Outcomes of an event or experiment are equally likely if each has the same of occurring.

• Probability of an event

• Probability of an event can have a value from 0 to 1.

• Probability of an impossible event is 0.

19

MCQ

1. The range of the data: 6,14,20,16,6,5,4,18,25,15, and 5 is

I. 4

II. 21

III. 25

IV. 20

2. The class mark of the class 20-30 is

I. 20

II. 30

III. 25

IV. 10

3. The difference between the highest and the lowest value of the observations in a data is

called:

I. Mean

II. Range

III. Total frequency

IV. Sum of observation

4. In the interval 35-45, 45 is called

I. Upper limit

II. Lower limit

III. Range

IV. Frequency

5. The number of times a particular observation occurs in a given data is called:

I. Its frequency

II. Its range

III. Its mean

IV. None of these

6. Is history which of the following is represented by the heights of the rectangles?

I. Frequency

II. Class interval

III. Class size

IV. Range

7. Tally marks are used to find which of the following?

I. Frequency

II. Lower limits

III. Upper limits

IV. Class marks

8. Which of the following is the probability of an impossible event?

I. 0

II. 1

III. 2

IV. None of these

9. Which of the following is the probability of a sure event?

I. 0

II. 1

III. 2

IV. None of these

20

10. A coin is tossed. Which of the following is the probability of getting a head or tail?

I. 0

II. 1

III. 1/2

IV. None of these

SET-1

1. What is the number of students of Class VIII whose marks obtained in an examination are

expressed in the following frequency distribution.

2. The marks scored by 20 students in a test are given below:

84, 57, 53, 89, 41, 57, 47, 64, 58, 44, 53, 72, 51, 78, 71, 62, 56, 68, 54, 42

Complete the following frequency table:

(i) What is the upper limit of 40–50?

(ii) What is the lower limit of 70–80?

(iii) What is the class size?

3. Number of workshops organized by a school in different areas during the last six years

are as follows:

Draw a histogram representing the data.

21

4. Draw a histogram for the daily earnings of 30 general stores given in the following table

5. The number of students admitted in different faculties of a college are given below.

Represent the above information by a pie-chart.

6. Draw a pie-chart for the following data of expenditure on various items in a family.

7. The following pie-chart represents the marks scored by a students. If he obtained 540 as

total marks, answer the following questions:

(i) In which subject did the student score 120 marks?

(ii) What is the difference in the marks obtained in Maths and English?

(iii) In which subject did he get minimum marks?

8. A die is thrown. What is the probability of getting:

(i) an even number?

(ii) an odd number?

(iii) A number between 3 and 6?

9. What is the probability of a number selected from the numbers 1, 2, 3, ....., 20 such that it

is a prime number?

10. A bag contains 3 blue and 2 red balls. A ball is drawn at random. What is the probability

of drawing a red ball?

22

Set 2

1. Read the following bar-graph and answer the following questions.

(i) What is the information given by the bar graph?

(ii) In which year is the increase in the number of students is maximum?

(iii) In which year is the number of students maximum?)

(iv) Is the number of students during 2005-06 twice that of 2003-04?

2. Read the following bar graph and answer the following questions.

(i) What is the information given by the double bar graph?

(ii) In which subject has the performance improved the most?

(ii) In which subject the performance deteriorated?

(iv) In which subject is the performance at par?

3. A group of students was asked for their favorite subject. The results were listed as under:

Art, Mathematics, Science, English, Mathematics, Art. English, Mathematics, English; Art,

Science ,Art, Science, Science, Mathematics, Art. English. Art., Science, Mathematics. , Art.

Answer the following questions:

(i) Which is the most liked subject?

(ii) Which is the least liked subject?

23

4. Read the following histogram and answer the questions given below:

5. Look at the following circle graph and answer the questions given below:

(i) Find the fraction of the circle representing each of these given information.

(ii) What is the central angle corresponding to the activities “Play and Home work”?

MCQ ANSWERS 1. II

2. III

3. II

4. I

5. I

6. I

7. I

8. I

9. II

10. II

SET 1 ANWERS

1) 40 2) (i) 50 (ii) 70 (iii) 10 7) (i) English (ii) 15

(iii) Hindi

8 ) (i) 1/2 (ii) 1/2

(iii) 1/3

9) 2/5 10) 2 /5

SET 2 ANWERS

1) (i) number of students of class 8 of a school from 2003 to 2008 (ii) 2004 – 2005

(iii) 2007-2008 (iv) Yes

2) (i) marks of various subjects in the year 2005 – 6 and 2006 – 7 (ii) Maths

(iii) English (iv) Hindi

3) (i) Art (ii) English

24

Ch-6: Squares and Square Roots

4.1 Square of a number.

If a natural number m can be expressed as n2 (where n is a natural number), then m is the

square root or perfect square.

i.e. if m = n2 (m, n – natural numbers)

E.g. 81 = 3 × 3 × 3 × 3

= 32 × 3

2 = (3 × 3)

2 = 9

2

Hence, 9 is the square root of 81.

4.2 Properties of Square Root.

Below is the table that has squares of numbers from 1 to 10.

If we see the above results carefully, we can conclude that numbers ending with 0, 1, 4, 5, 6,

or 9 at units place are perfect squares None of these end with 2, 3, 7 or 8, So numbers that

end with 2, 3, 7, 8 are not perfect squares.

Thus, numbers like 122, 457, 183, 928 are not perfect squares.

4.3 One’s digit in square of a number.

(1). The ones digit in the square of number can be determined if the ones digit of the number

is known.

(2). The number of zeros at the end of a perfect square is always even and double the number

of zeros at the end of the number

E.g.

Double zero 7002 = 490000 four zero (even)

(3). The square of an even number is always an even number and square of an odd number is

always an odd number.

E.g.

4.4 Interesting patterns of Square Root.

25

Number between square numbers

There are „2a‟ non perfect square numbers between the square of two Consecutive natural

numbers n + (n + 1)

Between 22 = 4 & 32 = 9 → 5, 6, 7, 8

2 × 2 = 4 non square numbers

Between

32 = 9 & 4

2 = 16 → 10, 11, 12, 13, 14, 15

2 × 3 = 6 non square numbers.

Adding Consecutive odd numbers.

So, we can conclude that the sum of first in odd natural numbers in n2 or, we can say if the

number is a square number, it has to be the sum of successive odd numbers.

E.g. 36

Successively subtract 1, 3, 5, 7 ... from 36

36 – 1 = 35

35 – 3 = 32

32 – 5 = 27

27 – 7 = 20

20 – 9 = 11

11 – 11 = 0

∴ 36 is a perfect square

4.5 Short Cut Method of Squaring a number:

We can use the below to methods for Calculating square of any natural number

(a + b)2 = a

2 + b

2 + 2ab

(a ï¿½ b)2 = a

2 + b

2ï¿½ 2ab

The logic behind using these formulae is to convert square of unknown number into square of

a known number.

E.g.

(53)2 = (50 + 3)

2

= 502 + 3

2 + 2.50.3

= 250 + 9 + 300

= 2809

992 = (100 – 1)

2

= 1002 + 1

2 – 2.100.1

= 10000 + 1 – 200

= 9801

4.6 Square Roots

Finding the Square root is inverse operation of squaring.

If 9 × 9 = 81 i.e, 92 = 81

Then the square root of 81 = 9

In other words, if p = q2, the q is called the square root of p

The Square root of a number is the number which when multiplied with itself gives the

number as the product. The square root of a number is denoted by the

symbol

26

Some Properties of square root

1. The Square root of an even perfect square is even and that of an odd Perfect square is

odd.

2. Since there is no number whose square is negative the square root of a negative number is

not defined.

3. If a number ends with an odd number of zeroes, then it cannot have a square root which is

a natural number.

4. If the units digit of a number is 2, 3, 7 or 8 then square root of that number (in natural

numbers) is not possible.

5. If m is not a perfect square, then there is no integer n such that square root of m is n.

Finding square root through repeated subtraction Recall the pattern formed while adding

Consecutive odd numbers.

1 + 3 + 5 + 7 + 9 = 52 = 25

1 + 3 + 5 + 7 + 9 + 11 = 62 = 36

Sum of first n odd numbers = n2

The above Pattern can be used to find the square root of the given number.

1. Obtain the given perfect square whose square root is to the calculated let the number be a.

2. Subtract from it successively 1, 3, 5, 7, 9 till you get zero.

3. Count the number of times the subtraction is performed to arrive at zero let the number be

n.

4. Write

Finding square root through Prime factorization. In order to find the square root of a perfect

square by prime factorization.

We follow the following steps.

1. Obtain the given number

2. Resolve the given number into prime factors by successive division.

3. Make pairs of prime factors such that both the factors in each pair are equal.

4. Take one factor from each pair and find their product.

5. The product obtained is the required square root.

E.g.

Find the square root of 400

27

Finding the square root by division Method when the numbers are large even the method of

finding square root by prime factorization becomes slightly difficult. To over come this

problem we use long division method.

(1). Consider the following steps to find the square root of 784

Step 1: Place a bar over every pair of digits starting from the digit at oneï¿½s place. If the

number of digits is odd, then the left most single digit too will have a bar. Thus we have

Step 2: Find the largest number whose square is less than or equal to the number under the

extreme left bar (22 < 7 < 3

2). Take this number as the divisor and the quotient with the

number under the extreme left bar as the dividend (here 7) Divide and get the remainder (5 in

this case)

Step 3: Bring down the number under the next bar (i.e, 84 in this case) to the right of the

remainder. So the new divided is 384.

Step 4: Double the divisor and enter it with a blank on its right

Step 5: Gives a largest possible digit to fill the blank which will also become the new digit in

the quotient, such that when the new divisor is multiplied to the new quotient the product is

less than or equal to the dividend.

In this case 47 × 7 = 329

As 48 × 8 = 384 so we choose

the new digit as 8. Get the remainder

Step 6: Since the remainder is 0 and no digits are left in the given number,

therefore

MCQ

1. Which of the following can be a perfect square?

(i) A number ending in 3 or 7

(ii) A number ending with odd number of zeros

(iii) A number ending with even number of zeros

(iv) A number ending in 2.

2. Which of the following can be the square of a natural number „n‟?

28

(i) sum of the squares of first n natural numbers

(ii) sum of the first n natural numbers

(iii) sum of first (n – 1) natural numbers

(iv) sum of first „n‟ odd natural numbers.

3. Which of the following is the number non-perfect square numbers‟ between the square of

the numbers n and n + 1?

(i) n + 1

(ii) n

(iii) 2n

(iv) 2n + 1

4. Which of the following is the difference between the squares of two consecutive natural

number is:

(i) sum of the two numbers

(ii) difference of the numbers

(iii) twice the sum of the two numbers

(iv) twice the difference between the two numbers.

5. How many non-perfect square number lies between (17)2 and (18)

2?

(i) 613

(ii) 35

(iii) 34

(iv) 70

6. Which of the following is the difference between the squares of 21 and 22?

(i) 21

(ii) 22

(iii) 42

(iv) 43

7. Which of the following is the number of zeros in the square of 900?

(i) 3

(ii) 4

(iii) 5

(iv) 2

8. If a number of n-digits is a perfect square and „n‟ is an even number, then which of the

following is the number of digits of its square root?

(i)

(ii)

(iii)

(iv) 2n

9. If a number of n-digits is perfect square and „n‟ is an odd number, then which of the

following is the number of digits of its square root?

(i)

(ii)

(iii)

(iv) 2n

29

10. Which of the following is a Pythagorean-triplet?

(i) n, (n+1) and (n2 + 1)

(ii) (n – 1), (n2 – 1) and (n

2 + 1)

(iii) (n + 1), (n2 – 1) and (n

2 + 1)

(iv) 2n, (n2 – 1) and (n

2 + 1)

SET 1

1. A perfect square number can never have the digits ... at the units place.

2. Find

3. Find the value of (23)2 using column method.

4. Find the value

5. Write a Pythagorean triplet whose smaller member is 6.

6. What is the sum of first n odd natural numbers?

7. A number ending in an odd number of zeros is never a

8. If m, n, p are natural numbers such that

(m2 + n

2) = p

2, then (m, n, p) is called

9. Express 49 as the sum of seven odd numbers.

10. Without adding, find the sum.

(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17)

11. Find the value of

12. Write the unit digit of square of 799.

SET 2

1. Find the square root of 144 by the method of repeated subtraction.

2. Find the smallest number by which 1800 must be multiplied so that it becomes a perfect

square. Also find the square root of the perfect square so obtained.

3. Is 2352 a perfect square? if not, find the smallest number by which 2352 must be

multiplied so that the product is a perfect square. Find the square root of new number.

4. The area of a square field is 8281 m2. Find the length of its side.

5. Find the square root of

6. Simplify:

7. Factorise: p2 – 10p + 25.

8. 1225 plants are to be planted in a garden in such a way that each row contains as many

plants as the number of rows. Find the number of rows and the number of plants in each row.

9. Find the smallest number by which 3645 should be divided so as to get a perfect square.

Also, find the square root of the number so obtained.

10. For each of the following numbers, find the smallest number by which we divide it so as

to get a perfect square. Also find the square root of the square numbers so obtained.

(a) 37845 (b) 2800 (c) 45056

11. The students of Class VIII of a school donated Rs 2401 for Prime Minister National

Relief Fund. Each student donated as many rupees as the number of students in the Class.

Find the number of students in the Class.

12. There are 500 children in a school. For a P.T. drill they have to stand in such a manner

that the number of rows is equal to number of columns. How many children would be left out

in this arrangement?

13. A school collected Rs 2304 as fees from its students. If each student paid as many paise

as there were students in the school, how many students were there in the school?

30

14. 2025 plants are to be planted in a garden in such a way that each row contains as many

plants as the number of rows. Find the number of rows and the number of plants in each row.

15. 10404 students are sitting in a lecture room in such a manner that there are as many

students in a row as there are rows in a lecture room. How many students are there in each

row of a lecture room?

MISCELLANEOUS

1. Write a Pythagorean triplet whose one member is 14.

2. Find the square root of 50625 by prime factorisation method.

3. Find the square root of 49 by repeated subtraction.

4. Find the square root of 11025 by prime factorisation method.

5. Find the smallest whole number with which 1458 should be multiplied so as to get a

perfect square. Also, find the square root of the square number so obtained.

6. Find the smallest whole number by which 1620 should be divided so as to get a perfect

Square. Find the square root of the square number so obtained.

7. The product of two numbers is 972 and their quotient is 4

3 . Find the numbers.

8. Find the smallest square number which is divisible by each of the numbers 8, 15 and 20.

9. The product of two numbers is 10625. If one number is 17 times the other number, find

the numbers.

10. Find the square roof the following by division method:

(i)4489 (ii) 54756

11. Find the least number which must be subtracted from 4216 so as to get a perfect

square. Also find the square root of this perfect square so obtained.

12. Find the least number which must be added to 1500 so as to get a perfect square. Also

find the square root of the perfect square number so obtained.

13. Find the greatest 5 digit number which is a perfect square.

14. The area of a square field is 23394

729 sq. m. Find its side.

31

15.Find a number whose one-fourth multiplied by its one-sixth becomes 486.

MCQ ANSWERS

1. (III)

2. (IV)

3. (III)

4. (I)

5. (III)

6. (IV)

7. (II)

8. (II)

9. (III)

10. (IV)

SET 1 ANSWERS

1. 2,3,7 or 8 2. 75 3. 529 4. 30 5. 6,8,10 6. n2 7. Perfect Square

8. Pythagorean Triplet 9. (7)2 10. 81 11. 21 12. 28, 26

SET 2 ANSWERS

1. 12 2. 2, 60 3. 3, 84 4. 91m 5. 29/13 6. 999 7. (p – 5) ( p - 5 )

8. 35 9. 5, 29 10. (a) 5, 87 (b) 7, 20 (c) 11, 64 11. 49

12. 16 13. 48 14. 45 15. 102


1. 14, 48, 50 2. 225 3. 7 4. 105 5. 2, 54 6. 5, 18 7. 36 and 27

8. 3600 9. 25 and 425 10. (i) 67 (ii) 234 11. 120, 64

12. 21, 39 13. 99856 14. 4 23

27 m 15. 108

32

Ch-7 : Cubes and Cube Roots

1. When we multiply a number by itself three times, the product so obtained is called the

perfect cube of that number.

2. There are only 10 perfect cubes from 1 to 1000.

3. Cubes of even numbers are even and those of odd numbers are odd

4. The cube of a negative number is always negative.

5. If the digit in ones place of a number is 0, 1, 4, 5, 6 or 9, then its cube will end in the

same digit.

6. If the digit in ones place of a number is 2, then its cube will end in 8 and vice-versa.

7. If the digit in ones place of a number is 3, then its cube will end in 7 and vice-versa.

8. If the prime factor of a number can not be made into groups of 3, it is not a perfect square.

9. The symbol 𝑥3 denotes the cube root of a number x

MCQ

1. Which of the following is correct?

I. Cube of a negative number is always positive.

II. Cube of a negative number is always negative.

III. Cube of a negative number may be positive or negative.

IV. All of the above

2. If the digit in one‟s place of a number is 2, then the last digit of its cube will be:

I. 2

II. 4

III. 6

IV. 8


I. 3

II. 6

III. 7

IV. 9


I. 6

II. 3

III. 2

IV. 8

5. The volume of a cubical box is 64 cm3. Which of the following is its side?

I. 2 cm

II. 4 cm

III. 6 cm

IV. 8 cm

6. Which of the following is a perfect cube?

I. 10000

II. 243

III. 343

IV. 270000

7. If a number is doubled then which of the following is a correct statement?

I. Its cube is two times the cube of the given number.

II. Its cube is three times the cube of the given number.

III. Its cube is six times the cube of the given number.

IV. Its cube is eight times the cube of the given number.

33

8. Which of the following is equal to its own cube?

I. -1

II. -2

III. -3

IV. -9

9. Which of the following is the cube root of 27000?

I. 30

II. 300

III. 3000

IV. None of these

10. Which of the following is the cube root of -64/243?

I. 7/4

II. -7/4

III. 4/7

IV. -4/7

SET 1

1. What is the smallest number by which 288 must be multiplied so the product is a perfect

cube?

2. Find the cube of 4/5.

3. Show that 0.001728 is a cube root of a rational number.

4. Find the sides of a cubical box whose volume is 64 cm3.

5. If the surface area of a cube is 486 cm2, find its volume.

6. Find the volume of a cube whose surface area is 96 cm2.

7. Write all the digits that would appear as the last digits of their respective cubes.

8. Show that if a number is doubled, then it cube becomes eight times the cube of the given

number. SET 2

1. Is 9000 a perfect cube?

2. By which smallest number should 42592 be divided so that the quotient is a perfect cube?

3. Show that 46656 is a perfect cube.

4. By which smallest number should 704 be divided to obtain a perfect cube?

5. Find the cube root of 9197.

6. Show that 384 is not a perfect cube.

7. By which smallest number should 648 be multiplied so that the product is a perfect cube?

8. Find the number whose cube is 27000

MCQ ANSWERS

1) II 2) IV 3) III 4) I 5) II 6) III 7) IV 8) I 9) I 10) IV

SET 1 ANSWERS

1. 6 2. 64/125 3. 0.12 4. 4m 5. 729cm3 6. 64cm

3 7. 0,1,4,5,6 and 9)

SET 2 ANSWERS

1. No 2. 4 4. 11 5. 13 7. 9 8. 30

34

Ch-8 :COMPARING QUANTITIES

Introduction

• S.P. is selling price

• C.P. is cost price

• The overhead expenses are a part of CP.

• Profit or loss is always calculated on CP

• For profit, SP>CP and for loss CP>SP

• Discount is always calculated on the marked price.

• VAT is not a part of selling price. Although it is calculated on the SP but it is excluded

from the SP while calculating profit or loss.

• Formula for compound interest gives

Amount=P(1+R/100)n

then CI=A-P

• If the interest is compounded half yearly the time period becomes twice as much as the

numbers of years. The rate of interest is reduced to half.

• We compare two quantites by division.It is called ratio. Quantities written in same unit.

Ratio has no unit

• Another way to compare quantity is to express in percentages.

• Profit= Profit/CP X 100, Loss= Loss/Cp X 100

MCQ

1. On what a discount is calculated?

a. S.P

b. M.P

c. marked price

d. none of these

2. On which figure the VAT of a product is calculated?

a. S.P

b. C.P.

c. market price

d. none of these

3. On which of the following percent profit or profit loss is calculated?

a. S,P.

b. C.P.

c. market price

d. none of these

4. If an article sold for Rs 100 then there is a gain of Rs 20, which of the following is the

gain percent?

a. 25%

b. 22%

c. 20%

d. 16 . %

5. An article is at 10% more than the CP. If discount of 10% is allowed then which of the

following is right?

a. 1% gain

b. 1% loss

c. no gain no loss

d. 1.1% loss

35

6. A building worth Rs a is depreciated by R% per annum. Which of the following is true?

a. P[1- 5/100]

b. P [1+5/100]

c. P[(1+5/100)-1]

d. P[1-(1-5/100)]

7. If MP of a box is Rs 10 and a discount of 10% is allowed then what should be the sale

price?

a. Rs 10

b. rs 9

c. Rs 11

d. none of these

8. What should be the rate of interest per annum if interest is calculated quarterly?

a. reduced to half

b. reduced to one fourth

c. is doubled

d. becomes four times

9. What time period is taken when interest is calculated half yearly?

a. twice as much as the number of given years

b. half as much as the number of given years

c. same as the number of given years

d. none of these

10. what should be percentage gain on a product when it is sold for Rs 120 with a gain of Rs

20.

a. 20%

b. 25%

c. 22%

d. 16.25%

SET 1

1. A shopkeeper purchased 200 bulbs for Rs.10 each. However 5 bulbs were fused and had

to be thrown away. The remaining were sold at Rs 12 each. Find the gain or loss per cent.

2. A second hand Tv is for Rs 2500. And then Rs 500 was spent on its repair and sold for Rs

3300. Find profit/loss percent.

3. What amount has to paid to on a loan of Rs 12000 for 1½ years at 10% per annum

compounded half yearly.

4. The population of a city was Rs20,000 in 1997. It increased at the rate of 5% per annum.

Find the population of city at the end of the year 2000.

5. A fan is marked at Rs 15600 and it is available for Rs 12480. Find the discount given and

discount percent

6. Convert 12

15 as percentage.

7. What should be the current price of the box which was Rs 25000 last year and it increased

by 20% this year.

8. Find the population of a city after 2 years, which is at present is 20 lacs if the rate of

increase is 5%p.a.

36

SET 2

1. If customers pay VAT to the shopkeeper in addition to selling price, is the VAT a part of

profit?

2. What is the formula to calculate the compound interest.

3. Anything spend on an item say cost spent is Rs 15 on its transportation. And the item

purchased for Rs 100. What is the actual cost price of the item for the shopkeeper ?

4. On which amount the overhead charges are added CP or SP.

5. What should be the sales tax when cost of doll us Rs750 and sales tax is charged 5%. Also

find bill amount.

6. A sum of Rs 10,000 is borrowed at a rate of interest 15%p.a. for 2 years. Find simple

interest and the amount to be paid at the end of 2 years.

7. Mario invested Rs8,000 in a domain. She would be paid interest ar 5% per annum

calculated annually. Find: (i) the amount credited against his name at end of third year.

8. The cost of shoes was Rs. 700. The sales tax charged was 12%. Find the bill amount.

9. What should be price before VAT of a pair of shoes for Rs 300 including a tax of 10%.

10. Vanshika got 150 out of 200 and Sakshi got 120 marks out of 180. Whose performance is

better?

MCQ ANSWERS

1. c

2. a

3. b

4. a

5. b

6. a

7. b

8. b

9. a

10. a

SET 1 ANSWERS SET 2 ANSWERS

1. 17% 1. No

2. 10% 2. Amount=P(1+R/100)n

3. 13891.5 3. Rs 115

4. 2315.5 Approx 4. C.P

5. 2% 5. Rs 37.5, Rs 787.5

6. 80% 6. Rs 3000, Rs 13000

7. 30000 7. Rs 9261

8. 220500 8. Rs 784

9. Rs 272.7 Approx

10. Vanshika 75%)

37

Ch-9: Algebraic Expressions and Identities

Algebraic expressions are expressions formed from the variables and the constants. A

variable can take any value. The value of an expression changes with the value chosen for

variables it contains.

A number line has infinite number of points. A variable can take position on number line.

Expressions containing one, two or three terms are called monomial, binomial and

trinomial respectively.

Any expression having one or more terms is called polynomial.

A monomial is obtained on multiplying any monomial with another monomial.

The numerical factor of a term is called its coefficient.

An identity is a standard equality which is true for all the values of the variables in the

equality.

Few commonly used identities are

I. (a + b)2 = a

2 + b

2 + 2ab

II. (a – b)2 = a

2 + b

2 – 2ab

III. (a + b)(a – b) =a2 – b

2

IV. (x + a)(x + b) = x2 + (a+b)x + ab

Note – Mainly the above formulas are used to solve all the problems of this chapter.

MCQ

1. Which of the following is the numerical coefficient of x2y

2?

I. 0

II. 1

III. x2

IV. y2

2. Which of the following is the numerical coefficient of -5xy?

I. 5

II. -x

III. -5

IV. -y

3. pqr is what type of polynomial?

I. Monomial

II. Binomial

III. Trinomial

IV. None of these

4. The value of x2 - 5 at x= -1 is-

I. -2

II. -1

III. -4

IV. -5

5. a2-b

2 is a product of

I. (a+b)(a-b)

II. (a+b)(a+b)

III. (a-b)(a-b)

IV. None of these

38

6. Which of the following is the value of (x+ 1/x)2?

I. x2 + 1/x

2

II. x2 - 1/x

2

III. x2 + 1/x

2 + 2

IV. x2+ 1/x

2 + 2x

7. Which of the following is obtained by subtracting x2-y

2 from y

2- x

2?

I. -2(x2-y

2)

II. -2(x2 + y

2)

III. 2(x2+ y

2)

IV. 2(x2- y

2)

8. What degree does x3 - x

2y

2 - 8y

2+ 2 have?

I. 2

II. 3

III. 4

IV. 7

9. What is the value of 5x25

- 3x32

+ 2x-12

at x=1?

I. 0

II. 2

III. 4

IV. None of these

10. What is the product of (x+a) and (x+b)?

I. x2+ (a-b)x + ab

II. x2 + (a+b)x - ab

III. x2 + (a+b)x - ab

IV. x2 + (a+b)x + ab

SET 1

1. Find (2x + 3y)2 using algebraic identities.

2. Using suitable identities find (109)2.

3. Using suitable identity , find (5a – 7b)2.

4. Find 194 * 206 using suitable identity.

5. Use a suitable identity to find the product of (3a2 + 1/3)(3a

2 – 1/3).

6. The length and breadth of a rectangle are (3x2 + 2) and 2x+ 5 respectively. Find its area.

7. Find the value of: x2 – 1/5 at x= – 1.

8. What is the value of x2 + y

2 – 10 at x = 0 and y = 0?

9. Find the product of 9a, 4ab and -2a.

10. Simplify (a – b + c)(a – b – c).

11. Using identities evaluate: 8.56 X 11.60.

12. Using identities evaluate: (99)2.

13. Simplify x2(3x – 5) – 4 and find its value at x = – 2.

14. Evaluate the value of (95)2 using identities.

SET 2

1. Add: a + b + ab; b – c + bc and c + a + ac.

2. Verify the identity (x + a)(x + b) = x2 + (a + b)x + ab for a = 2, b = 3 and x = 4.

3. Find the volume of cuboid whose dimensions are (x2 – 2); (2x + 4) and (x - 3).

4. Write the terms and coefficients of 3 – xy + yz – xz.

39

5. Simplify: (a + b +c)(a + b – c).

6. Simplify the expression x(2x-1) + 5 and its value at x = – 2. (

7. Using suitable identities find (xy + 3p)2.

8. Subtract 5x2 – 6y

2 + 8y – 5 from 7x

2 – 5xy + 10y

2 + 5x – 4y.

MCQ ANSWERS

1. II

2. III

3. I

4. III

5. I

6. III

7. I

8. III

9. III

10. I

SET 1 ANSWERS

1. 4x2 + 12xy + 9y

2

2. 11881

3. 25a2 –70ab + 49b

2

4. 39964

5. 9a4 – 1/9

6. 6x3 + 9x

2 + 4x + 6

7. –4/5

8. –10

9. –72a2b

10. a2 – 2ab – b

2 –c

2

11. 99.296

12. 9801

13. 3x3 – 5x

2 – 4 , – 48

14. 9025

SET 2 ANSWERS

1. 2a + 2b + ab + bc + ac

3. 2x4 – 2x

3 – 16x

2 + 4x + 24

4. – 1, – 1 , – 1 )

5. a2

+ b2

+ 2ab – c2

6. 2x2 – x + 5 , 15

7. X2y

2 + 6xyp + 9p

2 )

8. 2x2 – 5xy + 16y

2 +5x – 12xy + 5

40

Ch-10: Visualising Solid Shapes

FACTS THAT MATTER

• The numbers with negative exponents also obey the following laws:

• Perspective is the an of representing solid objects on a flat surface to show its height,

width, distance, etc.

• 3-D bodies have different views from different positions.

• For a polyhedron,

F + V - E = 2

Where F stands for number of faces,

V stands for number of vertices and

E stands for number of edges.

• The relation F + V - E = 2 is called Euler‟s formula.

WE KNOW THAT Plane shapes have two dimensions (measurements) like length and breadth. That is why they

are called two dimensional (2-D) shapes. On the other hand, a solid object having three

dimensions like length, breadth, height (or depth) is called 3-D object. Triangles, rectangles,

circles, trapeziums, etc. are 2-D Figures while cuboids, cubes, cylinders, spheres, etc. are 3-D

objects. A solid is bounded by one or more surface and it always occupies some space. Its

surface (or face) can be a plane or a curved surface. When any two faces of a solid meet

together, we get a line segment called an edge. When three more faces of a solid meet at one

point, then the point is called a vertex of the solid.

Note: Perspective is an art of representing solid objects on flat surfaces to convey the

impression of distance, depth, width and height, using oblique lines and shading.

MCQ

1. How many number of faces does a solid sphere has?

I. 1

II. 2

III. Many

IV. None

2. How many number of vertices does a cone has?

I. 1

II. 2

III. 3

IV. None of these

3. How many number of faces does a hemisphere has?

I. 1

II. 2

III. Many

IV. None of these

4. Which of the following is a triangular pyramid having all the faces as equilateral triangle?

I. Rectangular pyramid

II. Square pyramid

III. Tetrahedron

IV. None of these

5. Which of the following is the number of vertices of sphere?

I. 0

41

II. 1

III. 2

IV. 4

6. Which of the following can be other name of a cylinder?

I. A triangular prism

II. A rectangular prism

III. A vertical prism

IV. A circular prism

7. If the base of a prism is a polygon of ï¿½nï¿½ sides, then which of the following is the

number of faces of the prism?

I. n+2

II. n+1

III. n

IV. n-1

8. If F, E and V represent the faces, edges and vertices respectively of a polyhedral then

which of the following is the Eulerï¿½s formula?

I. F - V + E = 2

II. F + V + E = 2

III. F + V - E = 2

IV. F + V =2 - E

9. Which of the following is the base of a tetrahedron?

I. A square

II. A rectangle

III. A circle

IV. A triangle

10. Which of the following is the other name of a cube?

I. A tetrahedron

II. A regular hexahedron

III. A square antiprism

IV. A cuboctanedron

SET 1

1. A polyhedron has 30 edges and 12 vertices. How many faces does it have?

2. A polyhedron has 5 faces and 6 vertices. How many edges does it have?

3. What is Euler‟s formula? Verify the Euler‟s formula for a pentagonal prism.

4. What is a least number of planes that can enclose a solid?

5. Name the simplest regular polyhedron and verify Euler‟s formula for it.

6. An icosahedrons is having 20 triangular faces and 12 vertices. Find the number of its

edges.

SET 2

1. How many vertices are there of a sphere?

2. How many vertices are there in a cone?

3. Which of the following is not a polyhedron? A cube, a prism, a cone or a cuboid?

4. How many faces, edges and vertices does a triangular prism have?

5. What is a triangular pyramid? What is a pyramid called if it has a square base?

6. A dice is a cube in which the number on the opposite faces must total 7. Draw its net.

42

7. The number of face of a pyramid is 5. Find the number of its vertices when its edges are

eight.

8. Draw the net of a triangular prism whose base is an equilateral triangle.

9. What is Euler‟s Formula? Using it find the number of faces of tetrahedron having vertices

as 4 and 6 edges.

MCQ ANSWERS

1. I

2. I

3. II

4. III

5. I

6. IV

7. II

8. III

9. IV

10. II

SET 1 ANSWERS

1. 20

2. 9

3. Euler‟s formula F + V - E)

4. 4

5. Triangular Pyramid

6. 30

SET 2 ANSWERS

1. 0

2. 1

3. Cone

4. F = 5, V = 6, E = 9

5. Square Pyramid

7. 5

43

Ch-11:Mensuration

STANDARD UMTS OF VOLUME

1 dm = 10 cm

1 dm3 or litre = 1 dm × 1 dm × 1 dm = (10 × 10 × 10) cm

3 = 1000 cm

3

1 cm = 10 mm

1m =100 cm

1m3 = (100 × 100 × 100) cm

3 = 1000000 cm

3

1 m3 = 1000 × 1000 cm

3 = 1000 litre

1 kilolitre = 1m3 = 1000 litre

(mm) millimeter, ( cm) centimetre, (dm) decimetre, (m) metre

Plane figures

The geometrical figures which have only two dimensions are called as the plane figures.

Booster 1

A square with sides of 1 cm has an area of 1 cm2.

Find the area of the shaded shape.

Explanation

The shape covers 11 squares, so its area is 11 cm2.

Booster 2

Find the area of the shaded triangle.

Explanation

The triangle covers 6 full squares marked F, and 4 half squares marked H. Area = 6 + 2 = 8

cm2.

44

Booster 3

Estimate the area of the shape shaded in the diagram.

Explanation

This is a much more complicated problem as there are only 9 full squares marked F, but

many other part square. You need to combine part squares that approximately make a whole

square. For example, the squares marked make about 1 full square; the squares marked ï¿½

make about 1 full square; the squares marked + make about 1 full square; the squares marked

• make about 1 full square. Thus the total area is approximately

9 + 4 = 13 cm2.

Triangle

(i)

Area of triangle

(ii) Area of an equilateral triangle

(iii) Area of an isosceles triangle :- base = b, equal side = a

45

Booster 4

The parallel sides of a trapezium are 20 cm and 10 cm. Its non-parallel sides are both equal,

each being 13 cm. Find the area of the trapezium.

Explanation

Let ABCD be a trapezium such that,

AB = 20 cm, CD = 10 cm and AD = BC = 13 cm

Draw CL || AD and CM || AB.

Now, CL || AD and CD || AB.

∴ ALCD is a parallogram.

⇒ AL = CD = 10 cm and CL = AD = 13 cm

In ΔCLB, we have CL = CB = 13 cm

∴ΔCLB is an isosceles triangle.

[∵BL= AB – AL = (20 – 10) cm = 10 cm]

Applying Pythagoras theorem in ∆CML, we have

CL2 = CM

2 + LM

2

132 = CM

2 + 5

2

CM2 = 169 – 25 = 144

Area of parallelogram ALCD = AL × CM = (10 × 12) = 120 cm2

Hence, Area of trapezium ABCD = Area of parallelogram ALCD + Area of ∆CLB = (120 +

60) cm2 = 180 cm

2

Booster 5

If the area of a rhombus be 24 cm2 and one of its diagonals be 4 cm, find the perimeter of the

rhombus.

Explanation

Let ABCD be a rhombus such that its one diagonal AC = 4 cm. Suppose the diagonals AC

and BD intersect at O.

Area of rhombus ABCD = 24 cm2

Thus, we have AC = 4 cm and BD = 12 cm

46

Since the diagonals of a rhombus bisect each other at right angle. Therefore, ∆OAB is right

triangle, right angled at 0.

Using Pythagoras theorem in ∆OAB, we have

AB2 = OA

2 + OB

2

AB2 = 22 + 62 = 40

Hence, perimeter of rhombus ABCD

Booster 6

Find the area of a regular octagon each of whose sides measures 4 cm.

Explanation

Area of the octagon

Booster 7

The diagram shows a lorry.

Find the volume of the load-carrying part of the lorry.

Explanation

The load-carrying part of the lorry is represented by a cuboid, so its volume is given by

V = 2 × 2.5 × 4 = 20m3.

Cube

A cuboid whose length, breadth and height are all equal is called a cube.

(i) Surface area of a cube . Since all the faces of a cube are squares of the same size i.e., for

a cube we have l = b = h. Thus if l cm is the length of the edge of side of a cube, then

Total Surface area of the cube = 2(l × l + l × l + l × l)

= 2 × 3l2 = 6l

2 = 6(Edge)

2

(ii) Lateral surface area of the cube = 2(l × l + l × l)

= 2(l2 + l

2) = 4l

2 = 4(Edge)

2

(iii) Volume of a cube = l × l × l = l3

Booster 8

Find the total surface of a hollow cylinder open at ends, if the length is 12 cm, the external

diameter 10 cm and thickness 2 cm.

47

Explanation

The outer radius (R) = 5 cm, Thickness = 2 cm

∴. Inner radius (r) = 5 cm – 2 cm = 3 cm

Outer curved surface of the cylinder

= 2πrh = 2 × π × 5 × 12 = 120πcm2

Inner curved surface of the cylinder 27πrh

= 2 × π × 3 × 12 = 72 π cm2.

Both ends of the cylinder will be of the shape, as shown in figure (ii).

∴. Area of one of end of the cylinder = πR2 – πr

2

= π × 52 – π × 3

2 = 25π – 9π = 16π cm

2.

Area of both ends = (2 × 16π) cm2 = 32π cm

2

∴Total surface of the cylinder = External curved surface + Internal curved surf ace + 2 (Area

of the base of the ring)

Remark : It is advisable to put the value of π is the end in such calculations.

Circular cone

There are many objects around us which are conical in shape like an ice-cream cone, a

conical, tent a birthday cap etc. These objects are of the shape of a right circular cone.

AOC is right angled at O.

By Pythagoras theorem,

AC2 = AO

2 + OC

2

i.e. l2 = h

2 + r

2

(i) Area of shaded region

(ii) Total surface area of cone = Curved surface area of cone + Area of the circular base

= πrl + πr2

= πr (r + l) sq. units.

(iii) Volume of a cone (where „r‟ r is the radius and „h‟ is the height of cone).

Sphere and hemisphere

48

Sphere: The set of all the points in space which are equidistant from a fixed point is called a

sphere.

Hemisphere: A lane through the centre of a sphere divides the sphere into two equal parts and

each part is called a hemisphere.

(i) Curved surface area of hemisphere = 2πr2

(ii) Total surface area of hemisphere 2πr2 + πr

2 = 3πr

2

Booster 9

If radius of the base of a cone is 140 dm and its slant height is 9 m. Find the

(i) curved surface area

(ii) total surface area

Explanation

Radius of the base of the cone (r) = 140 dm = 14 m

Slant height (l) = 9 m

(i) Curved surf ace area of the cone

(ii) Total surface area of cone = πrl + πr2

Question 12

The surface area of a sphere is 2464 dm2. Find its diameter.

Solution

Surface area of a sphere = 2464 dm2

Therefore, 4πr2 = 2464

Diameter of the given sphere = 2 × 14 = 28 dm.

Booster 10

The dome of a building is in the form of a hemisphere of radius 63 dm.

Find the cost if it is to be painted at the rate of Rs 5 per m2.

Explanation

Radius of the hemisphere = 63 dm

49

Surface area of hemisphere = 2πr2

Booster 11

The earth taken out while digging a pit, is evenly spread over a rectangular field of length 90

m, width 60 m.

If the volume of the earth dug is 3078 m3, find the height of the field raised.

Explanation

3078 m3 = 90 m × 60m × h

height of field raised = 0.57 m

MULTIPLE CHOICE QUESTIONS

1. Which of the following is the area of a rhombus?

(i) Product of its diagonals

(ii) 1/2 (sum of its diagonals)

(iii) 1/2 (Product of its diagonals)

(iv) 2 (Product of its diagonals)

2. If the edge of a cube is 1 cm then which of the following is its volume?

(i) 6 m3

(ii) 3 m3

(iii) 1 m3

(iv) none of these

3. If the parallel sides of a parallelogram are 2 cm apart and their sum is 10 cm then its area

is:

(i) 20 cm2

(ii) 5 cm2

(iii) 10 cm2

(iv) none of these

4. Which of the following has its area and perimeter numerically equal?

(i) an equilateral triangle of side 1 cm

(ii) a square of side 1 cm

(iii) a square of side 1 cm

(iv) a regular pentagon of side 1 cm.

5. If the edge of a cube is 1 cm then which of the following is its total surface area?

(i) 1 cm2

(ii) 4 cm2

(iii) 6 cm2

(iv) none of these

6. Which of the following is equal to 1 kilolitre?

(i) 1000 milliliters

(ii) 100 dm3

(iii) 1 dm3

(iv) 1000 dm3

50

7. If the dimensions of a room are I, b and h, (∴. l → length, l → breadth and h → height)

them which of the following is the area of its four walls?

(i) 2 h(1 + b)

(ii) 2 h (1 + h)

(iii) 2 1(h + h)

(iv) 2 h + 1 + b

8. If the dimensions of a room are 2 m, 3 and 4 m then which of the following is the number

of cubes of size which can he placed is the room?

(i) 960

(ii) 672

(iii) 676

(iv) 576

9. If base area of a room 12 m2 and height is 3 m then its volume is:

(i) 4 m3

(ii) 36 m3

(iii) 12 m3

(iv) 18 m3

10. Two identical cubes each of total surface area of 6 cm2 are joined end to end. Which of

the following is the total surface area of the cuboid so formed?

(i) 12 cm2

(ii) 18 cm2

(iii) 10 cm2

(iv) 8 cm2

SET 1

1. The length, breadth and height of a cuboid are 20 cm, 15 cm, 10 cm respectively.

Find its total surface area.

2. In a building there are 24 cylindrical pillars with each having a radius 28 cm and height

4 m. Find the cost of painting the curved surface area of all pillars at the rate of

Rs. 8 per meter square.

3. Find the height of cylinder whose radius is 7 cm and TSA is 968 cm2.

4. A box is in the form of cuboid of dimensions (80 x 30 x 40) cm. The base, the side faces

and back faces are to be covered with a colored paper.

Find the area of paper needed.

5. The LSA of a hollow cylinder is 4224 cm2. It is cut along its height and formed

rectangular sheet of width 33 cm. find the perimeter of rectangular sheet.

6. A roller takes 750 complete revolutions to move once over a level of road. Find the area

of road if the diameter of the roller is 84 cm and length is 1 m.

SET 2

1. If each side of a cube is doubled then how many times will its surface area increase?

2. Find the height of a cuboid whose base area is 180 cm2 and volume is 900 cm

3.

3. A cuboid is of dimensions (60 x 48 x 30)cm. How many small cubes with side 6 cm

can be placed in the given cuboid?

51

4. Find the height of the cylinder whose volume is 1.54 m3 and diameter of base is 140

cm.

5. Find the area of trapezium where length of parallel sides are 15 cm and 25 cm and the

height is 12 cm.

6. Find the area of rhombus whose diagonals are 8cm and 10cm.

7. If each side of a cube is doubled, how many times will its volume increase?

8. A rectangular sheet of paper is having measures 11 cm x 4 cm. it is folded without

overlapping to make a cylinder of height 4 cm. Find the volume of the cylinder

MCQ ANSWERS

SET 1 ANSWERS

1. 1300cm2

2. Rs 1351.68

3. h=15cm

4. 8000cm2

5. 322cm

6. 1980cm2

SET 2 ANSWERS

1. 4 times

2. 5cm

3. 400

4. h = 1cm

5. 140 cm2

6. 40 cm2

7. 8 times

8. 38.5 cm3

52

Ch-12: Exponents and Powers

FACTS THAT MATTER

• The numbers with negative exponents also obey the following laws:

(i) xm × x

n = x

m + n

(ii) xm ÷ x

n = x

m – n

(iii) xm × b

m = (x

b)

m

(iv) x0 = 1

• A number is said to be in the standard Form, if it is expressed as the product of a number

between 1 and 10 and the integral power of 10.

• Very small numbers can be expressed in standard form using negative exponents.

WE KNOW THAT

When we write 54, it means 5 × 5 × 5 × 5, i.e. 5 is multiplied 4 times. So 5 is the base and 4 is

the exponent.

We express very small or very large numbers in standard form (i.e scientific notation)

for example:

POWER WITH NEGATIVE EXPONENTS

For a non-zero integer x, we have

MCQ

1. What is the value of (-1)-1

?

I. 0

II. -1

III. 1

IV. None of these

2. Which of the following is the value of 'm' in 6m

/ 6-3

= 65?

I. -3

II. -2

III. 3

IV. 2

3. Which of the following is the standard form of 0.00001275?

I. 1.275 x 10-5

II. 1.275 x 105

III. 127.5 x 10-7

IV. 127.5 x 107

4. Which of the following is used as a form of 5.05 * 106?

I. 505000

II. 505000000

III. 5050000

53

IV. 50500000

5. For which of the following is m= 8?

I. 2m

= 256

II. m + 7 = 8

III. 5m = 625

IV. None of the above

6. 1 micron = 1/1000000 m. which of the following is its standard form?

I. 1.1 x 10-5

II. 1.6 x 10-5

III. 0.1 x 10-6

IV. 1.0 x 10-6

7. [(1 / 2)-1

+ (2 /3 )2 - (3/4)

0]

-2 is equal to:

I. 81/484

II. 81/169

III. 169/81

IV. 16/81

8. Which of the following = (100 - 990) x 100?

I. 10000

II. 100

III. 9900

IV. 99000

9. What is the reciprocal of (-3 / 4)0?

I. -1

II. 1

III. -4/3

IV. 4/3

10. Which of the following is the value of (4 / 5)-9

/ (4 / 5)-9

?

I. (4/5)18

II. 4/5

III. 1

IV. (5/4)9

SET 1

1. Simplify (1 / 32)

3

2. Evaluate: (5-1

x 82) / ( 2

-3 x 10

-1)

3. Find the value of 'm' for which 6m / 6

-3 = 6

5?

4. Evaluate [(1/2)-1

- (1/3)-1

]-1

5. Simplify: (-3)5 x (5/3)

5.

6. Add 7 x 10-6

and 129 x 10-7

7. The size of a plant cell is 0.00001275 m. express it in standard form.

8. If the thickness of a paper sheet is 0.0016 cm, find the thickness of 100 sheets.

Express the answer in standard form.

54

SET 2

1. Find the value of 5-3

x 1/53

2. Simplify 25 / 2

-6

3. Express 4-3

as a power with base 2.

4. Simplify and write the answer in exponential form: (25 / 2

8)

5 x 2

-5

5. Find m so that (-3)m+1

x (-3)5 = (-3)

7

6. Find the value of (2/3)-2

.

7. Simplify: (5/8)-7

x (8/5)-5

8. Simplify (-4)-10

x (-4)5

MCQ ANSWERS

1. II

2. IV

3. I

4. III

5. I

6. IV

7. II

8. III

9. II

10. III

SET 1 ANSWERS

1. 1/729

2. 128

3. m=2

4. -1)

5. -55

6. 19.9 x 10-6

7. 1.275 x 10-5

8. 1.6 x 10-

SET 2 ANSWERS

1. 1/3125)

2. 211

3. 2-6

4. ½20

)

5. m=1

6. 9/4

7. 64/25

8. 1/-45

55

Ch-13:Direct and Inverse Proportions

FACTS THAT MATTER

• If two quantities x and y vary (change) together in such a manner that the ratio of their

corresponding values remains constant, then x and y are said to be in direct proportion.

• if two quantities x and y vary (change) in such a manner that an increase in x causes a

proportional decrease in y (and vice versa), then x and y are said to be inverse proportion.

• If x and y are in a direct proportion, then

• If x and y are in an inverse variation, then xy = constant.

WE KNOW THAT

The value of a variable is not constant and keeps on changing. There are many quantities

whose value varies as per the circumstances. Some quantities have u relation with oilier

quantities such that when one changes the other also changes. Such quantities are inter-

redated. This is gilled a variation. Variation is of two types: (i) Direct variation and (ii)

Inverse variation.

DIRECT PROPORTION

If two quantities are related in such a way that an increase in one quantity leads a

corresponding increase in the other and vice versa, then this is a ease of direct variation. Also,

a decrease in one quantity brings a corresponding decrease in the other.

Two quantities x and y are said to be in direct proportion, if

Note: I. In a direct proportion two quantities x and y vary with each other such that

remains constant.

II. is always a positive number.

III. or k is called the constant of variation.

MULTIPLE CHOICE QUESTIONS 1. If „x‟ and „y‟ are in a direct proportion then which of the following is correct?

(i) x – y = constant

(ii) x + y = constant

(iii) x × y = constant

(iv)

2. If „x‟ and „y‟ are in an inverse variation then which of the following is correct?

(i) x – y = constant

(ii) x + y = constant

(iii) xy = constant

(iv)

3. If „A‟ can finish a work in „n‟ days then part of work finished in 1 day is:

(i) 1 – n

(ii)

(iii) n – 1

(iv) none of these

56

4. If amount of work completed by „A‟ in one day is then the whole work will be finished

by „A‟ is:

(i) n days

(ii) 1 – n days

(iii) n – 1 days

(iv) none of these.

5. If an increase in one quantity brings about a corresponding decrease in the other and ice

versa, then the two quantities vary:

(i) directly

(ii) inversely

(iii) sometimes directly and sometimes inversely

(iv) none of these.

6. “If speed is increased then time to cover a fixed distance would be less”. This is a case of:

(i) inverse variation

(ii) direct variation

(ii) direct variation

(iv) none of the above.

8. Which of the following is not a case of direct variation?

(i) Number of sheets of some kind are increased then their total weight its increased

(ii) More quantity of petrol is required to travel more distance with a fixed speed

(iii) More fees would be collected if number of students are increased in a class

(iv) Time taken will be less if number of workers are increased to complete the same work.

9. Which of the following is ease of direct variation;

(i) If the length of radius is increased the circumference will be increased

(ii) If number of students is a hostel are increased then the fixed food provision will last for

less days

(iii) For fixed duration, more the periods, lesser will be the duration of one period

(iv) In case of a cylindrical vessel, lesser the diameter more is the level of water in it.

10. If x and y vary inversely. Then using the following table?

The value of x for y = 10 is

(i) 10

(ii) 40

(iii) 15

(iv) 20

SET 1

1. A contractor estimates that 5 persons complete a task in 4 days. If he uses 4 persons

instead of 5, how long should they take to complete the task?

2. A school has 9 periods a day each of 50 minutes duration. How many period will

there be, if the duration of every period is reduced by 5 minutes?

3. A machine can fill 420 bottles of mineral water in 3 hours. How many bottles can be

57

filled in 5 hours?

4. In a model of a ship. the mast is 9 cm high, while the mast of the actual ship is 12 m

high. If the length of the model ship is 21 cm, then how long is the actual ship?

5. If a kg of sugar contains 2.25 x 107 crystals. How many sugar crystals are there in

2 kg of sugar?

SET 2

1. 6 pipes are required to fill a rank in 1 hour 20 minutes. How long will it take if only 5

pipes of the same type are used?

2. There are 100 students in a hostel. Food provision for them is for 20 days. How long

will these provisions last, if 25 more students join the group?

3. If 15 workers can build a wall in 48 hours. How many workers will be required to do

the same work in 30 hours?

4. The principal sanctioned a certain amount to the librarian to purchase some

Mathematics books for the school library. She could buy 80 books casting Rs 90 each

from the local book seller. There she approached to the publisher who offered her a

20% discount, Find the number of copies of Mathematics books which she could buy

from the publisher for the sanctioned money.

5. A mixture of paint is prepared by mixing 1 part of green pigments with 6 parts of the

base. In the following table, find the parts of base needed to be added.

6. A machine fills 540 bottles in six hours. How many bottles will it fill in five hours?

7. Jagmeet has a road map with a scale of 1 cm = 20 km. He drives on a road for 72 km.

What would be his distance covered in the map?

8. In a PG House, the food provision for 20 persons is for 10 days. How long would the

food provision last if there were 5 more persons in that PG house?

MCQ ANSWERS

SET 1 ANSWERS

1) 5 2) 10 3) 700 4) 28m 5 ) 4.45 x 107

SET 2 ANSWERS

1) 96 min 2) 16 days 3) 24 workers 4) 100 books 5 ) 24,30,36 6) 450 bottles

7) 3.6 cm 8) 8 days

58

Ch-14 ;Factorisation

FACTS THAT MATTER

• Factorisation means to write an expression as a product of' its factors.

• Like prime factors, an irreducible factor, a factor which cannot be expressed further as a

product of factors.

• Some expression can easily be factorised using these identities:

I. a2 + 2ab + b

2 = (a + b)

2

II. a2 – 2ab + b

2 = (a – b)

2

III. a2 – b

2 = (a – b)(a + b)

IV. x2 + (a + b)x + ab = (x + a)(x + b)

• The number 1 is a factor of every algebraic term also, but it is shown only when needed.

• When factorisation of x2 + (a + b)x + ab is done by splitting the middle term, the two

numbers which give the product ab and (a + b) as the coefficient of x have to be chosen very

carefully with correct sign.

Note: In case of factorisation of a term of an expression, the word 'irreducible' is used in

place of 'prime'.

For example, 6pq = 2 × 3 × pq is not the irreducible corm because pq can further be

factorised as p × q, i.e. the irreducible forn of 6pq = 2 × 3 × p × q.

Example: Write 10xy as irreducible factor form.

Solution: We have 10 = 2 × 5

xy = x × y

10xy = 2 × 5 × x × y

MULTIPLE CHOICE QUESTIONS

1. Which of the following is the common factor of 21 x2y and 35 xy

2?

(i) 7

(ii) xy

(iii) 7 xy

(iv) none of these.

2. Which of the following arc the factors of 1 – x2?

(i) (x + l) (x – I)

(ii) (1 – x) (1 + x)

(iii) (1 – x) (1 – x)

(iv) (1 + x) (1 + x).

3. Which of the following is the common factor of:5xy, 3pqr and 40 xyz?

(i) 5

(ii) 0

(iii) xy

(iv) 1

4. Which of the following is quotient obtained on dividing –18 xyz2 by –3 xz?

(i) 6 Yz

(ii) –6 yz

(iii) 6 xy2

(iv) 6 xy

5. Which of the following is quotient obtained on dividing (x2 – b) (x – a) by –(x – a)?

59

(i) (x2 – b)

(ii)

(iii) –(x2 – b)

(iv) – (x + a)

6. Which of the following are the factors of a2 + ab + bc + ca

(i) ab – a – b + 1 = (1 – a)(1 – b)

(ii) ab – a – b + 1 = (a – 1)(b – 1)

(iii) ab – a – b + 1 = (1 – a)(b – 1)

(iv) ab – a – b + 1 = (a – 1)(1 – b)

7. (y – x) (y + x) is equal to which of the following:

(i) y2 – yx

(ii) yx – x2

(iii) y2 – x

2

(iv) x2 – y

2

8. Which of the following are the factors of a2 + ab +bc + ca

(i) (b + c) (c + a)

(ii) (a + b) (a + c)

(iii) a(a + b + c)

(iv) (a + b) (b + c).

9. Which of the following is the factorisation of x3 – x?

(i) x(x – x2)

(ii) x[(1+ x) (1 – x)]

(iii) x(x2 – x)

(iv) x[(x + 1) (x – 1)]

10. Which of the following is equal to x3 – 225x

(i) x(1 – 15x) (1 + 15x)

(ii) x(x – 15) (x + 15)

(iii) x(1 – 15x) (1 – 15x)

(iv) x(1 + 15x) (1 – 15x).

SET 1

1. Simplify: –45p3 ÷ 9p

2

2. Simplify: 4x2y

2(3z – 24), 36xy(z – 8)

3. Divide: 81x3(50x

2 – 98) by 27x

2 (5x + 7)

4. What is the remainder when z(5z2 – 80) is divided by 5z(z – 4):

5. What is the quotient when 44(x4 – 5x

3 – 24x

2) is divided by 22x(x – 8):

6. Which of the following is factorization of (1 – x4)

7. By what should a4 – b

4 be divided to get quotient (a

2+ b

2) (a – b) and, remainder as 0.

8. Is (a – 1) (b – 1) the factorization of (ab – a – b +1) . Explain

60

SET 2

1. Factorise: 54x2 + 24x

3 – 30x

4

2. Factorise: 2x2yz + 2xy

2z + 2xyz

3. Factorise: 30xy – 12x + 10y– 4

4. Regroup the terms and factorise: z – 19 + 19xy – xyz

5. Factorise: 100x2 – 80xy + 16y

2

6. Factorise: 16x4 – 81y

4

7. Factorise: x2 + 6x + 8

8. Factorise: 49y2 – 1

9. Divide 10(x3y

2x

2 + x

2y

3z

2 + x

2y

2z

3) by 5x

2y

2z

2

10. Simplify: 12(y2 + 7y + 10) ÷ 6(y + 5)

MCQ ANSWERS

SET 1 ANSWERS SET 2 ANSWERS

1. - 5p 1. 6x2 (5x -9) (x -1)

2. xy/3) 2. 2xyz (x + y + z)

3. 6x(5x -7) 3. 2 (5y-2) (3x +1)

4. z + 4) 4. (1 – xy) ( z -19)

5. 2x(x+3) 5. 4 ( 5x -2y)2

6. (1 + x2) ( 1 – x) ( 1 + x) 6. (4x

2 + 9y

2) (2x + 3y) (2x – 3y)

7. a + b) 7. (x + 4)( x + 2)

8. (7y + 1) (7y -1)

9. 2(x + y +z )

10. 2 (y + 2)

61

Ch-15: Introduction to Graphs

FACTS THAT MATTER

• Pictorial and graphical representation are easier of understand.

• Bar graphs and pie graphs are used for comparing categories and parts respectively.

• A histogram is a bar graph that shows data in intervals.

• A line graph displays data that change continuously over periods of time.

• A line graph being wholly an unbroken line is called a linear graph.

• For fixing a point the graph sheet we take two mutually perpendicular lines called axes.

• The horizontal line of the axes is called the x-axis and the vertical line is called the y-axis.

WE KNOW THAT

The purpose of drawing a graph is to show numerical facts in usual form so that they can

be understood quickly, easily and clearly. We also know that a bar graph is used to show a

comparison among categories. A bar graph may consists of two or more parallel horizontal or

vertical bars of equal width.

A pie-graph is used to compare various parts of a whole using a circle. A histogram is a

bar-graph that shows data in intervals. It has adjacent bars over the intervals as shown below.

A line-graph is called a linear-graph and it displays data that changes continuously over

periods of time.

MCQ

1. (o, y) are the co-ordinates of a point lying on which of the following?

(i) origin

(ii) x-axis

(iii) y-axis

(iv) none of these.

2. The point (3, 2) is nearer to:

(i) x-axis

(ii) y-axis

(iii) origin

(iv) none of these.

3. The point (–5, 6) is nearer it:

(i) x-axis

(ii) y-axis

(iii) origin

(iv) none of these.

4. The point (–3, –3) is

62

(i) nearer to x-axis

(ii) y-axis

(iii) near to origin

(iv)equidistant from x-axis and y-axis.

5. The point (0, 4) lies on which of the following:

(i) x-axis

(ii) y-axis

(iii) origin

(iv) none of these.

6. The point (–3, 0) lies on which of the following?

(i) x-axis

(ii) y-axis

(iii) origin

(iv) none of these.

7. The points (–3, 2) and (2 , –3) represent:

(i) different points

(ii) same point

(iii) the origin

(iv) none of these.

8. By joining (–1, –1), (0, 0) and (3, 3) represent:

(i) a triangle

(ii) a curved line

(iii) a straight line passing through origin

(iv) a straight line not passing through origin.

9. By joining (–3 , 2) , (–3 , –3) and (–3, 4), which of the following is obtained?

(i) a triangle

(ii) A straight line not passing through origin

(iii) A straight line passing through origin

(iv) none of these.

10. Which of the following points lies on y-axis?

(i) (–4 , 0)

(ii) (4 ,0)

(iii) (0 , –4)

(iv)(–4 , 4)

SET 1

1. Draw the points (5, 4) and (4, 5). Do they represent the same point?

2. Draw a line passing through (2, 1) and (1, 2). Find the coordinates of the points at which

this line meets the x-axis and y-axis.

3. Draw the graph for the following table of values of time (in hours) and distances (in km)

covered by a car.

Time (in hours) 7:00 8:00 9:00 10:00

Distance (in km) 60 120 180 240

From the graph, find:

(i) The distance covered by the car during the period 7:00 to 8:00.

(ii) At what time the car would have covered 180 km?

4. Find the coordinates of the vertices of ï¿½ï¿½ABC given in graph. Draw a triangle by

taking vertices as A(5, 2), B(1, 2) and C(1, 5).

63

5. Following graph describes the movement of a car from a town A to town D. Study the

graph a l answer the following questions:

(i) What is the distance between town A and town D?

(ii) What did the car start from town A?

(iii) Where did the car stop and for what duration?

(iv) How long did it take to go from town C to town D?

6. Read the following `time-temperature' graph of a place and answer the questions given

below.

(i) What was the temperature at 7 a.m.?

(ii) When the temperature was maximum?

(iii) When was the temperature 40°C?

(iv) During which period, the temperature remained constant?

64

SET 2

1. The graph shows the temperature of a patient recorded before noon. Read it and answer

the following questions.

(i) What was patients temperature at 9 a.m.?

(ii) What the highest temperature of the patient?

(iii) When was the patient's temperature lowest?

(iv) During which period, the patient's temperature remained constant?

2. The graph shows the yearly sales figure of a shoe manufacturing company.

(i) What were the sales in 2000?

(ii) In which year the sales were maximum?

(iii) What is the difference between the sales in the year 2003 and 2005?

3. Draw a linear graph for the following data:

Month May June July August

Rainfall. (in cm) 5 7 4 6

4. Plot the points on a graph: A(4. 9); B(6, 0); C(7, 7); D(2, 4)

5. Plot the points A(4, 3). B(4, 0), (4, ï¿½2), (4, 6) and join them. Do they lie on the same

line?

MCQ ANSWERS

65

Ch-16: Playing with Numbers

6.1 Numbers in general form

A 2 digit number can always be written as a combination of 2 different numbers.

for eg:–

65 = 10 × 6 + 5 → 6 is at tens place and 5 is at ones place.

23 = 10 × 2 + 3 → 2 is at tens place and 3 is at ones place

Thus, any two digit number can be written in a general form as 10 × x + y

Similarly,

572 = 5 × 100 + 7 × 10 + 2

123 = 1 × 100 + 2 × 10 + 3

The three digit number xyz can be written as 100 × x + 10 × y + z

6.2 Puzzles & games

Puzzles and games are a source of entertainment and education that makes interesting and

challenging situations.

Reversing the digits of a two-digit number

Addition

Step-1 : Choose any 2-digit number of the form 10 x + y.

Step-2 : Reverse the digits to get a new number i.e.,

Step-3: Add the reversed number to the original number.

(10x + y) + (10y +x) = 11x + 11y = 11(x +y)

Step-4 : Divide the answer by 11.

11(x + y) ÷ 11 = (x + y)

Result: There is no remainder.

Remark: The sum of a two-digit number and the number formed by reversing its digits is

exactly divisible by 11 and the quotient obtained is the sum of the digits of the original 2-

digit number.

Adding both the number, we get 36 + 63 = 99, which is exactly divisible by 11

∴ Hence proved.

Subtraction

Step-1 : Choose a two digit number in the form 10x + y.

Step-2 : Reverse the digits to get a new number in the form 10y + x.

Step-3 : Subtract both the numbers.

(10y + x) – (10x + y) = 9y – 9x = (9 (y – x)


9(y – x) ÷ 9 = (y – x)


Remark: The difference of a two digit number and its reversed number is exactly divisible

by 9 and the quotient obtained is either the difference of the digits of the original 2-digit

number or 0.

Reversing the digits of a three-digit number

Addition

Step-1 : Choose any three-digit number xyz in the form 100x + 10y + z

Step-2 : From 2 more numbers in a way

66

yzx = 100z + 10x + y.

Step-3 : Add all the three numbers

(100x + 10y + z) + (100y + 10z + x) + (100z + 10x + y)


111 (x + y + z) ÷ 111 = (x + y + z).


Remark: The sum of a 3-digit number and the number formed by arranging its digits in such

a way that each digit occupies a place value only once, is exactly divisible by 111 and the

quotient obtained is the sum of the digits of the original 3-digit number.

Step-1 : Take any three-digit number xyz in the form 100x + 10y + z.

Step-2 : Reverse the digits : zyx = 100z + 10y + x.

Step-3 : Substract both the numbers.

(100x + 10y + z) – (100z + 10y + x) = 99x – 99z = 99(x – z)


Remark: The difference of a 3-digit number and the number formed by reversing the digits

is exactly divisible by 99 and the quotient so obtained is either the difference between the

hundredth digit and the ones digit of the original 3-digit number or 0.

6.3 Letter for digits

Every game has same rules. So, there are some rules for such puzzles also. There are two

rules for solving them.

(i) Each letter in the puzzle must stand for just one digit. Each digit must be represented by

just one letter.

(ii) The first digit of a number cannot be zero.

Numerical Ability 6.1

Solve for Q:

Solution: From the addition above, we can say Q + 3 = 1. For this, Q must be equal to 8. So, the puzzle

becomes:

Numerical Ability 6.2

Find the digits A, B and C.

Solution:

Since the oneï¿½s digit of B × 3 is B, it must be B = 0 or B = 5.

Now, for A

67

If A = 1

These two are not possible because C cannot be zero.

If A = 2

These two are not possible because C cannot be zero.

If A = 3

If is not possible because A is not zero and C cannot be zero.

If A = 5

6.4 Tests of divisibility

Divisible by 2

A number is divisible by 2, if its unit digit is 0 or divisible by 2 i.e., 2, 4, 6, 8

DIY.

Find the condition when a two-digit number xy and a 3-digit number xyz will be exactly

divisibly by 2.

Explanation 2-digit number xy can be written as 10x + y. 2 will always divide 10x.

So, 10x + y will be exactly divisible by 2 if y = 0, 2, 4, 6 or 8.

A 3-digit number xyz can be written as 100x + 10y + z. We can say, 2 will always divide

100x and 10y. So,

100x + 10y + z will be divisible by 2 if z = 0, 2, 4, 6 or 8.

Divisible by 3 A number is divisible by 3, if the sum of its digits is divisible by 3.

68

Divisible by 4 A number is divisible by 4, if the sum its last 2 digits is divisible by 4.

Divisible by 5 A number is divisible by 5, if the digit in its units place is 5 or zero.

Divisible by 9 A number is divisible by 9, if the sum of its digits is divisible by 9.

Divisible by 10 If number is divisible by 10, if the digit at unit‟s place is zero.

MCQ

1. If M is a number such that M ÷ 5 gives a remainder of 1, then which of the following is

the one‟s digit of M?

(i) 1

(ii) 6

(iii) 1 or 6

(iv) none of these.

2. A number divisible by 9 is also divisible by:

(i) 3

(ii) 6

(iii) 11

(iv) none of these.

3. If [3X 74] is a number divisible by 9, then the least value of X is:

(i) 1

(ii) 2

(iii) 3

(iv) 4

4. If [1X 2Y 6Z] is a number divisible by 9, then the least value of X + Y + Z is:

(i) 0

(ii) 1

(iii) 6

(iv) 9

5. The number 28221 is divisible by which of the following:

(i) 2

(ii) 3

(iii) 6

(iv) 9

6. Which of the following is one‟s digit of a number, when divided by 5 gives a remainder

of 3?

(i) 8

(ii) 3

(iii) 3 or 8

(iv) none of these.

69

7. If the 4-digit number 2X Y7 is exactly divisible by 3, then which of the following is the

least value of (X + Y)?

(i) 3

(ii) 4

(iii) 6

(iv) 6

8. If a number is divisible by 2, then which of the following cannot be a one‟s digit in it?

(i) 0

(ii) 1

(iii) 2

(iv) 4

9. If a number is divisible by 5, then which of the following can be its one‟s digit?

(i) 2

(ii) 3

(iii) 4

(iv) 5

10. If a number is divisible by 10, then which of the following can be its one‟s digit?

(i) 0

(ii) 1

(iii) 3

(iv) 5

SET 1

1. Check the divisibility by 21436587 by 9.

2. Check the divisibility of 152875 by 9.

3. If the three digit number 24x is divisible by 9, what is the value of x?



SET 2

1. If 42x5 is a multiple of 9 and x is a digit, then find the value of x.

2. Is 10011 divisibly by 3?

3. If 3x12 is a multiple of 3 and x is digit, then find the value of x.

4. If 35x is a multiple of 9 and x is digit, then find the value of x.

5. The usual form of the number 9 × 100 + 7 × 1

(a) 97

(b) 9007

(c) 907

(d) 16

6. A is a digit and 3A15 is a multiple of 9. Which of the following can be the value of A?

(a) 1 or 9

(b) 0 or 8

(c) 0 or 7

(d) 0 or 9

70

7. The value of A and B in

(a) A = 7, B = 6

(b) A = 7, B = 7

(c) A = 7, B = 5

(d) A = 7, B = 4

8. The value of A and B in

(a) A = 9, B = 9

(b) A = 7, B = 9

(c) A = 7, B = 7

(d) A = 9, B = 7

MCQ ANSWERS

SET 1 ANSWERS

1) It is divisible by 9 2) Not divisible 3) 3 4) Divisible by 3 5 ) Not divisible

SET 2 ANSWERS

1) 8 2) Yes 3) 0,3,6,9 4) 1 5 ) c 6) d 7) a 8) b

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DON BOSCO SCHOOL - ALAKNANDA - NEW DELHI- 110 019 CLASS 8 SAMPLE PAPER M.M : 50 1. Write: (2 marks) (a) The rational number that does not have a reciprocal. (b) The rational numbers those are equal to their reciprocals. (c) The rational number that is equal to its negative. (d) The additive inverse of ‘’ a ‘’ , where a is a rational number

2. Find 10 rational numbers between 3

5 and

3

4 . (2 marks)

3. Solve for x, if x – 3 = 9. (2 marks) 4 .Draw and find the number of diagonals of a regular pentagon. (2 marks)

5. How many sides does a regular polygon have if each of its interior angles is 1700 ?

6. Find 2

5 x

3

7 –

1

14 –

3

7 x

3

5 . (3 marks)

7. Multiply 6

7 by the reciprocal of

−5

31 . (3 marks)

8. Represent −3

5 on the number line. (3 marks)

9. The sum of three consecutive multiples of 8 is 888. Find the multiples. (3 marks) 10. Two adjacent angles of parallelogram ABCD are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram ABCD. (3 marks) 11. ABCD is a rectangle. Its diagonals meet at O. If AO = 4x + 5 and DO = 3x + 6, find x and also the lengths of the diagonals AC and BD. (5 marks) 12. Sum of the digits of a two digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two digit number? (5 marks)

13. Solve for m : If m – 𝑚−1

2 = 1 –

𝑚−2

3 . (5 marks)

14. Construct Quadrilateral JUMP, in which JU = 3.5cm, UM = 4cm, MP = 5cm, PJ = 4.5cm and PU = 6.5cm. Also write the steps briefly. (5 marks) 15. Construct a rhombus whose diagonals are 5.2cm and 6.4cm long. Write the steps briefly. (5 marks)

72

DON BOSCO SCHOOL - ALAKNANDA - NEW DELHI- 110 019

CLASS 8 SAMPLE PAPER M.M : 50

PART A ( each question carries 2 marks )

1 .Write: (i)The multiplicative inverse of 31

3

(ii)a rational number that is equal to its negative.

2. Find ten rational numbers between 2

5 and

1

2

3. Solve for x, if 𝑥

3 + 1 =

7

15

4. Identify all the quadrilaterals that have: (i)four sides of equal length, (ii) four right angles 5. Can a quadrilateral ABCD be a parallelogram if :- ∠ B + ∠ D = 1800 , (ii) AB = DC = 8cm , AD = 4cm and BC= 4.4cm. Give reasons in each case. PART B ( each question carries 3 marks )

6. Find:- 3

7 +

−6

11 +

−8

21 +

5

22

7. Represent −5

6 and

3

6 on the same number line.

8. Find using properties: 2

5 X

−3

7 –

1

14 –

3

7 X

3

5

9. The ages of Rahul and Haroon are in the ratio 7 : 5.Four years later the sum of their ages will be 56 years. What are their present ages? 10. HELP is a parallelogram whose diagonals intersect at O. Given that OE = 4cm and HL is 5cm more than PE. Find OH .Draw a neat figure and also state the properties you used. Part C ( each question carries 5 marks ) 11.ABCD is a parallelogram whose diagonals intersect at O. If OA = 16cm, OC = ( x + y)cm, OB =( y + 7)cm and OD = 20cm, find the lengths of x and y. Draw a neat figure and also write properties used. 12. Lakshmi is a cashier in a bank. She has currency notes of denominations Rs.100, Rs.50 and Rs10, respectively. The ratio of the number of these notes is 2 : 3 : 5. The total cash with Lakshmi is Rs.4,00,000. How many notes of each denomination does she have?

13.Slove and check: 6𝑥+1

3 + 1 =

𝑥−3

6

14. Half of a herd of deer are grazing in the field and three fourth of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the number of deer in the herd. 15. Construct quadrilateral GOLD in which, OL = 7.5cm, GL = 6cm, GD = 6cm, LD = 5cm, OD = 10cm. Write the steps briefly and also find the length of the fourth side of quadrilateral .

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DON BOSCO SCHOOL - ALAKNANDA - NEW DELHI- 110 019 CLASS 8 SAMPLE PAPER M.M : 80

General instructions: (i) All questions are compulsory (ii)This question paper consists of 30 questions divided into four sections – A, B,C &D (iii)Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and section D contains 8 questions of 4 marks each.

SECTION A ( 1 mark each)

1. Are rational numbers closed under division? Give reason. 2. Find the solution of 2x – 3 = 7 3. What is the sum of the measures of the interior angles of a polygon with 18 sides? 4. Name the quadrilaterals whose diagonals are equal. 5. When a die is thrown, find the probability of getting a prime number. 6. Convert 1:5 into a percentage. SECTION B (2 marks each)

7. Find five rational numbers between 2

3 and

4

5

8. Represent −5

6 on the number line.

9. Sum of two numbers is 95. If one exceeds the other by 15, find the numbers. 10. Write a Pythagorean triplet whose one member is 12. 11 . How many sides does a regular polygon have if each of its interior angle is 1600? 12.(i) What is the width or size of a class interval? (ii) Define the frequency of a class. SECTION C (3 marks each)

13. Find: 2

3 ×

−3

7−

1

14−

3

7 ×

3

5

14. What should be added to twice the rational number −7

3 to get

3

7 ?

15. How many numbers lie between the squares of 12 and 14? Justify your answer. 16. Find the least number that must be subtracted from 5607 so as to get a perfect square. 17. The difference of two whole numbers is 66. The ratio of the two numbers is 2 : 5 . What are the two numbers? 18. HELP is a parallelogram in which the diagonals PE & HL intersect at O. If OE = 4cm and HL is 5cm more than PE. Find HL. Draw a neat Figure and label it. Give reasons for your answer. 19. Find the smallest square number divisible by each of the numbers 8, 15 & 20. 20. Find the square root of 9261 by the prime factorisation method. 21. Numbers 1 to 10 are written on ten separate slips ( one number on one slip), kept in a box and mixed well. One slip is chosen from the box without looking into it. What is the probability of :

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(i)getting a number 6. (ii)getting a number less than 5. (iii) getting a two digit number. 22. A shop keeper purchased 200 bulbs for Rs. 10 each. However 5 bulbs were fused and had to be thrown away. The remaining bulbs were sold at Rs.12 each. Find the gain or loss % . SECTION D (4 marks each) 23. During a sale, a shop offered a discount of 10% on the marked prices of all the items. What would a customer have to pay for a pair of jeans marked at Rs. 1450 and two shirts marked at Rs. 850 each. 24. If Chamely had Rs. 600 left after spending 75% of her money, how much did she have in the beginning? 25. Find the amount and compound interest on Rs. 18000 for 2 years 6 months at 10% per annum compounded annually. 26. A scooter was bought at Rs. 45000. Its value depreciated at the rate of 10% per annum. Find its value after one year. 27. Find the cube root of 110592 by prime factorisation method. 28. Construct a quadrilateral ABCD, in which AB = 4.5cm, BC = 5.5cm, CD = 4cm, AD = 6cm & AC = 7cm. Write the steps very briefly. 29. Construct a rhombus whose diagonals are 6cm and 8cm long. Write the steps very briefly. 30. On a particular day, the sales (in rupees) of different items of a baker’s shop are given below. Draw a pie chart for this data.

Ordinary bread Rs. 320

Fruit bread Rs. 80

Cakes and pastries

Rs.160

biscuits Rs. 120

others Rs. 40

Total Rs. 720

75

DON BOSCO SCHOOL -ALAKNANDA- NEW DELHI- 110 019

CLASS 8 Sample Paper MM-50

General Instructions:

1.Section A consists of 5 questions of 1 mark each. 2. Section B consists of 5 questions of 2 marks each. 3. Section C consists of 5 questions of 3 marks each. 4. Section C consists of 5 questions of 4 marks each.

SECTION A

1. Evaluate x(x-3) + 2 for x=1 2. Find the value of (2-1 +3-2+ 4-2)0

3. A square has side 12.5cm. Find its perimeter. 4. Find the height of a cuboid whose volume is 275cm3 and base area is 25cm2. 5. Expess 0.083 in standard form.

SECTION B 6. Add: 2m(3 + m ) and 2m2 – 3m 7. The diagonals of a rhombus are 7.5cm and 9.5 cm. Find its area. 8. Find the side of a cube whose surface area is 2400cm2. 9. Can a polyhedron have 10 faces, 20 edges and 15 vertices. Explain. 10. Write 3.678 x 10-2 in usual form

SECTION C

11. Simplify: (5

8 ) -2 x (

8

5 ) - 3

12. Subtract: 3pq(p-q) from 2pq(p+q) 13. Simplify: ( x2 – 5) (x + 5) +25 14.Find the height of a cylinder whose radius is 7cm and total surface area is 968cm2.

15.Using Euler’s formula find the unknown.

Faces ? 5 20

Vertices 6 ? 12

Edges 12 9 ?

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SECTION D 16. Evaluate using the suitable identity: 78 x 82. Also write the identity used. 17. A rectangular piece of paper 11cm x 4cm is folded without overlapping to make a cylinder of height 4 cm. Find the volume of the cylinder. 18. The area of a trapezium shaped field is 34m2, and the length of one of the parallel sides is 10m and its height is 4m. Find the length of the other parallel side. 19. Find m so that ( – 3)m + 6 x ( – 3)5 = ( – 3 )17

20. Simplify: (25 x t-4) ÷ (5-3 x 10 x t-8)

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DON BOSCO SCHOOL - ALAKNANDA- NEW DELHI- 110 019

CLASS 8 SAMPLE PAPER MM-50

General Instructions:

1.Section A consists of 5 questions each of marks 2. 2. Section B consists of 5 questions each of marks 3. 3. Section C consists of 5 questions each of marks 5. SECTION A 1. Add: 5m(3 – m ) and 6m2 – 13m 2. The diagonals of a rhombus are 7.5cm and 12cm. Find its area. 3. Find the side of a cube whose surface area is 600cm2.

4. Find the value of ( 1

2 )-2 + (

1

3 )-2 + (

1

4 )-2

5. Write in standard form: The size of a bacteria is 0. 0000005m SECTION B

6. Simplify: (5

8 ) - 7 x (

8

5 ) - 4

7. Subtract 3p (p – 4m + 5n ) from 4p ( 10n – 3m + 2p ) 8. Simplify: ( 1.5x – 4y )( 1.5x + 4y + 3) – 4.5x +12y 9.Find the height of a cuboid whose base area is 180cm2 and volume is 900cm3

10.Using Euler’s formula find the unknown.

SECTION C 11. Evaluate using the suitable identity: 1.05 x 9.5 Also write the identity used. 12. Water is pouring into a cuboidal reservoir at the rate of 100 litres per minute. If the volume of the reservoir is 108m3, find the number of hours it will take to fill the reservoir. 13. The area of a trapezium shaped field is 480m2, the distance between two parallel sides is 15m and one of the parallel sides is 20m. Find the other parallel side. 14. Find m so that ( – 3)m + 1 x ( – 3)5 = ( – 3 )7

15. Simplify: −35𝑋 10−5 𝑋 125

5−7 𝑋 6−5 . (write the answer in standard form)

Faces ? 5 20

Vertices 6 ? 12

Edges 12 9 ?

78

DON BOSCO SCHOOL - ALAKNANDA - NEW DELHI- 110 019

CLASS 8 ANNUAL EXAMINATION

Time : 2 𝟏

𝟐 hours MATHEMATICS M.M : 80

General instructions: (i) All questions are compulsory (ii)This question paper consists of 30 questions divided into four sections : A, B,C &D (iii)Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and section D contains 8 questions of 4 marks each. SECTION A ( 1 MARK EACH) 1. Write a rational number that is equal to its negative. 2. Solve: 14y – 8 = 13 3. Write a Pythagorean triplet whose one member is 14 4. Obtain the product of : m, – mn and mnp 5. Find the value of 3 – 2

6. If 21y5 is a multiple of 9, where y is a digit, what is the value of y? SECTION B ( 2 MARKS EACH)

7. Find 10 rational numbers between 2

5 and

1

2

8. Find the square root of 6400 by prime factorisaton method

9. Find 64 3

x 7293

10. Add: 4y (3y2 + 5y – 7) and 2(y3 – 4y2 + 5) 11. Using Euler’s formula find the unknown.

Faces F 5 20

Vertices 6 V 12

Edges 12 9 E

12. The diagonals of a rhombus are 7.5cm and 12cm. Find its area. SECTION C (3 MARKS EACH) 13. 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used?

79

14. Find the smallest square number which is divisible by each of the numbers 4, 9 and 10. 15. Mass of the Earth is 5.97 x 1024kg and mass of the Moon is 7.35 X 1022kg. What is the total mass? Express the answer in the standard form. 16. Find ‘m’ so that ( – 3 )m + 1 ÷ ( – 3 )5 = ( – 3 )9

17. Simplify: (1.5x – 4y) (1.5x + 4y + 3) – 4.5x + 12y 18. Find the smallest six – digit number which is a perfect square. 19. Simplify: (7m – 8n)2 + (7m + 8n)2

20. The age of A is one- third of the age of B. After 15 years, the age of A will be half that of the age of B. Find their present ages.

21. By what number should we multiply −5

14 so as the product may be

9

16 ?

22. How many bricks will be required for a wall which is 9m long, 8m high and 25cm broad, ieach brick measures 20 cm X 18 cm X 10cm. SECTION D ( 4 MARKS EACH) 23. The lateral surface area of a hollow cylinder is 4224cm2. It is cut along its height and formed a rectangular sheet of width 33cm. Find the perimeter of the rectangular sheet. 24. Factorise: 25a2 – 4b2 + 28bc – 49c2

25. Divide the sum of −9

7 and

7

3 by the difference of

3

5 and

2

7 .

26. A farmer has enough food to feed 20 animals in his cattle for 2 weeks. How long would the food last if there were 8 more animals in his cattle?

27. Solve : 6𝑥 + 1

2 =

7𝑥−3

3 – 1

28. On a graph paper draw the line passing through (2,3) and (3,2). (i)Find the coordinates of the point at which this line meets the x – axis. (ii) Find the coordinates of the point at which this line meets the y – axis. 29. The product of two numbers is 10625. If one number is 17 times the other number, find the Numbers. 30. Three numbers are in the ratio 2: 3: 4 and the sum of their cubes is 12375. Find the numbers.

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