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Mathematics
Published by :
WITH COMPLETE SOLUTIONSQUESTION BANK
OSWAAL
OSWAAL BOOKS0562-2857671, 25277811/11, Sahitya Kunj, M.G. Road, Agra -282002 (UP) India
0562-2854582 [email protected] www.OswaalBooks.com
Class
8
( iii )
CONTENTS
l Syllabus vii - viii
1. Rational Numbers 1 - 11
2. Linear Equations in One Variable 12 - 25
3. Understanding Quadrilaterals 26 - 37
4. Practical Geometry 38 - 48
5. Data Handling 49 - 61
6. Squares and Square Roots 62 - 71
7. Cubes and Cube Roots 72 - 84
8. Comparing Quantities 85 - 99
9. Algebraic Expressions and Identities 100 - 109
10. Visualising Solid Shapes 110 - 117
11. Mensuration 118 - 131
12. Exponents and Powers 132 - 142
13. Direct and Inverse Proportions 143 - 154
14. Factorisation 155 - 162
15. Introduction to Graphs 163 - 176
16. Playing with Numbers 177 - 184
ll
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Mathematics Syllabus
Number System (50 hrs)
(i) Rational Numbers:
w Properties of rational numbers. (including identities).
w Using general form of expression to describe properties
w Consolidation of operations on rational numbers.
w Representation of rational numbers on the number line
w Between any two rational numbers there lies another rational number (Making children see that if we take two rational numbers then unlike for whole numbers, in this case you can keep finding more and more numbers that lie between them.)
w Word problem (higher logic, two operations, including ideas like area)
(ii) Powers
w Integers as exponents.
w Laws of exponents with integral powers
(iii) Squares, Square roots, Cubes, Cube roots.
w Square and Square roots
w Square roots using factor method and division method for numbers containing (a) no more than total 4 digits and (b) no more than 2 decimal places
w Cubes and cubes roots (only factor method for numbers containing at most 3 digits)
w Estimating square roots and cube roots. Learning the process of moving nearer to the required number.
(iv) Playing with numbers
w Writing and understanding a 2 and 3 digit number in generalized form (100a + 10b + c , where a, b, c can be only digit 0-9) and engaging with various puzzles concerning this. (Like finding the missing numerals represented by alphabets in sums involving any of the four operations.) Children to solve and create problems and puzzles.
w Number puzzles and games
w Deducing the divisibility test rules of 2, 3, 5, 9, 10 for a two or three-digit number expressed in the general form.
Algebra (20 hrs)
(i) Algebraic Expressions
w Multiplication and division of algebraic exp.(Coefficient should be integers)
w Some common errors (e.g. 2 + x ≠ 2x, 7x + y ≠ 7xy )2 2 2 2 2
w Identities (a ± b) = a ± 2ab + b , a – b = (a – b) (a + b)
w Factorisation (simple cases only) as examples the following types 2 2 2
w a(x + y), (x ± y) , a – b , (x + a).(x + b)
w Solving linear equations in one variable in contextual problems involving multiplication and division (word problems) (avoid complex coefficient in the equations)
Ratio and Proportion (25 hrs)
w Slightly advanced problems involving applications on percentages, profit & loss, overhead expenses, Discount, tax.
w Difference between simple and compound interest (compounded yearly up to 3 years or half-yearly up to 3 steps only), Arriving at the formula for compound interest through patterns and using it for simple problems. Direct variation – Simple and direct word problems Inverse variation – Simple and direct word problems.
w Time & work problems– Simple and direct word problems
( vii )
w Geometry (40 hrs)
(i) Understanding shapes:
• Properties of quadrilaterals – Sum of angles of a quadrilateral is equal to 3600 (By verification)
• Properties of parallelogram (By verification)
(i) Opposite sides of a parallelogram are equal,
(ii) Opposite angles of a parallelogram are equal,
(iii) Diagonals of a parallelogram bisect each other. [Why (iv), (v) and (vi) follow from (ii)]
(iv) Diagonals of a rectangle are equal and bisect each other.
(v) Diagonals of a rhombus bisect each other at right angles.
(vi) Diagonals of a square are equal and bisect each other at right angles.
(ii) Representing 3-D in 2-D
w Identify and Match pictures with objects [more complicated e.g. nested, joint 2-D and 3-D shapes (not more than 2)].
w Drawing 2-D representation of 3-D objects (Continued and extended)
w Counting vertices, edges & faces & verifying Euler's relation for 3-D figures with flat faces (cubes, cuboids, tetrahedrons, prisms and pyramids)
(iii) Construction:
Construction of Quadrilaterals:
w Given four sides and one diagonal
w Three sides and two diagonals. Three sides and two included angles. Two adjacent sides and three angles.
Mensuration (15 hrs)
(i) Area of a trapezium and a polygon.
(ii) Concept of volume, measurement of volume using a basic unit, volume of a cube, cuboid and cylinder
(iii) Volume and capacity (measurement of capacity)
(iv) Surface area of a cube, cuboid, cylinder.
Data handling (15 hrs)
(i) Reading bar-graphs, ungrouped data, arranging it into groups, representation of grouped data through bar-graphs, constructing and interpreting bar-graphs.
(ii) Simple Pie charts with reasonable data numbers
(iii) Consolidating and generalising the notion of chance in events like tossing coins, dice etc. Relating it to chance in life events. Visual representation of frequency outcomes of repeated throws of the same kind of coins or dice. Throwing a large number of identical dice/coins together and aggregating the result of the throws to get large number of individual events. Observing the aggregating numbers over a large number of repeated events. Comparing with the data for a coin. Observing strings of throws, notion of randomness
Introduction to graphs (15 hrs)
PRELIMINARIES:
(i) Axes (Same units), Cartesian Plane
(ii) Plotting points for different kind of situations (perimeter vs length for squares, area as a function of side of a square, plotting of multiples of different numbers, simple interest vs number of years etc.)
(iii) Reading off from the graphs Reading of linear graphs Reading of distance vs time graph
________________
( viii )
LET’S REVISE Natural Numbers : Counting numbers starting from 1 are known as natural numbers and denoted
by N.i.e., N = {1, 2, 3, 4, 5...............}
Whole Numbers : All natural numbers together with 0 are called whole numbers and denotedby W.i.e., W = {0, 1, 2, 3, 4, 5..............}
Integers : All natural numbers and negatives of natural numbers included with 0 are calledintegers.i.e., ............. – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5...................etc. are all integersWe can represent the integers on the number line as shown below :
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6XX'
Rational Number : A number is called rational if we can write the number in the form of ,pq
where p and q are integers and q 0
i.e.,
31 2 0 5 71 , 2 , 0 and , , are all rational numbers.
1 1 1 8 14 15
Between two rational numbers x and y, there exists a Rational number 2
·x y
The idea of ‘mean’ helps us to find rational numbers between two given rational numbers. We can find countless rational numbers between two rational numbers.
– xy is called the additive inverse of
xy and vice-versa.
yx
is called the multiplicative inverse or reciprocal of ·xy
The rational number 0 is the additive identity for all rational numbers because a number doesnot change when 0 is added to it.
The rational number 1 is the multiplicative identity for all rational numbers because on multiplyinga rational number with 1, its value does not change.
Rational numbers can be represented on a number line. The denominator of the rational number indicates the number of equal parts into which the first
unit has been divided whereas the numerator indicates as to how many of these parts are to betaken into consideration.
Rational Numbers
1CHAPTERUnit I
Number System
OSWAAL CBSE Question Bank, Mathematics - VIII2 ]
[A] Objective Type Questions [1 mark each]
(i) Multiple Choice Questions
1. Which of the following numbers is the additive inverse of 7
29 :
(A) 297 (B) –
297 (C) –
729 (D)
729
2. Which of the following numbers is the multiplicative inverse of 1531
:
(A) 3115 (B) –
3115 (C) –
1531 (D)
1531
3. Which of the following numbers has no multiplicative inverse :(A) 0 (B) 1 (C) – 1 (D) none of these
4. Which of the following numbers is the product of 6
13 and –
263
:
(A) 1 (B) – 4 (C) –266133 (D)
266133
5. Which of the following numbers is its own reciprocal :
(A) 10 (B) zero (C) 15 (D) 1
6. Which of the following numbers is the decimal form of 14
:
(A) – 0·25 (B) 2·5 (C) 0·25 (D) – 2·5
7. Which of the following numbers lies in the middle of 34
and 74
:
(A) 5·0 (B) 3·0 (C) 2·5 (D) 1·258. Which pair of following numbers are respectively the additive and multiplicative
identities :(A) 2 and 0 (B) 1 and – 1 (C) – 1 and 0 (D) 0 and 1
9. Which of the following numbers is the simplest form of 3 1 5 :4 4 4
(A) 94
(B) –34
(C) –94
(D) 74
10. Which of the following properties indicates the given operation
1 3 1 1 3 15 5 7 5 5 7
(A) commutative (B) associative (C) distributive (D) none of these
11. What should be added to 34
to get ‘– 1’ ?
(A) 14
(B) 14
(C) 1 (D) 34
12. For rational numbers, multiplicative identity is :(A) 0 (B) – 1 (C) 2 (D) 1
13. The value of 3 4 15 145 7 16 9
is equal to :
(A) 14
(B) 12
(C) 18
(D) 16
14. The reciprocal of
45
is :
(A) 45 (B)
54
(C) 54
(D) 45
15. Which of the following numbers has no reciprocal ?(A) – 3 (B) – 2 (C) – 1 (D) 0
16. Which of the following is the product of 7
8 and
2?
21
(A) 1
12 (B) 12 (C)
6316
(D) 16147
17. What should be subtracted from 35
to get – 2 ?
(A) 75
(B) 13
5(C)
135
(D) 75
18. Additive inverse of 59
is :
(A) 95
(B) 34
(C) 59
(D) 95
19. A rational number between 23
and 14
is :
(A) 5
12(B)
512
(C) 5
24(D)
524
20.
4 112 7?
3 5 15 20
(A) 15
(B) 415
(C) 1360
(D) 730
(ii) Fill in the Blanks : [NCERT]1. Zero has .................... reciprocal.2. The numbers .................... and ................ are their reciprocals.3. The reciprocal of –5 is ....................
4. Reciprocal of 1,
x wherex 0 is ...................
5. The product of two rational numbers is always a ....................
(iii) True/False :
1. The additive inverse of 2 2
is ·3 3
2.12
is a natural number.
3. The multiplicative inverse of 12
is 2.
RATIONAL NUMBERS [ 3
OSWAAL CBSE Question Bank, Mathematics - VIII4 ]
4. The negative of 2 is 1
.2
5. If a and b are two consecutive rational numbers, then
.2
a bb [NCERT]
ANSWERS
[A] Objective Type Questions
(i) Multiple Choice Questions :
1. (C) 729
2. (A) 3115
3. (B) 1 4. (B) – 4
5. (D) 1 6. (C) 0·25 7. (D) 1·25 8. (B) 1 and – 1
9. (B) 34
10. (B) associative 11. (B) 14 12. (D) 1
13. (B) 12
14. (C) 54
15. (D) 0 16. (A) 1
12
17. (D) 75 18. (C)
59 19. (D)
524
20. (C) 13
·60
(ii) Fill in the Blanks :1. no 2. 1 and – 1 3. 1
5 4. x5. rational number.
(iii) True/False :1. True 2. False 3. True 4. False5. True
[B] Very Short Answer Type Questions [1 mark each]
Q. 1. Is 1 the multiplicative identity for integers ? Also for whole numbers.Ans. Yes, 1 is the multiplicative identity for integers as well as for whole numbers. 1
Q. 2. Write the additive inverse of 5
·9 [NCERT]
Ans. Additive inverse of 5 5
is ·9 91
Q. 3. Write the multiplicative inverse of 13
·19[NCERT]
Ans. The multiplicative inverse of 1319
is 19
·131
Q 4. Write the rational number that does not have a reciprocal.Ans. The rational number ‘0’ does not have a reciprocal. 1Q 5. Use sign < or > to fill in the box :
10 5
7 7
Ans. 10 5
7 7 1
Q. 6. Is 0·7 the multiplicative inverse of 3
1 ?7
Give reasons. [NCERT]
Ans. Yes, since 0·7 = 7
10½
Multiplicative inverse of 7
10 =
107
= 3
1 ·7
½
Q. 7. The rational numbers that are equal to their reciprocals.Ans. The rational numbers 1 and (– 1) are equal to their reciprocals respectively. 1Q. 8. The rational number that is equal to its negative.Ans. The rational number 0 is equal to its negative. 1Q. 9. How many integers are there between – 1 and 1 ?Ans. There is only one integer between – 1 and 1. It is 0. 1Q. 10. How many integers are there between – 9 and – 10 ?Ans. There is no integer between – 9 and – 10. 1
[C] Short Answer Type Questions [2 marks each]
Q. 1. Find using distributivity :
37 7 55 12 5 12
Sol.
37 7 55 12 5 12 =
37 55 12 12
1
=3 57
5 12
= 7 2 7 1 7
·5 12 5 6 301
Q. 2. Multiply 6
13 by the reciprocal of 7
·16
Sol. Reciprocal of 7 16
is ·16 7 1
Now,6 16
13 7 =
6 ( 16) 96·13 7 91
1
Q. 3. Simplify : 16 939 26
Sol. We have
16 939 ( 26) =
91639 26
½
Now, the LCM of 39 and 26 is 78.
Rewriting 1639
and 926
in such a manner they have the same denominator 78.
1639
=
16 239 2
= 3278
½
926
=
9 3
26 3 =
2778
16 939 26
=
32 ( 27)78 78
½
= 32 ( 27)78
=32 27
78 =
578
. ½
RATIONAL NUMBERS [ 5
OSWAAL CBSE Question Bank, Mathematics - VIII6 ]
Q. 4. Verify the following :
5 38 5
=53
5 8
Sol. Verification : L.H.S. = 5 38 5
1
= 5 5 3 8
40
=
25 24 1
40 40
R.H.S. =53
5 8
1
= 3 8 ( 5) 5
40
=
24 25 1
40 40 L.H.S. = R.H.S. Verified.
Q. 5. Subtract 3 5
from ·8 7
Sol. The additive inverse of 3 3 is ·8 8
1
5 37 8
= 5 37 8
= ( 5) 8 3 756
= 40 2156
=19
56
. 1
Q. 6. What should be subtract from – 34 , so as to get
56 ? [NCERT]
Sol. Suppose x is the rational number to be subtracted from 34
to get 5
.6
Then ½
34
x =56
3 54 6
= x
x =3 54 6
x =( 3) 3 ( 5) 2
12
1½{ LCM of 4 and 6 is 12}
x =9 ( 10)
12
x =19
12
. ½
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