studymaterial for classx maths sa2 -2...- 5 - 6. find the roots of the quadratic equation 6x2 −x...

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CHAPTER:1.QUADRATIC EQUATIONS

VSA(1) SA(2) SA(3) LA(4) TOTAL(4)

1(1)=1 2(1)=2 3(1)=3 4(1)=4 10M

Points to be remembered:

1. A quadratic equation in the variable x is of the form ax2 + bx +c = 0, where a,b

and c are real numbers and a ≠0.

2. Quadratic Formula: x =a

acbb

2

42 −±−.

3. A quadratic equation ax2 + bx +c = 0, has

a) Two distinct real roots if, b2 – 4ac >0

b) Two equal roots if , b2 – 4ac=0

c) No real roots if b2 – 4ac<0.

Q1.The roots of the equation x2 + x – p(p+1) = 0 , where p is a constant, are (1)

a) P , p+1 b) - p , p +1 c) p , - p – 1 d) – p , - p – 1.

Ans: c) , since x2 + x - p(p+1) = 0 implies x

2 - p

2 + x – p =0

Or, (x + p)(x – p) +(x – p) = 0

Or, (x- p) ( x+ p +1) = 0

Either x - p = 0 , x = p , or , x + p + 1 = 0 , x = - p – 1.

Q2. The value of k for which equation 9 x2 + 8xk + 4= 0has equal roots is : (1)

a) 3/2 b) - 3/2 c) 3/2, - 3/2 d) 9/2

Ans: c) , since for equal roots b2 -4ac = 0 , (8k)

2 – 4x4x9=0, 64k

2 – 144=0

K2 = 144/64 , k

2 = 9/4,k = - 3/2 , 3/2.

Q3. The value of k for which x = - 2 is a root of the quadratic equation kx2 + x – 6 =0 ,

is (1)

a) – 1 b) – 2 c) 2 d) – 3/2

Ans: c) , since k.22 - 2 – 6=0, 4k - 8 =0, k = 8/4=2

Q4. If the difference of roots of the quadratic equation x2 + kx +12 = 0 is 1 , the

positive value of k is :

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a) – 7 b) 7 c) 4 d) 8 (1)

Ans: b), since if u and v be the roots of the equation then , u – v = 1(given)

Also here, uv = product of the roots = c/a = 12

Hence , (u + v)2 = ( u – v )

2 + 4uv = 1 + 4x12 = 49

U + v = ±7

Here, u + v = sum of roots = - b/a = - k

Hence - k = ±7. Thus the positive value of k is 7.

Q5. Find the value of p so that the quadratic equation px( x – 3) + 9 =0 has two equal

roots. (2)

Ans: here, px(x – 3) + 9 = 0, px2 – 3px + 9 = 0, a = p , b = - 3p , c = 9

Since roots are equal , discriminant = 0

B2 - 4ac = 0, ( -3p)

2 – 4xpx9 = 0, 9p

2 - 36p=0, 9p = 36, since p ≠0

P = 36/9 = 4.

Q6. Find the roots of the quadratic equation 6x2 –x -2 = 0 (2)

Ans: 6x2 –x -2 = 0

Or, 6x2 +3x -4x -2 = 0

Or, 3x(2x+1) -2(2x+1)=0

Or, (2x+1)(3x-2) =0

Or, 2x+1=0 or 3x-2 =0

Or, x = -1/2 or x= 2/3.

Q.7. Find the roots of the quadratic equation 3x2 - 5x + 2 =0. (2)

Ans: Here, a =3 , b = - 5 , c =2.

X= 6

15

6

15

32

324)5()5(

2

4 22 ±=±=×

××−−±−−=−±−

a

acbb

X= 6/6=1 or x = 4/6=2/3

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Q8: Find the roots of the quadratic equation: 3x2 - 62 x + 2 =0 (3)

Soln: 3x2 - 62 x + 2 =0

3x2 - 6 x - 6 x + 2 =0, 3 x( 3 x - 2 ) - 2 ( 3 x - 2 ) = 0

( 3 x - 2 )( 3 x - 2 )=0 , Hence , x = 3

2

Q9. The sum of the squares of two consecutive natural numbers is 421. Find the

numbers. (3)

Soln: Let the two numbers be x, x+1.

A/q, x2 + ( x+1)

2 = 421

X2 + x

2 + 2x +1 =421

X2 +x – 210 = 0

X2 + 15x – 14x – 210 =0

X(x + 15) - 14(x + 15) = 0.

(x +15)(x – 14) = 0, x = -15 or x = 14. Since numbers are positive. Hence the two

numbers be 14, 14+1=15.

Q9.A shopkeeper buys a number of books for Rs. 1200. If he had bought 10 more

books for the same amount , each book would have cost him Rs 20 less. How many

books did he buy? (4)

Ans: Let the number of books brought = x.

A/q, 1200/x - 1200/(x+10) = 20.

20)10(

1200)10(1200 =+

−+xx

xx

X2 + 10x – 600 = 0

X2 + 30x-20x – 600 = 0

X(x+30) – 20(x + 30) = 0

(x+30)(x – 20) = 0

Either, x = -30 , x =20

Hence, number of books bought=20

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Q10. The denominator of a fraction is two more than its numerator .If the sum of the

fraction and its reciprocal is 34/15, find the fraction. (4)

Soln: Let the numerator be x and denominator is x +2.

A/q, x/(x+2) + (x+2)/x = 34/15

15

34

)2(

)2( 22

=+++

xx

xx

15(2x2+4x+4)=34(x

2 + 2x)

4x2+8x – 60 = 0

X2 + 2x – 15 =0

X2 + 5x – 3x – 15 =0

X(x + 5) – 3(x+5) =0

(x+5)(x – 3)=0

X = 3 , hence the fraction is 3/5.

Q11. A takes 6 days less than the time taken by B to finish a piece of work. If both A

and B together can finish it in 4 days, find the time taken by B to finish the work. (4)

Ans: Let B alone takes to finish the work in x days and A alone takes x -6 days .

A/q, 1/x +1/(x – 6) = ¼, ( for 1 day work together).

X2 – 14x + 24 = 0

(x – 12)( x -2) = 0

X = 12 or x = 2. Here x cannot be less than 6. Hence, x =12.Thus B can finish the

work in 12 days.

VVVVSA QuestionsSA QuestionsSA QuestionsSA Questions::::

1. Give an example of a quadratic equation.

2. Solution of 0352 2 =−− xx is x=?

3. If the disarmament of a quadratic equation is 0 than the root of that equn. are

________ and _______ .

4. In a quadratic equation 02 =++ cbxax ; sum of the roots is equal to _______ and

_______

5. Find the nature of root of quadratic equation 0352 2 =+− xx

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6. Find the roots of the quadratic equation 026 2 =−− xx

7. find two numbers whose sum is 27 and product is 182

8. Solve the following quadratic equations:

a) 09124 2 =+− xx b) 0126 2 =−+ xx

9. Find the roots of 0534 2 =++ xx by the method of completing the square.

10. The sum of the squares of two consecutive natural number is 421. Find the

numbers.

11. Find the value of K for which the roots of the equation 01)1()4( 2 =++++ xkxk

are real and equal.

12. A train travels 360 Km at a uniform speed. If the speed had been 5Km/hr more.

It would have taken 1 Hr less for the same journey. Find the speed of the train.

CHAPTER: 2. ARITHMETIC PROGRESSION (AP)


1(1)=1 2(1)=2 3(2)=6 4(1)=4 13M

POINTS TO BE REMEMBERED:

1. The general form of an A is a, a+d,a+2d, a+3d,..........., where

a is called 1st

term and d is called common difference(cd).

2. General term of and AP is given by an = a + ( n – 1 ) d. This is also called

general term.

3. The sum of the first terms of and AP is given by :

Sn = n/s[2a + (n – 1 )d]

4. Also , Sn = n/2( a + l), where l is the last term of the A.P.

Q1. The 10th

term of the A.P. 2,7,12,............is (1)

a) 30 b) 35 c) 37 d) 47

Ans:d), since, a10 = a + 9d = 2 + 9x5 = 47.

Q2. The sum of the first 22 terms of the AP: 8,3, - 2 , ...... (1)

a) -579 b) -679 c) -879 d) – 979.

Ans: d) , here, a = 8 , d = 3 – 8 = - 5 .

S22 = 22/2[ 2x8 + (22 – 1)( - 5)] = 11(- 89) = - 979.

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Q3. If the common difference of an AP is 5, then a18 – a13 is : (1)

a) 5 b) 20 c) 25 d) 30.

Ans:c), since , a18 – a13 = a + 17d - ( a +12d) = 5d = 5x5 = 25.

Q4. The 10th

term of an AP - 1.0, - 1.5, - 2.0,..................is: (1)

a) 3.5 b)5.5 c) – 5.5 d) – 6.5

Ans: c) , since a10 = a + 9d = - 1 + 9( - .5) = - 1 – 4.5 = - 5.5 .

Q5. If the numbers x + 3, 2x + 1 and x – 7 are in AP , find the value of x. (2)

Ans: Since the terms are in AP, hence

2nd

term – 1st

term = 3rd

term – 2nd

term

(2x + 1) – ( x + 3) = ( x – 7) – ( 2x + 1)

X – 2 = - x – 8 or 2x = -6 or x = -3.

Q6. Which term of the AP 3,10,17,............will be 84 more than its 13th

term? (2)

Ans: Here a = 3 , d = 10 – 3 = 7.

Now , an = a13 + 84

Or, a + ( n – 1) d = a + 12d + 84

Or, (n – 1)7 = 12x7 + 84

Or, (n – 1) = 12 + 12 or, n = 25.

Q7. Find if 102 is a term of the AP 25, 28 , 31 ............or not. (3)

Ans: Here, a = 25 , d = 3.

Let, an = 102 or, a + ( n -1)d = 102

Or, 25 + ( n -1) 3 = 102 , or (n – 1) = 77/3 or, n = 80/3.

Since n is not a natural number, hence 102 is not a term of the AP.

Q8. Find the 8th

term from the end of the AP 7, 10 ,13,........184. (2)

Ans:The AP can be written as 184, ..........13,10,7.

Now , a = 184 , d = 7 – 10 = - 3 .

Hence , a8 = a + 7d = 184 + 7(

Hence the 8th

term from the end of the AP is 163.

Q9. Find the sum of the integers between 100 a

Ans: The series as per question is 102, 108 , 114, ..........198.

Let, an = 198 or, a + ( n –

Or, ( n -1) 6 = 96 or, ( n

S17 = 17/2( 102 + 198) = 2550. [ using S

Q10. The sum of first n terms of an AP is given by S

the 12th

term.

Ans: Sn = 3n2 - 4n

S1 = 3(1)2 - 4(1) =

a = S1 = - 1 , a2 =S

d = a2 - a = 5 – (

Hence, the AP is -1, 5, 11,17 , 23 ,........

a12 = a + 11d = - 1 + 11(6) = 65.

Q11.

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= a + 7d = 184 + 7( - 3) = 184 – 21 = 163.

term from the end of the AP is 163.

Q9. Find the sum of the integers between 100 and 200 that are divisible by 6.


– 1) d = 198 or 102 + ( n -1) 6 = 198

1) 6 = 96 or, ( n -1) = 96/6 = 16 or, n =17 .

= 17/2( 102 + 198) = 2550. [ using Sn = n/2( a + l)].

Q10. The sum of first n terms of an AP is given by Sn = 3n2 - 4n.Determine the AP and

4(1) = - 1. S2 = 3(2)2 - 4(2) = 4.

=S2 - S1 = 4 + 1 = 5.

( - 1) = 6.

1, 5, 11,17 , 23 ,........

1 + 11(6) = 65.

nd 200 that are divisible by 6. (3)


4n.Determine the AP and

(3)

(4)

Q12.

Q13.

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(4)

(3)

Q14. Find the sum of first 24 terms of the list of numbers whose an = 3 + 2n

Q15. The sum of first 14 terms of an AP is 1050 and its first term is

term of the A.P. Soln:

910 = 91or,

Therefore, a20 = 10 + (20

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Find the sum of first 24 terms of the list of numbers whose nth term is given by

The sum of first 14 terms of an AP is 1050 and its first term is

910 = 91d or, d = 10

= 10 + (20 – 1) × 10 = 200, i.e. 20th term is 200.

th term is given by (4)

The sum of first 14 terms of an AP is 1050 and its first term is 10, find the 20th

(3)

1) × 10 = 200, i.e. 20th term is 200.

QUESTIONS FOR PRACTICE:

1. Which term of the A.P. 4,9,14,19 ........... is 109?

2. How many terms are there in the A.P. 7, 10, 13, ..........151?

3. what is the sum of all natural numbers from 1 to 100

4. Find the values of x for which 95x+2), (4x

5. Find the sum of 51 terms of the AP whose second term is 2 and the 4

6. If the 5th

and 12th

terms of an AP are

first 20 terms of the AP.

CHAPTER

VSA(1) SA(2)

1(2)=2 2(1)=2

Q1. The distance between the points P ( 0,y) and Q(x,0) is given by:

a) X2 + y

2

Ans: c), since distance =

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Which term of the A.P. 4,9,14,19 ........... is 109?

terms are there in the A.P. 7, 10, 13, ..........151?

what is the sum of all natural numbers from 1 to 100

Find the values of x for which 95x+2), (4x-1) and ( x+2) are in AP.

Find the sum of 51 terms of the AP whose second term is 2 and the 4

terms of an AP are -4 and -18 respectively. Find the sum of

first 20 terms of the AP.

HAPTER3: CO- ORDINATE GEOMETRY.

SA(3) LA(4)

3(1)=3 4(1)=4

POINTS TO BE REMEMBERED:

between the points P ( 0,y) and Q(x,0) is given by:

b) 22 yx − c) 22 yx +

Ans: c), since distance = 22 )0()0( −+− yx = 22 yx +

1) and ( x+2) are in AP.

Find the sum of 51 terms of the AP whose second term is 2 and the 4th

term is 8.

18 respectively. Find the sum of

TOTAL(5)

11M

between the points P ( 0,y) and Q(x,0) is given by: (1)

d) xy

Q2. The perpendicular distan

a) 13 units

Ans: b), since perpendicular distance of the point A from y axis = x

ordinate = 5 unit.

Q3. The distance between the points P(2,

10 units

Ans: b), since distance =

Q4. Find a point on y axis which is equidistant from the points A(6, 5) and B( Soln: Let the point on y axis be P(0,y)

Hence the point is P(0,9).

Q5.

Ans:Let the vertices be A(5, -

Q6.

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The perpendicular distance of A(5,12) from the y-axis

b) 5 units c) 12 units d ) 17 units

Ans: b), since perpendicular distance of the point A from y axis = x

ordinate = 5 unit.

The distance between the points P(2,- 3) and Q(2,2) is

b) 5 units c) 15 units d ) 9 units

Ans: b), since distance = 22 )32()22( ++− = 22 50 +

which is equidistant from the points A(6, 5) and B(

Let the point on y axis be P(0,y)

Hence the point is P(0,9).

- 2) ,B(6,4) and C( 7, - 2).

(1)

d ) 17 units

Ans: b), since perpendicular distance of the point A from y axis = x – co

3) and Q(2,2) is (1)

d ) 9 units

which is equidistant from the points A(6, 5) and B(-4,3). (2)

(2)

(2)

- 12 -

Q7. (2)

Q8. (3)

- 13 -

Q9. (4)

Ans:

Q10.

(4)

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1. Write the coordinates of a point lying on x-axis and y-axis separately

2. Find the distance between the points (a,b) and (-a,-b).

3. Write the formula for mid- point of a line segment joining points A(m,n) and

B(m2,n2).

4. Find the value of k if the distance between the points (2,k) and (4,3) be 8.

5. Show that the points (1,7), (4,2), (-1,-1) and (-4,4) are the vertices of a

square.

6. Find a relation between x and y such that the points (x,y) is equidistant from

the points (7,1) and (3,5)

7. Find the distance between the points

−2,

5

8 and

2,5

2.

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CHAPTER:4.APPLICATION OF TRIGONOMETRY


1(1)=1 - 3(1)=3 4(1)=4 8M

POINTS TO REMEMBER:

(i) The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. (ii) The angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level, i.e., the case when we raise our head to look at the object. (iii) The angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level, i.e., the case when we lower our head to look at the object. 2. The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

Q1. If the altitude of the sun is at 600 , then the height of the vertical tower that will

cast a show of length 30 m is

a) 15m b) 30√3 m c) 15√2m d) 30/√2m

Ans: b) , since, height = base xtan 600 = 30 √3 m

Q2. The height of the tower is 100m . When the angle of elevation of sun is 300 , then

the shadow of the tower is

a) 100√3 m b) 100m c)100(√3 -1)m d) 100/√3 m

Ans:a), since base = height x tan300= 100√3 m .

Q3. The length of a kite string flying at 100m above the ground with angle of elevation

450

is: a) 100m b) 100√2 m c)25√3 m d) 100√3 m

Ans: b) , since length of the string= height x cosec450 = 100√2 m

Q4. (3)

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Ans:

Q5 (3)

Ans:

- 17 -

Q6. (4)

Ans:

Q7 The angles of depression of the top and the bottom of an 8 from the top of a multiheight of the multi-storeyed building and the distance between the two buildings.

Ans: PC denotes the m

building. We are interested to determine the height of the multibuilding, i.e., PC and the distance between the two buildings, i.e., AC. Look at the figure carefully. Observe that PB is a transverBD. Therefore, QPB and PBD = 30°. Similarly,

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The angles of depression of the top and the bottom of an 8 from the top of a multi-storeyed building are 30° and 45°, respectively. Find the

storeyed building and the distance between the two buildings.

PC denotes the multistoryed building and AB denotes the 8 m tall

building. We are interested to determine the height of the multibuilding, i.e., PC and the distance between the two buildings, i.e., AC. Look at the figure carefully. Observe that PB is a transversal to the parallel lines PQ and

QPB and PBD are alternate angles, and so are equal. So PBD = 30°. Similarly, ∠ PAC = 45°.

The angles of depression of the top and the bottom of an 8 m tall building storeyed building are 30° and 45°, respectively. Find the

storeyed building and the distance between the two buildings. (4)

ultistoryed building and AB denotes the 8 m tall

building. We are interested to determine the height of the multi-storeyed building, i.e., PC and the distance between the two buildings, i.e., AC. Look at

sal to the parallel lines PQ and PBD are alternate angles, and so are equal. So ∠

1. A pole 6m heigh cast a shadow 2

elevation is :

a) 600 b)45

2 . The shadow of a tower standing on a level ground is found to be 40 m longerwhen the Sun’s altitude is 30° than when it is 60°. Find the height of the tower.

3. An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney? 4.A boy of height 1.7m is standing 30m away from a flag staff on the same level

ground . He observed that the angle of elevation of the top of the flag staff is

300. Calculate the height of the flag staff.

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PROBLEMS FOR PRACTICE:

A pole 6m heigh cast a shadow 2√3 m long on the ground, then the sun

b)450 c)30

0 d)90

0

2 . The shadow of a tower standing on a level ground is found to be 40 m longerwhen the Sun’s altitude is 30° than when it is 60°. Find the height of the tower.

r 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the chimney?

4.A boy of height 1.7m is standing 30m away from a flag staff on the same level

d . He observed that the angle of elevation of the top of the flag staff is

. Calculate the height of the flag staff.

m long on the ground, then the sun

2 . The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun’s altitude is 30° than when it is 60°. Find the height of the tower.

r 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is 45°. What is the height of the

4.A boy of height 1.7m is standing 30m away from a flag staff on the same level

d . He observed that the angle of elevation of the top of the flag staff is

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CHAPTER:5. CIRCLES.


1(2)=2 2(1)=2 3(1)=3 4(1)=4 10M

Points to Remember:

1.The tangent to a circle is perpendicular to the radius through the point of contact. 2. The lengths of the two tangents from an external point to a circle are equal.

3.The common point of the circle and the tangent is called the point of

contact.

Q1. (1)

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Q2. (1)

If PQ and PR are two tangents to a circle with centre O inclined at an angle 300,

Then angle of QOR is: a) 1200

b) 300 c) 150

0 d) 60

0

Ans: c), since angle QPR + angleQOR = 1800, hence QOR∠ = 180

0 – 30

0 = 150

0.

Q3. (1)

The length of the tangent drawn from a point 8cm away from the centre of a

circle of radius 6cm is : a) √7cm b)2 √7cm c) 10cm d) 5cm.

Ans: b), since length of the tangent =

22 )()inttan( radiuscenterfrompotheofcedis − = 22 68 − =2√7��

Q4. (1)

A tangent is perpendicular to the radius at the ................

a) Point of contact. b) centre c) infinity d) core.

Ans: a), since The tangent to a circle is perpendicular to the radius through the point of contact.

Q5. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. (3) Ans:

Q6. The length of a tangent from a point A at distance 5 cm from the centre of the circle

is 4. Find the radius of the circle.

Ans:

Q7. Two tangents TP and TQ are drawn point T.Prove that ∠ Solution : We are given a circle with centre O, an external point T and two tangents TP and TQ to the circle, where P,

We need to prove that Let ∠ PTQ = ϴ. Ηere, TP = TQ.

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The length of a tangent from a point A at distance 5 cm from the centre of the circle

is 4. Find the radius of the circle.

Two tangents TP and TQ are drawn to a circle with centre O from an external PTQ = 2∠ OPQ.

We are given a circle with centre O, an external point T and two

tangents TP and TQ to the circle, where P, Q are the points of contact.

We need to prove that ∠ PTQ = 2 ∠ OPQ ere, TP = TQ.

The length of a tangent from a point A at distance 5 cm from the centre of the circle

(3)

to a circle with centre O from an external (3)

We are given a circle with centre O, an external point T and two Q are the points of contact.

Q8. Prove that the angle between the two tangents drawn from an external point to acircle is supplementary to the angle subtended by the linecontact at the centre.

Ans:

- 23 -

Prove that the angle between the two tangents drawn from an external point to acircle is supplementary to the angle subtended by the line-segment joining the points of

Prove that the angle between the two tangents drawn from an external point to a segment joining the points of

(4)

- 24 -

Q9 . Prove that the parallelogram circumscribing a circle is a rhombus. (4)

Ans: Since ABCD is a parallelogram, hence AB = CD ......(1)

Q10.

Ans:

(4)

- 25 -

PROBLEM FOR PRACTICE:

1. The length of triangle drawn from an external point of the circle are ___

2. The tangents drawn at the end of a diameter of a circle are ________

3. Who introduced the word tangent?

4. A tangent PQ at a point p of a circle of radius 5cm meets a line through the

centre o at a point Q so that OQ=12 cm. What is the length of PQ?

5. From a point Q, the length of the tangent to a circle is 24 cm and the distance of

Q from the centre is 25 cm. Find the radius of the circle.

6. If TP and TQ are the two tangents to a circle with centre o so that 0110=∠POQ .

Find the value of PTQ∠

7. Prove that the tangents line at any point of a circle is perpendicular to the radius

through the point of contact.

8. Prove that tangents drawn from an external point of a circle are equal.

- 26 -

CHAPTER:6. CONSTRUCTION.


- - 3(1)=3 4(1)=4 7M

POINTS TO REMEMBER:

1. To divide a line segment in a given ratio. 2. To construct a triangle similar to a given triangle as per a given scale factor

which may be less than 1 or greater than 1. 3.To construct the pair of tangents from an external point to a circle.

Q1. (3)

- 27 -

Q2. (4)

Ans:

- 28 -

Q3. (4)

.

- 29 -

Q4. (4).

Ans:

1.Draw a line segment AB = 7 cm and divide it in the ratio 2:3

2 . Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. 3.Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are triangle.

CHAPTER

VSA(1) SA(2)

1(1)=1 2(1)=2

Q1.

The diameter of a wheel is 1.26m .The distance covered in 500

revolutions is

a) 2670m b) 2880m

Ans:c), Distance covered = 500x2

Q2.

The perimeter of the

a) 32cm b) 23cm

Ans: a) , since perimeter of the sector = 2r + arc length of the sector

= 2x10.5+60/360x 2

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Problems for practice:

a line segment AB = 7 cm and divide it in the ratio 2:3

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by

3.Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then nother triangle whose sides are 3/2times the corresponding sides of the isosceles

HAPTER:7. AREA RELATED TO A CIRCLE.

SA(3) LA(4)

3(1)=3 4(2)=8

POINTS TO REMEMBER:


b) 2880m c) 1980m d) 1596m

Ans:c), Distance covered = 500x2�r = 500x2x22/7x0.63 = 1980m

The perimeter of the sector with radius 10.5cm and sector angle 60

b) 23cm c) 41cm d) 11cm


= 2x10.5+60/360x 2�� = 21cm + 11cm= 32cm.

a line segment AB = 7 cm and divide it in the ratio 2:3

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by

3.Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then times the corresponding sides of the isosceles

REA RELATED TO A CIRCLE.

TOTAL(5)

14M

(1)


d) 1596m

r = 500x2x22/7x0.63 = 1980m

(1)

sector with radius 10.5cm and sector angle 600 is


= 21cm + 11cm= 32cm.

Q3.

Ans: D). Since area of a

Q4.

If the radius of a circle is doubled , its area becomes

a) 2 times

Ans: b) , since new area =

Q5.

Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.

Ans: Area of the sector =

Q6. Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major sector (Use Ans:

:

- 31 -

Ans: D). Since area of a sector of angle p is = p/3600 x

If the radius of a circle is doubled , its area becomes

b) 4 times c) 8 times d) 16 times.

b) , since new area = �2�2 = 4� r

2= 4x area of the circle.

Find the area of a sector of a circle with radius 6 cm if angle of the sector is

Area of the sector = �/3600 x �r2 = 60/360 x 22/7 x 6

Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major sector (Use � = 3.14).

(1)

x �r

2 =p/720

0x2 �r

2.

(1)

d) 16 times.

= 4x area of the circle.

(2)

Find the area of a sector of a circle with radius 6 cm if angle of the sector is

= 60/360 x 22/7 x 62 = 132/7 cm

2

Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find (3)

Q7.Find the area of the segment AYB if radius of the circle is 21 cm and

°. (Use 22/7 ).

- 32 -

Find the area of the segment AYB if radius of the circle is 21 cm and

Find the area of the segment AYB if radius of the circle is 21 cm and ∠ AOB = 120

(3)

Q8. Find the area of the shaded region in , where ABCD is a square of side 14 cm. Solution : Area of square ABCD = 14 × 14 cm

Diameter of each circle = 14/2 cm = 7 cmSo, radius of each circle = 7 cm .

Q9. Find the area of the shaded design in, where ABCD is a square osemicircles are drawn with each side of the square as diameter.

Ans: Let us mark the four unshaded region as I,II,III and IV.

- 33 -

e shaded region in , where ABCD is a square of side 14 cm.

Area of square ABCD = 14 × 14 cm2 = 196 cm2 Diameter of each circle = 14/2 cm = 7 cm

radius of each circle = 7 cm .

Find the area of the shaded design in, where ABCD is a square oe drawn with each side of the square as diameter.

mark the four unshaded region as I,II,III and IV.

e shaded region in , where ABCD is a square of side 14 cm. (3)

Find the area of the shaded design in, where ABCD is a square of side 10 cm and (4)

Q10. In Fig., AB and CD are two dieach other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region..

Ans: Diameter of the shaded circle =OD= OA = 7cm Radius of the shaded circle = 7/2 cm Area of the shaded circle = Area of the shaded segments = Area of the semi circle AOBC

∆ABC.

Total are of the shaded region = (38.5+ 28) = 66.5 cm

- 34 -

In Fig., AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of

Ans: Diameter of the shaded circle =OD= OA = 7cm Radius of the shaded circle = 7/2 cm Area of the shaded circle = �r

2= 22/7 x (7/2)

2 = 77/2 = 38.5 cm

Area of the shaded segments = Area of the semi circle AOBC

= 1/2x 22/7x72 - ½ x ABxOC

= 77 – 1/2x 14x7 = 28 cm2.

Total are of the shaded region = (38.5+ 28) = 66.5 cm2

ameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of

(4)

= 77/2 = 38.5 cm2.

Area of the shaded segments = Area of the semi circle AOBC – Area of

2.

- 35 -

PROBLEM FOR PRACTICE:

1. Find the perimeter of a semi-circle of radius 7 cm.

2. What will be the area of a circle if radius of a circle is doubled.

3. Find the area of a quadrant of a circle of radius 14 cm.

4. The length of wire is 66 m. Find how many circles of circumference 1.32 cm can

be made from this wire.

5. A copper wire when bent in the form of a square enclosed an area of 121 cm2. If

the same wire is bent in the form of a circle, Find the area of the circle.

6. Find the area of a sector of a circle with radius 10 cm and subtends a right angle

at the centre.

7. A garden roller has a circumference of 3 meters. How many revolutions does it

make in moving 21 meters.

8. The difference between the circumference and radius of a circle is 37 cm. Find

the area of the circle.

9. In a fig, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn

with BC as diameter. Find the area of shaded region.

10. The cost of fencing a circular filed at the rate of Rs. 24 per meter is Rs. 5280. The

field is to be played at the rate of Rs. 0.50 per m2. Find the cost of plugging the

field.

A B

C

- 36 -

CHAPTER:8. SURFACE AREA AND VOLUME.


1(1)=1 2(1)=2 3(1)=3 4(1)=4 10M

POINTS TO REMEMBER:

1To determine the surface area of an object formed by combining any two of the basic

solids, namely, cuboids, cone, cylinder, sphere and hemisphere.

2.To find the volume of objects formed by combining any two of a cuboids, cone,

cylinder, sphere and hemisphere.

Q1. The radius of the base of a cone is 5cm and height is 12 cm. Its curve surface area is : (1)

a) 30 � cm2 b) 65 � cm2 c) 80 � cm2 d) none of these.

Ans: b) , since , slant height l = 22 hr + = 22 125 + =13 cm. CSA of the cone = �rl= �x5x13 = 65 � cm2.

Q2. 2 cubes each of volume 64 cm3 are joined end to end. The surface area of the resulting cuboid is: (1)

a) 130 cm2 b) 140 cm2 c) 150 cm2 d) 160 cm2 Ans: d). Since

Q3. If two solid hemi spheres of same base radius r are joined together along bases, then curve surface of the new solid is:

a) 4� r2 b)6Ans: a) , since the resulting solid will be a sphere of radius r. Hence its curve surface area = 4�r2. Q4. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 1inner surface area of the vessel.

Ans:

- 37 -

If two solid hemi spheres of same base radius r are joined together along bases, then curve surface of the new solid is:

b)6�r2 c) 3�r2 d)8�r2 Ans: a) , since the resulting solid will be a sphere of radius r. Hence its curve surface

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

If two solid hemi spheres of same base radius r are joined together along their (1)

Ans: a) , since the resulting solid will be a sphere of radius r. Hence its curve surface

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The 4 cm and the total height of the vessel is 13 cm. Find the

(3)

Q5. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of remaining solid to the nearest cm

Ans:

Q6. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost of the canvas of the tent at the rate of Rs 500 per m2. (Note that the base of the tent will not be covered with canvas.) Ans:

- 38 -

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of

solid to the nearest cm2.

of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost

of the tent at the rate of Rs 500 per m2. (Note that the base of the tent will not be covered with canvas.)

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the

(4)

of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. Also, find the cost

of the tent at the rate of Rs 500 per m2. (Note that the base of the tent will (4)

Q7. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, onewater flows out. Find the number of lead shots dropped in the vessel. Ans:

- 39 -

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, onewater flows out. Find the number of lead shots dropped in the vessel.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel. (4)

- 40 -

Q8. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere. (2)

Ans:

Q9. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform. (4)

Ans:

- 41 -

Q10. The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum. (3)

Ans:

Q11. A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container, at the rate of Rs 20 per liter. Also find the cost of metal sheet used to make the container, if it costs Rs 8 per 100 cm2. (4)

Ans:

Q1. The volume of a right circular cylinder of base radius 7 cm and height 10cm is

a) 1540 cm3

Q2. The ratio of the total surface area of a solid hemisphere to the square of its radius is a) 2� :1 Q3. A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 cm of uniform thickness . find the thickness of the wire .

Q4 A 20 m deep wall with diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of unifor

Q5 How many silver coins , 1.75 cm in diameter and of thickness 2mm, must be melted to form a cuboids of dimensions 5.5 cm X 10 cm X 3.5 cm .

- 42 -

PROBLEMS FOR PRACTICE:

Q1. The volume of a right circular cylinder of base radius 7 cm and height 10cm

b) 770cm3 c) 1155 cm3

Q2. The ratio of the total surface area of a solid hemisphere to the square of its b)3� :1 c) 4� :1 d) 1:

A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 cm of uniform thickness . find the thickness of the wire .

Q4 A 20 m deep wall with diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.


Q1. The volume of a right circular cylinder of base radius 7 cm and height 10cm

d) 154 cm3

Q2. The ratio of the total surface area of a solid hemisphere to the square of its d) 1: 4�

A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 cm of uniform thickness . find the thickness of the wire .

Q4 A 20 m deep wall with diameter 1 cm and length 8 cm is drawn into a m thickness. Find the thickness of the wire.


- 43 -

Q6.Water in a canal , 6 m wide and 1.5 m deep is flowing with a speed of 10

km/h . How much area will it irrigate in 30 minutes , if 8 cm of standing water is

needed.

Q7. Two cubes each of volume 64 cm3 are joined end to end . Find the surface

area of the resulting cuboids.

Q8 . If right triangle , whose sides are 15 cm and 20 cm is made to revolve about

its hypotenuse . Find the volume and surface area of the double cone so formed.

Q9. A right angle triangle, whose remaining angles are300

and 600 revolves

about the hypotenuse, which is 84 cm long. Find the volume of double cone so

formed.

Q10. The radii of the internal and external surfaces of hollow metallic sphere are

3 cm and 5 cm respectively . It is melted and its material is recast into a solid

cylinder of height 10 3

4cm . Find the diameter of the cylinder.

VSA(1) SA(2)

1(1)=1 -

Q1.. Which of the following can

(A) 2 (B) Ans: D), since probability of an event cannot be less than 0 and more than

1. Q2. A card is accidently dropped from a pack of 52 playing cards. The probability that it is a red que

Ans: D), Probability ( red queen) = no. Of re queen/ total no of cards. Q3. A die is thrown once. The probability of getting a number 3 or 4 is :

A) 1/3 B) 2/3Ans: A), since p( getting 3 or 4 ) = 2/6=1/3.

Q4. If the probability of happening of an event is 5/9, then the probability of nonhappening of this event is: A) 0 Ans: C), since, P(nonQ5. If two coins are tossed simultaneously , then the is

A) ½ b) 1/3Ans: A). Since, P( getting 1 head )= P(HT or TH) = 2/4=1/2 .

- 44 -

CHAPTER:9.PROBABILITY

SA(3) LA(4)

3(1)=3 4(1)=4

POINTS TO REMEMBER:

Which of the following can be the probability of an event?(B) –1.5 (C) 17 (D) 0.7

Ans: D), since probability of an event cannot be less than 0 and more than

Q2. A card is accidently dropped from a pack of 52 playing cards. The probability that it is a red queen is : A) ¼ B) 1/13 C) 1/52

s: D), Probability ( red queen) = no. Of re queen/ total no of cards.

= 2/52 = 1/26. Q3. A die is thrown once. The probability of getting a number 3 or 4 is :

B) 2/3 C) 0 D) 1 Ans: A), since p( getting 3 or 4 ) = 2/6=1/3.

bability of happening of an event is 5/9, then the probability of nonhappening of this event is: A) 0 B) 1 C) 4/9

Ans: C), since, P(non-happening ) = 1 – P( happening) = 1 Q5. If two coins are tossed simultaneously , then the probability of getting 1 head

b) 1/3 C) ¼ D) 1 Ans: A). Since, P( getting 1 head )= P(HT or TH) = 2/4=1/2 .

TOTAL(3)

8M

be the probability of an event? (1)

Ans: D), since probability of an event cannot be less than 0 and more than

Q2. A card is accidently dropped from a pack of 52 playing cards. The C) 1/52 D) 1/26.

(1) s: D), Probability ( red queen) = no. Of re queen/ total no of cards.

Q3. A die is thrown once. The probability of getting a number 3 or 4 is : (1)

bability of happening of an event is 5/9, then the probability of non-D) 2/3 (1)

P( happening) = 1 – 5/9 = 4/9. probability of getting 1 head

(1) Ans: A). Since, P( getting 1 head )= P(HT or TH) = 2/4=1/2 .

Q6 . A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is thetaken out will be (i) red ? (ii) white ? (iii) not green?. Ans: Total number of marbles = 5+8+4 =17.

i) P(red marbles)= number of red marbles/ total number of marbles=5/17

ii) P( white marbles) = 8/17iii) P( not green)=P( red or

Q7 (i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is t he probability that this bulb is defective?(ii) Suppose the bulb drawn in (i ) is not defective and i s not replaced. Now one bulb israndom from the rest. What is the probability that this bulb is not

Ans: Q8

. A carQ8.

Q8.consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Jimmy, a trader, will only accept the trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton. What is the probability that

(i) it is acceptable to Jimmy? (ii) it is acceptable to Sujatha? Ans:

- 45 -

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One out of the box at random. What is the probability that the marble (i) red ? (ii) white ? (iii) not green?.

Ans: Total number of marbles = 5+8+4 =17. P(red marbles)= number of red marbles/ total number of marbles=5/17 P( white marbles) = 8/17 P( not green)=P( red or white) = 13/17.

Q7 (i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is t he probability that this bulb is defective?(ii) Suppose the bulb drawn in (i ) is not defective and i s not replaced. Now one bulb israndom from the rest. What is the probability that this bulb is not

consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Jimmy, a trader, will only accept the shirts which are good, but Sujatha, another trader, will only reject the shirts which have major defects. One shirt is drawn at random from the carton. What is the probability that

(i) it is acceptable to Jimmy? (ii) it is acceptable to Sujatha?

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One probability that the marble

(3)

P(red marbles)= number of red marbles/ total number of

Q7 (i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is t he probability that this bulb is defective?(ii) Suppose the bulb drawn in (i ) is not defective and i s not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective? (4)

consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major shirts which are good, but Sujatha, another

trader, will only reject the shirts which have major defects. One shirt is drawn at

1. A die is thrown once. What is the probability that it shows

ii) an odd number and

than 4 ?

2. Seventeen cards numbered 1,2,3,4,....

throughly. One person draws a card from the box. Find the Probability

getting a prime number card

3. A box containing 5 red marbles, 10 white marbles and 15 green

one marble is taken out of the box

marble taken out will be i) red 9 ii) white?

4. There are 850 tickets sold in a raffle. Brahma brought five tickets and shilpy

brought four tickets. W

winning tickets. ii) Silpi has winning ticket?

5. It is known that a box of 200 electric bulbs contains 16 defective bulbs.

bulbs is taken out at

bulb drawn is

i) Defective

- 46 -

Problems for practice:

. A die is thrown once. What is the probability that it shows

ii) an odd number and iv) an even number v) a number greater

2. Seventeen cards numbered 1,2,3,4,.........,16,17 are put in a box

person draws a card from the box. Find the Probability

number card.

3. A box containing 5 red marbles, 10 white marbles and 15 green

one marble is taken out of the box at random. What is the probability that the

marble taken out will be i) red 9 ii) white? iii) not green?

There are 850 tickets sold in a raffle. Brahma brought five tickets and shilpy

What is the probability that i) Brah

ii) Silpi has winning ticket?

5. It is known that a box of 200 electric bulbs contains 16 defective bulbs.

random from the box. What is the probability that the

Defective ii) non defective.

. A die is thrown once. What is the probability that it shows i) a ‘3’ ii) a ‘4’

v) a number greater

.....,16,17 are put in a box and mixed

person draws a card from the box. Find the Probability of

3. A box containing 5 red marbles, 10 white marbles and 15 green marbles.

at random. What is the probability that the

There are 850 tickets sold in a raffle. Brahma brought five tickets and shilpy

i) Brahma has a

5. It is known that a box of 200 electric bulbs contains 16 defective bulbs. One

hat is the probability that the

- 47 -

- 48 -

CHAPTER:1.QUADRATIC EQUATION

1.The quadratic equation with real coefficients whose one root is 2+√3 is

A) x2 – 2x +1 = 0 B) x

2 – 4x +1 = 0 C) x

2 –4x +3 = 0 D) x

2 – 4x +4 = 0

2.If x2 – 5x +1 = 0 then the value of x + 1/x is

A) - 5 B) 5 C)- 2 D) 3 .

3.If the difference of the roots of the quadratic equation x2 +kx +12 = 0 is 1 , the value

of k is :

A) – 7 B) 7 C) 4 D) 8.

4.If px2 +3x +q = 0 has two roots - 1 and - 2 , the value of q – p is

A) – 1 B) 1 C) 2 D) - 2

SHORT ANSWER/LONG ANSWER QUESTIONS

1. Give an example of a quadratic equation. (1)

2. Solution of 0352 2 =−− xx is x=? (1)

3. If the disarmament of a quadratic equation is 0 than the root of that equn. are

________ and _______ . (1)

4. In a quadratic equation 02 =++ cbxax ; sum of the roots is equal to _______ and

_______ . (1)

5.Find the nature of root of quadratic equation 0352 2 =+− xx (1)

6.Find the roots of the quadratic equation 026 2 =−− xx (2)

7.find two numbers whose sum is 27 and product is 182. (2)

8.Find the volume of K of the given quadratic equations, so that they have two equal

roots 032 2 =++ Kxx (2)

9.Find the discriminant of the quadratic equation 0342 2 =+− xx and hence find the

nature of its roots. (3)

10.Find the value of K for which the roots of the equation 01)1()4( 2 =++++ xkxk are

real and equal. (3)

11.Two water tap together can fill a tank in 8

39 hrs. The tap of longer diameter taken

10 hours less than the smaller one to fill the tank separately. find the time in which

each tap can separately fill the tank (4)

12. A motor boat whose speed is 18Km/h in still water takes 1 hr more to go 24km

up stream than to return downstream to the same spot. Find the speed of the steam.

(4)

13.A train travels 360 Km at a uniform speed. If the speed had been 5Km/hr more. It

would have taken 1 Hr less for the same journey. Find the speed of the train. (4)

- 49 -

CHAPTER 2. ARITHEMETIC PROGRESSION:

MCQ(1)

1. If 18, a,b, - 3 are in AP then a +b = ? (1)

A) 19 B) 7 C) 11 D) 15

2. If sum of n terms of an AP id 2n2 + 5n, then its 4

th term is: (1)

A) 52 B) 33 C) 19 D)16

3. Which term of the AP 5,2,-1,......is - 49? (1)

A) 19th

B)15th

C) 16th

D)20th

.

4.Which term of an AP 100,90,80,........is 0? (1)

A)5th

B) 6th

C)10th

D)11th

5 .Which term of the AP 113,108,103,....is the first negative term? (1)

A) 22nd

B)24th

C) 26th

D) 28th


6.Which term of the sequence 99, 94, 89, 84, ..... is the first negative term? (3)

7.The 8th

term of an A.P. is 17 and its 14th

term is 29. the common difference of A.P. is

______. (2)

8. ?100..........................................535251 =++++ (3)

9. If the pth

term of an AP is q and its qth

term is P then show that its (p+q)th

term is

zero. (3)

10. Find the sum of all natural numbers less than 100 which are divisible by 6. (2)

11. Prove that the sum of n terms of an AP in which first term=a, common

difference=d and last term=l is given by

)(2

lan

Sn += and [ ]dnan

Sn )1(22

−+= (4)

12. The sum of the first n terms of an AP is given by )52( 2 nnSn += Find the nth

term of

the AP. (2)

13. If the 5th

and 12th

terms of an AP are -4 and -18 respectively. Find the sum of first

20 terms of the AP. (4)

14.If the sum of first n, 2n and 3n terms of an AP be S1, S2 and S3 respectively, then

prove that )(3 123 SSS −= (3)

- 50 -

CHAPTER:3.CO-ORDINATE GEOMETRY

MCQ(1)

1.The points ( -4, 0) ,(4,0) ,(0,3) are vertices of a

A. right triangle B) equilateral triangle C) isosceles D) none of these.

2.AOBC is a rectangle whose vertices are A(0,3),O(0,0),B(5,0).The length of its

diagonal is: A) 5 B) 3 C) √34 D)4.

3. If the points (1,x),(5,2),(9,5) are collinear, then value of x is :

A)5/2 B) – 5/2 C) – 1 D) 1.

4.If the point P(2,1) lies on the line segment joining points A(4,2) and B(8,4) then :A)

AP = 1/3 AB B) AP= PB C) PB = 1/3AB D) AP= ½ AB.

5.The point which divide the line segment joining the points (7, - 6) and ( 3,4) in the

ratio 1:2 internally lies in the :

A) I quadrant B) II quadrant C) IIIrd

quadrant D) IV quadrant.


1. If A and B are (-2 , 2 ) and ( 2 , -4 ) respectively , Find the coordinates of P such that

(2)

2. The line joining the points (2,1) and (5,-8) is trisected at the points P and Q

.If P lies on the line 2x-y+k=0 .Find the value of k . (3)

3. If A(4,-6),B(3,-2)and C(5,2)are the vertices of a triangle ABC then verify that a

median of the triangle divides it into two triangles of equal areas.

(4)

4. Find a point on the Y axis which is equidistant from the point A (6, 5) and B (-4, 3).

(3)

5.Show that the points (1, 7), (4, 2), (–1, –1) and (– 4, 4) are the vertices of a square.

(3)

6.Find the ratio in which the points (m,6) divides the join of A(-4,3) and B (2,8) also find

the value of m. (3)

7.Prove that the points A (4,5) B (7,6) C (6,3) and D ( 3,2) are the vertices of a

parallelogram. Is it a rhombus? (4)

3

7

AP

AB=

- 51 -

8.Do the following points (3,2) (-2,-3) and (2,3) form a triangle ? If so , Name the type

of the triangle so formed . (3)

9. If the area of the triangle formed by the points (k, 4/3) , (-2, 6) and (3, 1) is 5 sq units

, then k is a) 3 (b) 5 (c) 2/3 (d) 3/5

(1)

10. Find the ratio in which the line 2x + y - 4 = 0 divides the line segment joining A (2, -

2) and B (3, 7). (3)

CHAPTER:4.SOME APPLICATION TRIGONOMETRY.

MCQ (1MARKS)

1. If the length of shadow on the ground of a pole 60m high is 20√3m

,the sun’s altitude is :A) 600 B) 40

0 C)30

0 D) 15

0

2. The angle of elevation of the top of a tower from a point P on the ground is ∝ .

After walking a distance d towards the foot of the tower, the angle of elevation

is found to be 6 ,then

A) ∝< 6 B) ∝> 6 C) ∝= 6 D) none of these.

3.The ratio of the length of a tree and its shadow is 1:1/√3, the angle of sun’s

elevation is: A) 600 B) 45

0 C)30

0 D) 90

0

4.An observer 1.5m tall is 28.5m away from a tower 30m high. The angle of

elevation of the top of the tower from eye is

A) 600 B) 45

0 C)30

0 D) 90

0

LONG ANS QUESTIONS(3/4 MARKS):

1.The angle of depression of the top and bottom of 8 m tall building from height of

the multi-storied building and the distance between the buildings.

2.Two pillars of equal heights stands on either side of a road which is 150m wide. At a

point on the road between the pillars, the angles of elevation of the tops of the pillars

are 60° and 30°. Find the height of each pillar and the position of the point on the

road.

3. A ladder 10 meters long reaches a point 10 meters below the top of a vertical

flagstaff. From the foot of the ladder, the elevation of the flagstaff is 60o. Find the

height of the flagstaff.

- 52 -

4.The angle of elevation of a chord from a point 60m above a lake is 300 and the angle

of depression of its reflection in the lake is 600. Find the height of the cloud.

5.The shadow of a tower standing on a level ground is found to be 40m longer when

the Sun’s altitude is 300 than when it is 60

0. Find the height of the tower.

6.Two poles of equal height are standing opposite each other on the either side of the

road which is 80 m wide from a point between them on the road, the angle of

elevation of the top of the poles are 600 and 30

0 respectively. Find the height of the

poles and the distance of the point from the poles.

7.At the foot of a mountain the angle of elevation of its summit is 450. After ascending

1000 m towards the top of the mountain at an angle 300, the elevation is found to be

600. Find the height of the mountain.

8.From a window , h meters high above the ground of a house in a street . The angles

of elevation and depression of the top and foot of another house on the opposite side

of the street are θ and φ respectively . Show that height of the opposite house is

h ( 1+tanθ cotφ) .

CHAPTER:5.CIRCLES

MCQ

1.If PQ and PR are two tangents to a circle with center O inclined at an angle of 300 , then angle QOR is:

A) 1200 B) 150

0 C)30

0 D)60

0

2.If the legth of tangent from a point A at a distance of 26 cm from the centre of

the circle is 10cm, then the radius of the circle is:

A) 22cm B) 24cm C)21 cm D) 23cm.

3.To draw a pair of tangents to a circle which are inclined to each other at an

angle of 300 , it is required to draw tangents at end points of two radii of circle , the

angle between which should be:

A) 600 B) 120

0 C)30

0 D) 150

0

4.From a point P, which is at a distance of 13 cm from the centre O of a circle of

radius 5cm, the pair of tangents PQ and PR are drawn to the circle, then the area of

the quadrilateral PQOR( in cm2) is :

A) 60 B) 65

C)30 D) 325

- 53 -

SHORT ANSWER/LONG ANSWER QUESTIONS:

1. If tangents PA and PB from a point P to a circle with centre o are inclined to each

other at an angle of 800, find the value of POA∠ (1)

2. Prove that the tangents drawn at the end points of a diameter are parallel.

(2)

3. Prove that the length of tangents drawn from an external points to a circle are

equal . (2)

4. Prove that “ In a circle the tangent and the radius drawn at the point of contact

of the tangent are perpendicular to each other . Using this theorem solve the

following :

In two concentric circles AB is a chord of the larger circle touching the inner

circle at P . Prove that AP = B P. (4)

5. Prove that opposite sides of a quadrilateral circumscribing a circle subtend

supplementary angles at the centre of the circle. (3)

6. Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of

the larger circle which touches the smaller circle. (3)

7. The length of a tangent from a point A at distance 5cm from the centre of the

circle is 4 cm. find the radius of the circle. (1)

8. What is the distance between the parallel tangents of a circle of the radius 4

cm? (1)

9. Two tangents’ TP and TQ are drawn to a circle with O from an external point T.

Prove that OPQPTQ ∠=∠⊥ 2 (3)

10. PQ is a chord of length 8 cm of a circle of radius 5 cm. the tangent at P and Q

intersect at a point T. Find the length TP. (2)

11. Prove that The parallelogram circumscribing a circle is a rhombus. (3)

12. prove that the tangents drawn at the ends of a diameter of a circle are parallel.

(2)

13. A quadrilateral ABCD is drawn to circumscribed a circle prove that AB+CD=AD +

BC (3)

14. Two concentric circles are of radii 5cm and 3cm. find the length of the chord of

the larger circle which touching the smaller circle. (3)

- 54 -

CHAPTER: 6.CONSTRUCTION

LONG ANSWER QUESTION (3/4 MAKS)

1. Draw a line segment AB of length 10 cm and divide it internally with ratio 3:4.

2. Construct a triangle similar to given ABC in which AB =4cm . BC =6cm and ∠ABC

=600 , such that each side of new triangle is

4

8 of given ∆ ABC .

3. Draw a right triangle in which the sides other than hypotenuse are of lengths 4 cm

and 3 cm then construct another triangle whose sides are 9

4 times the corresponding

sides of the given triangle .

4.Draw a circle of radius 4 cm . at a point P on the circle , draw a tangent without

using centre .

5. Draw a pairs of tangent to a circle of radius 5 cm which are inclined to each other at

an angle of 600.

6Construct one isosceles triangle whose base is 8 cm and altitude 4 cm and then

construct another triangle whose sides are 1 :

3 times the corresponding sides of the

isosceles triangle .

7Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of

radius 6cm and measure its length . Also verify the measurement by actual calculation.

8Draw a pair of tangents to a circle of radius 4 cms which are inclined to each other at

an angle of 45°.

9Draw a right triangle in which the sides are of the lengths cm and 3 cm .Then

construct another triangle whose sides are 3

5 times the corresponding sides of the

given triangle .

- 55 -

CHAPTER: 7.AREA RELATED TO CIRCLE

MCQ(1MARKS)

1.The area of the largest square that can be inscribed in a circle of radius 12 cm

is :A) 24 cm2 B) 249 cm 2 C) 288 cm 2 D)196√2 CM 2

2.The perimeter of the sector with radius 10.5 cm and sector angle 60 0 is : A) 32

cm B) 23 cm C) 41 cm D) 11cm

3. The area of the largest circle that can be inscribed in a square of side 10cm is :

A) 40� cm2 B) 30 � cm 2 C) 100 � cm 2 D)25 � cm2

4.If the perimeter and the area of the circle are numerically equal , then the

radius of the circle is :

A) 2 units B) � ;<=>? C) 4 units D) 7 units

5.A steel wire when bent in the form of a square encloses an area of 121 cm2. If

the same wire is bent in the form of a circle, then the circumference of the circle

is :

A) 88cm B) 44cm C) 22cm D) 11cm


1. Find the area of a quadrant of a circle of radius 14 cm. (1)

2.If the ratio of radii of two circles are in the ratio 3:5 then what will be the ratio of

their circumferences. (1)

3.What will be the area of sector of angle ө and radius r. (1)

4.Find the area of a sector of a circle with radius 10 cm and subtends a right angle at

the centre. (1)

5.A garden roller has a circumference of 3 meters. How many revolutions does it make

in moving 21 meters. (2)

6.The difference between the circumference and radius of a circle is 37 cm. Find the

area of the circle. (3)

7.In a fig, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with

BC as diameter. Find the area of shaded region. (3)

A B

C

8.The cost of fencing a circular filed at the rate of Rs. 24 per meter is Rs. 5280. The

field is to be played at the rate of Rs. 0.50 per m2. Find the cost of

9.From a copper plate, which is a square of side 12.5 cm, circular disc of diameter 7 cm

is cut off. Find out the weight of remaining part, if 1 Sq. cm of plate weigh 0.8 gm. (π =

7

22).

10.Three equal circles, each of radius 6cm, touch one another. Find the area enclosed

between them. ( Take π =3.14 and

11.Three horses are tied with 7

having sides 20 m, 34 m and 42 m. find the area of the plot which can be grazed by the

horse. Also find the area of the plot which remains ungrazed.

12.Find the area of the sector of a circle with radius 7 cm and making an of angle 30°

at the centre. Also, find the perimeter of the shaded region.

CHAPTER: 8.SURFACE AREA AND VOLUME

1.The ratio total surface area of a solid hemi sphere to the square of its radius is :

A) 2�: 1 B) 3 �: 1

2.The volumes of two spheres are in the ratio 64:27.The ratio of their surface area is :

A) 3:4 B)4:3 C) 9:16 D)16:9

3.If a cone is cut into two parts by a horizontal plane passing through the

its axis, the ratio of the volumes of the upp

A) 1:2 B) 1:4 C)1:6 D) 1:8

4.A frustum of a right circular cone of height 16 cm with radii of its circular ends as 8

cm and 20 cm has its slant height equal to :

A) 18 cm B) 16cm

- 56 -

The cost of fencing a circular filed at the rate of Rs. 24 per meter is Rs. 5280. The

field is to be played at the rate of Rs. 0.50 per m2. Find the cost of

From a copper plate, which is a square of side 12.5 cm, circular disc of diameter 7 cm


Three equal circles, each of radius 6cm, touch one another. Find the area enclosed

between them. ( Take π =3.14 and 3 =1.73 ).

Three horses are tied with 7 meter long rope at three corners of a triangular field


horse. Also find the area of the plot which remains ungrazed.

Find the area of the sector of a circle with radius 7 cm and making an of angle 30°

at the centre. Also, find the perimeter of the shaded region.


MCQ (1MARKS)

The ratio total surface area of a solid hemi sphere to the square of its radius is :

C)4 �: 1 D) 1:4 �

two spheres are in the ratio 64:27.The ratio of their surface area is :

A) 3:4 B)4:3 C) 9:16 D)16:9

3.If a cone is cut into two parts by a horizontal plane passing through the

its axis, the ratio of the volumes of the upper part and the cone is :

A) 1:2 B) 1:4 C)1:6 D) 1:8


cm and 20 cm has its slant height equal to :

A) 18 cm B) 16cm C) 20cm D) 24cm.

The cost of fencing a circular filed at the rate of Rs. 24 per meter is Rs. 5280. The

field is to be played at the rate of Rs. 0.50 per m2. Find the cost of pouching the field.

(4)

From a copper plate, which is a square of side 12.5 cm, circular disc of diameter 7 cm


(4)

Three equal circles, each of radius 6cm, touch one another. Find the area enclosed

(3)

long rope at three corners of a triangular field


(4)

Find the area of the sector of a circle with radius 7 cm and making an of angle 30°

(3)


The ratio total surface area of a solid hemi sphere to the square of its radius is :

two spheres are in the ratio 64:27.The ratio of their surface area is :

3.If a cone is cut into two parts by a horizontal plane passing through the mid point of

er part and the cone is :


5. A solid sphere of radius r is melted and recast into the shape of a solid cone of

height r . The radius of the base cone is:

A) 2r B) 3r C)r D) 4r


1. Water in a canal , 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h . How

much area will it irrigate in 30 minutes , if 8 cm of standing water is needed.

2. 2 cubes each of volume 64 cm3 are joined end to end . Fi

resulting cuboids.

3.A cylinder whose height is equal to its diameter has the same volume as a sphere of

radius 4 cm , calculate the radius of the base of cylinder.

4.A solid metallic right cone is melted and a number of solid r

with the material . Find the number of cylinder if the radius of the base of each

cylinder is half the radius of

height of the cone.

5.A spherical ball of iron has been melte

each smaller ball is one –fourth of the radius of original

be made.

6.The slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends

are 18 cm and 6 cm . Find the curved surface area of the frustum.

7.A right angle triangle , whose remaining angles are30

hypotenuse , which is 84 cm long. Find the volume of double cone so formed.

8.Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no

colour on it. He wanted to colour it with his crayons. The top is shaped like a cone

surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the

top is 3.5 cm. Find the area he has to colour. (Take

9.An open metal bucket is in the shape of a frustum of a cone

bounded on a hollow cylindrical base made of same metallic

- 57 -


height r . The radius of the base cone is:

B) 3r C)r D) 4r

NSWER/LONG ANSWER QUESTIONS:

Water in a canal , 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h . How


2 cubes each of volume 64 cm3 are joined end to end . Find the surface area of the

A cylinder whose height is equal to its diameter has the same volume as a sphere of

radius 4 cm , calculate the radius of the base of cylinder.

A solid metallic right cone is melted and a number of solid right cylinders are made


cylinder is half the radius of the cone and the height of each cylinder one third of the

A spherical ball of iron has been melted and made into smaller balls . If the radius of

fourth of the radius of original one, how many such balls can

The slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends

Find the curved surface area of the frustum.

A right angle triangle , whose remaining angles are300

and 60


Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no



op is 3.5 cm. Find the area he has to colour. (Take 7/22=Π )

An open metal bucket is in the shape of a frustum of a cone

bounded on a hollow cylindrical base made of same metallic


Water in a canal , 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h . How


nd the surface area of the

A cylinder whose height is equal to its diameter has the same volume as a sphere of

ight cylinders are made


and the height of each cylinder one third of the

d and made into smaller balls . If the radius of

how many such balls can

The slant height of a frustum of a cone is 4 cm and the perimeter of its circular ends

and 60 0 revolves about the


Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no



- 58 -

sheet . The diameter of the two circular ends of the bucket are 45cm and 25cm , The

total vertical height of the bucket is 40 cm and that of the cylindrical base is 6cm. Find

the are of the metallic sheet used to make the bucket. Also find the volume of the

bucket .(Do no take account the handle of the bucket).

10. 500 men took a dip in an 80 m long and 50 m broad tank. What is the rise in the

water level if the average displacement of water by a man is 4 sq m?

11. A hemispherical tank of radius 7

4meter is full of water . It is connected with a

pipe which empties it at the rate of 7 liters per second . How long would it take to

empty the tank completely .

12. Water flows through a cylindrical pipe of internal diameter 7 cm at 36 kmph

calculate the time in minutes , it would take to fill the cylindrical tank the radius of

whose base is 35 cm and height 1m .

CHAPTER:9.PROBABILITY. MCQ(1MARKS)

1.In a throw of a pair of dice , the probability of getting a doublet is

A)1/2 B) 1/3 C)1/6 D)5/12

2.The probability of getting 53 Fridays in a leap year is ;

A)1/7 B)2/7 C)4/7 D) 5/7

3.If three coins are tossed simultaneously , then probability of getting no head is

; A) ¾ B) 3/8 C)1/8 D)1/4

4.The probability of getting a sum of 9 , when two dice are thrown

simultaneously is: A) 1/36 B) 1/9 C)1/27 D)2/9

5. The probability of getting a red face card from a well shuffled pack of 52 card

is : A)3/26 B) 1/13 C) 1/26 D) 2/13

- 59 -


1. A game of chance consists of spinning an arrow which comes to rest pointing at one

of the members 1,2,3,4,5,6,7 and there are equally likely outcomes what is the

probability that it will point at

i) 8 ii) an odd number

iii) a number greater than 2?

iv) a number less than 9?

v) a number greater than 3 and less than 5.

2. There are 850 tickets sold in a raffle. Brahma brought five tickets and shilpy brought

four tickets. what is the probability that

i) Brahma has a winning tickets. ii) Silpi has winning ticket?

3 .One card is drawn from well shuffled deck of 52 playing cards . Find the probability

of getting (i) a non face cards (ii) a black king .

4. An alphabet is chosen from the English alphabets. What is the probability of getting

a letter of the word

( i ) CHILD (ii) MATHEMATICS.

5. A bag contains 4 red , 5 black and 6 white balls a ball is drawn from the bag at

random . Find the probability that the ball drawn is (i) Black or white (ii) not black .

6. Savita and Hamida are friends .What is the probability that (i) Both will have same

birthdays (ii) different birthdays (Ignoring the leap years).

7. Fine the probability of getting a head when a coin is tossed once . after find the

probability of getting a tail .

8.In a cricket match , a batsman hits the boundary 5 times out of 40 balls played by

him . Find the Probability that the boundary is not.hit by the ball.

9.A child has a die whose six focus show the letters as given below:

10.A bag contains 5 red balls and some blue balls . If the Probability of drawing a blue

ball is double that of a red ball determine the number of blue balls in the bag.?

8

7

6

5 4

3

2

1

A B C D E A

- 60 -

11. One cord is drawn from a well-shuffled deck of 52 cards. Find the probability of

getting

i) a space ii)the green of diamonds iii) a red face card

12.In a survey of 200 ladies , it was that142 like coffee, while 58 dislike it.

Find the probability that a lady chosen at random

i) Likes coffee ii) dislike coffee.

BLUE PRINT

SUB:MATHS CLASS:X (SA-2).2013-14

SL.N. TOPIC VSA 1M SA 2M SA 3M LA 6M TOTAL

1

A.QUADRATIC

EQUATION

B.ARITHMETIC

PROGRESSION

1(1)

1(1)

2(1)

2(1)

3(1)

3(2)

4(1)

4(1)

10(4)

13(5)

2. A.CIRCLES

B.CONSTRUCTION

1(1)

-

2(1)

-

3(1)

3(1)

4(1)

4(1)

10(4)

7(2)

3. A.SOME APPLICATION OF

TRIGONOMETRY

1(1) - 3(1) 4(1) 8(3)

4 PROBABILITY 1(1) - 3(1) 4(1) 8(3)

5 CO-ORDINATE

GEOMETRY

1(2) 2(1) 3(1) 4(1) 11(5)

6 A.AREA RELATED TO

CIRCLE

B.SURFACE AREA AND

VOLUME.

1(1)

-

2(1)

2(1)

3(1)

3(1)

4(2)

4(1)

14(5)

9(3)

TOTAL 1(8) 2(6) 3(10) 4(10) 90(34)

- 61 -

Sample Paper- 2013 -14

Subject: Mathematics

Class- Xth

……………………………………………………………………………………………

General Instructions:

(i) All questions are compulsory. There are three sections A, B, C and D in the

question paper.

(ii) Section A: Q. Nos. 1 to 8 carry 1 mark each.

Section B: Q. Nos.9 to 14 carry 2 marks each.

Section C: Q. Nos. 15 to 24 carry 3 mark each.

Section D: Q. Nos. 25 to 34 carry 4 mark each.

(iii) There is no overall choice, however internal choices has been provided in

one question of 2 marks, three questions of 3 marks and 2 questions of 4

marks. You have to attempt only one of the alternatives in all such

questions.

--------------------------------------------------------------------------------------------- Section A

1. If the equation x2 + 4x + k = 0 has real and distinct roots, then

(a) K< 4 (b) k > 4 (c) k ≥ 4 (d) k ≤ 4

2. If x > y > 0, x2 + y2 = 13 and x y = 6, then y =

(a) 4 (b) 3 (c) 2 (d) none of these

3. If the area of the triangle formed by the points (k, 4/3) , (-2, 6) and (3, 1)

is 5 sq units , then k is

(a) 3 (b) 5 (c) 2/3 (d) 3/5

4. The sum of n term of an AP is 3n2 + 5n, then 164 is its

(a) 24th term (b) 27th term (c) 29th term (d) none of these

5. If first term of an AP is a and nth term is b, then its common difference is

(a) (b-a)/n+1 (b) (b-a)/n-1 (c) (b-a)/n (d) none of these

6. The height of a tower is 100√3 m. the angle of elevation of its top from a

point 100 m away from its foot is

(a) 30o (b)450 (c)600 (d) None of these

- 62 -

7. Which was the first book on Probability?

(a) Dealing with possibilities

(b) World of chances

(c ) Book on games of chance

(d) None of the above

8. A letter is chosen at random from the letters of the word ‘ASSASSINATION’. Find the probability that the letter chosen is a consonant. (a) 1/13 (b) 2/13 (c) 7/13 (d) 6/13

Section B

9. Find the value of K so that the sum of the roots of the equation 3x2 +

(2k+1) x – k – 5 = 0 is equal to the product of roots.

10.Show that the roots of the equation. (x - a) (x - b) + (x - b) (x - c) + (x - c)

(x - a) = 0 are always real and they cannot be equal unless a = b = c.

11.Solve for x, 4√6 x2 -13 x -2√6 = 0 by using a completing the square.

12.Prove that the tangents drawn at the ends of a diameter of a circle are

parallel.

13.Find the distance between the points P (-4, 0) and Q (2,-5). Or

Show that the points A(1,2), B(5,4), C(3,8) and D(-1,6) are the vertices of a square.

14.Divide a line segment of length 8 cm internally in the ratio 4:5. Also, give justification of the construction.

Section C 15.Find the centre of a circle passing through the points (6, -6), (3, -7) and (3, 3). Also find the radius.

16.One natural number is 3 times the other number. Sum of their squares exceeds 13 times of the greatest number by 4. Find both the numbers.

- 63 -

17.What is the probability that a leap year, selected at random will contain 53 Sundays? 18.Find the 103th term of the AP 4, 4 ½, 5, 5 ½, 6 ……..

Or

The fourth term of an AP is 0. Prove that its 25th term is triple its 11th term. 19.Prove that am + n + am - n =2am.

20.For what value of n, the nth term of the following two A.P.’s are equal? 21.20, 25, 30, 35… And -17, -10, -3, 4… 22.Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle. 23.Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. 24.A copper wire 0.4 cm in diameter is evenly wound about a cylinder whose length is 24 cm and diameter 20 cm so as to cover the whole surface. Find the length and weight of the wire assuming the specific gravity to be 10 gm/cm3. 25.500 men took a dip in an 80 m long and 50 m broad tank. What is the rise in the water level if the average displacement of water by a man is 4 sq m?

Section D

10. If the sides of a right angled triangle are x, x + 1 and x - 1, find the hypotenuse.

11. Find the ratio in which the line 2x + y - 4 = 0 divides the line segment joining A (2, -2) and B (3, 7)

12. A metallic cylinder has radius 3 mm and height 5 mm. It is made of a metal A. to reduce its weight, a conical hole is drilled in the cylinder as shown in the figure and

it completely filled with a lighter metal B. the conical hole has a radius of 1.5 mm and its depth is 8/9 mm. calculate the ratio of the volume of the metal A to the volume of metal B in the solid.

- 64 -

13. The adjoin figure shows the cross section of an ice cream consisting of a cone surmounted by a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 10.5 cm. the outer shell ABCDFE is shaded and it is not filled with ice

Cream. AE = DC = 0.5 cm, and AB is parallel to EF, BC is parallel to FD Calculate:

(i) The volume of the ice cream in the cone (the un shaded Portion including the hemi sphere)

(ii) The volume of the outer shell (the shaded portion)

14. Two circles touch externally. The sum of their areas is 130 π sq cm and the distance between the centers is 14 cm. find the radii of the circles.

15. In the given figure, a crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD = 9 cm and at EF it is 5 cm. find the radii of two circles and the area of the shaded region.

16.Two pillars of equal heights stands on either side of a road which is 150m

wide. At a point on the road between the pillars, the angles of elevation of the

tops of the pillars are 60° and 30°. Find the height of each pillar and the

position of the point on the road.

Or

A ladder 10 metres long reaches a point 10 meters below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the flagstaff is 60o. Find the height of the flagstaff.

17.Construct a triangle of sides 4 cm, 5cm and 6cm and then a triangle similar to it whose side’s are2/3 of the corresponding sides of the first triangle.

Give the justification of the construction.

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18.A bag contains only black and white balls. The probability of picking at random a black ball from the bag is 7/10. (i) What is the probability of picking a white ball from the bag? (ii) Can you say how many black and white balls are in the bag?

CLASS - X

SUB: MATHEMATICS

SAMPLE PAPER.

SECTION A

1. For the quadratic eqn. 0122 =+− xx the value of x

x1+

(i)-1 (ii) 1 (iii) 2 (iv) -2.

2. If there are p terms in an AP, then the nth term from the end is___th term.

3. If tangents PA & PB are drawn to a circle such that 030=∠APB & chord AC is

drawn || to the tangent PB then =∠ABC

(i) 300

(ii) 450 (iii) 60

0 (iv) 90

0

4. If the equation 0142 =++ xkx has real & distinct roots, then

(i) k<4 (ii) k>4 (iii) 4≤k (iv) 4≥k

5. A chess board has 64 equal squares & the area of each square is 6.25cm2. A

border around the board is 2cm wide. Find the length of the side of the chess

board.

(i) 24cm (ii) 25cm (iii) 26cm (iv) 20cm

6. The S.A. of a cube is equal to the S.A. of a sphere, then the ratio of their volumes

is

(i) 11 : 231 (ii) 11 : 15 (iii) 15 : 11 (iv) 6 : Π.

7. A cone is cut into 2 parts by the horizontal plane passing through the mid point

of its axis, the ratio of the volumes of the upper part & the cone is

(i) 1:2 (ii) 1:4 (iii) 1:6 (iv) 1:8

8. If the radius of the circle is diminished by 10% then the area will be diminished

by

(i) 10% (ii) 19% (iii) 20 % (iv) 36 %.

9. There is a square field whose side is 44 m. A

centre leaving a gravel path all around the flower bed. The total cost of laying

the flower bed & gravelling the path @

4904. Find the width of the gravel path.

10. The sum of the first 6 terms of the AP is 42. The ratio of its 10

term is 1 : 3. Calculate the 1

11. PO ⊥ QO. The tangents to the circle at P & Q intersect at point T. Prove that PQ

& OT are right bisectors of each other.

12. ABC is an equilateral ∆ inscribed in a circle of radius 4cm with centre O Find the

area of the shaded region.

13. A cylindrical pipe has inner diameter 7 cm & water flows through it @ 192.5

litres per minute. Find the rate of flow in kilome

14. If P (x, y) is any point on the line joining the points A (a, 0) & B (0, B). Then show

that 1=+b

y

a

x.

15. The sum of the squares of two consecutive natural numbers is 421. Find the

numbers.

16. Find the 8th

term from the end of the AP 7, 10 ,13,........184.

17. Find the points on the y axis whose distances from the points (6, 7) and (4,

are in the ratio 1:2.

18. A number x is selected from the numbers 1, 2, 3 & then a second number y is

randomly selected from th

product xy of the two numbers will be less than 9.

19. Two pipes running together can fill a cistern in 6 minutes. If one pipe takes 5

minutes more than the other to fill the cistern, find the time in which e

would fill the cistern.

If the roots of the equation

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SECTION B

There is a square field whose side is 44 m. A square flowerbed is prepared in its


the flower bed & gravelling the path @ 2.75 & 1.50 per sqm. respectively is

4904. Find the width of the gravel path.

first 6 terms of the AP is 42. The ratio of its 10

term is 1 : 3. Calculate the 1st

& the 13th

term of the A.P.

QO. The tangents to the circle at P & Q intersect at point T. Prove that PQ

& OT are right bisectors of each other.

∆ inscribed in a circle of radius 4cm with centre O Find the

area of the shaded region.

A cylindrical pipe has inner diameter 7 cm & water flows through it @ 192.5

litres per minute. Find the rate of flow in kilometers per hour.

If P (x, y) is any point on the line joining the points A (a, 0) & B (0, B). Then show

SECTION:C

The sum of the squares of two consecutive natural numbers is 421. Find the


Find the points on the y axis whose distances from the points (6, 7) and (4,

A number x is selected from the numbers 1, 2, 3 & then a second number y is

randomly selected from the numbers 1, 4, 9. What is the probability that the

product xy of the two numbers will be less than 9.

Two pipes running together can fill a cistern in 6 minutes. If one pipe takes 5

minutes more than the other to fill the cistern, find the time in which e

OR

If the roots of the equation 0512 2 =++ mxx are in the ratio 3:2, then m= ?

square flowerbed is prepared in its


1.50 per sqm. respectively is

first 6 terms of the AP is 42. The ratio of its 10th

term to its 30th

QO. The tangents to the circle at P & Q intersect at point T. Prove that PQ

∆ inscribed in a circle of radius 4cm with centre O Find the

A cylindrical pipe has inner diameter 7 cm & water flows through it @ 192.5

ters per hour.

If P (x, y) is any point on the line joining the points A (a, 0) & B (0, B). Then show

The sum of the squares of two consecutive natural numbers is 421. Find the


Find the points on the y axis whose distances from the points (6, 7) and (4,-3)

A number x is selected from the numbers 1, 2, 3 & then a second number y is

e numbers 1, 4, 9. What is the probability that the

Two pipes running together can fill a cistern in 6 minutes. If one pipe takes 5

minutes more than the other to fill the cistern, find the time in which each pipe

are in the ratio 3:2, then m= ?

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20. If an A.P. consists of n terms with 1st

term a & nth

term l show that the sum of

the mth

term from the beginning & mth

term from the end is la +

(may ignore this question)

21. Two tangents TP & TQ are drawn to a circle with centre O from an external point

T. Prove that OPQPTQ ∠=∠ 2 .

22. Let ABC be a right ∆ in which AB=3 cm & 090=∠B BD is the perpendicular from B

on AC The circle through B, C, D is drawn. Construct tangents from this point to

the circle.

23. Three horses are tied to three vertices of a ∆ having sides 10m 12m & 14m with

ropes of length 7m each, find the area grazed by the 3 horses.

24. Water is flowing @ 5 km/hr through a pipe of diameter 14 cm into a rectangular

tank which is 50m long & 44m wide. Determine the time in which the level of

the water in the tank will rise by 7cm.

SECTIO:D

25. If the angle of elevation of a cloud from a point h meters above a lake is α & the

angle of depression of its reflection in the lake be β Prove that the distance of

the cloud from the point of observation is αβ

αtantan

sec2

−h

.

26. Let the opposite angular points of a square be (3, 4) & (1, -1). Find the

coordinates of the remaining angular points.

27. If P(2-1)Q(34)R(-23)S(-3-2) be four points in a plane, show that PQRS is a

rhombus but not a square. Also, find the area of the rhombus.

28. Two dice are thrown simultaneously. What is the probability that (i) 5 will not

come up on either of them. (ii) 5 will come up on at least once (iii) 5 will come

up at both dice.

29. The perimeter of right ∆ is 60cm. Its hypotenuse is 25cm. Find the area of the ∆.

OR

A swimming pool is filled by 3 pipes with uniform flow, The first two pipes

operating simultaneously, fill the pool in the same time during which the pool is

filled by the 3rd

pipe alone. The 2nd

pipe fills the pool 5 hours faster than the 1st

pipe & 4 hours slower than the 3rd

pipe. Find the time required by each pipe to

fill the pool separately.

30. Find the sum of numbers from 1 to 100 which are neither divisible by 2 nor by 5.

31. In two concentric circles with centre O, PQ is diameter of outer circle & QS is the

tangent line to the inner circle touching it in R & outer circle in S. Find the length

of PR, if radii of two circles are 13 cm & 8 cm.

32. The total height of a

total surface area.

(Hint : plumb line is a combination of cone & hemisphere)

33. A milk container is made of metal sheet in the shape of frustum of a cone whose

volume is 3

7

310459 cm The radii o

respectively. Find the cost of metal sheet used in making the container @

1.40per cm2.

34. From the top of a building AB, 60 m high, the angles of depression of the top &

bottom of a vertical lamp post CD are observed to be 30

Find

(i) The horizontal distance b/w AB & CD.

(ii) The height of the lamp post.

(iii) The difference b/w the heights of the building & the lamp post.

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In two concentric circles with centre O, PQ is diameter of outer circle & QS is the

ner circle touching it in R & outer circle in S. Find the length

of PR, if radii of two circles are 13 cm & 8 cm.

The total height of a plumb line is 14 cm, its radius is 7cm. Find its volume &

is a combination of cone & hemisphere)

A milk container is made of metal sheet in the shape of frustum of a cone whose

The radii of the lower & the upper ends is


From the top of a building AB, 60 m high, the angles of depression of the top &

bottom of a vertical lamp post CD are observed to be 300

The horizontal distance b/w AB & CD.

The height of the lamp post.

The difference b/w the heights of the building & the lamp post.

In two concentric circles with centre O, PQ is diameter of outer circle & QS is the

ner circle touching it in R & outer circle in S. Find the length

is 14 cm, its radius is 7cm. Find its volume &

A milk container is made of metal sheet in the shape of frustum of a cone whose

f the lower & the upper ends is 8 cm & 20 cm


From the top of a building AB, 60 m high, the angles of depression of the top & 0 & 60

0 respectively.

The difference b/w the heights of the building & the lamp post.

studymaterial for classx maths sa2 -2...- 5 - 6. find the roots of the quadratic equation 6x2 −x...

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