# Solved question paper for Math Mar-2018 (PSEB 10th)

## MATHEMATICS

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### Question paper 1

1. 1.  Select the corect ansrver in the following:

Arca of a sector of angle p (in degrees) of a circle rvith radius R is :

a) $${P\over180}*2nR$$     b)$${P \over 180} * nR^2$$   c) $${P\over360}*2nR$$   d) $${P \over 360} * 2nR^2$$

c) $${P\over360}*2nR$$

2. 2.  Which of the following cannot be the probability of an event :

a) $${2\over3}$$    b)-1.5   c)15%   d) 0.7

-1.5 cannot be the probability of an event

Probability never be neagtive

3. 3. Every composite number can be expressed (factorized) as a product of prirnes, (True/False)

True

4. 4. Find the first term a and the conrmon difference d of A.P: - 5, - I, 3 7,______

First Term = -5

Common difference = -1 -(-5) = -1 + 5 = 4

5. 5. Write the formula for finding volume of a firustum of a cone.

volume of a firustum of a cone

V = $${\pi \over 3 }( R^2 + Rr + r^2)$$

6. 6. If the area of a triangle is O square units then the vertices of a triangle are _______

Collinear

7. 7. sin (A+B)=sinA+sinB                                                                         (Write Ture/False)

False

8. 8. A polynomial of degree _______ is called a linear polynomial.

one

9. 9. If tangents PA and PB from a point P to a circle with centre 0 are inclined to each other at angle of 800. then find the valne of POA

Angle POA = ?

Sun of angle of triangle is = 1800

LP + LO + LA = 1800

40 + 90 + LPOA = 1800

LPOA = 180 -130

= 50

10. 10. A child has a die whose six faces shorw the letters as given below :

A B C D E A

The die is tlrown once. What is tlre probability of getting

(i) A? (ii) D ?

P(A) = 2 / 6 =  1/ 3

P(D) = 1/ 6

11. 11. Use Euclid's division algorithm to find the H.C.F. of 420 and 130'

12. 12. Solve the pair of lirrear equation 2x + 3y = 11 and 2x - 4y = -24

13. The wickets taken by a bowler in l0 cricket matches are as follows :

2 6 4 5 0 2 1 3 2 3

Find the mode of the data

2 6 4 5 0 2 1 3 2 3

Mode = 2

2 occur three times which is greater than Every Number

14. 14. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their top

Let AC and BE Two towers 12m apart

In CED , DC2 = CE2 + DE2

= 122 + 52

= 144 + 25

= 169

DC = 13

Distance between their Poles = 13 m

15. 15. Find the discriminant of the quadratic equation 2x2 6x + 3 = 0. and hence find the nature of its roots

2x2 6x + 3 = 0

Here a = 2 b = 6 c = 3

D = b- 4ac  (-6)2 - 4. 2. 3

= 36 -24 =12

16. 16. Divide the polynomial p (x) = x3 - 3x2 + 5x - 3 by the polynomial g(x) = x2 -2 Find the quotient and remainder.

17. 17. The angle of elevation of the top of a tower from a point on the ground. which is 30 m away front the foot of the tower, is 300. Find fhe heiglrt of the tower

BC be a tower  with hight R

In Triangle ABC ,

Tan30 = R/AB

$$1 \over \sqrt 3$$ = R/ 30

h = $${30 \over \sqrt 3} * {\sqrt3 \over \sqrt 3} = { 30 \sqrt 3 \over 3}$$

h = $$10\sqrt 3$$

18. 18. In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 crn, find tlre area of the

19. 19. Prove that opposite sides ofn quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

or

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that : AE2 + BD2 = AB2 + DE2

20. 20. In a class test, the sum of Shefali's rnarks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Let Marks in Math = x

Marks in English = y

given x + y = 30

(x+2) * (y-3) = 210

From  x + y = 30

y = 30 - x

(x+2)(30-x-3) = 210

(x+2)(27-x) = 210

27x - x2 + 54 -2x =210

- x +25x = 156

x2 -25x +156 = 0

x2  -13x - 12x + 156 = 0

(x-12) (x-13) = 0

x = 12 , x = 13

y = 30-12 = 18

y = 30-13 = 17

21. 21. Consider TringleACB , right-angled at C. in which AB = 29 units, BC = 21 units and ZABC = 0 (see figure). Detemrine the value of sin2 0 + cos2 0

or

Prove that :

$${l+sec \over sec} = {sin2 A \over 1-cosA}$$

$${l+sec \over sec} = {sin2 A \over 1-cosA}$$

$${l+secA \over secA}$$ = $$1 + {1 \over cosA} \over {1 \over cosA}$$  = $$cos A + 1 \over cos A$$$$cosA \over 1$$ = 1 + cos A

By Rationalizing

$${l+secA \over secA}$$ = $$1 + cos A \over 1$$$$(1 - cosA )\over (1-cos A)$$

$$1^2 - cos ^2 A \over 1- cos A$$

$$Sin^2A \over 1- cos A$$

LHS = RHS

22. 22. An A.P. consists of 50 terms of which Srd term is 12 and the last term is 106. Find the 29th term

Given  n = 50

a3 = 12

a50 = 106

a29 = ?

an = a + (n-1)d

a3 = a + (3-1)d

12 = a + 2d

-47d = -94

d = 97/47 =2

put

a + 2d = 12

a+ 2(2) = 12

a = 12-4 = 8

a =8

a29 = a + (29-1)d

= 8 + (28)2

= 8 + 56

a29 =  64

23. 23. If A and B are (-2, -2) and (2, -4), respectively, find the coordinates of P such that  AP = $${3\over7}$$ AB and P lies on the line segment AB

Given A(-2, -2) , B(2 -4) be

two Points

AB be line joinin those points P be any Point on line Let coordinate of P be (x,y)

given AP = 3/7 AB

$${AP \over AB} = {3 \over 7}$$

$${AP \over PB} = {3 \over 4}$$

Here $$(x_1 y_1) = (-2 , -2) , (x_2 y_2) = (2, -4)$$

$$m_1 = 3, m_2 = 4$$

By Section Formula

x = $${m_1 x_2 + m_2 x_1 \over m_1 + m_1 } = {3(2) + 4(-2) \over 3+4}$$

x = $${6 - 8 \over 7} = {-2 \over 7}$$

y = $${m_1 y_2 + m_2 y_1 \over m_1 + m_1 } = {3(-4) + 4(-2) \over 3+4}$$

y = $${-12 - 8 \over 7} = {-20 \over 7}$$

P = ($${-2\over 7} , {-20 \over 7}$$)

24. 24. A well of diameter 3 m is dug 14 m cleep. The earth taken ont of it has been spreacl evenly all arouud it in the slrape of a circular ring of rvidth 4 nr to fonrr an embanklrrent. Fincl the height of the embankment.

Diameter of well = 3m

depth H = 14m

Vol of well = $$\pi r^2 H$$

$$\pi * (1.5)^2 * 14$$

Volume of inner embankment = $$\pi r^2 h$$

$$\pi * (1.5)^2 * h$$

Volume of outter embankment = $$\pi * (5.5)^2 * h$$

Volume of well = Volume of inner embankment - Volume of outter embankment

$$\pi * (1.5)^2 * 14$$ = $$\pi * (1.5)^2 * h$$$$\pi * (5.5)^2 * h$$

h = $${(1.5)^2 * 14 \over (5.5)^2 - (1.5)^2} = {2.25 * 14 \over 30.25 - 2.25} = 1.125 m$$

Height of the embankment = 1.125m

### Question paper 2

1. Part-A

1. Find the first term a and the conrmon difference d of A.P: - 5, - 1, 3 7,______

First Term = -5

Common difference = -1 -(-5) = -1 + 5 = 4

2. 2. sin (A + B) = sin A +sin B  (Write True/False)

False

3. 3.  Which of the following cannot be the probability of an event :

a) $${2\over3}$$    b)-1.5   c)15%   d) 0.7

-1.5 cannot be the probability of an event

Probability never be neagtive

4. 4. Every composite number can be expressed (factorized) as a product of primes. (True/False)

True

5. 5. If the area of a triangle is 0 square units then the vertices of a triangle are _________     (Fill in the blanks)

Collinear

6. 6. Write the formula for finding volume of a frustum of a cone

volume of a firustum of a cone

V = $${\pi \over 3 }( R^2 + Rr + r^2)$$

7. 7. A polynomial of degree is called a linear polynomial                       (Fill in the blanks)

One

8. 8.  Select the corect ansrver in the following:

Arca of a sector of angle p (in degrees) of a circle rvith radius R is :

a) $${P\over180}*2nR$$     b)$${P \over 180} * nR^2$$   c) $${P\over360}*2nR$$   d) $${P \over 360} * 2nR^2$$

c) $${P\over360}*2nR$$

9. Part-B

9. Find the discriminant of the quadratic equation 2x2 - 6x + 3 = 0, and hence find the nature of its roots.

2x2 6x + 3 = 0

Here a = 2 b = 6 c = 3

D = b- 4ac  (-6)2 - 4. 2. 3

= 36 -24 =12

10. 10. If tangents PA and PB from a point P to a circle with centre o are inclined to each other at angle of 80°, then find the value of LPOA.

Angle POA = ?

Sun of angle of triangle is = 1800

LP + LO + LA = 1800

40 + 90 + LPOA = 1800

LPOA = 180 -130

= 50

11. 11. A child has a die whose six faces shorw the letters as given below :

A B C D E A

The die is tlrown once. What is tlre probability of getting

(i) A? (ii) D ?

P(A) = 2 / 6 =  1/ 3

P(D) = 1/ 6

12. 12. Use Euclid's division algorithm to find the H.C.F. of 420 and 130.

13. 13. Solve the pair of linear equation 2x + 3y = 11 and 2x - 4y = -24,

14. 14. The wickets taken by a bowler in 10 cricket matches are as follows:

2  6  4  5  0  2  1  3  2  3

Find the mode of the data.

2 6 4 5 0 2 1 3 2 3

Mode = 2

2 occur three times which is greater than Every Number

15. 15. Divide the polynomial p(x) = x 3x +5x-3  by the polynomial g(x)= x2 -2. Find the quotient and remainder.

16. 16. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops

Let AC and BE Two towers 12m apart

In CED , DC2 = CE2 + DE2

= 122 + 52

= 144 + 25

= 169

DC = 13

Distance between their Poles = 13 m

17. Part-C

17. If A and B are (-2,-2) and (2,-4), respectively, find the coordinates of P such that AP= 3/7 AB and P lies on the line segment AB.

Given A(-2, -2) , B(2 -4) be

two Points

AB be line joinin those points P be any Point on line Let coordinate of P be (x,y)

given AP = 3/7 AB

$${AP \over AB} = {3 \over 7}$$

$${AP \over PB} = {3 \over 4}$$

Here $$(x_1 y_1) = (-2 , -2) , (x_2 y_2) = (2, -4)$$

$$m_1 = 3, m_2 = 4$$

By Section Formula

x = $${m_1 x_2 + m_2 x_1 \over m_1 + m_1 } = {3(2) + 4(-2) \over 3+4}$$

x = $${6 - 8 \over 7} = {-2 \over 7}$$

y = $${m_1 y_2 + m_2 y_1 \over m_1 + m_1 } = {3(-4) + 4(-2) \over 3+4}$$

y = $${-12 - 8 \over 7} = {-20 \over 7}$$

P = ($${-2\over 7} , {-20 \over 7}$$)

18. 18. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 30°. Find the height of the tower.

BC be a tower  with hight R

In Triangle ABC ,

Tan30 = R/AB

$$1 \over \sqrt 3$$ = R/ 30

h = $${30 \over \sqrt 3} * {\sqrt3 \over \sqrt 3} = { 30 \sqrt 3 \over 3}$$

h = $$10\sqrt 3$$

19. 19. Consider TringleACB , right-angled at C. in which AB = 29 units, BC = 21 units and ZABC = 0 (see figure). Detemrine the value of sin2 0 + cos2 0

or

Prove that :

$${l+sec \over sec} = {sin2 A \over 1-cosA}$$

$${l+sec \over sec} = {sin2 A \over 1-cosA}$$

$${l+secA \over secA}$$ = $$1 + {1 \over cosA} \over {1 \over cosA}$$  = $$cos A + 1 \over cos A$$$$cosA \over 1$$ = 1 + cos A

By Rationalizing

$${l+secA \over secA}$$ = $$1 + cos A \over 1$$$$(1 - cosA )\over (1-cos A)$$

$$1^2 - cos ^2 A \over 1- cos A$$

$$Sin^2A \over 1- cos A$$

LHS = RHS

20. 20. In the given figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 crn, find tlre area of the

21. 21. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle

or

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that : AE? + BD = ABS + DE

22. 22. Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

23. 23. In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Let Marks in Math = x

Marks in English = y

given x + y = 30

(x+2) * (y-3) = 210

From  x + y = 30

y = 30 - x

(x+2)(30-x-3) = 210

(x+2)(27-x) = 210

27x - x2 + 54 -2x =210

- x +25x = 156

x2 -25x +156 = 0

x2  -13x - 12x + 156 = 0

(x-12) (x-13) = 0

x = 12 , x = 13

y = 30-12 = 18

y = 30-13 = 17

24. 24. An A.P. consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29th term.

Given  n = 50

a3 = 12

a50 = 106

a29 = ?

an = a + (n-1)d

a3 = a + (3-1)d

12 = a + 2d

-47d = -94

d = 97/47 =2

put

a + 2d = 12

a+ 2(2) = 12

a = 12-4 = 8

a =8

a29 = a + (29-1)d

= 8 + (28)2

= 8 + 56

a29 =  64

25. Part-D

25. In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. Prove it.

or

The lengths of the tangents drawn from an external point to a circle are equal. Prove it.