## MATHEMATICS

### Previous year question paper with solutions for MATHEMATICS Mar-2018

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### Question paper 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) \({P \over 360} * 2nR^2 \)

Answer:

c) \({P\over360}*2nR\)

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

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

Answer:

-1.5 cannot be the probability of an event

Probability never be neagtive

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

Answer:

True

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

Answer:

First Term = -5

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

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

Answer:

volume of a firustum of a cone

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

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

Answer:

Collinear

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

Answer:

False

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

Answer:

one

9. If tangents PA and PB from a point P to a circle with centre 0 are inclined to each other at angle of 80

^{0}. then find the valne of POAAnswer:

Angle POA = ?

Sun of angle of triangle is = 180

^{0}LP + LO + LA = 180

^{0}40 + 90 + LPOA = 180

^{0}LPOA = 180 -130

= 50

^{0 }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 ?

Answer:

P(A) = 2 / 6 = 1/ 3

P(D) = 1/ 6

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

Answer:

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

Answer:

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

Answer:

2 6 4 5 0 2 1 3 2 3

Mode = 2

2 occur three times which is greater than Every Number

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

Answer:

Let AC and BE Two towers 12m apart

In CED , DC

^{2}= CE^{2}+ DE^{2}= 12

^{2}+ 5^{2}= 144 + 25

= 169

DC = 13

Distance between their Poles = 13 m

15. Find the discriminant of the quadratic equation 2x

^{2}6x + 3 = 0. and hence find the nature of its rootsAnswer:

2x

^{2}6x + 3 = 0Here a = 2 b = 6 c = 3

D = b

^{2 }- 4ac (-6)^{2}- 4. 2. 3= 36 -24 =12

16. Divide the polynomial p (x) = x

^{3}- 3x^{2}+ 5x - 3 by the polynomial g(x) = x^{2}-2 Find the quotient and remainder.Answer:

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 30

^{0}. Find fhe heiglrt of the towerAnswer:

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. 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

(i) Quadrant OACB (ii) Shaded region.

Answer:

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 : AE

^{2}+ BD^{2}= AB^{2}+ DE^{2}Answer:

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.

Answer:

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 - x

^{2}+ 54 -2x =210- x

^{2 }+25x = 156x

^{2}-25x +156 = 0x

^{2}-13x - 12x + 156 = 0(x-12) (x-13) = 0

x = 12 , x = 13

y = 30-12 = 18

y = 30-13 = 17

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

^{2}0 + cos^{2}0or

Prove that :

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

Answer:

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

\({l+secA \over secA} \) = \(1 + {1 \over cosA} \over {1 \over cosA}\) = \(cos A + 1 \over cos A\)* \(cosA \over 1\) = 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. An A.P. consists of 50 terms of which Srd term is 12 and the last term is 106. Find the 29th term

Answer:

Given n = 50

a

_{3}= 12a

_{50 }= 106a

_{29 }= ?a

_{n}= a + (n-1)da

_{3}= a + (3-1)d12 = a + 2d

-47d = -94

d = 97/47 =2

put

a + 2d = 12

a+ 2(2) = 12

a = 12-4 = 8

a =8

a

_{29 }= a + (29-1)d= 8 + (28)2

= 8 + 56

a

_{29 }= 6423. 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

Answer:

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. 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.

Answer:

Diameter of well = 3m

Radius = 1.5 m

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

Part-A

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

Answer:

First Term = -5

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

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

Answer:

False

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

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

Answer:

-1.5 cannot be the probability of an event

Probability never be neagtive

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

Answer:

True

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

Answer:

Collinear

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

Answer:

volume of a firustum of a cone

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

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

Answer:

One

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) \({P \over 360} * 2nR^2 \)

Answer:

c) \({P\over360}*2nR\)

Part-B

9. Find the discriminant of the quadratic equation 2x

^{2}- 6x + 3 = 0, and hence find the nature of its roots.Answer:

2x

^{2}6x + 3 = 0Here a = 2 b = 6 c = 3

D = b

^{2 }- 4ac (-6)^{2}- 4. 2. 3= 36 -24 =12

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.

Answer:

Angle POA = ?

Sun of angle of triangle is = 180

^{0}LP + LO + LA = 180

^{0}40 + 90 + LPOA = 180

^{0}LPOA = 180 -130

= 50

^{0 }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 ?

Answer:

P(A) = 2 / 6 = 1/ 3

P(D) = 1/ 6

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

Answer:

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

Answer:

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.

Answer:

2 6 4 5 0 2 1 3 2 3

Mode = 2

2 occur three times which is greater than Every Number

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

Answer:

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

Answer:

Let AC and BE Two towers 12m apart

In CED , DC

^{2}= CE^{2}+ DE^{2}= 12

^{2}+ 5^{2}= 144 + 25

= 169

DC = 13

Distance between their Poles = 13 m

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.

Answer:

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. 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.

Answer:

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

^{2}0 + cos^{2}0or

Prove that :

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

Answer:

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

\({l+secA \over secA} \) = \(1 + {1 \over cosA} \over {1 \over cosA}\) = \(cos A + 1 \over cos A\)* \(cosA \over 1\) = 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. 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

(i) Quadrant OACB (ii) Shaded region.

Answer:

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

orD 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

Answer:

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.

Answer:

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.

Answer:

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 - x

^{2}+ 54 -2x =210- x

^{2 }+25x = 156x

^{2}-25x +156 = 0x

^{2}-13x - 12x + 156 = 0(x-12) (x-13) = 0

x = 12 , x = 13

y = 30-12 = 18

y = 30-13 = 17

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

Answer:

Given n = 50

a

_{3}= 12a

_{50 }= 106a

_{29 }= ?a

_{n}= a + (n-1)da

_{3}= a + (3-1)d12 = a + 2d

-47d = -94

d = 97/47 =2

put

a + 2d = 12

a+ 2(2) = 12

a = 12-4 = 8

a =8

a

_{29 }= a + (29-1)d= 8 + (28)2

= 8 + 56

a

_{29 }= 64Part-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.Answer: