maths part 2

1. The equation 2x + 3x = 9 has :

(A) Two real roots and one imaginary roots (B) One real & one imaginary

(C) Two imaginary roots (D) Two real roots

2. Rahul and Sameer solved a quadratic equation while solving it, Rahul made a mistake in the constant term

and obtained the roots as 5, � 3. Where as sameer made a mistake in the coefficient of x and obtained the

roots as 1, � 3. The correct roots of quadratic equation are :

(A) 1, 3 (B) � 1, 3 (C) � 1, � 3 (D) 1, � 1

3. If f(x) = ax2 + bx + c, g(x) = � ax2 + bx + c, where ac 0, then f(x).g(x) = 0, has :

(A) at least three real roots (B) no real roots

(C) at least two real roots (D) two real roots and two imaginary roots

4. If a3 = b3 and a b, then the sum of the roots of equation x2 � (a2 + ab + b2) x + k = 0 is equal to :

(A) 0 (B) k (C) a2 (D) b2

5. The number of roots satisfying the equation x5 = x x5 is/are :

(A) 1 (B) 3 (C) 2 (D) 4

6. The possible values of the coefficient �a� for which x2 + ax + 1 = 0 and x2 + x + a = 0 have at least one

common root are :

(A) a = 1 & 2 (B) a = 1 & � 2 (C) a = � 1 & 2 (D) a = � 1 & � 2

7. If 4a � 5 + b = 2a + b × 2b × 2a � 4 � 63, then find the sum of a and b.

(A) 2 (B) 3 (C) 4 (D) 5

8. (2x2 + 3x + 5)1/2 + (2x2 + 3x + 20)1/2 = 15, therefore x is :

(A)

38

(B)

514

(C) 4 &

211

(D) 4

9. Find the sum of the series )73(1

+ )117(1

+ )1511(1

+ .....

(A) 31

(B) 61

(C) 121

(D) 241

10. The roots of the equation 12x2 + mx + 5 = 0 will be in the ratio 3 : 2, if m equals :

(A) 121

(B) 10

125(C)

12

105(D) ± 105

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SESSION-2011-12

Subject : Mathematics Topic : ALGEBRA AND SETS DPP No. 01

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11. What is the maximum possible value of yx

for which (x � 2)2 = 9 and (y � 3)2 = 25 ?

(A) 21

(B) 85

(C) 81

(D) 25

12. The set {x : (x � 3)(x � 5) > 0} is equal to :

(A) {x : 3 < x < 5} (B) {x : x < 3} {x : x < 5}

(C) {x : x < 3} (x : x > 5} (D) none of these

13. If A is the set of all integral multiples of 3 and B is the set of all integral multiples of 5, then A B is the

set of all integral multiples of :

(A) 3 + 5 (B) 5 � 3 (C) GCD (3, 5) (D) LCM (3, 5)

14. If A has 3 elements and B has 6 elements, then the minimum number of elements in A B is :

(A) 3 (B) 6 (C) 9 (D) 18

15. If the sets A and B are defined as

A =

Rx0 ,x

1y ; y,x , B = Rx ,xy ; y,x , then

(A) ABA (B) BBA (C) BA (D) None of these

16. Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the

total, 64 played both basketball and hockey ; 80 played cricket and basketball and 40 played cricket and

hockey 24 player all the three games. The number of boys who did not play any game is

(A) 128 (B) 216 (C) 240 (D) 160

17. If a2 + b2 + c2 = 1, then which of the following cannot be a value of (ab + bc + ca) ?

(A) 0 (B) 21

(C) 41

(D) � 1

18. 13213

33

xy)xy(yx

yx

(A) x + y (B) y � x (C) x1� y

1(D)

x1

+ y1

19. If a1/3 + b1/3 + c1/3 = 0, then :(A) a + b + c = 0 (B) a + b + c = 3abc (C) a3 + b3 + c3 = 0 (D) (a + b + c)3 = 27abc

20. In a group of 500 people, 200 can speak Hindi alonewhile only 125 can speak English alone. The numberof people who can speak both Hindi and English is :(A) 175 (B) 325 (C) 300 (D) 375

1. In the given figure, calculate the measure of POR ; where O is the center of the circle.

(A) 120º (B) 140º (C) 80º (D) 60º

2. In the given figure, the value of x is :

(A) 30º (B) 40º (C) 45º (D) 60º

3. What is the value of �d� in the given figure ?

(A) 150º (B) 60º (C) 105º (D) 90º

4. ABCD is a cyclic quadrilateral such that ADB = 30º and DCA = 80º. Find the value of DAB.

(A) 70º (B) 100º (C) 120º (D) 150º

5. It is given that AB and AC are the equal sides of an isosceles ABC, in which an equilateral DEF is

inscribed. As shown in the figure, BFD = a and ADE = b, and FEC = c. Then :

D E

A

BF C

a

b

c

(A) a = 2

cb (B) b =

2ca

(C) c = 2a + 2b (D) a = 3

cb



SESSION-2011-12

Subject : Mathematics Topic : GEOMETRY DPP No. 02


6. In a right-angled triangle, the product of two sides is equal to the half of the square of the third side, i.e.

hypotenuse. One of the acute angles may be

(A) 60º (B) 30º (C) 45º (D) 15º

7. In the given parallelogram ABCD, if 3(BE) = 2(DC) and the area of DQC is 36 square unit, then find

the area of BQE (sq. unit) :A E B

Q

CD(A) 16 (B) 20 (C) 24 (D) 18

8. In the following figure, if MN || BC, MN divides the triangle into two equal parts, then the value of the ratio

of MA and AB will be :

M N

A

B C

(A) 2 (B) 2

1(C)

2

12 (D)

2

12

9. In the given triangle ABC, points P, Q and R divide the sides BC, CA and AB in the ratios 1 : 2, 3 : 2 and

3 : 2 respectively. Find the ratio of the area of quadrilateral ARPQ to the area of the triangle ABC.A

R

BP

Q

C

(A) 157

(B) 32

(C) 52

(D) 7538

10. As shown in the figure, OD = 36 cm, OA = 20 cm and AB = 25 cm. Find the length of chord BC.

B C

A

D

36

25 20O

(A) 48 cm (B) 68 cm (C) 56 cm (D) 63 cm

11. Two circles of radius r and centres O1 and O

2 are moved towards one another and AB is the common

chord. The line joining O1 O

2 when extended meets the circumference of one of circles at P. What is the

maximum possible area of APB ?

O1 O2

A

B

P

(A) r2 (B) 34

r2 (C) 433

r2 (D) 2r2

12. P and Q are the points on the side BC, R and S are on the side CA, and T is on the side AB of a

ABC such that P and Q trisect BC, and CR : RS : SA = 1 : 1 : 2. T bisects AB. If area of the triangle

ABC = M sq. units, the area of pentagon PQRST is :

(A) 3M

(B) 4M

(C) 3M2

(D) 2M

13. RM is the direct common tangent to the circles with centres C1 and C

2. Points C

1, P, Q, C

2 K and N are

six distinct points on the same straight line. The radii of both the circles are integral multiples of a cm

and radius of the circle with C1 as the center is greater than that of the circle with C

2 as the center. If

PQ = 2cm and RM = 8 cm, then find the length of KN.

C1 C2

KQP

MR

N

(A) 59

(B) 3 cm (C) 411

(D) 2 cm

14. The inscribed circle of right angled triangle ABC touches the sides AB, BC and CA at D, E and F

respectively. If AD = 6 cm and BE = 5 cm, then find the length of AC.A

D

F

EB C5

6

(A) 59 cm (B) 57 cm (C) 55 cm (D) 61 cm

15. In PQR, PQ = PR, S and T are points on PR and PQ respectively such that RQ = QS = ST = TP.

PTS equals :P

T

S

RQ

(A) 7

(B) 72

(C) 7

3(D)

75

16. In the figure, AB and CD are diameters of the circle. AB is perpendicular to CD and chord DF intersects

AB at E. If DE = 6 units and EF = 2 units, then the area of the circle is :C

F

BOA

D

E

(A) 32 (B) 22 (C) 36 (D) 24

17. In the adjoining figure, ABCD is a parallelogram. AD is parallel to FE and FBAF

is 32

then find GDBG

.

A B

CDE

F

G

(A) 53

(B) 83

(C) 21

(D) 52

18. Consider the rectangle ABCD as shown, E is a point on CD su ch that AE = 3 unit, BE = 4 unit and

AE BE. The area of rectangle ABCD is :

A

D C

B

34

E

(A) 8 sq. unit (B) 10 sq. unit (C) 12 sq. unit (D) 14 sq. unit

1. A cow is tied by a rope to one of the vertices of a square of side 14cm. The length of the rope is 7 cm.

What percentage of the field is grazed by the cow ?

(A) 15% (B) 25% (C) 18% (D) 19.6%

2. A solid sphere is cut into 16 identical with 5 cuts. What is the percentage increase in the combined total

surface area of all the pieces over that of the original sphere ?

(A) 350% (B) 150% (C) 200% (D) 250%

3. There is a cage in hemisphere-shape in which a canary sleeps at the centre of the base. It wakes up,

flies to the top-most point in the cage, then in a straight line to the cage door at the intersection of the

curved surface and the base. It covers a total distance of 241 yards. Find out the approximate radius of

the hemispheric cage.

(A)120.5 yards (B) 140 yards (C) 50 yards (D) 100 yards

4. If the perimeter of a rectangle is p and its diagonal is d, then the difference between the length and width

of the rectagle is

(A) 2

pd8 22

(B) 2

pd8 22

(C) 2

pd6 22

(D) 4

pd8 22

5. On a square kilometre land, 2 cm of rain has fallen. Assuming that 50% fo the raindrops could have

been collected and contained in a pool having a (100 × 10) m2 base, find to what level the water level in

the pool has increased. (You have to assume that base of the pool is horizontal.

(A) 9.86 m (B) 15 m (C) 10 m (D) 20 m

6. A large solid sphere is melted and moulded to form identical right circular cones wih base radius and

height same as the radius of the sphere. One of these cones is melted and moulded to from a smaller

solid sphere. What is the ratio of the surface area of the smaller sphere to the surface area of the larger

sphere ?

(A) 1 : 34/3 (B) 1 : 23/2 (C) 1 : 33/2 (D) 1 : 24/3

7. A closed wooden box measures externally 9 cm long, 7 cm broad and 6 cm high. If the thickness of the

wood is half a centimetre, find the capacity of the box.

(A) 278 cm3 (B) 215 cm3 (C) 224 cm3 (D) 240 cm3

8. An equilateral triangle of side 6 cm is cut into smaller equilateral triangle of side 2 cm. What is the

greatest number of the smaller triangles that can be formed ?

(A) 9 (B) 6 (C) 12 (D) 15

9. A goat is tied at A, one of the vertices of a building which is in the shape of an equilateral triangular

prism. The length of each side of the building is 6 cm. The length of the rope to which the goat is tied

with is 12m. There is a huge lawn on the unshaded part of the compound. What is the area of the lawn

that the goat can graze on ? (Figure shows the view from the top.)

A

(A) 144 (B) 128 (C) 84 (D) 60



SESSION-2011-12

Subject : Mathematics Topic : MENSURATION DPP No. 03


10. The diameter of a circle is 10 cm. The radius of this circle is taken as a diameter and another circle isdrawn. A third circle is drawn with the radius of the second circle as its diameter. This process isrepeated n times until the diameter of the nth circle is less than 0.01 cm. What is the value of n ?(A) 9 (B) 10 (C) 11 (D) 12

11. Let A0 A

1 A

2 A

3 A

4 A

5 be a regular hexagon inscribed in a circle of unit radius.Then the product of

the lengths of the line segments A0 A

1, A

0 A

2 & A

0 A

4 is:

(A) 43

(B) 3 3 (C) 3 (D) 233

12. Two cylinders of same volume have their heights in the ratio 1 : 3, find the ratio of their radii �

(A) 3 : 1 (B) 2 : 1 (C) 5 : 2 (D) 2 : 5

13. If the radii of the ends of a bucket, 45 cm high are 28 cm, and 7 cm, determine it's surface area.(A) 555 cm2 (B) 545.50 cm2 (C) 561.49 cm2 (D) 567.49 cm2

14. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its

volume be 271

of the volume of the given cone, then the height above the base at which section has

been made is�(A) 10 cm (B) 15 cm (C) 20 cm (D) 25 cm

15. If h be the height and the semi vertical angle of a right circular cone, then its volume is given by -

(A) 31h3 tan2 (B)

31h2 tan2 (C)

31h2 tan3 (D)

31h3 tan3

16. The slant height of a cone is increased by P%. If radius remains same, the curved surface area is increased by �(A) P% (B) P2% (C) 2 P% (D) None

17. A hollow spherical ball whose inner radius is 4 cm is full of water. Half of the water is transferred to a conical cupand it completely filled the cup. If the height of the cup is 2 cm, then the radius of the base of cone in cm is :

(A) 4 (B) 8 (C) 8 (D) 16

18. A square and an equilateral triangle are inscribed incircle of radius 2007 cm. The ratio of the squares oftheir sides is(A) 2 : 3 (B) 3 : 2 (C) 3 : 4 (D) 4 : 3

19. In the diagram ABC is right angled at C. Also M, N and P are the mid points of sides BC, AC andAB, respectively. If the area of APN is 2 sq. cm, then the area of ABC, in sq. cm is :

(A) 8 (B) 12 (C) 16 (D) 4

20. Two circles occupy the positions shown with respect to the two squares. The two circles are eachinscribed in their square and the small square is inscribed in the large circle. What is the differencein the areas of the two shaded regions if the small square has sides which measure 8 cm ?

(A) 8 � 2 (B) 16 � 4 (C) 32 � 8 (D) 64 � 16

1. The number of zeros at the end of the product of the expression 10 × 100 × 1000 × 10000 ×

....10000000000 is :

(A) 10 (B) 100 (C) 50 (D) 55

2. Find the unit digit of the product of all the odd prime numbers.

(A) 2 (B) 3 (C) 5 (D) 7

3. A leading chocolate producing company produces �abc� chocolates per hour (abc is a three digit positive

number). In how many hours it will produce �abcabc� chocolates ?

(A) abc (B) 101 (C)1001 (D) can�t be determined

4. Total number of factors of the expression 623 � 543 � 83 is :

(A) 60 (B) 62 (C) 46 (D) can�t be determined

5. A number �p� is such that it is divisible by 7 but not by 2. Another number �q� is divisible by 6 but not by 5,

then the following expression which necessarily be an integer is :

(A) 42

q6p7 (B)

71q6p5

(C) 42

q7p6 (D) none of these

6. If 223 + 233 + 243 + .. + 873 + 883 is divided by 110 then the remainder will be :

(A) 55 (B) 1 (C) 0 (D) 44

7. If (x � 5)(y + 6)(z � 8) = 1331, then the minimum value of x + y + z is :

(A) 40 (B) 3 (C) 19 (D) not unique

8. The quotient when L.C.M. is divided by the H.C.F. of a G.P. with first term �a� and common ratio �r� is :

(A) rn � 1 (B) rn (C) a�1rn�2 (D) (rn � 1)

9. Number of pairs of positive integers which satisfy the equation b

a1

b1

a

= 11 where a + b 100

(A) 6 (B) 7 (C) 8 (D) 9

10. Convert (231)8 into decimal system :

(A) 163 (B) 153 (C) 123 (D) 113



SESSION-2011-12

Subject : Mathematics Topic : NUMBER SYSTEM DPP No. 04


11. The expression 2.1

1 +

3.21

+ 4.3

1 + ...+ )1n(n

1

for any natural number n, is :

(A) always greater than 1 (B) always less than 1(C) always equal to 1 (D) not definite

12. The digit at the 100th place in the decimal representation of 76

, is :

(A) 1 (B) 2 (C) 4 (D) 5

13. In how many ways can 576 be expressed as the product of two distinct factors ?(A) 10 (B) 11 (C) 12 (D) 13

14. Find the highest power of 63 whcih can exactly divide 6336!.(A) 2050 (B) 1054 (C) 1020 (D) 2120

15. Find the unit digit of the product of all the odd prime numbers.(A) 1 (B) 5 (C) 0 (D) None of these

1. If sin (A � B) = 21

and cos (A + B) = 21

, 0º < A + B 90º, then A and B are

(A) 45º, 15º (B) 60º, 30º (C) 30º, 15º (D) 45º, 30º

2. If sec A + tan A = p , then the value of sin A is

(A) 2p1

p2�1

(B)

1p

1�p2

2

(C) 2

2

p�12

p1(D)

1�p

1p2

2

3. If AD = 21

BD, then the value of sin is

(A) 22 x5y9

x

(B) 22 x5�y9

x2(C) 22 x5�y3

x(D) 22 x4�y3

x2

4. If sin 3 = cos (� 6º) , where 3and � 6º are acute angles, find the value of

(A) 42º (B) 21º (C) 24º (D) None of these

5.

sin1sin1

+

sin1sin1

is equal to :

(A) � 2 sec (B) 2 sec (C) 2 cosec (D) 2 tan

6. sin6A + cos6 A is equal to :(A) 1 � 3 sin2 A cos2A (B) 1 � 3sin A cos A (C) 1 + 3 sin2 A cos2A (D) 1

7. If sec x = P, cosec x = Q, then :(A) P2 + Q2 = PQ (B) P2 + Q2 = P2Q2 (C) P2 � Q3 = P2Q2 (D) P2 + Q2 = � P2Q2

8. ABCD is a parallelogram, where AB = 6 3 cm, BC = 6 cm and ABC = 120º. The bisectors of the

angles A, B, C and D form a quadrilateral PQRS. Find the area of PQRS (in cm2)

(A) 318 (B) )32( (C) 3

36(D) )32(318

9. The angle of elevation of the top of a tower as observed from a point on the horizontal ground is �x�. If we

move a distance �d� towards the foot of the tower, the angle of elevation increases to �y�, then the height of the

tower is :

(A) xtan�ytanytanxtand

(B) d(tan y + tan x) (C) d(tan y � tan x) (D) xtanytanytanxtand

10. Let be an acute angle such that sec2 +tan2 = 2. The value of (cosec2 + cot2), is(A) 9 (B) 5 (C) 4 (D) 2

11. If x = r sin cos , y = r sin sin , z = r cos then the value of x2 + y2 + z2 is(A) 0 (B) 1 (C) r2 (D) None



SESSION-2011-12

Subject : Mathematics Topic : TRIGONOMETRY DPP No. 05


12. Given 3 sin + 5 cos = 5, then the value of (3 cos � 5 sin )2 is equal to

(A) 9 (B) 59

(C) 31

(D) 91

13. If tan4 + tan2 = 1, then cos4 + cos2 has the value equal to :

(A) 21

(B) 1 (C) 41

(D) 23

14. On the level ground, the angle of elevation of the top of a tower is 30º. On moving 20 m nearer, the angleof elevation is 60º. The height of the tower is-

(A) 10 m (B) 15 m (C) 10 3 m (D) 20 m

15. The shadow of a pole standing on a horizontal plane is a metre longer when the sun�s elevation is

than when it is . The height of the pole of will be :

(A)

coscoscos

a m (B)

sinsinsin

a m (C)

sincossin

a m (D)

coscossin

a m

16. An aeroplane flying horizontally 1 km above the ground is observed by person on his right side at anelevation of 60º. If after 10 seconds the elevation is observed to be 30º, from the same point and in the

same direction, then uniform speed per hour (in km) of the aeroplane is (neglect the height of theperson for computations).

(A) 360 3 (B) 3

720(C) 720 (D) 720 3

maths part 2

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