Maths Paper & Solutions - Career Point

MATHEMATICS 

NDA-2012 EXAMINATION 

QUESTIONS & SOLUTIONS 

CAREER POINT 

Q.1 If the roots of a quadratic equation are m + n and m – n, then the quadratic equation will be 

(A) x 2 + 2mx + m 2 – mn + n 2 = 0 (B) x 2 + 2mx + (m – n) 2 = 0 

(C) x 2 – 2mx + m 2 – n 2 = 0 (D) x 2 + 2mx + m 2 – n 2 = 0 

Solution [C] 

Quadratic equation whose roots are m + n, m – n is 

x 2 –((m + n) + (m – n))x + (m + n) + (m – n) = 0 

x 2 – 2mx + m 2 – n 2 = 0 

Q.2 If α, β are the roots of x 2 + px – q = 0 and γ, δ are roots of x 2 – px + r = 0 then what is (β + γ) (β + δ) equal 

to? 

(A) p + r (B) p + q (C) q + r (D) p – q 

Solution [C] 

x 2 + px – q = 0 

α + β = – p ...(1) 

αβ = –q ...(2) 

x 2 – px + r = 0 

γ + δ = p ...(3) 

γδ = r ...(4) 

Now 

(β + γ) (β + δ) 

= β 2 + (γ + δ)β + γδ 

= β 2 + pβ + r using (3) & (4) 

= q + r Q (β is root of x 2 + px – q = 0 

⇒ β 2 + pβ – q = 0 

⇒ β 2 + pβ = q) 

Q.3 Consider the following statements : 

1. The sum of cubes of first 20 natural numbers is 44400. 

2. The sum of squares of first 20 natural numbers is 2870. 

Which of the above statements is/are correct ? 

(A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2 

Solution [B] 

Q 1 3 + 2 3 + 3 3 + ..... + n 3 ⎛ n(n + 1) ⎞ 

= ⎜ ⎟ 

⎝ 2 ⎠ 

∴ 1 2 + 2 3 + 3 3 + .... + 20 3 ⎛ 20.21⎞ 

= ⎜ ⎟ 

⎝ 2 ⎠ 

= (10 × 21) 2 

= 44100 

1 2 + 2 2 + 3 2 + ........ + n 2 = 6 

1 n(n + 1)(2n + 1) 

2 

2 

CAREER POINT, CP Tower, Road No.1, IPIA, Kota (Raj.) Ph.: 0744-3040000 

Website : www.careerpointgroup.com, Email: info@careerpointgroup.com 

1 / 33

MATHEMATICS 


CAREER POINT 

∴ 1 2 + 2 2 + ........ + 20 2 = 6 

1 20.21.41 = 2870 

∴ (B) is correct 


1. (ω 10 + 1) 7 + ω = 0 

2. (ω 105 + 1) 10 = p 10 for some prime number p 

where ω ≠ 1 is a cubic root of unity. Which of the above statements is/are correct ? 


Solution [B] 

Q ω is cube root of unity 

∴ ω 3 = 1 

and 1 + ω + ω 2 = 0 

Now 

(ω 10 + 1) 7 + ω 

= ((ω 3 ) 3 ω + 1) 7 + ω 

= (ω + 1) 7 + ω Q ω 3 = 1 

= –ω 14 + ω Q 1 + ω + ω 2 = 0 

= –(ω 3 ) 4 ω 2 + ω 

= –ω 2 + ω ∴ ω 3 = 1 

≠ 0 

and 

(ω 105 + 1) 10 

= ((ω 3 ) 35 + 1) 10 

= (1 + 1) 10 Q ω 3 = 1 

= 2 10 

∴ (ω 105 + 1) 10 = 2 10 and 2 is prime 


1 1 1 

Q.5 What is the sum of first eight terms of the series 1 − + − + ... ? 

2 4 8 

89 

(A) 128 

57 

(B) 384 

85 

(C) 128 

(D) none of the above 

Solution [C] 

1 1 1 

1 − + − + ...... 

2 4 8 

Sum of first 'n' terms of G.P. 

⎛ 

8 

⎟ ⎟ ⎞ 

⎜ ⎛ 1 ⎞ 

1 − ⎜ − ⎟ 1 

⎜ 

S n = 

⎝ ⎝ 2 ⎠ 1− 

1. 

⎠ 

= 256 

⎛ 1 ⎞ 1 

1 − ⎜ − ⎟ 1+ 

⎝ 2 ⎠ 2 

= 

255 

256 

3 

2 

85 

= 128 



2 / 33

MATHEMATICS 


CAREER POINT 

Q.6 The number of permutations that can be formed from all the letters of the word 'BASEBALL' is : 

(A) 540 (B) 1260 (C) 3780 (D) 5040 

Solution [D] 

BASEBALL 

AA, BB, E, LL, S 

No. of permutation using all letters 

= 

8! 

2!2!2! 

= 5040 

Q.7 The relation 'has the same father as' over the set of children is : 

(A) only reflexive 

(B) only symmetric 

(C) only transitive 

(D) an equivalence relation 

Solution [D] 

Reflexive 

aRb = given 

⇒ a has same father as b 

aRa = a is same father as a is true 

∴ it is reflexive 

Transitive 

aRb is true 

bRc true 

∴ aRc is also true because if a has same father as b & b has same father as c then a will have same father as c 

Symmetric 

aRc is given 

bRa is true 

Q If a has same father as a then be has same father as b 

Q.8 If the roots of the quadratic equation 3x 2 – 5x + p = 0 are real and unequal, then which one of the following is 

correct ? 

Solution [B] 

25 

(A) p = 12 

Q 3x 2 – 5x + p = 0 

has real and distinct roots 

∴ D > 0 

(–5) 2 – 4(3) × p > 0 

25 

25 – 12p > 0 ⇒ p < 12 

25 

(B) p < 12 

25 

(C) p > 12 

25 

(D) p ≤ 12 

Q.9 The decimal representation of the number (1011) 2 in binary system is : 

(A) 5 (B) 7 (C) 9 (D) 11 

Solution [D] 

(1011) 2 = 1 × 2 3 + 0 × 2 2 + 1 + 2 1 + 1 × 2 0 

= 8 + 0 + 2 + 1 

= (11) 10 = 11 



3 / 33

MATHEMATICS 


CAREER POINT 

Q.10 The decimal number (57.375) 10 when converted to binary number takes the form : 

(A) (111001.011) 2 (B) (100111.110) 2 (C) (110011.101) 2 (D) (111011.011) 2 

Solution [A] 

(57.375) 10 = (111001.011) 2 

(57) 10 = (111001) 2 

(375) 10 = (.011) 2 

Q.11 If (log 3 x) (log x 2x) (log 2x y) = log x x 2 , then what is y equal to ? 

(A) 4.5 (B) 9 (C) 18 (D) 27 

Solution [B] 

log x log 2x 

. . 

log3 log x 

∴ log 3 y = 2 

⇒ y = 9 

log y 

log 2x 

= 2 

Q.12 Let P = {1, 2, 3) and a relation on set P is given by the set R = {(1, 2), (1, 3), (2, 1), (1, 1), (2, 2), (3, 3), (2, 3)}. 

Then R is : 

(A) Reflexive, transitive but not symmetric (B) Symmetric, transitive but not reflexive 

(C) Symmetric, reflexive but not transitive (D) None of the above 

Solution [A] 

(1, 3) not (3, 1) 

so not symmetric 

(1, 2) (2, 3) (1, 3) 

(2, 1) (1, 3) (7, 3) 

Transitive 

13 

n n+ 

1) 

Q.13 The value of the sum ∑( i + i where i = − 1 is : 

n= 

1 

(A) i (B) – i (C) 0 (D) i – 1 

Solution [D] 

13 

∑ 

n= 

1 

( i 

n 

+ i 

n+ 

1) 

= (i + i 2 ) + (i 2 + i 3 ) ...... (i 13 + i 14 ) 

= i + 2{i 2 + i 3 + .... + i 13 } + i 14 

= i + i 14 = i – 1 Q (i 2 + i 3 .... i 13 = 0) 

FOR THE NEXT TWO (02) QUESTION THAT FOLLOW : 

The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively ? 

Q.14 What is its first term ? 

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

Q.15 What is the common difference ? 

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



4 / 33

MATHEMATICS 

Solution [14(B) & 15(C)] 

S 10 = 120, S 20 = 440 

10 (2a + 9d) = 120 

2 

2a + 9d = 24 ...(1) 

20 (2a + 19d) = 440 

2 

2a + 19d = 44 ...(2) 

Solving (1) and (2) we get 

a = 3, d = 2 


CAREER POINT 

Q.16 If a non-empty set A contains n elements, then its power set contains how many elements ? 

(A) n 2 (B) 2 n (C) 2n (D) n + 1 

Solution [B] 

Power set is set of all subsets 

∴ no of subsets will be 2 n 

Q.17 Let A = {x ∈ W, the set of whole numbers and x < 3}, B = {x ∈ N, the set of natural numbers and 2 ≤ x < 4} 

and C = {3, 4}, then how many elements will (A ∪ B) x C contains ? 

(A) 6 (B) 8 (C) 10 (D) 12 

Solution [B] 

A = {0, 1, 2}, B = {2, 3}, C = {3, 4} 

∴ A ∪ B = {0, 1, 2, 3} 

∴ no of elements in (A ∪ B)x C will be 8 

Q.18 What is the modulus of 

2 + i where i = − 1 ? 

2 − i 

(A) 3 (B) 1/2 (C) 1 (D) None of the above 

Solution [C] 

2 + i 

2 − i 

= 

3 = 1 

3 

Q.19 What is the number of diagonals which can be drawn by joining the angular points of a polygon of 100 sides? 

(A) 4850 (B) 4950 (C) 5000 (D) 10000 

Solution [A] 

C 2 – n 

100 C 2 – 100 

100 × 99 

= – 100 

2 

= 4950 – 100 

= 4850 



5 / 33

MATHEMATICS 


CAREER POINT 

Q.20 The angles of a triangle are in AP and the least angle is 30º. What is the greatest angle (in radian) ? 

π 

π 

π 

(A) (B) (C) (D) π 

2 3 4 

Solution [A] 

2B = A + C ...(1) 

A + B + C = 180º ...(2) 

⇒ B = 60º 

Least angle = 30º 

∴ Greatest angle = 2 

π 

Q.21 If each element in a row of a determinant is multiplied by the same factor r, then the value of the 

determinant? 

(A) is multiplied by r 3 

(B) in increased by 3r 

(C) remains unchanged 

(D) is multiplied by r 

Solution [A] 

Q.22 The inverse of a diagonal matrix is a : 

(A) symmetric matrix 

(C) diagonal matrix 

Solution [C] 

(B) skew-symmetric matrix 

(D) none of the above 

Q.23 If A = 

⎡3 

⎢ 

⎢ 

5 

⎢⎣ 

7 

4⎤ 

⎥ ⎡3 

6 

⎥ 

and B = ⎢ 

8⎥ 

⎣4 

⎦ 

5 

6 

7⎤ 

8 

⎥ , then which one of the following is correct ? 

⎦ 

(A) B is the inverse of A (B) B is the adjoint of A (C) B is the transpose of A (D) None of the above 

Solution [C] 

Q.24 

⎡x⎤ 

⎡y⎤ 

⎡z⎤ 

⎡10⎤ 

If the sum of matrices 

⎢ ⎥ 

⎢ 

x 

⎥ 

, 

⎢ ⎥ 

⎢ 

y 

⎥ 

and 

⎢ ⎥ 

⎢ 

0 

⎥ 

is the matrix 

⎢ ⎥ 

⎢ 

5 

⎥ 

, then what is the value of y ? 

⎢⎣ 

y⎥⎦ 

⎢⎣ 

z⎥⎦ 

⎢⎣ 

0⎥⎦ 

⎢⎣ 

5 ⎥⎦ 

(A) –5 (B) 0 (C) 5 (D) 10 

Solution [B] 

x + y + z = 10 ...(1) 

x + y = 5 ...(2) 

y + z = 5 ...(3) 

⇒ z = 5 

⇒ y = 0 

Q.25 If the matrix AB is zero matrix, then which one of the following is correct ? 

(A) A must be equal to zero matrix or B must be equal to zero matrix 

(B) A must be equal to zero matrix and B must be equal to zero matrix 

(C) It is not necessary that either A is zero matrix or B is zero matrix 

(D) None of the above 



6 / 33

MATHEMATICS 

Solution [C] 

AB = 0 does not implies that 

either A is zero or B is zero 

ex. 

⎡1 

⎢ 

⎣0 

0⎤ 

0 

⎥ ⎦ 

⎡0 

⎢ 

⎣0 

0⎤ 

⎡0 

0⎤ 

1 

⎥ = ⎢ ⎥ 

⎦ ⎣0 

0 ⎦ 


CAREER POINT 

Q.26 

⎡ α 2 2⎤ 

If the matrix 

⎢ ⎥ 

⎢ 

− 3 0 4 

⎥ 

is not invertible, then : 

⎢⎣ 

1 −1 

1⎥⎦ 

(A) α = –5 (B) α = 5 (C) α = 0 (D) α = 1 

Solution [A] 

⎡ α 2 2⎤ 

⎢ ⎥ 

⎢ 

− 3 0 4 

⎥ 

= 0 

⎢⎣ 

1 −1 

1⎥⎦ 

α(4) – 2(–3 – 4) + 2(3) = 0 

4α + 14 + 6 = 0 

α = –5 

Q.27 The value of the determinant 

x 

y 

z 

2 

2 

2 

1 

1 

1 

y 

z 

x 

2 

2 

2 

+ z 

+ x 

2 

+ y 

2 

2 

is : 

(A) 0 (B) x 2 + y 2 + z 2 (C) x 2 + y 2 + z 2 + 1 (D) None of the above 

Solution [A] 

x 

y 

z 

2 

2 

2 

1 

1 

1 

y 

z 

x 

2 

2 

2 

+ z 

+ x 

2 

+ y 

C 3 → C 3 + C 1 

x 

y 

z 

2 

2 

2 

1 

1 

1 

x 

x 

x 

2 

2 

2 

2 

+ y 

+ y 

+ y 

2 

2 

2 

2 

+ z 

+ z 

+ z 

2 

2 

2 

= (x 2 + y 2 + z 2 ) 

x 

y 

z 

2 

2 

2 

1 

1 

1 

1 

1 

1 

= 0 

Q.28 A square matrix [a ij ] such that a ij = 0 for i ≠ j and a ij = k where k is a constant for i = j is called : 

(A) diagonal matrix, but not scalar matrix 

(B) scalar matrix 

(C) unit matrix 


Solution [B] 



7 / 33

MATHEMATICS 


CAREER POINT 

Q.29 What is the value of sin 15º ? 

(A) 

2 

3 −1 

2 

(B) 

Solution [A] 

sin 15º = sin(45º – 30º) 

= sin 45º . cos 30º – cos 45º . sin 30º 

2 

3 +1 

2 

(C) 

3 −1 

3 + 1 

(D) 

3 + 1 

3 −1 

= 

1 3 1 1 

. – . 

2 2 2 2 

= 

2 

3 −1 

2 

Q.30 If 4 sin 2 θ = 1, where 0 < θ < 2π, how many values does θ take ? 

(A) 1 (B) 2 (C) 4 (D) None of these 

Solution [C] 

4 sin 2 θ = 1 

sin 2 1 

θ = 4 

⇒ sin θ = ± 2 

1 

⇒ sin θ = 2 

1 , sin θ = – 2 

1 

π 5π 7π 11π 

⇒ θ = , , , 6 6 6 6 

{Q 0 < θ < 2π} 

Q.31 What is the value of sin 18º cos 36º equal to ? 

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

Solution [C] 

sin 18º . cos 36º = 

5 −1 

5 + 1 

. = 

4 4 

5 − 1 = 1 

4 

⎡ −1⎛ 

3 ⎞ −1⎛ 

4 ⎞⎤ 

Q.32 What is sin ⎢sin 

⎜ ⎟ + sin ⎜ ⎟⎥ equal to ? 

⎣ ⎝ 5 ⎠ ⎝ 5 ⎠ ⎦ 

(A) 0 (B) 1/2 (C) 1 (D) 2 

Solution [C] 

⎡ −1⎛ 

3 ⎞ −1⎛ 

4 ⎞⎤ 

sin ⎢sin 

⎜ ⎟ + sin ⎜ ⎟⎥ ⎣ ⎝ 5 ⎠ ⎝ 5 ⎠ ⎦ 

⎛ − 3 ⎞ 

= sin ⎜sin 1 ⎛ − 4 ⎞ 

⎟ . cos ⎜sin 1 ⎛ − 3 ⎞ 

⎟ + cos ⎜sin 1 ⎛ − 4 ⎞ 

⎟ . sin ⎜sin 1 ⎟ 

⎝ 5 ⎠ ⎝ 5 ⎠ ⎝ 5 ⎠ ⎝ 5 ⎠ 

= 

3 

. 

5 

3 

5 

4 4 

+ . = 1 

5 5 



8 / 33

MATHEMATICS 


CAREER POINT 

13 

Q.33 If sec α = where 270º < α < 360º, then what is sin α equal to ? 

5 

(A) 13 

5 

12 

(B) 13 

12 

(C) – 13 

13 

(D) – 12 

Solution [C] 

13 5 

sec α = ⇒ cos α = 5 

13 

Q 270º < α < 360º 

12 

⇒ sin α = – 13 

Q.34 What is tan(–585º) equal to ? 

(A) 1 (B) –1 (C) – 2 (D) – 3 

Solution [B] 

tan(–585º) 

= – tan 585º 

= –tan(720º – 135º) 

= {–tan(135º)} 

= tan 135º 

= –1 


1. The value of cos 46º – sin 46º is positive 

2. The value of cos 44º – sin 44º is negative 



Solution [D] 

cos 46º – sin 46º is negative 

cos 44º – sin 44º is positive 

Hence (D) is correct. 

Q.36 The line making an angle (–120º) with x-axis is situated in the : 

(A) first quadrant (B) second quadrant (C) third quadrant (D) fourth quadrant 

Solution 

Data insufficient 

Q.37 The angle subtended at the centre of a circle of radius 3 cm by an arc of length 1 cm is : 

30º 

(A) 

π 

Solution [B] 

arc 

angle = radius 

1 60º 

= radian ⇒ 3 π 

(B) 

60º 

π 

(C) 60º 




9 / 33

MATHEMATICS 


CAREER POINT 

Q.38 If sin A = 

2 and cos B = 

1 

5 

10 

where A and B are acute angles, then what is A + B equal to ? 

(A) 135º (B) 90º (C) 75º (D) 60º 

Solution [A] 

sin A = 

2 1 

, cos B = 

5 

10 

∴ cos A = 

∴ sin (A + B) = 

∴ A + B = 135º 

1 3 , sin B = 

5 10 

2 

50 

+ 

3 

50 

= 

5 

50 

= 

1 

2 

Q.39 The top of a hill observed from the top and bottom of a building of height h is at angles of eleveation α and β 

respectively. The height of the hill is : 

h cotβ 

h cot α 

h tan α 

(A) 

(B) 

(C) 


cotβ − cot α 

cot α − cotβ 

tan α − tanβ 

Solution [B] 

A 

E 

α 

C 

x 

h 

β 

D 

d 

Let height is x 

from ∆ABD 

B 

x = tan β ...(1) 

d 

from ∆ACE 

x − h 

= tan α ...(2) 

d 

from (1) & (2) 

tan α 

x – h = d tan α = x 

tanβ 

x(tan β – tan α) = h tan β 

h tanβ 

∴ x = 

tanβ − tan α 

= 

h cot α 

cot α − cotβ 



10 / 33

MATHEMATICS 


CAREER POINT 

Q.40 From the top of a lighthouse 70 m high with its base at sea level, the angle of depression of a boat is 15º. The 

distance of the boat from the foot of the lighthouse is : 

(A) 70(2 – 3 )m (B) 70(2 + 3 )m (C) 70(3 – 3 )m (D) 70(3 + 3 )m 

Solution [B] 

70 

x 

from triangle 

15º 

70 = tan 15º 

x 

= 

x = 

3 − 1 = 2 – 3 

3 + 1 

70 

2 − 

3 

= 70(2 + 3 ) 

Q.41 The locus of a point equidistant from three collinear points is : 

(A) a straight line (B) a pair of points (C) a point (D) the null set 

Solution [D] 

Q.42 The equation of the locus of a point which is always equidistant from the points (1, 0) and (0, –2) is : 

(A) 2x + 4y + 3 = 0 (B) 4x + 2y + 3 = 0 (C) 2x + 4y – 3 = 0 (D) 4x + 2y – 3 = 0 

Solution [A] 

Let point is (h, k) 

∴ (h – 1) 2 + (k – 0) 2 = (h – 0) 2 + (k + 2) 2 

h 2 + 1 – 2h + k 2 = h 2 + k 2 + 4 + 4k 

2h + 4k + 3 = 0 

∴ Locus is 2x + 4y + 3 = 0 

Q.43 The points (5, 1), (1, –1) and (11, 4) are : 

(A) collinear 

(C) vertices of equilateral triangle 

Solution [A] 

∴ A(5, 1), B(1, –1), C(11, 4) 

m AB = m BC = m AC 

slope are equal 

(B) vertices of right angled triangle 

(D) vertices of an isosceles triangle 

Q.44 What is the perpendicular distance between the parallel lines 3x + 4y = 9 and 9x + 12 y + 28 = 0 ? 

(A) 7/3 units (B) 8/3 units (C) 10/3 units (D) 11/3 units 

Solution [D] 

3x + 4y – 9 = 0 ……. (1) 



11 / 33

MATHEMATICS 


CAREER POINT 

3x + 4y + 

28 = 0 ……..(2) 

3 

distance between parallel lines = 

55 11 

⇒ ⇒ units 

15 3 

28 

+ 9 

3 

3 

2 

+ 4 

2 

Q.45 Let p, q, r, s be the distances from origin of the points (2, 6), (3, 4), (4, 5) and (–2, 5) respectively. Which one 

of the following is a whole number ? 

(A) p (B) q (C) r (D) s 

Solution [B] 

p = 

q = 

r = 

s = 

2 

2 

2 + 6 ⇒ 40 

2 

2 

3 + 4 = 5 

2 

2 

4 + 5 41 

2 

2 

2 + 5 = 29 

Q.46 From the point (4, 3) a perpendicular is dropped on the x-axis as well as on the y-axis. If the lengths of 

perpendiculars are p, q respectively, then which one of the following is correct ? 

(A) p = q (B) 3p = 4q (C) 4p = 3q (D) p + q = 5 

Solution [C] 

p = 3, q = 4 

⇒ 4p = 3q 

Q.47 What is the value of λ if the straight line (2x + 3y + 4) + λ(6x – y + 12) = 0 is parallel to y-axis ? 

(A) 3 (B) – 6 (C) 4 (D) – 3 

Solution [A] 

⎛ 6λ + 2 ⎞ 

Slope = – ⎜ ⎟ 

⎝ 3 – λ ⎠ 

For slope not to be defined λ = 3 

Q.48 The line y = 0 divides the line joining the points (3, – 5) and (–4, 7) in the ratio - 

(A) 3 : 4 (B) 4 : 5 (C) 5 : 7 (D) 7 : 9 

Solution [A] 

–4λ + 3 

= 0 

λ + 1 

λ = 4 

3 ⇒ 3 : 4 



12 / 33

MATHEMATICS 


CAREER POINT 

Q.49 The sum of the focal distances of a point on the ellipse 

2 

2 

x y 

+ = 1 is - 

4 9 

(A) 4 units (B) 6 units (C) 8 units (D) 10 units 

Solution [B] 

PS + PS' = 2b = 2 × 3 = 6 

Q.50 The eccentricity e of an ellipse satisfies the condition - 

(A) e < 0 (B) 0 < e < 1 (C) e = 1 (D) e > 1 

Solution [B] 

0 < e < 1 

Q.51 The equation of a straight line which makes an angle 45° with the x-axis with y-intercept 101 units is - 

(A) 10x + 101y = 1 (B) 101x + y = 1 (C) x + y – 101 = 0 (D) x – y + 101 = 0 

Solution [D] 

y = mx + c 

y = x + 101 

Q.52 If the points (2, 4), (2, 6) and (2 + 3 , k) are the vertices of an equilateral triangle, then what is the value of k ? 

(A) 6 (B) 5 (C) – 3 (D) 1 

Solution [B] 

AB = BC ⇒ (k – 4) 2 + 3 = 4 

k – 4 = ± 1 

k = 5, 3 

AC = BC 

3 + (k – 6) 2 = 4 

(k – 6) 2 = 1 

(k – 6) = ± 1 

k = 7 , 5 

∴ k = 5 

Q.53 If the distance between the points (7, 1, – 3) and (4, 5, λ) is 13 units, then what is one of the values of λ ? 

(A) 20 (B) 10 (C) 9 (D) 8 

Solution [C] 

2 

2 

( 7 – 4) + (1 – 5) + ( λ + 3) = 13 

9 + 16 + λ 2 + 9 + 6λ = 169 

λ 2 + 6λ – 135 = 0 

λ = 9, – 15 

2 



13 / 33

MATHEMATICS 


CAREER POINT 

Q.54 If a line OP of length r (where 'O' is the origin) makes an angle α with x-axis and lies in the xz-plane, then 

what are the coordinates of P ? 

(A) (r cos α, 0, r sin α) (B) (0, 0, r sin α) (C) (r cos α, 0, 0) (D) (0, 0, r cos α) 

Solution [A] 

The coordinate of P will be (r cos α, 0, r sin α) 

Q y coordinate will be zero 

Q.55 What is the distance of the point (1, 2, 0) from yz-plane is - 

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

Solution [A] 

Distance of (1, 2, 0) from yz-plane is 1. 

Q.56 What are the direction cosines of a line which is equally inclined to the positive directions of the axes ? 

1 1 1 

1 1 1 

1 1 1 

1 1 1 

(A) , , (B) – , , (C) – ,– , (D) , , 

3 3 3 

3 3 3 

3 3 3 3 3 3 

Solution [A] 

If a line makes angle α, β, γ 

with axes then direction cosines cos α, cos β, cos γ such that cos 2 α + cos 2 β + cos 2 γ = 1 

if α = β = γ 

cos 2 α + cos 2 α + cos 2 α = 1 

3cos 2 α = 1 

⇒ cos α = 

1 

3 

Q.57 What is the angle between the lines 

(A) 2 

π 

(B) 3 

π 

x – 2 

1 

Solution [A] 

x – 2 y + 1 z + 2 

= = 

1 – 2 1 

Direction ratios are proportional to 1, –2, 1 

x –1 

1 

= 

3 

y + 

2 

3 

2 

= 

z + 5 

2 

Direction ratios are proportional to 1, 2 

3 , 2 

= 

y + 1 z + 2 = 

– 2 1 

(C) 6 

π 

and 

x –1 

1 

= 

2 y + 3 

3 

= 

z + 5 ? 

2 

(D) None of these 



14 / 33

MATHEMATICS 


CAREER POINT 

cos θ = 

a 

= 

1 

π 

⇒ θ = 2 

2 

1 

2 

a a 

1 

2 

2 

1 

+ b 

+ (–2) 

+ b b 

1 

2 

1 

+ c 

2 

2 

a 

2 

+ 1 

+ c c 

2 

2 

1 

+ b 

1 

2 

2 

2 

2 

+ c 

2 

2 

⎛ 3 ⎞ 

(1)(1) + (–2) ⎜ ⎟ + (1)(2) 

⎝ 2 ⎠ 

⎛ 2 ⎞ 

+ ⎜ ⎟ 

⎝ 3 ⎠ 

2 

2 

+ 1 

= 0 

Q.58 What is the equation to the plane through (1, 2, 3) parallel to 3x + 4y – 5z = 0 ? 

(A) 3x + 4y + 5z + 4 = 0 (B) 3x + 4y – 5z + 14 = 0 (C) 3x + 4y – 5z + 4 = 0 (D) 3x + 4y – 5z – 4 = 0 

Solution [C] 

Let equation of plane parallel to 

3x + 4y – 5z = 0 is 

3x + 4y – 5z + λ = 0 

Q it passes through (1, 2, 3) 

∴ 3(1) + 4(2) – 5(3) + λ = 0 

⇒ 3 + 8 – 15 + λ = 0 

⇒ λ = 4 

∴ equation of plane is 3x + 4y – 5z + 4 = 0 

Q.59 What are the direction ratios of the line of intersection of the planes x = 3z + 4 and y = 2z – 3 

(A) 〈1, 2, 3〉 (B) 〈2, 1, 3〉 (C) 〈3, 2, 1〉 (D) 〈1, 3, 2〉 

Solution [C] 

x = 3z + 4 

and y = 2z –3 

⇒ 

⇒ 

x – 4 

3 

x – 4 

3 

= z and 

= 

y + 3 

2 

y + 3 

2 

= 1 

z 

= z 

∴ Direction ratios are proportional to 〈3, 2, 1〉 

Q.60 What is the equation to the straight line passing through (a, b, c) and parallel to z-axis ? 

(A) 

(C) 

x – a 

1 

x – a 

0 

= 

= 

y – b 

0 

y – b 

1 

= 

= 

z – c 

0 

z – c 

0 

(B) 

(D) 

x – a 

0 

x – a 

0 

Solution [B] 

Equation of line passing through (a, b, c) and parallel to z-axis is 

x – a 

0 

= 

y – b 

0 

= 

z – c 

1 

= 

= 

y – b 

0 

y – b 

1 

= 

= 

z – c 

1 

z – c 

1 



15 / 33

MATHEMATICS 


CAREER POINT 

Q.61 What is 1+ x –1 

equal to ? 

x→0 

x 

(A) 0 (B) 1/2 (C) 1 (D) –1/2 

Solution [B] 

lim 

x→0 

1+ x – 1 

x 

= 

( 1 + x – 1)( 1 + x + 1) 

lim 

x→ 0 x( 1 + x + 1 ) 

= lim (1 + x – 1) 

x→ 0 x( 1+ 

x + 1 ) 

= lim 1 

x→ 0 ( 1+ 

x + 1 ) 

1 

= 2 

Q.62 What is 2(1 – cos x) 

x→0 

2 

x 

equal to ? 

(A) 0 (B) 1/2 (C) 1/4 (D) 1 

Solution [D] 

x lim →0 

= 

= 

x lim →0 

x lim →0 

(1 – cos x) 

2. 

2 

x 

⎛ 2 x ⎞ 

⎜2sin 

⎟ 

2 

2. 

⎝ ⎠ 

2 

x 

⎛ 

⎜ sin 

4. ⎜ 

⎜ x 

⎝ 2 

x 

2 

⎞ 

2 

⎟ 

⎟ 

⎟ 

⎠ 

× 4 

1 = 1 

Q.63 Consider the following ? 

1. 

2. 

lim x→0 

x lim →0 

1 exists. 

x 

1 

e 

x 

does not exist 

which of the above is/are correct ? 


Solution [B] 

lim 

x→0 

1 does not exist 

x 



16 / 33

MATHEMATICS 


CAREER POINT 

Q L.H.L. = 

lim 

– 

x→0 

1 = – ∞ 

x 

R.H.L. = 

1 

lim 

+ 

= + ∞ 

x→0 x 

∴ limit does not exist 

x lim →0 

1 

e 

x 

does not exist 

Q L.H.L. = 

lim 

– 

x→0 

1 

e 

x 

= e –∞ = 0 

R.H.L. = 

1 

lim 

x 

+ 

x→0 

∴ limit does not exist 


e = e +∞ = ∞ 

Q.64 If x m + y m dy x 

= 1 such that = – , then what should be the value of m ? 

dx y 

(A) 0 (B) 1 (C) 2 (D) None of the above 

Solution [C] 

mx m–1 + my m–1 dy = 0 

dx 

m–1 

dy x = – 

m– 1 

dx y 

∴ m –1 = 1 

m = 2 

= – y 

x 

x 2 

Q.65 Which one of the following is correct in respect of the function f(x) = for x ≠ 0 and f(0) = 0 ? 

| x | 

(A) f(x) is discontinuous every where 

(B) f(x) is continuous every where 

(C) f(x) is continuous at x = 0 only 

(D) f(x) is discontinuous at x = 0 only 

Solution [B] 

x 2 

f(x) = , x ≠ 0 

| x | 

f(0) = 0 

⎧– 

x 

⎪ 

f(x) = ⎨ x 

⎪ 

⎩ 0 

for 

for 

x < 0 

x > 0 

x = 0 

∴ f(x) is continuous every where 



17 / 33

MATHEMATICS 


CAREER POINT 

Q.66 What is 

lim 

x→ 2 

2 

x – 4 

x – 2 

lim 

x→ 2 2 

x – 4 

equal to ? 

(A) 0 (B) 1/4 (C) 1/2 (D) 1 

Solution [B] 

x – 2 

= lim x – 2 

x→ 2 (x – 2)(x + 2 ) 

⇒ 

1 

lim 

x→ 2 (x + 2 ) 

= 4 

1 

Q.67 The radius of a circle is uniformly increasing at the rate of 3 cm/s. What is the rate of increase in area, when 

the radius is 10 cm ? 

(A) 6π cm 2 /s (B) 10π cm 2 /s (C) 30π cm 2 /s (D) 60π cm 2 /s 

Solution [D] 

dr = 3 

dt 

A = πr 2 

dA dr = π2r 

dt dt 

= π.2.10.3 

= 60π cm 2 /s 

Q.68 

x + 5 

Let f: R → R be a function whose inverse is . What is f(x) equal to ? 

3 

(A) f(x) = 3x + 5 (B) f(x) = 3x – 5 (C) f(x) = 5x – 3 (D) f(x) does not exist 

Solution [B] 

x + 5 

y = 

3 

y is inverse of f(x) 

3y = x + 5 

∴ f(x) will be inverse of y 

x = 3y – 5 

f(x) = 3x – 5 

Q.69 Consider the following statements – 

dy 

1. If y = ln(sec x + tan x), then = sec x dx 

dy 

2. If y = ln (cosec x – cot x), then = cosec x 

dx 

Which of the above is/are correct ? 


Solution [C] 



18 / 33

MATHEMATICS 


CAREER POINT 

1. y = ln(sec x + tan x) 

dy 1 = (sec x tan x + sec 2 x) 

dx (secx + tan x) 

= secx 

2. y = ln (cosec x – cot x) 

dy 1 = [–cosec x cot x + cosec 2 x] 

dx (cosecx – cot x) 

⇒ cosec x 

= cosec x 

[cosecx – cot x] 

(cosecx – cot x) 

Q.70 If f(x) = 2 sin x , then what is the derivative of f(x) ? 

(A) 2 sin x ln 2 (B) (sin x)2 sin x–1 (C) (cosx)2 sin x– 1 (D) None of the above 

Solution [D] 

f(x) = 2 sin x 

f '(x) = 2 sinx . ln 2. cosx 

Q.71 The function f(x) = x 3 – 3x 2 + 6 is an increasing function for - 

(A) 0 < x < 2 (B) x < 2 (C) x > 2 or x < 0 (D) all x 

Solution [C] 

f(x) = x 3 – 3x 2 + 6 

f '(x) = 3x 2 – 6x 

= 3x(x – 2) 

∴ f '(x) > 0 ∀ x ∈ (–∞, 0] ∪ [2, ∞) 

Q.72 Consider the following statements: 

1. If f(x) = x 3 and g(y) = y 3 then f = g 

2. Identity function is not always a bijection. 



Solution [A] 

f(x) = x 3 

f(y) = y 3 

f and g are identical function 

⇒ f = g 

f(x) = x is an identity function and identity function is always one-one-onto 



19 / 33

MATHEMATICS 


CAREER POINT 

Q.73 Let A = {x ∈ R, |x ≥ 0|}. A function f: A → A is defined by f(x) = x 2 . Which one of the following is correct ? 

(A) The function does not have inverse 

(B) f is its own inverse 

(C) The function has an inverse but f is not its own inverse 


Solution [C] 

Q.74 If y = ln(e mx + e –mx dy 

), then what is at x = 0 equal to ? 

dx 

(A) –1 (B) 0 (C) 1 (D) 2 

Solution [B] 

y = log e (e mx + e –mx ) 

dy 1 = (e mx .m + e –mx (–m)) 

–mx 

dx (e 

+ e ) 

= 

(e 

m 

mx + 

e 

–mx 

(e mx – e –mx ) 

) 

⎛ dy ⎞ 

⎜ ⎟ 

⎝ dx ⎠ 

x= 

0 

= 0 

Q.75 What is the minimum value of |x| ? 

(A) – 1 (B) 0 (C) 1 (D) 2 

Solution [D] 

min. value of |x| is 0 

Q.76 What is 

∫ 

a e dx 

x 

x 

equal to ? 

Solution [C] 

x 

x 

a e 

(A) + c (B) a x e x + c (C) 

ln a 

Where c is the constant of integration. 

∫ 

a 

x 

e 

x 

dx 

x 

∫ 

(ae ) dx = 

(ae) 

+ c 

log (ae) 

e 

x 

x 

x 

a e 

ln(ae) 

+ c (D) None of the above 



20 / 33

MATHEMATICS 


CAREER POINT 

1 

Q.77 What is 

∫ 

x | x | dx equal to ? 

–1 

(A) 2 (B) 1 (C) 0 (D) –1 

Solution [C] 

Let f(x) = x|x| 

1 

I = 

∫ 

–1 

x 

| x | dx 

Q f(x) is an odd function 

∴ I = 0 

1 

tan x 

Q.78 What is 

∫ 

dx equal to ? 

2 

1+ 

x 

(A) 

Solution [B] 

2 

π 

1 

8 

0 

–1 

–1 

tan x 

I = 

∫ 

dx, 

2 

1+ 

x 

0 

tan –1 x = t 

1 

dx = dt 

2 

1+ 

x 

π/ 

4 

I = 

∫ 

t dt 

0 

2 

π 

(B) 32 

(C) 4 

π 

(D) 8 

π 

⎡ 

2 

t ⎤ 

= ⎢ ⎥ 

⎣ 2 ⎦ 

π / 4 

0 

= 2 

1 

⎛ 

2 

π ⎞ 

⎜ ⎟ 

⎝ 16 ⎠ 

2 

π 

= 32 

π/ 

2 

Q.79 What is 

∫sin 2x ln(cot x) dx equal to ? 

0 

(A) 0 (B) π ln 2 (C) –π ln2 (D) 

Solution [A] 

π/ 

2 

I = 

∫sin 

2x ln(cot x) dx 

0 

⎛ π ⎞ 

Q f ⎜ – x ⎟ = – f(x) 

⎝ 2 ⎠ 

⎪⎧ 

2a 

⎪⎫ 

⎨∫ 

f (x) dx = 0, if f (2a – x) = –f (x) ⎬ 

⎪⎩ 0 

⎪⎭ 

∴ I = 0 

πln 2 

2 



21 / 33

MATHEMATICS 


CAREER POINT 

Q.80 What is the area of the portion of the curve y = sin x, lying between x = 0, y = 0 and x = 2π ? 

(A) 1 square unit (B) 2 square unit (C) 4 square unit (D) 8 square unit 

Solution [C] 

π 

Area = 2 

∫sin 

x dx 

0 

π 

= 2[– 

cos x] 0 

= –2(–1 –1) 

= 4 

ln x 

Q.81 What is 

∫ dx equal to ? 

x 

(A) 

2 

(ln x) 

2 

+ c (B) 

Where c is constant of integration 

Solution [A] 

ln x 

I = 

∫ 

dx 

x 

put 

ln x = t 

1 dx = dt 

x 

I = 

∫ t dt 

t 2 

= 2 

(ln x) 

2 

+ c (C) (ln x) 2 + c (D) None of the above 

= 

2 

(ln x) 

2 

+ c 

Q.82 What is the area of the region bounded by the lines y = x, y = 0 and x = 4 ? 

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

Solution [B] 

(4, 4) 

(0, 0) (4, 0) 

y = x 

x = 4 

Area = 2 

1 × 4 × 4 = 8 square unit 



22 / 33

MATHEMATICS 


CAREER POINT 

⎛ 1 1 ⎞ 

Q.83 What is 

∫ ⎜ – ⎟dx 

equal to ? 

2 2 

⎝ cos x sin x ⎠ 

(A) 2 cosec 2x + c (B) –2cot 2x + c (C) 2sec 2x + c (D) –2tan 2x + c 

Where c is constant of integration 

Solution [A] 

⎛ 1 1 ⎞ 

∫⎜ 

– ⎟dx 

2 2 

⎝ cos x sin x ⎠ 

= 

∫ 

sec 2 x dx – 

∫ 

cosec x dx 

= tan x + cot x + c 

= 

sin x 

cos x 

2 

sin x 

+ 

cos x 

sin x 

2 

cos x 

+ c 

+ 

= 

+ c 

sin x.cos x 

1 

= 

+ c 

sin x.cos x 

2 

= + c 

sin 2x 

= 2 cosec 2x + c 

where c is constant of integration 

2 

Q.84 

3 

d y ⎛ 

2 

d y ⎞ 

What is the degree of the differential equaltion + 2 ⎜ ⎟ 

3 

2 

dx 

dx 

⎝ ⎠ 

dy 

– + y = 0 dx 

(A) 6 (B) 3 (C) 2 (D) 1 

Solution [D] 

3 

d y ⎛ 

2 

d y ⎞ 

+ 2 ⎜ ⎟ 

3 

2 

dx 

dx 

⎝ ⎠ 

degree = 1 

2 

dy 

– + y = 0 dx 

Q.85 Consider a differential equation of order m and degree n. Which one of the following pairs is not feasible ? 

(A) (3, 2) (B) (2, 3/2) (C) (2, 4) (D) (2, 2) 

Solution [B] 

Degree of differential equation cannot be fraction 

Q.86 The differential equation respresenting the family of curves y = a sin(λx + a) is - 

(A) 

d 

2 

dx 

y 

2 

+ λ 2 y = 0 (B) 

Solution [A] 

y = a sin (λx + α) 

dy = a cos (λx + α). λ 

dx 

= aλ cos (λx + α) 

d 

2 

dx 

y 

2 

– λ 2 y = 0 (C) 

2 

2 

d y 

+ λy = 0 (D) none of the above 

2 

dx 



23 / 33

MATHEMATICS 


CAREER POINT 

d 

2 

dx 

y 

2 

= aλ(–sin(λx + α)).λ 

= – λ 2 y sin (λx + α) 

= –λ 2 y 

d 

2 

dx 

y 

2 

+ λ 2 y = 0 

Q.87 

dy 

The differential equation y + x = a where 'a' is any constant represents - 

dx 

(A) A set of straight lines (B) A set of ellipses (C) A set of circles (D) None of the above 

Solution [C] 

dy 

y. + x = a dx 

∫ y . dy = ∫ 

( a – x) dx 

y 2 x 2 

= ax – + c 

2 2 

x 2 + 

2 

y 2 – ax – c = 0 

2 

x 2 + y 2 – 2ax – 2c = 0 

Which is a circle 

2 

Q.88 

⎛ dy ⎞ ⎛ dy ⎞ 

For the differential equation ⎜ ⎟⎠ –x ⎜ ⎟ + y = 0, which one of the following is not its solution ? 

⎝ dx ⎝ dx ⎠ 

(A) y = x – 1 (B) 4y = x 2 (C) y = x (D) y = – x –1 

Solution [C] 

2 

⎛ dy ⎞ ⎛ dy ⎞ 

⎜ ⎟⎠ – x. ⎜ ⎟ + y = 0 

⎝ dx ⎝ dx ⎠ 

dy 

if y = x – 1 then = 1 dx 

⎛ dy ⎞ 

2 ⎛ dy ⎞ 

⎜ ⎟ − x⎜ 

⎟ y 

⎝ dx ⎠ ⎝ dx ⎠ 

+ 

= (1) 2 – x (1) + (x – 1) 

= 1 – x + x – 1 

= 0 

⇒ y = x – 1 is a solution 

if 4y = x 2 

dy dy x 

4 ⋅ = 2x ⇒ = 

dx dx 2 

∴ 

⎛ dy ⎞ 

2 ⎛ dy ⎞ 

⎜ ⎟ − x⎜ 

⎟ y 

⎝ dx ⎠ ⎝ dx ⎠ 

+ 

⎛ x ⎞ 

= ⎜ ⎟ 

⎝ 2 ⎠ 

2 

2 

x x 

− x ⋅ + 

2 4 



24 / 33

MATHEMATICS 


CAREER POINT 

= 0 ⇒ 4y = x 2 is a solution of differential equation 

if y =x 

dy 

then = 1 

dx 

2 

⎛ 

∴ 

dy ⎞ ⎛ dy ⎞ 

⎜ ⎟ − x⎜ 

⎟ + y = 1 

2 − x ⋅1+ 

x 

⎝ dx ⎠ ⎝ dx ⎠ 

= 1 ≠ 0 

⇒ y = x is not its solution 

Hence (C) is correct. 

Q.89 What is the general solution of the differential equation x 2 dy + y 2 dx = 0 ? 

(A) x + y = c 

(B) xy = c 

(C) c(x + y) = xy 

(D) None of these 

Which c is the constant of integration 

Solution [C] 

x 2 dy + y 2 dx = 0 

x 2 dy = – y 2 dx 

dx dy 

+ = 0 

2 2 

x y 

on integrating, we get 

1 1 

− − = C 1 

where C 1 is constant of integration 

x y 

x + y 

= −C 1 

xy 

x + y = – C 1 (xy) 

1 

⇒ (x + y) = xy 

− C 

1 

⇒ C(x + y) = xy 

1 

where C = − 

Q.90 What is the general solution of the differential equation e x tanydx + (1 – e x ) sec 2 ydy = 0 ? 

(A) sin y = c(1 – e x ) (B) cos y = c(1– e x ) 

(C) cot y = c(1 – e x ) 


where c is the constant of integration 

Solution [D] 

e x tan y dx + (1 – e x ) sec 2 y dy = 0 

e x tan y dx = (e x – 1) sec 2 y dy 

x 

2 

e sec y 

dx = dy 

x 

e −1 

tan y 

Integrating both side 

∫ 

dx 

x = 

∫ 

ln 

e 

e 

x 

−1 

e x −1 

tan y 

2 

C 1 

sec y 

dy ⇒ ln (e x –1) = ln tan y + ln C 

tan y 

e x −1 

= ln C ⇒ = C 

tan y 

⇒ (1 – e x ) = C 1 tany 



25 / 33

MATHEMATICS 


CAREER POINT 

Q.91 EFGH is a rhombus such that the angle EFG is 60°. The magnitude of vectors FH and { EG} 

Solution [D] 

H 

where m is a scalar. What is the value of m ? 

(A) 3 (B) 1.5 (C) 2 (D) 3 

G 

m are equal 

E 

Let FH = λ & EG = G 

from ∆EFG 

2 2 2 

a + a − k 

cos60° 

= 

2 

2a 

2 2 

1 2a − k 

= 

2 

2 2a 

k 2 = a 2 

∴ k = a 

from ∆EFH 

2 2 2 

a + a − λ 

cos120° 

= 

2.a.a 

1 2 2 

2a − 

− = 

λ 

2 

2 2a 

–a 2 = 2a 2 –λ 2 

λ 2 = 3a 2 

∴ λ = 3a 

Q | FH | = m | EG | 

3 a = ma 

∴ m = 3 

F 

r r r r r 

Q.92 If a .b = 0 and a × b = 0 then which one of the following is correct ? 

(A) a r is parallel to b r 

(B) a r is perpendicular to b r 

r r r 

(C) a = 0 or b = 0 


Solution [C] 

a⋅b = 0 

a & b are perpendicular 

a × b = 0 

a & b are parallel 

r r 

it is possible only when a = 0 or b = 0 

r r r 

Q.93 The vectors a ( b × a) 

(A) a r only 

(C) both a r and b r 

Solution [C] 

× is coplanar with 

(B) b r only 

(D) Neither a r or b r 



26 / 33

MATHEMATICS 


r r r 

a × (b× 

a) 

r r r rr r 

⇒ (a.a)b − (ab) a 

r r 

⇒ λ b − µ a = coplanar with a r & b r 

CAREER POINT 

Q.94 Consider the following 

1. 4 î × 3î = 0 

2. 

4 î 4 = 

3î 3 

Which of the following above is/are correct ? 

(A) 1 only 

(B) 2 only 

(C) Both 1 and 2 (D) Neither 1 nor 2 

Solution [A] 

1. 4 î × 3î = 0 

2. 

4 î 4 = 

3î 3 

Q division of vector not applicable 

∴ only 1 st is correct 

Q.95 What is the value of λ for which ( λ î + ĵ − kˆ) × (3î − 2ĵ + 4kˆ) = (2î −11ĵ 

− 7kˆ ) ? 

(A) 2 (B) –2 (C) 1 (D) 7 

Solution [A] 

( λ î + ĵ − k) × (3î − 2ĵ + 4k) = (2î −11ĵ 

− 7k) 

î 

λ 

3 

ĵ 

1 

− 2 

from above 

4λ + 3 = 11 

∴ λ = 2 

k 

−1 

= 2î −11ĵ− 

7kˆ 

4 

Q.96 The magnitude of the scalar p for which the vector p ( − 3î − 2ĵ + 13kˆ ) is of unit length is 

(A) 1/8 (B) 1/64 (C) 182 (D) 

1 

182 

Solution [D] 

| p( − 3î − 2ĵ + 13kˆ) | = 1 

∴ p 9 + 4 + 169 = 1 

∴ 

p = 

1 

182 

Q.97 The vector 2ĵ 

− kˆ lies 

(A) in the plane of XY (B) in the plane of YZ (C) in the Plane of XZ (D) along the X-axis 

Solution [B] 

given vector is 2ĵ 

− kˆ 

hence it lies in yz plane only 



27 / 33

MATHEMATICS 


Q.98 ABCD is a parallelogram. If AB = a 

r , BC = b 

r 

, then what is BD equal to ? 

(A) a 

r r 

r r 

+ b 

(B) a − b 

(C) a 

r − b 

r 

r r 

− (D) − a + b 

Solution [D] 

D 

C 

CAREER POINT 

b r 

A a r 

BC = AD 

AB + BD = AD 

∴ BD = AD − AB 

r 

⇒ b a 

r − 

B 

Q.99 What is the geometric mean of the sequence 1, 2, 4, 8, .........., 2 n ? 

(A) 2 n/2 (B) 2 (n+1)/2 (C) 2 (n + 1) –1 (D) 2 (n–1) 

Solution [A] 

G = al 

n 

= 1× 2 

= 2 n/2 

Q.100 The mean of 10 observations is 5. If 2 added to each observation and then multiplied by3, then what will be 

the new mean ? 

(A) 5 (B) 7 (C) 15 (D) 21 

Solution [D] 

xi 

xi 

x = 5 = 

∑ = 

∑ 

n 10 

∑ x i = 10× 

5 

= 50 

if 2 is added to each number 

new sum = ∑ x i + n × 2 

= 50 + 10 × 2 

= 70 

70 

New mean x = = 7 

10 

Since, 3 is multiplied to each number mean is also multiplied by 3 

New mean = 7 × 3 = 21 

Q.101 What is the mean of first n odd natural numbers ? 

(A) n (B) (n + 1) / 2 

(C) n (n + 1) /2 (D) n + 1 

Solution [A] 

Sum of first n odd natural number 



28 / 33

MATHEMATICS 


CAREER POINT 

S = 1 + 3 + 5 + ............ + (2n – 1) 

n 

2 

S = (1 + 2n −1) 

= n 

2 

sum of observation n 2 

mean = = = n 

number of observation n 

Q.102 The arithmetic mean of numbers a, b, c, d, e is M. What is the value of 

(a – M) + (b – M) + (c – M) + (d – M) + (e – M) ? 

(A) M (B) a + b + c + d + e (C) 0 (D) 5M 

Solution [C] 

a + b + c + d + e 

Mean = 

= M 

5 

( a − M) + (b − M) + (c − M) + (d − M) + (e − m) 

New mean = 

5 

⎛ a + b + c + d + e ⎞ 

= ⎜ 

⎟ − M 

⎝ 5 ⎠ 

= M– M 

new mean = 0 

Q.103 The algebraic sum of the deviations of 20 observations measured from 30 is 2. What would be the mean of 

the observations ? 

(A) 30 (B) 32 (C) 30.2 (D) 30.1 

Solution [D] 

Standard deviation = 30 

algebraic sum with deviation =∑ x i h = 

∑ − h 2 

x i 

Mean = + h = + 30 

20 20 

= 30 + 0.1 

= 30.1 

− 2 

Q.104 The median of 27 observations of a variable is 18, three more observations are made and the values of these 

observations are 16, 18 and 50. What is the median of these 30 observations ? 

(A) 18 (B) 19 

(C) 25.5 

(D) Can not be determined due to insufficient data 

Solution [A] 

Median of 27 observations = 18 

th 

⎛ 27 + 1⎞ 

then , ⎜ ⎟ observation = 18 

⎝ 2 ⎠ 

14 th observation = 18 

Then, observation whose value is 16 come before 18, 

i.e. 14 th observation 

Now, number of observation is even = 30 observation 

th 

th 

⎛ 30 ⎞ ⎛ 30 ⎞ 

⎜ ⎟ + ⎜ + 1⎟ 

2 2 

median = 

⎝ ⎠ ⎝ ⎠ 

2 



29 / 33

MATHEMATICS 

th 

15 obs + 16 

= 

2 

18 + 18 

= = 18 

2 

th 

obs 


CAREER POINT 

Q.105 Frequency curve may be 

(A) symmetrical (B) positive skew (C) negative skew (D) all the above 

Solution [D] 

Frequency curve may be 

(1) positive skew 

(2) negative skew 

(3) Symmetrical 

Q.106 The monthly family expenditure (in percentage) on different items are as follows 

Food Rent Cloth transport Education Others 

38 19 18 - - - - - - 9 6 

If the total monthly expenditure is Rs 9000, What is the expenditure on transport ? 

(A) Rs 180 (B) Rs 1000 

(C) Rs. 900 (D) Rs. 360 

Solution [C] 

Food Rent Cloth transport Education Others total 

38 19 18 - - - - - - 9 6 100% 

Then, Transport = 100 – (38 + 19 + 18 + 9 + 6) 

= 100 – (90) 

= 10% 

Total expenditure = Rs. 9000 

10 

Expenditure on Transport = × 9000 

100 

= Rs .900 

Q.107 If the mean of few observations is 40 and standard deviation is 8, then what is the coefficient of variation ? 

(A) 1% (B) 10% (C) 20% (D) 30% 

Solution [C] 

x = 40 

σ = 8 

coefficient of variance = 

6 × 100 

x 

8 

⇒ × 100 = 20% 

40 

Q.108 What is the standard deviation of 7, 9, 11, 13, 15 ? 

(A) 2.4 (B) 2.5 (C) 2.7 (D) 2.8 

Solution [D] 

x x – x (x – x ) 2 

7 – 4 16 

9 –2 14 

11 0 0 

13 2 4 

15 4 16 



30 / 33

MATHEMATICS 

55 

x = = 11 

5 

2 

∑(x 

− x) 40 

σ = 

= ≈ 2. 8 

n 5 


CAREER POINT 

Q.109 Which one of the following is a measure of dispersion ? 

(A) Mean (B) Median (C) Mode (D) Standard deviation 

Solution [D] 

Measure of dispersion is standard deviation 

Q.110 Let X and Y be two related variables. The two regression lines are given by x – y + 1 = 0 and 2x – y + 4 = 0 . 

The two regression lines pass through the point 

(A) (–4, –3) (B) (–6, –5) (C) (3, –2) (D) (–3, –2) 

Solution [D] 

Lines passing through ( ( x, y) 

which is intersection point of two lines 

Q.111 If P(E) denotes the probability of an event E, then E is called certain event if 

(A) P(E) = 0 (B) P(E) = 1 (C) P(E) is either 0 or 1 (D) P(E) = 1/2 

Solution [B] 

Since E is certain event, 

P(E) = 1 

Q.112 What is the probability that a leap year selected at random will contain 53 Mondays ? 

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

Solution [A] 

1 leap year = 365 days 

364 = 52 × 7 

so 

366 – 367 = 2 days 

two possibilities are there for two monday out of seven possibilities 

2 

Hence, probability is = 7 

3 

1 2 

Q.113 If A and B are two events such that P (A ∪ B) = , P (A ∩ B) = , P (A) = where A is the complement 

4 

4 3 

of A, then what is P(B) equal to ? 

(A) 1/3 (B) 2/3 (C) 1/9 (D) 2/9 

Solution [B] 

2 

P (A) = 

3 

2 1 

then P (A) = 1− 

= 

3 3 

3 1 1 

= + P(B) − 

4 3 4 

3 1 1 

P(B) 

= + − 

4 4 3 

4 1 1 2 

= − = 1− 

= 

4 3 3 3 



31 / 33

MATHEMATICS 


CAREER POINT 

Q.114 Three coins are tossed simultaneously . What is the probability that they will fall two heads and one tail ? 

(A) 1/3 (B) 1/2 (C) 1/4 (D) 3/8 

Solution [D] 

Sample space = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH} 

3 

P(two head and one tail) = 8 

Q.115 Which one of the following is correct ? 

(A) An event having no sample point is called an elementary event 

(B) An event having one sample point is called an elementary event 

(C) An event having two sample points is called an elementary event 

(D) An event having many sample points is called an elementary event 

Solution [B] 

Q.116 What is the most probable number of successes in 10 trials with probability of success 2/3 ? 

(A) 10 (B) 7 (C) 5 (D) 4 

Solution [B] 

p(r) = n C r ⋅ p r ⋅ q n–r 

n = 10, p = 2/3, q = 1/3 

FOR THE NEXT TWO (02) QUESTIONS THAT FOLLOW 

An urn contains one black ball and one green ball. A second urn contains one white and one green ball. One 

ball is drawn at random from each urn. 

Q.117 What is the probability that both balls are of same colour ? 

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

Solution [C] 

Since, ball should be of same colour 

So, colour is green 

ways of selecting a green ball from 1 st urn = 1/2 

ways of selecting a green ball from II nd urn = ½ 

1 1 

Now, probability that both of should be of same colour = = 

2× 

2 4 

Q.118 What is the probability of getting at least one green ball ? 

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

Solution [D] 

P(at least one green ball) = 1 – P (no ball is green) 

= 1 – P (i.e. ball is white and black) 

= 

1 1 

1 − × 

2 2 

1 

= 1− 

4 

3 

= 4 



32 / 33

MATHEMATICS 


CAREER POINT 

FOR THE NEXT TWO (02) QUESTIONS THAT FOLLOW 

Two dice each numbered from 1 to 6 are thrown together. Let A and B be two events given by 

A : event number on the first die 

B : number on the second die is greater than 4 

Q.119 What is P(A ∪ B) equal to ? 

(A) 1/2 (B) 1/4 

(C) 2/3 (D) 1/6 

Solution [C] 

A : even numbers = {2, 4, 6} 

B : number > 4 = {5, 6} A ∩ B = {6} 

1 

P (A ∩ B) = 

6 

3 1 

P (A) = = 

6 2 

2 1 

P (B) = = 

6 3 

P(A∪B) = P(A) + P(B) –P(A∩B) 

1 1 1 

= + − 

2 3 6 

3 + 2 −1 

= = 

6 

4 = 

6 

2 

3 

Q.120 What is P(A∩B) equal to ? 

(A) 1/2 (B) 1/4 

(C) 2/3 (D) 1/6 

Solution [D] 

1 

P (A ∩ B) = 

6 

A ∩ B = {6} 



33 / 33

Maths Paper & Solutions - Career Point

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?