NDA II 2015 Mathematics

• Question 1
  Let X be the set of all the persons living in Delhi. The persons, a and b, in X are said to be related if the difference in their ages is at most 5 years. The relation is:

  Option A: an equivalence relation

  Option B: reflexive and transitive but not symmetric

  Option C: symmetric and transitive but not reflexive

  Option D: reflexive and symmetric but not transitive

  VIEW SOLUTION

• Question 2
  The matrix $A=\left[\begin{array}{ccc}1& 3& 2\\ 1& x-1& 1\\ 2& 7& x-3\end{array}\right]$ will have inverse for every real number x except for

  Option A: $x=\frac{11\pm \sqrt{5}}{2}$

  Option B: $x=\frac{9\pm \sqrt{5}}{2}$

  Option C: $x=\frac{11\pm \sqrt{3}}{2}$

  Option D: $x=\frac{9\pm \sqrt{3}}{2}$

  VIEW SOLUTION

• Question 3
  If the value of the determinant $\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive, where a ≠ b ≠ c, then the value of abc

  Option A: cannot be less than 1

  Option B: is greater than –8

  Option C: is less than –8

  Option D:  must be greater than 8

  VIEW SOLUTION

• Question 4
  Consider the following statements with respect to the determinant  
  $\left|\begin{array}{cc}{\cos }^2\frac{\alpha }{2}& {\sin }^2\frac{{\displaystyle \alpha }}{{\displaystyle 2}}\\ {\sin }^2\frac{{\displaystyle \beta }}{{\displaystyle 2}}& {\cos }^2\frac{{\displaystyle \beta }}{{\displaystyle 2}}\end{array}\right|$
  where α, β are complementary angles
  1. The value of the determinant is $\frac{1}{\sqrt{2}}\cos \left(\frac{\alpha -\beta }{2}\right)$ .
  2. The maximum value of the determinant is $\frac{1}{\sqrt{2}}$.
  Which of the above statements is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 5
  What is (1000000001)₂ – (0.0101)₂ equal to?

  Option A: (512.6775)₁₀

  Option B: (512.6875)₁₀

  Option C: (512.6975)₁₀

  Option D: (512.0909)₁₀

  VIEW SOLUTION

• Question 6
  If  $A=\left[\begin{array}{ccc}1& 0& -2\\ 2& -3& 4\end{array}\right],$ then the matrix X for which 2X + 3A = 0 holds true is

  Option A: $\left[\begin{array}{ccc}-\frac{3}{2}& 0& -3\\ -3& -\frac{9}{2}& -6\end{array}\right]$

  Option B: $\left[\begin{array}{ccc}\frac{3}{2}& 0& -3\\ 3& -\frac{9}{2}& -6\end{array}\right]$

  Option C: $\left[\begin{array}{ccc}\frac{3}{2}& 0& 3\\ 3& \frac{9}{2}& 6\end{array}\right]$

  Option D: $\left[\begin{array}{ccc}-\frac{3}{2}& 0& 3\\ -3& \frac{9}{2}& -6\end{array}\right]$

  VIEW SOLUTION

• Question 7
  If z₁ and z₂ are complex numbers with |z₁| = |z₂|, then which of the following is/are correct?
  1.  z₁ = z₂
  2. Real part of z₁ = Real part of z₂
  3. Imaginary part of z₁ = Imaginary part of z₂
  Select the correct answer out of the given options:

  Option A: 1 only

  Option B: 2 only

  Option C: 3 only

  Option D: None

  VIEW SOLUTION

• Question 8
  If $A=\left[\begin{array}{ccc}1& 1& -1\\ 2& -3& 4\\ 3& -2& 3\end{array}\right]and B=\left[\begin{array}{ccc}-1& -2& -1\\ 6& 12& 6\\ 5& 10& 5\end{array}\right],$ then which of the following is/are correct?
   1. A and B commute
   2. AB is a null matrix
  Select the correct answer out of the following options:

  Option A: 1 only

  Option B: 2 only

  Option C: 
  Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 9
  The number of real roots of the equation x² – 3 |x| + 2 = 0 is:  

  Option A: 4

  Option B: 3

  Option C: 2

  Option D: 1

  VIEW SOLUTION

• Question 10
  If the sum of the roots of the equation ax² + bx + c = 0 is equal to the sum of their squares, then

  Option A: a² + b² = c²

  Option B: a² + b² = a + b

  Option C: ab + b² = 2ac

  Option D: ab – b² = 2ac

  VIEW SOLUTION

• Question 11
  If $A=\left\{x\in IR:x^2+6x-7<0\right\} and B=\left\{x\in IR:x^2+9x+14>0\right\},$then which of the following is/are correct ?
  1. $\left(A\cap B\right)=\left(-2, 1\right)$
  2.$\left(A/B\right)=\left(-7, -2\right)$
  Select the correct answer using the code give below:

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 12
  A, B, C and D are four sets such that $A\cap B=C\cap D=\varphi .$ Consider the following:
  1. $A\cup C and B\cup D$ are always disjoint.
  2. $A\cap C and B\cap D$ and are always disjoint.
  Which of the above statements is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 13
  If A is an invertible matrix of order n and k is any positive real number, then the value of [det(kA)]^–1 det A is

  Option A: k^–n

  Option B: k^–1

  Option C: kⁿ

  Option D: nk

  VIEW SOLUTION

• Question 14
  The value of the infinite product $6^{\frac{1}{2}}\times 6^{\frac{1}{2}}\times 6^{\frac{3}{8}}\times 6^{\frac{1}{4}}\times ...$ is

  Option A: 6

  Option B: 36

  Option C: 216

  Option D: $\infty$

  VIEW SOLUTION

• Question 15
  If the roots of the equation x² – nx + m = 0 differ by 1, then

  Option A: n² – 4m – 1 = 0

  Option B: n² + 4m – 1 = 0

  Option C: m² + 4n + 1 = 0

  Option D: m² – 4n – 1 = 0

  VIEW SOLUTION

• Question 16
  If different words are formed with all the letters of the word ‘AGAIN’ and are arranged alphabetically among themselves as in a dictionary, the word at the 50^th place will be:

  Option A: NAAGI

  Option B: NAAIG

  Option C: IAAGN

  Option D: IAANG

  VIEW SOLUTION

• Question 17
  The number of ways in which a cricket team of 11 players can be chosen out of a batch of 15 players, so that the captain of the team is always included would be:

  Option A: 165

  Option B: 364

  Option C: 1001

  Option D: 1365

  VIEW SOLUTION

• Question 18
  In the expansion of , ${\left(\sqrt{x}+\frac{1}{3x^2}\right)}^{10}$ the value of the constant term (independent of x) is :

  Option A: 5

  Option B: 8

  Option C: 45

  Option D: 90

  VIEW SOLUTION

• Question 19
  The value of sin²5º + sin²10º + sin²15º + sin²20º + …… + sin²90º is:

  Option A: 7

  Option B: 8

  Option C: 9

  Option D: $\frac{19}{2}$

  VIEW SOLUTION

• Question 20
  On simplifying , $\frac{{\sin }^{3}A+\sin 3A}{\sin A}+\frac{{\displaystyle {\cos }^3}{\displaystyle A}{\displaystyle -}{\displaystyle \cos }{\displaystyle 3}{\displaystyle A}}{\cos {\displaystyle A}},$ we get: 

  Option A: sin3A

  Option B: cos3A

  Option C: sinA + cosA

  Option D: 3

  VIEW SOLUTION

• Question 21
  The value of $\tan \left(2{\tan }^{-1}\frac{1}{5}-\frac{\mathrm{\pi }}{4}\right)$ is :

  Option A: $-\frac{7}{17}$

  Option B: $\frac{5}{16}$

  Option C: $\frac{5}{4}$

  Option D: $\frac{7}{17}$

  VIEW SOLUTION

• Question 22
  Two poles are 10 m and 20 m high. The line joining their tops makes an angle of 15º with the horizontal. The distance between the poles is approximately equal to:

  Option A: 36.3 m

  Option B: 37.3 m

  Option C: 38.3 m

  Option D: 39.3 m

  VIEW SOLUTION

• Question 23
  If $g\left(x\right)=\frac{1}{f\left(x\right)}and f\left(x\right)=x, x\ne 0,$ , then which one of the following is correct?

  Option A: f(f(f(g(g(f(x)))))) = g(g(f(g(f(x)))))

  Option B: f(f(g(g(g(f(x)))))) = g (g (f(g(f(x)))))

  Option C: f(g(f(g(g(f(g(x)))))) = g(g (f (g(f(x)))))

  Option D:  f(f(f(g(g(f(x)))))) = f(f(f(g(f(x)))))

  VIEW SOLUTION

• Question 24
  Consider the following:
  $$\begin{gathered} 1. {\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{3}{5}=\frac{\mathrm{\pi }}{2} \\ 2. {\tan }^{-1}\sqrt{3}+{\tan }^{-1}1=-{\tan }^{-1}\left(2+\sqrt{3}\right) \end{gathered}$$
  Which of the above statements is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 25
  If A is an orthogonal matrix of order 3 and $B=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right],$ then which of the following is/are correct? 
  1. $\left|AB\right|=\pm 47$
  2. AB = BA
  Select the correct answer using the code given below:

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 26
  If a, b, c are the sides of  triangle ABC, then $a^{\frac{1}{p}}+b^{\frac{1}{p}}-c^{\frac{1}{p}}$ where p > 1, is:    

  Option A: always negative

  Option B: always positive

  Option C: always zero

  Option D: positive if 1 < p < 2 and negative if p > 2

  VIEW SOLUTION

• Question 27
  If a, b, c are real numbers, then the value of the determinant $\left|\begin{array}{ccc}1-a& a-b-c& b+c\\ 1-b& b-c-a& c+a\\ 1-c& c-a-b& a+b\end{array}\right|$ is

  Option B: (a – b) (b – c) (c – a)

  Option C: (a + b + c)²

  Option D: (a + b + c)³

  VIEW SOLUTION

• Question 28
  If the point z₁ = 1 + i where $i=\sqrt{-1}$  is the reflection of a point z₂ = x + iy in the line $i\overline{z}-iz=5,$, then the point z₂would be:       

  Option A: 1 + 4i

  Option B: 4 + i

  Option C: 1 – i

  Option D: – 1 –i

  VIEW SOLUTION

• Question 29
  If sinx + siny = a and cosx + cosy = b, then ${\tan }^2\left(\frac{x+y}{2}\right)+{\tan }^2\left(\frac{x-y}{2}\right)$ is equal to:

  Option A: $\frac{a^4+b^4+4b^2}{a^2{b}^2+b^4}$

  Option B: $\frac{a^4-b^4+4b^2}{a^2{b}^2+b^4}$

  Option C: $\frac{a^4-b^4+4a^2}{a^2{b}^2+a^4}$

  Option D: None of these

  VIEW SOLUTION

• Question 30
  A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3 m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are 30º and 45º respectively. Which one of the following gives the best approximation to the height of the tower?

  Option A: 3.90 m

  Option B: 4.00 m

  Option C: 4.10 m

  Option D: 4.25 m

  VIEW SOLUTION

• Question 31
  Consider the expansion of (1 + x)²ⁿ⁺¹
  If the coefficients of x^r and x^r+1 are equal in the expansion, then r is equal to:    

  Option A: n

  Option B: $\frac{2n – 1}{2}$

  Option C: $\frac{2n+1}{2}$

  Option D: n + 1

  VIEW SOLUTION

• Question 32
  Consider the expansion of (1 + x)²ⁿ⁺¹
  The average of the coefficient of the two middle terms in the expansion is:

  Option A: ${}^{2n+1}C_{n+2}$

  Option B: ${}^{2n+1}C_n$

  Option C: ${}^{2n+1}C_{n-1}$

  Option D: ${}^{2n}C_{n+1}$

  VIEW SOLUTION

• Question 33
  Consider the expansion of (1 + x)²ⁿ⁺¹
  The sum of the coefficients of all the terms in the expansion is:

  Option A: 2^2n–1

  Option B: 4^n–1

  Option C: 2 × 4ⁿ

  Option D: None of the above

  VIEW SOLUTION

• Question 34
  The n^th term of an A.P is $\frac{3+n}{4},$ then the sum of first 105 terms is:

  Option A: 270

  Option B: 735

  Option C: 1409

  Option D: 1470

  VIEW SOLUTION

• Question 35
  A polygon has 44 diagonals. The number of its sides is:

  Option A: 11

  Option B: 10

  Option C: 8

  Option D: 7

  VIEW SOLUTION

• Question 36
  If p, q, r are in one geometric progression and a, b, c are in another geometric progression, then ap, bq, cr are in:

  Option A: Arithmetic progression

  Option B: Geometric progression

  Option C: Harmonic progression

  Option D: None of the above

  VIEW SOLUTION

• Question 37
  Consider  triangle ABC satisfying
   $2a {\sin }^2\left(\frac{C}{2}\right)+2c {\sin }^2\left(\frac{A}{2}\right)=2a+2c-3b$
  The sides of the triangle are in
   

  Option A: G.P.

  Option B: A.P.

  Option C: H.P.

  Option D: Neither in G.P. nor in A.P. nor in H.P.

  VIEW SOLUTION

• Question 38
  Consider  triangle ABC satisfying
   $2a {\sin }^2\left(\frac{C}{2}\right)+2c {\sin }^2\left(\frac{A}{2}\right)=2a+2c-3b$    
  sin A, sin B, sin C are in

  Option A: G.P.

  Option B: A.P.

  Option C: H.P.

  Option D: Neither in G.P. nor in A.P. nor in H.P.

  VIEW SOLUTION

• Question 39
  If  $p=\tan \left(-\frac{11\mathrm{\pi }}{6}\right),q=\tan \left(\frac{21\mathrm{\pi }}{4}\right) and r= cot\left(\frac{283\mathrm{\pi }}{6}\right),$ then which of the following is/are correct?
  1. The value of p × r is 2.
  2. p, q and r are in G.P.
  Select the correct answer using the code given below:

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 40
  The number of ways in which 3 holidays tickets can be given to 20 employees of an organisation if each employee is eligible for any one or more of the tickets, is

  Option A: 1140

  Option B: 3420

  Option C: 6840

  Option D: 8000

  VIEW SOLUTION

• Question 41
  What is the sum of n terms of the series $\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+...?$

  Option A: $\frac{n\left(n-1\right)}{\sqrt{2}}$

  Option B: $\sqrt{2}n\left(n+1\right)$

  Option C: $\frac{n\left(n+1\right)}{\sqrt{2}}$

  Option D: $\frac{n\left(n-1\right)}{2}$

  VIEW SOLUTION

• Question 42
  The coefficient of x⁹⁹ in the expansion of (x – 1)(x – 2)(x – 3) ……. (x – 100) is

  Option A: 5050

  Option B: 5000

  Option C: – 5050

  Option D: –5000

  VIEW SOLUTION

• Question 43
  $z\overline{z}+\left(3-i\right)z+\left(3+i\right)\overline{z}+1=0$ represents a circle with

  Option A: Centre (–3, –1) and radius 3

  Option B: Centre (–3, 1) and radius 3

  Option C: Centre (–3, –1) and radius 4

  Option D: Centre (–3, 1) and radius 4

  VIEW SOLUTION

• Question 44
  The number of 3-digit even number that can be formed from the digits 0, 1, 2, 3, 4 and 5, repetition of digits being not allowed, is

  Option A: 60

  Option B: 56

  Option C: 52

  Option D: 48

  VIEW SOLUTION

• Question 45
  If${\log }_{8}m+{\log }_8\frac{1}{6}=\frac{2}{3},$ then m is equal to

  Option A: 24

  Option B: 18

  Option C: 12

  Option D: 4

  VIEW SOLUTION

• Question 46
  The area of the figure formed by the lines $ax+by+c=0, ax-by+c=0, ax+by-c=0 and ax-by-c=0$ is  

  Option A: $\frac{c^2}{ab}$

  Option B: $\frac{2c^2}{ab}$

  Option C: $\frac{c^2}{2ab}$

  Option D: $\frac{c^2}{4ab}$

  VIEW SOLUTION

• Question 47
  If a line is perpendicular to the line 5x – y = 0 and forms a triangle of area 5 square units with co-ordinate axes, then its equation is      

  Option A: $x+5y\pm 5\sqrt{2}=0$

  Option B: $x-5y\pm 5\sqrt{2}=0$

  Option C: $5x+y\pm 5\sqrt{2}=0$

  Option D: $5x-y\pm 5\sqrt{2}=0$

  VIEW SOLUTION

• Question 48
  Consider any point P on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ in the first quadrant. Let r and s represents its distance from (4, 0) and (–4, 0) respectively, then (r + s) is equal to

  Option A: 10 unit

  Option B: 9 unit

  Option C: 8 unit

  Option D: 6 unit

  VIEW SOLUTION

• Question 49
  A straight line x = y + 2 touches the circle $4\left(x^2+y^2\right)=r^2.$ The value of r is

  Option A: $\sqrt{2}$

  Option B: $2\sqrt{2}$

  Option C: 2

  Option D: 1

  VIEW SOLUTION

• Question 50
  The three lines 4x + 4y = 1, 8x – 3y =2, y = 0 are

  Option A: the sides of an isosceles triangle

  Option B: concurrent

  Option C: mutually perpendicular

  Option D: the sides of an equilateral triangle

  VIEW SOLUTION

• Question 51
  The line $3x+4y-24=0$ intersects the x-axis at A and y-axis at B. Then the circumcentre of triangle OAB where O is the origin is

  Option A: (2, 3)

  Option B: (3, 3)

  Option C: (4, 3)

  Option D: None of these

  VIEW SOLUTION

• Question 52
  The eccentricity of the hyperbola 16x² – 9y² = 1 is

  Option A: $\frac{3}{5}$

  Option B: $\frac{5}{3}$

  Option C: $\frac{4}{5}$

  Option D: $\frac{5}{4}$

  VIEW SOLUTION

• Question 53
  The product of the perpendiculars from two points (± 4, 0) to the line $3x\cos \varphi +5y\sin \varphi =15$ is

  Option A: 25

  Option B: 16

  Option C: 9

  Option D: 8

  VIEW SOLUTION

• Question 54
  If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively are

  Option A: 3 units and 4 units

  Option B: 6 units and 4 units

  Option C: 3 units and 8 units

  Option D: 6 units and 8 units

  VIEW SOLUTION

• Question 55
  The lines 2x = 3y = – z and 6x = – y = –4z

  Option A: are perpendicular

  Option B: are parallel

  Option C: intersect at an angle 45º

  Option D: intersect at an angle 60º

  VIEW SOLUTION

• Question 56
  Two straight lines passing through the point A(3, 2) cut the line 2y = x + 3 and x-axis perpendicularly at P and Q, respectively. The equation of the line PQ is

  Option A: 7x + y – 21 = 0

  Option B: x + 7y + 21 = 0

  Option C: 2x + y – 8 = 0

  Option D: x + 2y + 8 = 0

  VIEW SOLUTION

• Question 57
  The radius of the sphere 3x² + 3y² + 3z² – 8x + 4y + 8z – 15 = 0 is

  Option A: 2

  Option B: 3

  Option C: 4

  Option D: 5

  VIEW SOLUTION

• Question 58
  The direction ratios of the line perpendicular to the lines with direction ratios <1, –2, –2 > and < 0, 2, 1 > are

  Option A: < 2, –1, 2 >

  Option B: < –2, 1, 2 >

  Option C: < 2, 1, –2 >

  Option D: < –2, –1, –2 >

  VIEW SOLUTION

• Question 59
  What are the co-ordinates of the foot of the perpendicular drawn from the point (3, 5, 4) on the plane z = 0?

  Option A: (0, 5, 4)

  Option B: (3, 5, 0)

  Option C: (3, 0, 4)

  Option D: (0, 0, 4)

  VIEW SOLUTION

• Question 60
  The length of the intercepts on the coordinate axes made by the plane 5x + 2y + z – 13 = 0 are

  Option A: 5, 2, 1 unit

  Option B: $\frac{13}{5},\frac{13}{2},13 \mathrm{unit}$

  Option C: $\frac{5}{13},\frac{2}{13},\frac{1}{13}\mathrm{unit}$

  Option D: 1, 2, 5 unit

  VIEW SOLUTION

• Question 61
  The area of the square, one of whose diagonals is $3\hat{i}+4\hat{j}$is

  Option A: 12 square unit

  Option B: 12.5 square unit

  Option C: 25 square unit

  Option D: 156.25 square unit

  VIEW SOLUTION

• Question 62
  ABCD is a parallelogram and P is the point of intersection of the diagonals. If O is the origin, then $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$  is equal to

  Option A: $4\overrightarrow{OP}$

  Option B: $2\overrightarrow{OP}$

  Option C: $\overrightarrow{OP}$

  Option D: Null vector

  VIEW SOLUTION

• Question 63
  If  $\overrightarrow{b}$and $\overrightarrow{c}$ are the position vectors of the point B are C, respectively, then the position vector of the point D such that $\overrightarrow{BD}=4\overrightarrow{BC}$ is

  Option A: $4\left(\overrightarrow{c}-\overrightarrow{b}\right)$

  Option B: $–4\left(\overrightarrow{c}-\overrightarrow{b}\right)$

  Option C: $4\overrightarrow{c}-3\overrightarrow{b}$

  Option D: $4\overrightarrow{c}+3\overrightarrow{b}$

  VIEW SOLUTION

• Question 64
  If the position vector $\overrightarrow{a}$ of the point (5, n) is such that $\left|\overrightarrow{a}\right|=13$, then the value/values of n can be

  Option A: ± 8

  Option B: ± 12

  Option C: 8 only

  Option D: 12 only

  VIEW SOLUTION

• Question 65
  If $\left|\overrightarrow{a}\right|=2 \mathrm{and}\left|\overrightarrow{b}\right|=3, \mathrm{then} {\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2+ {\left|\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}\right|}^2$ is equal to

  Option A: 72

  Option B: 64

  Option C: 48

  Option D: 36

  VIEW SOLUTION

• Question 66
  Consider the following inequalities in respect of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$:
  $$\begin{gathered} 1. \left|\overrightarrow{a}+\overrightarrow{b}\right|⩽\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right| \\ 2. \left|\overrightarrow{a}–\overrightarrow{b}\right|⩾\left|\overrightarrow{a}\right|-\left|\overrightarrow{b}\right| \end{gathered}$$    
  Which of the above is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 67
  If the magnitude of difference of two unit vectors is $\sqrt{3}$, then the magnitude of sum of the two vectors is

  Option A: $\frac{1}{2}\mathrm{unit}$

  Option B:   1 unit

  Option C: 2 unit

  Option D: 3 unit

  VIEW SOLUTION

• Question 68
  If the vectors $a\hat{i}+a\hat{j}+\gamma \hat{k}, \hat{i}+\widehat{k }\mathrm{and} \mathrm{\gamma }\hat{\mathit{i}}+ \mathrm{\gamma }\hat{\mathit{j}}+\mathrm{\beta }\hat{k}$ lie on a plane, where $\alpha , \beta \mathrm{and} \gamma$ are distinct non-negative numbers, then $\gamma$ is

  Option A: Arithmetic mean of $\alpha$ and $\beta$

  Option B: Geometric mean of $\alpha$ and $\beta$

  Option C: Harmonic mean of α and $\beta$

  Option D: None of the above

  VIEW SOLUTION

• Question 69
  The vectors $\overrightarrow{a}, \overrightarrow{b}, \stackrel{\rightharpoonup }{ c} \mathrm{and} \overrightarrow{d}$ are such that $\overrightarrow{a}\times \stackrel{\rightharpoonup }{b}=\overrightarrow{c}\times \overrightarrow{d} \mathrm{and} \overrightarrow{\mathit{a}}\mathit{\times }\stackrel{\mathit{\rightharpoonup }}{\mathit{c}}\mathit{=}\overrightarrow{\mathit{b}}\mathit{\times }\overrightarrow{\mathit{d}}\mathit{.}$Which of the above is/are correct?
  $$\begin{gathered} 1. \left(\stackrel{\rightharpoonup }{a}-\overrightarrow{d}\right)\times \left(\overrightarrow{b}-\overrightarrow{c}\right)=\overrightarrow{0} \\ 2. \left(\stackrel{\rightharpoonup }{a}\times \overrightarrow{d}\right)\times \left(\overrightarrow{c}\times \overrightarrow{d}\right)=\overrightarrow{0} \end{gathered}$$
  Select the correct answer using the code give below:

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 70
  The value of  ${\int }_a^b\frac{x^7+\sin x}{\cos }dx$ where a + b = 0 is

  Option A: 2b – asin (b – a)

  Option B: a + 3bcos (b – a)

  Option C: sin a – (b – a) cos b

  VIEW SOLUTION

• Question 71
  If , $f\left(x\right)=\sqrt{25-x^2},$then what is  $\underset{x\to 1}{\lim }\frac{f\left(x\right)-f\left(1\right)}{x-1}$ equal to ?

  Option A: $\frac{1}{5}$

  Option B: $\frac{1}{24}$

  Option C: $\sqrt{24}$

  Option D: $-\frac{{\displaystyle 1}}{{\displaystyle 24}}$

  VIEW SOLUTION

• Question 72
  Consider the function
  $f\left(x\right)=\left\{\begin{array}{ll}ax-2& \mathrm{for} -2<x<-1\\ -1& \mathrm{for} -1\le x\le 1\\ a+2{\left(x-1\right)}^2& \mathrm{for} 1 < x\mathit{ }< 2\end{array}\right.$      
  What is the value of a for which f(x) is continuous at x = –1 and x = 1?

  Option A: –1

  Option B: 1

  Option D: 2

  VIEW SOLUTION

• Question 73
  The function $f\left(x\right)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}$ is not defined at x = π. The value of f(π) so that f(x) is continuous at x = π is

  Option A: $-\frac{1}{2}$

  Option B: $\frac{1}{2}$

  Option C: –1

  Option D: 1

  VIEW SOLUTION

• Question 74
  Consider the following functions:
  $$\begin{gathered} 1. f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{x}& \mathrm{if} x\ne 0\\ 0& \mathrm{if} x=0\end{array}\right. \\ 2. f\left(x\right)=\left\{\begin{array}{ll}2x+5& \mathrm{if} x>0\\ x^2+2x+5& \mathrm{if} x\le 0\end{array}\right. \end{gathered}$$        
  Which of the above functions is/are derivable at x = 0?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 75
  The domain of the function $f\left(x\right)=\frac{1}{\sqrt{\left|x\right|-x}}$is

  Option A: [0, ∞)

  Option B: (–∞, 0)

  Option C: [1, ∞)

  Option D: (–∞, 0]

  VIEW SOLUTION

• Question 76
  Consider the following statements:
  1. The function f(x) = x² + 2cosx is increasing in the interval (0, π)
  2. The function $f\left(x\right)=\mathrm{In} \left(\sqrt{1+x^2}-x\right)$  is decreasing in the interval (–∞, ∞)
  Which of the above statements is/are correct?

  Option A: Only 1

  Option B: Only2

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 77
  The derivative of ln(x + sinx) with respect to (x + cosx) is

  Option A: $\frac{1+\cos x}{\left(x+\sin x\right)\left(1-\sin x\right)}$

  Option B: $\frac{{\displaystyle 1-\cos x}}{{\displaystyle \left(x+\sin x\right)\left(1+\sin x\right)}}$

  Option C: $\frac{{\displaystyle 1-\cos x}}{{\displaystyle \left(x-\sin x\right)\left(1+\cos x\right)}}$

  Option D: $\frac{{\displaystyle 1+\cos x}}{{\displaystyle \left(x-\sin x\right)\left(1-\cos x\right)}}$

  VIEW SOLUTION

• Question 78
  If $y=co{t}^{-1}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right],$  where  $0<x<\frac{\pi }{2},$ then $\frac{dy}{dx}$ is equal to

  Option A: $\frac{1}{2}$

  Option B: 2

  Option C: sin x  + cos x

  Option D: sin x – cos x

  VIEW SOLUTION

• Question 79
  The function $f\left(x\right)=\frac{x^2}{e^x}$ is monotonically increasing it

  Option A: x < 0 only

  Option B: x > 2 only

  Option C: 0 < x < 2

  Option D: $x\in \left(-\infty ,0\right)\cup \left(2,\infty \right)$

  VIEW SOLUTION

• Question 80
  If x^ay^b = (x – y) ^a+b, then the value of $\frac{dy}{dx}-\frac{y}{x}$ is equal to

  Option A: $\frac{a}{b}$

  Option B: $\frac{b}{a}$

  Option C: 1

  VIEW SOLUTION

• Question 81
  If $f: IR\to IR, g: IR \to IR$ be two functions given by
  f(x) = 2x – 3 and g(x) = x³ + 5, then (fog)^–1(x) is equal to

  Option A: ${\left(\frac{x+7}{2}\right)}^{\frac{1}{3}}$

  Option B: ${\left(\frac{x-7}{2}\right)}^{\frac{1}{3}}$

  Option C: ${\left(x-\frac{7}{2}\right)}^{\frac{1}{3}}$

  Option D: ${\left(x+\frac{7}{2}\right)}^{\frac{1}{3}}$

  VIEW SOLUTION

• Question 82
  If 0 < a < b, then  is equal${\int }_a^b\frac{\left|x\right|}{x}dx$ is equal to

  Option A: $\left|b\right|-\left|a\right|$

  Option B: $\left|a\right|-\left|b\right|$

  Option C: $\left|\frac{b}{a}\right|$

  VIEW SOLUTION

• Question 83
   ${\int }_0^{2\pi }{\sin }^5\left(\frac{x}{4}\right)dx$ is equal to

  Option A: $\frac{8}{15}$

  Option B: $\frac{16}{15}$

  Option C: $\frac{32}{15}$

  VIEW SOLUTION

• Question 84
  If $f\left(x\right)=\frac{\sin \left(e^{x-2}-1\right)}{\mathrm{In}\left(x-1\right)},$  then  $\underset{x\to 2}{\lim }f\left(x\right)$ is equal to

  Option A: –2

  Option B: –1

  Option D: 1

  VIEW SOLUTION

• Question 85
  Consider the following statements:
  1. f(x) = ln x is an increasing function on (0, ∞).
  2. $f\left(x\right)=e^x-x(\mathrm{In} x)$ is an increasing function on (1, ∞).
  Which of the above statements is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 86
  If $s=\sqrt{t^2+1},$ then $\frac{d^{2}s}{d{t}^2}$ is equal to

  Option A: $\frac{1}{s}$

  Option B: $\frac{1}{s^2}$

  Option C: $\frac{1}{s^3}$

  Option D: $\frac{1}{s^4}$

  VIEW SOLUTION

• Question 87
  Consider the following statements:
  Statement 1: The function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ such that f(x) = x³ for all $x\in \mathrm{ℝ}$ is one-one.
  Statement 2: $f\left(a\right)=f\left(b\right)\Rightarrow a=b$ for all $a, b \in \mathrm{ℝ}$ if the function f is one-one.
  Which one of the following is correct in respect of the above statements?

  Option A: Both the statements are true and Statement 2 is the correct explanation of Statement 1.

  Option B: Both the statements are true and Statement 2 is not the correct explanation of Statement 1.

  Option C: Statement 1 is true but Statement 2 is false.

  Option D: Statement 1 is false but Statement 2 is true.

  VIEW SOLUTION

• Question 88
  $\int \frac{dx}{1+e^{-x}}$ is equal to where c is the constant of integration

  Option A: 1 + e^x + c

  Option B: ln (1 + e^–x) + c

  Option C: ln (1 + e^x) + c

  Option D: 2 ln (1 + e^–x) + c

  VIEW SOLUTION

• Question 89
  ${\int }_{-1}^{1}x\left|x\right|dx$ is equal to

  Option B: $\frac{2}{3}$

  Option C: 2

  Option D: –2

  VIEW SOLUTION

• Question 90
  The area bounded by the coordinate axes and the curve $\sqrt{x}+\sqrt{y}=1,$ is

  Option A: 1 square unit

  Option B: $\frac{1}{2}$ square unit

  Option C: $\frac{1}{3}$ square unit

  Option D: $\frac{1}{6}$ square unit

  VIEW SOLUTION

• Question 91
  Consider the function $f\left(x\right)={\left(\frac{1}{x}\right)}^{2x^2},$ where x > 0
  At what value of x, does the function attain maximum value?

  Option A: e

  Option B: $\sqrt{e}$

  Option C: $\frac{1}{\sqrt{e}}$

  Option D: $\frac{1}{e}$

  VIEW SOLUTION

• Question 92
  Consider the function $f\left(x\right)={\left(\frac{1}{x}\right)}^{2x^2},$ where x > 0
  The maximum value of the function is

  Option A: e

  Option B: $e^{\frac{2}{e}}$

  Option C: $e^{\frac{1}{e}}$

  Option D: $\frac{1}{e}$

  VIEW SOLUTION

• Question 93
  Consider $f\text{'}\left(x\right)=\frac{x^2}{2}-kx+1$ such that f(0) = 0 and f(3) = 15
  The value of k is

  Option A: $\frac{5}{3}$

  Option B: $\frac{3}{5}$

  Option C: $-\frac{5}{3}$

  Option D: $-\frac{3}{5}$

  VIEW SOLUTION

• Question 94
  Consider $f\text{'}\left(x\right)=\frac{x^2}{2}-kx+1$ such that f(0) = 0 and f(3) = 15
  $f"\left(-\frac{2}{3}\right)$ is equal to

  Option A: –1

  Option B: $\frac{1}{3}$

  Option C: $\frac{1}{2}$

  Option D: 1

  VIEW SOLUTION

• Question 95
  Consider the function f(x) = 2x³ – 9x² – 12x + 1
  The function f(x) is an increasing function in the interval

  Option A: (–2, –1)

  Option B: (–∞, –2)

  Option C: (–1, 2)

  Option D: (–1, ∞)

  VIEW SOLUTION

• Question 96
  Consider the function f(x) = 2x³ – 9x² – 12x + 1
  The function f(x) is a decreasing function in the interval

  Option A: (–2, –1)

  Option B: (–∞, –2) only

  Option C: (–1, ∞) only

  Option D: $\left(-\infty ,-2\right)\cup \left(-1, \infty \right)$

  VIEW SOLUTION

• Question 97
  Consider the integrals
  $A={\int }_0^{\pi }\frac{\sin xdx}{\sin x+\cos x}\mathrm{and} B\mathit{=}{\mathit{\int }}_{\mathit{0}}^{\mathit{\pi }}\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{d}\mathit{x}}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{-}\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$
  
  Which one of the following is correct?

  Option A: A = 2B

  Option B: B = 2A

  Option C: A = B

  Option D: A = 3B

  VIEW SOLUTION

• Question 98
  $A={\int }_0^{\mathrm{\pi }}\frac{\sin xdx}{\sin x+\cos x}\mathrm{and} B\mathit{=}{\int }_0^{\mathrm{\pi }}\frac{\sin xdx}{\sin x\mathit{-}\cos x}$
  
  What is the value of B?

  Option A: $\frac{\pi }{4}$

  Option B: $\frac{{\displaystyle \pi }}{{\displaystyle 2}}$

  Option C: $\frac{{\displaystyle 3\pi }}{{\displaystyle 4}}$

  Option D: $\pi$

  VIEW SOLUTION

• Question 99
  Consider the function
    $f\left(x\right)=\left\{\begin{array}{ll}-2\sin x& \mathrm{if} x\mathit{⩽}\mathit{-}\frac{\pi }{\mathit{2}}\\ A\sin x+B& \mathrm{if} -\frac{\pi }{2}<x<\frac{\pi }{2}\\ \cos x& \mathrm{if} x\mathit{⩾}\frac{\pi }{\mathit{2}}\end{array}\right.$             
  Which is continuous everywhere.
  
  The value of A is

  Option A: 1

  Option C: –1

  Option D: –2

  VIEW SOLUTION

• Question 100
  $f\left(x\right)=\left\{\begin{array}{ll}-2\sin x& \mathrm{if} x⩽-\frac{\mathrm{\pi }}{2}\\ A\sin x+B& \mathrm{if} -\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}\\ \cos x& \mathrm{if} x⩾\frac{\mathrm{\pi }}{2}\end{array}\right.$
  Which is continuous everywhere.
  
  The value of B is

  Option A: 1

  Option C: –1

  Option D: –2

  VIEW SOLUTION

• Question 101
  The degree of the differential equation $\frac{dy}{dx}-x={\left(y-x\frac{dy}{dx}\right)}^{-4}$  is

  Option A: 2

  Option B: 3

  Option C: 4

  Option D: 5

  VIEW SOLUTION

• Question 102
  The solution of $\frac{dy}{dx}=\sqrt{1-x^2-y^2+x^2{y}^2}$ is where c is an arbitrary constant

  Option A: ${\sin }^{-1}y={\sin }^{-1}x+c$

  Option B: $2{\sin }^{-1}y=\sqrt{1-x^2}+{\sin }^{-1}x+c$

  Option C: $2{\sin }^{-1}y=x\sqrt{1-x^2}+{\sin }^{-1}x+c$

  Option D: $2{\sin }^{-1}y=x\sqrt{1-x^2}+{\cos }^{-1}x+c$

  VIEW SOLUTION

• Question 103
  The differential equation of the family of circles passing through the origin and having centres on the x-axis is

  Option A: $2xy\frac{dy}{dx}=x^2-y^2$

  Option B: $2xy\frac{dy}{dx}=y^2-x^2$

  Option C: $2xy\frac{dy}{dx}=x^2+y^2$

  Option D: $2xy\frac{dy}{dx}+x^2+y^2=0$

  VIEW SOLUTION

• Question 104
  The order and degree of the differential equation of parabolas having vertex at the origin and focus at (a, 0) where a > 0, are respectively    

  Option A: 1, 1

  Option B: 2, 1

  Option C: 1, 2

  Option D: 2, 2

  VIEW SOLUTION

• Question 105
  $f\left(xy\right)=f\left(x\right)+f\left(y\right)$ is true for all

  Option A: Polynomial functions f

  Option B: Trigonometric functions f

  Option C: Exponential function f

  Option D: Logarithmic functions f

  VIEW SOLUTION

• Question 106
  Three digits are chosen at random from 1, 2, 3, 4, 5, 6, 7, 8 and 9 without repeating any digit. What is the probability that the product is odd?

  Option A: $\frac{2}{3}$

  Option B: $\frac{7}{48}$

  Option C: $\frac{5}{42}$

  Option D: $\frac{5}{108}$

  VIEW SOLUTION

• Question 107
  Two events A and B are such that P(not B) = 0.8, P(A∪B) = 0.5 and P(A|B) = 0.4. Then P(A) is equal to

  Option A: 0.28

  Option B: 0.32

  Option C: 0.38

  Option D: None of these

  VIEW SOLUTION

• Question 108
  If the mean and variance of a Binomial variate X are 2 and 1, respectively, then the probability that X takes a value greater than 1 is

  Option A: $\frac{2}{3}$

  Option B: $\frac{4}{5}$

  Option C: $\frac{7}{8}$

  Option D: $\frac{11}{16}$

  VIEW SOLUTION

• Question 109
  Seven unbiased coins are tossed 128 times. In how many throws would you find at least three heads?

  Option A: 99

  Option B: 102

  Option C: 103

  Option D: 104

  VIEW SOLUTION

• Question 110
  A coin is tossed five times. What is the probability that heads are observed more than three times?

  Option A: $\frac{3}{16}$

  Option B: $\frac{5}{16}$

  Option C: $\frac{1}{2}$

  Option D: $\frac{3}{32}$

  VIEW SOLUTION

• Question 111
  The geometric mean of the observations x₁, x₂, x₃,…….,x_n is G₁. The geometric mean of the observations y₁, y₂, y₃,……y_n is G₂. The geometric mean of the observations $\frac{x_1}{y_1},\frac{{\displaystyle x_2}}{{\displaystyle y_2}},\frac{{\displaystyle x_3}}{{\displaystyle y_3}},...\frac{{\displaystyle x_n}}{{\displaystyle y_n}}$ is

  Option A: G₁G₂

  Option B: ln (G₁G₂)

  Option C: $\frac{G_1}{G_2}$

  Option D: $\mathrm{In}\left(\frac{{\displaystyle G_1}}{{\displaystyle G_2}}\right)$

  VIEW SOLUTION

• Question 112
  The arithmetic mean of 1, 8, 27, 64, … Up to n terms is given by

  Option A: $\frac{n(n+1)}{2}$

  Option B: $\frac{{\displaystyle n(n+1{)}^2}}{{\displaystyle 2}}$

  Option C: $\frac{{\displaystyle n(n+1{)}^2}}{{\displaystyle 4}}$

  Option D: $\frac{{\displaystyle n^2(n+1{)}^2}}{{\displaystyle 4}}$

  VIEW SOLUTION

• Question 113
  An unbiased coin is tossed until the first head appears or until four tosses are completed, whichever happens earlier. Which of the following statements is/are correct?
  1. The probability of no head observed is $\frac{1}{16}.$
  2. The probability that the experiment ends with three tosses is $\frac{1}{8}.$
  Select the correct answer using the code given below:

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 114
  If $x\in [0, 5]$, then what is the probability that x² – 3x + 2 ≥ 0?

  Option A: $\frac{4}{5}$

  Option B: $\frac{1}{5}$

  Option C: $\frac{2}{5}$

  Option D: $\frac{3}{5}$

  VIEW SOLUTION

• Question 115
  A bag contains 4 white and 2 black balls and another bag contains 3 white and 5 black balls. If one ball is drawn from each bag, then the probability that one ball is white and one ball is black is

  Option A: $\frac{5}{24}$

  Option B: $\frac{13}{24}$

  Option C: $\frac{1}{4}$

  Option D: $\frac{2}{3}$

  VIEW SOLUTION

• Question 116
  The problem in statistics is given to three students A, B and C whose chances of solving it independently are $\frac{1}{2},\frac{1}{3}\mathrm{and} \frac{1}{4}$ respectively. The probability that the problem will be solved is

  Option A: $\frac{1}{12}$

  Option B: $\frac{11}{12}$

  Option C: $\frac{1}{2}$

  Option D: $\frac{3}{4}$

  VIEW SOLUTION

• Question 117
  An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident involving a scooter driver, car driver and a truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. The probability that the person is a scooter driver is

  Option A: $\frac{1}{52}$

  Option B: $\frac{3}{52}$

  Option C: $\frac{15}{52}$

  Option D: $\frac{19}{52}$

  VIEW SOLUTION

• Question 118
  A coin is tossed 5 times. The probability that tails appear an odd number of times, is

  Option A: $\frac{1}{2}$

  Option B: $\frac{1}{3}$

  Option C: $\frac{2}{5}$

  Option D: $\frac{1}{5}$

  VIEW SOLUTION

• Question 119
  The regression coefficients of a bivariate distribution are –0.64 and –0.36. Then the correlation coefficient of the distribution is

  Option A: 0.48

  Option B: – 0.48

  Option C: 0.50

  Option D: – 0.50

  VIEW SOLUTION

• Question 120
  What is the probability that the sum of any two different single digit natural numbers is a prime number?

  Option A: $\frac{5}{27}$

  Option B: $\frac{7}{18}$

  Option C: $\frac{1}{3}$

  Option D: None of the these

  VIEW SOLUTION