NDA II 2017 Mathematics

• Question 1
  If x + log₁₀ (1 + 2^x) = xlog₁₀5 + log₁₀6 then x is equal to

  Option A: 2, –3

  Option B: 2 only

  Option C: 1

  Option D: 3

  VIEW SOLUTION

• Question 2
  The remainder and the quotient of the binary division (101110)₂ ÷ (110)₂ are respectively

  Option A: (111)₂ and (100)₂

  Option B: (100)₂ and (111)₂

  Option C: (101)₂ and (101)₂

  Option D: (100)₂ and (100)₂

  VIEW SOLUTION

• Question 3
  The matrix A has x rows and x + 5 columns. The matrix B has y rows and 11 – y columns. Both AB and BA exist. What are the values of x and y respectively?

  Option A: 8 and 3

  Option B: 3 and 4

  Option C: 3 and 8

  Option D: 8 and 8

  VIEW SOLUTION

• Question 4
  If $S_n=nP+\frac{n(n-1)Q}{2},$ where S_n denotes the sum of the first n terms of an AP, then the common difference is

  Option A: P + Q

  Option B: 2P + 3Q

  Option C: 2Q

  Option D: Q

  VIEW SOLUTION

• Question 5
  The roots of the equation
  (q – r)x² + (r – p) x + (p – q) = 0 are

  Option A: (r – p)/ (q – r), 1/2

  Option B: (p – q) / (q – r), 1

  Option C: (q – r) / (p – q), 1

  Option D: (r – p) / (p – q), 1/2

  VIEW SOLUTION

• Question 6
  If E is the universal set and A = B ∪ C, then the set E – (E – (E – (E – (E – A)))) is the same as the set

  Option A: B’∪ C’

  Option B: B ∪ C

  Option C: B’∩ C’

  Option D: B ∩ C

  VIEW SOLUTION

• Question 7
  If A = {x : x is a multiple of 2}, B = {x : x is a multiple of 5} and C = {x : x is a multiple of 10}, then A ∩ (B ∩ C) is equal to

  Option A: A

  Option B: B

  Option C: C

  Option D: {x : x is a multiple of 100}

  VIEW SOLUTION

• Question 8
  If α and β are the roots of equation 1 + x + x² = 0, then the matrix product $\left[\begin{array}{cc}1& \beta \\ \alpha & \alpha \end{array}\right] \left[\begin{array}{cc}\alpha & \beta \\ 1& \beta \end{array}\right]$ is equal to

  Option A: $\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$

  Option B: $\left[\begin{array}{cc}-1& -1\\ -1& 2\end{array}\right]$

  Option C: $\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$

  Option D: $\left[\begin{array}{cc}-1& -1\\ -1& -2\end{array}\right]$

  VIEW SOLUTION

• Question 9
  If |a| denotes the absolute value of an integer, then which of the following are correct?
  1. |ab| = |a| |b|
  2. |a + b| ≤ |a| + |b|
  3. |a – b| ≥ ||a| – |b||
  Select the correct answer using the code given below.

  Option A: 1 and 2 only

  Option B: 2 and 3 only

  Option C: 1 and 3 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 10
  How many different permutations can be made out of the letters of the word 'PERMUTATION'?

  Option A: 19958400

  Option B: 19954800

  Option C: 19952400

  Option D: 39916800

  VIEW SOLUTION

• Question 11
  If  $A=\left[\begin{array}{cc}4i-6& 10i\\ 14i& 6+4i\end{array}\right]$ and $k=\frac{1}{2i},$ where $i=\sqrt{-1},$  then kA is equal to

  Option A: $\left[\begin{array}{cc}2+3i& 5\\ 7& 2-3i\end{array}\right]$

  Option B: $\left[\begin{array}{cc}2-3i& 5\\ 7& 2+3i\end{array}\right]$

  Option C: $\left[\begin{array}{cc}2-3i& 7\\ 5& 2+3i\end{array}\right]$

  Option D: $\left[\begin{array}{cc}2+3i& 5\\ 7& 2+3i\end{array}\right]$

  VIEW SOLUTION

• Question 12
  The sum of all real roots of equation ${\left|x-3\right|}^2+\left|x-3\right|-2=0$ is

  Option A: 2

  Option B: 3

  Option C: 4

  Option D: 6

  VIEW SOLUTION

• Question 13
  It is given that the roots of equation x² – 4x – log₃ P = 0 are real. For this, the minimum value of P is

  Option A: $\frac{1}{27}$

  Option B: $\frac{1}{64}$

  Option C: $\frac{1}{81}$

  Option D: 1

  VIEW SOLUTION

• Question 14
  If A is a square matrix, then the value of adjA^T – (adj A)^T is equal to

  Option A: A

  Option B: 2|A|I, where I is the identity matrix

  Option C: null matrix whose order is the same as that of A

  Option D: unit matrix whose order is the same as that of A

  VIEW SOLUTION

• Question 15
  The value of the product $6^{\frac{1}{2}}\times 6^{\frac{1}{4}}\times 6^{\frac{1}{8}}\times 6^{\frac{1}{16}}\times ...$ up to infinite terms is

  Option A: 6

  Option B: 36

  Option C: 216

  Option D: 512

  VIEW SOLUTION

• Question 16
  The value of determinant
  $\left|\begin{array}{cc}{\cos }^2\frac{\theta }{2}& {\sin }^2\frac{\theta }{2}\\ {\sin }^2\frac{\theta }{2}& {\cos }^2\frac{\theta }{2}\end{array}\right|$
  
  for all the values of θ, is

  Option A: 1

  Option B: cos θ

  Option C: sin θ

  Option D: cos2θ

  VIEW SOLUTION

• Question 17
  The number of terms in the expansion of ${(x+a)}^{100}+{(x-a)}^{100}$ after simplification is

  Option A: 202

  Option B: 101

  Option C: 51

  Option D: 50

  VIEW SOLUTION

• Question 18
  In the expansion of (1 + x)⁵⁰, the sum of coefficients of odd powers of x is

  Option A: 2²⁶

  Option B: 2⁴⁹

  Option C: 2⁵⁰

  Option D: 2⁵¹

  VIEW SOLUTION

• Question 19
  If a, b, c are non-zero real numbers, then the inverse of matrix
  
  $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
  
  is equal to

  Option A: $\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$

  Option B: $\frac{1}{abc}\left[\begin{array}{ccc}{a}^{-1}& 0& 0\\ 0& b^{-1}& 0\\ 0& 0& c^{-1}\end{array}\right]$

  Option C: $\frac{1}{abc}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

  Option D: $\frac{1}{abc}\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$

  VIEW SOLUTION

• Question 20
  A person is to count 4500 notes. Let a_n denote the number of notes that he counts in the n^th minute. If a₁= a₂ = a₃ = … = a₁₀ = 150, and a₁₀, a₁₁, a₁₂, … are in AP with the common difference –2, then the time taken by him to count all the notes is

  Option A: 24 minutes

  Option B: 34 minutes

  Option C: 125 minutes

  Option D: 135 minutes

  VIEW SOLUTION

• Question 21
  The smallest positive integer n, which ${\left(\frac{1+i}{1-i}\right)}^n=1$, is

  Option A: 1

  Option B: 4

  Option C: 8

  Option D: 16

  VIEW SOLUTION

• Question 22
  If we define a relation R on the set N × M as (a, b) R (c, d) ⇔ a + d = b + c for all (a, b), (c, d) ∈ N × N, then the relation is

  Option A: symmetric only

  Option B: symmetric and transitive only

  Option C: equivalence relation

  Option D: reflexive only

  VIEW SOLUTION

• Question 23
  If y = x + x² + x³+ … up to infinite terms, where x < 1, then which of the following is correct?

  Option A: $x=\frac{y}{1+y}$

  Option B: $x=\frac{y}{1-y}$

  Option C: $x=\frac{1+y}{y}$

  Option D: $x=\frac{1-y}{y}$

  VIEW SOLUTION

• Question 24
  If α and β are the roots of equation 3x² + 2x + 1 = 0, then the equation whose roots are α + β^–1 and β + α^–1 is

  Option A: 3x² + 8x + 16 = 0

  Option B: 3x²– 8x – 16 = 0

  Option C: 3x² + 8x – 16 = 0

  Option D: x² + 8x + 16 = 0

  VIEW SOLUTION

• Question 25
  The value of $\frac{1}{{\log }_{3}e}+\frac{1}{{\log }_3{e}^2}+\frac{1}{{\log }_3{e}^4}+...$ up to infinite terms is

  Option A: log_e9

  Option C: 1

  Option D: log_e3

  VIEW SOLUTION

• Question 26
  A tea party is arranged for 16 people along the two sides of a long table with eight chairs on each side. Four particular men wish to sit on one particular side and two particular men on the other side. The number of ways they can be seated is

  Option A: 24 × 8! × 8!

  Option B: (8!)³

  Option C: 210 × 8! × 8!

  Option D: 16!

  VIEW SOLUTION

• Question 27
  The system of equations kx + y + z = 1, x + ky + z = k and x + y + kz = k² has no solution if k equals

  Option B: 1

  Option C: –1

  Option D: –2

  VIEW SOLUTION

• Question 28
  If $1.3+2{.3}^2+3{.3}^3+\dots +n.3^n=\frac{(2n-1)3^a+b}{4}$ then a and b are respectively

  Option A: n, 2

  Option B: n, 3

  Option C: n + 1, 2

  Option D: n + 1, 3

  VIEW SOLUTION

• Question 29
  In ∆PQR, $\angle R=\frac{\mathrm{\pi }}{2}$. If tan$\left(\frac{P}{2}\right)$ and tan$\left(\frac{Q}{2}\right)$ are the roots of equation ax² + bx + c = 0, then which of the following is correct?

  Option A: a = b + c

  Option B: b = c + a

  Option C: c = a + b

  Option D: b = c

  VIEW SOLUTION

• Question 30
  If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of |z| is equal to

  Option A: $1+\sqrt{3}$

  Option B: $1+\sqrt{5}$

  Option C: $1-\sqrt{5}$

  Option D: $\sqrt{5}-1$

  VIEW SOLUTION

• Question 31
  The angle of elevation of stationary cloud from the point 25 m above a lake is 15° and the angle of depression of its image in the lake is 45°. The height of the cloud above the lake level is

  Option A: 25 m

  Option B: $25\sqrt{3}$ m

  Option C: 50 m

  Option D: $50\sqrt{3}$ m

  VIEW SOLUTION

• Question 32
  The value of tan 9° – tan 27° – tan 63° + tan 81° is equals to

  Option A: –1

  Option C: 1

  Option D: 4

  VIEW SOLUTION

• Question 33
  The value of $\sqrt{3}$ cosec 20° – sec 20° is equal to

  Option A: 4

  Option B: 2

  Option C: 1

  Option D: –4

  VIEW SOLUTION

• Question 34
  Angle α is divided into two parts A and B such that A – B = x and tan A : tan B = p : q. The value of sin x is equal to

  Option A: $\frac{\left(p+q\right) \sin \alpha }{p-q}$

  Option B: $\frac{p\sin \alpha }{p+q}$

  Option C: $\frac{p\sin \alpha }{p-q}$

  Option D: $\frac{\left(p-q\right) \sin \alpha }{p+q}$

  VIEW SOLUTION

• Question 35
  The value of ${\sin }^{-1}\left(\frac{3}{5}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)$ is equal to

  Option B: $\frac{\mathrm{\pi }}{4}$

  Option C: $\frac{\mathrm{\pi }}{3}$

  Option D: $\frac{\mathrm{\pi }}{2}$

  VIEW SOLUTION

• Question 36
  The angles of the elevation of the top of a tower from the top and the foot of a pole are respectively 30° and 45°. If h_T is the height of the tower and h_P is the height of the pole, then which of the following are correct?
  
  1. $\frac{2h_P{h}_T}{3+\sqrt{3}}=h_P^2$
  
  2. $\frac{h_T-h_P}{\sqrt{3}+1}=\frac{h_P}{2}$
  
  3. $\frac{2\left(h_{P+}{h}_T\right)}{h_P}=4+\sqrt{3}$
  Select the correct answer using the code given below

  Option A: 1 and 3 only

  Option B: 2 and 3 only

  Option C: 1 and 2 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 37
  In a triangle ABC, a – 2b + c = 0. The value of $\cot \left(\frac{A}{2}\right) \cot \left(\frac{C}{2}\right)$ is

  Option A: $\frac{9}{2}$

  Option B: 3

  Option C: $\frac{3}{2}$

  Option D: 1

  VIEW SOLUTION

• Question 38
  $\sqrt{1+sin A}=-\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)$ is true if

  Option A: $\frac{3\pi }{2}<A<\frac{5\pi }{2}$ only

  Option B: $\frac{\pi }{2}<A<\frac{3\pi }{2}$ only

  Option C: $\frac{3\pi }{2}<A<\frac{7\pi }{2}$

  Option D: $0<A<\frac{3\pi }{2}$

  VIEW SOLUTION

• Question 39
  In triangle ABC, if $\frac{{\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C}{{\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C}=2$ then the triangle is

  Option A: right-angled

  Option B: equilateral

  Option C: isosceles

  Option D: obtuse-angled

  VIEW SOLUTION

• Question 40
  The principal value of sin^–1 x lies in the interval

  Option A: $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

  Option B: $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$

  Option C: $\left[0,\frac{\pi }{2}\right]$

  Option D: [0, π]

  VIEW SOLUTION

• Question 41
  The points (a, b), (0, 0), (–a, –b) and (ab, b²) are

  Option A: the vertices of parallelogram

  Option B: the vertices of a rectangle

  Option C: the vertices of a square

  Option D: collinear

  VIEW SOLUTION

• Question 42
  The length of the normal from origin to the plane x + 2y – 2z = 9 is equal to

  Option A: 2 units

  Option B: 3 units

  Option C: 4 units

  Option D: 5 units

  VIEW SOLUTION

• Question 43
  If α, β and γ are the angles which the vector $\overrightarrow{OP}$ (O being the origin) makes with positive direction of coordinate axes, then which of the following are correct?
  1. cos²α + cos² β = sin² γ
  2. sin² α + sin² β = cos² γ
  3. sin² α + sin² β + sin² γ  = 2
  Select the correct answer using the code given below

  Option A: 1 and 2 only

  Option B: 2 and 3 only

  Option C: 1 and 3 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 44
  The angle between the lines x + y – 3 = 0 and x – y + 3 = 0 is α and the acute angle between the lines $x –\sqrt{3}y+2\sqrt{3}=0$ and $\sqrt{3}x–y+1=0$ is β. Which one of the following is correct?

  Option A: α = β

  Option B: α > β

  Option C: α < β

  Option D: α = 2β

  VIEW SOLUTION

• Question 45
  Let $\overrightarrow{\alpha }=\hat{i}+2\hat{j}-\hat{k}, \overrightarrow{\beta }=2\hat{i}-\hat{j}+3\hat{k}$ and $\overrightarrow{\gamma }=2\hat{i}+\hat{j}+6\hat{k}$ be three vectors. If  $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are both perpendicular to the vector $\overrightarrow{\mathrm{\delta }}$ and $\overrightarrow{\mathrm{\delta }}·\stackrel{\rightharpoonup }{\gamma }=10$, then what is the magnitude of $\overrightarrow{\mathrm{\delta }}$?

  Option A: $\sqrt{3}$ units

  Option B: $2\sqrt{3}$ units

  Option C: $\frac{\sqrt{3}}{2}$ unit

  Option D: $\frac{1}{\sqrt{3}}$ unit

  VIEW SOLUTION

• Question 46
  If $\hat{a}$ and $\hat{b}$ are two unit vectors, then the vector $\left(\hat{a}+\hat{b}\right)\times \left(\hat{a}\times \hat{b}\right)$ is parallel to

  Option A: $\left(\hat{a}-\hat{b}\right)$

  Option B: $\left(\hat{a}+\hat{b}\right)$

  Option C: $\left(2\hat{a}-\hat{b}\right)$

  Option D: $\left(2\hat{a}+\hat{b}\right)$

  VIEW SOLUTION

• Question 47
  A force $\overrightarrow{F}=\hat{i}+3\hat{j}+2\hat{k}$ acts on a particle to displace it from the point $A\left(\hat{i}+2\hat{j}-3\hat{k}\right)$ to the point $B\left(3\hat{i}-\hat{j}+5\hat{k}\right)$. The work done by the force will be

  Option A: 5 units

  Option B: 7 units

  Option C: 9 units

  Option D: 10 units

  VIEW SOLUTION

• Question 48
  For any vector $\overrightarrow{a}$ ${\left|\overrightarrow{a}\times \hat{i}\right|}^2+{\left|\overrightarrow{a}\times \hat{j}\right|}^2+{\left|\overrightarrow{a}\times \hat{k}\right|}^2$ is equal to

  Option A: ${\left|\overrightarrow{a}\right|}^2$

  Option B: $2{\left|\overrightarrow{a}\right|}^2$

  Option C: $3{\left|\overrightarrow{a}\right|}^2$

  Option D: $4{\left|\overrightarrow{a}\right|}^2$

  VIEW SOLUTION

• Question 49
  A man running around a race course notes that the sum of the distance of two flag-posts from him is always 10 m and the distance between the flag-posts is 8 m. The area of the path enclosed is

  Option A: 18π square metres

  Option B: 15π square metres

  Option C: 12π square metres

  Option D: 8π square metres

  VIEW SOLUTION

• Question 50
  The distance of point (1,3) from the line 2x + 3y = 6, measured parallel to the line 4x + y = 4, is

  Option A: $\frac{5}{\sqrt{13}}$ units

  Option B: $\frac{3}{\sqrt{17}}$ units

  Option C: $\sqrt{17}$ units

  Option D: $\frac{\sqrt{17}}{2}$ units

  VIEW SOLUTION

• Question 51
  If the vectors $a\hat{i}+\hat{j}+\hat{k}, \hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k} \left(a, b, c\ne 1\right)$ are coplanar, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

  Option B: 1

  Option C: a + b + c

  Option D: abc

  VIEW SOLUTION

• Question 52
  The point of intersection of the line joining the points (–3, 4, –8) and (5, –6, 4) with the XY-plane is

  Option A: $\left(\frac{7}{3}, -\frac{8}{3}, 0\right)$

  Option B: $\left(-\frac{7}{3}, -\frac{8}{3}, 0\right)$

  Option C: $\left(-\frac{7}{3}, \frac{8}{3}, 0\right)$

  Option D: $\left(\frac{7}{3}, \frac{8}{3}, 0\right)$

  VIEW SOLUTION

• Question 53
  If the angle between the lines whose direction ratios are (2, –1, 2) and (x, 3, 5) is $\frac{\mathrm{\pi }}{4}$, then the smaller value of x is

  Option A: 52

  Option B: 4

  Option C: 2

  Option D: 1

  VIEW SOLUTION

• Question 54
  The position of the point (1, 2) relative to the ellipse $2x^2+7y^2=20$ is

  Option A: outside the ellipse

  Option B: inside the ellipse but not at the focus

  Option C: on the ellipse

  Option D: at the focus

  VIEW SOLUTION

• Question 55
  The equation of a straight line which cuts off an intercept of 5 units on negative direction of y-axis and makes an angle 120° with positive direction of x-axis is

  Option A: $y+\sqrt{3}x+5=0$

  Option B: $y-\sqrt{3}x+5=0$

  Option C: $y+\sqrt{3}x-5=0$

  Option D: $y-\sqrt{3}x-5=0$

  VIEW SOLUTION

• Question 56
  The equation of the line passing through the point (2, 3) and the point of intersection of lines 2x – 3y + 7 = 0 and 7x + 4y + 2 = 0 is

  Option A: 21x + 46y – 180 = 0

  Option B: 21x – 46y + 96 = 0

  Option C: 46x + 21y – 155 = 0

  Option D: 46x – 21y – 29 = 0

  VIEW SOLUTION

• Question 57
  The equation of the ellipse whose centre is at the origin, major axis is along x-axis with eccentricity $\frac{3}{4}$ and latus rectum 4 units is

  Option A: $\frac{x^2}{1024}+\frac{7y^2}{64}=1$

  Option B: $\frac{49x^2}{1024}+\frac{7y^2}{64}=1$

  Option C: $\frac{7x^2}{1024}+\frac{49y^2}{64}=1$

  Option D: $\frac{x^2}{1024}+\frac{y^2}{64}=1$

  VIEW SOLUTION

• Question 58
  The equation of the circle which passes through the points (1, 0), (0, –6) and (3, 4) is

  Option A: $4x^2+4y^2+142x+47y+140=0$

  Option B: 2$4x^2+4y^2-142x-47y+138=0$

  Option C: $4x^2+4y^2-142x+47y+138=0$

  Option D: $4x^2+4y^2+150x-49y+138=0$

  VIEW SOLUTION

• Question 59
  A variable plane passes through a fixed point (a, b, c) and cuts the axes in A, B and C respectively. The locus of the centre of the sphere OABC, O being the origin, is

  Option A: $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

  Option B: $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=1$

  Option C: $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2$

  Option D: $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=2$

  VIEW SOLUTION

• Question 60
  The equation of the plane passing through the line of intersection of the planes x + y + z = 1, 2x + 3y + 4z = 7, and perpendicular to the plane x – 5y + 3z = 5 is given by

  Option A: x + 2y + 3z – 6 = 0

  Option B: x + 2y + 3z + 6 = 0

  Option C: 3x + 4y + 5z – 8 = 0

  Option D: 3x + 4y + 5z + 8 = 0

  VIEW SOLUTION

• Question 61
  The inverse of the function y = 5^{ln x} is

  Option A: $x=y^{\frac{1}{\ln 5}}, y>0$

  Option B: $x=y^{\ln 5}, y>0$

  Option C: $x=y^{\frac{1}{\ln 5}}, y<0$

  Option D: $x=5 \ln y, y>0$

  VIEW SOLUTION

• Question 62
  A function is defined as follows:
  $f\left(x\right)=\left\{\begin{array}{cc}-\frac{x}{\sqrt{x^2}},& x\ne 0\\ 0,& x=0\end{array}\right.$
  Which one of the following is correct in respect of the above function?

  Option A: f(x) is continuous at x = 0 but not differentiable at x = 0

  Option B: f(x) is continuous as well as differentiable at x = 0

  Option C: f(x) is discontinuous at x = 0

  Option D: None of the above

  VIEW SOLUTION

• Question 63
  If $y={(\cos x)}^{{(\cos x)}^{{{(\cos x)}^{{.}^{{.}^{.}}}}^{\infty }}}$, then $\frac{dy}{dx}$ is equal to

  Option A: $-\frac{y^2 \tan x}{1-y \ln (\cos x)}$

  Option B: $\frac{y^2 \tan x}{1+y \ln (\cos x)}$

  Option C: $\frac{y^2 \tan x}{1-y \ln (\sin x)}$

  Option D: $\frac{y^2 \sin x}{1+y \ln (\sin x)}$

  VIEW SOLUTION

• Question 64
  Consider the following:
  1. x + x² is continuous at x = 0
  2. x + cos $\frac{1}{x}$ is discontinuous at x = 0
  3. x² + cos $\frac{1}{x}$ is continuous at x = 0
  Which of the above are correct?

  Option A: 1 and 2 only

  Option B: 2 and 3 only

  Option C: 1 and 3 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 65
  Consider the following statements:
  1. dy/dx at a point on the curve gives slope of the tangent at that point.
  2. If a(t) denotes acceleration of a particle, then $\int a\left(t\right)dt+c$ gives velocity of the particle.
  3. If s(t) gives displacement of a particle at time t, then ds/dt gives its acceleration at that instant.
  Which of the above statements is/are correct?

  Option A: 1 and 2 only

  Option B: 2 only

  Option C: 1 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 66
  If $y={\sec }^{-1}\left(\frac{x+1}{x-1}\right)+{\sin }^{-1}\left(\frac{x-1}{x+1}\right),$ then $\frac{dy}{dx}$ is equal to

  Option B: 1

  Option C: $\frac{x-1}{x+1}$

  Option D: $\frac{x+1}{x-1}$

  VIEW SOLUTION

• Question 67
  What is$\int {\tan }^{-1}(\sec x+\tan x)dx$ equal to?

  Option A: $\frac{\pi x}{4}+\frac{x^2}{4}+c$

  Option B: $\frac{\pi x}{2}+\frac{x^2}{4}+c$

  Option C: $\frac{\pi x}{4}+\frac{\pi x^2}{4}+c$

  Option D: $\frac{\pi x}{4}-\frac{x^2}{4}+c$

  VIEW SOLUTION

• Question 68
  A function is defined in (0, ∞) by
  
  $f\left(x\right)=\left\{\begin{array}{c}1-x^2 \mathrm{for} 0<x\le 1\\ \ln x \mathrm{for} 1<x\le 2\\ \ln 2-1+0.5x \mathrm{for} 2<x<\mathrm{\infty }\end{array}\right\}$
  
  Which of the following is correct in respect of the derivation of the function, i.e., f’(x)?

  Option A: f’(x) = 2x for 0 < x ≤ 1

  Option B:  f’(x) = –2x for 0 < x ≤ 1

  Option C:  f’(x) = –2x for 0 < x < 1

  Option D: f’(x) = 0 for 0 < x < ∞

  VIEW SOLUTION

• Question 69
  Which of the following is correct in respect of the function f(x) = x(x – 1)(x+1)?

  Option A: The local maximum value is larger than local minimum value

  Option B: The local maximum value is smaller than local minimum value

  Option C: The function has no local maximum

  Option D: The function has no local minimum

  VIEW SOLUTION

• Question 70
  Consider the following statements:
  1. Derivative of f(x) may not exist at some point.
  2. Derivative of f(x) may exist finitely at some point.
  3. Derivative of f(x) may be infinite (geometrically) at some point.
  Which of the above statements are correct?

  Option A: 1 and 2 only

  Option B: 2 and 3 only

  Option C: 1 and 3 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 71
  The maximum value of $\frac{\ln x}{x}$ is

  Option A: e

  Option B: $\frac{1}{e}$

  Option C: $\frac{2}{e}$

  Option D: 1

  VIEW SOLUTION

• Question 72
  The function $f\left(x\right)=\left|x\right|-x^3$ is

  Option A: odd

  Option B: even

  Option C: both even and odd

  Option D: neither even nor odd

  VIEW SOLUTION

• Question 73
  If
  $$\begin{gathered} l_1=\frac{d}{dx}\left(e^{\sin x}\right) \\ {l}_2=\underset{h\to 0}{\lim }\frac{e^{\sin (x+h)}-e^{\sin x}}{h} \\ {l}_3=\int e^{\sin x}\cos x dx \end{gathered}$$
  
  then which one of the following is correct?

  Option A: $l_1\ne l_2$

  Option B: $\frac{d}{dx}\left(l_3\right)=l_2$

  Option C: $\int l_{3}dx=l_2$

  Option D: $l_2=l_3$

  VIEW SOLUTION

• Question 74
  The general solution of
  $\frac{dy}{dx}=\frac{ax+h}{by+k}$
  represents a circle only when

  Option A: a = b = 0

  Option B: a = –b ≠ 0

  Option C: a = b ≠ 0, h = k

  Option D: a = b ≠ 0

  VIEW SOLUTION

• Question 75
  If $\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x}{x}=l$ and $\underset{x\to \infty }{\lim }\frac{\cos x}{x}=m$, then which of the following is correct?

  Option A: l = 1, m = 1

  Option B: $l=\frac{2}{\pi }, m=\mathrm{\infty }$

  Option C: $l=\frac{2}{\pi }, m=0$

  Option D: l = 1, m = ∞

  VIEW SOLUTION

• Question 76
  What is $\underset{0}{\overset{2\pi }{\int }}\sqrt{1+\sin \frac{x}{2}}dx$ equal to?

  Option A: 8

  Option B: 4

  Option C: 2

  VIEW SOLUTION

• Question 77
  The area bounded by the curve $\left|x\right|+\left|y\right|=1$ is

  Option A: 1 square unit

  Option B: $2\sqrt{2}$ square units

  Option C: 2 square units

  Option D: $2\sqrt{3}$ square units

  VIEW SOLUTION

• Question 78
  If x is any real number, then $\frac{x^2}{1+x^4}$ belongs to which of the following intervals?

  Option A: (0,1)

  Option B: $\left(0,\frac{1}{2}\right]$

  Option C: $\left[0,\frac{1}{2}\right]$

  Option D: [0,1]

  VIEW SOLUTION

• Question 79
  The left-hand derivative of
  
  f(x) = [x] sin (πx) at x = k
  
  where k is an integer and [x] is the greatest integer function, is

  Option A: ${(-1)}^k(k-1)\pi$

  Option B: ${(-1)}^{k-1}(k-1)\pi$

  Option C: ${(-1)}^{k}k\pi$

  Option D: ${(-1)}^{k-1}k\pi$

  VIEW SOLUTION

• Question 80
  If $f\left(x\right)=\frac{x}{2}-1,$ then on the interval [0, π] which of the following is correct?

  Option A: tan [f(x)], where [·] is the greatest integer function, and $\frac{1}{f\left(x\right)}$ are both continuous

  Option B: tan [f(x)], where [·] is the greatest integer function, and f^–1(x) are both continuous

  Option C: tan [f(x)], where [·] is the greatest integer function, and $\frac{1}{f\left(x\right)}$ are both discontinuous

  Option D: tan [f(x)], where [·] is the greatest integer function, is discontinuous but $\frac{1}{f\left(x\right)}$ is continuous

  VIEW SOLUTION

• Question 81
  The order and degree of the differential equation ${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3={\rho }^2{\left[\frac{d^{2}y}{d{x}^2}\right]}^2$ are respectively

  Option A: 3 and 2

  Option B: 2 and 2

  Option C: 2 and 3

  Option D: 1 and 3

  VIEW SOLUTION

• Question 82
  If $y={\cos }^{-1}\left(\frac{2x}{1+x^2}\right),$ then $\frac{dy}{dx}$ is equal to

  Option A: $-\frac{2}{1+x^2}$for all $\left|x\right|<1$

  Option B: $-\frac{2}{1+x^2}$ for all $\left|x\right|>1$

  Option C: $\frac{2}{1+x^2}$ for all $\left|x\right|<1$

  Option D: None of the above

  VIEW SOLUTION

• Question 83
  The set of all points, where the function $f\left(x\right)=\sqrt{1-e^{-x^2}}$ is differentiable, is

  Option A: (0, ∞)

  Option B: (–∞, ∞)

  Option C: (–∞, 0) ∪ (0, ∞)

  Option D: (–1, ∞)

  VIEW SOLUTION

• Question 84
  Match List-I with List-II and select the correct answer using the code given below the lists:
   

  List–I (Function)		List–II (Maximum value)
  A.	sin x + cos x	1.	$\sqrt{10}$
  B.	3 sin x + 4 cos x	2.	$\sqrt{2}$
  C.	2 sin x + cos x	3.	5
  D.	sin x + 3 cos x	4.	$\sqrt{5}$

  
  Code:

  Option A: 

  A	B	C	D
  2	3	1	4

  Option B: 

  A	B	C	D
  2	3	4	1

  Option C: 

  A	B	C	D
  3	2	1	4

  Option D: 

  A	B	C	D
  3	2	4	1

  VIEW SOLUTION

• Question 85
  If $f\left(x\right)=x(\sqrt{x}-\sqrt{x+1}),$ then f(x) is

  Option A: continuous but not differentiable at x = 0

  Option B: differentiable at x = 0

  Option C: not continuous at x = 0

  Option D: None of the above

  VIEW SOLUTION

• Question 86
  Which of the following graphs represents the function $f\left(x\right)=\frac{x}{x}, x\ne 0$?

  Option A: 

  Option B: 

  Option C: 

  Option D: None of the above

  VIEW SOLUTION

• Question 87
  Let $f\left(n\right)=\left[\frac{1}{4}+\frac{n}{1000}\right]$ , where [x] denotes the integral part of x. Then the value of $\sum _{n=1}^{1000}f\left(n\right)$ is

  Option A: 251

  Option B: 250

  Option C: 1

  VIEW SOLUTION

• Question 88
  $\int {(\ln x)}^{-1}dx-\int {(\ln x)}^{-2}dx$ is equal to

  Option A: x(ln x)^–1 + c

  Option B: x(ln x)^–2 + c

  Option C: x(ln x) + c

  Option D: x(ln x)² + c

  VIEW SOLUTION

• Question 89
  A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is

  Option A: 1

  Option B: 2

  Option C: 3

  Option D: 4

  VIEW SOLUTION

• Question 90
  The value of $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\sqrt{\tan x}dx+\underset{0}{\overset{\frac{\pi }{4}}{\int }}\sqrt{\cot x}dx$ is equal to

  Option A: $\frac{\mathrm{\pi }}{4}$

  Option B: $\frac{\mathrm{\pi }}{2}$

  Option C: $\frac{\mathrm{\pi }}{2\sqrt{2}}$

  Option D: $\frac{\mathrm{\pi }}{\sqrt{2}}$

  VIEW SOLUTION

• Question 91
  Let g be the greatest integer function. Then the function f(x) = (g(x))² – g(x) is discontinuous at

  Option A: all integers

  Option B: all integers except 0 and 1

  Option C: all integers except 0

  Option D: all integers except 1

  VIEW SOLUTION

• Question 92
  The differential equation of the minimum order by eliminating the arbitrary constants A and C in the equation y = A[sin(x + C) + cos(x + C)] is

  Option A: y'' + (sin x + cos x)y' = 1

  Option B: y'' = (sin x + cos x)y'

  Option C: y'' = (y')² + sinxcosx

  Option D: y'' + y = 0

  VIEW SOLUTION

• Question 93
  Consider the following statements:
  Statements I:
  x > sin x for all x > 0
  Statement II:
  f(x) = x – sin x is an increasing function for all x > 0
  Which one of the following is correct in respect of the above statements?

  Option A: Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.

  Option B: Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I

  Option C: Statement I is true but Statement II is false

  Option D: Statement I is false but Statement II is true

  VIEW SOLUTION

• Question 94
  The solution of the differential equation $\frac{dy}{dx}=\frac{y\varphi \text{'}\left(x\right)-y^2}{\varphi \left(x\right)}$ is

  Option A: $y=\frac{x}{\varphi \left(x\right)+c}$

  Option B: $y=\frac{\varphi \left(x\right)}{x}+c$

  Option C: $y=\frac{\varphi \left(x\right)+c}{x}$

  Option D: $y=\frac{\varphi \left(x\right)}{x+c}$

  VIEW SOLUTION

• Question 95
  If $f\left(x\right)=\frac{4x+x^4}{1+4x^3}$ and $g\left(x\right)=\ln \left(\frac{1+x}{1-x}\right)$, then what is the value of f$\circ$$g\left(\frac{e-1}{e+1}\right)$ equal to?

  Option A: 2

  Option B: 1

  Option D: $\frac{1}{2}$

  VIEW SOLUTION

• Question 96
  The value of the determinant $\left|\begin{array}{ccc}1-\alpha & \alpha -{\alpha }^2& {\alpha }^2\\ 1-\beta & \beta -{\beta }^2& {\beta }^2\\ 1-\gamma & \gamma -{\gamma }^2& {\gamma }^2\end{array}\right|$ is equal to

  Option A: $(\alpha -\beta )(\beta -\gamma )(\alpha -\gamma )$

  Option B: $(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )$

  Option C: $(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )(\alpha +\beta +\gamma )$

  VIEW SOLUTION

• Question 97
  The adjoint of matrix $A=\left[\begin{array}{ccc}1& 0& 2\\ 2& 1& 0\\ 0& 3& 1\end{array}\right]$ is

  Option A: $\left[\begin{array}{ccc}-1& 6& 2\\ -2& 1& -4\\ 6& 3& 1\end{array}\right]$

  Option B: $\left[\begin{array}{ccc}1& 6& -2\\ -2& 1& 4\\ 6& -3& 1\end{array}\right]$

  Option C: $\left[\begin{array}{ccc}6& 1& 2\\ 4& -1& 2\\ 6& 3& -1\end{array}\right]$

  Option D: $\left[\begin{array}{ccc}-6& 2& 1\\ 4& -2& 1\\ 3& 1& -6\end{array}\right]$

  VIEW SOLUTION

• Question 98
  If $A=\left(\begin{array}{cc}-2& 2\\ 2& -2\end{array}\right)$ , then which of following is correct?

  Option A: A² = −2A

  Option B: A² = −4A

  Option C: A² = −3A

  Option D: A² = 4A

  VIEW SOLUTION

• Question 99
  Geometrically, Re (z² – i) = 2, where $i=\sqrt{-1}$ and Re is the real part, represents

  Option A: circle

  Option B: ellipse

  Option C: rectangular hyperbola

  Option D: parabola

  VIEW SOLUTION

• Question 100
  If p + q + r = a + b + c = 0, then the determinant $\left|\begin{array}{ccc}pa& qb& rc\\ qc& ra& pb\\ rb& pc& qa\end{array}\right|$ equals

  Option B: 1

  Option C: pa + qb + rc

  Option D: pa + qb + rc + a + b + c

  VIEW SOLUTION

• Question 101
  A committee of two persons is selected from two men and two women. The probability that the committee will have exactly one woman is

  Option A: $\frac{1}{6}$

  Option B: $\frac{2}{3}$

  Option C: $\frac{1}{3}$

  Option D: $\frac{1}{2}$

  VIEW SOLUTION

• Question 102
  Let a die be loaded in such a way that even faces are twice likely to occur as the odd faces. What is the probability that a prime number will show up when the die is tossed?

  Option A: $\frac{1}{3}$

  Option B: $\frac{2}{3}$

  Option C: $\frac{4}{9}$

  Option D: $\frac{5}{9}$

  VIEW SOLUTION

• Question 103
  Let the sample space consist of non-negative integers up to 50. X denotes the numbers which are multiples of 3 and Y denotes the odd numbers. Which of the following is/are correct?
  1. $P\left(X\right)=\frac{8}{25}$
  
  2. $P\left(Y\right)=\frac{1}{2}$
  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 104
  For two events A and B, let $P\left(A\right)=\frac{1}{2},$ $P(A\cup B)=\frac{2}{3}$ and $P(A\cap B)=\frac{1}{6}$.  What is $P\left(\overline{A}\cap B\right)$ equal to?

  Option A: $\frac{1}{6}$

  Option B: $\frac{1}{4}$

  Option C: $\frac{1}{3}$

  Option D: $\frac{1}{2}$

  VIEW SOLUTION

• Question 105
  Consider the following statements:
  1. Coefficient of variation depends on the unit of measurement of the variable.
  2. Range is a measure of dispersion.
  3. Mean deviation is the least when measured about median.
  Which of the above statements are correct?

  Option A: 1 and 2 only

  Option B: 2 and 3 only

  Option C: 1 and 3 only

  Option D: 1, 2 and 3

  VIEW SOLUTION

• Question 106
  Given that the arithmetic mean and standard deviation of a sample of 15 observations are 24 and 0, respectively. Then which one of the following is the arithmetic mean of the smallest five observations in the data?

  Option B: 8

  Option C: 16

  Option D: 24

  VIEW SOLUTION

• Question 107
  Which of the following can be considered as the appropriate pair of values of regression coefficient of y on x and regression coefficient of x on y?

  Option A: (1, 1)

  Option B: (–1, 1)

  Option C: $\left(-\frac{1}{2}, 2\right)$

  Option D: $\left(\frac{1}{3}, \frac{10}{3}\right)$

  VIEW SOLUTION

• Question 108
  Let A and B be two events with P(A) = $\frac{1}{3}$ , P(B) = $\frac{1}{6}$ and $P(A\cap B)=\frac{1}{12}$. What is $\mathrm{P}\left(B|\overline{A}\right)$ equal to?

  Option A: $\frac{1}{5}$

  Option B: $\frac{1}{7}$

  Option C: $\frac{1}{8}$

  Option D: $\frac{1}{10}$

  VIEW SOLUTION

• Question 109
  In a binomial distribution, the mean is $\frac{2}{3}$ and the variance is $\frac{5}{9}$. What is the probability that X = 2?

  Option A: $\frac{5}{36}$

  Option B: $\frac{25}{36}$

  Option C: $\frac{25}{216}$

  Option D: $\frac{25}{54}$

  VIEW SOLUTION

• Question 110
  The probability that a ship safely reaches a port is $\frac{1}{3}$. The probability that out of 5 ships, at least 4 ships would arrive safely is

  Option A: $\frac{1}{243}$

  Option B: $\frac{10}{243}$

  Option C: $\frac{11}{243}$

  Option D: $\frac{13}{243}$

  VIEW SOLUTION

• Question 111
  What is the probability that at least two persons out of a group of three persons were born in the same month (disregard the year)?

  Option A: $\frac{33}{144}$

  Option B: $\frac{17}{72}$

  Option C: $\frac{1}{144}$

  Option D: $\frac{2}{9}$

  VIEW SOLUTION

• Question 112
  It is given that $\overline{X}$ = 10, $\overline{Y}$ = 90, σ_X = 3, σ_Y = 12 and r_XY = 0.8. The regression equation of X and Y is

  Option A: Y = 3.2X + 58

  Option B: X = 3.2Y + 58

  Option C: X = −8 + 0.2Y

  Option D: Y = –8 + 0.2X

  VIEW SOLUTION

• Question 113
  If P(B) = $\frac{3}{4}$, $P\left(A\cap B\cap \overline{C}\right)=\frac{1}{3}$ and $P\left(\overline{A}\cap B\cap \overline{C}\right)=\frac{1}{3}$, then what is P(B ∩ C) equal to?

  Option A: $\frac{1}{12}$

  Option B: $\frac{3}{4}$

  Option C: $\frac{1}{15}$

  Option D: $\frac{1}{9}$

  VIEW SOLUTION

• Question 114
  The following table gives the monthly expenditure of two families:
   

  	Expenditure (in Rs)
  Items	Family A	Family B
  Food	3,500	2,700
  Clothing	500	800
  Rent	1,500	1,000
  Education	2,000	1,800
  Miscellaneous	2,500	1,800

  
  In constructing a pie diagram to the above data, the radii of the circles are to be chosen by which of the following ratios?

  Option A: 1 : 1

  Option B: 10 : 9

  Option C: 100 : 91

  Option D: 5 : 4

  VIEW SOLUTION

• Question 115
  If a variable takes values 0, 1, 2, 3, ..… , n with frequencies 1, C(n, 1), C(n, 2), C(n, 3), ….. , C(n, n) respectively, then the arithmetic mean is

  Option A: 2n

  Option B: n + 1

  Option C: n

  Option D: $\frac{n}{2}$

  VIEW SOLUTION

• Question 116
  In a multiple choice test, an examinee either knows the correct answer with probability p, or guesses with probability 1 – p. The probability of answering a question correctly is $\frac{1}{m}$, if he or she merely guesses. If the examinee answers a question correctly, the probability that he or she really knows the answer is

  Option A: $\frac{mp}{1+mp}$

  Option B: $\frac{mp}{1+\left(m-1\right)p}$

  Option C: $\frac{\left(m-1\right)p}{1+\left(m-1\right)p}$

  Option D: $\frac{\left(m-1\right)p}{1+mp}$

  VIEW SOLUTION

• Question 117
  If x₁ and x₂ are positive quantities, then the condition for the difference between arithmetic mean and the geometric mean to be greater than 1 is

  Option A: $x_1+x_2>2\sqrt{x_1{x}_2}$

  Option B: $\sqrt{x_1}+\sqrt{x_2}>\sqrt{2}$

  Option C: $\left|\sqrt{x_1}-\sqrt{x_2}\right|>\sqrt{2}$

  Option D: $x_1+x_2<2\left(\sqrt{x_1{x}_2}+1\right)$

  VIEW SOLUTION

• Question 118
  Consider the following statements:
  1. Variance is unaffected by the change of origin and change of scale.
  2. Coefficient of variance is independent of the unit of observations.
  Which of the statements given 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 119
  Five sticks of lengths 1, 3, 5, 7 and 9 feet are given. Three of these sticks are selected at random. What is the probability that the selected sticks can form a triangle?

  Option A: 0.5

  Option B: 0.4

  Option C: 0.3

  VIEW SOLUTION

• Question 120
  The coefficient of correlation when coefficients of regression are 0.2 and 1.8 is

  Option A: 0.36

  Option B: 0.2

  Option C: 0.6

  Option D: 0.9

  VIEW SOLUTION