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# NDA II 2017 Mathematics

This test contains 120 question. Each question comprises four responses (answers). You need to select only ONE response for each question.

All questions carry equal marks.

Each question for which a wrong answer has been marked, one-third of the marks assigned to that question will be deducted as penalty.

If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to
that question.

If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
• Question 1
If x + log10 (1 + 2x) = 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 ${S}_{n}=nP+\frac{n\left(n-1\right)Q}{2},$ where Sn 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
(qr)x2 + (rp) x + (pq) = 0 are
Option A: (rp)/ (qr), 1/2
Option B: (pq) / (qr), 1
Option C: (q – r) / (pq), 1
Option D: (rp) / (pq), 1/2
VIEW SOLUTION
• Question 6
If E is the universal set and A = BC, then the set E – (E – (E – (E – (E – A)))) is the same as the set
Option A: B’∪ C
Option B: BC
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 (BC) 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 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 adjAT – (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 ${6}^{\frac{1}{2}}×{6}^{\frac{1}{4}}×{6}^{\frac{1}{8}}×{6}^{\frac{1}{16}}×...$ 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
$\left|\begin{array}{cc}{\mathrm{cos}}^{2}\frac{\theta }{2}& {\mathrm{sin}}^{2}\frac{\theta }{2}\\ {\mathrm{sin}}^{2}\frac{\theta }{2}& {\mathrm{cos}}^{2}\frac{\theta }{2}\end{array}\right|$

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 ${\left(x+a\right)}^{100}+{\left(x-a\right)}^{100}$ 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 an denote the number of notes that he counts in the nth 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 $\frac{1}{{\mathrm{log}}_{3}e}+\frac{1}{{\mathrm{log}}_{3}{e}^{2}}+\frac{1}{{\mathrm{log}}_{3}{e}^{4}}+...$ up to infinite terms is
Option A: loge9
Option C: 1
Option D: loge3
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 $1.3+2{.3}^{2}+3{.3}^{3}+\dots +n.{3}^{n}=\frac{\left(2n-1\right){3}^{a}+b}{4}$ 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 AB = x and tan A : tan B = p : q. The value of sin x is equal to
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 35
The value of ${\mathrm{sin}}^{-1}\left(\frac{3}{5}\right)+{\mathrm{tan}}^{-1}\left(\frac{1}{7}\right)$ 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 hT is the height of the tower and hP is the height of the pole, then which of the following are correct?

1. $\frac{2{h}_{P}{h}_{T}}{3+\sqrt{3}}={h}_{P}^{2}$

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  is
Option A: $\frac{9}{2}$
Option B: 3
Option C: $\frac{3}{2}$
Option D: 1
VIEW SOLUTION
• Question 38
is true if
Option A: $\frac{3\pi }{2} only
Option B: $\frac{\pi }{2} only
Option C: $\frac{3\pi }{2}
Option D: $0
VIEW SOLUTION
• Question 39
In triangle ABC, if $\frac{{\mathrm{sin}}^{2}A+{\mathrm{sin}}^{2}B+{\mathrm{sin}}^{2}C}{{\mathrm{cos}}^{2}A+{\mathrm{cos}}^{2}B+{\mathrm{cos}}^{2}C}=2$ 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 $\stackrel{\to }{OP}$ (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 xy + 3 = 0 is α and the acute angle between the lines  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  and $\stackrel{\to }{\gamma }=2\stackrel{^}{i}+\stackrel{^}{j}+6\stackrel{^}{k}$ be three vectors. If  $\stackrel{\to }{\alpha }$ and $\stackrel{\to }{\beta }$ are both perpendicular to the vector $\stackrel{\to }{\mathrm{\delta }}$ and $\stackrel{\to }{\mathrm{\delta }}·\stackrel{⇀}{\gamma }=10$, then what is the magnitude of $\stackrel{\to }{\mathrm{\delta }}$?
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 $\stackrel{^}{a}$ and $\stackrel{^}{b}$ are two unit vectors, then the vector $\left(\stackrel{^}{a}+\stackrel{^}{b}\right)×\left(\stackrel{^}{a}×\stackrel{^}{b}\right)$ is parallel to
Option A: $\left(\stackrel{^}{a}-\stackrel{^}{b}\right)$
Option B: $\left(\stackrel{^}{a}+\stackrel{^}{b}\right)$
Option C: $\left(2\stackrel{^}{a}-\stackrel{^}{b}\right)$
Option D: $\left(2\stackrel{^}{a}+\stackrel{^}{b}\right)$
VIEW SOLUTION
• Question 47
A force $\stackrel{\to }{F}=\stackrel{^}{i}+3\stackrel{^}{j}+2\stackrel{^}{k}$ acts on a particle to displace it from the point $A\left(\stackrel{^}{i}+2\stackrel{^}{j}-3\stackrel{^}{k}\right)$ to the point $B\left(3\stackrel{^}{i}-\stackrel{^}{j}+5\stackrel{^}{k}\right)$. 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 $\stackrel{\to }{a}$ ${\left|\stackrel{\to }{a}×\stackrel{^}{i}\right|}^{2}+{\left|\stackrel{\to }{a}×\stackrel{^}{j}\right|}^{2}+{\left|\stackrel{\to }{a}×\stackrel{^}{k}\right|}^{2}$ is equal to
Option A: ${\left|\stackrel{\to }{a}\right|}^{2}$
Option B: $2{\left|\stackrel{\to }{a}\right|}^{2}$
Option C: $3{\left|\stackrel{\to }{a}\right|}^{2}$
Option D: $4{\left|\stackrel{\to }{a}\right|}^{2}$
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  and  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:
Option B:
Option C:
Option D:
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 $2{x}^{2}+7{y}^{2}=20$ 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: $\frac{{x}^{2}}{1024}+\frac{7{y}^{2}}{64}=1$
Option B: $\frac{49{x}^{2}}{1024}+\frac{7{y}^{2}}{64}=1$
Option C: $\frac{7{x}^{2}}{1024}+\frac{49{y}^{2}}{64}=1$
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: $4{x}^{2}+4{y}^{2}+142x+47y+140=0$
Option B: 2$4{x}^{2}+4{y}^{2}-142x-47y+138=0$
Option C: $4{x}^{2}+4{y}^{2}-142x+47y+138=0$
Option D: $4{x}^{2}+4{y}^{2}+150x-49y+138=0$
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 = 5ln x is
Option A:
Option B:
Option C:
Option D:
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 , then $\frac{dy}{dx}$ is equal to
Option A:
Option B:
Option C:
Option D:
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 $y={\mathrm{sec}}^{-1}\left(\frac{x+1}{x-1}\right)+{\mathrm{sin}}^{-1}\left(\frac{x-1}{x+1}\right),$ 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$\int {\mathrm{tan}}^{-1}\left(\mathrm{sec}x+\mathrm{tan}x\right)dx$ 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

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  is
Option A: e
Option B: $\frac{1}{e}$
Option C: $\frac{2}{e}$
Option D: 1
VIEW SOLUTION
• Question 72
The function $f\left(x\right)=|x|-{x}^{3}$ is
Option A: odd
Option B: even
Option C: both even and odd
Option D: neither even nor odd
VIEW SOLUTION
• Question 73
If

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 $\underset{x\to \frac{\pi }{2}}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}=l$ and , then which of the following is correct?
Option A: l = 1, m = 1
Option B:
Option C:
Option D: l = 1, m = ∞
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• Question 76
What is $\underset{0}{\overset{2\pi }{\int }}\sqrt{1+\mathrm{sin}\frac{x}{2}}dx$ equal to?
Option A: 8
Option B: 4
Option C: 2
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• 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
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• 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: $\left(0,\frac{1}{2}]$
Option C: $\left[0,\frac{1}{2}\right]$
Option D: [0,1]
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• 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: ${\left(-1\right)}^{k}\left(k-1\right)\pi$
Option B: ${\left(-1\right)}^{k-1}\left(k-1\right)\pi$
Option C: ${\left(-1\right)}^{k}k\pi$
Option D: ${\left(-1\right)}^{k-1}k\pi$
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• 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
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• 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
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• Question 82
If $y={\mathrm{cos}}^{-1}\left(\frac{2x}{1+{x}^{2}}\right),$ then $\frac{dy}{dx}$ is equal to
Option A: $-\frac{2}{1+{x}^{2}}$for all $|x|<1$
Option B: $-\frac{2}{1+{x}^{2}}$ for all $|x|>1$
Option C: $\frac{2}{1+{x}^{2}}$ for all $|x|<1$
Option D: None of the above
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• 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, ∞)
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• 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
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• Question 85
If $f\left(x\right)=x\left(\sqrt{x}-\sqrt{x+1}\right),$ 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
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• Question 86
Which of the following graphs represents the function ?
Option A:
Option B:
Option C:
Option D: None of the above
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• 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
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• Question 88
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
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• 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
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• Question 90
The value of 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: $\frac{\mathrm{\pi }}{\sqrt{2}}$
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• Question 91
Let g be the greatest integer function. Then the function f(x) = (g(x))2g(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
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• 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
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• 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
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• 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: $y=\frac{\varphi \left(x\right)}{x+c}$
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• Question 95
If $f\left(x\right)=\frac{4x+{x}^{4}}{1+4{x}^{3}}$ and $g\left(x\right)=\mathrm{ln}\left(\frac{1+x}{1-x}\right)$, 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: $\frac{1}{2}$
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• 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: $\left(\alpha -\beta \right)\left(\beta -\gamma \right)\left(\alpha -\gamma \right)$
Option B: $\left(\alpha -\beta \right)\left(\beta -\gamma \right)\left(\gamma -\alpha \right)$
Option C: $\left(\alpha -\beta \right)\left(\beta -\gamma \right)\left(\gamma -\alpha \right)\left(\alpha +\beta +\gamma \right)$
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• 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: $\left[\begin{array}{ccc}-6& 2& 1\\ 4& -2& 1\\ 3& 1& -6\end{array}\right]$
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• 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
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• Question 99
Geometrically, Re (z2i) = 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
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• 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
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• 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: $\frac{1}{2}$
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• 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: $\frac{5}{9}$
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• 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
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• Question 104
For two events A and B, let $P\left(A\right)=\frac{1}{2},$ $P\left(A\cup B\right)=\frac{2}{3}$ and $P\left(A\cap B\right)=\frac{1}{6}$.  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: $\frac{1}{2}$
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• 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
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• 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
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• 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:
Option D:
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• Question 108
Let A and B be two events with P(A) = $\frac{1}{3}$ , P(B) = $\frac{1}{6}$ and $P\left(A\cap B\right)=\frac{1}{12}$. 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: $\frac{1}{10}$
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• 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: $\frac{25}{54}$
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• 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: $\frac{13}{243}$
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• 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: $\frac{2}{9}$
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• Question 112
It is given that $\overline{X}$ = 10, $\overline{Y}$ = 90, σX = 3, σY = 12 and rXY = 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
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• 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: $\frac{1}{9}$
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• 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
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• 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: $\frac{n}{2}$
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• 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: $\frac{\left(m-1\right)p}{1+mp}$
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• Question 117
If xand 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: ${x}_{1}+{x}_{2}<2\left(\sqrt{{x}_{1}{x}_{2}}+1\right)$
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• 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
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• 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
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• 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
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