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# NDA II 2015 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
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: $x=\frac{11±\sqrt{5}}{2}$
Option B: $x=\frac{9±\sqrt{5}}{2}$
Option C: $x=\frac{11±\sqrt{3}}{2}$
Option D: $x=\frac{9±\sqrt{3}}{2}$
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 ≠ bc, 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
$\left|\begin{array}{cc}{\mathrm{cos}}^{2}\frac{\alpha }{2}& {\mathrm{sin}}^{2}\frac{\alpha }{2}\\ {\mathrm{sin}}^{2}\frac{\beta }{2}& {\mathrm{cos}}^{2}\frac{\beta }{2}\end{array}\right|\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$
where α, β are complementary angles
1. The value of the determinant is $\frac{1}{\sqrt{2}}\mathrm{cos}\left(\frac{\alpha -\beta }{2}\right)$ .
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 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 then which of the following is/are correct ?
1.
2.
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.  are always disjoint.
2.  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: kn
Option B: k–1
Option C: kn
Option D: nk
VIEW SOLUTION
• Question 14
The value of the infinite product ${6}^{\frac{1}{2}}×{6}^{\frac{1}{2}}×{6}^{\frac{3}{8}}×{6}^{\frac{1}{4}}×...$ is
Option A: 6
Option B: 36
Option C: 216
Option D: $\infty$
VIEW SOLUTION
• Question 15
If the roots of the equation x2nx + 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 50th 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 , ${\left(\sqrt{x}+\frac{1}{3{x}^{2}}\right)}^{10}$ 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 , $\frac{{\mathrm{sin}}^{3}A+\mathrm{sin}3A}{\mathrm{sin}A}+\frac{{\mathrm{cos}}^{3}A-\mathrm{cos}3A}{\mathrm{cos}A},$ we get:
Option A: sin3A
Option B: cos3A
Option C: sinA + cosA
Option D: 3
VIEW SOLUTION
• Question 21
The value of $\mathrm{tan}\left(2{\mathrm{tan}}^{-1}\frac{1}{5}-\frac{\mathrm{\pi }}{4}\right)$ 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  , 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:

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. $\left|AB\right|=±47$
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: (ab) (bc) (ca)
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 ${\mathrm{tan}}^{2}\left(\frac{x+y}{2}\right)+{\mathrm{tan}}^{2}\left(\frac{x-y}{2}\right)$ is equal to:
Option A: $\frac{{a}^{4}+{b}^{4}+4{b}^{2}}{{a}^{2}{b}^{2}+{b}^{4}}$
Option B: $\frac{{a}^{4}-{b}^{4}+4{b}^{2}}{{a}^{2}{b}^{2}+{b}^{4}}$
Option C: $\frac{{a}^{4}-{b}^{4}+4{a}^{2}}{{a}^{2}{b}^{2}+{a}^{4}}$
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 xr and xr+1 are equal in the expansion, then r is equal to:
Option A: n
Option B:
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: 22n–1
Option B: 4n–1
Option C: 2 × 4n
Option D: None of the above
VIEW SOLUTION
• Question 34
The nth 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 abc 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

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

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   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${\mathrm{log}}_{8}m+{\mathrm{log}}_{8}\frac{1}{6}=\frac{2}{3},$ 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  is
Option A: $\frac{{c}^{2}}{ab}$
Option B: $\frac{2{c}^{2}}{ab}$
Option C: $\frac{{c}^{2}}{2ab}$
Option D: $\frac{{c}^{2}}{4ab}$
VIEW SOLUTION
• Question 47
If a line is perpendicular to the line 5xy = 0 and forms a triangle of area 5 square units with co-ordinate axes, then its equation is
Option A: $x+5y±5\sqrt{2}=0$
Option B: $x-5y±5\sqrt{2}=0$
Option C: $5x+y±5\sqrt{2}=0$
Option D: $5x-y±5\sqrt{2}=0$
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 $3x\mathrm{cos}\varphi +5y\mathrm{sin}\varphi =15$ 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:
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 $3\stackrel{^}{i}+4\stackrel{^}{j}$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 $\stackrel{\to }{OA}+\stackrel{\to }{OB}+\stackrel{\to }{OC}+\stackrel{\to }{OD}$  is equal to
Option A: $4\stackrel{\to }{OP}$
Option B: $2\stackrel{\to }{OP}$
Option C: $\stackrel{\to }{OP}$
Option D: Null vector
VIEW SOLUTION
• Question 63
If  $\stackrel{\to }{b}$and $\stackrel{\to }{c}$ are the position vectors of the point B are C, respectively, then the position vector of the point D such that $\stackrel{\to }{BD}=4\stackrel{\to }{BC}$ is
Option A: $4\left(\stackrel{\to }{c}-\stackrel{\to }{b}\right)$
Option B: $–4\left(\stackrel{\to }{c}-\stackrel{\to }{b}\right)$
Option C: $4\stackrel{\to }{c}-3\stackrel{\to }{b}$
Option D: $4\stackrel{\to }{c}+3\stackrel{\to }{b}$
VIEW SOLUTION
• Question 64
If the position vector $\stackrel{\to }{a}$ of the point (5, n) is such that $\left|\stackrel{\to }{a}\right|=13$, 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 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 $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$:

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 lie on a plane, where 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 are such that Which of the above is/are correct?

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  ${\int }_{a}^{b}\frac{{x}^{7}+\mathrm{sin}x}{\mathrm{cos}}dx$ where a + b = 0 is
Option A: 2basin (ba)
Option B: a + 3bcos (ba)
Option C: sin a – (ba) cos b
VIEW SOLUTION
• Question 71
If , $f\left(x\right)=\sqrt{25-{x}^{2}},$then what is  $\underset{x\to 1}{\mathrm{lim}}\frac{f\left(x\right)-f\left(1\right)}{x-1}$ equal to ?
Option A: $\frac{1}{5}$
Option B: $\frac{1}{24}$
Option C: $\sqrt{24}$
Option D: $-\frac{1}{24}$
VIEW SOLUTION
• Question 72
Consider the function

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 $f\left(x\right)=\frac{1-\mathrm{sin}x+\mathrm{cos}x}{1+\mathrm{sin}x+\mathrm{cos}x}$ 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:

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   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: $\frac{1+\mathrm{cos}x}{\left(x+\mathrm{sin}x\right)\left(1-\mathrm{sin}x\right)}$
Option B: $\frac{1-\mathrm{cos}x}{\left(x+\mathrm{sin}x\right)\left(1+\mathrm{sin}x\right)}$
Option C: $\frac{1-\mathrm{cos}x}{\left(x-\mathrm{sin}x\right)\left(1+\mathrm{cos}x\right)}$
Option D: $\frac{1+\mathrm{cos}x}{\left(x-\mathrm{sin}x\right)\left(1-\mathrm{cos}x\right)}$
VIEW SOLUTION
• Question 78
If $y=co{t}^{-1}\left[\frac{\sqrt{1+\mathrm{sin}x}+\sqrt{1-\mathrm{sin}x}}{\sqrt{1+\mathrm{sin}x}-\sqrt{1-\mathrm{sin}x}}\right],$  where  $0 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 xayb = (xy) 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
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• Question 81
If  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: ${\left(x+\frac{7}{2}\right)}^{\frac{1}{3}}$
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• 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: $\left|\frac{b}{a}\right|$
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• Question 83
${\int }_{0}^{2\pi }{\mathrm{sin}}^{5}\left(\frac{x}{4}\right)dx$ is equal to
Option A: $\frac{8}{15}$
Option B: $\frac{16}{15}$
Option C: $\frac{32}{15}$
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• Question 84
If $f\left(x\right)=\frac{\mathrm{sin}\left({e}^{x-2}-1\right)}{\mathrm{In}\left(x-1\right)},$  then  $\underset{x\to 2}{\mathrm{lim}}f\left(x\right)$ is equal to
Option A: –2
Option B: –1
Option D: 1
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• Question 85
Consider the following statements:
1. f(x) = ln x is an increasing function on (0, ∞).
2. 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
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• 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: $\frac{1}{{s}^{4}}$
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• Question 87
Consider the following statements:
Statement 1: The function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ such that f(x) = x3 for all $x\in \mathrm{ℝ}\phantom{\rule{0ex}{0ex}}$ is one-one.
Statement 2: $f\left(a\right)=f\left(b\right)⇒a=b$ for all  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.
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• Question 88
$\int \frac{dx}{1+{e}^{-x}}$ is equal to where c is the constant of integration
Option A: 1 + ex + c
Option B: ln (1 + ex) + c
Option C: ln (1 + ex) + c
Option D: 2 ln (1 + ex) + c
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• Question 89
${\int }_{-1}^{1}x\left|x\right|dx$ is equal to
Option B: $\frac{2}{3}$
Option C: 2
Option D: –2
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• 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
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• Question 91
Consider the function $f\left(x\right)={\left(\frac{1}{x}\right)}^{2{x}^{2}},$ 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: $\frac{1}{e}$
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• Question 92
Consider the function $f\left(x\right)={\left(\frac{1}{x}\right)}^{2{x}^{2}},$ 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: $\frac{1}{e}$
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• 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: $-\frac{3}{5}$
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• 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
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• 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, ∞)
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• 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:
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• Question 97
Consider the integrals

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

What is the value of B?
Option A: $\frac{\pi }{4}$
Option B: $\frac{\pi }{2}$
Option C: $\frac{3\pi }{4}$
Option D: $\pi$
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• Question 99
Consider the function

Which is continuous everywhere.

The value of A is
Option A: 1
Option C: –1
Option D: –2
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• Question 100

Which is continuous everywhere.

The value of B is
Option A: 1
Option C: –1
Option D: –2
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• 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
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• 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: ${\mathrm{sin}}^{-1}y={\mathrm{sin}}^{-1}x+c$
Option B: $2{\mathrm{sin}}^{-1}y=\sqrt{1-{x}^{2}}+{\mathrm{sin}}^{-1}x+c$
Option C: $2{\mathrm{sin}}^{-1}y=x\sqrt{1-{x}^{2}}+{\mathrm{sin}}^{-1}x+c$
Option D: $2{\mathrm{sin}}^{-1}y=x\sqrt{1-{x}^{2}}+{\mathrm{cos}}^{-1}x+c$
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• 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: $2xy\frac{dy}{dx}+{x}^{2}+{y}^{2}=0$
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• 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
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• 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
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• 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
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• 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: $\frac{11}{16}$
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• 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
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• 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: $\frac{3}{32}$
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• Question 111
The geometric mean of the observations x1, x2, x3,…….,xn is G1. The geometric mean of the observations y1, y2, y3,……yn is G2. The geometric mean of the observations $\frac{{x}_{1}}{{y}_{1}},\frac{{x}_{2}}{{y}_{2}},\frac{{x}_{3}}{{y}_{3}},...\frac{{x}_{n}}{{y}_{n}}$ is
Option A: G1G2
Option B: ln (G1G2)
Option C: $\frac{{G}_{1}}{{G}_{2}}$
Option D: $\mathrm{In}\left(\frac{{G}_{1}}{{G}_{2}}\right)$
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• Question 112
The arithmetic mean of 1, 8, 27, 64, … Up to n terms is given by
Option A: $\frac{n\left(n+1\right)}{2}$
Option B: $\frac{n\left(n+1{\right)}^{2}}{2}$
Option C: $\frac{n\left(n+1{\right)}^{2}}{4}$
Option D: $\frac{{n}^{2}\left(n+1{\right)}^{2}}{4}$
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• 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
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• Question 114
If , 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: $\frac{2}{3}$
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• Question 116
The problem in statistics is given to three students A, B and C whose chances of solving it independently are  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: $\frac{3}{4}$
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• 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: $\frac{19}{52}$
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• 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: $\frac{1}{5}$
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• 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
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• 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
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