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# NDA I 2017 Mathematic

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 S be the set of all persons living in Delhi. We say that x, y in S are related if the y were born in Delhi on the same day. Which one of the following is correct?
Option A: The relation is an equivalent relation
Option B: The relation is not reflective but it is symmetric and transitive
Option C: The relation is not symmetric bout it is reflexive and transitive
Option D: The relation is not transitive but it is reflexive ands ymmetric
VIEW SOLUTION
• Question 2
Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing two or three elements is
Option A: 45
Option B: 120
Option C: 165
Option D: 330
VIEW SOLUTION
• Question 3
3. The value of  ${i}^{2n}+{i}^{2n+1}+{i}^{2n+2}+{i}^{2n+3}$, where ,is $i=\sqrt{-1}$
Option A: 0
Option B: 1
Option C: i
Option D: i
VIEW SOLUTION
• Question 4
If the difference between the roots of the equation ${x}^{2}+kx+1=0$ is strictly less than $\sqrt{5}$, where |k| ≥ 2, then k can be any element of the interval
Option A: $\left(-3,-2\right]\cup \left[2,3\right)$
Option B: (–3, 3)
Option C: $\left[-3,-2\right]\cup \left[2,3\right]$
Option D: None of these
VIEW SOLUTION
• Question 5
If the roots of the equation ${x}^{2}+px+q=0$ are in the same ratio as those of the equation ${x}^{2}+lx+m=0$, then which one of the following is correct?
Option A: ${p}^{2}m={l}^{2}q$
Option B: ${m}^{2}p={l}^{2}q$
Option C: ${m}^{2}p={q}^{2}l$
Option D: ${m}^{2}{p}^{2}={l}^{2}q$
VIEW SOLUTION
• Question 6
The value of ${\left(\frac{-1+i\sqrt{3}}{2}\right)}^{n}+{\left(\frac{-1-i\sqrt{3}}{2}\right)}^{n}$where n is not a multiple of 3 and $i=\sqrt{-1}$ is
Option A: 1
Option B: –1
Option C: i
Option D: i
VIEW SOLUTION
• Question 7
Three-digit numbers are formed from the digits 1, 2 and 3 in such a way that the digits are not repeated. What is the sum of such 3 digit numbers?
Option A: 1233
Option B: 1322
Option C: 1323
Option D: 1332
VIEW SOLUTION
• Question 8
What is the sum of the series 0.3 + 0.33 + 0.333 +….n terms?
Option A: $\frac{1}{3}\left[n-\frac{1}{9}\left(1-\frac{1}{{10}^{n}}\right)\right]$
Option B: $\frac{1}{3}\left[n-\frac{2}{9}\left(1-\frac{1}{{10}^{n}}\right)\right]$
Option C: $\frac{1}{3}\left[n-\frac{1}{3}\left(1-\frac{1}{{10}^{n}}\right)\right]$
Option D: $\frac{1}{3}\left[n-\frac{1}{9}\left(1+\frac{1}{{10}^{n}}\right)\right]$
VIEW SOLUTION
• Question 9
If 1, ω, ω2 are the cube roots of unity, then $\left(1+\omega \right)\left(1+{\omega }^{2}\right)\left(1+{\omega }^{3}\right)\left(1+\omega +{\omega }^{2}\right)$ is equal to
Option A: –2
Option B: –1
Option D: 2
VIEW SOLUTION
• Question 10
If the sum of m terms of an AP is n and the sum of n terms is m, then the sum of (m + n) term is
Option A: mn
Option B: m + n
Option C: 2(m + n)
Option D: – (m + n)
VIEW SOLUTION
• Question 11
The modulus and the principal argument of the complex numbers $\frac{1+2i}{1-{\left(1-i\right)}^{2}}$ are respectively
Option A: 1, 0
Option B: 1, 1
Option C: 2, 0
Option D: 2, 1
VIEW SOLUTION
• Question 12
If the graph of a quadratic polynomial lies entirely above the x-axis, then which one of the following is correct?
Option A: Both the roots are real
Option B: One root is real and the other is complex
Option C: Both the roots are complex
Option D: Cannot say
VIEW SOLUTION
• Question 13
If $|z+4|\le 3$, then the maximum value of |z + 1| is
Option B: 4
Option C: 6
Option D: 10
VIEW SOLUTION
• Question 14
The number of roots of the equation ${z}^{2}=2\overline{z}$ is
Option A: 2
Option B: 3
Option C: 4
VIEW SOLUTION
• Question 15
If cot α and cot β are the roots of the equation ${x}^{2}+bx+c=0$ with b ≠ 0 then the value of cot (α + β) is
Option A: $\frac{c-1}{b}$
Option B: $\frac{1-c}{b}$
Option C: $\frac{b}{c-1}$
Option D: $\frac{b}{1-c}$
VIEW SOLUTION
• Question 16
The sum of the roots of the equation ${x}^{2}+bx+c=0$ (where b and c are non-zero) is equal to the sum of the reciprocals of their square. Then are in
Option A: AP
Option B: GP
Option C: HP
Option D: None of the above
VIEW SOLUTION
• Question 17
The sum of the roots of the equation $a{x}^{2}+x+c=0$ (where a and c are non-zero) is equal to the sum of the reciprocals of their squares. Then a, ca2, c2 are in
Option A: AP
Option B: GP
Option C: HP
Option D: None of the above
VIEW SOLUTION
• Question 18
The value of  [C(7, 0) + C(7, 1)] + [C(7, 1) + C(7, 2) ] + ……+ [C(7, 6) + C(7, 7)] is
Option A: 254
Option B: 255
Option C: 256
Option D: 257
VIEW SOLUTION
• Question 19
The number of different words (eight-letter words) ending and beginning with a consonant which can be made out of the letters of the word ‘EQUATION’ is
Option A: 5200
Option B: 4320
Option C: 3000
Option D: 2160
VIEW SOLUTION
• Question 20
The fifth term of an AP of n terms, whose sum is n2 – 2n, is
Option A: 5
Option B: 7
Option C: 8
Option D: 15
VIEW SOLUTION
• Question 21
The sum of all the two-digit odd numbers is
Option A: 2475
Option B: 2530
Option C: 4905
Option D: 5049
VIEW SOLUTION
• Question 22
The sum of the first n terms of the series  $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+....$ is equal to
Option A: ${2}^{n}-n-1$
Option B: 1 – 2–n
Option C: ${2}^{-n}+n-1$
Option D: ${2}^{n}-1$
VIEW SOLUTION
• Question 23
Consider the following in respect of sets A and B:
1. $\left(A-B\right)\cup B=A$
2. $\left(A-B\right)\cup A=A$
3. $\left(A-B\right)\cap B=\varphi$
4. $A\subseteq B⇒A\cup B=B$
Which of the above are correct?
Option A: 1, 2 and 3
Option B: 2, 3 and 4
Option C: 1, 3 and 4
Option D: 1, 2 and 4
VIEW SOLUTION
• Question 24
In the binary equation ${\left(1p101\right)}_{2}+{\left(10q1\right)}_{2}={\left(100r00\right)}_{2}$ where p, q and r are binary digits, what are the possible value of p, q and r, respectively?
Option A: 0, 1, 0
Option B: 1, 1, 0
Option C: 0, 0, 1
Option D: 1, 0, 1
VIEW SOLUTION
• Question 25
If , then S is
Option A: {– 1}
Option B: {0}
Option C: {1}
Option D: An empty set
VIEW SOLUTION
• Question 26
The expansion of  is done in the descending powers of x. If the sum of the fifth and sixth terms is zero, then $\frac{x}{y}$ is equal to
Option A: $\frac{n-5}{6}$
Option B: $\frac{n-4}{5}$
Option C: $\frac{5}{n-4}$
Option D: $\frac{6}{n-5}$
VIEW SOLUTION
• Question 27
If  $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ and $\mathrm{det}\left({A}^{3}\right)=125$ then α is equal to
Option A: ± 1
Option B: ± 2
Option C: ± 3
Option D: ± 5
VIEW SOLUTION
• Question 28
If B is a non-singular matrix and A is a square matrix, then the value of $\mathrm{det}\left({B}^{-1}AB\right)$ is equal to
Option A: det(B)
Option B: det(A)
Option C: det(B–1)
Option D: det(A–1)
VIEW SOLUTION
• Question 29
If a ≠ b ≠ c, then one value of x which satisfies the equation $\left|\begin{array}{ccc}0& x-a& x-b\\ x+a& 0& x-c\\ x+b& x+c& 0\end{array}\right|=0$ is given by  ;
Option A: a
Option B: b
Option C: c
VIEW SOLUTION
• Question 30
$A=\left[\begin{array}{cc}\mathrm{cos}\alpha & \mathrm{sin}\alpha \\ -\mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]$ then what is AAT equal to (where AT is the transpose of A)?
Option A: Null matrix
Option B: Identity matrix
Option C: A
Option D: –A
VIEW SOLUTION
• Question 31
The equations
x + 2y + 3z = 1
2x + y + 3z = 2
5x + 5y + 9z = 4
Option A: have the unique solution
Option B: have Infinitely many solutions
Option C: are in inconsistent
Option D: None of these
VIEW SOLUTION
• Question 32
.
If AB = C, then what is A2 equal to?
Option A:
Option B: $\left[\begin{array}{cc}4& -4\\ 8& -16\end{array}\right]$
Option C:
Option D:
VIEW SOLUTION
• Question 33
What is the value of the determinant $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+xyz& 1\\ 1& 1& 1+xyz\end{array}\right|$?
Option A: 1 + x + y + z
Option B: 2xyz
Option C: x2 y2 z2
Option D: 2x2 y2 z2
VIEW SOLUTION
• Question 34
If then $\left|\begin{array}{ccc}x& y& 0\\ 0& x& y\\ y& 0& x\end{array}\right|$ = 0, then which one of the following is correct?
Option A: $\frac{x}{y}$ is one of the cube root of unity
Option B: x is one of the cube roots of Unity
Option C: y is one of the cube roots of unity
Option D: $\frac{x}{y}$ is one of the cube root of –1
VIEW SOLUTION
• Question 35
Consider the set A of all matrices of order 3 × 3 with entire 0 or 1 only. Let B be the subset of A consisting of all matrices whose determinant is 1. Let C be the subset of A consisting of all matrices whose determinant is –1. Then which one of the following is correct?
Option A: C is empty
Option B: B has as many elements as C
Option C: A = BC
Option D: B has thrice as many elements as C
VIEW SOLUTION
• Question 36
If ; then what is A3 equal to?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 37
What is the order of $\left[\begin{array}{ccc}x& y& z\end{array}\right]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$?
Option A: 3 × 1
Option B: 1 × 1
Option C: 1 × 3
Option D: 3 × 3
VIEW SOLUTION
• Question 38
If $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, then the value of A4 is
Option A: $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
Option B: $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
Option C: $\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$
Option D: $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$
VIEW SOLUTION
• Question 39
If , then is equal to
Option A: $\frac{1}{\sqrt{10}}$
Option B: $-\sqrt{\frac{3}{10}}$
Option C: $\frac{\sqrt{3}}{\sqrt{10}}$
Option D: None of these
VIEW SOLUTION
• Question 40
What is equal to?
Option B: 1
Option C: 2
Option D: 4
VIEW SOLUTION
• Question 41
From the top of a lighthouse, 100 m high, the angle of depression of a boat is ${\mathrm{tan}}^{-1}\left(\frac{5}{12}\right)$. What is the distance between the boat and the lighthouse?
Option A: 120 m
Option B: 180 m
Option C: 240 m
Option D: 360 m
VIEW SOLUTION
• Question 42
The maximum value of  in the interval is attained at
Option A: $\frac{\pi }{12}$
Option B: $\frac{\pi }{6}$
Option C: $\frac{\pi }{3}$
Option D: $\frac{\pi }{2}$
VIEW SOLUTION
• Question 43
If $K=\mathrm{sin}\left(\frac{\pi }{18}\right)\mathrm{sin}\left(\frac{5\pi }{18}\right)\mathrm{sin}\left(\frac{7\pi }{18}\right)$, then what is the value of K?
Option A: $\frac{1}{2}$
Option B: $\frac{1}{4}$
Option C: $\frac{1}{8}$
Option D: $\frac{1}{16}$
VIEW SOLUTION
• Question 44
The expression   is equal to
Option A: $\mathrm{tan}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)$
Option B: $\mathrm{cot}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)$
Option C: $\mathrm{sin}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)$
Option D: $\mathrm{cos}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)$$\mathrm{cos}\left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)$
VIEW SOLUTION
• Question 45
If sin θ = 3sin (θ + 2α), then the value of tan(θ + α) + 2 tanα is equal to
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 46
What is the value of tan18°?
Option A: $\frac{\sqrt{5}-1}{\sqrt{10+2\sqrt{5}}}$
Option B: $\frac{\sqrt{5}-1}{\sqrt{10+\sqrt{5}}}$
Option C: $\frac{\sqrt{10+2\sqrt{5}}}{\sqrt{5-1}}$
Option D: $\frac{\sqrt{10+\sqrt{5}}}{\sqrt{5-1}}$
VIEW SOLUTION
• Question 47
Let x, y, z be positive real numbers such that x, y, z are in GP and are in AP. Then which one of the following is correct?
Option A: x = y = z
Option B: xz = 1
Option C: x ≠ y and y = z
Option D: x = y and y ≠ z
VIEW SOLUTION
• Question 48
If tan(α + β) = 2 and tan (α – β) = 1, then tan (2α) is equal to
Option A: –3
Option B: –2
Option C: $-\frac{1}{3}$
Option D: 1
VIEW SOLUTION
• Question 49
Consider the following of triangle ABC:
(a) $\mathrm{sin}\left(\frac{B+C}{2}\right)=\mathrm{cos}\left(\frac{A}{2}\right)$
(b) $\mathrm{tan}\left(\frac{B+C}{2}\right)=\mathrm{cot}\left(\frac{A}{2}\right)$
(c) sin (B + C) = cos A
(d) tan (B + C) = –cot A

Which of the above are correct?
Option A: a and c
Option B: a and b
Option C: a and d
Option D: b and c
VIEW SOLUTION
• Question 50
If sec θ – cosec θ = $\frac{4}{3}$, then what is (sin θ – cos θ) equal to?
Option A: –2 only
Option B: $\frac{1}{2}$ only
Option C: Both –2 and $\frac{1}{2}$
Option D: Neither $\frac{1}{2}$ nor – 2
VIEW SOLUTION
• Question 51
If a vertex of a triangle is (1, 1) and the midpoints of two sides of the triangle through this vertex are (–1, 2) and (–3, 2), then the centroid of the triangle is
Option A: $\left(-\frac{1}{3},\frac{7}{3}\right)$
Option B: $\left(-1,\frac{7}{3}\right)$
Option C: $\left(\frac{1}{3},\frac{7}{3}\right)$
Option D: $\left(1,\frac{7}{3}\right)$
VIEW SOLUTION
• Question 52
The incentre of a triangle with vertices A(1, $\sqrt{3}$), B(0, 0), and C(2, 0) is
Option A: $\left(1,\frac{\sqrt{3}}{2}\right)$
Option B:
Option C:
Option D: $\left(1,\frac{1}{\sqrt{3}}\right)$
VIEW SOLUTION
• Question 53
If the three consecutive vertices of a parallelogram are (–2,  –1), (1, 0) and (4, 3), then what are the coordinates of the fourth vertex?
Option A: (1, 2)
Option B: (1, 0)
Option C: (0, 0)
Option D: (1, –1)
VIEW SOLUTION
• Question 54
The two circles ${x}^{2}+{y}^{2}={r}^{2}$ and ${x}^{2}+{y}^{2}-10x+16=0$ intersect at two distinct points. Then which one of the following is correct?
Option A: 2 < r < 8
Option B: r = 2 or r = 8
Option C: r < 2
Option D: r > 2
VIEW SOLUTION
• Question 55
What is the equation of the circle which passes through the points (3, –2) and (–2, 0) and having its centre on the line 2xy – 3 = 0?
Option A: ${x}^{2}+{y}^{2}+3x+2=0$
Option B: ${x}^{2}+{y}^{2}+3x+12y+2=0$
Option C: ${x}^{2}+{y}^{2}+2x=0$
Option D: ${x}^{2}+{y}^{2}=5$
VIEW SOLUTION
• Question 56
What is the ratio in which the point  divides the line joining the points A(–2, –2) and B(2, –4)?
Option A: 1 : 3
Option B: 3 : 4
Option C: 1 : 2
Option D: 2 : 3
VIEW SOLUTION
• Question 57
What is the equation of the ellipse having foci (±2, 0) and the eccentricity $\frac{1}{4}$?
Option A: $\frac{{x}^{2}}{64}+\frac{{y}^{2}}{60}=1$
Option B: $\frac{{x}^{2}}{60}+\frac{{y}^{2}}{64}=1$
Option C: $\frac{{x}^{2}}{20}+\frac{{y}^{2}}{24}=1$
Option D: $\frac{{x}^{2}}{24}+\frac{{y}^{2}}{20}=1$
VIEW SOLUTION
• Question 58
What is the equation of the straight line parallel to 2x + 3y + 1 = 0 and passes through the point (–1, 2)?
Option A: 2x + 3y – 4 = 0
Option B: 2x + 3y – 5 = 0
Option C: x + y – 1 = 0
Option D: 3x – 2y + 7 = 0
VIEW SOLUTION
• Question 59
What is the acute angle between the pair of straight lines ?
Option A: ${\mathrm{tan}}^{-1}\left(\frac{1}{2\sqrt{6}}\right)$
Option B: ${\mathrm{tan}}^{-1}\left(\frac{1}{\sqrt{2}}\right)$
Option C: ${\mathrm{tan}}^{-1}\left(3\right)$
Option D: 4${\mathrm{tan}}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
VIEW SOLUTION
• Question 60
If the centroid of a triangle formed by (7, x), (y, –6) and (9, 10) is (6, 3), then the values of x and y are respectively
Option A: 5, 2
Option B: 2, 5
Option C: 1, 0
Option D: 0, 0
VIEW SOLUTION
• Question 61
A straight line with direction cosines  is
Option A: parallel to x-axis
Option B: parallel to y-axis
Option C: parallel to z-axis
Option D: equally inclined to all the axes
VIEW SOLUTION
• Question 62
(0, 0 , 0), (a, 0, 0), (0, b, 0) and (0, 0, c) are four distinct points. What are the coordinates of the point, which is equidistant from the four points?
Option A:
Option B: (a, b, c )
Option C:
Option D:
VIEW SOLUTION
• Question 63
The point P(3, 2, 4), Q(4, 5, 2), R(5, 8, 0) and S(2, –1, 6) are
Option A: vertices of a rhombus which is not a square
Option B: non-coplanar
Option C: collinear
Option D: coplanar but not collinear
VIEW SOLUTION
• Question 64
The line passing through the points (1, 2, –1) and (3, –1, 2) meets the yz-plane at which one of the following points?
Option A: $\left(0,-\frac{7}{2},\frac{5}{2}\right)$
Option B: $\left(0,\frac{7}{2},\frac{1}{2}\right)$
Option C: $\left(0,-\frac{7}{2},-\frac{5}{2}\right)$
Option D: $\left(0,\frac{7}{2},-\frac{5}{2}\right)$
VIEW SOLUTION
• Question 65
Under which one of the following conditions, are the lines x = ay + b; z = cy + d and x = ey + f; z = gy + h perpendicular?
Option A: ae + cg – 1 = 0
Option B: ae + bf – 1 = 0
Option C: ae + cg + 1 = 0
Option D: ag + ce – 1 = 0
VIEW SOLUTION
• Question 66
If , are three coplanar vectors and $|\stackrel{\to }{c}|=\sqrt{6}$, then which one of the following is correct?
Option A: m = 2 and n = ±1
Option B: m = ±2 and n = –1
Option C: m = 2 and n = –1
Option D: m = ±2 and n = 1
VIEW SOLUTION
• Question 67

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin. What is $\stackrel{\to }{OA}+\stackrel{\to }{OB}+\stackrel{\to }{OC}+\stackrel{\to }{OD}$ equal to?

Option A: $2\stackrel{\to }{OP}$
Option B: $4\stackrel{\to }{OP}$
Option C: $6\stackrel{\to }{OP}$
Option D: $8\stackrel{\to }{OP}$
VIEW SOLUTION
• Question 68
ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?
Option A: $\stackrel{\to }{BA}+\stackrel{\to }{CD}=\stackrel{\to }{AC}+\stackrel{\to }{DB}$
Option B: $\stackrel{\to }{BA}+\stackrel{\to }{CD}=\stackrel{\to }{BD}+\stackrel{\to }{CA}$
Option C: $\stackrel{\to }{BA}+\stackrel{\to }{CD}=\stackrel{\to }{AC}+\stackrel{\to }{BD}$
Option D: $\stackrel{\to }{BA}+\stackrel{\to }{CD}=\stackrel{\to }{BC}+\stackrel{\to }{AD}$
VIEW SOLUTION
• Question 69
If , then which one of the following is correct?
Option A: $\stackrel{\to }{a},\stackrel{\to }{b},\stackrel{\to }{c}$ are orthogonal in pairs and $|\stackrel{\to }{a}|=|\stackrel{\to }{c}|\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}|\stackrel{\to }{b}|=1$
Option B: $\stackrel{\to }{a},\stackrel{\to }{b},\stackrel{\to }{c}$ are non-orthogonal to each other
Option C: $\stackrel{\to }{a},\stackrel{\to }{b},\stackrel{\to }{c}$ are orthogonal in pairs but $|\stackrel{\to }{a}|\ne |\stackrel{\to }{c}|$
Option D: $\stackrel{\to }{a},\stackrel{\to }{b},\stackrel{\to }{c}$ are orthogonal in pairs but $|\stackrel{\to }{b}|\ne 1$
VIEW SOLUTION
• Question 70
If are perpendicular, then what is the value of  λ?
Option A: 2
Option B: 3
Option C: 4
Option D: 5
VIEW SOLUTION
• Question 71
What is $\underset{x\to 0}{\mathrm{lim}}\frac{{e}^{x}-\left(1+x\right)}{{x}^{2}}$ equal to?
Option B: $\frac{1}{2}$
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 72
What is ${\int }_{0}^{\frac{\pi }{2}}\frac{d\theta }{1+\mathrm{cos}\theta }$ equal to?
Option A: $\frac{1}{2}$
Option B: 1
Option C: $\sqrt{3}$
Option D: None of these
VIEW SOLUTION
• Question 73
What is $\int \frac{dx}{x\left({x}^{7}+1\right)}$ equal to?
Option A: $\frac{1}{2}\mathrm{In}\left|\frac{{x}^{7}-1}{{x}^{7}+1}\right|+c$
Option B: $\frac{1}{7}\mathrm{In}\left|\frac{{x}^{7}+1}{{x}^{7}}\right|+c$
Option C: $\mathrm{In}\left|\frac{{x}^{7}-1}{7x}\right|+c$
Option D: $\frac{1}{7}\mathrm{In}\left|\frac{{x}^{7}}{{x}^{7}+1}\right|+c$
VIEW SOLUTION
• Question 74
The function f : X → Y defined by f(x) = cos x, where xX, is one-one and onto if X and Y are respectively equal to
Option A: [0, π] and [–1, 1]
Option B: $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ and [–1, 1]
Option C: [0, π] and [–1, 1]
Option D: [0, π] and [0, 1]
VIEW SOLUTION
• Question 75
If (x) = $\frac{x}{x-1}$, then what is $\frac{f\left(a\right)}{f\left(a+1\right)}$ equal to?

Option A: $f\left(-\frac{a}{a+1}\right)$
Option B: $f\left({a}^{2}\right)$
Option C: $f\left(\frac{1}{a}\right)$
Option D:  f(–a)
VIEW SOLUTION
• Question 76
What is $\int \frac{\left({x}^{e-1}+{e}^{x-1}\right)dx}{{x}^{e}+{e}^{x}}$ equal to?
Option A: $\frac{{x}^{2}}{2}+c$
Option B: In (x + e) + c
Option C:
Option D:
VIEW SOLUTION
• Question 77
Let $f:\left[-6,6\right]\to \mathrm{ℝ}$ be defined by $f\left(x\right)={x}^{2}-3$. Consider the following:

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 78
Let f(x) = px + q and g(x) = mx + n. Then f(g(x)) = g(f(x)) is equivalent to
Option A: f(p)  = g(m)
Option B: f(q)  = g(n)
Option C: f(n)  = g(q)
Option D: f(m) = g(p)
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• Question 79
If $F\left(x\right)=\sqrt{9-{x}^{2}}$ = then what is $\underset{x\to 1}{\mathrm{lim}}\frac{F\left(x\right)-F\left(1\right)}{x-1}$ equal to?
Option A: $-\frac{1}{4\sqrt{2}}$
Option B: $\frac{1}{8}$
Option C: $-\frac{1}{2\sqrt{2}}$
Option D: $\frac{1}{2\sqrt{2}}$
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• Question 80
What is $\frac{{d}^{2}x}{d{y}^{2}}$ equal to?
Option A: $-{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-3}$
Option B: ${\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{-1}{\left(\frac{dy}{dx}\right)}^{-2}$
Option C: $-\left(\frac{{d}^{2}y}{d{x}^{2}}\right){\left(\frac{dy}{dx}\right)}^{-3}$
Option D: ${\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{-1}$
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• Question 81
Let
If is

Option A: one-one and into
Option B: neither one-one nor onto
Option C: many one and onto
Option D: one-one and onto
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• Question 82
What is the length of the longest interval in which the function is increasing?
Option A: $\frac{\pi }{3}$
Option B: $\frac{\pi }{2}$
Option C: $\frac{3\pi }{2}$
Option D: π
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• Question 83
If x dy = y(dx + y dy); y(1) = 1 and y(x) > 0, then what is y(–3) equal to?
Option A: 3
Option B: 2
Option C: 1
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• Question 84
What is the maximum value of the function f(x) = 4 sin2 x + 1?
Option A: 5
Option B: 3
Option C: 2
Option D: 1
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• Question 85
Let f(x) be an indefinite integral of sin2 x.
Consider the following statements:

Statement 1: The function f(x) satisfies f(x + π) = f(x) for all real x.
Statement 2: ${\mathrm{sin}}^{2}\left(x+\pi \right)={\mathrm{sin}}^{2}x$ for all real x.

Which one of the following is correct with respect to 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 but 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 86
What are the degree and order respectively of the differential equation $y=x{\left(\frac{dy}{dx}\right)}^{2}+{\left(\frac{dx}{dy}\right)}^{2}?$
Option A: 1, 2
Option B: 2, 1
Option C: 1, 4
Option D: 4, 1
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• Question 87
What is the differential equation corresponding to ${y}^{2}-2ay+{x}^{2}={a}^{2}$ by eliminating a?
Option A:
Option B:
Option C:
Option D:
Where $p=\frac{dy}{dx}$
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• Question 88
What is the general solution of the differential equation $ydx-\left(x+2{y}^{2}\right)dy=0$?
Option A: $x={y}^{2}+cy$
Option B: $x=2c{y}^{2}$
Option C: $x=2{y}^{2}+cy$
Option D: None of these
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• Question 89
Let f(x + y) = f(x) f(y) for all x and y. Then what is f ’(5) equal to [where f ’(x) is the derivative of f(x)]?
Option A: f(5) f ’(0)
Option B: f(5) – f ’(0)
Option C: f(5) f(0)
Option D: f(5) + f ’(0)
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• Question 90
If f(x) and g(x) are continuous functions satisfying f(x) = f(a – x) and g(x) + g(a – x) = 2, then what is ${\int }_{0}^{a}f\left(x\right)g\left(x\right)dx$ equal to?
Option A: ${\int }_{0}^{a}g\left(x\right)dx$
Option B: ${\int }_{0}^{a}f\left(x\right)dx$
Option C: $2{\int }_{0}^{a}f\left(x\right)dx$
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• Question 91
What is the solution of the differential equation $\mathrm{ln}\left(\frac{dy}{dx}\right)-a=0$?
Option A: $y=x{e}^{a}+c$
Option B: $x=y{e}^{a}+c$
Option C: y = ln x + c
Option D: x = ln y + c
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• Question 92
Let f(x) be defined as follows:

Which one of the following statements is correct with respect to the above function?
Option A: It is discontinuous at x = –2 but continuous at every other point.
Option B: It is continuous only in the interval (–3, –2).
Option C: It is discontinuous at x = 0 but continuous at every other point.
Option D: It is discontinuous at every point
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• Question 93
Consider the following statements
1. If both exist, then $\underset{x\to a}{\mathrm{lim}}\left\{f\left(x\right)g\left(x\right)\right\}$ exists.
2. If $\underset{x\to a}{\mathrm{lim}}\left\{f\left(x\right)g\left(x\right)\right\}$ exists, then both  and  must exist.

Which of the above statement 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 94
Which one of the following functions is neither even nor odd?
Option A: ${x}^{2}-1$
Option B: $x+\frac{3}{x}$
Option C: |x|
Option D: ${x}^{2}\left(x-3\right)$
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• Question 95
What is the derivative of ${\mathrm{log}}_{10}\left(5{x}^{2}+3\right)$ with respect to x?
Option A: $\frac{x{\mathrm{log}}_{10}e}{5{x}^{2}+3}$
Option B: $\frac{2x{\mathrm{log}}_{10}e}{5{x}^{2}+3}$
Option C: $\frac{10x{\mathrm{log}}_{10}e}{5{x}^{2}+3}$
Option D: $\frac{10x{\mathrm{log}}_{10}10}{5{x}^{2}+3}$
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• Question 96
Let $f\left(a\right)=\frac{a-1}{a+1}$.
Consider the following:
1. f(2a) = f(a) + 1
2. $f\left(\frac{1}{a}\right)=-f\left(a\right)$
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
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• Question 97
What is the maximum area of a triangle that can be inscribed in a circle of radius a?
Option A: $\frac{3{a}^{2}}{4}$
Option B: $\frac{{a}^{2}}{2}$
Option C: $\frac{3\sqrt{3}{a}^{2}}{4}$
Option D: $\frac{\sqrt{3}{a}^{2}}{4}$
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• Question 98
Let . Which one of the following is correct?
Option A: f(x) fluctuates in the interval
Option B: f(x) increases in the interval
Option C: f(x) decreases in the interval
Option D: None of these
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• Question 99
Suppose the function  is differentiable for all x. Then n can be any element of the interval
Option A: [1, ∞)
Option B: (0, ∞)
Option C:
Option D: None of these
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• Question 100
What is equal to?
Option A: $\frac{3}{2}$
Option B: $\frac{5}{2}$
Option C: 3
Option D: 4
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• Question 101
The variance of 20 observations is 5. If each observation is multiplied by 3, then what is the new variance of the resulting observations?
Option A: 5
Option B: 10
Option C: 15
Option D: 45
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• Question 102
The mean of a group of 100 observations was found to be 20. Later it was found that four observations were incorrect, which were recorded as 21, 21, 18 and 20. What is the mean if the incorrect observations are omitted?
Option A: 18
Option B: 20
Option C: 21
Option D: 22
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• Question 103
A committee of two persons is constituted from two men and two women. What is the probability that the committee will have only women?
Option A: $\frac{1}{6}$
Option B: $\frac{1}{3}$
Option C: $\frac{1}{2}$
Option D: $\frac{2}{3}$
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• Question 104
A question is given to three students A, B and C whose chance of solving it are respectively. What is the probability that the question will be solved?
Option A: $\frac{1}{24}$
Option B: $\frac{1}{4}$
Option C: $\frac{3}{4}$
Option D: $\frac{23}{24}$
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• Question 105
The mean weight of 150 students in a certain class is 60 kg. The mean weight of boys in the class is 70 kg and that of girls is 55 kg. What is the number of boys in the class?
Option A: 50
Option B: 55
Option C: 60
Option D: 100
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• Question 106
For two dependent events, A and E, it is given that P(A) = 0.2 and P(B) = 0.5. If $A\subseteq B$, then the value of conditional probabilities are respectively
Option A: $\frac{2}{3},\frac{3}{5}$
Option B: $\frac{2}{5},1$
Option C: $1,\frac{2}{5}$
Option D: Information is insufficient
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• Question 107
A point is chosen at random inside a circle. What is the probability that the point is closer to the centre of the circle then to its boundary?
Option A: $\frac{1}{5}$
Option B: $\frac{1}{4}$
Option C: $\frac{1}{3}$
Option D: $\frac{1}{2}$
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• Question 108
If two regression lines between height (x) and weight (y) are 4y – 15x + 410 = 0 and 30 x – 2y – 825 = 0, then what will be the correlation coefficient between height and weight?
Option A: $\frac{1}{3}$
Option B: $\frac{1}{2}$
Option C: $\frac{2}{3}$
Option D: $\frac{3}{4}$
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• Question 109
In an examination, 40% of candidates get second class. When the data are represented by a pie chart, what is the angle corresponding to the second class?
Option A: 40°
Option B: 90°
Option C: 144°
Option D: 320°
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• Question 110
Consider the following statements:
Statement 1: Range is not a good measure of dispersion
Statement 2: Range is highly affected by the existence of extreme values

Which one of the following is correct in respect of the above statements?
Option A: Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation of Statement 1
Option B: Both Statement 1 and Statement 2 are correct but Statement 2 is not the correct explanation of Statement 1
Option C: Statement 1 is correct but Statement 2 is not correct
Option D: Statement 2 is correct but Statement 1 is not correct
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• Question 111
A card is drawn from a well shuffled ordinary deck of 52 cards. What is the probability that it is an ace?
Option A: $\frac{1}{13}$
Option B: $\frac{2}{13}$
Option C: $\frac{3}{13}$
Option D: $\frac{1}{52}$
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• Question 112
If the data is moderately non-symmetrical then which one of the following empirical relationships is correct?
Option A: 2 × Standard deviation = 5 × Mean deviation
Option B: 5 × Standard deviation = 2 × Mean deviation
Option C: 4 × Standard deviation = 5 × Mean deviation
Option D: 5 × Standard deviation = 4 × Mean deviation
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• Question 113
Data can be represented in which of the following forms
I. Textual form
II. Tabular form
III Graphical form

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
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• Question 114
For given statistical data, the graphs for less than ogive and more than ogive are drawn. If the point at which the two curves intersect is P, then abscissa of point P gives the value of which one of the following measures of central tendency?
Option A: Median
Option B: Mean
Option C: Mode
Option D: Geometric mean
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• Question 115
Consider the following statements:
1. Two events are mutually exclusive if the occurrence of one event prevents the occurrence of the other.
2. The probability of the union of two mutually exclusive events is the sum of their individual probabilities

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 116
If the regression coefficient of x on y and y on x are respectively, then what is the correlation coefficient between x and y?
Option A: $-\frac{1}{4}$
Option B: $-\frac{1}{16}$
Option C: $\frac{1}{16}$
Option D: $\frac{1}{4}$
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• Question 117
A sample of 5 observations has mean 32 and median 33. Later it is found that an observation was recorded incorrectly as 40 instead of 35. If we correct the data, then which one of the following is correct?
Option A: The mean and median remain the same
Option B: The median remains the same but the mean will decrease
Option C: The mean and median both will decrease
Option D: The mean remains the same but median will decrease
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• Question 118
If two fair dice are thrown, then what is the probability that the sum is neither 8 nor 9?
Option A: $\frac{1}{6}$
Option B: $\frac{1}{4}$
Option C: $\frac{3}{4}$
Option D: $\frac{5}{6}$
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• Question 119
Let A and B are two mutually exclusive events with P(A) = $\frac{1}{3}$ and P(B) = $\frac{1}{4}$. What is the value of $P\left(\overline{A}\cap \overline{B}\right)$
Option A: $\frac{1}{6}$
Option B: $\frac{1}{4}$
Option C: $\frac{1}{3}$
Option D: $\frac{5}{12}$
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• Question 120
The mean and standard deviation of a binomial distribution are 12 and 2 respectively. What is the number of trials?
Option A: 2
Option B: 12
Option C: 18
Option D: 24
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