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# NDA I 2016 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
Suppose ω is a cube root of unity with ω ≠ 1. Suppose P and Q are the points on the complex plane defined by ω and ω2. If O is the origin, then what is the angle between OP and OQ?
Option A: 60°
Option B: 90°
Option C: 120°
Option D: 150°
VIEW SOLUTION
• Question 2
Suppose there is a relation * between the positive numbers x and y given by x * y if and only if xy2. Then which one of the following is correct?
Option A: * is reflexive but not transitive and symmetric
Option B: * is transitive but not reflexive and symmetric
Option C: * is symmetric and reflexive but not transitive
Option D: * is symmetric but not reflexive and transitive
VIEW SOLUTION
• Question 3
If x2px + 4 > 0 for all real values of x, then which one of the following is correct?
Option A: |p| < 4
Option B: |p| ≤ 4
Option C: |p| > 4
Option D: |p| ≥ 4
VIEW SOLUTION
• Question 4
If ,  then what is the fundamental amplitude of $\frac{z-\sqrt{2}}{z-i\sqrt{2}}$?
Option A: π
Option B: $\frac{\pi }{2}$
Option C: $\frac{\pi }{3}$
Option D: $\frac{\pi }{4}$
VIEW SOLUTION
• Question 5
If $f\left({x}_{1}\right)-f\left({x}_{2}\right)=f\left(\frac{{x}_{1}-{x}_{2}}{1-{x}_{1}{x}_{2}}\right)$ for x1, x2 ∈ (−1, 1), then what is f(x) equal to?
Option A: $\mathrm{ln}\left(\frac{1-x}{1+x}\right)$
Option B: $\mathrm{ln}\left(\frac{2+x}{1-x}\right)$
Option C: ${\mathrm{tan}}^{-1}\left(\frac{1-x}{1+x}\right)$
Option D: ${\mathrm{tan}}^{-1}\left(\frac{1+x}{1-x}\right)$
VIEW SOLUTION
• Question 6
What is the range of the function $y=\frac{{x}^{2}}{1+{x}^{2}}$ where, x R?
Option A: [0, 1)
Option B: [0, 1]
Option C: (0, 1)
Option D: (0, 1]
VIEW SOLUTION
• Question 7
A straight line intersects x and y axes at P and Q, respectively. If (3, 5) is the middle point of PQ, then what is the area of the triangle OPQ?
Option A: 12 square units
Option B: 15 square units
Option C: 20 square units
Option D: 30 square units
VIEW SOLUTION
• Question 8
If a circle of radius b units with centre at (0, b) touches the line $y=x-\sqrt{2}$, then what is the value of b?
Option A: $2+\sqrt{2}$
Option B: $2-\sqrt{2}$
Option C: $2\sqrt{2}$
Option D: $\sqrt{2}$
VIEW SOLUTION
• Question 9
Consider the function $f\left(\theta \right)=4\left({\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{4}\theta \right)$

What is the maximum value of the function f(θ)?
Option A: 1
Option B: 2
Option C: 3
Option D: 4
VIEW SOLUTION
• Question 10
Consider the function $f\left(\theta \right)=4\left({\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{4}\theta \right)$

What is the minimum value of the function f(θ)?
Option B: 1
Option C: 2
Option D: 3
VIEW SOLUTION
• Question 11
Consider the function $f\left(\theta \right)=4\left({\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{4}\theta \right)$
Consider the following statements:
1. f(θ) = 2 has no solution.
2. $f\left(\mathrm{\theta }\right)=\frac{7}{2}$ has no solution.
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 12
Consider the curves

Where do the curves intersect?
Option A: At (2, 3) only
Option B: At (−1, −2) only
Option C: At (2, 3) and (−1, −2)
Option D: Neither at (2, 3) nor at (−1, −2)
VIEW SOLUTION
• Question 13
Consider the curves

What is the area bounded by the curves?
Option A: $\frac{17}{6}$ square units
Option B: $\frac{8}{3}$ square units
Option C: 2 square units
Option D:  $\frac{1}{3}$ square units
VIEW SOLUTION
• Question 14
Consider the curves
$f\left(x\right)=\frac{27\left({x}^{2/3}-x\right)}{4}$
How many solutions does the function f(x) = 1 have?
Option A: One
Option B: Two
Option C: Three
Option D: Four
VIEW SOLUTION
• Question 15
Consider the curves
$f\left(x\right)=\frac{27\left({x}^{2/3}-x\right)}{4}$
How many solutions does the function f(x) = −1 have?
Option A: One
Option B: Two
Option C: Three
Option D: Four
VIEW SOLUTION
• Question 16
Consider the functions
where, [.] is the greatest integer function.
What is ${\int }_{\frac{1}{3}}^{\frac{1}{2}}g\left(x\right)dx$ equal to?
Option A: $\frac{1}{6}$
Option B: $\frac{1}{3}$
Option C: $\frac{5}{18}$
Option D: $\frac{5}{36}$
VIEW SOLUTION
• Question 17
Consider the functions
where, [.] is the greatest integer function. What is ${\int }_{\frac{1}{3}}^{1}f\left(x\right)dx$ equal to?
Option A: $\frac{37}{72}$
Option B: $\frac{2}{3}$
Option C: $\frac{17}{72}$
Option D: $\frac{37}{144}$
VIEW SOLUTION
• Question 18
Consider the function
f(x) = | x − 1| + x2
where, xR.
Which one of the following statements is correct?
Option A: f(x) is continuous but not differentiable at x = 0
Option B: f(x) is continuous but not differentiable at x = 1
Option C: f(x) is differentiable at x = 1
Option D: f(x) is not differentiable at x = 0 and x = 1
VIEW SOLUTION
• Question 19
Consider the function
f(x) = | x − 1| + x2
where, xR.
Which one of the following statements is correct?
Option A: f(x) is increasing in and decreasing in
Option B: f(x) is decreasing in  and increasing in
Option C: f(x) is increasing in (− ∞, 1) and decreasing in (1, ∞)
Option D: f(x) is decreasing in (− ∞, 1) and increasing in (1, ∞)
VIEW SOLUTION
• Question 20
Consider the function
f(x) = | x − 1| + x2
where, xR. Which one of the following statements is correct?
Option A: f(x) has local minima at more than one point in (−∞, ∞)
Option B: f(x) has local maxima at more than one point in (−∞, ∞)
Option C: f(x) has local minimum at one point only  in (−∞, ∞)
Option D: f(x) has neither maxima nor minima in (−∞, ∞)
VIEW SOLUTION
• Question 21
Consider the function
f(x) = | x − 1| + x2
where, xR. What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = $\frac{1}{2}$ and x = 1?
Option A: $\frac{5}{12}$ square unit
Option B: $\frac{5}{6}$ square unit
Option C: $\frac{7}{6}$ square units
Option D: 2 square units
VIEW SOLUTION
• Question 22
Consider the function
f(x) = | x − 1| + x2
where, xR. What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1 and x$\frac{3}{2}$?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 23
Given that ${a}_{n}={\int }_{0}^{\pi }\frac{{\mathrm{sin}}^{2}\left\{\left(n+1\right)x\right\}}{\mathrm{sin}2x}dx$

Consider the following statements:
1. The sequence {a2n} is in AP with common difference zero.
2. The sequence {a2n+1} is in AP with common difference zero.
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 24
Given that ${a}_{n}={\int }_{0}^{\pi }\frac{{\mathrm{sin}}^{2}\left\{\left(n+1\right)x\right\}}{\mathrm{sin}2x}dx$. What is an − 1an − 4 equal to?
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 25
Consider the equation x + |y| = 2y.
Which of the following statements are not correct?
1. y as a function of x is not defined for all real x
2. y as a function of x is not continuous at x = 0.
3. y as a function of x is differentiable for all x.

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 26
Consider the equation x + |y| = 2y.

What is the derivative of y as a function of x with respect to x for x < 0?
Option A: 2
Option B: 1
Option C: $\frac{1}{2}$
Option D: $\frac{1}{3}$
VIEW SOLUTION
• Question 27
Consider the lines y = 3x, y = 6x and y = 9. What is the area of the triangle formed by these lines?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 28
Consider the lines y = 3x, y = 6x and y = 9. The centroid of the triangle is at which one of the following points?
Option A: (3, 6)
Option B:
Option C: (3, 3)
Option D:
VIEW SOLUTION
• Question 29
Consider the function f(x) = (x − 1)2 (x + 1) (x − 2)3

What is the number of points of local minima of the function f(x)?
Option A: None
Option B: One
Option C: Two
Option D: Three
VIEW SOLUTION
• Question 30
Consider the function f(x) = (x − 1)2 (x + 1) (x − 2)3

What is the number of points of local maxima of the function f(x)?
Option A: None
Option B: One
Option C: Two
Option D: Three
VIEW SOLUTION
• Question 31
Let f(x) and g(x) be twice differentiable functions on [0, 2] satisfying f ''(x) = g''(x), f '(1) = 4, g'(1) = 6, f(2) = 3 and g(2) = 9. Then what is f(x) − g(x) at x = 4 equal to?
Option A: –10
Option B: –6
Option C: –4
Option D: 2
VIEW SOLUTION
• Question 32
Consider the curves y = |x − 1| and | x | = 2. What is/are the points(s) of intersection of the curves?
Option A: (−2, 3) only
Option B: (2, 1) only
Option C: (−2, 3) and  (2, 1)
Option D: Neither (−2, 3) nor (2, 1)
VIEW SOLUTION
• Question 33
Consider the curves y = |x − 1| and | x | = 2

What is the area of the region bounded by the curves and x-axis?
Option A: 3 square units
Option B: 4 square units
Option C: 5 square units
Option D: 6 square units
VIEW SOLUTION
• Question 34
Consider the function $f\left(x\right)=\left|\begin{array}{ccc}{x}^{3}& \mathrm{sin}x& \mathrm{cos}x\\ 6& -1& 0\\ p& {p}^{2}& {p}^{3}\end{array}\right|$ where, p is a constant. What is the value of f´(0)?
Option A: p3
Option B: 3p3
Option C: 6p3
Option D: − 6p3
VIEW SOLUTION
• Question 35
Consider the function $f\left(x\right)=\left|\begin{array}{ccc}{x}^{3}& \mathrm{sin}x& \mathrm{cos}x\\ 6& -1& 0\\ p& {p}^{2}& {p}^{3}\end{array}\right|$ where, p is a constant

What is the value of p for which f"(0) = 0?
Option A:
Option B: –1 or 0
Option C:
Option D: –1 or 1
VIEW SOLUTION
• Question 36
Consider  triangle ABC in which  . What is the value of $\mathrm{sin}\frac{\mathrm{A}}{2}\mathrm{sin}\frac{\mathrm{B}}{2}\mathrm{sin}\frac{\mathrm{C}}{2}$?
Option A: $\frac{1}{2}$
Option B: $\frac{1}{4}$
Option C: $\frac{1}{8}$
Option D: $\frac{1}{16}$
VIEW SOLUTION
• Question 37
Consider triangle ABC in which  . What is the value of $\mathrm{cos}\left(\frac{A+B}{2}\right)\mathrm{cos}\left(\frac{B+C}{2}\right)\mathrm{cos}\left(\frac{C+A}{2}\right)$?
Option A: $\frac{1}{4}$
Option B: $\frac{1}{2}$
Option C: $\frac{1}{16}$
Option D: None of the above
VIEW SOLUTION
• Question 38
Given that tan α and tan β are the roots of the equation x2 + bx + c = 0 with b ≠ 0. What is tan(α + β) equal to?
Option A: b(c − 1)
Option B: c(b − 1)
Option C: c(b − 1) −1
Option D: b(c − 1)−1
VIEW SOLUTION
• Question 39
Given that tan α and tan β are the roots of the equation x2 + bx + c = 0 with b ≠ 0. What is sin (α + β) sec α sec β equal to?
Option A: b
Option B: b
Option C: c
Option D: c
VIEW SOLUTION
• Question 40
Consider the two circles
(x − 1)2 + (y − 3)2 = r2 and x2 + y2 − 8x + 2y + 8 = 0. What is the distance between the centres of the two circles?
Option A: 5 units
Option B: 6 units
Option C: 8 units
Option D: 10 units
VIEW SOLUTION
• Question 41
Consider the two circles (x − 1)2 + (y − 3)2 = r2 and x2 + y2 − 8x + 2y + 8 = 0. If the circles intersect at two distinct points, then which one of the following is correct?
Option A: r = 1
Option B: 1 < r < 2
Option C: r = 2
Option D: 2 < r < 8
VIEW SOLUTION
• Question 42
Consider the two lines
x + y + 1 = 0  and  3x + 2y + 1 = 0. What is the equation of the line passing through the point of intersection of the given lines and parallel to x-axis?
Option A: y + 1 = 0
Option B: y − 1 = 0
Option C: y − 2 = 0
Option D: y + 2 = 0
VIEW SOLUTION
• Question 43
Consider the two lines
x + y + 1 = 0  and  3x + 2y + 1 = 0. What is the equation of the line passing through the point of intersection of the given lines and parallel to y-axis?
Option A: x + 1 = 0
Option B: x − 1 = 0
Option C: x − 2 = 0
Option D: x + 2 = 0
VIEW SOLUTION
• Question 44
Consider the equation  k sin x + cos 2x = 2k − 7. If the equation possesses solutions, then what is the minimum value of k?
Option A: 1
Option B: 2
Option C: 4
Option D: 6
VIEW SOLUTION
• Question 45
Consider the equation  k sin x + cos 2x = 2k − 7. If the equation possesses solution, then what is the maximum value of k?
Option A: 1
Option B: 2
Option C: 4
Option D: 6
VIEW SOLUTION
• Question 46
Consider the function $f\left(x\right)=\frac{{a}^{\left[x\right]+x}-1}{\left[x\right]+x}$. Where [.] denotes the greatest integer function. What is  equal to?
Option A: 1
Option B: ln a
Option C: 1 − a−1
Option D: Limit does not exit
VIEW SOLUTION
• Question 47
Consider the function $f\left(x\right)=\frac{{a}^{\left[x\right]+x}-1}{\left[x\right]+x}$. What is  equal to?
Option B: ln a
Option C: 1 − a−1
Option D:  Limit does not exist
VIEW SOLUTION
• Question 48
Let z1, z2 and z3 be non-zero complex numbers satisfying z2 = $i\overline{z}$, where, $i=\sqrt{-1}$.
What is z1 + z2 + z3 eqaul to?
Option A:  i
Option B: i
Option D: 1
VIEW SOLUTION
• Question 49
Let z1, z2 and z3 be non-zero complex numbers satisfying z2 = $i\overline{z}$, where, $i=\sqrt{-1}$.
Consider the following statements:
1. z1z2z3 is purely imaginary.
2. z1z2 + z2z3 + z3z1 is purely real.
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 50
Given that logx y, logz x, logy z are in GP, xyz = 64 and x3, y3, z3 are in AP.
Which one of the following is correct?
x, y and z are
Option A: in AP only
Option B: in GP only
Option C: in both AP and GP
Option D: neither in AP nor in GP
VIEW SOLUTION
• Question 51
Given that logx y, logz x, logy z are in GP, xyz = 64 and x3, y3, z3 are in AP.
Which one of the following is correct?
xy, yz and zx are
Option A: in AP only
Option B: in GP only
Option C: in both AP and GP
Option D: neither in AP nor in GP
VIEW SOLUTION
• Question 52
Let z be a complex number satisfying
What is |z| equal to?
Option A: 6
Option B: 12
Option C: 18
Option D: 36
VIEW SOLUTION
• Question 53
Let z be a complex number satisfying
What is $\left|\frac{z-6}{z+6}\right|$ equal to?
Option A: 3
Option B: 2
Option C: 1
VIEW SOLUTION
• Question 54
A function f(x) is defined as follows:

Consider the following statements:
1. The function f(x) is continuous at x = 0.
2. The function f(x) is continuous at $x=\frac{\pi }{2}$.
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 55
A function f(x) is defined as follows:

Consider the following statements:
1. The function f(x) is differentiable at x = 0.
2. The function f(x) is differentiable at $x=\frac{\pi }{2}$.
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 56
Let α and β (α < β) be the roots of the equation x2 + bx + c = 0, where, b > 0 and c < 0.
Consider the following:
1. β < –αa
2. β < |α|
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 57
Let α and β (α < β) be the roots of the equation x2 + bx + c = 0, where, b > 0 and c < 0.
Consider the following:
1. α + β + αβ > 0
2. α2β + β2α > 0|
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 58
Consider a parallelogram whose vertices are A(1, 2), B(4, y), C(x, 6) and D(3, 5) taken in order.
What is the value of AC2BD2?
Option A: 25
Option B: 30
Option C: 36
Option D: 40
VIEW SOLUTION
• Question 59
Consider a parallelogram whose vertices are A(1, 2), B(4, y), C(x, 6) and D(3, 5) taken in order.
What is the point of intersection of the diagonals?
Option A:
Option B: (3, 4)
Option C:
Option D: (3, 5)
VIEW SOLUTION
• Question 60
Consider a parallelogram whose vertices are A(1, 2), B(4, y), C(x, 6) and D(3, 5) taken in order.
What is the area of the parallelogram?
Option A: $\frac{7}{2}$square units
Option B: 4 square units
Option C: $\frac{11}{2}$ square units
Option D: 7 square units
VIEW SOLUTION
• Question 61

Let be a function such that . What is f(1) equal to?

Option A: –2
Option B: –1
Option D: 4
VIEW SOLUTION
• Question 62
Let be a function such that .
What is f ´(1) equal to?
Option A: –6
Option B: –5
Option C: 1
VIEW SOLUTION
• Question 63
Let be a function such that .
What is f ''' (10) equal to?
Option A: 1
Option B: 5
Option C: 6
Option D: 8
VIEW SOLUTION
• Question 64
Let be a function such that .
Consider the following:
1.  f(2) = f(1) − f(0)
2. f '' (2) – 2f ' (1) = 12
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 65
A plane P passes through the line of intersection of the planes 2xy + 3z = 2, x + yz = 1 and the point (1, 0, 1). What are the direction ratios of the line of intersection of the given planes?
Option A: $\left(2,-5,-3\right)$
Option B: $\left(1,-5,-3\right)$
Option C:
Option D:
VIEW SOLUTION
• Question 66
A plane P passes through the line of intersection of the planes 2x y + 3z = 2, x + y − z = 1 and the point (1, 0, 1). What is the equation of the plane P?
Option A: 2x + 5y − 2 = 0
Option B: 5x + 2y − 5 = 0
Option C: x + z − 2 = 0
Option D: 2x − y − 2z = 0
VIEW SOLUTION
• Question 67
A plane P passes through the line of intersection of the planes 2x y + 3z = 2, x + y − z = 1 and the point (1, 0, 1). If the plane P touches the sphere x2 + y2 + z2 = r2, then what is r equal to?
Option A: $\frac{2}{\sqrt{29}}$
Option B: $\frac{4}{\sqrt{29}}$
Option C: $\frac{5}{\sqrt{29}}$
Option D: 1
VIEW SOLUTION
• Question 68
Consider the function $f\left(x\right)=|{x}^{2}-5x+6|$. What is f '(4) equal to?
Option A: −4
Option B: −3
Option C: 3
Option D: 2
VIEW SOLUTION
• Question 69
Consider the function $f\left(x\right)=|{x}^{2}-5x+6|$. What is ${f}^{\text{'}\text{'}}\left(2.5\right)$  equal to?
Option A: –3
Option B: –2
Option D: 2
VIEW SOLUTION
• Question 70
Let, f(x) be the greatest integer function and g(x) be the modulus function.
What is $\left(g\circ f\right)\left(-\frac{5}{3}\right)-\left(f\circ g\right)\left(-\frac{5}{3}\right)\underset{x\to \infty }{\mathrm{lim}}$ equal to?
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 71
Let, f(x) be the greatest integer function and g(x) be the modulus function.
What is  equal to?
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 72
Consider a circle passing through the origin and the points (a, b) and (–b, –a).
On which line, does the centre of the circle lie?
Option A: x + y = 0
Option B: xy = 0
Option C: x + y = a + b
Option D: xy = a2b2
VIEW SOLUTION
• Question 73
Consider a circle passing through the origin and the points (a, b) and (–b, –a).
What is the sum of the squares of the intercepts cut off by the circle on the axes?
Option A: ${\left(\frac{{a}^{2}+{b}^{2}}{{a}^{2}-{b}^{2}}\right)}^{2}$
Option B: $2{\left(\frac{{a}^{2}+{b}^{2}}{a-b}\right)}^{2}$
Option C: $4{\left(\frac{{a}^{2}+{b}^{2}}{a-b}\right)}^{2}$
Option D: None of the above
VIEW SOLUTION
• Question 74
Let,  be two unit vectors and θ be the angle between them.
What is $\mathrm{cos}\left(\frac{\theta }{2}\right)$ equal to?
Option A: $\frac{|\stackrel{^}{a}-\stackrel{^}{b}|}{2}$
Option B: $\frac{|\stackrel{^}{a}+\stackrel{^}{b}|}{2}$
Option C: $\frac{|\stackrel{^}{a}-\stackrel{^}{b}|}{4}$
Option D: $\frac{|\stackrel{^}{a}+\stackrel{^}{b}|}{4}$
VIEW SOLUTION
• Question 75
Let,  be two unit vectors and θ be the angle between them.
What is $\mathrm{sin}\left(\frac{\theta }{2}\right)$ equal to?
Option A: $\frac{|\stackrel{^}{a}-\stackrel{^}{b}|}{2}$
Option B: $\frac{|\stackrel{^}{a}+b|}{2}$
Option C: $\frac{|\stackrel{^}{a}-\stackrel{^}{b}|}{4}$
Option D: $\frac{|\stackrel{^}{a}+\stackrel{^}{b}|}{4}$
VIEW SOLUTION
• Question 76
Consider the following statements:
1. There exists $\mathrm{\theta }\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ for which .
2. ${\mathrm{sin}}^{-1}\left(\frac{1}{3}\right)-{\mathrm{sin}}^{-1}\left(\frac{1}{5}\right)={\mathrm{sin}}^{-1}\left(\frac{2\sqrt{2}\left(\sqrt{3}-1\right)}{15}\right)$
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 77
Consider the following statements:
1. ${\mathrm{tan}}^{-1}x+{\mathrm{tan}}^{-1}\left(\frac{1}{x}\right)=\pi$
2. There exists .
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 78
What are the order and degree, respectively, of the differential equation whose solution is $y=cx+{c}^{2}-3{c}^{3/2}+2$, where c is a parameter?
Option A: 1, 2
Option B: 2, 2
Option C: 1, 3
Option D: 1, 4
VIEW SOLUTION
• Question 79
What is equal to, where [.] is greatest integer function?
Option B: 1
Option C: 2
Option D: 4
VIEW SOLUTION
• Question 80
If then what is${\int }_{-2}^{0}f\left(x\right)dx$ equal to?
Option A: –3
Option B: 2
Option C: 3
Option D: 5
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• Question 81
If , where a ≠ 0, then what is equal to?
Option A: a2
Option B: a−2
Option C: a2
Option D: a
VIEW SOLUTION
• Question 82
What is  equal to?
Option B: 1
Option C: –1
Option D: Limit does not exist
VIEW SOLUTION
• Question 83
If A is a square matrix, then what is adj(A−1) − (adj A)−1 equal to?
Option A: 2 |A|
Option B: Null matrix
Option C: Unit matrix
Option D: None of the above
VIEW SOLUTION
• Question 84
What is the binary equivalent of the decimal number 0.3125?
Option A: 0.0111
Option B: 0.1010
Option C: 0.0101
Option D: 0.1101
VIEW SOLUTION
• Question 85
Let R be a relation on the Set N of natural numbers defined by $\text{'}nRm⇔n$ is a factor of m’. Then which one of the following is correct?
Option A: R is reflexive, symmetric but not transitive
Option B: R is transitive, symmetric but not reflexive
Option C: R is reflexive, transitive but not symmetric
Option D: R is an equivalence relation
VIEW SOLUTION
• Question 86
What is  equal to?
Option B: 2
Option C: 4
Option D: 8
VIEW SOLUTION
• Question 87
What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?
Option A: 860
Option B: 854
Option C: 840
Option D: 824
VIEW SOLUTION
• Question 88
(a, 2b) is the mid-point of the line segment joining the points (10, −6) and (k, 4). If a − 2b = 7, then what is the value of k?
Option A: 2
Option B: 3
Option C: 4
Option D: 5
VIEW SOLUTION
• Question 89
Consider the following statements:
1. If ABC is an equilateral triangle, then 3 tan(A + B) tanC = 1
2. If ABC is a triangle in which A = 78º, B = 66º, then
3. If ABC is any triangle, then
Which of the above statements is/are correct?
Option A: 1 only
Option B: 2 only
Option C: 1 and 2
Option D: 2 and 3
VIEW SOLUTION
• Question 90
If  A = (cos 12º − cos 36º) (sin 96º + sin 24º) and B = (sin 60º − sin 12º) (cos 48º − cos 72º), then what is  $\frac{A}{B}$ equal to?
Option A: –1
Option C: 1
Option D: 2
VIEW SOLUTION
• Question 91
What is the mean deviation from the mean of the numbers 10, 9, 21, 16, 24?
Option A: 5.2
Option B: 5.0
Option C: 4.5
Option D: 4.0
VIEW SOLUTION
• Question 92
Three dices are thrown simultaneously. What is the probability that the sum of the three faces is at least 5?
Option A: $\frac{17}{18}$
Option B: $\frac{53}{54}$
Option C: $\frac{103}{108}$
Option D: $\frac{215}{216}$
VIEW SOLUTION
• Question 93
Two independent events A and B have . What is the probability that exactly one of the two events A or B occurs?
Option A: $\frac{1}{4}$
Option B: $\frac{5}{6}$
Option C: $\frac{5}{12}$
Option D: $\frac{7}{12}$
VIEW SOLUTION
• Question 94
A coin is tossed three times. What is the probability of getting head and tail alternately?
Option A: $\frac{1}{8}$
Option B: $\frac{1}{4}$
Option C: $\frac{1}{2}$
Option D: $\frac{3}{4}$
VIEW SOLUTION
• Question 95
If the total number of observations is 20, , then what is the variance of the distribution?
Option A: 1500
Option B: 1600
Option C: 1700
Option D: 1800
VIEW SOLUTION
• Question 96
A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is queen of spade?
Option A: $\frac{1}{52}$
Option B: $\frac{1}{13}$
Option C: $\frac{1}{4}$
Option D: $\frac{1}{8}$
VIEW SOLUTION
• Question 97
If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?
Option A: $\frac{13}{18}$
Option B: $\frac{5}{6}$
Option C: $\frac{11}{12}$
Option D: $\frac{35}{36}$
VIEW SOLUTION
• Question 98
A certain type of missile hits the target with probability p = 0.3. What is the least number of missiles should be fired so that there is at least an 80% probability that the target is hit?
Option A: 5
Option B: 6
Option C: 7
Option D: None of the above
VIEW SOLUTION
• Question 99
For two mutually exclusive events A and B, P(A) = 0.2 and . What is equal to $P\left(A|\left(A\cup B\right)\right)$ equal to?
Option A: $\frac{1}{2}$
Option B: $\frac{2}{5}$
Option C: $\frac{2}{7}$
Option D: $\frac{2}{3}$
VIEW SOLUTION
• Question 100
What is the probability of 5 Sundays in the month of December?
Option A: $\frac{1}{7}$
Option B: $\frac{2}{7}$
Option C: $\frac{3}{7}$
Option D: None of the above
VIEW SOLUTION
• Question 101
If m is the geometric mean of then what is the value of m?
Option A: 1
Option B: 3
Option C: 6
Option D: 9
VIEW SOLUTION
• Question 102
A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?
Option A: $\frac{2}{3}$
Option B: $\frac{1}{3}$
Option C: $\frac{1}{4}$
Option D: $\frac{2}{5}$
VIEW SOLUTION
• Question 103
The mean of the series x1, x2,...,xn is $\overline{X}$. If x2 is replaced by λ, then what is the new mean?
Option A: $\overline{X}-{x}_{2}+\lambda$
Option B: $\frac{\overline{X}-{x}_{2}-\lambda }{n}$
Option C: $\frac{\overline{X}-{x}_{2}+\lambda }{n}$
Option D: $\frac{n\overline{X}-{x}_{2}+\lambda }{n}$
VIEW SOLUTION
• Question 104
For the data  3, 5, 1, 6, 5, 9, 5, 2, 8, 6 the mean, median and mode are x, y and z, respectively. Which one of the following is correct?
Option A: x = y ≠ z
Option B: xy = z
Option C: xyz
Option D: x = y = z
VIEW SOLUTION
• Question 105
Consider the following statements in respect to a histogram:
1. The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and the x-axis.
2. When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency.
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 106
A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?
Option A: $\frac{1}{2}$
Option B: $\frac{3}{8}$
Option C: $\frac{1}{4}$
Option D: $\frac{1}{8}$
VIEW SOLUTION
• Question 107
What is the number of ways in which 3 holidays travel tickets are to be given to 10 employees of an organization if each employee is eligible for any one or more of the tickets?
Option A: 60
Option B: 120
Option C: 500
Option D: 1000
VIEW SOLUTION
• Question 108
If one root of the equation (lm) x2 + lx + 1 = 0 is double the other and l is real, then what is the greatest value of m?
Option A: $-\frac{9}{8}$
Option B: $\frac{9}{8}$
Option C: $-\frac{8}{9}$
Option D: $\frac{8}{9}$
VIEW SOLUTION
• Question 109
What is the number of four-digit decimal numbers (< 1) in which no digit is repeated?
Option A: 3024
Option B: 4536
Option C: 5040
Option D: None of the above
VIEW SOLUTION
• Question 110
What is a vector of unit length orthogonal to both the vectors and  ?
Option A: $\frac{-4\stackrel{^}{i}+3\stackrel{^}{j}-\stackrel{^}{k}}{\sqrt{26}}$
Option B: $\frac{-4\stackrel{^}{i}+3\stackrel{^}{j}+\stackrel{^}{k}}{\sqrt{26}}$
Option C: $\frac{-3\stackrel{^}{i}+2\stackrel{^}{j}-\stackrel{^}{k}}{\sqrt{14}}$
Option D: $\frac{-3\stackrel{^}{i}+2\stackrel{^}{j}+\stackrel{^}{k}}{\sqrt{14}}$
VIEW SOLUTION
• Question 111
If   are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?
Option A: $\stackrel{\to }{a}+\stackrel{\to }{b}+\stackrel{\to }{c}=\stackrel{\to }{0}$
Option B:
Option C: $\stackrel{\to }{a}+\stackrel{\to }{b}=\stackrel{\to }{c}$
Option D: $\stackrel{\to }{a}=\stackrel{\to }{b}+\stackrel{\to }{c}$
VIEW SOLUTION
• Question 112
What is the area of the parallelogram having diagonals ?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 113
Consider the following in respect to the matrix :
1. A2 = − A
2. A3 = 4A
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 114
Which of the following determinants have value ‘zero’?
1. $\left|\begin{array}{ccc}41& 1& 5\\ 79& 7& 9\\ 29& 5& 3\end{array}\right|$

2. $\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$

3. $\left|\begin{array}{ccc}0& c& b\\ -c& 0& a\\ -b& -a& 0\end{array}\right|$

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 115
What is the acute angle between the lines represented by the equations ?
Option A: 30º
Option B: 45º
Option C: 60º
Option D: 75º
VIEW SOLUTION
• Question 116
The system of linear equations kx + y + z = 1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions?
Option A:
Option B:
Option C:
Option D:
VIEW SOLUTION
• Question 117
What is the number of different messages that can be represented by three 0's and two 1's?
Option A: 10
Option B: 9
Option C: 8
Option D: 7
VIEW SOLUTION
• Question 118
If loga(ab) = x, then what is logb(ab) equal to?
Option A: $\frac{1}{x}$
Option B: $\frac{x}{x+1}$
Option C: $\frac{x}{1-x}$
Option D: $\frac{x}{x-1}$
VIEW SOLUTION
• Question 119
If y = log10 x + logx 10 + logx x + log10 10 then what is ${\left(\frac{dy}{dx}\right)}_{x=10}$ equal to?
Option A: 10
Option B: 2
Option C: 1
VIEW SOLUTION
• Question 120
Suppose ω1 and ω2 are two distinct cube roots of unity different from 1. Then what is (ω1ω2)2 equal to?
Option A: 3
Option B: 1
Option C: –1
Option D: –3
VIEW SOLUTION
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