NDA I 2016 Mathematics

• 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°

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• Question 2
  Suppose there is a relation * between the positive numbers x and y given by x * y if and only if x ≤ y². 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

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• Question 3
  If x² − px + 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

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• Question 4
  If $z=x+iy={\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)}^{-25}, \mathrm{where} i=\sqrt{-1}$,  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: $\ln \left(\frac{1-x}{1+x}\right)$

  Option B: $\ln \left(\frac{2+x}{1-x}\right)$

  Option C: ${\tan }^{-1}\left(\frac{1-x}{1+x}\right)$

  Option D: ${\tan }^{-1}\left(\frac{1+x}{1-x}\right)$

  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]

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• 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

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• 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: $\sqrt{2}$

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• Question 9
  Consider the function $f\left(\theta \right)=4({\sin }^2\theta +{\cos }^4\theta )$
  
  What is the maximum value of the function f(θ)?

  Option A: 1

  Option B: 2

  Option C: 3

  Option D: 4

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• Question 10
  Consider the function $f\left(\theta \right)=4({\sin }^2\theta +{\cos }^4\theta )$
  
  What is the minimum value of the function f(θ)?

  Option B: 1

  Option C: 2

  Option D: 3

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• Question 11
  Consider the function $f\left(\theta \right)=4({\sin }^2\theta +{\cos }^4\theta )$
  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
  $f\left(x\right)= x\left|x\right|-1 \mathrm{and} g\left(x\right) =\left\{\begin{array}{l}\begin{array}{l}\frac{3x}{2}, x>0\\ 2x, x\le 0\end{array}\end{array}\right.$
  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
  $f\left(x\right)= x\left|x\right|-1 \mathrm{and} g\left(x\right) =\left\{\begin{array}{l}\begin{array}{l}\frac{3x}{2}, x>0\\ 2x, x\le 0\end{array}\end{array}\right.$
  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
   $f\left(x\right)=\frac{27(x^{2/3}-x)}{4}$
  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
   $f\left(x\right)=\frac{27(x^{2/3}-x)}{4}$
  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
  $f\left(x\right)=xg\left(x\right) \mathrm{and} g\left(x\right)=\left[\frac{1}{x}\right]$ where, [.] is the greatest integer function.
  What is ${\int }_{\frac{{\displaystyle 1}}{3}}^{\frac{1}{2}}g\left(x\right)dx$ 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
  $f\left(x\right)=xg\left(x\right) \mathrm{and} g\left(x\right)=\left[\frac{1}{x}\right]$ where, [.] is the greatest integer function. What is ${\int }_{\frac{{\displaystyle 1}}{3}}^{1}f\left(x\right)dx$ 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, x ∈R.
  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

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• Question 19
  Consider the function
  f(x) = | x − 1| + x²
  where, x ∈R.
  Which one of the following statements is correct?

  Option A: f(x) is increasing in $\left(-\mathrm{\infty }, \frac{1}{2}\right)$ and decreasing in $\left(\frac{1}{2}, \infty \right)$

  Option B: f(x) is decreasing in $\left(-\mathrm{\infty }, \frac{1}{2}\right)$ and increasing in $\left(\frac{1}{2}, \infty \right)$

  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, x ∈R. 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 (−∞, ∞)

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• Question 21
  Consider the function
  f(x) = | x − 1| + x²
  where, x ∈R. 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, x ∈R. 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: $\frac{5}{12} \mathrm{square} \mathrm{unit}$

  Option B: $\frac{7}{12} \mathrm{square} \mathrm{unit}$

  Option C: $\frac{2}{3} \mathrm{square} \mathrm{unit}$

  Option D: $\frac{11}{12} \mathrm{square} \mathrm{unit}$

  VIEW SOLUTION

• Question 23
  Given that $a_n={\int }_0^{\pi }\frac{{\sin }^2\left\{(n+1)x\right\}}{\sin 2x}dx$
   
  Consider the following statements:
   1. The sequence {a_2n} is in AP with common difference zero.
   2. The sequence {a_2n+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 $a_n={\int }_0^{\pi }\frac{{\sin }^2\left\{(n+1)x\right\}}{\sin 2x}dx$. What is a_{n − 1} − a_{n − 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: $\frac{27}{4} \mathrm{square} \mathrm{units}$

  Option B: $\frac{{\displaystyle 27}}{{\displaystyle 2}} \mathrm{square} \mathrm{units}$

  Option C: $\frac{{\displaystyle 19}}{{\displaystyle 4}} \mathrm{square} \mathrm{units}$

  Option D: $\frac{{\displaystyle 19}}{{\displaystyle 2}} \mathrm{square} \mathrm{units}$

  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: $\left(\frac{3}{2}, 6\right)$

  Option C: (3, 3)

  Option D: $\left(\frac{3}{2}, 9\right)$

  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 $f\left(x\right)=\left|\begin{array}{ccc}{x}^3& \sin x& \cos x\\ 6& -1& 0\\ p& p^2& p^3\end{array}\right|$ 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 $f\left(x\right)=\left|\begin{array}{ccc}{x}^3& \sin x& \cos x\\ 6& -1& 0\\ p& p^2& p^3\end{array}\right|$ where, p is a constant
  
  What is the value of p for which f"(0) = 0?

  Option A: $-\frac{1}{6} \mathrm{or} 0$

  Option B: –1 or 0

  Option C: $-\frac{1}{6} \mathrm{or} 1$

  Option D: –1 or 1

  VIEW SOLUTION

• Question 36
  Consider  triangle ABC in which  $\cos A+\cos B+\cos C=\sqrt{3} \sin \frac{\pi }{3}$. What is the value of $\sin \frac{\mathrm{A}}{2}\sin \frac{\mathrm{B}}{2}\sin \frac{\mathrm{C}}{2}$?

  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  $\cos A+\cos B+\cos C=\sqrt{3} \sin \frac{\pi }{3}$. What is the value of $\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{B+C}{2}\right)\cos \left(\frac{C+A}{2}\right)$?

  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

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• 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 $\underset{x\to 0^{+}}{\lim } f\left(x\right)$ 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 $\underset{x\to 0^{-}}{\lim } f\left(x\right)$ 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 log_x y, log_z x, log_y 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 log_x y, log_z x, log_y 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 $\left|\frac{z-4}{z-8}\right|=1 \mathrm{and} \left|\frac{z}{z-2}\right|=\frac{3}{2}$
  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 $\left|\frac{z-4}{z-8}\right|=1 \mathrm{and} \left|\frac{z}{z-2}\right|=\frac{3}{2}$
  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:
  $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{lll}x+\pi & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in [-\pi ,0)\\ \pi \cos x& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}& x\in \left[0,\frac{\pi }{2}\right]\\ {\left(x-\frac{\pi }{2}\right)}^2& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \left(\frac{\pi }{2},\pi \right)\end{array}\end{array}\right.$
  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:
  $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{lll}x+\pi & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in [-\pi ,0)\\ \pi \cos x& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}& x\in \left[0,\frac{\pi }{2}\right]\\ {\left(x-\frac{\pi }{2}\right)}^2& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{for}& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\in \left(\frac{\pi }{2},\pi \right)\end{array}\end{array}\right.$
  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 AC² − BD²?

  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: $\left(\frac{7}{2}, 4\right)$

  Option B: (3, 4)

  Option C: $\left(\frac{7}{2}, 5\right)$

  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 $f : \mathrm{ℝ}\to \mathrm{ℝ}$ be a function such that $f\left(x\right)=x^3+x^2 f\text{'}\left(1\right)+xf\text{'}\text{'}\left(2\right)+f\text{'}\text{'}\text{'}\left(3\right) \mathrm{for} x\in \mathrm{ℝ}$. What is f(1) equal to?

  Option A: –2

  Option B: –1

  Option D: 4

  VIEW SOLUTION

• Question 62
  Let $f : \mathrm{ℝ}\to \mathrm{ℝ}$ be a function such that $f\left(x\right)=x^3+x^2 f\text{'}\left(1\right)+xf\text{'}\text{'}\left(2\right)+f\text{'}\text{'}\text{'}\left(3\right) \mathrm{for} x\in \mathrm{ℝ}$.
  What is f ´(1) equal to?

  Option A: –6

  Option B: –5

  Option C: 1

  VIEW SOLUTION

• Question 63
  Let $f : \mathrm{ℝ}\to \mathrm{ℝ}$ be a function such that $f\left(x\right)=x^3+x^2 f\text{'}\left(1\right)+xf\text{'}\text{'}\left(2\right)+f\text{'}\text{'}\text{'}\left(3\right) \mathrm{for} x\in \mathrm{ℝ}$. 
  What is f ''' (10) equal to?

  Option A: 1

  Option B: 5

  Option C: 6

  Option D: 8

  VIEW SOLUTION

• Question 64
  Let $f : \mathrm{ℝ}\to \mathrm{ℝ}$ be a function such that $f\left(x\right)=x^3+x^2 f\text{'}\left(1\right)+xf\text{'}\text{'}\left(2\right)+f\text{'}\text{'}\text{'}\left(3\right) \mathrm{for} x\in \mathrm{ℝ}$. 
  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 2x − y + 3z = 2, x + y − z = 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: $\left(2, 5, 3\right)$

  Option D: $\left(1, 3, 5\right)$

  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 $f^{\text{'}\text{'}}(2.5)$  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 $\left(g\circ f\right)\left(-\frac{5}{3}\right)-\left(f\circ g\right)\left(-\frac{5}{3}\right)\underset{x\to \infty }{\lim }$ 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 $\left(f\circ f\right)\left(-\frac{9}{5}\right)+\left(g\circ g\right) \left(-2\right)$ 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: x − y = 0

  Option C: x + y = a + b

  Option D: x − y = a² − b²

  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, $\hat{a}, \hat{b}$ be two unit vectors and θ be the angle between them.
  What is $\cos \left(\frac{\theta }{2}\right)$ equal to?

  Option A: $\frac{|\hat{a}-\hat{b}|}{2}$

  Option B: $\frac{|\hat{a}+\hat{b}|}{2}$

  Option C: $\frac{|\hat{a}-\hat{b}|}{4}$

  Option D: $\frac{|\hat{a}+\hat{b}|}{4}$

  VIEW SOLUTION

• Question 75
  Let, $\hat{a}, \hat{b}$ be two unit vectors and θ be the angle between them.
  What is $\sin \left(\frac{\theta }{2}\right)$ equal to?

  Option A: $\frac{|\hat{a}-\hat{b}|}{2}$

  Option B: $\frac{|\hat{a}+b|}{2}$

  Option C: $\frac{|\hat{a}-\hat{b}|}{4}$

  Option D: $\frac{|\hat{a}+\hat{b}|}{4}$

  VIEW SOLUTION

• Question 76
  Consider the following statements:
   1. There exists $\mathrm{\theta }\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ for which ${\tan }^{-1}(\tan \mathrm{\theta })\ne \mathrm{\theta }$.
   2. ${\sin }^{-1}\left(\frac{1}{3}\right)-{\sin }^{-1}\left(\frac{1}{5}\right)={\sin }^{-1}\left(\frac{2\sqrt{2}(\sqrt{3}-1)}{15}\right)$
   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. ${\tan }^{-1}x+{\tan }^{-1}\left(\frac{1}{x}\right)=\pi$
  2. There exists $x,y\in [-1,1], \mathrm{where}, x\ne y \mathrm{such} \mathrm{that} {\sin }^{-1}x+{\cos }^{-1}y=\frac{\pi }{2}$.
  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 78
  What are the order and degree, respectively, of the differential equation whose solution is $y=cx+c^2-3c^{3/2}+2$, 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 ${\int }_{-2}^{2}xdx-{\int }_{-2}^2\left[x\right] dx$ equal to, where [.] is greatest integer function?

  Option B: 1

  Option C: 2

  Option D: 4

  VIEW SOLUTION

• Question 80
  If ${\int }_{-2}^{5}f\left(x\right)dx=4 \mathrm{and}{\int }_0^5\{1+f\left(x\right)\} dx=7$ 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

  VIEW SOLUTION

• Question 81
  If $\underset{x\to 0}{\lim } \varphi \left(x\right)=a^2$, where a ≠ 0, then what is $\underset{x\to 0}{\lim } \varphi \left(\frac{x}{a}\right)$ equal to?

  Option A: a²

  Option B: a⁻²

  Option C: − a²

  Option D: − a

  VIEW SOLUTION

• Question 82
  What is $\underset{x\to 0}{\lim } e^{-\frac{1}{x^2}}$ 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 $\text{'}nRm\iff n$ 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 ${\int }_0^{4\pi }|\cos x|dx$ 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 $\tan \left(\frac{A}{2}+C\right)<\tan A$
  3. If ABC is any triangle, then $\tan \left(\frac{A+B}{2}\right) \sin \left(\frac{C}{2}\right)<\cos \left(\frac{C}{2}\right)$
  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 $P\left(A\right)=\frac{1}{3} \mathrm{and} P\left(B\right)=\frac{3}{4}$. 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, $\Sigma x_i=1000 \mathrm{and} \Sigma x_i^2=84000$, 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 $P\left(\overline{A}\cap B\right)=0.3$. What is equal to $P\left(A|\left(A\cup B\right)\right)$ equal to?

  Option A: $\frac{1}{2}$

  Option B: $\frac{{\displaystyle 2}}{{\displaystyle 5}}$

  Option C: $\frac{{\displaystyle 2}}{{\displaystyle 7}}$

  Option D: $\frac{{\displaystyle 2}}{{\displaystyle 3}}$

  VIEW SOLUTION

• Question 100
  What is the probability of 5 Sundays in the month of December?

  Option A: $\frac{1}{7}$

  Option B: $\frac{{\displaystyle 2}}{{\displaystyle 7}}$

  Option C: $\frac{{\displaystyle 3}}{{\displaystyle 7}}$

  Option D: None of the above

  VIEW SOLUTION

• Question 101
  If m is the geometric mean of ${\left(\frac{y}{z}\right)}^{\log \left(yz\right)},{\left(\frac{z}{x}\right)}^{\log \left(zx\right)}\mathrm{and} {\left(\frac{x}{y}\right)}^{\log \left(xy\right)}$ 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: $\frac{{\displaystyle 1}}{{\displaystyle 3}}$

  Option C: $\frac{{\displaystyle 1}}{{\displaystyle 4}}$

  Option D: $\frac{{\displaystyle 2}}{{\displaystyle 5}}$

  VIEW SOLUTION

• Question 103
  The mean of the series x₁, x₂,...,x_n 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: $\frac{{\displaystyle n\overline{X}-x_2+\lambda }}{n}$

  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: x ≠ y = z

  Option C: x ≠ y ≠ z

  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 (l − m) 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 $\hat{i}+\hat{j}+\hat{k} \mathrm{and} 2\hat{i}+3\hat{i}-\hat{k}$ ?

  Option A: $\frac{-4\hat{i}+3\hat{j}-\hat{k}}{\sqrt{26}}$

  Option B: $\frac{-4\hat{i}+3\hat{j}+\hat{k}}{\sqrt{26}}$

  Option C: $\frac{-3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}$

  Option D: $\frac{-3\hat{i}+2\hat{j}+\hat{k}}{\sqrt{14}}$

  VIEW SOLUTION

• Question 111
  If  $\overrightarrow{a},\overrightarrow{b} \mathrm{and} \overrightarrow{c}$ 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: $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$

  Option B: $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\mathrm{unit} \mathrm{vector}$

  Option C: $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$

  Option D: $\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}$

  VIEW SOLUTION

• Question 112
  What is the area of the parallelogram having diagonals $3\hat{i}+\overrightarrow{j}-\overrightarrow{2k} \mathrm{and} \hat{i}-3\hat{j}+4\overrightarrow{k}$?

  Option A: $5\sqrt{5} \mathrm{square} \mathrm{units}$

  Option B: $4\sqrt{5} \mathrm{square} \mathrm{units}$

  Option C: $5\sqrt{3} \mathrm{square} \mathrm{units}$

  Option D: $15\sqrt{2} \mathrm{square} \mathrm{units}$

  VIEW SOLUTION

• Question 113
  Consider the following in respect to the matrix $A=\left[\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right]$:
  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 $y-\sqrt{3}x-5=0 \mathrm{and} \sqrt{3}y-x+6=0$?

  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: $k\ne 1 \mathrm{and} k\ne -2$

  Option B: $k\ne 1 \mathrm{and} k\ne 2$

  Option C: $k\ne -1 \mathrm{and} k\ne -2$

  Option D: $k\ne -1 \mathrm{and} k\ne 2$

  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 log_a(ab) = x, then what is log_b(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 + log_x 10 + log_x 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 (ω₁ − ω₂)² equal to?

  Option A: 3

  Option B: 1

  Option C: –1

  Option D: –3

  VIEW SOLUTION