NDA I 2015 Mathematics

• Question 1
  Let X be the set of all the persons living in a city.  Persons $x, y$ in X are said to be related as $x<y$ if $y$ is at least 5 years older than $x$. Which of the following is correct?

  Option A: The relation is an an equivalence relation on X

  Option B: The relation is transitive but neither reflexive nor symmetric

  Option C: The relation is reflexive but neither transitive nor symmetric

  Option D: The relation is symmetric but neither transitive nor reflexive

  VIEW SOLUTION

• Question 2
  Which one of the following matrices is an elementary matrix?

  Option A: $\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$

  Option B: $\left[\begin{array}{ccc}1& 5& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

  Option C: $\left[\begin{array}{ccc}0& 2& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$

  Option D: $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 5& 2\end{array}\right]$

  VIEW SOLUTION

• Question 3
  Consider the following statements in respect to the given equation :
  ${(x^2+2)}^2+8x^2=6x(x^2+2)$
  1. All the roots of the equation are complex.
  2. The sum of all the roots of the equation is 6.
  
  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 4
  In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds – 9 and –1 for roots. The correct equation is

  Option A: $x^2-10x+9=0$

  Option B: $x^2+10x+9=0$

  Option C: $x^2-10x+16=0$

  Option D: $x^2-8x-9=0$

  VIEW SOLUTION

• Question 5
  If $A=\left[\begin{array}{cc}2& 7\\ 1& 5\end{array}\right]$ then what is A + 3A^–1 equal to?

  Option A: 3I

  Option B: 5I

  Option C: 7I

  Option D: None of the above

  VIEW SOLUTION

• Question 6
  In a class of 60 students, 45 students like music, 50 students like dancing, 5 students like neither. Then the number of students in the class who like both music and dancing is

  Option A: 35

  Option B: 40

  Option C: 50

  Option D: 55

  VIEW SOLUTION

• Question 7
  If log₁₀ 2, log₁₀ (2^x – 1) and log₁₀ (2^x + 3) are three consecutive terms of an AP, then the value of x is

  Option A: 1

  Option B: Log₅ 2

  Option C: Log₂ 5

  Option D: Log₁₀ 5

  VIEW SOLUTION

• Question 8
  The matrix $\left[\begin{array}{cc}0& -4+i\\ 4+i& 0\end{array}\right]$  is

  Option A: Symmetric

  Option B: Skew-Symmetric

  Option C: Hermitian

  Option D: Skew-Hermitian

  VIEW SOLUTION

• Question 9
  Let Z be the set of integers and aRb, where $a, b\in Z$ if and only if (a – b) is divisible by 5.
  Consider the following statements:
  1. The relation R partitions Z into five equivalent classes.
  2. Any two equivalent classes are either equal or disjoint.
  
   Which of the above statements is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 10
  If $z=\frac{-2\left(1+2i\right)}{3+i}$ where $i=\sqrt{-1}$, then the argument θ (–π < θ ≤ π) of z is

  Option A: $\frac{3\pi }{4}$

  Option B: $\frac{\pi }{4}$

  Option C: $\frac{5\pi }{6}$

  Option D: $-\frac{3\pi }{4}$

  VIEW SOLUTION

• Question 11
  If m and n are the roots of the equation (x + p) (x + q) – k = 0, then the roots of equation  (x – m) (x – n) + k = 0 are

  Option A: p and q

  Option B: $\frac{1}{p} \mathrm{and} \frac{1}{q}$

  Option C: –p and –q

  Option D: p + q and p – q

  VIEW SOLUTION

• Question 12
  What is the sum of the series 0.5 + 0.55 + 0.555 + ... to n terms?

  Option A: $\frac{5}{9}\left[n-\frac{2}{9}\left(1-\frac{1}{{10}^n}\right)\right]$

  Option B: $\frac{1}{9}\left[5-\frac{2}{9}\left(1-\frac{1}{{10}^n}\right)\right]$

  Option C: $\frac{1}{9}\left[n-\frac{5}{9}\left(1-\frac{1}{{10}^n}\right)\right]$

  Option D: $\frac{5}{9}\left[n-\frac{1}{9}\left(1-\frac{1}{{10}^n}\right)\right]$

  VIEW SOLUTION

• Question 13
  If 1, ω, ω² are the cube roots of unity, then the value of  $(1+\omega )(1+{\omega }^2)(1+{\omega }^4)(1+{\omega }^8)$

  Option A: –1

  Option C: 1

  Option D: 2

  VIEW SOLUTION

• Question 14
  Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing exactly two elements is

  Option A: 20

  Option B: 40

  Option C: 45

  Option D: 90

  VIEW SOLUTION

• Question 15
  What is the square root of i, where $i=\sqrt{-1}$?

  Option A: $\frac{1+i}{2}$

  Option B: $\frac{1-i}{2}$

  Option C: $\frac{1+i}{\sqrt{2}}$

  Option D: None of the above

  VIEW SOLUTION

• Question 16
  The decimal number (127.25)₁₀, when converted to binary number, takes the form

  Option A: (1111111.11)₂

  Option B: (1111110.01)₂

  Option C: (1110111.11)₂

  Option D: (1111111.01)₂

  VIEW SOLUTION

• Question 17
  Consider the following in respect of two non-singular matrices A and B of same order:
  1. det (A + B) = det A + det B
  2. (A + B)^–1 = A^–1 + B^–1
  Which of the above statements is/are correct?

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 18
  If $X=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right], B=\left[\begin{array}{cc}5& 2\\ -2& 1\end{array}\right] \mathrm{and} A=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$ satisfies the equation AX = B, then the matrix A is equal to

  Option A: $\left[\begin{array}{cc}-7& 26\\ 1& -5\end{array}\right]$

  Option B: $\left[\begin{array}{cc}-7& 26\\ -6& 23\end{array}\right]$

  Option C: $\left[\begin{array}{cc}-7& -4\\ 26& 13\end{array}\right]$

  Option D: $\left[\begin{array}{cc}-7& 26\\ 6& 23\end{array}\right]$

  VIEW SOLUTION

• Question 19
  What is $\sum _{r=0}^1{}^{n+r}C_n$ equal to?

  Option A: ${}^{n+2}{C}_1$

  Option B: ${}^{n+2}{C}_n$

  Option C: ${}^{n+3}{C}_n$

  Option D: ${}^{n+2}{C}_{n+1}$

  VIEW SOLUTION

• Question 20
  How many words can be formed using all the letters of the word ‘NATION’ so that all the three vowels should never come together?

  Option A: 354

  Option B: 348

  Option C: 288

  Option D: None of the above

  VIEW SOLUTION

• Question 21
  (x³ – 1) can be factorised as
  where ω is one of the cube roots of unity.

  Option A: $(x-1)(x-\omega )(x+{\omega }^2)$

  Option B: $(x-1)(x-\omega )(x-{\omega }^2)$

  Option C: $(x-1)(x+\omega )(x+{\omega }^2)$

  Option D: $(x-1)(x+\omega )(x-{\omega }^2)$

  VIEW SOLUTION

• Question 22
  What is ${\left[\frac{\sin \frac{\pi }{6}+i\left(1-\cos \frac{\pi }{6}\right)}{\sin \frac{\pi }{6}-i\left(1-\cos \frac{\pi }{6}\right)}\right]}^3$ where $i=\sqrt{-1}$, equal to?

  Option A: 1

  Option B: –1

  Option C: i

  Option D: –i

  VIEW SOLUTION

• Question 23
  Let $A=\left[\begin{array}{cc}x+y& y\\ 2x& x-y\end{array}\right], B=\left[\begin{array}{c} 2\\ -1\end{array}\right] \mathrm{and} C=\left[\begin{array}{c}3\\ 2\end{array}\right]$
   If AB = C, then what is A² equal to?

  Option A: $\left[\begin{array}{cc}6& -10\\ 4& 26\end{array}\right]$

  Option B: $\left[\begin{array}{cc}-10& 5\\ 4& 24\end{array}\right]$

  Option C: $\left[\begin{array}{cc}-5& -6\\ -4& -20\end{array}\right]$

  Option D: $\left[\begin{array}{cc}-5& -7\\ -5& -20\end{array}\right]$

  VIEW SOLUTION

• Question 24
  The value of $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$  is

  Option A: x + y

  Option B: x – y

  Option C: xy

  Option D: 1 + x + y

  VIEW SOLUTION

• Question 25
  If A = {x  :x is a multiple of 3} and B = {x : x is a multiple of 4} and C = {x : x is a multiple of 12}, then which one of the following is a null set?

  Option A: (A \ B) ⋃ C

  Option B: (A \ B) \ C

  Option C: (A ⋂ B) ⋂ C

  Option D: (A ⋂ B) \ C

  VIEW SOLUTION

• Question 26
  If (11101011)₂ is converted to the decimal system, then the resulting number is

  Option A:  235

  Option B: 175

  Option C: 160

  Option D: 126

  VIEW SOLUTION

• Question 27
  What is the real part of (sin x + i cos x)³ where $i=\sqrt{-1}$ ?

  Option A: – cos 3x

  Option B: – sin 3x

  Option C: sin 3x

  Option D: cos 3x

  VIEW SOLUTION

• Question 28
  If  $E\left(\mathrm{\theta }\right)=\left[\begin{array}{cc}\cos \mathrm{\theta }& \sin \theta \\ -\sin \mathrm{\theta }& \cos \mathrm{\theta }\end{array}\right] \mathrm{then} E\left(\mathrm{\alpha }\right) E\left(\mathrm{\beta }\right)$ is equal to

  Option A: E(αβ)

  Option B: E(α – β)

  Option C: E(α + β)

  Option D: –E(α + β)

  VIEW SOLUTION

• Question 29
  If A = {x, y, z) and B = {p, q, r, s). What is the number of distinct relations from B to A?

  Option A: 4096

  Option B: 4094

  Option C: 128

  Option D: 126

  VIEW SOLUTION

• Question 30
  If 2p + 3q = 18 and 4p² + 4pq – 3q² – 36 = 0, then what is (2p + q) equal to?

  Option A: 6

  Option B: 7

  Option C: 10

  Option D: 20

  VIEW SOLUTION

• Question 31
  Let θ be a positive angle. If the number of degrees in θ is divided by the number of radians in θ, then an irrational number 180/ π results. If the number of degrees in θ is multiplied by the number of radians in θ, then an irrational number 125π/9 results. The angle θ must be equal to

  Option A: 30º

  Option B: 45º

  Option C: 50º

  Option D: 60º

  VIEW SOLUTION

• Question 32
  In a triangle ABC,  $a=(1+\sqrt{3})$ cm, b = 2 cm and angle C = 60º. Then the other two angles are

  Option A: 45º and 75º

  Option B: 30º and 90º

  Option C: 105º and 15º

  Option D: 100º and 20º

  VIEW SOLUTION

• Question 33
  Let α be the root of the equation 25cos²θ + 5cosθ – 12 = 0, where  $\frac{\pi }{2}<\alpha <\pi$.
  What is tanα equal to?

  Option A: $-\frac{3}{4}$

  Option B: $\frac{3}{4}$

  Option C: $-\frac{4}{3}$

  Option D: $-\frac{4}{5}$

  VIEW SOLUTION

• Question 34
  Let α be the root of the equation 25cos²θ + 5 cosθ – 12 = 0, where  $\frac{\pi }{2}<\alpha <\pi$.
  What is sin 2α equal to?

  Option A: $\frac{24}{25}$

  Option B: $-\frac{24}{25}$

  Option C: $-\frac{5}{12}$

  Option D: $-\frac{21}{25}$

  VIEW SOLUTION

• Question 35
  The angle of elevation of the top of a tower from a point 20 m away from its base is 45º. What is the height of the tower?

  Option A: 10 m

  Option B: 20 m

  Option C: 30 m

  Option D: 40 m

  VIEW SOLUTION

• Question 36
  The equation ${\tan }^{-1}(1+x)+{\tan }^{-1}(1-x)=\frac{\pi }{2}$ is satisfied by

  Option A: x = 1

  Option B: x = – 1

  Option C: x = 0

  Option D: $x=\frac{1}{2}$

  VIEW SOLUTION

• Question 37
  The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances 49 m and 36 m are 43º and 47º, respectively. What is the height of the tower?

  Option A:  40 m

  Option B: 42 m

  Option C: 45 m

  Option D: 47 m

  VIEW SOLUTION

• Question 38
  (1 – sin A + cos A)² is equal to

  Option A: 2(1 – cos A) (1 + sin A)

  Option B: 2(1 – sin A) (1 + cos A)

  Option C: 2(1 – cos A) (1 – sin A)

  Option D: None of the above

  VIEW SOLUTION

• Question 39
  What is $\frac{\cos \mathrm{\theta }}{1-\tan \mathrm{\theta }}+\frac{\sin \mathrm{\theta }}{1-\cot \mathrm{\theta }}$ equal to?

  Option A: sin θ – cos θ

  Option B: sinθ + cosθ

  Option C: 2sin θ

  Option D: 2cos θ

  VIEW SOLUTION

• Question 40
  Consider $x=4 {\tan }^{-1}\left(\frac{1}{5}\right), y={\tan }^{-1}\left(\frac{1}{70}\right) \mathrm{and} z={\tan }^{-1}\left(\frac{1}{99}\right)$.
  What is x equal to?

  Option A: ${\tan }^{-1}\left(\frac{60}{119}\right)$

  Option B: ${\tan }^{-1}\left(\frac{120}{119}\right)$

  Option C: ${\tan }^{-1}\left(\frac{90}{169}\right)$

  Option D: ${\tan }^{-1}\left(\frac{170}{169}\right)$

  VIEW SOLUTION

• Question 41
  Consider $x=4 {\tan }^{-1}\left(\frac{1}{5}\right), y={\tan }^{-1}\left(\frac{1}{70}\right) \mathrm{and} z={\tan }^{-1}\left(\frac{1}{99}\right)$.
  What is x – y equal to?

  Option A: ${\tan }^{-1}\left(\frac{828}{845}\right)$

  Option B: ${\tan }^{-1}\left(\frac{8287}{8450}\right)$

  Option C: ${\tan }^{-1}\left(\frac{8281}{8450}\right)$

  Option D: ${\tan }^{-1}\left(\frac{8287}{8471}\right)$

  VIEW SOLUTION

• Question 42
  Consider $x=4 {\tan }^{-1}\left(\frac{1}{5}\right), y={\tan }^{-1}\left(\frac{1}{70}\right) \mathrm{and} z={\tan }^{-1}\left(\frac{1}{99}\right)$.
  What is x – y + z equal to?

  Option A: $\frac{\pi }{2}$

  Option B: $\frac{\pi }{3}$

  Option C: $\frac{\pi }{6}$

  Option D: $\frac{\pi }{4}$

  VIEW SOLUTION

• Question 43
  Consider the triangle ABC with vertices A(–2, 3), B(2, 1) and C(1, 2).
  What is the circumcentre of triangle ABC?

  Option A: (–2, –2)

  Option B: (2, 2)

  Option C: (–2, 2)

  Option D: (2, –2)

  VIEW SOLUTION

• Question 44
  Consider the triangle ABC with vertices A(–2, 3), B(2, 1) and C(1, 2).
  What is the centroid of triangle ABC?

  Option A: $\left(\frac{1}{3}, 1\right)$

  Option B: $\left(\frac{1}{3}, 2\right)$

  Option C: $\left(1, \frac{2}{3}\right)$

  Option D: $\left(\frac{1}{2}, 3\right)$

  VIEW SOLUTION

• Question 45
  Consider the triangle ABC with vertices A(–2, 3), B(2, 1) and C(1, 2).
  What is the foot of the altitude from the vertex A of triangle ABC?

  Option A: (1, 4)

  Option B: (–1, 3)

  Option C: (–2, 4)

  Option D: (–1, 4)

  VIEW SOLUTION

• Question 46
  The point on the parabola y² = 4ax nearest to the focus has its abscissa.

  Option A: x = 0

  Option B: x = a

  Option C: $x=\frac{a}{2}$

  Option D: x = 2a

  VIEW SOLUTION

• Question 47
  A line passes through (2, 2) and is perpendicular to line 3x + y = 3. Its y-intercept is

  Option A: $\frac{3}{4}$

  Option B: $\frac{{\displaystyle 4}}{{\displaystyle 3}}$

  Option C: $\frac{{\displaystyle 1}}{{\displaystyle 3}}$

  Option D: 3

  VIEW SOLUTION

• Question 48
  The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$  passes through the point $(3\sqrt{5},1)$ and the length of its latus rectum is $\frac{4}{3}$ units.
  The length of the conjugate axis is

  Option A: 2 units

  Option B: 3 units

  Option C: 4 units

  Option D: 5 units

  VIEW SOLUTION

• Question 49
  The perpendicular distances between the straight lines 6x + 8y + 15 = 0 and 3x + 4y + 9 = 0 is

  Option A: $\frac{3}{2} \mathrm{units}$

  Option B: $\frac{3}{10} \mathrm{unit}$

  Option C: $\frac{3}{4} \mathrm{unit}$

  Option D: $\frac{2}{7} \mathrm{unit}$

  VIEW SOLUTION

• Question 50
  The area of a triangle, whose vertices are (3, 4), (5, 2) and the point of intersection of the lines x = a and y = 5, is 3 square units. What is the value of a?

  Option A: 2

  Option B: 3

  Option C: 4

  Option D: 5

  VIEW SOLUTION

• Question 51
  The length of perpendicular from the origin to a line is 5 units and the line makes an angle 120º with the positive direction of x-axis. The equation of the line is

  Option A: $x+\sqrt{3}y=5$

  Option B: $\sqrt{3}x+y=10$

  Option C: $\sqrt{3}x-y=10$

  Option D: None of the above

  VIEW SOLUTION

• Question 52
  The equation of the line joining the origin to the point of intersection of the lines  $\frac{x}{a}+\frac{y}{b}=1 \mathrm{and} \frac{x}{b}+\frac{y}{a}=1$ is

  Option A: x – y = 0

  Option B: x + y = 0

  Option C: x = 0

  Option D: y = 0

  VIEW SOLUTION

• Question 53
  The projections of a directed line segment on the coordinate axes are 12, 4, 3, respectively.
  What is the length of the line segment?

  Option A: 19 units

  Option B: 17 units

  Option C: 15 units

  Option D: 13 units

  VIEW SOLUTION

• Question 54
  The projections of a directed line segment on the coordinate axes are 12, 4, 3, respectively.
  What are the direction cosines of the line segment?

  Option A: $⟨\frac{12}{13}, \frac{4}{13}, \frac{3}{13}⟩$

  Option B: $⟨\frac{12}{13}, -\frac{4}{13}, \frac{3}{13}⟩$

  Option C: $⟨\frac{12}{13}, -\frac{4}{13}, -\frac{3}{13}⟩$

  Option D: $⟨-\frac{12}{13}, -\frac{4}{13}, \frac{3}{13}⟩$

  VIEW SOLUTION

• Question 55
  From the point P(3, –1, 11), a perpendicular is drawn on the line L given by the equation $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$.  Let Q be the foot of the perpendicular.
  What are the direction ratios of the line segment PQ?

  Option A: $⟨1, 6, 4⟩$

  Option B: $⟨-1, 6,-4⟩$

  Option C: $⟨-1,-6, 4⟩$

  Option D: $⟨2,-6, 4⟩$

  VIEW SOLUTION

• Question 56
  From the point P(3, –1, 11), a perpendicular is drawn on the line L given by the equation $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$.  Let Q be the foot of the perpendicular.
  What is the length of the line segment PQ?

  Option A: $\sqrt{47} \mathrm{units}$

  Option B: 7 units

  Option C: $\sqrt{53} \mathrm{units}$

  Option D: 8 units

  VIEW SOLUTION

• Question 57
  A triangular plane ABC with centroid (1, 2, 3) cuts the coordinate axes at A, B, C, respectively.
  What are the intercepts made by plane ABC on the axes?

  Option A: 3, 6, 9

  Option B: 1, 2, 3

  Option C: 1, 4, 9

  Option D: 2, 4, 6

  VIEW SOLUTION

• Question 58
  A triangular plane ABC with centroid (1, 2, 3) cuts the coordinate axes at A, B, C, respectively.
  What is the equation of plane ABC?

  Option A: x + 2y + 3z = 1

  Option B: 3x + 2y + z = 3

  Option C: 2x + 3y + 6z = 18

  Option D: 6x + 3y + 2z = 8

  VIEW SOLUTION

• Question 59
  A point P (1, 2, 3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes.
  What is the length of one of the diagonals of the cuboid?

  Option A: $\sqrt{10} \mathrm{units}$

  Option B: $\sqrt{14} \mathrm{units}$

  Option C: 4 units

  Option D: 5 units

  VIEW SOLUTION

• Question 60
  A point P (1, 2, 3) is one vertex of a cuboid formed by the coordinate planes and the planes passing through P and parallel to the coordinate planes.
  What is the equation of the plane passing through P(1, 2, 3) and parallel to xy-plane?

  Option A: x + y = 3

  Option B: x – y = –1

  Option C: z = 3

  Option D: x + 2y + 3z = 14

  VIEW SOLUTION

• Question 61
  If $G\left(x\right)=\sqrt{25-x^2}$, then what is $\underset{x\to 1}{\lim }\frac{G\left(x\right)-G\left(1\right)}{x-1}$ equal to?

  Option A: $-\frac{1}{2\sqrt{6}}$

  Option B: $\frac{{\displaystyle 1}}{{\displaystyle 5}}$

  Option C: $-\frac{1}{\sqrt{6}}$

  Option D: $\frac{1}{\sqrt{6}}$

  VIEW SOLUTION

• Question 62
  Consider the following statements:
   1. $y=\frac{e^x+e^{-x}}{2}$is an increasing function on [0, ∞).
   2. $y=\frac{e^x-e^{-x}}{2}$is an increasing function on (–∞, ∞).
   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 63
  For each non-zero real number x, let f(x) = $\frac{x}{\left|x\right|}$. The range of f is

  Option A: a null set

  Option B: a set consisting of only one element

  Option C: a set consisting of two elements

  Option D: a set consisting of infinitely many elements

  VIEW SOLUTION

• Question 64
  Consider the following statements:
  1. f(x) = [x], where [.] is the greatest integer function, is discontinuous at x = n, where $n \in Z$
  2. f(x) = cot x is discontinuous at x = nπ, where $n \in Z$.
   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 65
  What is the derivative of  ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to tan^–1 x?

  Option B: $\frac{1}{2}$

  Option C: 1

  Option D: x

  VIEW SOLUTION

• Question 66
  If $f\left(x\right)={\log }_e\left(\frac{1+x}{1-x}\right), g\left(x\right)=\frac{3x+x^3}{1+3x^2}$  and g$\circ$f(t) = g(f(t)), then what is $g\circ f\left(\frac{e-1}{e+1}\right)$ equal to?

  Option A: 2

  Option B: 1

  Option D: $\frac{1}{2}$

  VIEW SOLUTION

• Question 67
  Given a function $f\left(x\right)=\left\{\begin{array}{ccc}-1& \mathrm{if}& x\le 0\\ ax+b& \mathrm{if}& 0<x<1\\ 1& \mathrm{if}& x\ge 1\end{array}\right.$where a, b are constants, the function is continuous everywhere.
  
  What is the value of a?

  Option A: –1

  Option C: 1

  Option D: 2

  VIEW SOLUTION

• Question 68
  Given a function $f\left(x\right)=\left\{\begin{array}{ccc}-1& \mathrm{if}& x\le 0\\ ax+b& \mathrm{if}& 0<x<1\\ 1& \mathrm{if}& x\ge 1\end{array}\right.$where a, b are constants, the function is continuous everywhere.
  
  What is the value of b?

  Option A: –1

  Option B: 1

  Option D: 2

  VIEW SOLUTION

• Question 69
  Consider the following functions:
  1. $f\left(x\right)=x^3, x\in \mathrm{ℝ}$
  2. $f\left(x\right)=\sin x, 0<x<2\pi$
  3. $f\left(x\right)=e^x, x\in \mathrm{ℝ}$
  
  Which of the above functions have inverse defined on their ranges?

  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 70
  The integral $\int \frac{dx}{a \cos x+b \sin x}$ is of the form $\frac{1}{r} \ln \left[\tan \left(\frac{x+\alpha }{2}\right)\right].$
  
  What is r equal to?

  Option A: a² + b²

  Option B: $\sqrt{a^2+b^2}$

  Option C: a + b

  Option D: $\sqrt{a^2-b^2}$

  VIEW SOLUTION

• Question 71
  The integral $\int \frac{dx}{a \cos x+b \sin x}$ is of the form $\frac{1}{r} \ln \left[\tan \left(\frac{x+\alpha }{2}\right)\right].$
  
  What is α equal to?

  Option A: ${\tan }^{-1}\left(\frac{a}{b}\right)$

  Option B: ${\tan }^{-1}\left(\frac{b}{a}\right)$

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

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

  VIEW SOLUTION

• Question 72
  Consider the function $f\left(x\right)=\frac{x^2-1}{x^2+1},$ where $x\in \mathrm{ℝ}$.
  
  At what value of x does f(x) attain minimum value?

  Option A: –1

  Option C: 1

  Option D: 2

  VIEW SOLUTION

• Question 73
  Consider the function $f\left(x\right)=\frac{x^2-1}{x^2+1},$ where $x\in \mathrm{ℝ}$.
  
  What is the minimum value of f(x)?

  Option B: 1/2

  Option C: –1

  Option D: 2

  VIEW SOLUTION

• Question 74
  Consider the function $f\left(x\right)=\left\{\begin{array}{ccc}\frac{\alpha \cos x}{\pi -2x}& \mathrm{if}& x\ne \frac{\pi }{2}\\ 3& \mathrm{if}& x=\frac{\pi }{2}\end{array}\right.\begin{array}{}\\ \end{array}$which is continuous at $x=\frac{\pi }{2},$ where α is a constant.
  
  What is the value of α?

  Option A: 6

  Option B: 3

  Option C: 2

  Option D: 1

  VIEW SOLUTION

• Question 75
  Consider the function $fx=\left\{\begin{array}{ccc}\frac{\alpha \cos x}{\pi -2x}& \mathrm{if}& x\ne \frac{\pi }{2}\\ 3& \mathrm{if}& x=\frac{\pi }{2}\end{array}\right.\begin{array}{}\\ \end{array}$which is continuous at $x=\frac{\pi }{2},$ where α is a constant.
  
  What is $\underset{x\to 0}{\lim } f\left(x\right)$ equal to?

  Option B: 3

  Option C: $\frac{3}{\pi }$

  Option D: $\frac{6}{\pi }$

  VIEW SOLUTION

• Question 76
  Consider the line $x=\sqrt{3}y$ and the circle x² + y² = 4.
  
  What is the area of the region in the first quadrant enclosed by the x-axis, the line $x=\sqrt{3}$ and the circle?

  Option A: $\frac{\pi }{3}-\frac{\sqrt{3}}{2}$

  Option B: $\frac{\pi }{2}-\frac{\sqrt{3}}{2}$

  Option C: $\frac{\pi }{3}-\frac{1}{2}$

  Option D: None of the above

  VIEW SOLUTION

• Question 77
  Consider the line $x=\sqrt{3}y$ and the circle x² + y² = 4.
  
  What is the area of the region in the first quadrant enclosed by the x-axis, the line $x=\sqrt{3}y$ and the circle?

  Option A: $\frac{\pi }{3}$

  Option B: $\frac{\pi }{6}$

  Option C: $\frac{\pi }{3}-\frac{\sqrt{3}}{2}$

  Option D: None of the above

  VIEW SOLUTION

• Question 78
  Consider the curves y = sin x and y = cos x.
  
  What is the area of the region bounded by the above two curves and the lines x = 0 and $x=\frac{\pi }{4}$?

  Option A: $\sqrt{2}-1$

  Option B: $\sqrt{2}+1$

  Option C: $\sqrt{2}$

  Option D: 2

  VIEW SOLUTION

• Question 79
  Consider the curves y = sin x and y = cos x.
  
  What is the area of the region bounded by the above two curves and the lines $x=\frac{\pi }{4}$ and $x=\frac{\pi }{2}$?

  Option A: $\sqrt{2}-1$

  Option B: $\sqrt{2}+1$

  Option C: $2\sqrt{2}$

  Option D: 2

  VIEW SOLUTION

• Question 80
  Consider the function f(x) = 0·75x⁴ – x³ – 9x² + 7
  
  What is the maximum value of the function?

  Option A: 1

  Option B: 3

  Option C: 7

  Option D: 9

  VIEW SOLUTION

• Question 81
  Consider the function f(x) = 0·75x⁴ – x³ – 9x² + 7
  
  Consider the following statements :
  
  1. The function attains local minima at x = – 2 and x = 3.
  
  2. The function increases in the interval (–2, 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 82
  Consider the parametric equation
  
  $x=\frac{a(1-t^2)}{1+t^2}, y=\frac{2at}{1+t^2}$
  
  What does the equation represent?

  Option A: It represents a circle of diameter a

  Option B: It represents a circle of radius a

  Option C: It represents a parabola

  Option D: None of the above

  VIEW SOLUTION

• Question 83
  Consider the parametric equation
  
  $x=\frac{a(1-t^2)}{1+t^2}, y=\frac{2at}{1+t^2}$
  
  What is $\frac{dy}{dx}$ equal to?

  Option A: $\frac{y}{x}$

  Option B: $-\frac{y}{x}$

  Option C: $\frac{x}{y}$

  Option D: $-\frac{x}{y}$

  VIEW SOLUTION

• Question 84
  Consider the parametric equation
  
  $x=\frac{a(1-t^2)}{1+t^2}, y=\frac{2at}{1+t^2}$
  
  What is $\frac{d^{2}y}{d{x}^2}$ equal to?

  Option A: $\frac{a^2}{y^2}$

  Option B: $\frac{a^2}{x^2}$

  Option C: $-\frac{a^2}{x^2}$

  Option D: $-\frac{a^2}{y^3}$

  VIEW SOLUTION

• Question 85
  Consider the following statements:
  1. The general solution of $\frac{dy}{dx}=f\left(x\right)+x$ is of the form y = g(x) + c, where c is an arbitrary constant.
  2. The degree of ${\left(\frac{dy}{dx}\right)}^2=f\left(x\right)$ is 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 86
  What is$\int \frac{dx}{\sqrt{x^2+a^2}}$ equal to?
  
  where c is the constant of integration.

  Option A: $\ln \left|\frac{x+\sqrt{x^2+a^2}}{a}\right|+c$

  Option B: $\ln \left|\frac{x-\sqrt{x^2+a^2}}{a}\right|+c$

  Option C: $\ln \left|\frac{x^2+\sqrt{x^2+a^2}}{a}\right|+c$

  Option D: None of the above

  VIEW SOLUTION

• Question 87
  Consider the integral $I_m={\int }_0^{\pi }\frac{\sin 2mx}{\sin x}\hspace{0.17em}dx$, where m is a positive integer.
  What is I₁ equal to?

  Option B: 1/2

  Option C: 1

  Option D: 2

  VIEW SOLUTION

• Question 88
  Consider the integral $I_m={\int }_0^{\pi }\frac{\sin 2mx}{\sin x}\hspace{0.17em}dx$, where m is a positive integer.
  
  What is I₂ + I₃ equal to?

  Option A: 4

  Option B: 2

  Option C: 1

  VIEW SOLUTION

• Question 89
  Consider the integral $I_m={\int }_0^{\pi }\frac{\sin 2mx}{\sin x}\hspace{0.17em}dx$, where m is a positive integer.
  
  What is I_m equal to?

  Option B: 1

  Option C: m

  Option D: 2m

  VIEW SOLUTION

• Question 90
  Consider the integral $I_m={\int }_0^{\pi }\frac{\sin 2mx}{\sin x}\hspace{0.17em}dx$, where m is a positive integer.
  
  Consider the following:
  
  1. I_m – I_{m – 1} is equal to 0.
  
  2. I_2m > I_m
  
  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 91
  Given that $\frac{d}{dx}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=Ax+B.$
  
  What is the value of A?

  Option A: –1

  Option B: 1

  Option C: 2

  Option D: 4

  VIEW SOLUTION

• Question 92
  Given that $\frac{d}{dx}\left(\frac{1+x^2+x^4}{1+x+x^2}\right)=Ax+B.$
  
  What is the value of B?

  Option A: –1

  Option B: 1

  Option C: 2

  Option D: 4

  VIEW SOLUTION

• Question 93
  Given that$\underset{x\to \infty }{\lim } \left(\frac{2+x^2}{1+x}-Ax-B\right)=3.$
  
  What is the value of A?

  Option A: –1

  Option B: 1

  Option C: 2

  Option D: 3

  VIEW SOLUTION

• Question 94
  Given that$\underset{x\to \infty }{\lim } \left(\frac{2+x^2}{1+x}-Ax-B\right)=3.$
  
  What is the value of B?

  Option A: –1

  Option B: 3

  Option C: –4

  Option D: –3

  VIEW SOLUTION

• Question 95
  What is the solution of the differential equation
  
  $\frac{ydx-xdy}{y^2}=0?$
  
  Where c is an arbitrary constant.

  Option A: xy = c

  Option B: y = cx

  Option C: x + y = c

  Option D: x – y = c

  VIEW SOLUTION

• Question 96
  What is the solution of the differential equation
  
  $\sin \left(\frac{dy}{dx}\right)-a=0$
  
  Where c is an arbitrary constant.

  Option A: $y=x{\sin }^{-1}a+c$

  Option B: $x=y{\sin }^{-1}a+c$

  Option C: $y=x+x{\sin }^{-1}a+c$

  Option D: $y={\sin }^{-1}a+c$

  VIEW SOLUTION

• Question 97
  What is the solution of the differential equation
  
  $\frac{dx}{dy}+\frac{{\displaystyle x}}{{\displaystyle y}}-y^2=0?$
  
  Where c is the arbitrary constant.

  Option A: xy = x⁴ + c

  Option B: xy = y⁴ + c

  Option C: 4xy = y⁴ + c

  Option D: 3xy = y³ + c

  VIEW SOLUTION

• Question 98
  What is $\int \frac{x{e}^{x}dx}{{\left(x+1\right)}^2}$ equal to?
  
  Where c is the constant of integration.

  Option A: (x + 1)²e^x + c

  Option B: (x + 1) e^x + c

  Option C: $\frac{e^x}{x+1}+c$

  Option D: $\frac{e^x}{{(x+1)}^2}+c$

  VIEW SOLUTION

• Question 99
  The adjacent sides AB and AC of a triangle ABC are represented by the vectors $-2\hat{i}+3\hat{j}+2\hat{k}$ and $-4\hat{i}+5\hat{j}+2\hat{k}$ respectively. The area of triangle ABC is

  Option A: 6 square units

  Option B: 5 square units

  Option C: 4 square units

  Option D: 3 square units

  VIEW SOLUTION

• Question 100
  A force $\overrightarrow{F}=3\hat{i}+4\hat{j}-3\hat{k}$ is applied at the point P, whose position vector is $\overrightarrow{r}=2\hat{i}-2\hat{j}-3\hat{k}$. What is the magnitude of the moment of the force about the origin?

  Option A: 23 units

  Option B: 19 units

  Option C: 18 units

  Option D: 21 units

  VIEW SOLUTION

• Question 101
  Given that the vectors $\overrightarrow{\alpha }$ and $\overrightarrow{\beta }$ are non-collinear. The values of x and y for which $\overrightarrow{u}-\overrightarrow{v}=\overrightarrow{w}$ holds true if $\overrightarrow{u}=2x\overrightarrow{\alpha }+y\overrightarrow{\beta }, \overrightarrow{v}=2y\overrightarrow{\alpha }+3x\overrightarrow{\beta }$ and $\overrightarrow{w}=2\overrightarrow{\alpha }-5\overrightarrow{\beta },$ are

  Option A: x = 2, y = 1

  Option B: x = 1, y = 2

  Option C: x = –2, y = 1

  Option D: x = –2, y = –1

  VIEW SOLUTION

• Question 102
  If $\left|\overrightarrow{a}\right|=7,\left|\overrightarrow{b}\right|=11\hspace{0.17em}\mathrm{and}\hspace{0.17em}\left|\overrightarrow{a}+\overrightarrow{b}\right|=10\sqrt{3},$ then $\left|\overrightarrow{a}-\overrightarrow{b}\right|$ is equal to

  Option A: 40

  Option B: 10

  Option C: $4\sqrt{10}$

  Option D: $2\sqrt{10}$

  VIEW SOLUTION

• Question 103
  Let α, β, γ be distinct real numbers. The points with position vectors $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k},\beta \hat{i}+\gamma \hat{j}+\alpha \hat{k}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma \hat{i}+\alpha \hat{j}+\beta \hat{k}$

  Option A: are collinear

  Option B: form an equilateral triangle

  Option C: form a scalene triangle

  Option D: form a right-angled triangle

  VIEW SOLUTION

• Question 104
  If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0},$ then which of the following is/are correct?
  
  1. $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are coplanar.
  
  2. $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}$
  
  Select the correct answer using the code given below.

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 105
  If $\left|\overrightarrow{a}+\overrightarrow{b}\right|=\left|\overrightarrow{a}-\overrightarrow{b}\right|$, then which one of the following is correct?

  Option A: $\overrightarrow{a}=\lambda \overrightarrow{b}$ for some scalar λ

  Option B: $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$

  Option C: $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$

  Option D: $\overrightarrow{a}=\overrightarrow{b}=\overrightarrow{0}$

  VIEW SOLUTION

• Question 106
  The mean and variance of 10 observations are given to be 4 and 2, respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be, respectively

  Option A: 8 and 20

  Option B: 8 and 4

  Option C: 8 and 8

  Option D: 80 and 40

  VIEW SOLUTION

• Question 107
  Which one of the following measures of central tendency is used in construction of index numbers?

  Option A: Harmonic mean

  Option B: Geometric mean

  Option C: Median

  Option D: Mode

  VIEW SOLUTION

• Question 108
  The correlation coefficient between two variable X and Y is found to be 0·6. All the observations on X and Y are transformed using the transformation U = 2 – 3X and V = 4Y + 1. The correlation coefficient between the transformed variables U and V will be

  Option A: –0.5

  Option B: +0.5

  Option C: –0.6

  Option D: +0.6

  VIEW SOLUTION

• Question 109
  Which of the following statements is/are correct in respect of regression coefficients?
  1. It measures the degree of linear relationship between two variables.
  2. It gives the value by which one variable changes for a unit change in the other variable.
  Select the correct answer using the code given below.

  Option A: 1 only

  Option B: 2 only

  Option C: Both 1 and 2

  Option D: Neither 1 nor 2

  VIEW SOLUTION

• Question 110
  A set of annual numerical data, comparable over the years, is given for the last 12 years.
  Consider the following statements:
  1. The data is best represented by a broken line graph, each corner (turning point) representing the data of one year.
  2. Such a graph depicts the chronological change and also enables one to make a short-term forecast.
  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 111
  Two men hit at a target with probabilities 1/2 and 1/3, respectively. What is the probability that exactly one of them hits the target?

  Option A: 1/2

  Option B: 1/3

  Option C: 1/6

  Option D: 2/3

  VIEW SOLUTION

• Question 112
  Two similar boxes B_i (i = 1, 2) contain (i + 1) red and (5 – i – 1) black balls. One box is chosen at random and two balls are drawn randomly. What is the probability that both the balls are of different colours?

  Option A: 1/2

  Option B: 3/10

  Option C: 2/5

  Option D: 3/5

  VIEW SOLUTION

• Question 113
  In an examination, the probability of a candidate solving a question is 1/2. Out of the given 5 questions in the examination, what is the probability that the candidate was able to solve at least 2 questions?

  Option A: 1/64

  Option B: 3/16

  Option C: 1/2

  Option D: 13/16

  VIEW SOLUTION

• Question 114
  If $A\subseteq B$, then which one of the following is not correct?

  Option A: $P\left(A\cap \overline{B}\right)=0$

  Option B: $P\left(A|B\right)=\frac{P\left(A\right)}{P\left(B\right)}$

  Option C: $P\left(B|A\right)=\frac{P\left(B\right)}{P\left(A\right)}$

  Option D: $P\left(A|\left(A\cup B\right)\right)=\frac{P\left(A\right)}{P\left(B\right)}$

  VIEW SOLUTION

• Question 115
  The mean and the variance in a binomial distribution are found to be 2 and 1, respectively. The probability P(X = 0) is

  Option A: 1/2

  Option B: 1/4

  Option C: 1/8

  Option D: 1/16

  VIEW SOLUTION

• Question 116
  The mean of five numbers is 30. If one number is excluded, the mean becomes 28. The excluded number is

  Option A: 28

  Option B: 30

  Option C: 35

  Option D: 38

  VIEW SOLUTION

• Question 117
  If A and B are two events such that $P(A\cup B)=\frac{3}{4},\hspace{0.17em}\hspace{0.17em}P(A\cap B)=\frac{1}{4}\hspace{0.17em}\mathrm{and}\hspace{0.17em}P\left(\overline{A}\right)=\frac{2}{3}$, then what is P(B) equal to?

  Option A: 1/3

  Option B: 2/3

  Option C: 1/8

  Option D: 2/9

  VIEW SOLUTION

• Question 118
  The ‘less than’ ogive curve and the ‘more than’ ogive curve intersect at

  Option A: median

  Option B: mode

  Option C: arithmetic mean

  Option D: None of the above

  VIEW SOLUTION

• Question 119
  In throwing of two dice, the number of exhaustive events that ‘5’ will never appear on any one of the dice is

  Option A: 5

  Option B: 18

  Option C: 25

  Option D: 36

  VIEW SOLUTION

• Question 120
  Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. The probability of drawing two aces is

  Option A: 1/26

  Option B: 1/221

  Option C: 4/223

  Option D: 1/13

  VIEW SOLUTION