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# NDA I 2015 Mathematics

This test contains 120 question. Each question comprises four responses (answers). You need to select only ONE response for each question.

All questions carry equal marks.

Each question for which a wrong answer has been marked, one-third of the marks assigned to that question will be deducted as penalty.

If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens to be correct and there will be same penalty as above to
that question.

If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
• Question 1
Let X be the set of all the persons living in a city.  Persons  in X are said to be related as $x 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 :
${\left({x}^{2}+2\right)}^{2}+8{x}^{2}=6x\left({x}^{2}+2\right)$
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 (2x – 1) and log10 (2x + 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  if and only if (ab) 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  (xm) (xn) + k = 0 are
Option A: p and q
Option B:
Option C: p and –q
Option D: p + q and pq
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  $\left(1+\omega \right)\left(1+{\omega }^{2}\right)\left(1+{\omega }^{4}\right)\left(1+{\omega }^{8}\right)$
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 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: $\left(x-1\right)\left(x-\omega \right)\left(x+{\omega }^{2}\right)$
Option B: $\left(x-1\right)\left(x-\omega \right)\left(x-{\omega }^{2}\right)$
Option C: $\left(x-1\right)\left(x+\omega \right)\left(x+{\omega }^{2}\right)$
Option D: $\left(x-1\right)\left(x+\omega \right)\left(x-{\omega }^{2}\right)$
VIEW SOLUTION
• Question 22
What is ${\left[\frac{\mathrm{sin}\frac{\pi }{6}+i\left(1-\mathrm{cos}\frac{\pi }{6}\right)}{\mathrm{sin}\frac{\pi }{6}-i\left(1-\mathrm{cos}\frac{\pi }{6}\right)}\right]}^{3}$ where $i=\sqrt{-1}$, equal to?
Option A: 1
Option B: –1
Option C: i
Option D: i
VIEW SOLUTION
• Question 23
Let
If AB = C, then what is A2 equal to?
Option A:
Option B:
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   is
Option A: x + y
Option B: xy
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) â‹ƒ C
Option B: (A \ B) \ C
Option C: (A â‹‚ B) â‹‚ C
Option D: (A â‹‚ 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   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,  $a=\left(1+\sqrt{3}\right)$ 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 ${\mathrm{tan}}^{-1}\left(1+x\right)+{\mathrm{tan}}^{-1}\left(1-x\right)=\frac{\pi }{2}$ 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  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 .
What is x equal to?
Option A: ${\mathrm{tan}}^{-1}\left(\frac{60}{119}\right)$
Option B: ${\mathrm{tan}}^{-1}\left(\frac{120}{119}\right)$
Option C: ${\mathrm{tan}}^{-1}\left(\frac{90}{169}\right)$
Option D: ${\mathrm{tan}}^{-1}\left(\frac{170}{169}\right)$
VIEW SOLUTION
• Question 41
Consider .
What is xy equal to?
Option A: ${\mathrm{tan}}^{-1}\left(\frac{828}{845}\right)$
Option B: ${\mathrm{tan}}^{-1}\left(\frac{8287}{8450}\right)$
Option C: ${\mathrm{tan}}^{-1}\left(\frac{8281}{8450}\right)$
Option D: ${\mathrm{tan}}^{-1}\left(\frac{8287}{8471}\right)$
VIEW SOLUTION
• Question 42
Consider .
What is xy + 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:
Option B:
Option C:
Option D:
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: $\frac{4}{3}$
Option C: $\frac{1}{3}$
Option D: 3
VIEW SOLUTION
• Question 48
The hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$  passes through the point $\left(3\sqrt{5},1\right)$ 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:
Option B:
Option C:
Option D:
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   is
Option A: xy = 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:
Option B:
Option C:
Option D:
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:
Option B:
Option C:
Option D:
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:
Option B: 7 units
Option C:
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:
Option B:
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: xy = –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 $\underset{x\to 1}{\mathrm{lim}}\frac{G\left(x\right)-G\left(1\right)}{x-1}$ equal to?
Option A: $-\frac{1}{2\sqrt{6}}$
Option B: $\frac{1}{5}$
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) = $\frac{x}{|x|}$. 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
2. f(x) = cot x is discontinuous at x = , where .
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  ${\mathrm{tan}}^{-1}\left(\frac{\sqrt{1+{x}^{2}}-1}{x}\right)$ with respect to tan–1 x?
Option B: $\frac{1}{2}$
Option C: 1
Option D: x
VIEW SOLUTION
• Question 66
If   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}& 0where 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}& 0where 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.
2.
3.

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  is of the form

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  is of the form

What is α equal to?
Option A: ${\mathrm{tan}}^{-1}\left(\frac{a}{b}\right)$
Option B: ${\mathrm{tan}}^{-1}\left(\frac{b}{a}\right)$
Option C: ${\mathrm{tan}}^{-1}\left(\frac{a+b}{a-b}\right)$
Option D: ${\mathrm{tan}}^{-1}\left(\frac{a-b}{a+b}\right)$
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 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 which is continuous at $x=\frac{\pi }{2},$ where α is a constant.

What is  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) = 0·75x4x3 – 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) = 0·75x4x3 – 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
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• Question 82
Consider the parametric equation

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
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• Question 83
Consider the parametric equation

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

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: $-\frac{{a}^{2}}{{y}^{3}}$
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• 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:
Option B:
Option C:
Option D: None of the above
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• Question 87
Consider the integral ${I}_{m}={\int }_{0}^{\pi }\frac{\mathrm{sin}2mx}{\mathrm{sin}x} dx$, where m is a positive integer.
What is I1 equal to?
Option B: 1/2
Option C: 1
Option D: 2
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• Question 88
Consider the integral ${I}_{m}={\int }_{0}^{\pi }\frac{\mathrm{sin}2mx}{\mathrm{sin}x} dx$, where m is a positive integer.

What is I2 + I3 equal to?
Option A: 4
Option B: 2
Option C: 1
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• Question 89
Consider the integral ${I}_{m}={\int }_{0}^{\pi }\frac{\mathrm{sin}2mx}{\mathrm{sin}x} dx$, where m is a positive integer.

What is Im equal to?
Option B: 1
Option C: m
Option D: 2m
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• Question 90
Consider the integral ${I}_{m}={\int }_{0}^{\pi }\frac{\mathrm{sin}2mx}{\mathrm{sin}x} dx$, where m is a positive integer.

Consider the following:

1. ImIm – 1 is equal to 0.

2. I2m > Im

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

What is the value of A?
Option A: –1
Option B: 1
Option C: 2
Option D: 3
VIEW SOLUTION
• Question 94
Given that

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: xy = c
VIEW SOLUTION
• Question 96
What is the solution of the differential equation

Where c is an arbitrary constant.
Option A: $y=x{\mathrm{sin}}^{-1}a+c$
Option B: $x=y{\mathrm{sin}}^{-1}a+c$
Option C: $y=x+x{\mathrm{sin}}^{-1}a+c$
Option D: $y={\mathrm{sin}}^{-1}a+c$
VIEW SOLUTION
• Question 97
What is the solution of the differential equation

$\frac{dx}{dy}+\frac{x}{y}-{y}^{2}=0?$

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)2ex + c
Option B: (x + 1) ex + c
Option C: $\frac{{e}^{x}}{x+1}+c$
Option D: $\frac{{e}^{x}}{{\left(x+1\right)}^{2}}+c$
VIEW SOLUTION
• Question 99
The adjacent sides AB and AC of a triangle ABC are represented by the vectors $-2\stackrel{^}{i}+3\stackrel{^}{j}+2\stackrel{^}{k}$ and $-4\stackrel{^}{i}+5\stackrel{^}{j}+2\stackrel{^}{k}$ 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 $\stackrel{\to }{F}=3\stackrel{^}{i}+4\stackrel{^}{j}-3\stackrel{^}{k}$ is applied at the point P, whose position vector is $\stackrel{\to }{r}=2\stackrel{^}{i}-2\stackrel{^}{j}-3\stackrel{^}{k}$. 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
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• Question 101
Given that the vectors $\stackrel{\to }{\alpha }$ and $\stackrel{\to }{\beta }$ are non-collinear. The values of x and y for which $\stackrel{\to }{u}-\stackrel{\to }{v}=\stackrel{\to }{w}$ holds true if  and $\stackrel{\to }{w}=2\stackrel{\to }{\alpha }-5\stackrel{\to }{\beta },$ 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 $\left|\stackrel{\to }{a}\right|=7,\left|\stackrel{\to }{b}\right|=11 \mathrm{and} \left|\stackrel{\to }{a}+\stackrel{\to }{b}\right|=10\sqrt{3},$ then $\left|\stackrel{\to }{a}-\stackrel{\to }{b}\right|$ 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 $\alpha \stackrel{^}{i}+\beta \stackrel{^}{j}+\gamma \stackrel{^}{k},\beta \stackrel{^}{i}+\gamma \stackrel{^}{j}+\alpha \stackrel{^}{k} \mathrm{and} \gamma \stackrel{^}{i}+\alpha \stackrel{^}{j}+\beta \stackrel{^}{k}$
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 $\stackrel{\to }{a}+\stackrel{\to }{b}+\stackrel{\to }{c}=\stackrel{\to }{0},$ then which of the following is/are correct?

1.  are coplanar.

2. $\stackrel{\to }{a}×\stackrel{\to }{b}=\stackrel{\to }{b}×\stackrel{\to }{c}=\stackrel{\to }{c}×\stackrel{\to }{a}$

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 $\left|\stackrel{\to }{a}+\stackrel{\to }{b}\right|=\left|\stackrel{\to }{a}-\stackrel{\to }{b}\right|$, then which one of the following is correct?
Option A: $\stackrel{\to }{a}=\lambda \stackrel{\to }{b}$ for some scalar λ
Option B: $\stackrel{\to }{a}$ is parallel to $\stackrel{\to }{b}$
Option C: $\stackrel{\to }{a}$ is perpendicular to $\stackrel{\to }{b}$
Option D: $\stackrel{\to }{a}=\stackrel{\to }{b}=\stackrel{\to }{0}$
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
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• 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
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• 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 Bi (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
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• 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 $P\left(A\cup B\right)=\frac{3}{4}, P\left(A\cap B\right)=\frac{1}{4} \mathrm{and} P\left(\overline{)A}\right)=\frac{2}{3}$, then what is P(B) equal to?
Option A: 1/3
Option B: 2/3
Option C: 1/8
Option D: 2/9
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• 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
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• 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
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• 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
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