Q1
mcq
1 mark
Subtraction is not a binary operation in :
- A. N
- B. R
- C. Q
- D. Z
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Q2
mcq
1 mark
Suppose that X takes on one of the values 0, 1 and 2. If for some constant k , P(X=i)=k P(X=i−1) for i=1, 2 and 1 P(X 0) 7 = = , then the value of k is :
- A. 3
- B. 1
- C. 4
- D. 2
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Q3
mcq
1 mark
If A is a non-singular matrix of order 3×3 and |A|=5 then |A^{-1}| is :
- A. 5^{2}
- B. 5
- C. 1/5
- D. 1/5
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Q4
mcq
1 mark
A stone is thrown up vertically. The height it reaches at time t seconds is given by x=80 t−16 t^{2}. The stone reaches the maximum height in time t seconds is given by :
- A. 3
- B. 2
- C. 3.5
- D. 2.5
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Q5
mcq
1 mark
The order and degree of the differential equation \frac{d^{2}y}{dx^{2}} - 4 \frac{dy}{dx} - 7 = 0 are respectively :
- A. 1, 2
- B. 2, 1
- C. 2, 2
- D. 1, 1
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Q6
mcq
1 mark
If A = \begin{pmatrix} 2 & 3 \\ 5 & 2 \end{pmatrix} be such that \lambda A^{-1} = A, then \lambda is :
- A. 19
- B. 17
- C. 21
- D. 14
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Q7
mcq
1 mark
The slope at any point of a curve y=f(x) is given by \frac{dy}{dx} = 3x^{2} and it passes through (−1, 1). Then the equation of the curve is :
- A. y=3x^{3}+14
- B. y=x^{3}+12
- C. y=x^{3}+15
- D. y=3x^{2}+14
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Q8
mcq
1 mark
The domain of the function defined by f(x) = \sin^{-1} x - 1 is :
- A. [0, 1]
- B. [1, 2]
- C. [−1, 0]
- D. [−1, 1]
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Q9
mcq
1 mark
If u(x, y)=e^{x^{2}+y^{2}}, then \frac{\partial u}{\partial x} is equal to :
- A. x^{2}u
- B. e^{x^{2}+y^{2}}
- C. y^{2}u
- D. 2xu
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Q10
mcq
1 mark
The number of real numbers in [0, 2\pi] satisfying \sin^{4} x - 2\sin^{2} x + 1 = 0 is :
- A. 1
- B. 2
- C. \infty
- D. 4
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Q11
mcq
1 mark
The square root of i are :
- A. \pm \frac{1}{\sqrt{2}}(1+i)
- B. \pm \frac{1}{\sqrt{2}}(1+i)
- C. \pm \frac{1}{\sqrt{2}}(1-i)
- D. \pm \frac{1}{\sqrt{2}}(1-i)
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Q12
mcq
1 mark
The value of \sum_{n=1}^{13} (i^{n} + i^{n-1}) is :
- A. 1
- B. 11i
- D. i
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Q13
mcq
1 mark
If in 6 trials, X is a binomial variable which follows the relation 9P(X=4)=P(X=2), then the probability of success is :
- A. 0.375
- B. 0.125
- C. 0.75
- D. 0.25
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Q14
mcq
1 mark
The angle between the lines \frac{x-1}{3} = \frac{y-2}{2} = z and \frac{x-2}{1} = \frac{y-3}{3} = \frac{z-1}{2} is :
- A. \pi/3
- B. \pi/6
- C. \pi/2
- D. \pi/4
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Q14
mcq
The angle between the lines $\frac{x+1}{2} = \frac{y}{3} = \frac{z-1}{5}$ and $\frac{x+1}{1} = \frac{y}{3} = \frac{z-2}{2}$ is :
- A. \frac{\pi}{3}
- B. \frac{\pi}{6}
- C. \frac{\pi}{2}
- D. \frac{\pi}{4}
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Q15
mcq
1 mark
The point of inflection of the curve y=(x−1)^{3} is :
- A. (1, 0)
- B. (0, 0)
- C. (1, 1)
- D. (0, 1)
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Q16
mcq
1 mark
The value of \int_{0}^{2/3} \frac{dx}{\sqrt{4-9x^{2}}} is :
- A. \pi/4
- B. \pi/6
- C. \pi
- D. \pi/2
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Q17
mcq
1 mark
The volume of solid of revolution of the region bounded by y^{2}=x(a−x) about x-axis is :
- A. \frac{\pi a^{3}}{5}
- B. \pi a^{3}
- C. \frac{\pi a^{3}}{6}
- D. \frac{\pi a^{3}}{4}
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Q18
mcq
An ellipse has OB as semi minor axes, F and F' its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is :
- A. 1/4
- B. 1/2
- C. 1/3
- D. 1/2
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Q19
mcq
The volume of the parallelepiped with its edges represented by the vectors $\hat{i}+\hat{j}, \hat{j}+\hat{k}, \hat{k}+\hat{i}$ is :
- A. \pi
- B. \frac{\pi}{2}
- C. \frac{\pi}{4}
- D. \frac{\pi}{3}
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Q20
mcq
If f (x) > 0 for all x and g(x)=log(f (x)), then dg is :
- A. \frac{1}{f(x)} dx
- B. \frac{f'(x)}{f(x)} dx
- C. \frac{1}{x} dx
- D. \frac{1}{x} df(x)
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Q21
short answer
If $\text{adj } A = \begin{bmatrix} 1 & 2 & 2 \\ 1 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, find $A^{-1}$.
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Q22
short answer
If $z=x+iy$, then find $\text{Re}\left(\frac{1}{z}\right)$ in rectangular form.
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Q23
short answer
Find the value of $\tan^{-1}(-\sqrt{3})$.
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Q24
short answer
If $y=4x+c$ is a tangent to the circle $x^2+y^2=9$, find c.
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Q25
short answer
Find the slant (oblique) asymptote for the function $f(x) = \frac{x^2-6x+7}{x+5}$.
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Q26
short answer
Show that $F(x, y) = \frac{x^2+xy-5y^2}{3x+7y}$ is a homogeneous function of degree 1.
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Q27
short answer
Solve $\frac{dy}{dx} = \frac{1-y^2}{1-x^2}$.
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Q28
short answer
Find the constant C such that the function $f(x) = \begin{cases} Cx^2 & 0 < x < 4 \\ 0 & \text{otherwise} \end{cases}$ is a density function of X.
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Q29
short answer
Find a polynomial equation of minimum degree with rational coefficients having $i-2$ as a root.
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Q30
short answer
If $f(x)=\sin x$, then prove that $\int_{0}^{\pi} f(x) dx = 2 \int_{0}^{\pi/2} f(x) dx$.
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Q31
long answer
Solve the system of linear equations $2x+5y=-2, x+2y=-3$ by matrix inversion method.
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Q32
long answer
If $|z|=2$ show that $8 \le |z+6+8i| \le 12$.
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Q34
long answer
Prove that $\tan^{-1}(\frac{1}{11}) + \tan^{-1}(\frac{1}{24}) = \tan^{-1}(\frac{1}{2})$.
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Q34
short answer
Prove that $tan^{-1} (1/11) + tan^{-1} (2/24) + tan^{-1} (1/7) + tan^{-1} (1/2) = ...$
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Q35
long answer
If $\vec{a}, \vec{b}, \vec{c}$ are three vectors, prove that $[\vec{a}+\vec{c}, \vec{a}+\vec{b}, \vec{a}+\vec{b}+\vec{c}] = [\vec{a}, \vec{b}, \vec{c}]$.
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Q36
long answer
If $u(x, y)=x^2y+3xy^4$, $x=e^t$ and $y=\sin t$, find $\frac{du}{dt}$.
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Q37
long answer
Evaluate $\int_{0}^{2\pi} \frac{dx}{1+5\cos^2 x}$.
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Q38
short answer
A lottery with 600 tickets gives one prize of ` 200, four prizes of ` 100 and six prizes of ` 50. If the ticket cost is ` 2, find the expected profit amount of a ticket.
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Q39
short answer
Find the Taylor’s series about x=2 for f(x)=x^3-12x+11, (-∞ < x < ∞).
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Q42
long answer
Solve, by Cramer’s rule, the system of equations $x_1-x_2=3, 2x_1+3x_2+4x_3=17, x_2+2x_3=7$.
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Q43
long answer
Find the equation of tangent and normal to the curve given by $x=7 \cos t$ and $y=2 \sin t, t \in R$ at any point on the curve.
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Q44
long answer
If $\omega \neq 1$ is a cube root of unity, show that the roots of the equation $(z-1)^3+8=0$ are $-1, 1-2\omega, 1-2\omega^2$.
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Q45
long answer
Find the area of the region bounded by the parabola $y^2=x$ and the line $y=x-2$.
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Q46
long answer
Solve the equation $6x^4-5x^3-38x^2-5x+6=0$ if it is known that $1/3$ is a solution.
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Q48
long answer
A bridge has a parabolic arch that is 10 m high in the centre and 30 m wide at the bottom. Find the height of the arch 6 m from the centre, on either sides.
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Q49
long answer
Using vector method, prove that $cos(\alpha-\beta) = cos \alpha cos \beta + sin \alpha sin \beta$.
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Q50
long answer
Find the equation of the ellipse whose Foci are (2, 1), (-2, 1) and the length of the latus rectum is 6.
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Q51
long answer
Find the non-parametric form of Vector equation, and the Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines $\vec{r} = (i+2j-4k) + s(2i+3j+6k)$ and $\vec{r} = (i-3j+5k) + t(i+j-k)$.
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Q52
short answer
During war, 1 ship out of 9 was sunk on an average in making a certain voyage. What was the probability that : (i)Exactly 3 out of a convoy of 6 ships would arrive safely ? (ii)No ships arrive safely from a convoy of 4 ships.
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Q53
short answer
Find the non-parametric form of Vector equation, and the Cartesian equation of the plane passing through the point $(0, 1, -5)$ and parallel to the straight lines $\vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + s(2\hat{i} + 3\hat{j} + 6\hat{k})$ and $\vec{r} = (\hat{i} + 3\hat{j} + 5\hat{k}) + t(\hat{i} + \hat{j} - \hat{k})$
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Q54
short answer
The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple ?
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Q55
short answer
A hollow cone with base radius $a$ cm and height $b$ cm is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is $4/9$ times volume of the cone.
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Q56
short answer
Using truth table, prove that $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$
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