Q1
mcq
The area between $y^{2}=4x$ and its latus rectum is :
- A. 8/3
- B. 2/3
- C. 5/3
- D. 4/3
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Q2
mcq
The value of $\int_{0}^{a} (3a^{2}-x^{2})dx$ is :
- A. 2/3 a^{8}\pi
- B. a^{3}/16\pi
- C. 4/3 a^{8}\pi
- D. 4/3 a^{16}\pi
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Q3
mcq
If P(x, y) be any point on $16x^{2}-125y^{2}=400$ with foci $F_{1}(3, 0)$ and $F_{2}(-3, 0)$, then $PF_{1}-PF_{2}$ is :
- A. 10
- B. 8
- C. 12
- D. 6
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Q4
mcq
If $|z_{1}|=1, |z_{2}|=2, |z_{3}|=3$ and $|9z_{1}z_{2}+4z_{1}z_{3}+z_{2}z_{3}|=12$ then the value of $|z_{1}+z_{2}+z_{3}|$ is :
- A. 3
- B. 1
- C. 4
- D. 2
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Q5
mcq
The number of rows in the truth table of $(p\lor q) \land (p\lor r)$ is :
- A. 6
- B. 9
- C. 3
- D. 8
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Q6
mcq
If a vector $\vec{\alpha}$ lies in the plane of $\vec{\beta}$ and $\vec{\gamma}$, then :
- A. [\vec{\alpha}, \vec{\beta}, \vec{\gamma}] = 0
- B. [\vec{\alpha}, \vec{\beta}, \vec{\gamma}] = 1
- C. [\vec{\alpha}, \vec{\beta}, \vec{\gamma}] = 2
- D. [\vec{\alpha}, \vec{\beta}, \vec{\gamma}] = -1
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Q7
mcq
The differential equation of the family of curves $y=Ae^{x}+Be^{-x}$, where A and B are arbitrary constants is :
- A. d^{2}y/dx^{2}+y=0
- B. d^{2}y/dx^{2}+y=0
- C. d^{2}y/dx^{2}-y=0
- D. d^{2}y/dx^{2}-y=0
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Q8
mcq
The horizontal asymptote of $f(x) = 1/x$ is :
- A. x=c
- B. y=0
- C. y=c
- D. x=0
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Q9
mcq
If $f(x) = x/(x+1)$, then its differential is given by :
- A. dx/(x+1)
- B. -dx/(x+1)^{2}
- C. -dx/(x+1)
- D. dx/(x+1)^{2}
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Q10
mcq
If $(1+i)(1+2i)(1+3i) ... (1+ni)=x+iy$ then $2⋅5⋅10 ... (1+n^{2})$ is :
- A. x^{2}+y^{2}
- B. 1
- C. 1+n^{2}
- D. i
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Q11
mcq
The number given by the Rolle’s theorem for the function $x^{3}-3x^{2}$, $x∈[0, 3]$ is :
- A. 3/2
- B. 1
- C. 2
- D. 2
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Q12
mcq
The type of conic section for $x^{2}-3=5x+3y$ is :
- A. hyperbola
- B. ellipse
- C. circle
- D. parabola
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Q13
mcq
If A is a non-singular matrix such that $A^{-1} = \begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix}$, then $(A^{T})^{-1} = $
- A. \begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix}
- B. \begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix}
- C. \begin{bmatrix} 5 & 2 \\ 3 & 1 \end{bmatrix}
- D. \begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix}
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Q13
mcq
If A is a non-singular matrix such that $A \begin{bmatrix} 1 & 5 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$, then $(A^T)^{-1} =$
- A. $\begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix}$
- B. $\begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix}$
- C. $\begin{bmatrix} 5 & 2 \\ 3 & 1 \end{bmatrix}$
- D. $\begin{bmatrix} 5 & 3 \\ 2 & 1 \end{bmatrix}$
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Q14
mcq
The angle between the line $r=(i+2j+k)+t(i+j+k)$ and the plane $r⋅(4i+0j)=0$ is :
- A. 45^{\circ}
- B. 0^{\circ}
- C. 90^{\circ}
- D. 30^{\circ}
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Q14
mcq
The angle between the line $\vec{r} = (2\vec{i} - 3\vec{j} + \vec{k}) + t(2\vec{i} + \vec{j} + 3\vec{k})$ and the plane $\vec{r} \cdot (\vec{i} + 4\vec{j} + 0\vec{k}) = 0$ is :
- A. 45^\circ
- B. 0^\circ
- C. 90^\circ
- D. 30^\circ
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Q15
mcq
If $sin^{-1}(1/2) + cot^{-1}(1/2) = x$, then x is equal to :
- A. 2/5
- B. 1/2
- C. 3/2
- D. 1/5
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Q16
mcq
If α, β and γ are zeros of $x^3 + px^2 + qx + r$, then $\sum (1/\alpha)$ is :
- A. q/r
- B. -q/r
- C. -q/p
- D. -p/r
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Q18
mcq
The random variable X has binomial distribution with n=25 and p=0.8 then standard deviation of X is :
- A. 3
- B. 6
- C. 2
- D. 4
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Q19
mcq
If A, B and C are invertible matrices of some order, then which one of the following is not true ?
- A. det A^-1 = (det A)^-1
- B. adj A = |A|A^-1
- C. (ABC)^-1 = C^-1 B^-1 A^-1
- D. adj (AB) = (adj A)(adj B)
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Q22
short answer
If α and β are the roots of the quadratic equation $2x^2 - 7x + 13 = 0$, construct a quadratic equation whose roots are α^2 and β^2.
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Q23
short answer
Find df for $f(x) = x^2 + 3x$ and evaluate it for x=3 and dx=0.02.
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Q24
short answer
Find the differential equation for the family of all straight lines passing through the origin.
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Q25
short answer
For the random variable X with the given probability mass function $f(x) = 2(x-1)$ for $1 < x < 2$, $0$ otherwise. Find the Mean.
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Q26
short answer
Find the general equation of a circle with centre (-3, -4) and radius 3 units.
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Q27
short answer
Find the rank of the matrix $\begin{bmatrix} 1 & 3 \\ 4 & 7 \\ 3 & 4 \end{bmatrix}$.
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Q29
short answer
Evaluate : $\lim_{x \to 1} \frac{x^2 - 3x + 2}{x^2 - 4x + 3}$
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Q30
short answer
Show that the vectors $2\vec{i} - 3\vec{j} + \vec{k}$, $\vec{i} - \vec{j}$ and $3\vec{i} - 6\vec{j} + 3\vec{k}$ are coplanar.
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Q31
long answer
Show that $\cot^{-1} (1/\sqrt{x^2 - 1}) = \sec^{-1} x$, $|x| > 1$.
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Q31
short answer
Show that $\cot^{-1} \left( \frac{1}{\sqrt{x^2-1}} \right) = \sec^{-1} x$, $|x| > 1$.
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Q32
long answer
Find the equation of tangent and normal to the parabola $x^2 - 16x + 14y + 15 = 0$ at (1, -3).
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Q33
long answer
Prove that $[\vec{a}-\vec{b}, \vec{b}-\vec{c}, \vec{c}-\vec{a}] = 0$.
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Q34
long answer
If $u = \frac{x^2 + y^2}{x + y}$, prove that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = u$.
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Q35
long answer
Find two positive numbers whose sum is 12 and their product is maximum.
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Q36
long answer
Evaluate : $\int_{\pi/8}^{3\pi/8} \frac{1}{1 + \tan x} dx$
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Q36
short answer
Evaluate : $\int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \frac{1}{1 + \tan x} dx$.
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Q37
short answer
Simplify $\left( \frac{1+i}{1-i} \right)^3 - \left( \frac{1-i}{1+i} \right)^3$ into rectangular form.
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Q39
short answer
Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.
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Q40
short answer
If $A = \begin{bmatrix} 2 & 1 & 3 \\ 5 & 3 & 1 \\ -3 & 2 & 3 \end{bmatrix}$, then find $|adj(adj A)|$.
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Q45
long answer
Find the angle between the curves $y=x^2$ and $y=(x-3)^2$.
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Q46
long answer
Solve : $\tan^{-1}\left( \frac{x-1}{x-2} \right) + \tan^{-1}\left( \frac{x+1}{x+2} \right) = \frac{\pi}{4}$.
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A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws.
Q47
long answer
A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the probability mass function.
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Q48
long answer
If $z=x+iy$ is a complex number such that $\text{Im}\left( \frac{z-i}{z+1} \right) = 0$, show that the locus of z is $2x^2 + 2y^2 + x - 2y = 0$.
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Q49
long answer
A conical water tank with vertex down of 12 meters height has a radius of 5 meters at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?
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Q50
long answer
Prove by vector method that $\sin(\alpha-\beta)=\sin\alpha \cos\beta - \cos\alpha \sin\beta$.
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Q51
long answer
Assume that water issuing from the end of horizontal pipe, 7.5 m above the ground, describes a parabolic path. The vertex of the parabolic path is at the end of the pipe. At a position 2.5 m below the line of the pipe, the flow of water has curved outward 3 m beyond the vertical line through the end of the pipe. How far beyond this vertical line will the water strike the ground?
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Q52
long answer
Solve the Linear differential equation $\frac{dy}{dx} + \frac{y}{x} = \sin x$.
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Q53
long answer
The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours ?
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Q54
long answer
Find the vector and Cartesian equations of the plane containing $\frac{x-2}{2} = \frac{y-2}{3} = \frac{z-1}{3}$ and parallel to the line $\frac{x-1}{3} = \frac{y-1}{2} = \frac{z-1}{1}$.
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Q55
long answer
Find the area of the region bounded by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
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Q56
long answer
Find the vertex, focus and equation of the directrix of the parabola $y^2 - 4y - 8x + 12 = 0$.
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Q57
long answer
Show that $p \leftrightarrow q \equiv ((~p) \vee q) \wedge ((~q) \vee p)$
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Q58
long answer
Solve the system of linear equations by Cramer’s Rule. $x - 3y + 4z = 1$, $2x - y + 0z = 0$, $2x - 5y + 4z = 1$
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