Q1
mcq
1 mark
If $f(x) = \begin{cases} 2x & 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$ is a probability density function of a random variable, then the value of $a$ is :
- A. 3
- B. 1
- C. 4
- D. 2
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Q2
mcq
1 mark
Which one of the following is not true in the case of discrete random variable X ?
- A. $\lim_{x \to \infty} F(x) = 1$
- B. $0 \leq F(x) \leq 1$ for all $x \in R$
- C. $F(x)$ is real valued decreasing function.
- D. $\lim_{x \to -\infty} F(x) = 0$
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Q3
mcq
1 mark
If $f(x) = \frac{1}{x+1}$, then its differential is :
- A. $\frac{1}{x+1} dx$
- B. $-\frac{1}{(x+1)^2} dx$
- C. $-\frac{1}{x+1} dx$
- D. $\frac{1}{(1+x)^2} dx$
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Q4
mcq
1 mark
The value of $\int_{0}^{99} (1-x) dx$ is :
- A. $\frac{1}{10010}$
- B. $\frac{1}{11000}$
- C. $\frac{1}{10001}$
- D. $\frac{1}{10100}$
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Q5
mcq
1 mark
The principal value of $\cos^{-1} (\frac{\sqrt{3}}{2})$ is :
- A. $\frac{\pi}{2}$
- B. $\frac{\pi}{3}$
- C. $\frac{5\pi}{6}$
- D. $\frac{\pi}{6}$
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Q6
mcq
1 mark
If $A = \begin{bmatrix} 2 & 3 \\ 5 & 2 \end{bmatrix}$ 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
If $\alpha, \beta$ and $\gamma$ are the zeros of $x^3 + px^2 + qx + r$, then $\sum \frac{1}{\alpha}$ is :
- A. $\frac{q}{r}$
- B. $-\frac{q}{r}$
- C. $-\frac{q}{p}$
- D. $-\frac{p}{r}$
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Q8
mcq
1 mark
If $(1+i)(1+2i)(1+3i)...(1+ni) = x+iy$ then the value $2 \cdot 5 \cdot 10 ... (1+n^2)$ is :
- A. $x^2+y^2$
- B. 1
- C. $1+n^2$
- D. $i$
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Q9
mcq
1 mark
The minimum value of the function $f(x) = |3-x| + 9$ is :
- A. 6
- C. 9
- D. 3
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Q10
mcq
1 mark
The value of $\sum_{n=1}^{12} i^n$ is :
- B. 1
- C. -1
- D. $i$
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Q11
mcq
1 mark
If the vectors $2i-j+k, 3i+2j+k, m i+4j+k$ are coplanar, then the value of $m$ is :
- A. 2
- B. 3
- C. -2
- D. -3
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Q12
mcq
1 mark
The general equation of a circle with centre $(-3, -4)$ and radius 3 units is :
- A. $x^2+y^2-6x+8y-16=0$
- B. $x^2+y^2-6x-8y+16=0$
- C. $x^2+y^2+6x-8y+16=0$
- D. $x^2+y^2+6x+8y+16=0$
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Q13
mcq
1 mark
The solution of $\frac{dx}{dy} + p(x) = 0$ is :
- A. $x = c e^{-\int p dy}$
- B. $y = c e^{\int p dx}$
- C. $x = c e^{\int p dy}$
- D. $y = c e^{-\int p dx}$
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Q14
mcq
1 mark
The value of $\int_{0}^{\infty} x^3 e^{-x} dx$ is :
- A. 4/27
- B. 7/27
- C. 2/27
- D. 5/27
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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 angle between the lines $\frac{x-4}{2} = \frac{y+1}{-1} = \frac{z}{-2}$ and $\frac{x-1}{4} = \frac{y+1}{4} = \frac{z-2}{2}$ is :
- A. $\pi/2$
- B. $\pi/4$
- C. $2\pi/3$
- D. $\pi/3$
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Q17
mcq
1 mark
Which one of the following is a binary operation on N ?
- A. Multiplication
- B. Division
- C. Subtraction
- D. All the above
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Q18
mcq
1 mark
Which one of the following is incorrect ?
- A. If A is a square matrix of order n, and $\lambda$ is a scalar, then $Adj (\lambda A) = \lambda^n (Adj A)$.
- B. Adjoint of a symmetric matrix is also a symmetric matrix.
- C. $A(Adj A) = (Adj A)A = |A|I$.
- D. Adjoint of a diagonal matrix is also a diagonal matrix.
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Q19
mcq
1 mark
If $\sin x$ is the integrating factor of the linear differential equation $\frac{dy}{dx} + Py = Q$, then $P$ is :
- A. $\tan x$
- B. $\log \sin x$
- C. $\cot x$
- D. $\cos x$
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Q20
mcq
1 mark
The length of the latus rectum of the parabola $x^2=24y$ is :
- A. 8
- B. 24
- C. 6
- D. 12
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Q21
short answer
Prove the following properties: $Re(z) = \frac{z + \bar{z}}{2}$ and $Im(z) = \frac{z - \bar{z}}{2i}$.
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Q22
short answer
Find a polynomial equation of minimum degree with rational coefficients, having $2 - \sqrt{3}$ as a root.
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Q23
short answer
Find the principal value of $\tan^{-1}(\sqrt{3})$.
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Q24
short answer
Find the points on the curve $y=x^3-3x^2+x-2$ at which the tangent is parallel to the line $y=x$.
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Q25
short answer
Find $df$ for $f(x)=x^2+3x$ and evaluate it for $x=2$ and $dx=0.1$.
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Q26
long answer
Show that the differential equation of the family of curves $y=Ae^x+Be^{-x}$, where $A$ and $B$ are arbitrary constants, is $\frac{d^2y}{dx^2}-y=0$.
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Q27
short answer
Solve : $\frac{dy}{dx} = \frac{1-y}{1-x}$.
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Q28
short answer
A random variable X has the following probability mass function: $x: 1, 2, 3, 4, 5, 6$; $f(x): k, 2k, 6k, 5k, 6k, 10k$. Find $k$.
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Q29
short answer
X is the number of tails occurred when three fair coins are tossed simultaneously. Find the values of the random variable X and number of points in its reverse images.
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Q30
short answer
Show that the distance from the origin to the plane $3x+6y+2z+7=0$ is 1.
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Q31
long answer
Show that the rank of the matrix $\begin{bmatrix} 1 & 2 & 1 \\ 3 & 1 & 2 \\ 1 & -2 & 3 \\ 1 & -1 & 1 \end{bmatrix}$ is 3.
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Q32
short answer
Solve the following system of linear equations, using matrix inversion method: $5x+2y=3$, $3x+2y=5$.
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Q33
short answer
Which one of the points $10-8i, 11+6i$ is closest to $1+i$.
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Q34
short answer
Solve the equation $2x^3-9x^2+10x=3$, if 1 is a root, find the other roots.
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Q35
short answer
Find the magnitude and the direction cosines of the torque about the point $(2, 0, -1)$ of a force $2i+j-k$ whose line of action passes through the origin.
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Q36
short answer
Evaluate : $\lim_{x \to \infty} \frac{2x^2-3x+5}{x^2+3x-2}$.
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Q37
short answer
Assume that the cross section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an artery is increased from 2 mm to 2.1 mm, how much is cross-sectional area increased approximately.
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Q38
short answer
Show that $\int_{0}^{\pi/3} \frac{\sec x \tan x}{1+\sec x} dx = \tan(\pi/4) - 1$.
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Q39
short answer
Let * be defined on R by $(a*b) = a+b+ab-7$. Is * binary on R? If so, find $3 * (7/15)$.
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Q40
short answer
Prove that the general equation of the circle whose diameter is the line segment joining the points (-4, -2) and (-1, -1), is $x^2+y^2+5x+3y+6=0$.
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Q41
short answer
(a) Cramer’s rule is not applicable to solve the system $3x+y+z=2, x-3y+2z=1, 7x-y+4z=5$. Why?
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Q42
short answer
(b) Prove that the local minimum values for the function $f(x)=4x^6-6x^4$ attain at -1 and 1.
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Q43
short answer
(a) Show that the locus of $z=x+iy$ if $|z+i| = |z-1|$, is $x+y=0$.
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Q44
short answer
(b) Show that $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$.
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Q45
short answer
(a) Show that the equation of the parabola with focus (-2, 0) and directrix $x=2$ is $y^2=-8x$.
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Q46
short answer
(b) Find the value of $\cot^{-1}(1) + \sin^{-1}(1/2) - \sec^{-1}(2)$.
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Q47
short answer
(a) The maximum and minimum distances of the Earth from the Sun respectively are $152 \times 10^6$ km and $94.5 \times 10^6$ km. The Sun is at one focus of the elliptical orbit. Show that the distance from the Sun to the other focus is $575 \times 10^5$ km.
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Q48
short answer
(b) Prove by vector method $\sin(A+B) = \sin A \cos B + \cos A \sin B$.
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Q49
short answer
(a) Find the vector equation or Cartesian equation of a plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane $2x+6y+6z=9$.
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Q50
short answer
(b) Show that the angle between the curves $y=x^2$ and $x=y^2$ at (1, 1) is $\tan^{-1}(3)$.
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Q51
short answer
(a) The distribution function of a continuous random variable X is: $F(x) = 0$ if $x < 1$, $(x-1)/4$ if $1 \le x \le 5$, $1$ if $x > 5$. Find (i) $P(X < 3)$ (ii) $P(2 < X < 4)$ (iii) $P(3 \le X)$.
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Q52
short answer
(b) Show that the area of the region bounded by $3x-2y+6=0$, $x=-3$, $x=1$ and x-axis, is $15/2$.
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Q53
short answer
(a) Show that the solution of the differential equation $(1+x^2)dy/dx = 1+y^2$ is $\tan^{-1} y = \tan^{-1} x + C$ (or) $\tan^{-1} x = \tan^{-1} y + C$.
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Q54
short answer
(b) Prove $p \to (q \to r) \equiv (p \land q) \to r$ using truth table.
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Q55
mcq
If $f(x) = x/(1+x)$, then its differential is:
- A. $\frac{1}{1+x} dx$
- B. $\frac{-1}{(1+x)^2} dx$
- C. $\frac{1}{(1+x)^2} dx$
- D. $\frac{1}{1+x^2} dx$
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Q56
mcq
The solution of $dy/dx + py = 0$ is:
- A. $x=ce^{-\int pdy}$
- B. $y=ce^{\int pdx}$
- C. $x=ce^{\int pdy}$
- D. $y=ce^{-\int pdx}$
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Q57
mcq
1 mark
The value of $\int_{0}^{\frac{1}{99}} \frac{1}{x} d x$ is:
- A. $\frac{1}{10010}$
- B. $\frac{1}{11000}$
- C. $\frac{1}{10001}$
- D. $\frac{1}{10100}$
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Q58
mcq
1 mark
If the vectors $2i - 3j + k$, $3i + 2j - k$, $mi + 4j + k$ are coplanar, then the value of $m$ is:
- A. 2
- B. 3
- C. -2
- D. -3
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Q59
long answer
Show that $\int_{0}^{\pi/3} \frac{\sec x \tan x}{1 + \sec^2 x} dx = \tan^{-1}(2) - \frac{\pi}{4}$.
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Q60
long answer
Show that $\int_{0}^{\frac{\pi}{3}} \frac{\sec x \tan x}{1+\sec^2 x} dx = \tan^{-1}(2)$.
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Q61
long answer
Show that $\int_{0}^{a} \frac{f(x)}{f(x)+f(a-x)} dx = \frac{a}{2}$.
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Q62
long answer
Find the value of $\cot^{-1}(1) + \sin^{-1} \left(\frac{-1}{2}\right) - \sec^{-1} (2)$.
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Q63
long answer
The distribution function of a continuous random variable X is : $F(x) = \begin{cases} 0, & x < 1 \\ \frac{1}{4}(x^3-1), & 1 \leq x < 5 \\ 1, & x \geq 5 \end{cases}$. Find (i) $P(X < 3)$ (ii) $P(2 < X < 4)$ (iii) $P(3 \leq X)$.
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Q64
mcq
1 mark
If $\begin{bmatrix} 2 & 3 \\ 5 & 2 \end{bmatrix} A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ be such that $\lambda A^{-1} = A$, then $\lambda$ is :
- A. 19
- B. 17
- C. 21
- D. 14
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Q65
mcq
1 mark
The principal value of $\cos^{-1} \left(\frac{1}{2}\right)$ is :
- A. $\frac{\pi}{2}$
- B. $\frac{\pi}{3}$
- C. $\frac{5\pi}{6}$
- D. $\frac{\pi}{6}$
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Q66
mcq
If the vectors $2\hat{i}-3\hat{j}+\hat{k}$, $3\hat{i}+2\hat{j}-4\hat{k}$, $m\hat{i}+\hat{j}+\hat{k}$ are coplanar, then the value of m is :
- A. 2
- B. 3
- C. -2
- D. -3
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Q67
long answer
Show that $\int_{0}^{\pi/2} \frac{\sec x}{\sec x + \tan x} dx = \pi - 2$.
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Q68
short answer
Let * be defined on $\mathbb{R}$ by $(a*b)=a+b+ab-7$. Is * binary on $\mathbb{R}$ ? If so, find $3*(\frac{7}{15})$.
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