Q1
mcq
1 mark
A square matrix A of order n has inverse if and only if :
- A. ρ(A) > n
- B. ρ(A)=n
- C. ρ(A) ≠ n
- D. ρ(A) < n
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Q2
mcq
1 mark
Distance from the origin to the plane $3x−6y+12z-17=0$ is :
- A. 2
- C. 3
- D. 1
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Q3
mcq
1 mark
If $3 \cos^{-1}x = \cos^{-1}(4x^3-3x)$, then x is :
- A. [1/2, 1]
- B. [1/2, 1]
- C. x ∈ (−∞, 1]
- D. [1/2, 1)
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Q4
mcq
1 mark
The general solution of the differential equation $\frac{dy}{dx} = \frac{y}{x}$ is :
- A. y=kx
- B. xy=k
- C. logy=kx
- D. y=k logx
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Q5
mcq
1 mark
The number of normals that can be drawn from a point to the parabola $y^2=4ax$ is :
- A. 3
- B. 2
- D. 1
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Q6
mcq
1 mark
If $\vec{a}$ and $\vec{b}$ are parallel vectors then $[\vec{a}, \vec{c}, \vec{b}]$ is equal to :
- A. 1
- B. 2
- D. -1
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Q7
mcq
1 mark
The number of real numbers in $[0, 2\pi]$ satisfying $\sin^4x-2\sin^2x=1$ is :
- A. 1
- B. 2
- C. ∞
- D. 4
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Q8
mcq
1 mark
Suppose that X takes on one of the values 0, 1, 2. If for some constant k, $P(X=i) = kP(X=i-1)$ for i=1, 2 and $P(X=0)=1/7$, then the value of k is :
- A. 3
- B. 1
- C. 4
- D. 2
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Q9
mcq
1 mark
The maximum value of the function $x^2 e^{-2x}, x > 0$ is :
- A. 1/2e
- B. 1/e
- C. 4/4e
- D. 1/2e
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Q10
mcq
1 mark
The operation * defined by $a*b = \frac{ab}{7}$ is not a binary operation on :
- A. R
- B. Q
- C. C
- D. Z
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Q11
mcq
1 mark
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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Q12
mcq
1 mark
Angle between the curves $y^2=x$ and $x^2=y$ at the origin is :
- A. π/2
- B. tan⁻¹(3/4)
- C. π/4
- D. tan⁻¹(4/3)
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Q13
mcq
1 mark
|adj(adjA)| = |A|^16, then the order of the square matrix A is :
- A. 2
- B. 3
- C. 5
- D. 4
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Q14
mcq
1 mark
The value of $((1+i)/2)^8 + ((1-i)/2)^8$ is :
- A. 8
- B. 4
- C. 2
- D. 6
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Q14
mcq
The value of $\left( \frac{1}{2} i \right)^8 + \left( \frac{1}{2} i \right)^8$ is :
- A. 8
- B. 4
- C. 2
- D. 6
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Q15
mcq
1 mark
If |z|=1, then the value of $1+z/1+(1/z)$ is :
- A. 1/z
- B. z
- C. 1
- D. z
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Q15
mcq
If |z|=1, then the value of $z + \frac{1}{z}$ is :
- A. 1/z
- B. z
- C. 1
- D. z
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Q16
mcq
1 mark
The abscissa of the point on the curve $f(x) = x^8 - 2x$ at which the slope of the tangent is -0.25 ?
- A. -2
- B. -8
- D. -4
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Q17
mcq
1 mark
The value of $\int_{0}^{\pi/3} \tan x dx$ is :
- A. -log 2
- B. log 2
- C. -log 3
- D. log 3
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Q18
mcq
1 mark
The number of positive zeros of the polynomial $\sum_{r=0}^{n} C_r (-1)^r x^r$ is :
- A. < n
- C. r
- D. n
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Q19
mcq
1 mark
The Principal value of $\sin^{-1}(1/2)$ is :
- A. -π/6
- C. -π/2
- D. π/2
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Q20
mcq
Area of the greatest rectangle inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is :
- A. ab
- B. 2ab
- C. a/b
- D. ab
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Q22
short answer
If p and q are the roots of the equation $lx^2 + nx + n = 0$, show that $\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{n}{l}} = 0$
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Q23
short answer
If y=4x+c is a tangent to the circle $x^2 + y^2 = 9$, find c.
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Q24
short answer
If the radius of a sphere with radius 10 cm, has to decrease by 0.1 cm, approximately how much will its volume decrease ?
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Q25
short answer
Evaluate : $\int_{b}^{\infty} \frac{1}{a^2 + x^2} dx$, a > 0, b ∈ R.
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Q26
short answer
Find the vector equation of a plane which is at a distance of 7 units from the origin having 3, −4, 5 as direction ratios of a normal to it.
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Q27
short answer
Let $A = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ be any two Boolean matrices of the same type. Find $A \vee B$ and $A \wedge B$.
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Q28
short answer
Prove that $\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$ is orthogonal.
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Q29
short answer
Find the equation of tangent to the curve y=x^2 + 3x−2 at the point (1, 2).
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Q30
short answer
Express $e^{\cos \theta + i \sin \theta}$ in a+ib form.
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Q33
short answer
For what value of x, the inequality $\frac{2}{\pi} < \cos^{-1} (3x-1) < \pi$ holds ?
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Q34
short answer
Find the angle made by the straight line $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ with coordinate axes.
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Q35
short answer
Use the linear approximation to find an approximate value of $(123)^{2/3}$.
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Q36
short answer
Solve : $x \cos y dy = e^x (x \log x + 1) dx$
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Q37
long answer
If $F(\alpha) = \begin{bmatrix} \cos \alpha & 0 & \sin \alpha \\ 0 & 1 & 0 \\ \sin \alpha & 0 & \cos \alpha \end{bmatrix}$, show that $[F(\alpha)]^{-1}=F(-\alpha)$
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Q38
long answer
Show that $p \to q$ and $q \to p$ are not equivalent.
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Q40
long answer
If $a+b+c=0$ and a, b, c are rational numbers then, prove that the roots of the equation $(b+c-a)x^2 + (c+a-b)x + (a+b-c)=0$ are rational numbers.
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Q41
long answer
Solve the equation $z^3+8i=0$, where $z \in C$.
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Q43
long answer
Using vector method, prove that $\cos(\alpha-\beta)=\cos\alpha \cos\beta+\sin\alpha \sin\beta$.
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Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function of random variable X is $f(x) = k$ for $200 \le x \le 600$, and $0$ otherwise.
Q46
long answer
Find the probability that daily sales will fall between 300 litres and 500 litres.
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Q47
long answer
Identify the type of conic and find centre, foci and vertices of $18x^2+12y^2-144x+48y+120=0$.
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Q48
long answer
If $\cos^{-1}x + \cos^{-1}y + \cos^{-1}z = \pi$ and $0 < x, y, z < 1$, show that $x^2 + y^2 + z^2 + 2xyz = 1$.
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Q49
long answer
A boy is walking along the path $y=ax^2+bx+c$ through the points $(-6, 8), (-2, -12)$ and $(3, 8)$. He wants to meet his friend at $P(7, 60)$. Will he meet his friend ? (Use Gaussian Elimination method).
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Q50
long answer
Prove that the ellipse $x^2 + 4y^2 = 8$ and the hyperbola $x^2 - 2y^2 = 4$ intersect orthogonally.
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Q51
long answer
Find the parametric form of Vector equation and Cartesian equations of the plane containing the line $\vec{r} = (3\hat{i} + 2\hat{j} + 4\hat{k}) + t(2\hat{i} + \hat{j} + \hat{k})$ and perpendicular to the plane $\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 8$.
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Q52
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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Q53
long answer
Prove that $p \to (q \lor r) \equiv p \lor (q \lor r)$ using truth table.
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Q54
long answer
Suppose a person deposits ` 10,000 in a bank account at the rate of 5% per annum compounded continuously. How much money will be in his bank account 18 months later ?
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Q56
long answer
Find the area of the region common to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the straight line $\frac{x}{a} + \frac{y}{b} = 1$.
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