Q1
mcq
1 mark
If $A$ is a $3 \times 3$ non-singular matrix such that $AA^T=A^TA$ and $B=A^{-1}A^T$, then $BB^T=$
- A. $I_3$
- B. $A$
- C. $B^T$
- D. $B$
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Q2
mcq
1 mark
If $\rho(A)=\rho([A\,|\,B])$, then the system $AX=B$ of linear equations is:
- A. consistent and has infinitely many solutions
- B. consistent and has a unique solution
- C. inconsistent
- D. consistent
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Q3
mcq
1 mark
If $z$ is a complex number such that $z\in \mathbb{C}\setminus \mathbb{R}$ and $\dfrac{1}{z}\in \mathbb{R}^+$, then $z$ is:
- A. 2
- C. 3
- D. 1
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Q4
mcq
1 mark
The product of all four values of $\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}$ is:
- A. 1
- B. $-2$
- C. 2
- D. $-1$
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Q5
mcq
1 mark
If $f$ and $g$ are polynomials of degrees $m$ and $n$ respectively and if $h(x)=(f\circ g)(x)$, then the degree of $h$ is:
- A. $m^n$
- B. $mn$
- C. $n^m$
- D. $m^{1/n}$
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Q6
mcq
1 mark
If $x<0$, then $\tan^{-1}\!\left(\dfrac{1}{x}\right)$ is equal to:
- A. $-\pi - \cot^{-1}(x)$
- B. $\tan^{-1}(x)$
- C. $-\pi - \tan^{-1}x$
- D. $\cot^{-1}(x)$
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Q7
mcq
1 mark
The eccentricity of the circle is:
- A. $\dfrac{1}{2}$
- C. 2
- D. 1
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Q8
mcq
1 mark
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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Q9
mcq
1 mark
If the image of the point $A(1,2,3)$ with respect to the plane $\vec{r}\cdot(2\hat{i}+4\hat{j}+3\hat{k})=0$ is $A'(3,6,11)$, then the foot of the perpendicular from the point $A$ to the given plane is:
- A. $(2,5,7)$
- B. $(2,3,7)$
- C. $(2,-4,7)$
- D. $(2,4,7)$
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Q10
mcq
1 mark
One of the closest points on the curve $x^2-y^2=4$ to the point $(6,0)$ is:
- A. $(3,5)$
- B. $(2,0)$
- C. $(13,-3)$
- D. $(5,1)$
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Q11
mcq
1 mark
The value of $c$ satisfied by Rolle's theorem for the function $f(x)=x^3-3x^2$, $x\in[0,3]$ is:
- A. $\dfrac{3}{2}$
- B. 1
- C. 2
- D. 2
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Q12
mcq
1 mark
The percentage error of fifth root of 31 is approximately how many times the percentage error in 31?
- A. 5
- B. $\dfrac{1}{31}$
- C. 31
- D. $\dfrac{1}{5}$
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Q13
mcq
1 mark
Let $A=\{(x,y): a<x<b,\ c<y<d\}\subset \mathbb{R}^2$. If the function $u:A\to \mathbb{R}^2$ is harmonic in $A$, then:
- A. $\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0,\ \forall A$
- B. $\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=1,\ \forall A$
- C. $\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0,\ \forall A$
- D. $\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=1,\ \forall A$
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Q13
mcq
Let $A = \{(x, y) * a < x < b, c < y < d\} \subset R^2$.
If the function $u : A \to R^2$ is harmonic in $A$, then:
- A. $\left( \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,\ \forall A \right)$
- B. $\left( \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=1,\ \forall A \right)$
- C. $\left( \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,\ \forall A \right)$
- D. $\left( \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=1,\ \forall A \right)$
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Q14
mcq
If $\int_0^x t\cos t\,dt = f(x)$, then $\frac{d}{dx}f =$
- A. $x\cos x$
- B. $\cos x-x\sin x$
- C. $x\sin x$
- D. $\sin 1\,x\cos x$
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Q15
mcq
If $\int_1^{\sin x} \frac{e^u}{u}\,du = f(x)$ and $\int_{1/2}^{\;?} \bigl[f(x)\bigr]^2\,dx = a$, then one of the possible values of $a$ is:
- A. 9
- B. 3
- C. 5
- D. 6
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Q16
mcq
The solution of the differential equation $\frac{d^2y}{dx^2}-3y=x$ represents:
- A. Parabola
- B. Straight lines
- C. Ellipse
- D. Circles
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Q17
mcq
P is the amount of certain substance left in after time $t$. If the rate of evaporation of the substance is proportional to the amount remaining, then:
- A. $P=Ckt$
- B. $P=Ce^{kt}$
- C. $Pt=C$
- D. $P=Ce^{-kt}$
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Q18
mcq
If the function $f(x)=\frac{1}{12}$ for $a<x<b$, represents a probability density function of a continuous random variable $X$, then which of the following cannot be the value of $a$ and $b$?
- A. 7 and 19
- B. 0 and 12
- C. 16 and 24
- D. 5 and 17
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Q19
mcq
A rod of length $2l$ is broken into two pieces at random. The probability density function of the shorter of the two pieces is $f(x)=\begin{cases} \frac{1}{l}, & 0<x<l \\ 0, & l<x<2l \end{cases}$. The mean and variance of the shorter of the two pieces are respectively:
- A. $\frac{l}{2},\ \frac{l^2}{12}$
- B. $\frac{l}{2},\ \frac{l^2}{3}$
- C. $\frac{l}{2},\ \frac{l^2}{12}$
- D. $\frac{l}{2},\ \frac{l^2}{6}$
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Q20
mcq
The dual of $(p\lor q)\lor[p\lor(p\land r)]$ is:
- A. $(p\land q)\land[p\land(p\land r)]$
- B. $(p\land q)\land[p\lor(p\land r)]$
- C. $(p\land q)\land[p\land(p\lor r)]$
- D. $(p\land q)\land[p\land(p\lor r)]$
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Q21
long answer
If $A$ is a non-singular matrix of odd order, prove that adj $A$ is positive.
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Q23
short answer
If $x^2+12(k12)x19k=0$ has equal roots, find $k$.
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Q25
short answer
Obtain the equation of the circles with radius 5 cm and touching x-axis at the origin in general form.
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Q26
short answer
Prove that the function $f(x)=x^2-2x-3$ is strictly increasing in the interval $(2,\infty)$.
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Q27
short answer
If $f(x,y)=\cos\left(\frac{x}{y}\right)$, then show that $\frac{\partial^2 f}{\partial y^2}=\frac{x^2}{y^2-x^2}$.
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Q29
short answer
Determine the order and degree (if exists) of the differential equation $\left[\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)\right] + 1 = 0$.
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Q30
short answer
If $\operatorname{Var}(X)=\frac{1}{2}$ then, find the value of $\operatorname{Var}(2X13)$.
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Q42
short answer
Prove that $(p \land q) \lor (q \lor r)$ using truth table.
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Q42
short answer
Prove that $$\int_{0}^{\pi/4} \tan x\,dx + \int_{0}^{\pi/8} \log 2\,dx = \log 1$$
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Q43
short answer
Solve the equation $(x^{11})(x^{13})(x-2)(x-4)^{121}=0$
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Q44
short answer
Show that the line $x-y14=0$ is a tangent to the ellipse $x^{2}13y^{2}=12$. Also find the co-ordinates of the point of contact.
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Q44
short answer
Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a certain sample 10% of the original number of radioactive nuclei have undergone disintegration in a period of 100 years. What percentage of the original radioactive nuclei will remain after 1000 years ?
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Q45
short answer
By Vector method prove that: $\cos(\alpha1\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$
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Q45
short answer
$(x^{2}1y^{2})dy=xydx$. It is given that $y(1)=1$ and $y(x_{0})=e$. Find the value of $x_{0}$.
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Q46
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 (i)$P(2<X<6)$ (ii)$P(2\le X<5)$ (iii)$P(X\le 4)$ (iv)$P(3<X)$
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Q46
short answer
At a water fountain, water attains a maximum height of $4\,\text{m}$ at horizontal distance of $0.5\,\text{m}$ from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of $0.75\,\text{m}$ from the point of origin.
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Q47
short answer
A particle moves along a line according to the law $s(t)=2t^{3}-9t^{2}112t-4$, where $t/0$.
(i)At what times the particle changes direction ?
(ii)Find the total distance travelled by the particle in the first 4 seconds.
(iii)Find the particle’s acceleration each time the velocity is zero.
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Q47
short answer
Find the non-parametric form of Vector equation and Cartesian equation of the plane passing through the point $(1, -2, 4)$ and perpendicular to the plane $x12y-3z=11$ and parallel to the line $3\;7\;3\;1\;1\;yx\;z++= =−$
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