If $\sum ^{30}_{r=1} \frac{r^{2}(^{30}C_{r})}{^{30}C_{r}_{-1}}^{2} =\alpha \times 2^{29}$ , then $\alpha$ is equal to ______
✓ Accepted Answer:465
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Q2mcq
Let $\alpha ,\beta ,\gamma$ and $\delta$ be the coefficients of $x^{7},x^{5},x^{3}$ and $x$ respectively in the expansion of
$^{5}^{5} \alpha u+\beta v= 18$
$(x+ \sqrt{x^{3}-1}) +(x- \sqrt{x^{3}-1}) ,x> 1$ . If u and v satisfy the equations then $u+v$ equals :
$\gamma u+\delta v= 20$
a.
5
Correct Answer
b.
3
c.
4
d.
8
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Q3mcq
If $e^{x} \frac{xsin-1x}{\sqrt{1-x^{2}}} + \frac{sin^{-1}x}{(1-x^{2})^{3/2}} + \frac{x}{1-x^{2}} dx= g(x) + C$ , where C is the constant of integration, then $g \frac{1}{2}$ equals
$\int ( ) ( )$
:
a.
$\frac{\pi}{4}\sqrt{\frac{e}{3}}$
b.
$\frac{\pi}{6}\sqrt{\frac{e}{3}}$
Correct Answer
c.
$\frac{\pi}{4}\sqrt{\frac{e}{2}}$
d.
$\frac{\pi}{6}\sqrt{\frac{e}{2}}$
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Q4mcq
The perpendicular distance, of the line $\frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2}$ from the point $P(2, -10, 1)$ , is :
a.
6
b.
$5\sqrt{2}$
c.
$4\sqrt{3}$
d.
$3\sqrt{5}$
Correct Answer
No explanation saved for this question yet.
Q5nat
Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance
from one of them. An equilateral triangle $PQR$ is formed such that $Q$ lies on one of the parallel lines, while $R$
lies on the other. Then $(QR)^{2}$ is equal to _______ -.
✓ Accepted Answer:28
No explanation saved for this question yet.
Q6mcq
Let $A = {1, 2, 3, 4}$ and $B = {1, 4, 9, 16}$ . Then the number of many-one functions $f : A \rightarrow B\rightarrow$ such that
$1 \in f(\,A)$ is equal to :
a.
151
Correct Answer
b.
139
c.
163
d.
127
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Q7mcq
Suppose that the number of terms in an A.P. is $2k,k\in N$ . If the sum of all odd terms of the A.P. is 40 , the sum
of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then k is equal to :
a.
6
b.
5
Correct Answer
c.
8
d.
4
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Q8mcq
For a $3 \times 3$ matrix $M$ , let trace $(M)$ denote the sum of all the diagonal elements of $M$ . Let $A$ be a $3 \times 3$ matrix
such that $|A| = \frac{1}{2}$ and trace $(A) = 3$ . If $B= adj(adj(2A))$ , then the value of $|B|+$ trace (B) equals :
a.
56
b.
132
c.
174
d.
280
Correct Answer
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Q9mcq
The area of the region enclosed by the curves $y=x^{2} -4x+ 4$ and $y^{2} = 16 -8x$ is :
a.
$\frac{8}{3}$
Correct Answer
b.
$\frac{4}{3}$
c.
8
d.
5
No explanation saved for this question yet.
Q10mcq
Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$ , and be parallel to the vector $3\hat{i}+ 2\hat{j}+ 2\hat{k}$ . If the
distance of the point Q from the point $R(1, 3, 3)$ is 5 , then the square of the area of $\triangle PQR$ is equal to :
a.
148
b.
136
Correct Answer
c.
144
d.
140
No explanation saved for this question yet.
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$x+y+ 2z= 6$
If the system of linear equations : $2x+ 3y+ az= a + 1$ where $a,b\in R$ , has infinitely many solutions, then
$- x-3y+ bz= 2\,b$
$7a+ 3b$ is equal to :
a.
16
b.
12
c.
22
d.
9
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Let $f(x) =\int _{0}^{x^{2}} \frac{t^{2}-8t+15}{e^{t}} dt,x\in R$ . Then the numbers of local maximum and local minimum points of $f$ ,
respectively, are :
a.
2 and 3
b.
2 and 2
c.
3 and 2
d.
1 and 3
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Let $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{\,b^{2}} = 1, a > b$ and $H : \frac{x^{2}}{\,A^{2}} - \frac{y^{2}}{\,B^{2}} = 1$ . Let the distance between the foci of E and the foci of $H$
be $2\sqrt{3}$ . If $a- A= 2$ , and the ratio of the eccentricities of $E$ and $H$ is $\frac{1}{3}$ , then the sum of the lengths of their
latus rectums is equal to:
a.
10
b.
9
c.
8
d.
7
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Let $A(6, 8),B(10 cos\alpha , -10 sin\alpha )$ and $C(-10 sin\alpha , 10 cos\alpha )$ , be the vertices of a triangle. If $L(a, 9)$ and
$G(h,k)$ be its orthocenter and centroid respectively, then $(5a-3h+ 6k+ 100 sin 2\alpha )$ is equal to ______ -.
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If $A$ and $B$ are two events such that $P(A\cap B) = 0.1$ , and $P(A| B)$ and $P(B| A)$ are the roots of the equation
$-$
$12x^{2} -7x+ 1 = 0$ , then the value of $\frac{P(A\cup B)}{P(A\cap B)}$ is :
a.
$\frac{4}{3}$
b.
$\frac{7}{4}$
c.
$\frac{5}{3}$
d.
$\frac{9}{4}$
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Let $A= {1, 2, 3}$ . The number of relations on $A$ , containing $(1, 2)$ and $(2, 3)$ , which are reflexive and
transitive but not symmetric, is ______ -
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If $lim_{x\rightarrow \infty \rightarrow } ((\frac{e}{1-e} ) (\frac{1}{e} - \frac{x}{1+x} ))^{x} =\alpha$ , then the value of $\frac{log_{e}\alpha}{1+log_{e}\alpha}$ equals :
a.
$e^{-1}$
b.
$e^{2}$
c.
$e^{-2}$
d.
e
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Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$ . If $\lambda \vec{a}+ 2\vec{b}$ and $3\vec{a}- \lambda \vec{b}$ are
perpendicular to each other, then the number of values of $\lambda$ in $[-1, 3]$ is :
a.
2
b.
1
c.
0
d.
3
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Let $\alpha _{\theta }$ and $\beta _{\theta }$ be the distinct roots of $2x^{2} + (cos\theta )x-1 = 0,\theta \in (0, 2\pi )$ . If m and M are the minimum and
the maximum values of $\alpha ^{4}_{\theta } +\beta ^{4}_{\theta }$ , then $16(M +m)$ equals :
a.
24
b.
25
c.
17
d.
27
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Let $y=f(x)$ be the solution of the differential equation $\frac{dy}{\,dx} + \frac{xy}{x^{2}-1} = \frac{x^{6}+4x}{\sqrt{1-x^{2}}} , -1 <x< 1$ such that
$f(0) = 0$ . If $6\int _{-1/2}^{1/2} f(x)dx= 2\pi - \alpha$ then $\alpha ^{2}$ is equal to _______ .
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In a group of 3 girls and 4 boys, there are two boys $B_{1}$ and $B_{2}$ . The number of ways, in which these girls and
boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_{1}$ and $B_{2}$ are
not adjacent to each other, is :
a.
96
b.
144
c.
120
d.
72
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Let $P(4, 4)\sqrt{3}$ be a point on the parabola $y^{2} = 4ax$ and PQ be a focal chord of the parabola. If M and $N$ are the
foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the
quadrilateral PQMN is equal to :
a.
$17\sqrt{3}$
b.
$\frac{263\sqrt{3}}{8}$
c.
$\frac{34\sqrt{3}}{3}$
d.
$\frac{343\sqrt{3}}{8}$
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Let the curve $z(1 +i) + -z(1 - i) = 4,z\in C$ , divide the region $|z-3| \leq 1$ into two parts of areas $\alpha$ and $\beta$ .
Then $|\alpha - \beta |$ equals :
a.
$1 + \frac{\pi}{2}$
b.
$1 + \frac{\pi}{3}$
c.
$1 + \frac{\pi}{6}$
d.
$1 + \frac{\pi}{4}$
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If $x=f(y)$ is the solution of the differential equation $(1 +y^{2})+(x-2e^{tan^{-1}y}) \frac{dy}{\,dx} = 0,y\in (-\frac{\pi}{2}, \frac{\pi}{2})$ with
$f(0) = 1$ , then $f(\frac{1}{\sqrt{3}} )$ is equal to :
a.
$e^{\pi /12}$
b.
$e^{\pi /4}$
c.
$e^{\pi /3}$
d.
$e^{\pi /6}$
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A parallel plate capacitor of area $A= 16\,cm^{2}$ and separation between the plates 10 cm , is charged by a DC
current. Consider a hypothetical plane surface of area $A_{0} = 3.2\,cm^{2}$ inside the capacitor and parallel to the
plates. At an instant, the current through the circuit is 6A. At the same instant the displacement current through
$A_{0}$ is ________ mA .
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A rectangular metallic loop is moving out of a uniform magnetic field region to a field free region with a
constant speed. When the loop is partially inside the magnate field, the plot of magnitude of induced emf $(\epsilon )$
with time $(t)$ is given by
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A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The
velocity of the body at point $A$ is $10\,m/s$ . The ratio of its kinetic energies at point $B$ and $C$ is :
(Take acceleration due to gravity as $10\,m/s^{2}$ )
a.
$\frac{2+\sqrt{2}}{3}$
b.
$\frac{2+\sqrt{3}}{3}$
c.
$\frac{3+\sqrt{3}}{2}$
d.
$\frac{3-\sqrt{2}}{2}$
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A tube of length 1 m is filled completely with an ideal liquid of mass 2 M , and closed at both ends. The tube is
rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is
F then angular velocity of the tube is $\sqrt{\frac{F}{\alpha M}}$ in SI unit. The value of $\alpha$ is __________.
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For a short dipole placed at origin $O$ , the dipole moment $P$ is along $x$ -axis, as shown in the figure. If the
electric potential and electric field at A are $V_{0}$ and $E_{0}$ , respectively, then the correct combination of the electric
potential and electric field, respectively, at point $B$ on the $y$ -axis is given by
a.
$V_{0}$ and $\frac{E_{0}}{4}$
b.
zero and $\frac{E_{0}}{16}$
c.
zero and $\frac{E_{0}}{8}$
d.
$\frac{V_{0}}{2}$ and $\frac{E_{0}}{16}$
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A force $\vec{F}= 2\hat{i}+ b\hat{j}+ \hat{k}$ is applied on a particle and it undergoes a displacement $\hat{i}-2\hat{j}- \hat{k}$ . What will be the
value of $b$ , if work done on the particle is zero.
a.
0
b.
$\frac{1}{2}$
c.
2
d.
$\frac{1}{3}$
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A symmetric thin biconvex lens is cut into four equal parts by two planes $AB$ and $CD$ as shown in figure. If
the power of original lens is 4 D then the power of a part of the divided lens is
a.
D
b.
8D
c.
2D
d.
4D
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The maximum percentage error in the measurment of density of a wire is [Given, mass of wire
$= (0.60 \pm 0.003)g$ radius of wire $= (0.50 \pm 0.01)cm$ length of wire $= (10.00 \pm 0.05)cm]$
a.
8
b.
5
c.
4
d.
7
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The torque due to the force $(2\hat{i}+ \hat{j}+ 2\hat{k})$ about the origin, acting on a particle whose position vector is
$(\hat{i}+ \hat{j}+ \hat{k})$ , would be
a.
$\hat{i}- \hat{k}$
b.
$\hat{i}+ \hat{k}$
c.
$\hat{j}+ \hat{k}$
d.
$\hat{i}- \hat{j}+ \hat{k}$
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An electron projected perpendicular to a uniform magnetic field B moves in a circle. If Bohr's quantization is
applicable, then the radius of the electronic orbit in the first excited state is :
a.
$\sqrt{\frac{h}{\pi eeB}}$
b.
$\sqrt{\frac{2h}{\pi eB}}$
c.
$\sqrt{\frac{h}{2\pi eB}}$
d.
$\sqrt{\frac{4\,h}{\pi eB}}$
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Two long parallel wires $X$ and $Y$ , separated by a distance of 6 cm , carry currents of 5A and 4A, respectively,
in opposite directions as shown in the figure. Magnitude of the resultant magnetic field at point P at a distance
of 4 cm from wire Y is $x\times 10^{-5}\,T$ . The value of $x$ is__________ . Take permeability of free space as
$\mu _{0} = 4\pi \times 10^{-7}SI$ units.
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A series LCR circuit is connected to an alternating source of emf E. The current amplitude at resonant
frequency is $I_{0}$ . If the value of resistance R becomes twice of its initial value then amplitude of current at
resonance will be
a.
$2I_{0}$
b.
$I_{0}$
c.
$\frac{I_{0}}{2}$
d.
$\frac{I_{0}}{\sqrt{2}}$
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A small rigid spherical ball of mass M is dropped in a long vertical tube containing glycerine. The velocity of
the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the
viscous force acting on the ball will be (consider g as acceleration due to gravity)
a.
2 Mg
b.
Mg
c.
$\frac{3}{2}Mg$
d.
$\frac{Mg}{2}$
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For a diatomic gas, if $\gamma _{1} =(\frac{Cp}{Cv})$ for rigid molecules and $\gamma _{2} =(\frac{Cp}{Cv})$ for another diatomic molecules, but
also having vibrational modes. Then, which one of the following options is correct ? (Cp and Cv are specific
heats of the gas at constant pressure and volume)
a.
$\gamma _{2} =\gamma _{1}$
b.
$2\gamma _{2} =\gamma _{1}$
c.
$\gamma _{2} <\gamma _{1}$
d.
$\gamma _{2} >\gamma _{1}$
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A light source of wavelength $\lambda$ illuminates a metal surface and electrons are ejected with maximum kinetic
energy of 2 eV . If the same surface is illuminated by a light source of wavelength $\frac{\lambda}{2}$ , then the maximum
kinetic energy of ejected electrons will be (The work function of metal is 1 eV )
a.
3 eV
b.
2 eV
c.
6 eV
d.
5 eV
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A transparent film of refractive index, 2.0 is coated on a glass slab of refractive index, 1.45. What is the
minimum thickness of transparent film to be coated for the maximum transmission of Green light of
wavelength 550 nm . [Assume that the light is incident nearly perpendicular to the glass surface.]
a.
137.5 nm
b.
275 nm
c.
94.8 nm
d.
68.7 nm
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A ball of mass 100 g is projected with velocity $20\,m/s$ at $60^{\circ }$ with horizontal. The decrease in kinetic energy
of the ball during the motion from point of projection to highest point is
a.
5 J
b.
15 J
c.
20 J
d.
zero
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Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In Young's double slit experiment, the fringes produced by red light are closer as compared to
those produced by blue light. Reason (R): The fringe width is directly proportional to the wavelength of light.
In the light of the above statements, choose the correct answer from the options given below :
a.
Both $(A)$ and $(R)$ are true but $(R)$ is NOT the
correct explanation of $(A)$
b.
(A) is true but (R) is false
c.
Both (A) and (R) are true and (R) is the correct
explanation of (A)
d.
(A) is false but (R) is true
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Given are statements for certain thermodynamic variables, (A) Internal energy, volume (V) and mass (M) are
extensive variables. (B) Pressure (P), temperature (T) and density ( $\rho$ ) are intensive variables. (C) Volume (V),
temperature (T) and density ( $\rho$ ) are intensive variables. (D) Mass (M), temperature (T) and internal energy
are extensive variables. Choose the correct answer from the options given below :
a.
(B) and (C) Only
b.
(C) and (D) Only
c.
(D) and (A) Only
d.
(A) and (B) Only
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Which one of the following is the correct dimensional formula for the capacitance in F ? $M, L, T$ and C stand
for unit of mass, length, time and charge,
a.
$[F] =[C^{2}M^{-1}\,L^{-2}\,T^{2}]$
b.
$[F] =[C^{2}M^{-2}\,L^{2}\,T^{2}]$
c.
$[F] =[CM^{-2}\,L^{-2}\,T^{-2}]$
d.
$[F] =[CM^{-1}\,L^{-2}\,T^{2}]$
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A tube of length $L$ is shown in the figure. The radius of cross section at the point (1) is 2 cm and at the point
(2) is 1 cm , respectively. If the velocity of water entering at point (1) is $2\,m/s$ , then velocity of water leaving
the point (2) will be
a.
$4\,m/s$
b.
$2\,m/s$
c.
$6\,m/s$
d.
$8\,m/s$
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Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively,
than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R): The mass
of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose
the correct answer from the options given below :
a.
$(A)$ is false but (R) is true
b.
(A) is true but (R) is false
c.
Both $(A)$ and $(R)$ are true and $(R)$ is the correct
explanation of $(A)$
d.
Both (A) and (R) are true but (R) is NOT the
correct explanation of (A)
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A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of
$2 \times 10^{5}\,ms^{-1}$ . When the electric field is switched off, the proton moves along a circular path of radius 2 cm .
The magnitude of electric field is $x\times 10^{4}\,N/C$ . The value of $x$ is _______ Take the mass of the proton
$= 1.6 \times 10^{-27}\,kg$ .
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Given below are two statements : Statement (I) : Nitrogen, sulphur, halogen and phosphorus present in an
organic compound are detected by Lassaigne's Test. Statement (II) : The elements present in the compound are
converted from covalent form into ionic form by fusing the compound with Magnesium in Lassaigne's test. In
the light of the above statements, choose the correct anower from the options given below :
a.
Statement I is false but Statement II is true
b.
Both Statement I and Statement II are true
c.
Both Statement I and Statement II are false
d.
Statement I is true but Statement II is false
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Given below are two statements : Statement (I) : A spectral line will be observed for a $2p_{x} \rightarrow 2p\rightarrow_{y}$ transition.
Statement (II) : $2P_{x}$ and $2p_{y}$ are degenerate orbitals. In the light of the above statements, choose the correct
answer from the options given below :
a.
Both Statement I and Statement II are true
b.
Statement I is false but Statement II is true
c.
Both Statement I and Statement II are false
d.
Statement I is true but Statement II is false
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Identify the homoleptic complex(es) that is/are low spin. (A) $[Fe(CN)_{5}NO]^{2-}$ (B) $[CoF_{6}]^{3-}$ (C) $[Fe(CN)_{6}]^{4-}$
(D) $[Co(NH_{3})_{6}]^{3+}$ (E) $[Cr(H_{2}O)_{6}]^{2+}$ Choose the correct answer from the options given below :
a.
(C) only
b.
(C) and (D) only
c.
(A) and (C) only
d.
(B) and (E) only
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Match the Compounds (List - I) with the appropriate Catalyst/Reagents (List - II) for their reduction into
corresponding amines.
Choose the correct answer from the options given below :
a.
(A)-(II), (B)-(I), (C)-(III), (D)-(IV)
b.
(A)-(III), (B)-(II), (C)-(IV), (D)-(I)
c.
(A)-(II), (B)-(IV), (C)-(III), (D)-(I)
d.
(A)-(III), (B)-(IV), (C)-(II), (D)-(I)
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Given below are two statements : Statement (I) : Corrosion is an electrochemical phenomenon in which pure
metal acts as an anode and impure metal as a cathode. Statement (II) : The rate of corrosion is more in alkaline
medium than in acidic medium. In the light of the above statements, choose the correct answer from the
options given below :
a.
Both Statement I and Statement II are true
b.
Statement I is false but Statement II is true
c.
Statement I is true but Statement II is false
d.
Both Statement I and Statement II are false
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The compound with molecular formula $C_{6}H_{6}$ , which gives only one monobromo derivative and takes up four
moles of hydrogen per mole for complete hydrogenation has ________ $=$ electrons.
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Given below are two statements : Statement (I) : An element in the extreme left of the periodic table forms
acidic oxides. Statement (II) : Acid is formed during the reaction between water and oxide of a reactive
element present in the extreme right of the periodic table. In the light of the above statements, choose the
correct answer from the options given below :
a.
Statement I is true but Statement II is false
b.
Both Statement I and Statement II are true
c.
Both Statement I and Statement II are false
d.
Statement I is false but Statement II is true
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20 mL of 2 M NaOH solution is added to 400 mL of 0.5 M NaOH solution. The final concentration of the
solution is _______ $\times 10^{-2}M$ . (Nearest integer)
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