If $\sum ^{n}_{r=1} T_{r} = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$ , then $lim_{n\rightarrow \infty \rightarrow } \sum ^{n}_{r=1} (\frac{1}{T_{r}})$ is equal to :
a.
$0$
b.
$\frac{2}{3}$
Correct Answer
c.
$1$
d.
$\frac{1}{3}$
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Q2mcq
Let $L_{1} : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_{2} : \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$ be two lines. Then which of the following points lies
on the line of the shortest distance between $L_{1}$ and $L_{2}$ ?
a.
$(\frac{14}{3}, -3, \frac{22}{3})$
Correct Answer
b.
$(-\frac{5}{3} , -7, 1)$
c.
$(2, 3, \frac{1}{3})$
d.
$( \frac{8}{3}, -1, \frac{1}{3})$
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Q3mcq
Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x+ 2y= 2$ . If
the centroid of $\triangle PQR$ is the point $(\alpha ,\beta )$ , then $15(\alpha - \beta )$ is equal to :
a.
$19$
b.
$24$
c.
$21$
d.
$22$
Correct Answer
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Q4mcq
Let $x=x(y)$ be the solution of the differential equation $y^{2}\,dx+(x- \frac{1}{y})dy= 0$ . If $x(1) = 1$ , then $x( \frac{1}{2})$ is :
a.
$\frac{1}{2} + e$
b.
$3 +e$
c.
$3 - e$
Correct Answer
d.
$\frac{3}{2} +e$
No explanation saved for this question yet.
Q5nat
Let $A$ be a square matrix of order 3 such that $det(A) = -2$ and $det(3 adj(-6 adj(3A))) = 2^{m+n} \cdot 3^{mn}, m > n$
. Then $4\,m + 2n$ is equal to _______
✓ Accepted Answer:34
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Q6mcq
The product of all solutions of the equation $e^{5(log_{e}x)^{2}+3} =x^{8},x> 0$ , is :
a.
$e^{8/5}$
Correct Answer
b.
$e^{6/5}$
c.
$e^{2}$
d.
e
No explanation saved for this question yet.
Q7mcq
The number of non-empty equivalence relations on the set ${1, 2, 3}$ is :
a.
$6$
b.
$5$
Correct Answer
c.
$7$
d.
$4$
No explanation saved for this question yet.
Q8mcq
Let $A= {1, 2, 3, … , 10}$ and $B={\frac{m}{n} :m,n\in A,m<n$ and $gcd(m,n) = 1}$ . Then $n(B)$ is equal to :
a.
$36$
b.
$31$
Correct Answer
c.
$37$
d.
$29$
No explanation saved for this question yet.
Q9mcq
A circle $C$ of radius 2 lies in the second quadrant and touches both the coordinate axes. Let $r$ be the radius of a
circle that has centre at the point $(2, 5)$ and intersects the circle $C$ at exactly two points. If the set of all
possible values of r is the interval $(\alpha ,\beta )$ , then $3\beta -2\alpha$ is equal to :
a.
$10$
b.
$15$
Correct Answer
c.
$12$
d.
$14$
No explanation saved for this question yet.
Q10mcq
Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black
balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$ ,
where $gcd(m,n) = 1$ , then $m+n$ is equal to :
a.
$4$
b.
$14$
Correct Answer
c.
$13$
d.
$11$
No explanation saved for this question yet.
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Let $f(x)$ be a real differentiable function such that $f(0) = 1$ and $f(x+y) =f(x)f'(y) +f'(x)f(y)$ for all
$x,y\in R$ . Then $\sum ^{100}_{n=1} log_{e} f(n)$ is equal to :
a.
$2525$
b.
$5220$
c.
$2384$
d.
$2406$
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From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number
of ways, in which the middle letter is ' M ', is :
a.
$5148$
b.
$6084$
c.
$4356$
d.
$14950$
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Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum
values of $16((sec^{-1} x)^{2} +(cosec^{-1} x)^{2})$ is :
a.
$24\pi ^{2}$
b.
$22\pi ^{2}$
c.
$31\pi ^{2}$
d.
$18\pi ^{2}$
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Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+ 4\hat{k},\lambda > 0$ , on the vector $\vec{a}= \hat{i}+ 2\hat{j}+ 2\hat{k}$ . If $|\vec{a}+ \vec{c}| = 7$ , then the
area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________
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Let $f :R\rightarrow \rightarrow R$ be a twice differentiable function such that $f(x+y) =f(x)f(y)$ for all $x,y\in R$ . If
$f'(0) = 4a$ and $f$ satisfies $f''(x) -3af'(x) - f(x) = 0$ , $a > 0$ , then the area of the region
$R = {(x,y) |0 \leq y\leq f(ax), 0 \leq x\leq 2}$ is:
a.
$e^{2} -1$
b.
$e^{2} + 1$
c.
$e^{4} + 1$
d.
$e^{4} -1$
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Let the parabola $y=x^{2} + px-3$ , meet the coordinate axes at the points $P, Q$ and R . If the circle C with centre
at $(-1, -1)$ passes through the points $P,Q$ and $R$ , then the area of $\triangle PQR$ is :
a.
$7$
b.
$4$
c.
$6$
d.
$5$
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Let $z_{1},z_{2}$ and $z_{3}$ be three complex numbers on the circle $|z| = 1$ with $arg (z_{1}) = \frac{-\pi}{4}, arg (z_{2}) = 0$ and
$arg (z_{3}) = \frac{\pi}{4}$ . If $|z_{1}z-_{2}+z_{2}z-_{3}+z_{3}z-_{1}|^{2} =\alpha +\beta \sqrt{2},\alpha ,\beta \in Z$ , then the value of $\alpha ^{2} +\beta ^{2}$ is :
a.
$24$
b.
$29$
c.
$41$
d.
$31$
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Let $L_{1} : \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z+1}{0}$ and $L_{2} : \frac{x-2}{2} = \frac{y}{0} = \frac{z+4}{\alpha} ,\alpha \in R$ , be two lines, which intersect at the point $B$ . If
$P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_{2}$ , then the value of $26\alpha (\,PB)^{2}$ is _________
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$-3ax^{2}-2, x< 1$
Let the function, $f(x) ={$ be differentiable for all $x\in R$ , where $a> 1,b\in R$ . If the
$a^{2} +bx, x⩾ 1$
area of the region enclosed by $y=f(x)$ and the line $y= -20$ is $\alpha +\beta \sqrt{3},\alpha ,\beta \in Z$ , then the value of $\alpha +\beta$ is
________
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A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma ^{2}$ denote the
mean and variance of $X$ , then the value of $64(\mu +\sigma ^{2})$ is :
a.
$51$
b.
$64$
c.
$32$
d.
$48$
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A uniform circular disc of radius ' R ' and mass ' M ' is rotating about an axis perpendicular to its plane and
passing through its centre. A small circular part of radius $R/2$ is removed from the original disc as shown in
the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.
a.
$\frac{7}{32} MR^{2}$
b.
$\frac{9}{32}MR^{2}$
c.
$\frac{17}{32} MR^{2}$
d.
$\frac{13}{32}MR^{2}$
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An amount of ice of mass $10^{-3}\,kg$ and temperature $-10^{\circ }C$ is transformed to vapour of temperature $110^{\circ }C$ by
applying heat. The total amount of work required for this conversion is, (Take, specific heat of ice
$= 2100Jkg^{-1}\,K^{-1}$ , specific heat of water $= 4180Jkg^{-1}\,K^{-1}$ , specific heat of steam $= 1920Jkg^{-1}\,K^{-1}$ ,
Latent heat of ice $= 3.35 \times 10^{5}Jkg^{-1}$ and Latent heat of steam $= 2.25 \times 10^{6} Jkg^{-1}$ )
a.
3043 J
b.
3024 J
c.
3003 J
d.
3022 J
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The work functions of cesium (Cs) and lithium (Li) metals are 1.9 eV and 2.5 eV , respectively. If we incident
a light of wavelength 550 nm on these two metal surfaces, then photo-electric effect is possible for the case of
a.
Both Cs and Li
b.
Neither Cs nor Li
c.
Cs only
d.
Li only
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A small point of mass $m$ is placed at a distance $2R$ from the centre ' $O'$ of a big uniform solid sphere of mass
M and radius R . The gravitational force on ' m ' due to M is $F_{1}$ . A spherical part of radius $R/3$ is removed
from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found
to be $F_{2}$ . The value of ratio $F_{1} : F_{2}$ is
a.
$12 : 11$
b.
$11 : 10$
c.
$12 : 9$
d.
$16 : 9$
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A parallel-plate capacitor of capacitance $40\mu \,F$ is connected to a 100 V power supply. Now the intermediate
space between the plates is filled with a dielectric material of dielectric constant $K = 2$ . Due to the
introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor,
respectively, are
a.
4 mC and 0.2 J
b.
8 mC and 2.0 J
c.
2 mC and 0.4 J
d.
2 mC and 0.2 J
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Given below are two statements : Statement I : In a vernier callipers, one vernier scale division is always
smaller than one main scale division. Statement II : The vernier constant is given by one main scale division
multiplied by the number of vernier scale divisions. 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.
Statement I is false but Statement II is true
c.
Both Statement I and Statement II are false
d.
Both Statement I and Statement II are true
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An electron in the ground state of the hydrogen atom has the orbital radius of $5.3 \times 10^{-11}\,m$ while that for the
electron in third excited state is $8.48 \times 10^{-10}\,m$ . The ratio of the de Broglie wavelengths of electron in the
excited state to that in the ground state is
a.
3
b.
16
c.
9
d.
4
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$| ||$
Three conductors of same length having thermal conductivity $k_{1}, k_{2}$ and $k_{3}$ are connected as shown in figure.
Area of cross sections of $1^{st\,}$ and $2^{nd\,}$ conductor are same and for $3^{rd\,}$ conductor it is double of the $1^{st\,}$
conductor. The temperatures are given in the figure. In steady state condition, the value of $\theta$ is _______ $^{\circ }C$ .
(Given : $k_{1} = 60Js^{-1}\,m^{-1}\,K^{-1}, k_{2} = 120Js^{-1}\,m^{-1}\,K^{-1}, k_{3} = 135Js^{-1}\,m^{-1}\,K^{-1}$ )
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The position vectors of two 1 kg particles, (A) and (B), are given by $\vec{r}_{A} =(\alpha _{1}t^{2}\hat{i}+\alpha _{2}t\hat{j}+\alpha _{3}t\hat{k})m$ and
$\vec{r}_{B} =(\beta _{1}t\hat{i}+\beta _{2}t^{2}\hat{j}+\beta _{3}t\hat{k})m$ , respectively;
$(\alpha _{1} = 1\,m/s^{2},\alpha _{2} = 3nm/s,\alpha _{3} = 2\,m/s,\beta _{1} = 2\,m/\vec{s},\beta _{2} = -1\,m/s^{2},\beta _{3} = 4pm/s)$ , where t is time, n and p
$-$
are constants. At $t= 1\,s, V_{\vec{A}} = \vec{V}_{B}$ and velocities $V_{A}$ and $\vec{V}_{B}$ of the particles are orthogonal to each other.
At $t= 1\,s$ , the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is
$\sqrt{L}kgm^{2}\,s^{-1}$ . The value of L is _______ .
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A line charge of length $\frac{\,'\,a\,'\,}{2}$ is kept at the center of an edge $BC$ of a cube $ABCDEFGH$ having edge length '
$a$ ' as shown in the figure. If the density of line charge is $\lambda$ C per unit length, then the total electric flux through
all the faces of the cube will be . (Take, $\epsilon _{0}$ as the free space permittivity)
a.
$\frac{\lambda a}{2\epsilon _{0}}$
b.
$\frac{\lambda a}{4\epsilon _{0}}$
c.
$\frac{\lambda a}{16\epsilon _{0}}$
d.
$\frac{\lambda a}{8\epsilon _{0}}$
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Sliding contact of a potentiometer is in the middle of the potentiometer wire having resistance $R_{p} = 1\Omega$ as
shown in the figure. An external resistance of $R_{e} = 2\Omega$ is connected via the sliding contact. The electric
current in the circuit is :
a.
0.9 A
b.
1.35 A
c.
0.3 A
d.
1.0 A
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Given is a thin convex lens of glass (refractive index $\mu$ ) and each side having radius of curvature $R$ . One side
is polished for complete reflection. At what distance from the lens, an object be placed on the optic axis so that
the image gets formed on the object itself ?
a.
$R/\mu$
b.
$R/(2\mu -3)$
c.
$\mu R$
d.
$R/(2\mu -1)$
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Two spherical bodies of same materials having radii 0.2 m and 0.8 m are placed in same atmosphere. The
temperature of the smaller body is 800 K and temperature of the bigger body is 400 K . If the energy radiated
from the smaller body is E, the energy radiated from the bigger body is (assume, effect of the surrounding
temperature to be negligible),
a.
16 E
b.
E
c.
64 E
d.
256 E
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A particle is projected at an angle of $30^{\circ }$ from horizontal at a speed of $60\,m/s$ . The height traversed by the
particle in the first second is $h_{0}$ and height traversed in the last second, before it reaches the maximum height,
is $h_{1}$ . The ratio $h_{0} :h_{1}$ is _________ [Take, $g = 10\,m/s^{2}$ ]
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A closed organ and an open organ tube are filled by two different gases having same bulk modulus but
different densities $\rho _{1}$ and $\rho _{2}'$ , respectively. The frequency of $9^{th\,}$ harmonic of closed tube is identical with $4^{th\,}$
harmonic of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is
$\rho _{1} :\rho _{2} = 1 : 16$ , then the length of the open tube is :
a.
$\frac{15}{7} \,cm$
b.
$\frac{20}{7}\,cm$
c.
$\frac{15}{9} \,cm$
d.
$\frac{20}{9}\,cm$
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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) : If Young's double slit experiment is performed in an optically denser medium than air, then the
consecutive fringes come closer. Reason-(R) : The speed of light reduces in an optically denser medium than
air while its frequency does not change. In the light of the above statements, choose the most appropriate
answer from the options given below :
a.
Both $(A)$ and $(R)$ are true but $(R)$ is not the
correct explanation of $(A)$
b.
Both $(A)$ and $(R)$ are true and $(R)$ is the correct
explanation of $(A)$
c.
(A) is true but (R) is false
d.
$(A)$ is false but (R) is true
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A bob of mass $m$ is suspended at a point $O$ by a light string of length $l$ and left to perform vertical motion
(circular) as shown in figure. Initially, by applying horizontal velocity $v_{0}$ at the point ' A ', the string becomes
slack when, the bob reaches at the point ' D '. The ratio of the kinetic energy of the bob at the points $B$ and $C$ is
______ -.
a.
1
b.
2
c.
4
d.
3
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Two soap bubbles of radius 2 cm and 4 cm , respectively, are in contact with each other. The radius of
curvature of the common surface, in cm , is ______ .
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The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side
view mirror, which is a convex mirror with radius of curvature $R = 2\,m$ . Another car approaches him from
behind with a uniform speed of $90\,km/hr$ . When the car is at a distance of 24 m from him, the magnitude of
the acceleration of the image of the car in the side view mirror is ' a '. The value of 100 a is _______ $m/s^{2}$ .
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Given below are two statements : Statement-I : The equivalent emf of two nonideal batteries connected in
parallel is smaller than either of the two emfs. Statement-II : The equivalent internal resistance of two nonideal
batteries connected in parallel is smaller than the internal resistance of either of the two batteries. In the light
of the above statements, choose the correct answer from the options given below.
a.
Both Statement-I and Statement-II are false
b.
Statement-I is false but Statement-II is true
c.
Both Statement-I and Statement-II are true
d.
Statement-I is true but Statement-II is false
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In the diagram given below, there are three lenses formed. Considering negligible thickness of each of them as
compared to $|R_{1}|$ and $|R_{2}|$ , i.e., the radii of curvature for upper and lower surfaces of the glass lens, the power
of the combination is
An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal
plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component
of velocity $10^{6}\,m/s$ . If the magnitude of the electric field between the plates is $9.1\,V/cm$ , then the vertical
component of velocity of electron is (mass of electron $= 9.1 \times 10^{-31}\,kg$ and charge of electron
$= 1.6 \times 10^{-19}C$ )
a.
0
b.
$1 \times 10^{6}\,m/s$
c.
$16 \times 10^{6}\,m/s$
d.
$16 \times 10^{4}\,m/s$
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$A \rightarrow \,B\rightarrow$ The molecule A changes into its isomeric form B by following a first order kinetics at a temperature
of 1000 K . If the energy barrier with respect to reactant energy for such isomeric transformation is
$191.48\,kJ\,mol^{-1}$ and the frequency factor is $10^{20}$ , the time required for $50%$ molecules of A to become $B$ is
_________ picoseconds (nearest integer). $[R = 8.314\,J\,K^{-1}\,mol^{-1}]$
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In Carius method for estimation of halogens, 180 mg of an organic compound produced 143.5 mg of AgCl .
The percentage composition of chlorine in the compound is _______ \%. (Given : molar mass in $gmol^{-1}$ of
$Ag : 108, Cl : 35.5$ )
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A solution of aluminium chloride is electrolysed for 30 minutes using a current of 2 A . The amount of the
aluminium deposited at the cathode is [Given : molar mass of aluminium and chlorine are $27\,g\,mol^{-1}$ and
$35.5\,g\,mol^{-1}$ respectively. Faraday constant $= 96500Cmol^{-1}]$
a.
1.660 g
b.
0.336 g
c.
0.441 g
d.
1.007 g
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Given below are two statements : Statement I : One mole of propyne reacts with excess of sodium to liberate
half a mole of $H_{2}$ gas. Statement II : Four g of propyne reacts with $NaNH_{2}$ to liberate $NH_{3}$ gas which
occupies 224 mL at STP. In the light of the above statements, choose the most appropriate answer from the
options given below:
a.
Statement I is incorrect but Statement II is
correct
b.
Both Statement I and Statement II are correct
c.
Statement I is correct but Statement II is
incorrect
d.
Both Statement I and Statement II are incorrect
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A liquid when kept inside a thermally insulated closed vessel at $25^{\circ }C$ was mechanically stirred from outside.
What will be the correct option for the following thermodynamic parameters ?
a.
$\Delta U < 0, q = 0, w > 0$
b.
$\Delta U = 0, q = 0, w = 0$
c.
$\Delta U > 0, q = 0, w > 0$
d.
$\Delta U = 0, q < 0, w > 0$
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Match List-I with List-II.
$List- I cList- II$
$(A) Al^{3+} < Mg^{2+} < Na^{+} < F^{-} (I) IonisationEnthalpy$
$(B) B < C < O < N (II) Metalliccharacter$
$(C) B < Al < Mg < K (III) Electronegativity$
$(D) Si < P < S < Cl (IV) Ionicradii$
Choose the correct answer from the options given below :
a.
(A)-(IV), (B)-(I), (C)-(II), (D)-(III)
b.
(A)-(IV), (B)-(I), (C)- (III), (D)-(II)
c.
(A)-(III), (B)-(IV), (C)- (II), (D)-(I)
d.
(A)-(II), (B)-(III), (C)-(IV), (D)-(I)
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The number of molecules/ions that show linear geometry among the following is ________
$SO_{2}, BeCl_{2}, CO_{2}, \,N^{-}_{3} , NO_{2}, \,F_{2}O, XeF_{2}, NO^{+}_{2} , I^{-}_{3} , O_{3}$
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The compounds which give positive Fehling's test are : (A)
(B)
(C) $HOCH_{2}-CO -(CHOH)_{3}-CH_{2}-OH$ (D)
(E)
Choose the correct answer from the options given below :
a.
(A), (D) and (E) Only
b.
(C), (D) and (E) Only
c.
(A), (C) and (D) Only
d.
(A), (B) and (C) Only
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Some $CO_{2}$ gas was kept in a sealed container at a pressure of 1 atm and at 273 K . This entire amount of $CO_{2}$
gas was later passed through an aqueous solution of $Ca(OH)_{2}$ . The excess unreacted $Ca(OH)_{2}$ was later
neutralized with 0.1 M of 40 mL HCl . If the volume of the sealed container of $CO_{2}$ was $x$ , then $x$ is ________
$cm^{3}$ (nearest integer). [Given : The entire amount of $CO_{2}(\,g)$ reacted with exactly half the initial amount of
$Ca(OH)_{2}$ present in the aqueous solution.]
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Arrange the following solutions in order of their increasing boiling points. (i) $10^{-4}M\,NaCl$ (ii) $10^{-4}M\,$ Urea
(iii) $10^{-3}M\,NaCl$ (iv) $10^{-2}M\,NaCl$
a.
(i) < (ii) < (iii) < (iv)
b.
(iv) < (iii) < (i) < (ii)
c.
(ii) < (i $) \equiv ($ iii) < (iv)
d.
(ii) < (i) < (iii) < (iv)
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Given below are two statements : Statement I : $CH_{3}-O -CH_{2}-Cl$ will undergo $S_{N}1$ reaction though it is a
primary halide. Statement II :
will not undergo $S_{N}2$ reaction very easily though it is a primary halide. In the light of the above statements,
choose the most appropriate answer from the options given below :
a.
Both Statement I and Statement II are incorrect
b.
Both Statement I and Statement II are correct
c.
Statement I is incorrect but Statement II is
correct
d.
Statement I is correct but Statement II is
incorrect
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The incorrect statements regarding geometrical isomerism are : (A) Propene shows geometrical isomerism. (B)
Trans isomer has identical atoms/groups on the opposite sides of the double bond. (C) Cis-but-2-ene has higher
dipole moment than trans-but-2-ene. (D) 2-methylbut-2-ene shows two geometrical isomers. (E) Trans-isomer
has lower melting point than cis isomer. Choose the correct answer from the options given below :
a.
(A) and (E) Only
b.
(A), (D) and (E) Only
c.
(B) and (C) Only
d.
(C), (D) and (E) Only
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Lanthanoid ions with $4f^{7}$ configuration are : (A) $Eu^{2+}$ (B) $Gd^{3+}$ (C) $Eu^{3+}$ (D) $Tb^{3+}$ (E) $Sm^{2+}$ Choose the
correct answer from the options given below :
a.
(A) and (D) only
b.
(B) and (C) only
c.
(A) and (B) only
d.
(B) and (E) only
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A vessel at 1000 K contains $CO_{2}$ with a pressure of 0.5 atm . Some of $CO_{2}$ is converted into CO on addition
of graphite. If total pressure at equilibrium is 0.8 atm , then Kp is :
a.
1.8 atm
b.
0.3 atm
c.
3 atm
d.
0.18 atm
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