$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8 , and $B$
wins if he throws a sum of 8 before $A$ throws a sum of 5 . The probability, that $A$ wins if A makes the first
throw, is
a.
$\frac{8}{17}$
b.
$\frac{9}{19}$
Correct Answer
c.
$\frac{9}{17}$
d.
$\frac{8}{19}$
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Q2nat
If for some $\alpha ,\beta ;\alpha \leq \beta ,\alpha +\beta -8$ and $sec^{2} (tan^{-1} \alpha )+ cosec^{2} (cot^{-1} \beta )-36$ , then $\alpha ^{2} +\beta$ is_______.
✓ Accepted Answer:14
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Q3nat
Let $f$ be a differentiable function such that $2(x+ 2)^{2}f(x) -3(x+ 2)^{2} = 10\int _{0}^{x}(t+ 2)f(t)dt,x\geq 0$ . Then
$f(2)$ is equal to ______.
✓ Accepted Answer:19
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Q4mcq
$IfI(m,n) =\int _{0}^{1} x^{m-1}(1 - x)^{n-1}dx,m,n> 0$ , then $I(9, 14) +I(10, 13)$ is
a.
$I(19, 27)$
b.
$I(9, 1)$
c.
$I(1, 13)$
d.
$I(9, 13)$
Correct Answer
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Q5nat
$x$
$[ ]$
Let $A$ be a $3 \times 3$ matrix such that $X^{T}AX =O$ for all nonzero $3 \times 1$ matrices $X = y$ . If
$[ ]$
$z$
$1 1 1 0$
$[ ] [ ] [ ] [ ]$
$A 1 = 4 ,A 2 = 4$ , and $det(adj(2(A+1))) -2^{\alpha }3^{\beta }5^{\gamma },\alpha ,\beta ,\gamma \in N$ , then $\alpha ^{2} +\beta ^{2} +\gamma ^{2}$
$[ ] [ ] [ ] [ ]$
$1 -5 1 -8$
is_____.
✓ Accepted Answer:44
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Q6mcq
Let the lines $3x-4y- \alpha = 0, 8x-11y-33 = 0$ , and $2x-3y+\lambda = 0$ be concurrent. If the image of the
point $(1, 2)$ in the line $2x-3y+\lambda = 0$ is $(\frac{57}{13} , \frac{-40}{13})$ , then $|\alpha \lambda |$ is equal to
a.
$84$
b.
$113$
c.
$91$
Correct Answer
d.
$101$
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Q7mcq
The area of the region ${(x,y) :x^{2} + 4x+ 2 \leq y\leq |x+ 2|}$ is equal to
a.
$7$
b.
$5$
c.
$24/5$
d.
$20/3$
Correct Answer
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Q8mcq
For some $n\neq 10$ , let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of $(1 + x)^{n+4}$ be
in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:
Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ intersect the line
$\frac{x+2}{3} = \frac{y-3}{2} = \frac{z-4}{1}$ at the point $P$ . Then the distance of $P$ from the point $Q(4, -5, 1)$ is
a.
$5$
b.
$5\sqrt{5}$
c.
$5\sqrt{6}$
d.
$10$
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Let circle $C$ be the image of $x +y -2x+ 4y-4 = 0$ in the line $2x-3y+ 5 = 0$ and $A$ be the point on $C$
such that $OA$ is parallel to $x$ -axis and $A$ lies on the right hand side of the centre $O$ of $C$ . If $B(\alpha ,\beta )$ , with $\beta < 4$ ,
lies on $C$ such that the length of the are $AB$ is $(1/6)^{th\,}$ of the perimeter of $C$ , then $\beta - \sqrt{3}\alpha$ is equal to
a.
$3 + \sqrt{3}$
b.
$4$
c.
$4 - \sqrt{3}$
d.
$3$
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Consider the region $R={(x,y) :x\leq y\leq 9 - \frac{11}{3}x^{2},x\geq 0}$ . The area, of the largest rectangle of sides
parallel to the coordinate axes and inscribed in R , is:
a.
$\frac{730}{119}$
b.
$\frac{625}{111}$
c.
$\frac{821}{123}$
d.
$\frac{567}{121}$
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Let $S = {p_{1},p_{2}… . ,p_{10}}$ be the set of first ten prime numbers. Let $A=S\cup P$ , where $P$ is the set of all
possible products of distinct elements of $S$ . Then the number of all ordered pairs ( $x,y$ ), $x\in S$ , $y\in A$ , such
that $x$ divides $y$ , is ______.
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Let $S_{n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + …$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p
and common difference p is $\sqrt{2026\,S}_{2025}$ , then the absolute difference betwen $20^{th\,}$ and $15^{th\,}$ terms of the A.P.
is
a.
$20$
b.
$90$
c.
$45$
d.
$25$
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Let $f :R-{0} R$ be a function such that $f(x) -6f(\frac{1}{x} )= \frac{35}{3x} - \frac{5}{2}$ . If the
$lim_{x\rightarrow 0\rightarrow } (\frac{1}{\alpha x} +f(\vec{x}))=\beta ;\alpha ,\beta \in R$ , then $\alpha + 2\beta$ is equal to
a.
$5$
b.
$3$
c.
$4$
d.
$6$
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If $\alpha$ and $\beta$ are the roots of the equation $2z^{2} -3z-2i = 0$ , where $i = \sqrt{-1}$ , then
$16 \cdot Re( \frac{\alpha ^{19}+\beta ^{19}+\alpha ^{11}+\beta ^{11}}{\alpha ^{15}+\beta ^{15}})\cdot lm(\frac{\alpha ^{19}+\beta ^{19}+\alpha ^{11}+\beta ^{11}}{\alpha ^{15}+\beta ^{15}})$ is equal to
a.
$441$
b.
$398$
c.
$312$
d.
$409$
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For a statistical data $x_{1}, x_{2}, … , x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum ^{10}_{i=1} x^{2}_{i} = 371$ . He
later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8
, respectively. The variance of the corrected data is
a.
$9$
b.
$5$
c.
$7$
d.
$4$
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Let $\vec{a}= \hat{i}+ 2\hat{j}+ 3\vec{k}, \vec{b}= 3\hat{i}+ \hat{j}- \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$ . If the vector $\vec{C}$
is perpendicular to $\hat{b}$ and $\vec{a}\cdot \vec{c}= 5$ , then $|\vec{c}|$ is equal to
a.
$\sqrt{\frac{11}{6}}$
b.
$\frac{1}{3\sqrt{2}}$
c.
$16$
d.
$18$
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Let the product of the focal distances of the point $(\sqrt{3}, \frac{1}{2})$ on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a > b)$ , be $\frac{7}{4}$ . Then the
absolute difference of the eccentricities of two such ellipses is
a.
$\frac{1-\sqrt{3}}{\sqrt{2}}$
b.
$\frac{3-2\sqrt{2}}{2\sqrt{3}}$
c.
$\frac{3-2\sqrt{2}}{3\sqrt{2}}$
d.
$\frac{1-2\sqrt{2}}{\sqrt{3}}$
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Let $f(x) = \frac{2^{x}^{+2}+16}{2^{2}^{x}^{+1}+2^{x}^{+4}+32}$ . Then the value of $8(f(\frac{1}{15})+f(\frac{2}{15})+ … +f(\frac{59}{15}))$ is equal to
a.
$92$
b.
$118$
c.
$102$
d.
$108$
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Let in a $\triangle ABC$ , the length of the side $AC$ be 6 , the vertex $B$ be $(1, 2, 3)$ and the vertices $A,C$ lie on the line
$\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$ . Then the area (in sq. units) of $\triangle ABC$ is:
a.
$17$
b.
$21$
c.
$56$
d.
$42$
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A car of mass ' $m$ ' moves on a banked road having radius ' $r$ ' and banking angle $\theta$ . To avoid slipping from
banked road, the maximum permissible speed of the car is $v_{0}$ . The coefficient of friction $\mu$ between the wheels
of the car and the banked road is
The Young's double slit interference experiment is performed using light consisting of 480 nm and 600 nm
wavelengths to form interference patterns. The least number of the bright fringes of 480 nm light that are
required for the first coincidence with the bright fringes formed by 600 nm light is
a.
$5$
b.
$4$
c.
$6$
d.
$8$
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A uniform solid cylinder of mass ' $m$ ' and radius ' $r$ ' rolls along an inclined rough plane of inclination $45^{\circ }$ . If
it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be
a.
$\frac{1}{\sqrt{2}}\,g$
b.
$\frac{1}{3\sqrt{2}}\,g$
c.
$\frac{\sqrt{2}\,g}{3}$
d.
$\sqrt{2}g$
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An object of mass ' m ' is projected from origin in a vertical $xy$ plane at an angle $45^{\circ }$ with the x axis with an
initial velocity $v_{0}$ . The magnitude and direction of the angular momentum of the object with respect to origin,
when it reaches at the maximum height, will be [ g is acceleration due to gravity]
a.
$\frac{mv^{3}_{o}}{2\sqrt{2}g}$ along negative $z$ -axis
b.
$\frac{mv^{3}_{o}}{4\sqrt{2}g}$ along positive $z$ -axis
c.
$\frac{mv^{3}_{o}}{4\sqrt{2}g}$ along negative $z$ -axis
d.
$\frac{mv^{3}_{o}}{2\sqrt{2}g}$ along positive $z$ -axis
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A plano-convex lens having radius of curvature of first surface 2 cm exhibits focal length of $f_{1}$ in air. Another
plano-convex lens with first surface radius of curvature 3 cm has focal length of $f_{2}$ when it is immersed in a
liquid of refractive index 1.2 . If both the lenses are made of same glass of refractive index 1.5 , the ratio of $f_{1}$
and $f_{2}$ will be
a.
$1 : 2$
b.
$1 : 3$
c.
$3 : 5$
d.
$2 : 3$
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Consider a parallel plate capacitor of area A (of each plate) and separation 'd' between the plates. If $E$ is the
electric field and $\epsilon _{0}$ is the permittivity of free space between the plates, then potential energy stored in the
capacitor is
a.
$\epsilon _{0}E^{2}Ad$
b.
$\frac{1}{2}\epsilon _{0}E^{2}Ad$
c.
$\frac{1}{4}\epsilon _{0}E^{2}Ad$
d.
$\frac{3}{4}\epsilon _{0}E^{2}Ad$
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The temperature of 1 mole of an ideal monoatomic gas is increased by $50^{\circ }C$ at constant pressure. The total
heat added and change in internal energy are $E_{1}$ and $E_{2}$ , respectively. If $\frac{E_{1}}{E_{2}} = \frac{x}{9}$ then the value of $x$ is _____
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A thin plano convex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.2.
When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid
behaves like a concave mirror of focal length 0.2 m . The radius of curvature of the curved surface of the lens
is
a.
0.20 m
b.
0.25 m
c.
0.15 m
d.
0.10 m
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During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is
$2000$ Åand it becomes $6000$ Åwhen the electron jumps from state B to state C. Then the wavelength of the
radiation emitted during the transition of electrons from state A to state B is
a.
$4000$ Å
b.
$2000$ Å
c.
$3000$ Å
d.
$6000$ Å
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A satellite is launched into a circular orbit of radius ' R ' around the earth. A second satellite is launched into an
orbit of radius 1.03 R . The time period of revolution of the second satellite is larger than the first one
approximately by
a.
$9%$
b.
$3%$
c.
$4.5%$
d.
$2.5%$
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An electron of mass ' m ' with an initial velocity $\vec{v}= v_{0}\hat{i}(v_{0} > 0)$ enters an electric field $\vec{E}= -E_{o}\hat{k}$ . If the
initial de Broglie wavelength is $\lambda _{0}$ , the value after time t would be
The amount of work done to break a big water drop of radius ' $R$ ' into 27 small drops of equal radius is 10 J .
The work done required to break the same big drop into 64 small drops of equal radius will be
a.
15 J
b.
5 J
c.
20 J
d.
10 J
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A square loop of sides $a= 1\,m$ is held normally in front of a point charge $q = 1C$ . The flux of the electric
field through the shaded region is $\frac{5}{p} \times \frac{1}{\epsilon _{0}} \frac{Nm^{2}}{C}$ , where the value of p is _____.
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An air bubble of radius 0.1 cm lies at a depth of 20 cm below the free surface of a liquid of density
$1000\,kg/m^{3}$ . If the pressure inside the bubble is $2100\,N/m^{2}$ greater than the atmospheric pressure, then the
surface tension of the liquid in SI unit is (use $g = 10\,m/s^{2}$ )
a.
$0.1$
b.
$0.05$
c.
$0.02$
d.
$0.25$
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A force $F =\alpha +\beta x^{2}$ acts on an object in the x -direction. The work done by the force is 5 J when the object is
displaced by 1 m . If the constant $\alpha = 1\,N$ then $\beta$ will be
a.
$15\,N/m^{2}$
b.
$12\,N/m^{2}$
c.
$8\,N/m^{2}$
d.
$10\,N/m^{2}$
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A current of 5A exists in a square loop of side $\frac{1}{\sqrt{2}}\,m$ . Then the magnitude of the magnetic field $B$ at the centre
of the square loop will be $p\times 10^{-6}\,T$ . where, value of p is _____ . $[$ Take $\mu _{0} = 4\pi \times 10^{-7}TmA^{-1}]$ .
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For an experimental expression $y= \frac{32.3\times 1125}{27.4}$ , where all the digits are significant. Then to report the value of $y$
we should write
a.
$y= 1326.19$
b.
$y= 1330$
c.
$y= 1326.186$
d.
$y= 1326.2$
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The least count of a screw guage is 0.01 mm . If the pitch is increased by $75%$ and number of divisions on the
circular scale is reduced by $50%$ , the new least count will be _____ $\times 10^{-3}\,mm$
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Consider the following statements: A. The junction area of solar cell is made very narrow compared to a photo
diode. B. Solar cells are not connected with any external bias. C. LED is made of lightly doped p-n junction.
D. Increase of forward current results in continuous increase of LED light intensity. E. LEDs have to be
connected in forward bias for emission of light. Choose the correct answer from the options given below:
a.
B, E Only
b.
B, D, E Only
c.
A, C Only
d.
A, C, E Only
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An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly
with temperature. A. The work done by gas during the process is zero. B. The heat added to gas is different
from change in its internal energy. C. The volume of the gas is increased. D. The internal energy of the gas is
increased. E. The process is isochoric (constant volume process) Choose the correct answer from the options
given below:
a.
E Only
b.
A, B, C, D Only
c.
A, D, E Only
d.
A, C Only
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A parallel plate capacitor was made with two rectangular plates, each with a length of $l= 3\,cm$ and breath of
$b = 1\,cm$ . The distance between the plates is $3\mu \,m$ . Out of the following, which are the ways to increase the
capacitance by a factor of 10 ? A. $l= 30\,cm, \,b = 1\,cm, \,d = 1\mu \,m$ B. $l= 3\,cm, \,b = 1\,cm, \,d = 30\mu \,m$ C.
$l= 6\,cm, \,b = 5\,cm, \,d = 3\mu \,m$ D. $l= 1\,cm, \,b = 1\,cm, \,d = 10\mu \,m$ E. $l= 5\,cm, \,b = 2\,cm, \,d = 1\mu \,m$
Choose the correct answer from the options given below:
a.
A only
b.
C only
c.
B and D only
d.
$C$ and $E$ only
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A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the
total distance and displacement covered by the particle in 12.5 s , then $\frac{D}{d}$ is
a.
$\frac{16}{5}$
b.
$10$
c.
$\frac{15}{4}$
d.
$25$
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Aman has been asked to synthesise the molecule
(x).He thought of preparing the molecule using an aldol condensation reaction. He found a few cyclic alkenes
in his laboratory. He thought of performing ozonolysis reaction on alkene to produce a dicarbonyl compound
followed by aldol reaction to prepare " x ". Predict the suitable alkene that can lead to the formation of " $x$ ".
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$37.8\,g\,N_{2}O_{5}$ was taken in a 1 L reaction vessel and allowed to undergo the following reaction at 500 K
$2\,N_{2}O_{5(\,g)} \rightleftharpoons 2\,N_{2}O_{4(\,g)}+ O_{2(\,g)}$ The total pressure at equilibrium was found to be 18.65 bar. Then, $Kp =$
______ $\times 10^{-2}$ [nearest integer] Assume $N_{2}O_{5}$ to behave ideally under these conditions. Given: $R = 0.082$ bar
$Lmol^{-1}\,K^{-1}$
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Consider the given plots of vapour pressure (VP) vs temperature(T/K). Which amongst the following options
is correct graphical representation showing $\Delta T_{f}$ , depression in the freezing point of a solvent in a solution?
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Let us consider an endothermic reaction which is non-spontaneous at the freezing point of water. However, the
reaction is spontaneous at boiling point of water. Choose the correct option.
a.
Both $\Delta H$ and $\Delta S$ are (-ve)
b.
$\Delta H$ is (-ve) but $\Delta S$ is (+ve)
c.
$\Delta H$ is $(+ve)$ but $\Delta S$ is $(-ve)$
d.
Both $\Delta H$ and $\Delta S$ are (+ve)
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Consider the following reaction occurring in the blast furnace: $Fe_{3}O_{4(\,s)}+ 4CO_{(g)} \rightarrow 3Fe\rightarrow_{(l)}+ 4CO_{2(\,g)}$ ' $x$ '
kg of iron is produced when $2.32 \times 10^{3}\,kgFe_{3}O_{4}$ and $2.8 \times 10^{2}\,kgCO$ are brought together in the furnace.
The value of ' $x$ ' is _____. (nearest integer) Given: molar mass of $Fe_{3}O_{4} = 232\,g\,mol^{-1}$ molar mass of
$CO = 28\,g\,mol^{-1}$ molar mass of $Fe = 56\,g\,mol^{-1}}$
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Standard entropies of $X_{2}, Y_{2}$ and $XY_{5}$ are 70,50 and $110\,J\,K^{-1}\,mol^{-1}$ respectively. The temperature in
Kelvin at which the reaction $\frac{1}{2} X_{2}+ \frac{5}{2}Y_{2} \rightleftharpoons XY_{5}\Delta H^{Θ} = -35\,kJ\,mol^{-1}$ will be at equilibrium is_______.
(Nearest integer)
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One mole of the octahedral complex compound $Co(NH_{3})_{5}Cl_{3}$ gives 3 moles of ions on dissolution in water.
One mole of the same complex reacts with excess of $AgNO_{3}$ solution to yield two moles of $AgCl_{(s)}$ . The
structure of the complex is:
a.
$[Co(NH_{3})_{4}Cl_{2}] \cdot Cl \cdot NH_{3}$
b.
$[Co(NH_{3})_{3}Cl_{3}] \cdot 2NH_{3}$
c.
$[Co(NH_{3})_{5}Cl]Cl_{2}$
d.
$[Co(NH_{3})_{4}Cl] \cdot Cl_{2}\cdot NH_{3}$
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Which of the following linear combination of atomic orbitals will lead to formation of molecular orbitals in
homonuclear diatomic molecules [internuclear axis in $z$ -direction] ? A. $2p_{z}$ and $2p_{x}$ B. 2 s and $2p_{x}$ C. $3\,d_{xy}$
and $3\,d_{x^{2}-y^{2}}$ D. 2 s and $2p_{z}$ E. $2p_{z}$ and $3d_{x}^{2} - y^{2}$ Choose the correct answer from the options given below:
a.
A and B Only
b.
D Only
c.
E Only
d.
C and D Only
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Given below are two statements: Statement I: The conversion proceeds well in the less polar medium.
$-CH_{3}-CH_{2}-CH_{2}-CH_{2}-Cl^{HO}\rightarrow \rightarrow^{-} CH_{3}-CH_{2}-CH_{2}-CH_{2}-OH + Cl^{^{(-)}Statement II: The}$
conversion proceeds well in the more polar 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 true but Statement II is false
c.
Statement I is false but Statement II is true
d.
Both Statement I and Statement II are false
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Which of the following statements are NOT true about the periodic table? A. The properties of elements are
function of atomic weights. B. The properties of elements are function of atomic numbers. C. Elements having
similar outer electronic configurations are arranged in same period. D. An element's location reflects the
quantum numbers of the last filled orbital. E. The number of elements in a period is same as the number of
atomic orbitals available in energy level that is being filled. Choose the correct answer from the options given
below:
a.
$A, C$ and E Only
b.
A and E Only
c.
$B, C$ and E Only
d.
D and E Only
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Among the following cations, the number of cations which will give characteristic precipitate in their
identification tests with $K_{4}[Fe(CN)_{6}]$ is _____. $Cu^{2+}, Fe^{3+}, Ba^{2+}, Ca^{2+}, NH^{+}_{4} , Mg^{2+}, Zn^{2+}$
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Which of the following statement is true with respect to $H_{2}O, NH_{3}$ and $CH_{4}$ ? A. The central atoms of all the
molecules are $sp^{3}$ hybridized. B. The $H -O -H, H -N -H$ and $H -C -H$ angles in the above molecules
are $104.5^{\circ }, 107.5^{\circ }$ and $109.5^{\circ }$ , respectively. C. The increasing order of dipole moment is $CH_{4} < NH_{3} < H_{2}O$ .
D. Both $H_{2}O$ and $NH_{3}$ are Lewis acids and $CH_{4}$ is a Lewis base. E. A solution of $NH_{3}$ in $H_{2}O$ is basic. In this
solution $NH_{3}$ and $H_{2}O$ act as Lowry-Bronsted acid and base respectively. Choose the correct answer from the
options given below:
a.
A, B and C Only
b.
A, D and E Only
c.
C, D and E Only
d.
A, B, C and E Only
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Given below are two statements I and II. Statement I: Dumas method is used for estimation of "Nitrogen" in an
organic compound. Statement II: Dumas method involves the formation of ammonium sulphate by heating the
organic compound with conc $H_{2}SO_{4}$ . 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 false
c.
Statement I is false but Statement II is true
d.
Both Statement I and Statement II are true
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For a reaction, $N_{2}O_{5(\,g)} \rightarrow 2NO\rightarrow_{2(\,g)}+ \frac{1}{2}O_{2(\,g)}$ in a constant volume container, no products were present
initially. The final pressure of the system when $50%$ of reaction gets completed is
a.
5 times of initial pressure
b.
$5/2$ times of initial pressure
c.
$7/2$ times of initial pressure
d.
$7/4$ times of initial pressure
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The carbohydrate "Ribose" present in DNA, is A. A pentose sugar B. present in pyranose from C. in "D"
configuration D. a reducing sugar, when free E. in $\propto$ -anomeric form Choose the correct answer from the
options given below:
a.
A, D and E Only
b.
A, C and D Only
c.
A, B and E Only
d.
B, D and E Only
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