In the given figure, a body of mass $M$ is held between two massless springs, on a smooth inclined plane. The
free ends of the springs are attached to firm supports. If each spring has spring constant $k,$ the frequency of
oscillation of given body is:
A particle is projected with velocity $v_{0}$ along $x$ -axis. A damping force is acting on the particle which is
proportional to the square of the distance from the origin i.e. $ma= -\alpha x^{2}.$ The distance at which the particle
stops:
A soft ferromagnetic material is placed in an external magnetic field. The magnetic domains:
a.
have no relation with external magnetic field.
b.
decrease in size and changes orientation.
c.
increase in size but no change in orientation.
d.
may increase or decrease in size and change its
orientation.
Correct Answer
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Q4nat
A uniform thin bar of mass $6\, kg$ and length $2. 4$ meter is bent to make an equilateral hexagon. The moment of
inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is
___________ $\times 10^{-1}\, kg \,m^{2}.$
✓ Accepted Answer:8
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Q5nat
An electromagnetic wave of frequency $3\, GHz$ enters a dielectric medium of relative electric permittivity $2. 25$
from vacuum. The wavelength of this wave in that medium will be _________ $\times 10^{-2}\, cm.$
✓ Accepted Answer:667
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Q6nat
A uniform metallic wire is elongated by $0. 04\,m$ when subjected to a linear force $F.$ The elongation, if its
length and diameter is doubled and subjected to the same force will be _________ $cm.$
✓ Accepted Answer:2
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Q7mcq
An X-ray tube is operated at $1. 24$ million volt. The shortest wavelength of the produced photon will be:
a.
$10^{-3} nm$
Correct Answer
b.
$10^{-2} nm$
c.
$10^{-4} nm$
d.
$10^{-1} nm$
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Q8nat
A point charge of $+12\,\mu C$ is at a distance $6\, cm$ vertically above the centre of a square of side $12\, cm$ as shown
in figure. The magnitude of the electric flux through the square will be _______ $\times 10^{3}\,N\,m^{2}\,C^{-1}.$
✓ Accepted Answer:226
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Q9mcq
Two electrons each are fixed at a distance $2d.$ A third charge proton placed at the midpoint is displaced slightly
by a distance $x(x\ll d)$ perpendicular to the line joining the two fixed charges. Proton will execute simple
harmonic motion having angular frequency:
( $m=$ mass of charged particle)
A signal of $0. 1\, kW$ is transmitted in a cable. The attenuation of cable is $-5\, dB$ per $km$ and cable length is
$20\, km.$ The power received at the receiver is $10^{-x}\,W.$ The value of $x$ is _________. [Gain in
$dB = 10\, log_{10}(\frac{P_{O}}{P_{i}} )$ ]
✓ Accepted Answer:8
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If one mole of an ideal gas at $(P_{1},V_{1})$ is allowed to expand reversibly and isothermally ( $A$ to $B$ ) its pressure is
reduced to one-half of the original pressure (see figure). This is followed by a constant volume cooling till its
pressure is reduced to one-fourth of the initial value $(B\rightarrow \rightarrow C).$ Then it is restored to its initial state by a
reversible adiabatic compression ( $C$ to $A$ ). The net workdone by the gas is equal to:
a.
$0$
b.
$RTln(2)$
c.
$-\frac{RT}{2(\gamma -1)}$
d.
$RT [ln(2) - \frac{1}{2(\gamma -1)}]$
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Two cars are approaching each other at an equal speed of $7. 2\, km \, hr^{-1}.$ When they see each other, both blow
horns having a frequency of $676\, Hz.$ The beat frequency heard by each driver will be _______ $Hz.$ [Velocity of
sound in air is $340\,m\,s^{-1}.$
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The root-mean-square speed of molecules of a given mass of a gas at $27\,^{\circ}C$ and $1$ atmosphere pressure is
$200\,m\,s^{-1}.$ The root-mean-square speed of molecules of the gas at $127\,^{\circ}C$ and $2$ atmosphere pressure is
$\frac{x}{\sqrt{3}}\,m\,s^{-1}.$ The value of $x$ will be __________.
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Given below are two statements:
Statement I: $p- n$ junction diodes can be used to function as a transistor, simply by connecting two diodes,
back to back, which acts as the base terminal
Statement II: In the study of transistors, the amplification factor $\beta$ indicates ratio of the collector current to the
base current.
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 true but Statement II is false
c.
Statement I is false but Statement II is true
d.
Both Statement I and Statement II are true
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A body weighs $49\,N$ on a spring balance at the north pole. What will be its weight recorded on the same
weighing machine, if it is shifted to the equator?
[Use $g= \frac{GM}{R^{2}} = 9. 8\,m\,s^{-2}$ and radius of earth, $R= 6400\, km.$ ]
a.
$49. 17\,N$
b.
$48. 83\,N$
c.
$49. 83\,N$
d.
$49\,N$
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A cylindrical wire of radius $0. 5\, mm$ and conductivity $5 \times 10^{7}\,S\,m^{-1}$ is subjected to an electric field of
$10\, mV \,m^{-1}.$ The expected value of current in the wire will be $x^{3}\pi \, mA.$ The value of $x$ is _________.
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The period of oscillation of a simple pendulum is $T = 2\pi \sqrt{\frac{L}{g}}.$ Measured value of $L$ is $1. 0\,m$ from meter scale
having a minimum division of $1\, mm$ and time of one complete oscillation is $1. 95\,s$ measured from stopwatch
of $0. 01\,s$ resolution. The percentage error in the determination of $'g'$ will be:
a.
$1. 03%$
b.
$1. 33%$
c.
$1. 30%$
d.
$1. 13%$
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The figure shows a circuit that contains four identical resistors with resistance $R= 2. 0\,\Omega ,$ two identical
inductors with inductance $L= 2. 0\, mH$ and an ideal battery with E.M.F. $E = 9\,V.$ The current $i$ just after the
switch $S$ is closed will be:
a.
$9\,A$
b.
$3. 37\,A$
c.
$2. 25\,A$
d.
$3. 0\,A$
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Match List - I with List - II.
List-I List-II
(a) Source of microwave frequency (i) Radioactive decay of nucleus
(b) Source of infrared frequency (ii) Magnetron
(c) Source of Gamma Rays (iii) Inner shell electrons
(d) Source of $X$ -rays (iv) Vibration of atoms and molecules
(v) LASER
(vi) RC circuit
Choose the correct answer from the options given below:
a.
(a)-(ii), (b)-(iv), (c)-(vi), (d)-(iii)
b.
(a)-(vi), (b)-(iv), (c)-(i), (d)-(v)
c.
(a)-(vi), (b)-(v), (c)-(i), (d)-(iv)
d.
(a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)
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A series LCR circuit is designed to resonate at an angular frequency $\omega _{0} = 10^{5}\, rad \,s^{-1}.$ The circuit draws
$16\,W$ power from $120\,V$ source at resonance. The value of resistance $R$ in the circuit is ________ $\Omega .$
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A circular hole of radius $(\frac{a}{2})$ is cut out of a circular disc of radius $a$ as shown in figure. The centroid of the
remaining circular portion with respect to point $O$ will be:
a.
$\frac{2}{3}a$
b.
$\frac{5}{6}a$
c.
$\frac{1}{6}a$
d.
$\frac{10}{11} a$
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Two solids $A$ and $B$ of mass $1\, kg$ and $2\, kg$ respectively are moving with equal linear momentum. The ratio of
their kinetic energies $(K.E. )_{A} : (K.E. )_{B}$ will be $\frac{A}{1},$ so the value of $A$ will be _________.
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Sucrose hydrolyses in acid solution into glucose and fructose following first order rate law with a half-life of
$3. 33\,h$ at $25^{\circ}C.$ After $9\,h,$ the fraction of sucrose remaining is $f.$ The value of $log_{10}(\frac{1}{f})$ is ________ $\times 10^{-2}.$
(Rounded off to the nearest integer)
[Assume: In $10 = 2. 303, \, ln \,2 = 0. 693$ ]
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The solubility product of $PbI_{2}$ is $8. 0 \times 10^{-9}.$ The solubility of lead iodide in $0. 1$ molar solution of lead nitrate
is $x\times 10^{-6}\, mol /L.$ The value of $x$ is _________ (Rounded off to the nearest integer)
[Given $\sqrt{2}= 1. 41$ ]
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$1. 86\,g$ of aniline completely reacts to form acetanilide. $10%$ of the product is lost during purification. Amount
of acetanilide obtained after purification (in $g$ ) is ___________ $\times 10^{-2}.$
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The formula of a gaseous hydrocarbon which requires 6 times of its own volume of $O_{2}$ for complete oxidation
and produces $4$ times its own volume of $CO_{2}$ is $C_{x}H_{y}.$ The value of $y$ is ________.
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The magnitude of the change in oxidising power of the $MnO^{-}_{4} / Mn^{2+}$ couple is $x \times 10^{-4}\,V$ if the $H^{+}$
concentration is decreased from $1\,M$ to $10^{-4}M$ at $25^{\circ}C.$ (Assume concentration of $MnO^{-}_{4}$ and $Mn^{2+}$ to be
same on change in $H^{+}$ concentration). The value of $x$ is __________.
(Rounded off to the nearest integer)
[Given: $\frac{2.303 RT}{F} = 0. 059$ ]
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Match List - I with List - II.
List-I List-II
(a) Valium (i) Antifertility drug
(b) Morphine (ii) Pernicious anaemia
(c) Norethindrone (iii) Analgesic
(d) Vitamin $B_{12}$ (iv) Tranquilizer
a.
(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
b.
(a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)
c.
(a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)
d.
(a)-(iv), (b)-(iii), (c)-(i), (d)-(ii)
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Match List - I with List - II.
List-I (Salt) List-II (Flame colour wavelength)
(a) (i) $Br_{2}/ NaOH$
(b) (ii) $H_{2}/Pd -BaSO_{4}$
(c) (iii) $Zn(Hg)/ Conc. HCl$
(d) (iv) $Cl_{2}/ Red \,P, \,H_{2}O$
Choose the correct answer from the options given below:
a.
(a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)
b.
(a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)
c.
(a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
d.
(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)
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Given below are two statements:
Statement I: The value of the parameter "Biochemical Oxygen Demand (BOD)" is important for survival of
aquatic life.
Statement II: The optimum value of BOD is $6. 5\, ppm.$
In the light of the above statements, choose the most appropriate 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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$C_{6}H_{6}$ freezes at $5. 5^{\circ}C.$ The temperature at which a solution of $10\,g$ of $C_{4}H_{10}$ in $200\,g$ of $C_{6}H_{6}$ freeze is
_______ $^{\circ }C.$ (nearest integer value), (The molal freezing point depression constant of $C_{6}H_{6}$ is $5. 12^{\circ}C/m.$ )
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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: Hydrogen is the most abundant element in the Universe, but it is not the most abundant gas in the
troposphere.
Reason R: Hydrogen is the lightest element.
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.
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 true
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Match List - I with List - II.
List-I (Metal) List-II (Ores)
(a) Aluminium (i) Siderite
(b) Iron (ii) Calamine
(c) Copper (iii) Kaolinite
(d) Zinc (iv) Malachite
Choose the correct answer from the options given below:
a.
(a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)
b.
(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
c.
(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)
d.
(a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)
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The volume occupied by $4. 75\,g$ of acetylene gas at $50^{\circ}C$ and $740\, mm \, Hg$ pressure is ____________ $L.$
(Rounded off to the nearest integer)
[Given $R = 0. 0826\,L\, atm \,K^{-1}\, mol^{-1}$ ]
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According to Bohr's atomic theory:
(A) Kinetic energy of electron is $\propto \frac{Z^{2}}{n^{2}}.$
(B) The product of velocity (v) of electron and principal quantum number $(n), \,'vn'\propto Z^{2}.$
(C) Frequency of revolution of electron in an orbit is $\propto \frac{Z^{3}}{n^{3}}.$
(D) Coulombic force of attraction on the electron is $\propto \frac{Z^{3}}{n^{4}}.$
Choose the most appropriate answer from the options given below:
a.
(A), (C) and (D) only
b.
(A) and (D) only
c.
(C) only
d.
(A) only
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Among the following allotropic forms of sulphur, the number of allotropic forms, which will show
paramagnetism is _______.
(A) -sulphur$\alpha$
(B) -sulphur$\beta$
(C) $S_{2}$ -form
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Match List - I with List - II.
List-I (Salt) List-II (Flame colour wavelength)
(a) $LiCl$ (i) $455. 5\, nm$
(b) $NaCl$ (ii) $670. 8\, nm$
(c) $RbCl$ (iii) $780. 0\, nm$
(d) $CsCl$ (iv) $589. 2\, nm$
Choose the correct answer from the options given below:
a.
(a)-(iv), (b)-(ii), (c)-(iii), (d)-(i)
b.
(a)-(i), (b)-(iv), (c)-(ii), (d)-(iii)
c.
(a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)
d.
(a)-(ii), (b)-(iv), (c)-(iii), (d)-(i)
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Let $a,b\in R.$ If the mirror image of the point $P(a, 6, 9)$ with respect to the line $\frac{x-3}{7} = \frac{y-2}{5} = \frac{z-1}{-9}$ is
$(20, \,b, -a-9),$ then $|a+b|$ is equal to:
a.
$86$
b.
$90$
c.
$84$
d.
$88$
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The students $S_{1},S_{2}, … ,S_{10}$ are to be divided into $3$ groups $A, \,B$ and $C$ such that each group has at least one
student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups
is __________.
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$| |$
Let $f$ be a twice differentiable function defined on $R$ such that $f(0) = 1, \,f'(0) = 2$ and $f'(x) \neq 0$ for all
$f(x) f'(x)$
$x\in R.$ If $= 0,$ for all $x\in R,$ then the value of $f(1)$ lies in the interval
$f'(x) f''(x)$
a.
$(9, 12)$
b.
$(3, 6)$
c.
$(0, 3)$
d.
$(6, 9)$
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$21 21$
Let $i= .\sqrt{-1}$ If $\frac{(-1+i\sqrt{3})}{(1-i)^{24}} + \frac{(1+i\sqrt{3})}{(1+i)^{24}} =k,$ and $n= [|k|]$ be the greatest integral part of $|k|.$
Then $\sum ^{n}_{j=0}^{+5} (j+ 5)^{2}- \sum ^{n}_{j=0}^{+5}(j+ 5)$ is equal to ________.
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The sum of first four terms of a geometric progression $(G. P. )$ is $\frac{65}{12}$ and the sum of their respective reciprocals
is $\frac{65}{18}.$ If the product of first three terms of the $G. P.$ is $1,$ and the third term is $\alpha ,$ then $2\alpha$ is _________.
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If a curve $y=f(x)$ passes through the point $(1, \,2)$ and satisfies $x\frac{dy}{dx} +y=bx^{4},$ then for what value of
$b,\int _{1}^{2} f(x)dx= \frac{62}{5}?$
a.
$\frac{31}{5}$
b.
$10$
c.
$5$
d.
$\frac{62}{5}$
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Let $f :R\rightarrow \rightarrow R$ be defined as
$\{-55x, if \,x< -5$
$f(x) =2x^{3} -3x^{2} -120x, if -5 \leq x\leq 4$
$\{_{3}_{2}$
$2x -3x -36x-336, if \,x> 4$
Let $A= {x\in R:f$ is increasing}. Then $A$ is equal to:
a.
$(-5, \infty )$
b.
$(-5, \, -4) \cup (4, \,\infty )$
c.
$(-\infty , -5) \cup (-4, \infty )$
d.
$(-\infty , -5) \cup (4, \infty )$
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Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then
the system of linear equations $(A^{2} B^{2} - B^{2} A^{2})X =O,$ where $X$ is a $3 \times 1$ column matrix of unknown
variables and $O$ is a $3 \times 1$ null matrix, has
a.
exactly two solutions
b.
infinitely many solutions
c.
a unique solution
d.
no solution
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If the curve $y=ax^{2} +bx+c,x\in R,$ passes through the point $(1, \,2)$ and the tangent line to this curve at
origin is $y=x,$ then the possible values of $a, \,b, \,c$ are:
a.
$a= -1, \,b= 1, \,c= 1$
b.
$a= 1, \,b= 0, \,c= 1$
c.
$a= 1, \,b= 1, \,c= 0$
d.
$a= \frac{1}{2},b= \frac{1}{2},c= 1$
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Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c), (2,b)$ and $(a,b)$ be
$(\frac{10}{3}, \frac{7}{3} ).$ If $\alpha ,\beta$ are the roots of the equation $ax^{2} +bx+ 1 = 0,$ then the value of $\alpha ^{2} +\beta ^{2} - \alpha \beta$ is:
a.
$- \frac{71}{256}$
b.
$\frac{69}{256}$
c.
$\frac{71}{256}$
d.
$-\frac{69}{256}$
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Let $\lambda$ be an integer. If the shortest distance between the lines $x- \lambda = 2y-1 = -2z$ and $x=y+ 2\lambda =z- \lambda$
is $\frac{\sqrt{7}}{2\sqrt{2}},$ then the value of $|\lambda |$ is _______.
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Let a point $P$ be such that its distance from the point $(5, 0)$ is thrice the distance of $P$ from the point $(-5, 0).$ If
the locus of the point $P$ is a circle of radius $r,$ then $4r^{2}$ (in the nearest integer) is equal to __________.
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The angle of elevation of a jet plane from a point $A$ on the ground is $60^{\circ}.$ After a flight of $20$ seconds at the
speed of $432\, km / hour$ , the angle of elevation changes to $30^{\circ}.$ If the jet plane is flying at a constant height,
then its height is:
a.
$1200\sqrt{3}\,m$
b.
$2400\sqrt{3}\,m$
c.
$1800\sqrt{3}\,m$
d.
$3600\sqrt{3}\,m$
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$||$
$n^{n}C_{r}, if \,n\geq r\geq 0$
For integers $n$ and $r,$ let $( )={$ . The maximum value of $k$ for which the sum
$r 0, otherwise$
$_{k} 10 15_{k+1} 12 13$
$\sum ( )( )+\sum ( )( )$ is maximum, is equal to _________.
$^{i=0} i k- i^{i=0} i k+ 1 - i$
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If the area of the triangle formed by the $x$ -axis, the normal and the tangent to the circle
$(x-2)^{2} + (y-3)^{2} = 25$ at the point $(5, 7)$ is $A,$ then $24A$ is equal to _________.
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If $a+\alpha = 1,b+\beta = 2$ and $af(x) +\alpha f(\frac{1}{x} )=bx+ \frac{\beta}{x},x\neq 0,$ then the value of the expression $\frac{f(x)+f()}{x+^{1}}\frac{1}{x}$ is
$x$
___________.
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Let $f(x)$ be a differentiable function defined on $[0, \,2]$ such that $f'(x) =f'(2 - x)$ for all $x\in (0, 2),f(0) = 1$
and $f(2) =e^{2}.$ Then the value of $\int _{0}^{2} f(x)dx$ is
a.
$2(1 +e^{2})$
b.
$1 +e^{2}$
c.
$1 - e^{2}$
d.
$2(1 - e^{2})$
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The vector equation of the plane passing through the intersection of the planes $\vec{r}\cdot (ˆi +ˆj +ˆk)= 1$ and
$\vec{r}\cdot (ˆi -2ˆj)= -2,$ and the point $(1, \,0, \,2)$ is:
a.
$\vec{r}\cdot (ˆi + 7ˆj + 3ˆk)= 7$
b.
$\vec{r}\cdot (ˆi -7ˆj + 3ˆk)= \frac{7}{3}$
c.
$\vec{r}\cdot (ˆi + 7ˆj + 3ˆk)= \frac{7}{3}$
d.
$\vec{r}\cdot (3ˆi + 7ˆj + 3ˆk)= 7$
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For the statements $p$ and $q,$ consider the following compound statements:
$(a)\,(~q\wedge (p\rightarrow \rightarrow q)) \rightarrow ~\rightarrow p$
$(b)\,((p\vee q) \wedge ~p) \rightarrow \rightarrow q$
Then which of the following statements is correct?
a.
$(b)$ is a tautology but not $(a)$ .
b.
$(a)$ and $(b)$ both are tautologies.
c.
$(a)$ and $(b)$ both are not tautologies.
d.
$(a)$ is a tautology but not $(b)$ .
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For the system of linear equations:
$x-2y= 1, \,x- y+kz= -2, \,ky+ 4z= 6, \,k\in R$
Consider the following statements:
(A) The system has unique solution if $k\neq 2,k\neq -2.$
(B) The system has unique solution if $k= -2.$
(C) The system has unique solution if $k= 2.$
(D) The system has no-solution if $k= 2.$
(E) The system has infinite number of solutions if $k\neq -2.$
Which of the following statements are correct?
a.
$(A)$ and $(E)$ only
b.
$(B)$ and $(E)$ only
c.
$(A)$ and $(D)$ only
d.
$(C)$ and $(D)$ only
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