The dimension of magnetic field in $M, L, T$ and $C$ (Coulomb) is given as
a.
$MLT^{-1}C^{-1}$
b.
$MT^{2}C^{-2}$
c.
$MT^{-1}C^{-1}$
Correct Answer
d.
$MT^{-2}C^{-1}$
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Q2mcq
This question contains Statement $-1$ and Statement-2. Of the four choices given after the statements, choose
the one that best describes the two statements. Statement - I: Energy is released when heavy nuclei undergo
fission or light nuclei undergo fusion. and Statement - II For heavy nuclei, binding energy per nucleon
increases with increasing $Z$ while for light nuclei it decrease with increasing $Z$ .
a.
Statement $-1$ is false, Statement $-2$ is true.
b.
Statement - 1is true, Statement $-2$ is true;
Statement $-2$ is correct explanation for
Statement-1.
c.
Statement $-1$ is true, Statement $-2$ is true;
Statement $-2$ is not a correct explanation for
Statement-1.
d.
Statement $-1$ is true, Statement $-2$ is False.
Correct Answer
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Q3mcq
A body is at rest at $x= 0$ . At $t= 0$ , it starts moving in the positive $x$ -direction with a constant acceleration. At
the same instant another body passes through $x= 0$ moving in the positive $x$ direction with a constant speed.
The position of the first body is given by $x_{1}(t)$ after time ' $t$ ' and that of the second body by $x_{2}(t)$ after the
same time interval. Which of the following graphs correctly describes $(x_{1}- x_{2})$ as a function of time ' $t$ '?
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Q4mcq
A $5\,V$ battery with internal resistance $2\Omega$ and a $2\,V$ battery with internal resistance $1\Omega$ are connected to a $10\Omega$
resistor as shown in the figure. The current in the $10\Omega$ resistor is
a.
$0.27AP_{2}$ to $P_{1}$
b.
$0.03AP_{1}$ to $P_{2}$
c.
$0.03AP_{2}$ to $P_{1}$
Correct Answer
d.
$0.27\,APP_{1}$ to $P_{2}$
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Q5mcq
A working transistor with its three legs marked $P,Q$ and $R$ is tested using a multimeter. No conduction is
found between $P$ and $Q$ . By connecting the common (negative) terminal of the multimeter to $R$ and the other
(positive) terminal to $P$ or $Q$ , some resistance is seen on the multimeter. Which of the following is true for the
transistor?
a.
It is an npn transistor with $R$ as base
b.
It is a pnp transistor with $R$ as collector
Correct Answer
c.
It is a pnp transistor with $R$ as emitter
d.
It is an npn transistor with $R$ as collector
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Q6mcq
A thin spherical shell of radius $R$ has charge $Q$ spread uniformly over its surface. Which of the following
graphs most closely represents the electric field $E(r)$ produced by the shell in the range $0 \leq r< \infty$ , where $r$ is
the distance from the centre of the shell?
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Q7mcq
While measuring the speed of sound by performing a resonance column experiment, a student gets the first
resonance condition at a column length of $18\,cm$ during winter. Repeating the same experiment during
summer, she measures the column length to be $xcm$ for the second resonance. Then
a.
$18 >x$
b.
$x> 54$
Correct Answer
c.
$54 >x> 36$
d.
$36 >x> 18$
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Q8mcq
Paragraph:Wave property of electrons implies that they will show diffraction effects. Davisson and Germer
demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is
obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere
constructively (see in figure).
Question:If a strong diffraction peak is observed when electrons are incident at an angle 'i' from the normal to
the crystal planes with distance ' $d$ ' between them (see figure), de Broglie wavelength $\lambda _{dB}$ of electrons can be
calculated by the relationship ( $n$ is an integer)
a.
$dsini=n\lambda _{dB}$
b.
$2dcosi=n\lambda _{dB}$
c.
$2dsini=n\lambda _{dB}$
d.
$dcosi=n\lambda _{dB}$
Correct Answer
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Q9mcq
Suppose an electron is attracted towards the origin by a force $k/r$ where ' $k$ ' is a constant and ' $r$ ' is the
distance of the electron from the origin. By applying Bohr model to this system, the radius of the $n^{th\,}$ orbital of
the electron is found to be ' $r_{n}$ ' and the kinetic energy of the electron to be $T_{n}$ . Then which of the following is
true?
a.
$T_{n} \propto 1/n^{2}, r_{n} \propto n^{2}$
b.
$T_{n}$ independent of $n,r_{n} \propto n$
Correct Answer
c.
$T_{n} \propto 1/n, r_{n} \propto n$
d.
$T_{n} \propto 1/n, r_{n} \propto n^{2}$
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Q10mcq
Paragraph:Consider a block of conducting material of resistivity ' $\rho$ ' shown in the figure. Current 'l' enters at
'A' and leaves from ' $D$ '. We apply superposition principle to find voltage ' $\Delta V$ ' developed between ' $B$ ' and '
$C$ '. The calculation is done in the following steps: (i) Take current 'l' entering from 'A' and assume it to spread
over a hemispherical surface in the block. (ii) Calculate field $E(r)$ at distance ' $r$ ' from $A$ by using Ohm's law
$E =\rho j$ , where $j$ is the current per unit area at ' $r$ '. (iii) From the ' $r$ ' dependence of $E(r)$ , obtain the potential
$V(r)$ at $r$ . (iv) Repeat (i), (ii) and (iii) for current 'l' leaving ' $D$ ' and superpose results for ' $A$ ' and ' $D$ '.
Question: $\Delta V$ measured between $B$ and $C$ is
A wave travelling along the $x$ -axis is described by the equation $y(x,t) = 0.005 cos(\alpha x- \beta t)$ . If the
wavelength and the time period of the wave are $0.08\,m$ and $2.0\,s$ , respectively, then $\alpha$ and $\beta$ in appropriate
units are
Two full turns of the circular scale of a screw gauge cover a distance of $1\,mm$ on its main scale. The total
number of divisions on the circular scale is 50 . Further, it is found that the screw gauge has a zero error of
$-0.03\,mm$ while measuring the diameter of a thin wire, a student notes the main scale reading of $3\,mm$ and
the number of circular scale divisions in line with the main scale as 35 . The diameter of the wire is
a.
$3.32\,mm$
b.
$3.73\,mm$
c.
$3.67\,mm$
d.
$3.38\,mm$
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A horizontal overhead power line is at a height of $4\,m$ from the ground and carries a current of $100\,A$ from
east to west. The magnetic field directly below it on the ground is $(\mu _{0} = 4\pi \times 10^{-7}\,T\,m\,A^{-1})$
a.
$2.5 \times 10^{-7}\,T$ southward
b.
$5 \times 10^{-6}\,T$ northward
c.
$5 \times 10^{-6}$ T southward
d.
$2.5 \times 10^{-7}$ northward
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Consider a uniform square plate of side ' $a$ ' and mass ' $m$ '. The moment of inertia of this plate about an axis
perpendicular to its plane and passing through one of its corners is
a.
$\frac{5}{6}ma^{2}$
b.
$\frac{1}{12}ma^{2}$
c.
$\frac{7}{12}ma^{2}$
d.
$\frac{2}{3}ma^{2}$
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This question contains Statement $-1$ and Statement-2. Of the four choices given after the statements, choose the
one that best describes the two statements. Statement - I: For a mass $M$ kept at the centre of a cube of side ' $a$ ',
the flux of gravitational field passing through its sides is $4\pi$ GM. and Statement - II If the direction of a field
due to a point source is radial and its dependence on the distance ' $r$ ' for the source is given as $1/r^{2}$ , its flux
through a closed surface depends only on the strength of the source enclosed by the surface and not on the size
or shape of the surface
a.
Statement $-1$ is false, Statement $-2$ is true.
b.
Statement $-1$ is true, Statement $-2$ is true;
Statement $-2$ is correct explanation for
Statement-1.
c.
Statement $-1$ is true, Statement $-2$ is true;
Statement $-2$ is not a correct explanation for
Statement-1.
d.
Statement $-1$ is true, Statement $-2$ is False.
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A block of mass $0.50\,kg$ is moving with a speed of $2.00\,m/s$ on a smooth surface. It strikes another mass of
$1.00\,kg$ and then they move together as a single body. The energy loss during the collision is
a.
$0.16\,J$
b.
$1.00\,J$
c.
$0.67\,J$
d.
$0.34\,J$
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Paragraph:Consider a block of conducting material of resistivity ' $\rho$ ' shown in the figure. Current 'l' enters at
'A' and leaves from ' $D$ '. We apply superposition principle to find voltage ' $\Delta V$ ' developed between ' $B$ ' and '
$C$ '. The calculation is done in the following steps: (i) Take current 'l' entering from 'A' and assume it to spread
over a hemispherical surface in the block. (ii) Calculate field $E(r)$ at distance ' $r$ ' from $A$ by using Ohm's law
$E =\rho j$ , where $j$ is the current per unit area at ' $r$ '. (iii) From the ' $r$ ' dependence of $E(r)$ , obtain the potential
$V(r)$ at $r$ . (iv) Repeat (i), (ii) and (iii) for current 'l' leaving ' $D$ ' and superpose results for ' $A$ ' and ' $D$ '.
Question:For current entering at $A$ , the electric field at a distance ' $r$ ' from $A$ is
a.
$\frac{\rho l}{8\pi r^{2}}$
b.
$\frac{\rho l}{r^{2}}$
c.
$\frac{\rho l}{2\pi r^{2}}$
d.
$\frac{\rho l}{4\pi r^{2}}$
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Paragraph:Wave property of electrons implies that they will show diffraction effects. Davisson and Germer
demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is
obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere
constructively (see in figure).
Question:Electrons accelerated by potential $V$ are diffracted from a crystal. If $d= 1$ Åand $i= 30^{\circ },V$ should
be about $(h = 6.6 \times 10^{-34}Js, m_{e} = 9.1 \times 10^{-31}\,kg, e = 1.6 \times 10^{-19}C)$
a.
$2000\,V$
b.
$50\,V$
c.
$500\,V$
d.
$1000\,V$
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A parallel plate capacitor with air between the plates has a capacitance of $9pF$ . The separation between its
plates is ' $d$ '. The space between the plates is now filled with two dielectrics. One of the dielectrics has
dielectric constant $k_{1} = 3$ and thickness $\frac{d}{3}$ while the other one has dielectric constant $k_{2} = 6$ and thickness $\frac{2\,d}{3}$
. Capacitance of the capacitor is now
a.
$1.8pF$
b.
$45pF$
c.
$40.5pF$
d.
$20.25pF$
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Relative permittivity and permeability of a material are $\epsilon _{r}$ and $\mu _{r}$ , respectively. Which of the following values
of these quantities are allowed for a diamagnetic material?
a.
$\epsilon _{r} = 0.5,\mu _{r} = 1.5$
b.
$\epsilon _{r} = 1.5,\mu _{r} = 0.5$
c.
$\epsilon _{r} = 0.5,\mu _{r} = 0.5$
d.
$\epsilon _{r} = 1.5,\mu _{r} = 1.5$
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The speed of sound in oxygen $(O_{2})$ at a certain temperature is $460\,ms^{-1}$ . The speed of sound in helium (He) at
the same temperature will be (assumed both gases to be ideal)
a.
$460\,ms^{-1}$
b.
$500\,ms^{-1}$
c.
$650\,ms^{-1}$
d.
None of these
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A jar filled with two non mixing liquids 1 and 2 having densities $\rho _{1}$ and $\rho _{2}$ respectively. A solid ball, made of a
material of density $\rho _{3}$ , is dropped in the jar. It comes to equilibrium in the position shown in the figure. Which
of the following is true for $\rho _{1},\rho _{2}$ and $\rho _{3}$ ?
a.
$\rho _{3} <\rho _{1} <\rho _{2}$
b.
$\rho _{1} <\rho _{3} <\rho _{2}$
c.
$\rho _{1} <\rho _{2} <\rho _{3}$
d.
$\rho _{1} <\rho _{3} <\rho _{2}$
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A capillary tube $(A)$ is dropped in water. Another identical tube $(B)$ is dipped in a soap water solution. Which
of the following shows the relative nature of the liquid columns in the two tubes?
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Paragraph:Wave property of electrons implies that they will show diffraction effects. Davisson and Germer
demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is
obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere
constructively (see in figure).
Question:In an experiment, electrons are made to pass through a narrow slit of width 'd' comparable to their
de Broglie wavelength. They are detected on a screen at a distance ' $D$ ' from the slit (see figure).
Which of the following graph can be expected to represent the number of electrons ' $N$ ' detected as a function
of the detector position ' $y$ ' $(y= 0$ corresponds to the middle of the slit)?
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An insulated container of gas has two chambers separated by an insulating partition. One of the chambers has
volume $V_{1}$ and contains ideal gas at pressure $P_{1}$ and temperature $T_{1}$ . The other chamber has volume $V_{2}$ and
contains ideal gas at pressure $P_{2}$ and temperature $T_{2}$ . If the partition is removed without doing any work on the
gas, the final equilibrium temperature of the gas in the container will be
Two coaxial solenoids are made by winding thin insulated wire over a pipe of cross sectional area $A= 10\,cm^{2}$
and length $= 20\,cm$ . If one of the solenoids has 300 turns and the other 400 turns, their mutual inductance is
$(\mu _{0} = 4\pi \times 10^{-7}TmA^{-1})$
a.
$2.4\pi \times 10^{-5}H$
b.
$4.8\pi \times 10^{-4}H$
c.
$4.8\pi \times 10^{-5}H$
d.
$2.4\pi \times 10^{-4}H$
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A thin rod of length ' $L$ ' is lying along the $x$ -axis with its ends at $x= 0$ and $x=L$ . Its linear density
(mass/length) varies with $x$ ask $(\frac{x}{L} )^{n}$ , where $n$ can be zero or any positive number. If the position $x_{CM}$ of the
centre of mass of the rod is plotted against ' $n$ ', which of the following graphs best approximates the
dependence of $x_{CM}$ on $n$ ?
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A planet in a distant solar system is 10 times more massive than the earth and its radius is 10 times smaller.
Given that the escape velocity from the earth is $11kms^{-1}$ , the escape velocity from the surface of the planet
would be
a.
$1.1kms^{-1}$
b.
$11kms^{-1}$
c.
$110kms^{-1}$
d.
$0.11kms^{-1}$
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A student measures the focal length of convex lens by putting an object pin at a distance 'u' from the lens and
measuring the distance ' $v$ ' of the image pin. The graph between ' $u$ ' and ' $v$ ' plotted by the student should look
like
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A spherical solid ball of volume $V$ is made of a material of density $\rho _{1}$ . It is falling through a liquid of density
$\rho _{2}(\rho _{2} <\rho _{1})$ . Assume that the liquid applies a viscous force on the ball that is proportional to the square of its
speed v, i.e., $_{viscous\,} = -kv^{2}(k> 0)$ . The terminal speed of the ball is
a.
$\frac{F}{Vg(\rho _{1}-\rho _{2})}$
$\sqrt{\,}$
b.
$\frac{Vg\rho _{1}}{k}$
c.
$\frac{k}{V g\rho _{1}}$
$\sqrt{\,} _{k}$
d.
$\frac{Vg(\rho _{1}-\rho _{2})}{k}$
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The coordination number and the oxidation state of the element ' $E$ ' in the complex $[E(en)_{2}(C_{2}O_{4})]NO_{2}$
(where (en) is ethylene diamine) are, respectively,
a.
6 and 2
b.
4 and 2
c.
4 and 3
d.
6 and 3
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The vapour pressure of water at $20^{\circ }C$ is $17.5\,mmHg$ . If $18\,g$ of glucose $(C_{6}H_{12}O_{6})$ is added to $178.2\,g$ of
water at $20^{\circ }C$ , the vapour pressure of the resulting solution will be
a.
$17.675\,mmHg$
b.
$15.750\,mmHg$
c.
$16.500\,mmHg$
d.
$17.325\,mmHg$
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For a reaction $\frac{1}{2} A\rightarrow 2\rightarrow B$ , rate of disappearance of ' $A$ ' is related to the rate of appearance of 'B' by the
expression
a.
$-\frac{d[A]}{dt} = \frac{1}{2} \frac{d[B]}{dt}$
b.
$-\frac{d[A]}{dt} = \frac{1}{4} \frac{d[B]}{dt}$
c.
$-\frac{d[A]}{dt} = \frac{d[B]}{dt}$
d.
$-\frac{d[A]}{dt} = 4\frac{d[B]}{dt}$
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Four species are listed below i. $HCO^{-}_{3}$ ii. $H_{3}O^{+}$ iii. $HSO^{-}_{4}$ iv. $HSO_{3}\,F$ Which one of the following is the
correct sequence of their acid strength?
a.
iv $<$ ii $<$ iii $<$ i
b.
ii $<$ iii $<i<$ iv
c.
i $<$ iii $<$ ii $<$ iv
d.
iii $<$ i $<$ iv $<$ ii
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The $pK_{a}$ of a weak acid, $HA$ , is $4.80$ . The $pK_{b}$ of a weak base, $BOH$ , is $4.78$ . The $pH$ of an aqueous solution
of the corresponding salt, BA, will be
a.
$9.58$
b.
$4.79$
c.
$7.01$
d.
$9.22$
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In a compound atoms of element $Y$ from ccp lattice and those of element $X$ occupy $2/3^{rd\,}$ of tetrahedral voids.
The formula of the compound will be
a.
$X_{4}Y_{3}$
b.
$X_{2}Y_{3}$
c.
$X_{2}Y$
d.
$X_{3}Y_{4}$
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In the following sequence of reactions, the alkene affords the compound ' $B$ '
$CH_{3}CH = CHCH_{3} \longrightarrow ^{O_{3}} A^{H}Zn^{_{2}O} B$ . The compound $B$ is
a.
$CH_{3}CH_{2}CHO$
b.
$CH_{3}COCH_{3}$
c.
$CH_{3}CH_{2}COCH_{3}$
d.
$CH_{3}CHO$
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For the following three reactions $a,b$ and $c$ , equilibrium constants are given: a.
$CO(g) + H_{2}O(g)\rightleftharpoons CO_{2}(\,g) + H_{2}(\,g); K_{1}$ b. $CH_{4}(\,g) + H_{2}O(g)\rightleftharpoons CO(g) + 3H_{2}(\,g); K_{2}$ c.
$CH_{4}(\,g) + 2H_{2}O(g)\rightleftharpoons CO_{2}(\,g) + 4H_{2}(\,g); K_{3}$ Which of the following relations is correct?
a.
$K_{1}\sqrt{K}_{2} = K_{3}$
b.
$K_{2}\,K_{3} = K_{1}$
c.
$K_{3} = K_{1}\,K_{2}$
d.
$K_{3}\cdot K^{3}_{2} = K^{2}_{1}$
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The equilibrium constants $K_{P_{1}}$ and $K_{P_{2}}$ for the reactions $X \rightleftharpoons 2Y$ and $Z \rightleftharpoons P +Q$ , respectively are in the
ratio of $1 : 9$ . If the degree of dissociation of $X$ and $Z$ be equal then the ratio of total pressure at these
equilibria is
a.
$1 : 36$
b.
$1 : 1$
c.
$1 : 3$
d.
$1 : 9$
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At $80^{\circ }C$ , the vapour pressure of pure liquid ' $A$ ' is $520\,mmHg$ and that of pure liquid ' $B$ ' is $1000\,mm Hg$ . If
a mixture solution of ' $A$ ' and ' $B$ ' boils at $80^{\circ }C$ and $1\,atm$ pressure, the amount of ' $A$ ' in the mixture is
$(1\,atm = 760\,mmHg)$
a.
52 mol percent
b.
34 mol percent
c.
48 mol percent
d.
50 mol percent
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Amount of oxalic acid present in a solution can be determined by its titration with $KMnO_{4}$ solution in the
presence of $H_{2}SO_{4}$ . The titration gives unsatisfactory result when carried out in the presence of $HCl$ , because
$HCl$
a.
gets oxidised by oxalic acid to chlorine
b.
furnishes $H^{+}$ ions in addition to those from oxalic
acid
c.
reduces permanganate to $Mn^{2+}$
d.
oxidises oxalic acid to carbon dioxide and water
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$|||$
Which of the following factors is of no significance for roasting sulphide ores to the oxides and not subjecting
the sulphide ores to carbon reduction directly?
a.
Metal sulphides are thermodynamically more
stable than $CS_{2}$
b.
$CO_{2}$ is thermodynamically more stable than $CS_{2}$
c.
Metal sulphides are less stable than the
corresponding oxides
d.
$CO_{2}$ is more volatile than $CS_{2}$
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Toluene is nitrated and the resulting product is reduced with tin and hydrochloric acid. The product so obtained
is diazotised and then heated with cuprous bromide. The reaction mixture so formed contains
a.
mixture of $o-$ and p-bromotoluenes
b.
mixture of $o-$ and p-dibromobenzenes
c.
mixture of $o$ - and p-bromoanilines
d.
mixture of 0 - and $m$ -bromotoluenes
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Standard entropy of $X_{2},Y_{2}$ and $XY_{3}$ are 60,40 and $50JK^{-1}\,mol^{-1}$ , respectively. For the reaction,
$\frac{1}{2}X_{2}+ \frac{3}{2} Y_{2} \rightarrow \rightarrow XY_{3}, \Delta H = -30\,kJ$ , to be at equilibrium, the temperature will be
a.
$1250\,K$
b.
$500\,K$
c.
$750\,K$
d.
$1000\,K$
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The ionization enthalpy of hydrogen atom is $1.312 \times 10^{6}Jmol^{-1}$ . The energy required to excite the electron in
the atom from $n= 1$ to $n= 2$ is
a.
$8.51 \times 10^{5}Jmol^{-1}$
b.
$6.56 \times 10^{5}Jmol^{-1}$
c.
$7.56 \times 10^{5}Jmol^{-1}$
d.
$9.84 \times 10^{5}Jmol^{-1}$
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The electrophile, $E^{\oplus }$ attacks the benzene ring to generate the intermediate $\sigma$ -complex. Of the following, which
$\sigma$ -complex is of lowest energy?
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Oxidising power of chlorine in aqueous solution can be determined by the parameters indicated below:
$\frac{1}{2}Cl_{2}(\,g) \frac{1}{2}^{\Delta }\longrightarrow ^{_{ass}H^{\ominus }} Cl(g)^{\Delta }\longrightarrow ^{egH\ominus } Cl^{-}(g)^{\Delta }\longrightarrow ^{\longrightarrow monH\ominus } Cl^{-}(aq).$
The energy involved in the conversion of $\frac{1}{2}Cl_{2}(\,g)$ to $Cl^{-}(g)$ (using the data,
$\Delta _{diss\,}H_{Cl}^{\ominus }_{2} = 240kJmol^{-1}, \Delta _{eg}H^{\ominus }_{cl} = -349kJmol^{-1}, \Delta _{hyd}H^{\ominus }_{Cl} = -381kJmol^{-1})$ will be
a.
$+152kJmol^{-1}$
b.
$-610kJmol^{-1}$
c.
$-850kJmol^{-1}$
d.
$+120kJmol^{-1}$
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The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If
the terms of the geometric progression are alternately positive and negative, then the first term is
a.
$-4$
b.
$-12$
c.
12
d.
4
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$AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. $A$ man finds that the angle of elevation of the
point $A$ from a certain point $C$ on the ground is $60^{\circ }$ . He moves away from the pole along the line $BC$ to a
point $D$ such that $CD= 7\,m$ . From $D$ the angle of elevation of the point $A$ is $45^{\circ }$ . Then the height of the pole
is
a.
$\frac{7\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}-1}m$
b.
$\frac{7\sqrt{3}}{2} \cdot (\sqrt{3}+ 1)m$
c.
$\frac{7\sqrt{3}}{2} \cdot (\sqrt{3}-1)m$
d.
$\frac{7\sqrt{3}}{2} \cdot \frac{1}{\sqrt{3}+1}$
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In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement -1: The number
of different ways the child can buy the six ice-creams is $^{10}C_{5}$ . Statement $-2$ : The number of different ways
the child can buy the six ice-creams is equal to the number of different ways of arranging $6\,A^{'s\,}$ and 4 B's in a
row.
a.
Statement $-1$ is false, Statement $-2$ is true
b.
Statement $-1$ is true, Statement $-2$ is true,
Statement $-2$ is a correct explanation for
Statement $-1$
c.
Statement $-1$ is true, Statement $-2$ is true;
Statement $-2$ is not a correct explanation for
Statement $-1$ .
d.
Statement $-1$ is true, Statement $-2$ is false.
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Let $A$ be a $2 \times 2$ matrix with real entries. Let I be the $2 \times 2$ identity matrix. Denote by $tr(A)$ , the sum of
diagonal entries of $A$ . Assume that $A^{2} = 1$ . Statement -1: If $A\neq 1$ and $A\neq -1$ , then $detA= -1$ . Statement
$-2$ : If $A\neq 1$ and $A\neq -1$ , then $tr(A) \neq 0$ .
a.
Statement $-1$ is false, Statement $-2$ is true
b.
Statement $-1$ is true, Statement $-2$ is true,
Statement $-2$ is a correct explanation for
Statement $-1$
c.
Statement $-1$ is true, Statement $-2$ is true;
Statement $-2$ is not a correct explanation for
Statement $-1$
d.
Statement $-1$ is true, Statement $-2$ is false.
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The vector $\vec{a}=\alpha \hat{i}+ 2\hat{j}+\beta \hat{k}$ lies in the plane of the vectors $\vec{b}= \hat{i}+ \hat{j}$ and $\vec{c}= \hat{j}+ \hat{k}$ and bisects the angle
between $\vec{b}$ and $\vec{c}$ . Then which one of the following gives possible values of $\alpha$ and $\beta$ ?
a.
$\alpha = 2,\beta = 2$
b.
$\alpha = 1,\beta = 2$
c.
$\alpha = 2,\beta = 1$
d.
$\alpha = 1,\beta = 1$
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It is given that the events $A$ and $B$ are such that $P(A) = \frac{1}{4} ,P(\frac{A}{B})= \frac{1}{2}$ and $P(\frac{B}{A} )= \frac{2}{3}$ . Then $P(B)$ is
a.
$\frac{1}{6}$
b.
$\frac{1}{3}$
c.
$\frac{2}{3}$
d.
$\frac{1}{2}$
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$|| ||$
$|| ||$
The line passing through the points $(5, 1,a)$ and $(3,b, 1)$ crosses the $yz-$ plane at the point $(0, \frac{17}{2}, \frac{-13}{2})$ .
Then
a.
$a= 2,b= 8$
b.
$a= 4,b= 6$
c.
$a= 6,b= 4$
d.
$a= 8,b= 2$
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If the straight lines $\frac{x-1}{k} = \frac{y-2}{2} = \frac{z-3}{3}$ and $\frac{x-2}{3} = \frac{y-3}{k} = \frac{z-1}{2}$ intersect at a point, then the integer $k$ is equal
to
a.
$-5$
b.
5
c.
2
d.
$-2$
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Let $R$ be the real line. Consider the following subsets of the plane $R\times R$ . $S = {(x,y) :y=x+ 1$ and
$0 <x< 2},T = {(x,y) :x- y$ is an integer $}$ . Which one of the following is true?
a.
neither $S$ nor $T$ is an equivalence relation on $R$
b.
both $S$ and $T$ are equivalence relations on $R$
c.
$S$ is an equivalence relation on $R$ but $T$ is not
d.
$T$ is an equivalence relation on $R$ but $S$ is not
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Let $p$ be the statement " $x$ is an irrational number", $q$ be the statement " $y$ is a transcendental number", and $r$ be
the statement " $x$ is a rational number iff $y$ is a transcendental number". Statement $-1$ : $r$ is equivalent to either
$q$ or $p$ Statement $-2 :r$ is equivalent to $\sim (p\leftrightarrow \sim q)$ .
a.
Statement $-1$ is false, Statement $-2$ is true
b.
Statement $-1$ is true, Statement $-2$ is true,
Statement $-2$ is a correct explanation for
Statement $-1$
c.
Statement $-1$ is true, Statement $-2$ is true;
Statement $-2$ is not a correct explanation for
Statement $-1$ .
d.
Statement $-1$ is true, Statement $-2$ is false.
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Let $f :N Y$ be a function defined as $f(x) = 4x+ 3$ , where $Y = {y\in N :y= 4x+ 3$ for some $x\in N}$ .
Show that $\vec{f}$ is invertible and its inverse is
a.
$g(y) = \frac{3y+4}{3}$
b.
$g(y) = 4 + \frac{y+3}{4}$
c.
$g(y) = \frac{y+3}{4}$
d.
$g(y) = \frac{y-3}{4}$
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The quadratic equations $x^{2} -6x+a= 0$ and $x^{2} - cx+ 6 = 0$ have one root in common. The other roots of
the first and second equations are integers in the ratio $4 : 3$ . Then the common root is
a.
1
b.
4
c.
3
d.
2
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The non-zero verctors $\vec{a}, \vec{b}$ and $\vec{c}$ are related by $\vec{a}= 8\vec{b}$ and $\vec{c}= -7\vec{b}$ . Then the angle between $\vec{a}$ and $\vec{c}$ is
a.
0
b.
$\pi /4$
c.
$\pi /2$
d.
$\pi$
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A die is thrown. Let $A$ be the event that the number obtained is greater than 3 . Let $B$ be the event that the
number obtained is less than 5 . Then $P(A\cup B)$ is
a.
$\frac{3}{5}$
b.
0
c.
1
d.
$\frac{2}{5}$
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Let $a,b,c$ be any real numbers. Suppose that there are real numbers $x,y,z$ not all zero such that $x=$
$cy+bz,y=az+cx$ and $z=bx+ay$ . Then $a^{2} +b^{2} +c^{2} + 2abc$ is equal to
a.
2
b.
$-1$
c.
0
d.
1
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