A bullet of mass $50\,g$ is fired with a speed $100\,m/s$ on a plywood and emerges with $40\,m/s$ . The percentage
loss of kinetic energy is :
a.
$84%$
Correct Answer
b.
$16%$
c.
$32%$
d.
$44%$
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Q2nat
The refractive index of prism is $\mu = \sqrt{3}$ and the ratio of the angle of minimum deviation to the angle of prism
is one. The value of angle of prism is _______.
✓ Accepted Answer:60
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Q3nat
A particle is doing simple harmonic motion of amplitude $0.06\,m$ and time period $3.14\,s$ . The maximum
velocity of the particle is _______ $cm/s$ .
✓ Accepted Answer:12
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Q4mcq
The specific heat at constant pressure of a real gas obeying $PV^{2} =RT$ equation is:
a.
$\frac{R}{3} +C_{V}$
b.
$C_{V} +R$
c.
$C_{V} + \frac{R}{2V}$
Correct Answer
d.
$R$
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Q5nat
Radius of a certain orbit of hydrogen atom is $8.48$ Ã $…$ . If energy of electron in this orbit is $E/x$ . then $x=$
_____ (Given $a_{0} = 0.529$ Ã $… ,E =$ energy of electron in ground state).
✓ Accepted Answer:16
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Q6mcq
A light string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ (where $m_{2} >m_{1}$ ).
If the acceleration of the system is $\frac{g}{\sqrt{2}}$ , then the ratio of the masses $\frac{m_{1}}{m_{2}}$ is:
a.
$\frac{1+\sqrt{5}}{\sqrt{5}-1}$
b.
$\frac{\sqrt{2}-1}{\sqrt{2}+1}$
Correct Answer
c.
$\frac{1+\sqrt{5}}{\sqrt{2}-1}$
d.
$\frac{\sqrt{3}+1}{\sqrt{2}-1}$
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Q7mcq
Ques: $\sigma$ is the uniform surface charge density of a thin spherical shell of radius $R$ . The electric field at any
point on the surface of the spherical shell is :
a.
$\sigma /\epsilon _{o}R$
b.
$\sigma /\epsilon _{o}$
Correct Answer
c.
$\sigma /4\epsilon _{o}$
d.
$\sigma /2\epsilon _{o}$
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Q8mcq
A train starting from rest first accelerates uniformly up to a speed of $80\,km/h$ for time $t$ , then it moves with a
constant speed for time $3t$ . The average speed of the train for this duration of journey will be (in $km/h$ ) :
a.
40
b.
80
c.
30
d.
70
Correct Answer
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Q9nat
A circular coil having 200 turns, $2.5 \times 10^{-4}\,m^{2}$ area and carrying $100\mu A$ current is placed in a uniform
magnetic field of 1T. Initially the magnetic dipole moment $(\vec{M})$ was directed along $\vec{B}$ . Amount of work,
required to rotate the coil through $90^{\circ }$ from its initial orientation such that $\vec{M}$ becomes perpendicular to $\vec{B}$ , is
______ $\mu J$ .
✓ Accepted Answer:5
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Q10mcq
The value of unknown resistance $(x)$ for which the potential difference between $B$ and $D$ will be zero in the
arrangement shown, is :
a.
$3\Omega$
b.
$42\Omega$
c.
$9\Omega$
d.
$6\Omega$
Correct Answer
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While measuring diameter of wire using screw gauge the following readings were noted. Main scale reading is
$1\,mm$ and circular scale reading is equal to 42 divisions. Pitch of screw gauge is $1\,mm$ and it has 100
divisions on circular scale. The diameter of the wire is $\frac{x}{50} \,mm$ . The value of $x$ is :
a.
21
b.
142
c.
71
d.
42
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Given below are two statements: Statement I: In an LCR series circuit, current is maximum at resonance.
Statement II: Current in a purely resistive circuit can never be less than that in a series LCR circuit when
connected to same voltage source. In the light of the above statements, choose the correct from the options
given below:
a.
Statement I is false but Statement II is true
b.
Statement I is true but Statement II is false
c.
Both Statement I and Statement II are true
d.
Both Statement I and Statement II are false
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In photoelectric experiment energy of $2.48eV$ irradiates a photo sensitive material. The stopping potential was
measured to be $0.5\,V$ . Work function of the photo sensitive material is :
a.
$1.68eV$
b.
$2.48eV$
c.
$1.98eV$
d.
$0.5eV$
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A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets
to that of energy of big drop is $\frac{10}{x}$ . The value of $x$ is __________
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If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of
duration of the day of earth will be ______ hours 30 minutes.
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Which of the following phenomena does not explain by wave nature of light. A. reflection B. diffraction C.
photoelectric effect D. interference E. polarization Choose the most appropriate answer from the options given
below:
a.
E only
b.
B, D only
c.
C only
d.
A, C only
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To project a body of mass $m$ from earth's surface to infinity, the required kinetic energy is (assume, the radius of
earth is $R_{E},g=$ acceleration due to gravity on the surface of earth):
a.
$2mgR_{E}$
b.
$4mgR_{E}$
c.
$mgR_{E}$
d.
$1/2mgR_{E}$
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For three vectors $\vec{A}= (-x\hat{i}-6\hat{j}-2\hat{k}), \vec{B}= (-\hat{i}+ 4\hat{j}+ 3\hat{k})$ and $\vec{C} = (-8\hat{i}- \hat{j}+ 3\hat{k})$ , if $\vec{A}\cdot (\vec{B}\times \vec{C}) = 0$ ,
then value of $x$ is _________
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Electromagnetic waves travel in a medium with speed of $1.5 \times 10^{8}\,m\,s^{-1}$ . The relative permeability of the
medium is 2.0 . The relative permittivity will be:
a.
2
b.
4
c.
5
d.
1
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When a $dc$ voltage of $100\,V$ is applied to an inductor, a $dc$ current of $5\,A$ flows through it. When an ac voltage
of $200\,V$ peak value is connected to inductor, its inductive reactance is found to be $20\Omega \sqrt{3}$ . The power
dissipated in the circuit is _______ W.
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Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the
point $P$ is $\frac{x\sigma}{\epsilon _{o}}$ . The value of $x$ is _______ (all quantities are measured in SI units).
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A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho _{0}$ . After sometime, the ball falls
with constant velocity. The viscous force on the ball is :
a.
$mg(1 - \rho \rho _{0})$
b.
$mg(1 + \frac{\rho}{\rho _{0}})$
c.
$mg(\frac{\rho _{0}}{\rho} -1)$
d.
$mg(1 - \frac{\rho _{0}}{\rho})_{∇}$
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An element $\Delta l= \Delta x\hat{i}$ is placed at the origin and carries a large current $I = 10\,A$ . The magnetic field on the $y$ -
axis at a distance of $0.5\,m$ from the elements $\Delta x$ of $1\,cm$ length is:
a.
$4 \times 10^{-8}\,T$
b.
$10 \times 10^{-8}\,T$
c.
$8 \times 10^{-8}\,T$
d.
$12 \times 10^{-8}\,T$
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To find the spring constant $(k)$ of a spring experimentally, a student commits $2%$ positive error in the
measurement of time and $1%$ negative error in measurement of mass. The percentage error in determining value
of $k$ is :
a.
$5%$
b.
$1%$
c.
$3%$
d.
$4%$
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$\,LIST I\, \,LIST II\,$
$\frac{A.}{B.} Torque \frac{I.}{II.}\frac{[MLTA]}{[L^{2}A^{1}]}^{1}^{1}^{-2}^{-2}$
Match List I with List II $Magnetic field$ Choose the correct
$C. Magnetic moment III. \frac{[MTA]}{12-2}^{1}^{-2}^{-1}$
$D. Permeability of free space IV. [M L T ]$
answer from the options given below:
a.
A-III, B-I, C-II, D-IV
b.
A-IV, B-II, C-III, D-I
c.
A-IV, B-III, C-II, D-I
d.
A-I, B-III, C-II, D-IV
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A wire of resistance $R$ and radius $r$ is stretched till its radius became $r/2$ . If new resistance of the stretched
wire is $xR$ , then value of $x$ is _______
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The difference in the 'spin-only' magnetic moment values of $KMnO_{4}$ and the manganese product formed
during titration of $KMnO_{4}$ against oxalic acid in acidic medium is _______ BM. (nearest integer)
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$List - I List - II$
$(Precipitating reagent and conditions) \frac{(Cation)}{2+}$
$\,A.\, NH_{4}Cl + NH_{4}OH/ARKS \,I.\,$
Match List I with List II $\frac{Mn}{2+}$ Choose the
$\,B.\, NH_{4}OH + Na_{2}CO_{3} \,II.\, \frac{Pb}{3+}$
$\,C.\, NH_{4}OH + NH_{4}Cl + H_{2}\,S\,gas\, \,III.\, \frac{Al}{2+}$
$\,D.\, \,dilute\,HCl \,IV.\, Sr$
correct answer from the options given below:
a.
A-III, B-IV, C-I, D-II
b.
A-III, B-IV, C-II, D-I
c.
A-IV, B-III, C-II, D-I
d.
A-IV, B-III, C-I, D-II
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$9.3\,g$ of pure aniline upon diazotisation followed by coupling with phenol gives an orange dye. The mass of
orange dye produced (assume 100\% yield/conversion) is _______ g. (nearest integer)
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At $-20^{\circ }C$ and $1\,atm$ pressure, a cylinder is filled with equal number of $H_{2}, I_{2}$ and $HI$ molecules for the
reaction $H_{2}(\,g) + I_{2}(\,g)\rightleftharpoons 2HI(g)$ , the $K_{p}$ for the process is $x\times 10^{-1}$ . $x =$ _____ [Given :
$R = 0.082\,L\,atm\,K^{-1}\,mol^{-1}$ ]
a.
0.01
b.
10
c.
2
d.
1
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$List - I List - II$
$\frac{(Compound/Species)}{SF} \frac{(Shape/Geometry)}{\,Tetrahedral\,}$
Match List I with List II $\frac{\,A.\,}{\,B.\,}\frac{4}{BrF}\frac{\,I.\,}{\,II.\,}$ Choose the correct
$\frac{3}{-} \,Pyramidal\,$
$\frac{\,C.\,}{\,D.\,}\frac{BrO_{3}}{NH^{+}^{+}KS}\frac{III.\,}{\,IV.\,}\frac{\,See saw\,}{\,Bent T-Shape\,}$
$4$
answer from the options given below:
a.
A-II, B-III, C-I, D-IV
b.
A-II, B-IV, C-III, D-I
c.
A-III, B-IV, C-II, D-I
d.
A-III, B-II, C-IV, D-I
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he following complexes $[CoCl(NH_{3})_{5}]^{2+}, [Co(CN)_{6}]^{3-}, [Co(NH_{3})_{5}(H_{2}O)]^{3+}, [Cu(H_{2}O)_{4}]^{2+}$ The correct
$(A) (B) (C) (D)$
order of A, B, C and D in terms of wavenumber of light absorbed is :
a.
C $<$ D $<$ A $<$ B
b.
$B < C < A < D$
c.
A $<$ C $<$ B $<$ D
d.
D $<$ A $<$ C $<$ B
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Among $CrO, Cr_{2}O_{3}$ and $CrO_{3}$ , the sum of spin-only magnetic moment values of basic and amphoteric oxides
is _______ $10^{-2}BM$ (nearest integer). (Given atomic number of $Cr$ is 24)
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The density of ' $x$ ' $M$ solution (' $X$ ' molar) of $NaOH$ is $1.12\,g\,mL^{-1}$ , while in molality, the concentration of
the solution is $3\,m(3molal)$ . Then $x$ is (Given : Molar mass of $NaOH$ is $40\,g/mol$ )
a.
3.5
b.
3.8
c.
2.8
d.
3.0
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The electron affinity value are negative for A. $Be \rightarrow Be\rightarrow^{-}$ B. $N \rightarrow N\rightarrow^{-}$ C. $O \rightarrow O\rightarrow^{2-}$ D. $Na \rightarrow Na\rightarrow^{-}$ E.
$Al \rightarrow Al\rightarrow^{-}$ Choose the most appropriate answer from the options given below :
a.
D and E only
b.
A, B and C only
c.
A and D only
d.
A, B, D and E only
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Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atom is _______ $\times 10^{13}\,Hz$
$\,[Given :\,R_{H}(\,Rydberg constant\,) = 2.18 \times 10^{-18}\,J,h\,(Plank's\,$
(nearest integer).
$\,constant\,) = 6.6 \times 10^{-34}\,J.s.]\,$
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Given below are two statements : Statement I : Piciric acid is 2,4,6 - trinitrotoluene. Statement II : Phenol - 2,4
- disulphonic acid is treated with Conc. $HNO_{3}$ to get picric acid. 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 incorrect
c.
Statement I is correct but Statement II is
incorrect
d.
Both Statement I and Statement II are correct
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The major products from the following reaction sequence are product $A$ and product B.
The total sum of $\pi$ electrons in product $A$ and product $B$ are _______ (nearest integer)
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Which of the following statements are correct? A. Glycerol is purified by vacuum distillation because it
decomposes at its normal boiling point. B. Aniline can be purified by steam distillation as aniline is miscible in
water. C. Ethanol can be separated from ethanol water mixture by azeotropic distillation because it forms
azeotrope. D. An organic compound is pure, if mixed M.P. is remained same. Choose the most appropriate
answer from the options given below :
a.
A, B, C only
b.
A, C, D only
c.
A, B, D only
d.
B, C, D only
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$List - I List - II$
$\frac{(Hybridization)}{sp^{3}} \frac{(Orientation in Space)}{\,Trigonal bipyramidal\,}$
$\,A.\, \,I.\,$
Match List I with List II $_{2}$ Choose the correct
$\frac{\,B.\,}{\,C.\,}\frac{dsp}{sp^{3}\,d}\frac{\,II.\,}{\,III.\,}\frac{\,Octahedral\,}{\,Tetrahedral\,}$
$\,D.\, sp^{3}\,d^{2} \,IV.\, \,Square planar\,$
answer from the options given below:
a.
A-IV, B-III, C-I, D-II
b.
A-III, B-IV, C-I, D-II
c.
A-III, B-I, C-IV, D-II
d.
A-II, B-I, C-IV, D-III
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$List - I List - II$
$\frac{(Compound)}{\,Iodoform\,} \frac{(Uses)}{\,Fire extinguisher\,}$
Match List I with List II $\frac{\,A.\,}{\,B.\,} \frac{\,I.\,}{\,II.\,}$ Choose the correct
$\frac{\,Carbon tetrachloride\,}{\,CFC\,} \frac{\,Insecticide\,}{\,Antiseptic\,}$
$\frac{\,C.\,}{\,D.\,} \frac{\,III.\,}{\,IV.\,}$
$\,DDT\, \,Refrigerants\,$
answer from the options given below:
a.
A-I, B-II, C-III, D-IV
b.
A-III, B-I, C-IV, D-II
c.
A-II, B-IV, C-I, D-III
d.
A-III, B-II, C-IV, D-I
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$-$
An ideal gas, $C_{v} = \frac{5}{2}R$ , is expanded adiabatically against a constant pressure of 1 atm untill it doubles in
volume. If the initial temperature and pressure is $298\,K$ and $5\,atm$ , respectively then the final temperature is
$-$
_______ $K$ (nearest integer). [ $C_{v}$ is the molar heat capacity at constant volume]
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$List - I List - II$
$\frac{(Molecule/Species)}{SOCl} \frac{(Property/Shape)}{\,Paramagnetic\,}$
Match List I with List II $\frac{\,A.\,}{\,B.\,}\frac{22}{NO}\frac{\,I.\,}{\,II.\,}$ Choose the correct answer
$\,Diamagnetic\,$
$\frac{\,C.\,}{\,D.\,}\frac{NO_{2}}{I^{-}}^{-} \frac{III.\,}{IV.\,}\frac{\,Tetrahedral\,}{\,Linear\,}$
$3$
from the options given below:
a.
A-II, B-III, C-I, D-IV
b.
A-III, B-I, C-II, D-IV
c.
A-IV, B-I, C-III, D-II
d.
A-III, B-IV, C-II, D-I
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$-$ The major product of the following reaction is $P$ . $CH_{3}C = C -CH_{3}^{\,(i)\,Na/\,ing\,NH}_{\,(ii) dil.\,KMnO}\rightarrow \rightarrow_{4}^{3}''$ Number of oxygen
$273\,K$
atoms present in product ' $P$ ' is _______ (nearest integer)
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Given below are two statements: Statement I : Gallium is used in the manufacturing of thermometers.
Statement II : A thermometer containing gallium is useful for measuring the freezing point $(256\,K)$ of brine
solution. In the light of the above statements, choose the correct answer from the options given below :
a.
Both Statement I and Statement II are true
b.
Statement I is false but Statement II is true
c.
Both Statement I and Statement II are false
d.
Statement I is true but Statement II is false
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Consider the dissociation of the weak acid $HX$ as given below
$HX(aq)\rightleftharpoons H^{+}(aq) + X^{-}(aq), Ka = 1.2 \times 10^{-5} [K_{a}$ : dissociation constant $]$ The osmotic pressure of $0.03M$
aqueous solution of $HX$ at $300\,K$ is _______ $\times 10^{-2}$ bar (nearest integer). [Given :
$R = 0.083\,Lbarmol^{-1}\,K^{-1}$ ]
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A conductivity cell with two electrodes (dark side) are half filled with infinitely dilute aqueous solution of a
weak electrolyte. If volume is doubled by adding more water at constant temperature, the molar conductivity
of the cell will -
a.
decrease sharply
b.
increase sharply
c.
remain same or can not be measured accurately
d.
depend upon type of electrolyte
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Let a conic $C$ pass through the point $(4, -2)$ and $P(x,y),x\geq 3$ , be any point on $C$ . Let the slope of the line
touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3, -5)$ . If
the focal distance of the point $(7, 1)$ on $C$ is $d$ , then $12d$ equals ______
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Let $L_{1},L_{2}$ be the lines passing through the point $P(0, 1)$ and touching the parabola
$9x^{2} + 12x+ 18y-14 = 0$ . Let $Q$ and $R$ be the points on the lines $L_{1}$ and $L_{2}$ such that the $\triangle PQR$ is an
isosceles triangle with base $QR$ . If the slopes of the lines $QR$ are $m_{1}$ and $m_{2}$ , then $16(m^{2}_{1} +m^{2}_{2})$ is equal to
_______
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Let $f : (-\infty , \infty ) -{0} \rightarrow \rightarrow R$ be a differentiable function such that $f'(1) = lim_{a\rightarrow \infty \rightarrow } a^{2}f(\frac{1}{a})$ . Then
$lim_{a\rightarrow \infty \rightarrow } \frac{a(a+1)}{2}tan^{-1} (\frac{1}{a} )+a^{2}-2 log_{e}a$ is equal to
a.
$\frac{3}{2} + \frac{\pi}{4}$
b.
$\frac{3}{4} + \frac{\pi}{8}$
c.
$\frac{3}{8} + \frac{\pi}{4}$
d.
$\frac{5}{2} + \frac{\pi}{8}$
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Let $\vec{r}_{k} = \frac{\int _{0}^{1}(1-x^{7})dx}{_{\int }1_{(1-}_{x}_{7}_{)}k+1_{dx},}_{0}^{k} k\in N$ . Then the value of $\sum ^{10}_{k=1} \frac{1}{7(r_{k}-1)}$ is equal to________
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Let $y=y(x)$ be the solution of the differential equation $(2xlog_{e}x) \frac{dy}{dx} + 2y= \frac{3}{x} log_{e}x,x> 0$ and
$y(e^{-1})= 0$ . Then, $y(e)$ is equal to
a.
$-\frac{3}{e}$
b.
$-\frac{3}{2e}$
c.
$-\frac{2}{3e}$
d.
$-\frac{2}{e}$
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Let $C$ be the circle of minimum area touching the parabola $y= 6 - x^{2}$ and the lines $y= \sqrt{3}|x|$ . Then, which
one of the following points lies on the circle $C$ ?
a.
$(1, 2)$
b.
$(1, 1)$
c.
$(2, 2)$
d.
$(2, 4)$
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Let $x_{1},x_{2},x_{3},x_{4}$ be the solution of the equation $4x^{4} + 8x^{3} -17x^{2} -12x+ 9 = 0$ and
$(4 +x^{2}_{1}) (4 +x^{2}_{2}) (4 +x^{2}_{3}) (4 +x^{2}_{4})= \frac{125}{16}m$ . Then the value of $m$ is
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If $A(3, 1, -1),B(\frac{5}{3} , \frac{7}{3}, \frac{1}{3}),C(2, 2, 1)$ and $D( \frac{10}{3} , \frac{2}{3}, \frac{-1}{3})$ are the vertices of a quadrilateral $ABCD$ , then its
area is
a.
$\frac{2\sqrt{2}}{3}$
b.
$\frac{5\sqrt{2}}{3}$
c.
$2\sqrt{2}$
d.
$\frac{4\sqrt{2}}{3}$
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Let $P$ be the point $(10, -2, -1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1, 7, 6)$ on the
line passing through the points $(2, -5, 11)$ and $(-6, 7, -5)$ . Then the length of the line segment $PQ$ is equal to
________
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Let the first term of a series be $T_{1} = 6$ and its $r^{th\,}$ term $T_{r} = 3T_{r-1}+ 6^{r},r= 2, 3$ , $n$ . If the sum of the
first $n$ terms of this series is $\frac{1}{5}(n^{2}-12n+ 39)(4 \cdot 6^{n}-5 \cdot 3^{n}+ 1)$ , then $n$ is equal to______
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If the second, third and fourth terms in the expansion of $(x+y)^{n}$ are 135,30 and $\frac{10}{3}$ , respectively, then
$6(n^{3} +x^{2} +y)$ is equal to _______
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Let a variable line of slope $m> 0$ passing through the point $(4, -9)$ intersect the coordinate axes at the points
$A$ and $B$ . The minimum value of the sum of the distances of $A$ and $B$ from the origin is
a.
30
b.
25
c.
15
d.
10
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The mean and standard deviation of 20 observations are found to be 10 and 2 . respectively. On rechecking, it
was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is
a.
1.8
b.
1.94
c.
$\sqrt{3.96}$
d.
$\sqrt{3.86}$
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A company has two plants $A$ and $B$ to manufacture motorcycles. $60%$ motorcycles are manufactured at plant
$A$ and the remaining are manufactured at plant $B.80%$ of the motorcycles manufactured at plant $A$ are rated of
the standard quality, while $90%$ of the motorcycles manufactured at plant $B$ are rated of the standard quality. A
motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the
probability that it was manufactured at plant $B$ , then $126p$ is
a.
54
b.
66
c.
64
d.
56
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A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square
inscribed in this circle are $m$ and $n$ , respectively, then $m+n^{2}$ is equal to
a.
408
b.
414
c.
396
d.
312
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Let $\alpha ,\beta$ be the distinct roots of the equation $x^{2} - (t^{2} -5t+ 6)x+ 1 = 0,t\in R$ and $a_{n} =\alpha ^{n} +\beta ^{n}$ . Then the
minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is
a.
$-1/4$
b.
$-1/4$
c.
$-1/2$
d.
$1/4$
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Let $a= 2\hat{i}-3\hat{j}+ 4\hat{k}, \vec{b}= 3\hat{i}+ 4\hat{j}-5\hat{k}$ and a vector $\vec{c}$ be such that $\vec{a}\times (\vec{b}+ \vec{c}) + \vec{b}\times \vec{c}= \hat{i}+ 8\hat{j}+ 13\hat{k}$ . If
$\vec{a}\cdot \vec{c}= 13$ , then $(24 - \vec{b}\cdot \vec{c})$ is equal to_______
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Let the relations $R_{1}$ and $R_{2}$ on the set $X = {1, 2, 3, … , 20}$ be given by $R_{1} = {(x,y) : 2x-3y= 2}$ and
$R_{2} = {(x,y) : -5x+ 4y= 0}$ . If $M$ and $N$ be the minimum number of elements required to be added in $R_{1}$
and $R_{2}$ , respectively, in order to make the relations symmetric, then $M +N$ equals
a.
12
b.
16
c.
8
d.
10
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