A particle of mass $m$ moves on a straight line with its velocity increasing with distance according to the
equation $v=\alpha \sqrt{x}$ , where $\alpha$ is a constant. The total work done by all the forces applied on the particle during
its displacement from $x= 0$ to $x= d$ , will be :
a.
$\frac{m}{2\alpha ^{2}\,d}$
b.
$\frac{md}{2\alpha ^{2}}$
c.
$2\,m\alpha ^{2}\,d$
d.
$\frac{m\alpha ^{2}\,d}{2}$
Correct Answer
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Q2mcq
A light unstretchable string passing over a smooth light pulley connects two blocks of masses $m_{1}$ and $m_{2}$ . If the
acceleration of the system is $\frac{g}{8}$ , then the ratio of the masses $\frac{m_{2}}{m_{1}}$ is :
a.
$8 : 1$
b.
$5 : 3$
c.
$4 : 3$
d.
$9 : 7$
Correct Answer
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Q3mcq
Given below are two statements : Statement (I) : When currents vary with time, Newton's third law is valid
only if momentum carried by the electromagnetic field is taken into account. Statement (II) : Ampere's
circuital law does not depend on Biot-Savart's law. 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
Correct Answer
c.
Both Statement I and Statement II are false
d.
Statement I is false but Statement II is true
No explanation saved for this question yet.
Q4nat
At the centre of a half ring of radius $R = 10\,cm$ and linear charge density $4nCm^{-1}$ , the potential is $x\pi V$ . The
value of $x$ is _______
✓ Accepted Answer:36
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Q5mcq
A galvanmeter has a coil of resistance $200\Omega$ with a full scale deflection at $20\mu A$ . The value of resistance to be
added to use it as an ammeter of range $(0 -20)mA$ is ;
a.
$0.40\Omega$
b.
$0.20\Omega$
Correct Answer
c.
$0.50\Omega$
d.
$0.10\Omega$
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Q6nat
When a coil is connected across a $20\,V$ dc supply, it draws a current of $5\,A$ . When it is connected across
$20\,V, 50\,Hz$ ac supply, it draws a current of $4\,A$ . The self inductance of the coil is ______ $mH$ . $($ Take $\pi = 3$ )
✓ Accepted Answer:10
No explanation saved for this question yet.
Q7mcq
The equivalent resistance between $A$ and $B$ is :
a.
$18\Omega$
b.
$19\Omega$
Correct Answer
c.
$25\Omega$
d.
$27\Omega$
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Q8mcq
One main scale division of a vernier caliper is equal to $m$ units. If $n^{th\,}$ division of main scale coincides with
$(n+ 1)^{th\,}$ division of vernier scale, the least count of the vernier caliper is :
a.
$\frac{n}{(n+1)}$
b.
$\frac{1}{(n+1)}$
c.
$\frac{m}{(n+1)}$
Correct Answer
d.
$\frac{\vec{m}}{n(n+1)}$
No explanation saved for this question yet.
Q9nat
In a Young's double slit experiment, the intensity at a point is $( \frac{1}{4})^{th\,}$ of the maximum intensity, the minimum
distance of the point from the central maximum is ________ $\mu m$ . (Given :
$\lambda = 600\,nm, \,d = 1.0\,mm, D = 1.0\,m$ )
✓ Accepted Answer:200
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Q10mcq
The dimensional formula of latent heat is :
a.
$[ML^{2}\,T^{-2}]$
b.
$[M^{0}\,L^{2}\,T^{-2}]$
Correct Answer
c.
$[MLT^{-2}]$
d.
$[M^{\circ }LT^{-2}]$
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A heavy iron bar, of weight $W$ is having its one end on the ground and the other on the shoulder of a person.
The bar makes an angle $\theta$ with the horizontal. The weight experienced by the person is :
a.
$W cos\theta$
b.
$\frac{W}{2}$
c.
W
d.
$Wsin\theta$
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Given below are two statements : Statement (I) : When an object is placed at the centre of curvature of a
concave lens, image is formed at the centre of curvature of the lens on the other side. Statement (II) : Concave
lens always forms a virtual and erect image. 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.
Both Statement I and Statement II are false
c.
Statement I is true but Statement II is false
d.
Statement I is false but Statement II is true
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A sample of 1 mole gas at temperature $T$ is adiabatically expanded to double its volume. If adiabatic constant
for the gas is $\gamma = \frac{3}{2}$ , then the work done by the gas in the process is:
a.
$\frac{R}{T}[2 -]\sqrt{2}$
b.
$\frac{T}{R}[2 + ]\sqrt{2}$
c.
RT $[2 -]\sqrt{2}$
d.
$RT[2 + ]\sqrt{2}$
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A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure.
If the area of each stair is $\frac{A}{3}$ and the height is $d$ , the capacitance of the arrangement is :
a.
$\frac{13\epsilon _{o}A}{17d}$
b.
$\frac{11\epsilon _{o}A}{18\,d}$
c.
$\frac{18\epsilon _{o}A}{11\,d}$
d.
$\frac{11\epsilon _{o}A}{20\,d}$
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A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the
plates of the capacitor. The glow of the bulb:
a.
increases
b.
decreases
c.
remains same
d.
becomes zero
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A particle moving in a straight line covers half the distance with speed $6\,m/s$ . The other half is covered in two
equal time intervals with speeds $9\,m/s$ and $15\,m/s$ respectively. The average speed of the particle during the
motion is :
a.
$10\,m/s$
b.
$8\,m/s$
c.
$9.2\,m/s$
d.
$8.8\,m/s$
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A star has $100%$ helium composition. It starts to convert three $^{4}He$ into one $^{12}C$ via triple alpha process as
$^{4}He +^{4}He +^{4}He \rightarrow \rightarrow^{12}C + Q$ . The mass of the star is $2.0 \times 10^{32}\,kg$ and it generates energy at the rate of
$5.808 \times 10^{30}\,W$ . The rate of converting these $^{4}He$ to $^{12}C$ is $n \times 10^{42}\,s^{-1}$ , where $n$ is _________ [ Take, mass
of $^{4}He = 4.0026u$ , mass of $^{12}C = 12u$ ]
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A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is $1.42eV$ .
The wavelength of light emitted from the LED is :
a.
$1400\,nm$
b.
$650\,nm$
c.
$875\,nm$
d.
$1243\,nm$
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The position, velocity and acceleration of a particle executing simple harmonic motion are found to have
magnitudes of $4\,m, 2\,ms^{-1}$ and $16\,ms^{-2}$ at a certain instant. The amplitude of the motion is $\sqrt{x}, \,m$ where $x$ is
________
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An astronaut takes a ball of mass $m$ from earth to space. He throws the ball into a circular orbit about earth at an
altitude of $318.5\,km$ . From earth's surface to the orbit, the change in total mechanical energy of the ball is
$x \frac{GM_{e}m}{21R_{e}}$ . The value of $x$ is (take $R_{e} = 6370\,km)$ :
a.
10
b.
12
c.
9
d.
11
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A string is wrapped around the rim of a wheel of moment of inertia $0.40kgm^{2}$ and radius $10\,cm$ . The wheel is
free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $40\,N$ . The
angular velocity of the wheel after $10\,s$ is $xrad/s$ , where $x$ is _______
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Two persons pull a wire towards themselves. Each person exerts a force of $200\,N$ on the wire. Young's
modulus of the material of wire is $1 \times 10^{11}\,N\,m^{-2}$ . Original length of the wire is $2\,m$ and the area of cross
section is $2\,cm^{2}$ . The wire will extend in length by ______ $\mu m$ .
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A square loop of edge length $2\,m$ carrying current of $2\,A$ is placed with its edges parallel to the $x y$ axis. A
magnetic field is passing through the $x- y$ plane and expressed as $\vec{B}=B_{0}(1 + 4x)\hat{k}$ , where $B_{0} = 5\,T$ . The
net magnetic force experienced by the loop is _______ $N$ .
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If $\vec{a}$ and $\vec{b}$ makes an angle $cos^{-1} ( \frac{5}{9})$ with each other, then $|a+ \vec{b}| = |\sqrt{2} \vec{a}- \vec{b}|$ for $|\vec{a}| =n|\vec{b}|$ The integer
value of $n$ is ____
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A plane EM wave is propagating along $x$ direction. It has a wavelength of $4\,mm$ . If electric field is in $y$
direction with the maximum magnitude of $60Vm^{-1}$ , the equation for magnetic field is :
$($
A sphere of relative density $\sigma$ and diameter $D$ has concentric cavity of diameter $d$ . The ratio of $\frac{D}{d}$ , if it just
floats on water in a tank is :
a.
$(\frac{\sigma -2}{\sigma +2})^{1/3}$
b.
$( \frac{\sigma}{\sigma -1})^{1/3}$
c.
$(\frac{\sigma -1}{\sigma})^{1/3}$
d.
$( \frac{\sigma +1}{\sigma -1})^{1/3}$
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Molarity $(M)$ of an aqueous solution containing $x\,g$ of anhyd. $CuSO_{4}$ in $500\,mL$ solution at $32^{\circ }C$ is
$2 \times 10^{-1}M$ . Its molality will be ______ $\times 10^{-3}\,m$ . (nearest integer). [Given density of the solution
$= 1.25\,g/mL$ ]
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$0.05MCuSO_{4}$ when treated with $0.01MK_{2}Cr_{2}O_{7}$ gives green colour solution of $Cu_{2}Cr_{2}O_{7}$ . The two
solutions are separated as shown below : [SPM : Semi Permeable Membrane]
Due to osmosis :
a.
Molarity of $CuSO_{4}$ solution is lowered.
b.
Molarity of $K_{2}Cr_{2}O_{7}$ solution is lowered.
c.
Green colour formation observed on side Y.
d.
Green colour formation observed on side X.
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Given below are two statements : Statement (I) : The oxidation state of an element in a particular compound is
the charge acquired by its atom on the basis of electron gain enthalpy consideration from other atoms in the
molecule. Statement (II) : $p\pi -p\pi$ bond formation is more prevalent in second period elements over other
periods. In the light of the above statements, choose the most appropriate answer from the options given below
:
a.
Both Statement I and Statement II are correct
b.
Both Statement I and Statement II are incorrect
c.
Statement I is incorrect but Statement II is
correct
d.
Statement I is correct but Statement II is
incorrect
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Given below are two statements : Statement I: The rate law for the reaction $A+B\rightarrow \rightarrow C$ is rate $(r) =k[A]^{2}[B]$
. When the concentration of both A and B is doubled, the reaction rate is increased " $x$ " times. Statement II :
The figure is showing "the variation in concentration against time plot" for a " $y$ " order reaction. The Value of
$x+y$ is ______
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The heat of solution of anhydrous $CuSO_{4}$ and $CuSO_{4}\cdot 5H_{2}O$ are $-70\,kJ\,mol^{-1}$ and $+12\,kJ\,mol^{-1}$
respectively. The heat of hydration of $CuSO_{4}$ to $CuSO_{4}\cdot 5H_{2}O$ is $-x\,kJ$ . The value of $x$ is ______ (nearest
integer).
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When equal volume of $1MHCl$ and $1MH_{2}SO_{4}$ are separately neutralised by excess volume of $1M NaOH$
solution. $x$ and $y\,kJ$ of heat is liberated respectively. The value of $y/x$ is _______
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The standard reduction potentials at $298\,K$ for the following half cells are given below :
$Cr_{2}O^{2-}_{7} + 14H^{+} + 6e^{-} \rightarrow 2Cr\rightarrow^{3+} + 7H_{2}O, E^{\circ } = 1.33\,V$
$Fe^{3+}(aq) + 3e^{-} \rightarrow Fe\rightarrow E^{\circ } = -0.04\,V$
$Ni^{2+}(aq) + 2e^{-} \rightarrow Ni\rightarrow E^{\circ } = -0.25\,V$ Consider the given electrochemical reactions, The
$Ag^{+}(aq) + e^{-} \rightarrow Ag\rightarrow E^{\circ } = 0.80\,V$
$Au^{3+}(aq) + 3e^{-} \rightarrow Au\rightarrow E^{\circ } = 1.40\,V$
number of metal(s) which will be oxidized be $Cr_{2}O^{2-}_{7}$ , in aqueous solution is ______
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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) : Both rhombic and monoclinic sulphur exist as $S_{8}$ while oxygen exists as $O_{2}$ . Reason (R) :
Oxygen forms $p\pi -p\pi$ multiple bonds with itself and other elements having small size and high
electronegativity like $C, N$ , which is not possible for sulphur. In the light of the above statements, choose the
most appropriate answer from the options given below :
a.
(A) is correct but (R) is not correct
b.
(A) is not correct but (R) is correct
c.
Both (A) and (R) are correct and (R) is the
correct explanation of (A)
d.
Both (A) and (R) are correct but (R) is not the
correct explanation of (A)
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The total number of species from the following in which one unpaired electron is present, is _______
$N_{2}, O_{2}, C^{-}_{2} , O^{-}_{2} , O^{2-}_{2} , H^{+}_{2} , CN^{-}, He^{+}_{2}$
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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): $S_{N}2$ reaction of $C_{6}H_{5}CH_{2}Br$ occurs more readily than the $S_{N}2$ reaction of $CH_{3}CH_{2}Br$ . Reason
(R) : The partially bonded unhybridized p-orbital that develops in the trigonal bipyramidal transition state is
stabilized by conjugation with the phenyl ring. In the light of the above statements, choose the most
appropriate answer from the options given below :
a.
(A) is correct but (R) is not correct
b.
(A) is not correct but (R) is correct
c.
Both (A) and (R) are correct but (R) is not the
correct explanation of (A)
d.
Both (A) and (R) are correct and (R) is the
correct explanation of $(A)$
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The $F^{-}$ ions make the enamel on teeth much harder by converting hydroxyapatite (the enamel on the surface of
teeth) into much harder fluoroapatite having the formula.
a.
$[3 (Ca_{3}(PO_{4})_{2}) \cdot Ca(OH)_{2}]$
b.
$[3 (Ca_{3}(PO_{4})_{3}) \cdot CaF_{2}]$
c.
$[3 (Ca_{2}(PO_{4})_{2}) \cdot Ca(OH)_{2}]$
d.
$[3 (Ca_{3}(PO_{4})_{2}) \cdot CaF_{2}]$
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Total number of essential amino acid among the given list of amino acids is _______ Arginine, Phenylalanine,
Aspartic acid, Cysteine, Histidine, Valine, Proline
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Identify the incorrect statements regarding primary standard of titrimetric analysis. (A) It should be purely
available in dry form. (B) It should not undergo chemical change in air. (C) It should be hygroscopic and
should react with another chemical instantaneously and stoichiometrically. (D) It should be readily soluble in
water. (E) $KMnO_{4}\,&\,NaOH$ can be used as primary standard. Choose the correct answer from the options
given below :
a.
(A) and (B) only
b.
(C) and (E) only
c.
(B) and (E) only
d.
(C) and (D) only
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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) : The total number of geometrical isomers shown by $[Co(en)_{2}Cl_{2}]^{+}$ complex ion is three. Reason
(R): $[Co(en)_{2}Cl_{2}]^{+}$ complex ion has an octahedral geometry. In the light of the above statements, choose the
most appropriate answer from the options given below :
a.
Both (A) and (R) are correct but (R) is not the
correct explanation of (A)
b.
(A) is not correct but (R) is correct
c.
Both (A) and (R) are correct and (R) is the
correct explanation of (A)
d.
(A) is correct but (R) is not correct
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$---$
Let $O\vec{A}= 2\vec{a},O\vec{B}= 6\vec{a}+ 5\vec{b}$ and $O\vec{C} = 3\vec{b}$ , where $O$ is the origin. If the area of the parallelogram with
$--$
adjacent sides $O\vec{A}$ and $O\vec{C}$ is 15 sq. units, then the area (in sq. units) of the quadrilateral $OABC$ is equal to :
a.
32
b.
40
c.
38
d.
35
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Let the line $L$ intersect the lines $x-2 = -y=z-1, 2(x+ 1) = 2(y-1) =z+ 1$ and be parallel to the line
$\frac{x-2}{3} = \frac{y-1}{1} = \frac{z-2}{2}$ . Then which of the following points lies on L?
a.
$(-\frac{1}{3} , 1, -1)$
b.
$(-\frac{1}{3}, -1, 1)$
c.
$(-\frac{1}{3} , 1, 1)$
d.
$(-\frac{1}{3}, -1, -1)$
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Let $\int \frac{2-tanx}{3+tanx}\,dx= \frac{1}{2}(\alpha x+ log_{e}|\beta sinx+\gamma cosx|) +C$ , where $C$ is the constant of integration. Then $\alpha + \frac{\gamma}{\beta}$
is equal to :
a.
7
b.
4
c.
1
d.
3
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$3x+ 5y+\lambda z= 3$
Let $\lambda ,\mu \in R$ . If the system of equations $7x+ 11y-9z= 2$ has infinitely many solutions, then $\mu + 2\lambda$ is
$97x+ 155y-189z=\mu$
equal to :
a.
24
b.
25
c.
22
d.
27
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$|| ||$
$|| ||$
If a function $f$ satisfies $f(\,m + n) =f(\,m) +f(n)$ for all $m, n \in N$ and $f(1) = 1$ , then the largest natural
number $\lambda$ such that $\sum ^{2022}_{k=1} f(\lambda +k) \leq (2022)^{2}$ is equal to _________
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Let $\alpha ,\beta$ be the roots of the equation $x^{2} + 2\sqrt{2}x-1 = 0$ . The quadratic equation, whose roots are $\alpha ^{4} +\beta ^{4}$
and $\frac{1}{10}(\alpha ^{6}+\beta ^{6})$ , is :
a.
$x^{2} -190x+ 9466 = 0$
b.
$x^{2} -180x+ 9506 = 0$
c.
$x^{2} -195x+ 9506 = 0$
d.
$x^{2} -195x+ 9466 = 0$
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A variable line $L$ passes through the point $(3, 5)$ and intersects the positive coordinate axes at the points $A$ and
$B$ . The minimum area of the triangle $OAB$ , where $O$ is the origin, is :
a.
30
b.
25
c.
40
d.
35
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Let the set of all positive values of $\lambda$ , for which the point of local minimum of the function $(1 +x(\lambda ^{2} - x^{2}))$
satisfies $\frac{x^{2}+x+2}{x^{2}+5x+6} < 0$ , be $(\alpha ,\beta )$ . Then $\alpha ^{2} +\beta ^{2}$ is equal to _________
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Let $A$ be a non-singular matrix of order 3 . If $det(3 adj(2 adj((detA)A))) = 3^{-13} \cdot 2^{-10}$ and
$det(3 adj(2\,A)) = 2^{m} \cdot 3^{n}$ , then $|3\,m + 2n|$ is equal to
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Let $a, b$ and $c$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are
marked $1, 2, 3, 4$ . If the probability that $ax^{2} +bx+c= 0$ has all real roots is $\frac{m}{n}$ , $gcd(m, n) = 1$ , then $m + n$
is equal to ________
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$\{ \frac{tan 8x}{tan 7x,}$
$(\frac{8}{7}) 0 <x< \frac{\pi}{2}$
Let $f : (0,\pi ) R$ be a function given by $f(x) = a -8, x=$
$\frac{\pi}{2}$
$\{^{b}| tan_{x|}$
$(1 + | cotx|) , \frac{\pi}{2} <x<\pi$
where $a, b \in \vec{Z}$ . If $f$ is continuous at $x= \frac{\pi}{2}$ , then $a^{2} + b^{2}$ is equal to
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The coefficient of $x^{70}$ in $x^{2}(1 +x)^{98} +x^{3}(1 +x)^{97} +x^{4}(1 +x)^{96} + … +x^{54}(1 +x)^{46}$ is $^{99}C_{p}-^{46}C_{q}$ .
Then a possible value of $p+q$ is :
a.
55
b.
83
c.
61
d.
68
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Let $lim ( \frac{n}{\sqrt{n^{4}+1}} - \frac{2n}{(n2+1)\sqrt{n^{4}+1}} + \frac{n}{\sqrt{n^{4}+16}} - \frac{8n}{(n2+4)\sqrt{n^{4}+16}} + … + \frac{n}{\sqrt{n^{4}+n^{4}}} - \frac{2n\cdot n^{2}}{(n2+n2)\sqrt{n^{4}+n^{4}}}$ be $\frac{\pi}{k}$ ,
$n\rightarrow \infty \rightarrow )$
using only the principal values of the inverse trigonometric functions. Then $k^{2}$ is equal to ________
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Let $f(x) =x^{2} + 9,g(x) = \frac{x}{x-9}$ and $a =f^{\circ} g(10), b =g^{\circ} f(3)$ . If $e$ and $l$ denote the eccentricity and the
length of the latus rectum of the ellipse $\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1$ , then $8e^{2} +l^{2}$ is equal to.
a.
8
b.
16
c.
6
d.
12
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The solution curve, of the differential equation $2y\frac{dy}{\,dx} + 3 = 5\frac{dy}{\,dx}$ , passing through the point $(0, 1)$ is a conic,
whose vertex lies on the line:
a.
$2x+ 3y= 9$
b.
$2x+ 3y= -9$
c.
$2x+ 3y= -6$
d.
$2x+ 3y= 6$
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If the sum of the series $\frac{1}{1\cdot (1+d)} + \frac{1}{(1+d)(1+2\,d)} + … + \frac{1}{(1+9\,d)(1+10\,d)}$ is equal to 5 , then $50\,d$ is equal to :
a.
10
b.
5
c.
15
d.
20
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The frequency distribution of the age of students in a class of 40 students is given below.
$\frac{Age}{No ofStudents}\frac{15}{5}\frac{16}{8}\frac{17}{5}\frac{18}{12}\frac{19}{x}\frac{20}{y}$ If the mean deviation about the median is 1.25, then $4x+ 5y$
is equal to :
a.
46
b.
43
c.
44
d.
47
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Let $A= {2, 3, 6, 7}$ and $B= {4, 5, 6, 8}$ . Let $R$ be a relation defined on $A\times B$ by $(a_{1},b_{1})R(a_{2},b_{2})$ if and
only if $a_{1}+a_{2} =b_{1}+b_{2}$ . Then the number of elements in $R$ is _________
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Let a circle passing through $(2, 0)$ have its centre at the point $(h,k)$ . Let $(x_{c},y_{c})$ be the point of intersection of
the lines $3x+ 5y= 1$ and $(2 +c)x+ 5c^{2}y= 1$ . If $h = lim_{c\rightarrow 1\rightarrow } x_{c}$ and $k = lim_{c\rightarrow 1\rightarrow } y_{c}$ , then the equation of the
circle is :
a.
$25x^{2} + 25y^{2} -2x+ 2y-60 = 0$
b.
$5x^{2} + 5y^{2} -4x+ 2y-12 = 0$
c.
$5x^{2} + 5y^{2} -4x-2y-12 = 0$
d.
$25x^{2} + 25y^{2} -20x+ 2y-60 = 0$
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Let three vectors $\vec{a}=\alpha \hat{i}+ 4\hat{j}+ 2\hat{k}, \vec{b}= 5\hat{i}+ 3\hat{j}+ 4\hat{k}, \vec{c}=x\hat{i}+y\hat{j}+z\vec{k}$ form a triangle such that $\vec{c}= \vec{a}- \vec{b}$
and the area of the triangle is $5\sqrt{6}$ . If $\alpha$ is a positive real number, then $|\hat{c}|^{2}$ is equal to:
a.
16
b.
14
c.
12
d.
10
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Let the centre of a circle, passing through the points $(0, 0), (1, 0)$ and touching the circle $x^{2} +y^{2} = 9$ , be $(h,k)$
. Then for all possible values of the coordinates of the centre $(h,k), 4(h^{2} +k^{2})$ is equal to_________
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Let $|cos\theta cos(60 - \theta ) cos(60 +\theta )| \leq \frac{1}{8} ,\theta \epsilon [0, 2\pi ]$ . Then, the sum of all $\theta \epsilon [0, 2\pi ]$ , where $cos 3\theta$ attains its
maximum value, is :
a.
$15\pi$
b.
$18\pi$
c.
$6\pi$
d.
$9\pi$
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A ray of light coming from the point $P(1, 2)$ gets reflected from the point $Q$ on the $x$ -axis and then passes
through the point $R(4, 3)$ . If the point $S(h,k)$ is such that PQRS is a parallelogram, then $hk^{2}$ is equal to :
a.
70
b.
80
c.
60
d.
90
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