Two moles of an ideal monoatomic gas occupies a volume $V$ at $27^{o}C$ . The gas expands adiabatically to a
volume $2V$ . Calculate $(a)$ the final temperature of the gas and $(b)$ change in its internal energy.
a.
$(a)195\,K\,\,(b)2.7\, kJ$
b.
$(a)189\,K\,\,(b)2.7\, kJ$
c.
$(a)195\,K\,\,(b) -2.7\, kJ$
d.
$(a)189\,K\,\,(b) -2.7\, kJ$
Correct Answer
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Q2mcq
The dipole moment of a circular loop carrying a current I, is m and the magnetic field at the centre of the loop
is $B_{1}$ . When the dipole moment is doubled by keeping the current constant, the magnetic field at the centre of
the loop is $B_{2}$ . The ratio $\frac{B_{1}}{B_{2}}$ is:
a.
$\frac{1}{\sqrt{2}}$
b.
$2$
c.
$\sqrt{3}$
d.
$\sqrt{2}$
Correct Answer
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Q3mcq
In a potentiometer experiment, it is found that no current passes through the galvanometer when the terminals
of the cell are connected across 52 cm of the potentiometer wire. If the cell is shunted by a resistance of $5\,\Omega$ ,a
balance is found when the cell is connected across 40 cm of the wire. Find the internal resistance of the cell.
a.
$2.5\,\Omega$
b.
$1\,\Omega$
c.
$1.5\,\Omega$
Correct Answer
d.
$2\,\Omega$
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Q4mcq
For an RLC circuit driven with voltage of amplitude $v_{m}$ and frequency $\omega$ $_{0} = \frac{1}{\sqrt{LC}}$ the current exhibits
resonance. The quality factor, $Q$ is given by
a.
$\frac{CR}{\omega _{0}}$
b.
$\frac{\omega _{0}L}{R}$
Correct Answer
c.
$\frac{\omega _{0}R}{L}$
d.
$\frac{R}{(\omega _{0}C)}$
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Q5mcq
A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely
proportional to the $n^{th}$ power of R. If the period of rotation of the particle is T, then:
a.
$T \propto R^{n/2}$
b.
$T \propto R^{3/2}$ for any n
c.
$T \propto R\frac{n}{2}^{+1}$
d.
$T \propto R\frac{n+1}{2}$
Correct Answer
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Q6mcq
A particle is moving in a circular path of radius a under the action of an attractive potential $U = -\frac{k}{2r^{2}}.$ Its total
energy is:
a.
$-\frac{3}{2} \frac{k}{a^{2}}$
b.
$-\frac{k}{4a^{2}}$
c.
$\frac{k}{2a^{2}}$
d.
Zero
Correct Answer
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Q7mcq
If the series limit frequency of the Lyman series is $V_{L},$ then the series limit frequency of the Pfund series is:
a.
$\frac{V_{L}}{25}$
Correct Answer
b.
$25\,V_{L}$
c.
$16\,V_{L}$
d.
$\frac{V_{L}}{16}$
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Q8mcq
An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let
$\lambda _{n}, \,\lambda _{g}$ be the de Broglie wavelength of the electron in the $n^{th}$ state and the ground state respectively. Let $\wedge _{n}$
be the wavelength of the emitted photon in the transition from the $n^{th}$ state to the ground state. For large n, (A,
B are constants)
On interchanging the resistances, the balance point of a meter bridge shifts to the left by $10\, cm$ . The resistance
of their series combination is $1\,k\Omega$ . How much was the resistance on the left slot before interchanging the
resistances?
a.
$910\,\Omega$
b.
$990\,\Omega$
c.
$505\,\Omega$
d.
$550\,\Omega$
Correct Answer
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Q10mcq
The reading of the ammeter for a silicon diode in the given circuit is:
a.
$13 .5 \, mA$
b.
$0$
c.
$15\, mA$
d.
$11 .5 \, mA$
Correct Answer
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Unpolarized light of intensity $I$ passes through an ideal polariser $A$ . Another identical polariser $B$ is placed
behind $A$ . The intensity of light beyond B is found to be $\frac{I}{2}$ . Now another identical polariser $C$ is placed
between $A$ and $B$ . The intensity beyond B is now found to be $\frac{I}{8}$ . The angle between polariser $A$ and $C$ is
a.
$60^{\circ}$
b.
$0^{\circ}$
c.
$30^{\circ}$
d.
$45^{\circ}$
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In an $A.C.$ circuit, the instantaneous e.m.f. and current are given by, $E = 100 sin 30t,I = 20 sin(30t- \frac{$\pi$}{4})$ .
In one cycle of $A.C.$ , the average power consumed by the circuit (in watt) and the watt-less current (in ampere)
are, respectively:
a.
$50, 0$
b.
$50, 10$
c.
$\frac{1000}{\sqrt{2}}, \,10$
d.
$\frac{50}{\sqrt{2}}, \,0$
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From a uniform circular disc of radius R and mass 9 M, a small disc of radius $\frac{R}{3}$ is removed as shown in the
figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and
passing through centre of disc is:
a.
$\frac{37}{9}MR^{2}$
b.
$4MR^{2}$
c.
$\frac{40}{9}MR^{2}$
d.
$10MR^{2}$
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An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of
radii $r_{e}, \,r_{p}, \,r_{\alpha }$ respectively in a uniform magnetic field B. The relation between $r_{e}, \,r_{p}, \,r_{\alpha }$ is:
a.
$r_{e} <r_{\alpha } <r_{p}$
b.
$r_{e} >r_{P} =r_{\alpha }$
c.
$r_{e} <r_{p} =r_{\alpha }$
d.
$r_{e} <r_{p} <r_{\alpha }$
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A parallel plate capacitor of capacitance $90\, pF$ is connected to a battery of EMF $20\,V$ . If a dielectric material
of dielectric constant $K = \frac{5}{3}$ is inserted between the plates, the magnitude of the induced charge will be:
a.
$0. 9\, nC$
b.
$1. 2\, nC$
c.
$0. 3\, nC$
d.
$2. 4\, nC$
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The density of a material, in the shape of a cube, is determined by measuring three sides of the cube and its
mass. If the relative errors in measuring the mass and length are $1. 5%$ and $1%$ , respectively, the maximum error
in determining the density is:
a.
$6\,%$
b.
$2.5\,%$
c.
$3.5\,%$
d.
$4.5\,%$
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A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical
container. A massless piston of area $a$ floats on the surface of the liquid, covering entire cross-section of
cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the
fractional decrement in the radius of the sphere $(\frac{dr}{r})$ , is:
a.
$\frac{mg}{Ka}$
b.
$\frac{Ka}{mg}$
c.
$\frac{Ka}{3mg}$
d.
$\frac{mg}{3Ka}$
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Two batteries with e.m.f. $12\,V$ and $13\,V$ are connected in parallel across a load resistor of $10\,\Omega$ . The internal
resistance of the two batteries are $1\,\Omega$ and $2\,\Omega$ respectively. The voltage across the load lies between,
a.
$11. 7\,V$ and $11. 8\,V$
b.
$11. 6\,V$ and $11. 7\,V$
c.
$11. 5\,V$ and $11. 6\,V$
d.
$11. 4\,V$ and $11. 5\,V$
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Three concentric metal shells $A, \,B$ and $C$ of respective radii $a, \,b$ and $c\,(a\, < \,b\, < \,c)$ have surface charge
densities $+\sigma , \, - \sigma$ and $+\sigma$ respectively. The potential of shell $B$ is:
A telephonic communication service is working at a carrier frequency of $10\, GHz$ . Only $10%$ of it is utilized
for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a
bandwidth of $5\, kHz$ ?
a.
$2 \times 10^{6}$
b.
$2 \times 10^{3}$
c.
$2 \times 10^{4}$
d.
$2 \times 10^{5}$
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It is found that if a neutron suffers an elastic collinear collision with a deuterium at rest, the fractional loss of its
energy is $P_{d}$ , while for its similar collision with a carbon nucleus at rest, the fractional loss of energy is $P_{c}$ . The
values of $P_{d}$ and $P_{c}$ are respectively
a.
$0, \,1$
b.
$0. 89, \,0. 28$
c.
$0. 28, \,0. 89$
d.
$0, \,0$
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Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The
moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is:
a.
$\frac{181}{2} MR^{2}$
b.
$\frac{19}{2} MR^{2}$
c.
$\frac{55}{2}MR^{2}$
d.
$\frac{73}{2}MR^{2}$
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The mass of a hydrogen molecule is $3.32 \times 10^{-27\,} kg.$ If $10^{23}$ hydrogen molecules strike, per second, a fixed
wall of the area $2\, cm^{2}$ at an angle of $45^{o}$ to the normal, and rebound elastically with a speed of $10^{3}\,m\,s^{-1},$ then
the pressure on the wall is nearly:
a.
$4.70 \times 10^{2}\,N\,m^{-2}$
b.
$2.35 \times 10^{3}\,N\,m^{-2}$
c.
$4.70 \times 10^{3}\,N\,m^{-2}$
d.
$2.35 \times 10^{2}\,N\,m^{-2}$
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A granite rod of $60\, cm$ length is clamped at its middle point and is set into longitudinal vibrations. The density
of granite is $2.7 \times 10^{3}\, kg \,m^{-3}$ and its Young's modulus is $9.27 \times 10^{10}\, Pa$ . What will be the fundamental
frequency of the longitudinal vibrations?
a.
$7. 5\, kHz$
b.
$5\, kHz$
c.
$2. 5\, kHz$
d.
$10\, kHz$
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A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of $10^{12}\,s^{-1}$ .
What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver,
$= 108\,g\, mol^{-1}$ and Avogadro number $= 6.02 \times 10^{23}$ )
a.
$5.5\,N\,m^{-1}$
b.
$6.4\,N\,m^{-1}$
c.
$7. 1\,N\,m^{-1}$
d.
$2.2\,N\,m^{-1}$
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The angular width of the central maximum in a single slit diffraction pattern is $60^{\circ}$ . The width of the slit is
$1\,$ $\mu$ $m$ . The slit is illuminated by monochromatic plane waves. If another slit of the same width is made near it,
Young's fringes can be observed on a screen placed at a distance $50 cm$ from the slits. If the observed fringe
width is $1 cm$ , what is slit separation distance? (i.e., the distance between the centres of each slit.)
a.
$100\,$ $\mu$ $m$
b.
$25\,$ $\mu$ $m$
c.
$50\,$ $\mu$ $m$
d.
$75\,$ $\mu$ $m$
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Two masses $m_{1} = 5\, kg$ and $m_{2} = 10\, kg$ , connected by an inextensible string over a frictionless pulley, are
moving as shown in the figure. The coefficient of friction of horizontal surface is $0. 15$ . The minimum weight m
that should be put on top of $m_{2}$ to stop the motion is:
a.
$10 .3 \, kg$
b.
$18 .3 \, kg$
c.
$27 .3 \, kg$
d.
$43 .3 \, kg$
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In a collinear collision, a particle with an initial speed $v_{0}$ strikes a stationary particle of the same mass. If the
final total kinetic energy is $50%$ greater than the original kinetic energy, the magnitude of the relative velocity
between the two particles, after the collision, is
a.
$\frac{v_{0}}{\sqrt{2}}$
b.
$\frac{v_{0}}{4}$
c.
$\sqrt{\,2}\,v_{0}$
d.
$\frac{v_{0}}{2}$
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An EM wave from air enters a medium. The electric fields are $\vec{E}_{1} =E_{01}xˆcos[2\pi v( \frac{z}{c} - t)]$ in air and
$\vec{E}_{2} =E_{02}xˆcos[k(2z- ct)]$ in medium, where the wave number k and frequency $v$ refer to their values in the
air. The medium is non-magnetic. If $\epsilon _{r_{1}}$ and $\epsilon _{r_{2}}$ refer to relative permittivities of air and medium respectively,
which of the following options, is correct?
Phenol on treatment with $CO_{2}$ in the presence of NaOH followed by acidification produces compound X as
the major product. X on treatment with $(CH_{3}CO)_{2}$ O in the presence of catalytic amount of $H_{2}SO_{4}$ produces:
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The recommended concentration of fluoride ion in drinking water is up to 1 ppm as fluoride ion is required to
make teeth enamel harder by converting $[3 Ca_{3}(PO_{4})_{2}. Ca (OH)_{2}]$ to:
a.
$[3 Ca (OH)_{2}. CaF_{2}]$
b.
$[CaF_{2}]$
c.
$[3(CaF_{2}). Ca (OH)_{2}]$
d.
$[3 Ca_{3}(PO_{4})_{2}. CaF_{2}]$
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The ratio of mass percent of $C\, and \,H$ of an organic compound $(C_{X}H_{Y}O_{Z})$ is $6\, : \,1.$ If one molecule of the
above compound $(C_{X}H_{Y}O_{Z})$ contains half as much oxygen as required to burn one molecule of compound
$C_{X}H_{Y}$ completely to $CO_{2}$ and $H_{2}O.$ The empirical formula of the compound $C_{X}H_{Y}O_{Z}$ is
a.
$C_{2}H_{4}O_{3}$
b.
$C_{3}H_{6}O_{3}$
c.
$C_{2}H_{4}O$
d.
$C_{3}H_{4}O_{2}$
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The combustion of benzene $(l)$ gives $CO_{2}(g)$ and $H_{2}O(l).$ Given that heat of combustion of benzene at
constant volume is $-3263 .9 \, kJ \, mol^{-1}$ at $25^{\circ}C;$ the heat of combustion $(in \, kJ \, mol^{-1})$ of benzene at
constant pressure will be
$(R = 8 .314 \, JK^{-1} \, mol^{-1})$
a.
$-3267.6$
b.
$4152.6$
c.
$-452.46$
d.
$3260$
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How long (approximate) should water be electrolysed by passing through $100\,amperes \,$ current so that the
oxygen released can completely burn $27. 66\,g$ of diborane? (Atomic weight of $B = 10. 8\,u$ )
a.
$1. 6\, hours$
b.
$6. 4\, hours$
c.
$0. 8\, hours$
d.
$3. 2\, hours$
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An aqueous solution contains $0. 10\,M\,H_{2}S$ and $0. 20\,M\, HCl$ . If the equilibrium constant for the formation of
$HS^{-}$ from $H_{2}S$ is $1.0 \times 10^{-7}$ and that of $S^{2-}$ from $HS^{-}$ ions is $1.2 \times 10^{-13}$ , then, the concentration of $S^{2-}$
ions in the aqueous solution is:
a.
$5 \times 10^{-19}$
b.
$5 \times 10^{-8}$
c.
$3 \times 10^{-20}$
d.
$6 \times 10^{-21}$
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The oxidation states of $Cr$ in $[Cr (H_{2}O)_{6}] Cl_{3}, \,[Cr (C_{6}H_{6})_{2}]$ and $K_{2}[Cr (CN)_{2}(O)_{2}(O_{2})(NH_{3\,})]$ , respectively,
are:
a.
$+3, \,0$ and $+4$
b.
$+3, \, + 4$ and $+6$
c.
$+3, \, + 2$ and $+4$
d.
$+3, \,0$ and $+6$
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Consider the following reaction and statements:
$[Co (NH_{3})_{4}Br_{2}]^{+} + Br^{-} \rightarrow [Co (NH\rightarrow_{3})_{3}Br_{3}] + NH_{3}$
(i) Two isomers are produced if the reactant complex ion is a cis-isomer.
(ii) Two isomers are produced if the reactant complex ion is a trans-isomer.
(iii) Only one isomer is produced if the reactant complex ion is a trans-isomer.
(iv) Only one isomer is produced if the reactant complex ion is a cis-isomer.
The correct statements are
a.
(ii) and (iv)
b.
(i) and (ii)
c.
(i) and (iii)
d.
(iii) and (iv)
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Phenol reacts with methyl chloroformate in the presence of $NaOH$ to form product $A. \,A\,$ reacts with $Br_{2}$ to
form product $B. \,A\, and \,B$ are respectively
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At $518^{o}C,$ the rate of decomposition of a sample of gaseous acetaldehyde, initially at a pressure of $363 Torr$
was $1 .00 \, Torr \,s^{-1}$ when $5%$ had reacted and $0 .50 \, Torr \,s^{-1}$ when 33% had reacted. The order of the
reaction is:
a.
$0$
b.
$2$
c.
$3$
d.
$1$
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Hydrogen peroxide oxidises $[Fe (CN)_{6}]^{4-} to \,[Fe (CN)_{6}]^{3-}$ in acidic medium, but reduces
$[Fe (CN)_{6}]^{3-} to [Fe (CN)_{6}]^{4-}$ in alkaline medium. The other products formed are, respectively
a.
$H_{2}O\, and \,(H_{2}O + OH^{-})$
b.
$(H_{2}O + O_{2})\, and \,H_{2}O$
c.
$(H_{2}O + O_{2})\, and \,(H_{2}O + OH^{-})$
d.
$H_{2}O\, and \,(H_{2}O + O_{2})$
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An aqueous solution contains an unknown concentration of $Ba^{2+}$ . When $50\, mL \, of \,a\,1\,M$ solution of
$Na_{2}SO_{4}$ is added, $BaSO_{4}$ just begins to precipitate. The final volume is $500\, mL$ . The solubility product of
$BaSO_{4}$ is $1 \times 10^{-10}$ . What is the original concentration of $Ba^{2+}$ ?
a.
$1 .0 \times 10^{-10}M$
b.
$5 \times 10^{-9}M$
c.
$2 \times 10^{-9}M$
d.
$1 .1 \times 10^{-9}M$
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When metal $M\,$ is treated with $NaOH$ , a white gelatinous precipitate $X\,$ is obtained, which is soluble in excess
of $NaOH$ . Compound $X\,$ when heated strongly gives an oxide which is used in chromatography as an
adsorbent. The metal $M\,$ is
a.
$Fe$
b.
$Zn$
c.
$Ca$
d.
$Al$
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Tangents are drawn to the hyperbola $4x^{2} - y^{2} = 36$ at the points $P$ and $Q$ . If these tangents intersect at the
point $T(0, \,3)$ then the area (in sq. units) of $\Delta PTQ$ is:
a.
$36\sqrt{5}$
b.
$45\sqrt{5}$
c.
$54\sqrt{3}$
d.
$60\sqrt{3}$
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A straight line through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points $P$ and $Q$ . If $O$ is the
origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is:
a.
$3x+ 2y= 6xy$
b.
$3x+ 2y= 6$
c.
$2x+ 3y=xy$
d.
$3x+ 2y=xy$
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$|$
A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its color is observed and this
ball along with two additional balls of the same color are returned to the bag. If now a ball is drawn at random
from the bag, then the probability that this drawn ball is red, is:
a.
$\frac{3}{4}$
b.
$\frac{3}{10}$
c.
$\frac{2}{5}$
d.
$\frac{1}{5}$
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If $\sum ^{9}_{i=1}(x_{i}-5) = 9$ and $\sum ^{9}_{i=1} (x_{i}-5)^{2} = 45$ , then the standard deviation of the $9$ items $x_{1}, \,x_{2}, … . ,x_{9}$ is
a.
$3$
b.
$9$
c.
$4$
d.
$2$
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The integral $\int _{(sin^{5}x+cos^{3}xsin}^{sin}_{2}^{2}_{x}^{x}_{+sin}^{cos}_{3}^{2}_{x}^{x}_{cos^{2}x+cos^{5}x)^{2}}dx$ , is equal to
(where $C$ is the constant of integration).
a.
$\frac{-1}{1+cot^{3}x} +C$
b.
$\frac{1}{3(1+tan^{3}x)} +C$
c.
$\frac{-1}{3(1+tan^{3}x)} +C$
d.
$\frac{1}{1+cot^{3}x} +C$
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$12$
Let $a_{1}, \,a_{2}, \,a_{3}, … … ,a_{49}$ be in $A.P.$ such that $_{k}\Sigma _{=0}a_{4k+1} = 416$ and $a_{9}+a_{43} = 66.$ If
$a^{2}_{1} +a^{2}_{2} + … +a^{2}_{17} = 140m,$ then $m$ is equal to:
a.
$33$
b.
$66$
c.
$68$
d.
$34$
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Let the orthocentre and centroid of a triangle be $A(-3, \,5)$ and $B(3, \,3)$ respectively. If $C$ is the circumcentre
of this triangle, then the radius of the circle having line segment $AC$ as diameter, is:
a.
$\frac{3\sqrt{5}}{2}$
b.
$\sqrt{10}$
c.
$2\sqrt{10}$
d.
$3\sqrt{\frac{5}{2}}$
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$||$
If sum of all the solutions of the equation $8 cosx\cdot (cos(\frac{\pi}{6} +x)\cdot cos( \frac{\pi}{6} - x)- \frac{1}{2})= 1$ in $[0, \,\pi ]$ is $k\pi$ , then $k$
is equal to:
a.
$\frac{20}{9}$
b.
$\frac{2}{3}$
c.
$\frac{13}{9}$
d.
$\frac{8}{9}$
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For each $t\in R,$ let $[t]$ be the greatest integer less than or equal to $t$ . Then $_{x}lim_{\rightarrow 0\rightarrow }_{+}x([\frac{1}{x}]+[\frac{2}{x}]+ … +[\frac{15}{x}])$
a.
does not exist $(in \,R)$
b.
is equal to $0$
c.
is equal to $15$
d.
is equal to $120$
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$PQR$ is a triangular park with $PQ=PR= 200\,m$ . A T.V. tower stands at the mid-point of $QR$ . If the angles
of elevation of the top of the tower at $P,Q$ and $R$ are respectively, $45^{\circ}, \,30^{\circ}$ and $30^{\circ}$ , then the height of the
tower (in $m$ ) is:
a.
$50\sqrt{2}$
b.
$100$
c.
$50$
d.
$100\sqrt{3}$
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Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series
$1^{2} + 2 \cdot 2^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + 2 \cdot 6^{2} + …$
If $B-2A= 100\lambda ,$ then $\lambda$ is equal to :
a.
$496$
b.
$232$
c.
$248$
d.
$464$
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Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}= 2ˆi+ 3ˆj- ˆk$ and $\vec{b}= \,ˆj+ ˆk$ . If $\vec{u}$ is perpendicular to $\vec{a}$ and
$\vec{u}\cdot \vec{b}= 24$ , then $\vec{u}^{2}$ is equal to:
a.
$84$
b.
$336$
c.
$315$
d.
$256$
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If $L_{1}$ is the line of intersection of the planes $2x-2y+ 3z-2 = 0, \,x- y+z+ 1 = 0$ and $L_{2}$ is the line of
intersection of the planes $x+ 2y- z-3 = 0, \,3x- y+ 2z-1 = 0,$ then the distance of the origin from the
plane, containing the lines $L_{1}$ and $L_{2}$ is
a.
$\frac{1}{\sqrt{2}}$
b.
$\frac{1}{4\sqrt{2}}$
c.
$\frac{1}{3\sqrt{2}}$
d.
$\frac{1}{2\sqrt{2}}$
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Let $f(x) =x^{2} + \frac{1}{x^{2}}$ and $g(x) =x- \frac{1}{x}, \,x\in R-{-1, \,0, \,1}.$ If $h(x) = \frac{f(x)}{g(x)}$ , then the local minimum value
of $h(x)$ is:
a.
$2\sqrt{2}$
b.
$3$
c.
$-3$
d.
$-2\sqrt{2}$
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From $6$ different novels and $3$ different dictionaries, $4$ novels and $1$ dictionary are to be selected and arranged
in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is:
a.
At least $750$ but less than $1000$
b.
At least $1000$
c.
Less than $500$
d.
At least $500$ but less than $750$
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Let $y=y(x)$ be the solution of the differential equation $sinx\frac{dy}{dx} +ycosx= 4x, \,x\in (0, \,\pi ).$ If $y( \frac{\pi}{2})= 0,$
then $y(\frac{\pi}{6})$ is equal to
a.
$-\frac{4}{9}\pi ^{2}$
b.
$\frac{4}{9\sqrt{3}}\pi ^{2}$
c.
$\frac{-8}{9\sqrt{3}} \pi ^{2}$
d.
$-\frac{8}{9} \pi ^{2}$
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Let $g(x) = cosx^{2}, \,f(x) = \sqrt{x},$ and $\alpha ,\beta (\alpha <\beta )$ be the roots of the quadratic equation $18x^{2} -9\pi x+\pi ^{2} = 0$ .
Then the area (in sq. units) bounded by the curve $y= (gof)(x)$ and the lines $x=\alpha ,x=\beta$ and $y= 0,$ is
a.
$\frac{1}{2}(\sqrt{2}-1)$
b.
$\frac{1}{2}(\sqrt{3}-1)$
c.
$\frac{1}{2}(\sqrt{3}+ 1)$
d.
$\frac{1}{2}(\sqrt{3}- \sqrt{2})$
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If the system of linear equations
$x+ky+ 3z= 0$
$3x+ky-2z= 0$
$2x+ 4y-3z= 0$
has a non-zero solution $(x, \,y, \,z),$ then $\frac{xz}{y^{2}}$ is equal to:
a.
$30$
b.
$-10$
c.
$10$
d.
$-30$
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Tangent and normal are drawn at $P(16, 16)$ on the parabola $y^{2} = 16x$ , which intersect the axis of the parabola
at $A\,&B$ , respectively. If $C$ is the center of the circle through the points $P, \,A\,&B$ and $\angle CPB=\theta ,$ then a
value of $tan\theta$ is:
a.
$\frac{4}{3}$
b.
$\frac{1}{2}$
c.
$2$
d.
$3$
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