The maximum and minimum distances of a comet from the Sun are $1. 6 \times 10^{12}\,m$ and $8. 0 \times 10^{10}\,m$
respectively. If the speed of the comet at the nearest point is $6 \times 10^{4}\,m\,s^{-1}$ , the speed at the farthest point is
a.
$1. 5 \times 10^{3}\,m\,s^{-1}$
b.
$6. 6 \times 10^{3}\,m\,s^{-1}$
c.
$3. 0 \times 10^{3}\,m\,s^{-1}$
Correct Answer
d.
$4. 5 \times 10^{3}\,m\,s^{-1}$
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Q2mcq
A bar magnet of length $14\, cm$ is placed in the magnetic meridian with its north pole pointing towards the
geographic north pole. A neutral point is obtained at a distance of $18\, cm$ from the center of the magnet. If
$B_{H} = 0. 4\,G$ , the magnetic moment of the magnet is $(1\,G = 10^{-4}\,T)$
a.
$2. 880 \times 10^{3}\,J\,T^{-1}$
b.
$2. 880 \times 10^{2}\,J\,T^{-1}$
c.
$2. 880\,J\,T^{-1}$
Correct Answer
d.
$28. 80\,J\,T^{-1}$
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Q3nat
The resistance $R= \frac{V}{I}$ , where $V = (50 \pm 2)\,V$ and $I = (20 \pm 0. 2)\,A$ . The percentage error in $R$ is $x%$ . The
value of $x$ to the nearest integer is ________.
✓ Accepted Answer:5
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Q4mcq
For an electromagnetic wave travelling in free space, the relation between average energy densities due to
electric $(U_{e})$ and magnetic $(U_{m})$ fields is :
a.
$U_{e} =U_{m}$
Correct Answer
b.
$U_{e} >U_{m}$
c.
$U_{e} <U_{m}$
d.
$U_{e} \neq U_{m}$
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Q5mcq
The volume $V$ of an enclosure contains a mixture of three gases, $16\,g$ of oxygen, $28\,g$ of nitrogen and $44\,g$ of
carbon dioxide at absolute temperature $T$ . Consider $R$ as universal gas constant. The pressure of the mixture of
gases is :
a.
$\frac{88RT}{V}$
b.
$\frac{3RT}{V}$
c.
$\frac{5}{2} \frac{RT}{V}$
Correct Answer
d.
$\frac{4RT}{V}$
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Q6mcq
A block of $200\,g$ mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of
radius $20\, cm$ . If the block takes $40\,s$ to complete one round, the normal force by the side walls of the groove is:
a.
$0. 0314\,N$
b.
$9. 859 \times 10^{-2}\,N$
c.
$6. 28 \times 10^{-3}\,N$
d.
$9. 859 \times 10^{-4}\,N$
Correct Answer
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Q7mcq
Four equal masses, $m$ each are placed at the corners of a square of length $(l)$ as shown in the figure. The
moment of inertia of the system about an axis passing through $A$ and parallel to $DB$ would be :
a.
$ml^{2}$
b.
$2\,ml^{2}$
c.
$3\,ml^{2}$
Correct Answer
d.
$\sqrt{3}\,ml^{2}$
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Q8mcq
The angle of deviation through a prism is minimum when
(A) Incident ray and emergent ray are symmetric to the prism
(B) The refracted ray inside the prism becomes parallel to its base
(C) Angle of incidence is equal to that of the angle of emergence
(D) When angle of emergence is double the angle of incidence
Choose the correct answer from the options given below :
a.
Statements (A),(B) and (C) are true
Correct Answer
b.
Only statement (D) is true
c.
Only statements (A) and (B) are true
d.
Statements (B) and (C) are true
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Q9mcq
An $RC$ circuit as shown in the figure is driven by a $AC$ source generating a square wave. The output wave
pattern monitored by $CRO$ would look close to :
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Q10nat
Consider a $20\, kg$ uniform circular disk of radius $0. 2\,m$ . It is pin supported at its center and is at rest initially.
The disk is acted upon by a constant force $F = 20\,N$ through a massless string wrapped around its periphery
as shown in the figure.
Suppose the disk makes $n$ number of revolutions to attain an angular speed of $50\, rad \,s^{-1}$ . The value of $n$ , to
the nearest integer, is _______ .
[Given : In one complete revolution, the disk rotates by $6. 28\, rad$ ]
✓ Accepted Answer:20
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A sinusoidal voltage of peak value $250\,V$ is applied to a series $LCR$ circuit, in which $R= 8\,\Omega ,L= 24\, mH$
and $C = 60\,\mu F$ . The value of power dissipated at resonant condition is $xkW$ . The value of $x$ to the nearest
integer is _______.
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Consider a frame that is made up of two thin massless rods $AB$ and $AC$ as shown in the figure. A vertical
force $\vec{P}$ of magnitude $100\,N$ is applied at point $A$ of the frame.
Suppose the force is $\vec{P}$ resolved parallel to the arms $AB$ and $AC$ of the frame. The magnitude of the resolved
component along the arm $AC$ is $xN$ . The value of $x$ , to the nearest integer, is ________.
[Given : $sin(35^{\circ}) = 0. 573, cos(35^{\circ}) = 0. 819, \, sin(110^{\circ}) = 0. 939, cos(110^{\circ}) = -0. 342$ ]
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The first three spectral lines of $H$ -atom in the Balmer series are given $\lambda _{1},\lambda _{2},\lambda _{3}$ considering the Bohr atomic
model, the wave lengths of first and third spectral lines $(\frac{\lambda _{1}}{\lambda _{3}})$ are related by a factor of approximately $x$
$\times 10^{-1}$ . The value of $x$ , to the nearest integer, is ________.
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One main scale division of a vernier callipers is $acm$ and $n^{th}$ division of the vernier scale coincide with
$(n-1)^{th}$ division of the main scale. The least count of the callipers in $mm$ is :
a.
$\frac{10na}{(n-1)}$
b.
$\frac{10a}{(n-1)}$
c.
$\frac{(n-1)}{10n}$
d.
$\frac{10a}{n}$
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The velocity-displacement graph describing the motion of a bicycle is shown in the figure.
The acceleration-displacement graph of the bicycle's motion is best described by :
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A conducting bar of length $L$ is free to slide on two parallel conducting rails as shown in the figure
Two resistors $R_{1}$ and $R_{2}$ are connected across the ends of the rails. There is a uniform magnetic field $\vec{B}$
pointing into the page. An external agent pulls the bar to the left at a constant speed $v$ .
The correct statement about the directions of induced currents $I_{1}$ and $I_{2}$ flowing through $R_{1}$ and $R_{2}$
respectively is :
a.
Both $I_{1}$ and $I_{2}$ are in anticlockwise direction
b.
Both $I_{1}$ and $I_{2}$ are in clockwise direction
c.
$I_{1}$ is in clockwise direction and $I_{2}$ is in
anticlockwise direction
d.
$I_{1}$ is in anticlockwise direction and $I_{2}$ is in
clockwise direction
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For changing the capacitance of a given paralle plate capacitor, a dielectric material of dielectric constant $K$ is
used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is $\frac{3}{4}d$ , where $d$
is the separation between the plates of parallel plate capacitor. The new capacitance $C'$ in terms of original
capacitance $C_{0}$ is given by the following relation :
a.
$C' = \frac{3+K}{4K}C_{0}$
b.
$C' = \frac{4+K}{3}C_{0}$
c.
$C' = \frac{4K}{K+3} C_{0}$
d.
$C' = \frac{4}{3+K}C_{0}$
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A fringe width of $6\, mm$ was produced for two slits separated by $1\, mm$ apart. The screen is placed $10\,m$ away.
The wavelength of light used is $xnm$ . The value of $x$ to the nearest integer is ______.
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In the logic circuit shown in the figure, if input $A$ and $B$ are $0$ to $1$ respectively, the output at $Y$ would be $x$ .
The value of $x$ is _________.
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A ball of mass $10\, kg$ moving with a velocity $10\,m\,s\sqrt{3}^{-1}$ along $X$ -axis, hits another ball of mass $20\, kg$ which
is at rest. After the collision, the first ball comes to rest and the second one disintegrates into two equal pieces.
One of the pieces starts moving along $Y$ -axis at a speed of $10\,m\,s^{-1}$ . The second piece starts moving at a
speed of $20\,m\,s^{-1}$ at an angle $\theta$ (degree) with respect to the $X$ -axis. The configuration of pieces after the
collision is shown in the figure. The value of $\theta$ to the nearest integer is _________.
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A block of mass $m$ slides along a floor while a force of magnitude $F$ is applied to it at an angle $\theta$ as shown in
figure. The coefficient of kinetic friction is $\mu _{K}$ . Then, the block's acceleration $a$ is given by : ( $g$ is acceleration
due to gravity)
Time period of a simple pendulum is $T$ inside a lift when the lift is stationary. If the lift moves upwards with an
acceleration $\frac{g}{2}$ , the time period of pendulum will be :
a.
$\sqrt{3}T$
b.
$\frac{T}{\sqrt{3}}$
c.
$\sqrt{\frac{3}{2}}T$
d.
$\sqrt{\frac{2}{3}}T$
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A plane electromagnetic wave of frequency $500\, MHz$ is traveling in a vacuum along the $y-$ direction. At a
particular point in space and time, $\vec{B}= 8. 0 \times 10^{-8}ˆz\,T$ . The value of the electric field at this point is:(speed of
light $= 3 \times 10^{8}\, ms^{-1}$ ) $ˆx,ˆy,ˆz$ are unit vectors along $x,y$ and $z$ direction.
a.
$-24ˆx\,V\,m^{-1}$
b.
$2. 6ˆx\,\,V\,m^{-1}$
c.
$24ˆx\,\,V\,m^{-1}$
d.
$-2. 6ˆx\,\,V\,m^{-1}$
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The pressure acting on a submarine is $3 \times 10^{5}\, Pa$ at a certain depth. If the depth is doubled, the percentage
increase in the pressure acting on the submarine would be: (Assume that atmospheric pressure is $1 \times 10^{5}\, Pa$
density of water is $10^{3}\, kg \,m^{-3}, \,g = 10\, ms^{-2}$ )
a.
$\frac{200}{3} %$
b.
$\frac{200}{5} %$
c.
$\frac{5}{200} %$
d.
$\frac{3}{200} %$
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A conducting wire of length $l$ , area of crosssection $A$ and electric resistivity $\rho$ is connected between the
terminals of a battery. A potential difference $V$ is developed between its ends, causing an electric current. If the
length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current
would be___
a.
$\frac{1}{4} \frac{V A}{\rho l}$
b.
$\frac{3}{4} \frac{V A}{\rho l}$
c.
$\frac{1}{4} \frac{\rho l}{V A}$
d.
$4\frac{V A}{\rho l}$
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A $25\,m$ long antenna is mounted on an antenna tower. The height of the antenna tower is $75\,m$ . The
wavelength (in meter) of the signal transmitted by this antenna would be :
a.
$300$
b.
$400$
c.
$200$
d.
$100$
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The value of power dissipated across the zener diode $(V_{z} = 15\,V)$ connected in the circuit as shown in the
figure is $x\times 10^{-1}\,W$ .
The value of $x$ to the nearest integer is _______.
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Two salts $A_{2}X$ and $MX$ have the same value of solubility product of $4. 0 \times 10^{-12}$ . The ratio of their molar
solubilizes i.e. $\frac{S(A_{2}X)}{S(MX)} =$ ________.
(Round off to the Nearest Integer).
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Match List-I with List-II :
List-I List-II
Industrial process Application
(a) Haber's process (i) $HNO_{3}$ synthesis
(b) Ostwald's process (ii) Aluminium extraction
(c) Contact process (iii) $NH_{3}$ synthesis
(d) Hall-Heroult process (iv) $H_{2}SO_{4}$ synthesis
Choose the correct answer from the options given below:
a.
$(a) -(ii), (b) -(iii), (c) -(iv), (d) -(i)$
b.
$(a) -(iii), (b) -(iv), (c) -(i), (d) -(ii)$
c.
$(a) -(iii), (b) -(i), (c) -(iv), (d) -(ii)$
d.
$(a) -(iv), (b) -(i), (c) -(ii), (d) -(iii)$
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Given below are two statement : one is labelled as Assertion A and the other is labelled as Reason R :
Assertion A : Size of $Bk^{3+}$ ion is less than $Np^{3+}$ ion.
Reason R : The above is a consequence of the lanthanide contraction.
In the light of the above statements, choose the correct answer from the options given below :
a.
A is false but R is true
b.
Both A and R are true but R is not the correct explanation of A
c.
A is true but R is false
d.
Both A and R are true and R is the correct explanation of A
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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 $H -O -H$ bond angle in water molecule is $104. 5^{\circ}$ .
Reason R : The lone pair $-$ lone pair repulsion of electrons is higher than the bond pair $-$ bond pair repulsion.
a.
A is false but R is true
b.
Both A and R are true, but R is not the correct explanation of A
c.
A is true but R is false
d.
Both A and R are true, and R is the correct explanation of A
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$AB_{2}$ is $10%$ dissociated in water to $A^{2+}$ and $B^{-}$ . The boiling point of $10. 0$ molal aqueous solution of $AB_{2}$ is--
$^{\circ }C$ . (Round off to the Nearest Integer).
[Given : Molal elevation constant of water $K_{b} = 0. 5\,K\, kg \, mol^{-1}$ boiling point of pure water $= 100^{\circ}C$ ]
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$2 MnO^{-}_{4} + bC_{2}O^{2-}_{4} + cH^{+} \rightarrow xMn\rightarrow^{2+} + yCO_{2}+ zH_{2}O$
If the above equation is balanced with integer coefficients, the value of $c$ is _______ . (Round off to the
Nearest Integer).
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Complete combustion of $750\,g$ of an organic compound provides $420\,g$ of $CO_{2}$ and $210\,g$ of $H_{2}O$ . The
percentage composition of carbon and hydrogen in organic compound is $15. 3$ and ______ respectively.
(Round off to the Nearest Integer)
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The equivalents of ethylene diamine required to replace the neutral ligands from the coordination sphere of the
trans-complex of $CoCl_{3}. 4 NH_{3}$ is _______ . (Round off to the Nearest Integer).
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A $6. 50$ molal solution of $KOH(aq. )$ has a density of $1. 89\,g\, cm^{-3}$ . The molarity of the solution is $mol \, dm^{-3}$ .
(Round off to the Nearest Integer).
[Atomic masses: $K : 39. 0\,u; \,O : 16. 0\,u; \,H : 1. 0\,u$ ]
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For the reaction $A(g) \rightleftharpoons B(g)$ at $495\,K$ , $\Delta _{r}G^{^{\circ}} = -9. 478\, kJ \, mol^{-1}$
If we start the reaction in a closed container at $495\,K$ with $22$ millimoles of $A$ , the amount of $B$ is the
equilibrium mixture is ________ millimoles. (Round off to the Nearest Integer).
$[R = 8. 314\,J\, mol^{-1} \,K^{-1}; \ell n10 = 2. 303]$
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Given below are two statements:
Statement I : $H_{2}O_{2}$ can act as both oxidising and reducing agent in basic medium.
Statement II : In the hydrogen economy, the energy is transmitted in the form of dihydrogen. In the light of the
above statements, choose the correct answer from the options given below:
a.
Both statement I and statement II are false
b.
Both statement I and statement II are true
c.
Statement I is true but statement II is false
d.
Statement I is false but statement II is true
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Match List-I with List-II
List-I List-II
Name of oxo acid Oxidation state of ' $P$
(a) Hypophosphorous acid (i) $+5$
(b) Orthophosphoric acid (ii) $+4$
(c) Hypophosphoric acid (iii) $+3$
(d) Orthophosphorous acid (iv) $+2$
(v) $+1$
Choose the correct answer from the options given below :
Given below are two statements:
Statement I : Both $CaCl_{2}. 6H_{2}O$ and $MgCl_{2}. 8H_{2}O$ undergo dehydration on heating.
Statement II : $BeO$ is amphoteric, whereas the oxides of other elements in the same group are acidic.
In the light of the above statements, choose the correct answer from the options given below:
a.
Statement I is false but statement II is true
b.
Both statement I and statement II are false
c.
Both statement I and statement II are true
d.
Statement I is true but statement II is false
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A certain element crystallises in a bcc lattice of unit cell edge length 27 $A^{o}$ . If the same element under the same
conditions crystallises in the $fcc$ lattice, the edge length of the unit cell in $A^{o}$ will be __________. (Round off to
the Nearest Integer).
[Assume each lattice point has a single atom]
[Assume $\sqrt{3}= 1. 73, \, = 1. 41\sqrt{2}$ ]
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Given below are two statements :
Statement I : The $E^{\circ}$ value of $Ce^{4+} / Ce^{3+}$ is $+1. 74\,V$
Statement II : Ce is more stable in $Ce^{4+}$ state than $Ce^{3+}$ state.
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.
Statement I is incorrect but statement II is correct
c.
Both statement I and statement II are incorrect
d.
Statement I is correct but statement II is incorrect
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Assertion A : Enol form of acetone $[CH_{3}COCH_{3}]$ exists in $< 0. 1%$ quantity. However, the enol form of
acetyl acetone $[CH_{3}COCH_{2}OCCH_{3}]$ exists in approximately $15%$ quantity.
Reason R : enol form of acetyl acetone is stabilized by intramolecular hydrogen bonding, which is not possible
in enol form of acetone.
Choose the correct statement:
a.
A is false but R is true
b.
Both A and R are true and R is the correct
explanation of A
c.
Both A and R are true but R is not the correct
explanation of A
d.
A is true but R is false
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When light of wavelength $248\, nm$ falls on a metal of threshold energy $3. 0\, eV$ , the de-Broglie wavelength of
emitted electrons is ________ $A^{o}$ . (Round off to the Nearest Integer).
[Use :
$\sqrt{3}= 1. 73, \,h = 6. 63 \times 10^{-34}\, Js; \,m_{e} = 9. 1 \times 10^{-31}\, kg; \,c = 3. 0 \times 10^{8}\, ms^{-1}; \,1\,eV = 1. 6 \times 10^{-19}\,J]$
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The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at $300\,K$ is
$1. 0 \times 10^{-3}\,s^{-1}$ and the activation energy $E_{a} = 11. 488\, kJ \, mol^{-1}$ , the rate constant at $200\,K$ is _________
$\times 10^{-5}\,s^{-1}$ . (Round of to the Nearest Integer).
(Given $R = 8. 314\,J\, mol^{-1}\,K^{-1}$ )
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If $y=y(x)$ is the solution of the differential equation, $\frac{dy}{dx} + 2ytanx= sinx,y(\frac{\pi}{3})= 0$ , then the maximum
value of the function $y(x)$ over $R$ is equal to :
a.
$8$
b.
$\frac{1}{2}$
c.
$-\frac{15}{4}$
d.
$\frac{1}{8}$
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Let a complex number $z, \,|z| \neq 1$ , satisfy $log\frac{1}{\sqrt{2}} (\frac{|z|+11}{(|z|-1)^{2}})\leq 2$ . Then, the largest value of $|z|$ is equal to
_________.
a.
$8$
b.
$7$
c.
$6$
d.
$5$
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Let $f : (0, 2) R$ be defined as $f(x) = log_{2}(1 + tan(\frac{\pi x}{4}))$ .
Then, $_{n}lim_{\rightarrow \infty \rightarrow } \frac{2}{n}(\vec{f}(\frac{1}{n})+f( \frac{2}{n})+ … . +f(1))$ is equal to ________.
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Let a vector $\alpha$ $ˆi +$ $\beta$ $ˆj$ be obtained by rotating the vector $\sqrt{3}ˆi +ˆj$ by an angle $45^{\circ}$ about the origin in
counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices
$(\alpha ,\beta ), (0,\beta )$ and $(0, 0)$ is equal to
a.
$\frac{1}{2}$
b.
$1$
c.
$\frac{1}{\sqrt{2}}$
d.
$2\sqrt{2}$
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Let the curve $y=y(x)$ be the solution of the differential equation, $\frac{dy}{dx} = 2(x+ 1)$ . If the numerical value of
area bounded by the curve $y=y(x)$ and $x$ -axis is $\frac{4\sqrt{8}}{3}$ , then the value of $y(1)$ is equal to ________.
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Let $S_{k} =\sum ^{k}_{r=1} tan^{-1}(\frac{6^{r}}{2^{2}^{r}^{+1}+3^{2}^{r}^{+1}}),$ then $_{k}lim_{\rightarrow \infty \rightarrow } S_{k}$ is equal to :
a.
$tan^{-1}(\frac{3}{2})$
b.
$\frac{\pi}{2}$
c.
$cot^{-1}(\frac{3}{2} )$
d.
$tan^{-1}(3)$
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If for $a> 0$ , the feet of perpendiculars from the points $A(a, -2a, 3)$ and $B(0, 4, 5)$ on the plane
$lx+my+nz= 0$ are points $C(0, -a, -1)$ and $D$ respectively, then the length of line segment $CD$ is equal to
:
a.
$\sqrt{31}$
b.
$\sqrt{41}$
c.
$\sqrt{55}$
d.
$\sqrt{66}$
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A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The
probability that the missing card is not a spade, is :
a.
$\frac{3}{4}$
b.
$\frac{52}{867}$
c.
$\frac{39}{50}$
d.
$\frac{22}{425}$
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Let the position vectors of two points $P$ and $Q$ be $3ˆi -ˆj + 2ˆk$ and $ˆi + 2ˆj -4ˆk$ , respectively. Let $R$ and $S$ be
two points such that the direction ratios of lines $PR$ and $QS$ are $(4, -1, 2)$ and $(-2, 1, -2)$ , respectively. Let
$---$
lines $PR$ and $QS$ intersect at $T$ . If the vector $T\vec{A}$ is perpendicular to both $P\vec{R}$ and $Q\vec{S}$ and the length of vector
$-$
$T\vec{A}$ is $\sqrt{5}$ units, then the modulus of a position vector of $A$ is :
a.
$\sqrt{482}$
b.
$\sqrt{171}$
c.
$\sqrt{5}$
d.
$\sqrt{227}$
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$||$
Let $ABCD$ be a square of side of unit length. Let a circle $C_{1}$ centered at $A$ with unit radius is drawn. Another
circle $C_{2}$ which touches $C_{1}$ and the lines $AD$ and $AB$ are tangent to it, is also drawn. Let a tangent line from
the point $C$ to the circle $C_{2}$ meet the side $AB$ at $E$ . If the length of $EB$ is $\alpha + \sqrt{3}\beta$ , where $\alpha ,\beta$ are integers,
then $\alpha +\beta$ is equal to ________.
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Let $P$ be a plane $lx+my+nz= 0$ containing the line, $\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}$ . If plane $P$ divides the line
segment $AB$ joining points $A(-3, -6, 1)$ and $B(2, 4, -3)$ in ratio $k: 1$ then the value of $k$ is equal to :
a.
$1. 5$
b.
$3$
c.
$2$
d.
$4$
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The total number of $3 \times 3$ matrices $A$ having enteries from the set $(0, 1, 2, 3)$ such that the sum of all the
diagonal entries of $AA^{T}$ is $9$ , is equal to
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Let $f :R\rightarrow \rightarrow R$ be a continuous function such that $f(x) +f(x+ 1) = 2$ for all $x\in R$ . If $I_{1} =\int _{0}^{8} f(x)dx$ and
$I_{2} =\int _{-1}^{3} f(x)dx$ , then the value of $I_{1}+ 2I_{2}$ is equal to ________.
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Let $[x]$ denote greatest integer less than or equal to $x$ . If for $n\in N,(1 - x+x^{3})^{n} =\sum ^{3}_{j=0}^{n} a_{j}x^{j}$ , then
$\sum _{j}^{[}_{=0}\frac{3n}{2}^{]} a_{2j}+ 4\sum ^{[}_{j=0}\frac{3n-1}{2}^{]} a_{2j+1}$ is equal to :
a.
$2$
b.
$2^{n-1}$
c.
$1$
d.
$n$
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$-^{Let}z^{and}w^{be two complex numbers such that}w=zz-2z+ 2, \frac{z+i}{z-3i} = 1^{and}Re (w)^{has minimum value.}$
Then, the minimum value of $n\in N$ for which $w^{n}$ is real, is equal to _______.
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The range of $a\in R$ for which the function
$f(x) = (4a-3)(x+ log_{e}5) + 2(a-7) cot( \frac{x}{2} )sin^{2}(\frac{x}{2}),x\neq 2n\pi ,n\in N$ , has critical points, is :
a.
$(-3, 1)$
b.
$[-\frac{4}{3} , 2]$
c.
$[1, \infty )$
d.
$(-\infty , -1]$
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The locus of the midpoints of the chord of the circle, $x^{2} +y^{2} = 25$ which is tangent to the hyperbola,
$\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$ is :
Consider an arithmetic series and a geometric series having four initial terms from the set
${11, 8, 21, 16, 26, 32, 4}$ . If the last terms of these series are the maximum possible four digit numbers, then the
number of common terms in these two series is equal to _______.
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$-30 20 56 2 7 \omega ^{2}$
$[ ] [ ]$
Let $\vec{P} = 90 140 112$ and $A= -1 -\omega 1$ where $\omega = \frac{-1+i\sqrt{3}}{2}$ , and $I_{3}$ be the identity matrix
$[ ] [ ]$
$120 60 14 0 -\omega -\omega + 1$
of order $3$ . If the determinant of the matrix $(P^{-1}AP - I_{3})^{2}$ is $\alpha \omega ^{2}$ , then the value of $\alpha$ is equal to _________.
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Let the functions $f :R\rightarrow \rightarrow R$ and $g:R R$ be defined as :
$x+ 2, x< 0 x^{3}, x< 1$
$f(x) ={$ and $g(x) ={$
$x^{2}, x\geq 0 3\vec{x}-2, x\geq 1$
Then, the number of points in $R$ where $(fog)(x)$ is NOT differentiable is equal to :
a.
$3$
b.
$1$
c.
$0$
d.
$2$
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If for $x\in (0, \frac{\pi}{2}), log_{10} sinx+ log_{10} cosx= -1$ and $log_{10}(sinx+ cosx) = \frac{1}{2}(log_{10} n-1),n> 0$ , then the
value of $n$ is equal to :
a.
$20$
b.
$12$
c.
$9$
d.
$16$
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If the normal to the curve $y(x) =\int _{0}^{x}(2t^{2}-15t+ 10)dt$ at a point $(a,b)$ is parallel to the line
$x+ 3y= -5,a> 1$ , then the value of $|a+ 6b|$ is equal to ________.
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