The wavelength of the radiation emitted is $\lambda _{0}$ when an electron jumps from the second excited state to the first
excited state of hydrogen atom. If the electron jumps from the third excited state to the second orbit of the
hydrogen atom, the wavelength of the radiation emitted will be $\frac{20}{x}\lambda _{0}$ . The value of $x$ is________.
✓ Accepted Answer:10
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Q2mcq
A Carnot engine with efficiency $50%$ takes heat from a source at $600\,K$ . In order to increase the efficiency to
$70%$ , keeping the temperature of sink same, the new temperature of the source will be:
a.
$360\,K$
b.
$1000\,K$
c.
$900\,K$
Correct Answer
d.
$300\,K$
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Q3mcq
If $\vec{P} = 3ˆi + \sqrt{3}ˆj + 2ˆk$ and $\vec{Q}= 4ˆi + \sqrt{3}ˆj + 2. 5ˆk$ then, the unit vector in the direction of $\vec{P} \times \vec{Q}$ is
$\frac{1}{x}(\sqrt{3}ˆi +ˆj -2\sqrt{3}ˆk)$ . The value of $x$ is
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Q4mcq
The root mean square velocity of molecules of gas is
a.
Proportional to square of temperature ( $T^{2}$ ).
b.
Inversely proportional to square root
of temperature ( $\sqrt{\frac{1}{T}}$ ).
Correct Answer
c.
Proportional to square root of temperature $\sqrt{T}$ .
d.
Proportional to temperature $(T)$ .
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Q5mcq
A bowl filled with very hot soup cools from $98^{\circ}C$ to $86^{\circ }C$ in 2 minutes when the room temperature is $22^{\circ }C$ .
How long it will take to cool from $75^{\circ }C$ to $69^{\circ }C$ ?
a.
$2$ minute $1. 4$ minute
c.
$0. 5$ minute
d.
$1$ minute
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Q6mcq
A solenoid of $1200$ turns is wound uniformly in a single layer on a glass tube $2\,m$ long and $0. 2\,m$ in diameter.
The magnetic intensity at the center of the solenoid when a current of $2\,A$ flows through it is:
a.
$2. 4 \times 10^{3}\,A\,m^{-1}$
b.
$1. 2 \times 10^{3}\,A\,m^{-1}$
c.
$1\,A\,m^{-1}$
Correct Answer
d.
$2. 4 \times 10^{-3}\,A\,m^{-1}$
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Q7mcq
Match List I with List II
List - I List - II
(Current configuration) (Magnetic field at point $O$ )
A I. $B_{0} = \frac{\mu _{0}I}{4\pi r}[\pi + 2]$
B II. $B_{0} = \frac{\mu _{0}}{4} \frac{I}{r}$
C III. $B_{0} = \frac{\mu _{0}I}{2\pi r}[\pi -1]$
D IV. $B_{0} = \frac{\mu _{0}I}{4\pi r}[\pi + 1]$
Choose the correct answer from the option given below:
a.
A-III, B-IV, C-I, D-II
b.
A-I, B-III, C-IV, D-II
c.
A-III, B-I, C-IV, D-II
Correct Answer
d.
A-II, B-I, C-IV, D-III
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Q8mcq
Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the
earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass
of the particle is $100\,g$ . The time period of the motion of the particle will be (approximately)
(take $g= 10\, ms^{-2}$ , radius of earth $= 6400\, km$ )
a.
$24$ hours
b.
$1$ hour $24$ minutes
Correct Answer
c.
$1$ hour $40$ minutes
d.
$12$ hours
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Q9nat
All electromagnetic wave is transporting energy in the negative $z$ direction. At a certain point and certain time
the direction of electric field of the wave is along positive $y$ direction. What will be the direction of the
magnetic field of the wave at that point and instant?
✓ Accepted Answer:17
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Q10mcq
An object of mass $m$ initially at rest on a smooth horizontal plane starts moving under the action of force
$F = 2\,N$ . In the process of its linear motion, the angle $\theta$ (as shown in figure) between the direction of force and
horizontal varies as $\theta =kx$ , where $k$ is a constant and $x$ is the distance covered by the object from its initial
position. The expression of kinetic energy of the object will be $E = \frac{n}{k}sin \,\theta$ . The value of $n$ is ______.
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A uniform metallic wire carries a current $2\,A$ , when $3. 4\,V$ battery is connected across it. The mass of uniform
metallic wire is $8. 92 \times 10^{-3}\, kg$ , density is $8. 92 \times 10^{3}\, kg \,m^{-3}$ and resistivity is $1. 7 \times 10^{-8}\,\Omega -m$ The
length of wire is :
a.
$l= 6. 8\,m$
b.
$l= 10\,m$
c.
$l= 5\,m$
d.
$l= 100\,m$
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A ray of light is incident from air on a glass plate having thickness $\sqrt{3}\, cm$ and refractive index $\sqrt{2}$ . The angle
of incidence of a ray is equal to the critical angle for glass-air interface. The lateral displacement of the ray
when it passes through the plate is $____\times 10^{-2} cm$ . (given $sin \,15^{\circ}= 0. 26$ )
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As shown in the figure, in an experiment to determine Young's modulus of a wire, the extension-load curve is
plotted. The curve is a straight line passing through the origin and makes an angle of $45^{\circ}$ with the load axis.
The length of wire is $62. 8\, cm$ and its diameter is $4\, mm$ . The Young's modulus is found to be $x \times 10^{4}\,N\,m^{-2}$
The value of $x$ is _______.
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In an LC oscillator, if values of inductance and capacitance become twice and eight times, respectively, then
the resonant frequency of oscillator becomes $x$ times its initial resonant frequency $\omega _{0}$ . The value of $x$ is:
a.
$^{1}/_{4}$
b.
$16$
c.
$^{1}/_{16}$
d.
$4$
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A parallel plate capacitor has plate area $40\, cm^{2}$ and plates separation $2\, mm$ . The space between the plates is
filled with a dielectric medium of a thickness $1\, mm$ and dielectric constant $5$ . The capacitance of the system is:
a.
$24\epsilon _{0}\,F$
b.
$\frac{3}{10} \epsilon _{0}\,F$
c.
$\frac{10}{3}\epsilon _{0}\,F$
d.
$10\epsilon _{0}\,F$
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A car travels a distance of $x$ with speed $v_{1}$ and then same distance $x$ with speed $v_{2}$ in the same direction. The
average speed of the car is:
a.
$\frac{vv}{2(v_{1}+v_{2})}^{1}^{2}$
b.
$\frac{v+v}{2}^{1}^{2}$
c.
$\frac{2x}{v_{1}+v_{2}}$
d.
$\frac{2v_{1}v_{2}}{v_{1}+v_{2}}$
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Electron beam used in an electron microscope, when accelerated by a voltage of $20\, kV$ has a de-Broglie
wavelength of $\lambda _{0}$ . If the voltage is increased to $40\, kV$ then the de-Broglie wavelength associated with the
electron beam would be:
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In Young's double slits experiment, the position of 5thbright fringe from the central maximum is $5\, cm$ . The
distance between slits and screen is $1\,m$ and wavelength of used monochromatic light is $600\, nm$ . The
separation between the slits is:
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An object of mass $8\, kg$ is hanging from one end of a uniform rod $CD$ of mass $2\, kg$ and length $1\,m$ pivoted at
its end $C$ on a vertical wall as shown in figure. It is supported by a cable $AB$ such that the system is in
equilibrium. The tension in the cable is:
(Take $g= 10\,m\,s^{-2}$ )
a.
$240\,N$
b.
$90\,N$
c.
$300\,N$
d.
$30\,N$
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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: Photodiodes are used in forward bias usually for measuring the light intensity.
Reason R:For a $p- n$ junction diode, at applied voltage $V$ the current in the forward bias is more than the
current in the reverse bias for $|V_{z}| > \pm V \geq |V_{0}|$ where $V_{0}$ is the threshold voltage and $V_{z}$ is the breakdown
voltage.
In the light of the above statements, choose the correct answer from the options given below
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$I_{CM}$ is moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to
the plane of disc. $I_{AB}$ is its moment of inertia about an axis AB perpendicular to plane and parallel to axis CM at
a distance $\frac{2}{3} R$ from center, where R is the radius of the disc. The ratio of $I_{AB}$ and $I_{CM}$ is $x: 9$ . The value of $x$ is
______.
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$T$ is the time period of simple pendulum on the earth's surface. Its time period becomes $xT$ when taken to a
height $R$ (equal to earth's radius) above the earth's surface. Then, the value of $x$ will be:
a.
$4$
b.
$2$
c.
$\frac{1}{2}$
d.
$\frac{1}{4}$
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A car is moving with a constant speed of $20\,m\,s^{-1}$ in a circular horizontal track of radius $40\,m$ . A bob is
suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be :
(Take $g = 10\,m\,s^{-2}$ )
a.
$\frac{\pi}{6}$
b.
$\frac{\pi}{2}$
c.
$\frac{\pi}{4}$
d.
$\frac{\pi}{3}$
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The distance between two consecutive points with phase difference of $60^{\circ}$ in a wave of frequency $500\, Hz$ is
$6. 0\,m$ . The velocity with which wave is travelling is ______ $km \,s^{-1}$ .
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An LCR series circuit of capacitance $62. 5\, nF$ and resistance of $50\,\Omega$ , is connected to an A.C. source of
frequency $2. 0\, kHz$ . For maximum value of amplitude of current in circuit, the value of inductance is ____ $mH$ .
$(Take\,$ $\pi$ $^{2} = 10)$
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Match List I with List II
List - I List - II
A Surface tension I. $kg \,m^{-1}\,s^{-1}$
B Pressure II. $kg \,m\,s^{-1}$
C Viscosity III. $kg \,m^{-1}\,s^{-2}$
D Impulse IV. $kg \,s^{-2}$
Choose the correct answer from the options given below:
a.
A-IV, B-III, C-II, D-I
b.
A-IV, B-III, C-I, D-II
c.
A-III, B-IV, C-I, D-II
d.
A-II, B-I, C-III, D-IV
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A uniform electric field of $10\,N\,C^{-1}$ is created between two parallel charged plates (as shown in figure). An
electron enters the field symmetrically between the plates with a kinetic energy $0. 5\, eV.$ The length of each
plate is $10\, cm$ . The angle $(\theta )$ of deviation of the path of electron as it comes out of the field is ______(in
degree).
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In sulphur estimation. $0. 471\,g$ of an organic compound gave $1. 4439\,g$ of barium sulphate. The percentage of
sulphur in the compound is _______(Nearest Integer)
(Given: Atomic mass Ba: $137u : S : 32u, O : 16u$ )
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Consider the cell $Pt(s)|H_{2}(\,s)(latm)|H^{+}(aq, [H^{+}] = 1) Fe^{3+}(aq), Fe^{2+}(aq)|Pt(s)$
Given: $E_{Fe^{3+}/ Fe^{2+}}^{\circ}= 0. 771\,V$ and $E_{H^{+}/}\frac{1}{2}_{H_{2}}^{\circ}= 0\,V, \,T = 298\,K$
If the potential of the cell is $0. 712\,V$ the ratio of concentration of $Fe^{2+}$ to $Fe^{3+}$ is
(Nearest integer)
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A cubic solid is made up of two elements $X$ and $Y$ . Atoms of $X$ are present on every alternate corner and one at
the center of cube. $Y$ is at $\frac{1}{3}^{rd}$ of the total faces. The empirical formula of the compound is
a.
$X_{2}Y_{1.5}$
b.
$X_{1.5}Y$
c.
$XY_{2.5}$
d.
$X_{1.5}Y_{2}$
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How many of the following metal ions have similar value of spin only magnetic moment in gaseous state?
________
(Given: Atomic number $: V, 23; \, Cr, 24; \, Fe, 26; \, Ni, 28)$
$V^{3+}, \, Cr^{3+}, \, Fe^{2+}, \, Ni^{3+}$
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The osmotic pressure of solutions of PVC in cyclohexanone at $300\,K$ are plotted on the graph. The molar mass
of PVC is $g\, mol^{-1}$ (Nearest integer)
(Given : $R = 0. 083\,L\, atm \,K^{-1}\, mol^{-1}$ )
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The number of paramagnetic species from the following is
$[Ni(C\vec{N})_{4}]^{2-}, [\vec{N}i(CO)_{4}], [NiCl_{4}]^{2-}$
$[Fe(CN)_{6}]^{4-}, [Cu (NH_{3})_{4}]^{2+}$
$[Fe(CN)_{6}]^{3-}\,and\,[Fe (H_{2}O)_{6}]^{2+}$
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An athlete is given $100\,g$ of glucose $(C_{6}H_{12}O_{6})$ for energy. This is equivalent to $1800\, kJ$ of energy. The $50%$
of this energy gained is utilized by the athlete for sports activities at the event. In order to avoid storage of
energy, the weight of extra water he would need to perspire is $_____g$ (Nearest integer) Assume that there is no
other way of consuming stored energy.
Given : The enthalpy of evaporation of water is $45\, kJ \, mol^{-1}$
Molar mass of $C, H&O$ are $12.1$ and $16\,g\, mol^{-1}$ .
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Reaction of thionyl chloride with white phosphorus forms a compound $[A]$ , which on hydrolysis gives $[B]$ , a
dibasic acid. $[A]$ and $[B]$ are respectively
a.
$P_{4}O_{6}$ and $H_{3}PO_{3} PCl_{3}$ and $H_{3}PO_{3}$
c.
$PCl_{5}$ and $H_{3}PO_{4}$
d.
$POCl_{3}$ and $H_{3}PO_{4}$
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A litre of buffer solution contains $0. 1$ mole of each of $NH_{3}$ and $NH_{4}Cl$ . On the addition of $0. 02$ mole of $HCl$
by dissolving gaseous $HCl$ , the $pH$ of the solution is found to be $_____\times 10^{-3}$ (Nearest integer)
Given: $pK_{b}(NH_{3}) = 4. 745$
$log \,2 = 0. 301$
$log \,3 = 0. 477$
$T = 298\,K]$
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Match the List-I with List-II :
Cations Group reaction
$P \rightarrow Pb\rightarrow^{2+}, Cu^{2+} H_{2}\,S$ gas in presence of dilute $HCl$
$Q \rightarrow Al\rightarrow^{3+}, Fe^{3+} (NH_{4})_{2}CO_{3}$ in presence of $NH_{4}OH$
$R Co^{2+}, Ni^{2+} NH_{4}OH$ in presence of $NH_{4}CI$
$S \rightarrow Ba\rightarrow^{2+}, Ca^{2+} H_{2}\,S$ in presence of $NH_{4}OH$
a.
$\vec{P}\rightarrow i, Q \rightarrow iii, R \rightarrow ii, S \rightarrow iv\rightarrow \rightarrow \rightarrow \rightarrow$
b.
$P \rightarrow iv, Q \rightarrow ii, R \rightarrow iii, S \rightarrow i\rightarrow \rightarrow \rightarrow \rightarrow$
c.
$P iii, Q i, R \rightarrow iv, S \rightarrow ii\rightarrow \rightarrow$
d.
$P \rightarrow i, Q \rightarrow iii, R \rightarrow iv, S \rightarrow ii\rightarrow \rightarrow \rightarrow \rightarrow$
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Match items of Row I with those of Row II.
Row I:
(P)
(Q)
(R)
(S)
Row II:
(i) $\alpha$‐ $D$ ‐ $(-)$ Fructofuranose.
(ii) $\beta$‐ $D$ ‐ $(-)$ Fructofuranose.
(iii) $\alpha$‐ $D$ ‐ $($ ‐ $)$ Glucopyranose.
(iv) $\beta$‐ $D$ ‐ $($ ‐ $)$ Glucopyranose.
Correct match is
Given below are two statements: one is labelled as Assertion Aand the other is labelled as Reason R:
Assertion A:Acetal/Ketal is stable in basic medium.
Reason R:The high leaving tendency of alkoxide ion gives the stability to acetal/ketal in basic medium.
In the light of the above statements, choose the correct answer from the options given below:
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Compound A reacts with $NH_{4}Cl$ and forms a compound B. Compound B reacts with $H_{2}O$ and excess of $CO_{2}$
to form compound C which on passing through or reaction with saturated NaCl solution forms sodium
hydrogen carbonate. Compound A. B and C, are respectively.
a.
$CaCl_{2}, NH_{3}, NH_{4}HCO_{3}$
b.
$CaCl_{2}, NH^{+}_{4} , (NH_{4})_{2} CO_{3}$
c.
$Ca(OH)_{2}, NH_{3}, NH_{4}HCO_{3}$
d.
$Ca(OH)_{2}, NH^{+}_{4} , (NH_{4})_{2} CO_{3}$
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Match List I with List II
List I List II
Elements Colour imparted to the flame
A $K$ I Brick Red
B $Ca$ II Violet
C $Sr$ III Apple Green
D $Ba$ IV Crimson Red
Choose the correct answer from the options given below:
a.
A-II, B-I, C-III, D-IV
b.
A-II, B-IV, C-I, D-III
c.
A-II, B-I, C-IV, D-III
d.
A-IV, B-III, C-II, D-I
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The density of a monobasic strong acid (Molar mass $24. 2\,g\, mol)$ is $1. 21\, kg \,L$ . The volume of its solution
required for the complete neutralization of $25\, mL$ of $0. 24\,M\, NaOH$ is $10^{-2}\, mL$ (Nearest integer)
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For the first order reaction $A \rightarrow B\rightarrow$ the half life is $30\, min$ . The time taken for $75%$ completion of the reaction
is mm. (Nearest integer)
Given : $log \,2 = 0. 3010$
$log \,3 = 0. 4771$
$log \,5 = 0. 6989$
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Let $y = y(x)$ be the solution curve of the differential equation $\frac{dy}{dx} = \frac{y}{x}(1 +x^{2}(1 + log_{e} x))$ , $x > 0, y(1) = 3$ .
Then $\frac{y^{2}(x)}{9}$ is equal to :
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If the sum of all the solutions of
$tan^{-1}(\frac{2x}{1-x^{2}})+ cot^{-1}(\frac{1-x^{2}}{2x})= \frac{\pi}{3}$ , $-1 <x< 1,x\neq 0$ , is $\alpha - \frac{4}{\sqrt{3}}$ , then $\alpha$ is equal to _____ .
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Let M be the maximum value of the product of two positive integers when their sum is $66$ . Let the sample
space $S ={x\in Z :x(66 - x)\geq \frac{5}{9}M}$ and the event $A = {x \in S : x$ is a multiple of $3$ }. Then $P(A)$ is equal
to
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Let
$S ={\alpha : log_{2}(9^{2\alpha -4} + 13)-log_{2}( \frac{5}{2} \cdot 3^{2\alpha -4}+ 1)= 2}.$ Then the maximum value of $\beta$ for which the
equation $x^{2} -2(\sum _{\alpha \in s}\alpha )^{2}x +\sum _{a\in s}(\alpha + 1)^{2}\beta = 0$ has real roots, is _____ .
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If the area enclosed by the parabolas $P_{1} : 2y= 5x^{2}$ and $P_{2} :x^{2} - y+ 6 = 0$ is equal to the area enclosed by
$P_{1}$ and $y=\alpha x,\alpha > 0$ , then $\alpha ^{3}$ is equal to _____ .
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The mean and variance of the marks obtained by the students in a test are $10$ and $4$ respectively. Later, the
marks of one of the students is increased from $8$ to $12$ . If the new mean of the marks is $10. 2$ . then their new
variance is equal to:
a.
$4. 04$
b.
$4. 08$
c.
$3. 96$
d.
$3. 92$
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The vertices of a hyperbola $H$ are $(\pm 6, 0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$ . Let $N$ be the normal to $H$ at a point in the
first quadrant and parallel to the line $\sqrt{2}x+y= 2\sqrt{2}$ . If $d$ is the length of the line segment of $N$ between $H$
and the $y$ -axis then $d^{2}$ is equal to _____ .
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For some $a,b,c\in N$ , let $f(x) =ax-3$ and $g(x) =x^{b} +c,x\in R$ . If $(fog)^{-1}\,(x) =(\frac{x-7}{2} ) \frac{1}{3}$ , then
$(f^{\circ} g)(ac) + (g^{\circ} f)(b)$ is equal to _____ .
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The points of intersection of the line $ax\, + \,by= 0$ , $(a \neq b)$ and the circle $x^{2} + y^{2} -2x = 0$ are $A(\alpha , 0)$ and
$B(1,\beta )$ . The image of the circle with $AB$ as a diameter in the line $x + y + 2 = 0$ is :
a.
$x^{2} +y^{2} + 5x+ 5y+ 12 = 0$
b.
$x^{2} +y^{2} + 3x+ 5y+ 8 = 0$
c.
$x^{2} +y^{2} + 3x+ 3y+ 4 = 0$
d.
$x^{2} +y^{2} -5x-5y+ 12 = 0$
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Consider the lines $L_{1}$ and $L_{2}$ given by
$L_{1} : \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-2}{2}$
$L_{2} : \frac{x-2}{1} = \frac{y-2}{2} = \frac{z-3}{3}$
A line $L_{3}$ having direction ratios $1, -1, -2$ , intersects $L_{1}$ and $L_{2}$ at the points $P$ and $Q$ respectively. Then the
length of line segment $PQ$ is
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$||$
Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a\in R-{0}$ for which the system of linear equations
$ax+ 2ay-3az= 1$
$(2a+ 1)\,x+ (2a+ 3)\,y+ (a+ 1)z= 2$
$(3a+ 5)\,x+ (a+ 5)\,y+ (a+ 2)\,z= 3$
has unique solution and infinitely many solutions. Then
a.
$n(S_{1}) = 2$ and $S_{2}$ is an infinite set $S_{1}$ is an infinite set an $n(S_{2}) = 2$
c.
$S_{1} =\phi$ and $S_{2} =R-{0}$
d.
$S_{1} =R-{0}$ and $S_{2} =\phi$
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The distance of the point $P(4, 6, -2)$ from the line passing through the point $(-3, 2, 3)$ and parallel to a line
with direction ratios $3, 3, -1$ is equal to:
a.
$3$
b.
$\sqrt{6}$
c.
$2\sqrt{3}$
d.
$\sqrt{14}$
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Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b}\cdot \vec{c}= 0$ and $\vec{a}\times (\vec{b}\times \vec{c})= \frac{\vec{b}-\vec{c}}{2}$ . If be a vector such that $\vec{d}$
$\vec{b}\cdot \vec{d}= \vec{a}\cdot \vec{b}$ , then $(\vec{a}\times \vec{b})\cdot (\vec{c}\times \vec{d})$ is equal to
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The vector $\vec{a}= -ˆi+ 2ˆj+ˆk$ is rotated through a right angle, passing through the $y$ -axis in its way and the
resulting vector is $\vec{b}$ . Then the projection of $3\vec{a}+ \sqrt{2}\vec{b}$ on $\vec{c}= 5ˆi+ 4ˆj+ 3ˆk$ is
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Let $A_{1},A_{2},A_{3}$ be the three A.P. with the same common difference $d$ and having their first terms as
$A,A+ 1, \,A+ 2$ , respectively. Let $a,b,c$ be the $7^{th\,}, 9^{th\,}, 17^{th\,}$ terms of $A_{1}, \,A_{2}, \,A_{3}$ , respectively such that
$a 7 1$
$2b 17 1 + 70 = 0$ . If $a= 29$ , then the sum of first $20$ terms of an AP whose first term is $c- a- b$ and
$c 17 1$
common difference is $\frac{d}{12}$ , is equal to _____ .
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Let $f : (0, 1) \rightarrow \rightarrow R$ be a function defined by $f(x) = \frac{1}{1-e^{-}^{x}}$ , and $g(x) = (f(-x) - f(x))$ . Consider two
statements
(I) $g$ is an increasing function in $(0, 1)$
(II) $g$ is one-one in $(0, 1)$
Then,
a.
Only (I) is true
b.
Only (II) is true
c.
Neither (I) nor (II) is true
d.
Both (I) and (II) are true
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Let the equation of the plane passing through the line
$x-2y- z-5 = 0 =x+y+ 3z-5$ and parallel to the line $x+y+ 2z-7 = 0 = 2x+ 3y+z-2$ be
$ax+by+cz= 65$ . Then the distance of the point $(a, \,b, \,c)$ from the plane $2x+ 2y- z+ 16 = 0$ is _____ .
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Let $x= 2$ be a local minima of the function $f(x) = 2x^{4} -18x^{2} + 8x+ 12,x\in (-4, 4)$ . If $M$ is local
maximum value of the function $f$ in $(-4, 4)$ , then $M =$
a.
$12\sqrt{6}- \frac{33}{2}$
b.
$12\sqrt{6}- \frac{31}{2}$
c.
$18\sqrt{6}- \frac{33}{2}$
d.
$18\sqrt{6}- \frac{31}{2}$
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If $a_{r}$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1 +x)^{10}$ , then $\sum ^{10}_{r=1} r^{3}(\frac{a_{r}}{a_{r}_{-1}} )^{2}$ is equal to
a.
$4895$
b.
$1210$
c.
$5445$
d.
$3025$
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Let $x$ and $y$ be distinct integers where $1 \leq x\leq 25$ and $1 \leq y\leq 25$ . Then, the number of ways of choosing $x$
and $y$ , such that $x+y$ is divisible by $5$ , is _____ .
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