A tennis ball is dropped on to the floor from a height of $9. 8\,m$ . It rebounds to a height $5. 0\,m$ . Ball comes in
contact with the floor for $0. 2\,s$ . The average acceleration during contact is ______ $m\,s^{-2}$ . [Given $g= 10\,m\,s^{-2}$ ]
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Q2nat
Which of the following are true?
A. Speed of light in vacuum is dependent on the direction of propagation.
B. Speed of light in a medium is independent of the wavelength of light.
C. The speed of light is independent of the motion of the source.
D. The speed of light in a medium is independent of intensity.
Choose the correct answer from the question given below :
✓ Accepted Answer:300
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Q3nat
A radioactive element $^{242}_{92}X$ emits two $\alpha$ -particles, one electron and two positrons. The product nucleus is
represented by $^{234}_{P}Y$ . The value of $P$ is ______.
✓ Accepted Answer:10
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Q4nat
The threshold wavelength for photoelectric emission from a material is $5500\,A^{\circ }$ . Photoelectrons will be emitted,
when this material is illuminated with monochromatic radiation from a
A. $75\,W$ infra-red lamp
B. $10\,W$ infra-red lamp
C. $75\,W$ ultra-violet lamp
D. $10\,W$ ultra-violet lamp
Choose the correct answer from the options given below :
✓ Accepted Answer:120
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Q5mcq
A block of mass $m$ slides down the plane inclined at angle $30^{\circ}$ with an acceleration $\frac{g}{4}$ . The value of coefficient
of kinetic friction will be :
a.
$\frac{2+1\sqrt{3}}{2}$
b.
$\frac{1}{2\sqrt{3}}$
c.
$\frac{\sqrt{3}}{2}$
d.
$\frac{2\sqrt{3}-1}{2}$
Correct Answer
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Q6nat
A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to
a uniform magnetic field $B= 0. 8\,T$ . When released the radius of the loop starts shrinking at a constant rate of
$2\, cm \,s^{-1}$ . The induced emf in the loop at an instant when the radius of the loop is $10\, cm$ will be ______ $mV$ .
✓ Accepted Answer:120
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Q7mcq
A car is moving on a horizontal curved road with radius $50\,m$ . The approximate maximum speed of car will be,
if friction between tyres and road is $0. 34$ . [Take $g= 10\,m\,s^{-2}$ ]
a.
$3. 4\,m\,s^{-1}$
b.
$22. 4\,m\,s^{-1}$
Correct Answer
c.
$13\,m\,s^{-1}$
d.
$17\,m\,s^{-1}$
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Q8nat
If the height of transmitting and receiving antennas are $80\,m$ each, the maximum line of sight distance will be :
Given : Earth’s radius $= 6. 4 \times 10^{6}\,m$ .
✓ Accepted Answer:87
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Q9mcq
Two simple harmonic waves having equal amplitudes of $8\, cm$ and equal frequency of $10\, Hz$ are moving along
the same direction. The resultant amplitude is also $8\, cm$ . The phase difference between the individual waves is
_____ degree.
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Q10nat
In a metre bridge experiment the balance point is obtained if the gaps are closed by $2\,\Omega$ and $3\,\Omega$ . A shunt of
$X\,\Omega$ is added to $3\,\Omega$ resistor to shift the balancing point by $22. 5\, cm$ . The value of $X$ is ______.
✓ Accepted Answer:2
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A single current carrying loop of wire carrying current $I$ flowing in anticlockwise direction seen from $+ve z$
direction and lying in $xy$ plane is shown in figure. The plot of $ˆj$ component of magnetic field $(By)$ at a distance
$a$ (less than radius of the coil) and on $yz$ plane $vs z$ coordinate looks like
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A person observes two moving trains, $A$ reaching the station and $B$ leaving the station with equal speed of
$30\,m\,s^{-1}$ . If both trains emit sounds with frequency $300\, Hz$ , (Speed of sound : $330\,m\,s^{-1}$ ) approximate
difference of frequencies heard by the person will be :
a.
$33\, Hz$
b.
$55\, Hz$
c.
$80\, Hz$
d.
$10\, Hz$
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A stone is projected at angle $30^{\circ}$ to the horizontal. The ratio of kinetic energy of the stone at point of projection
to its kinetic energy at the highest point of flight will be :
a.
$1\, : \,2$
b.
$1\, : \,4$
c.
$4\, : \,1$
d.
$4\, : \,3$
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A bicycle tyre is filled with air having pressure of $270\, kPa$ at $27^{\circ}C$ . The approximate pressure of the air in the
tyre when the temperature increases to $36^{\circ}C$ is
a.
$270\, kPa$
b.
$262\, kPa$
c.
$278\, kPa$
d.
$360\, kPa$
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Given below are two statements. One is labelled as Assertion Aand the other is labelled as Reason R.
Assertion A :If $dQ$ and $dW$ represent the heat supplied to the system and the work done on the system
respectively. Then according to the first law of thermodynamics $dQ=dU - dW$ .
Reason R :First law of thermodynamics is based on law of conservation of energy.
In the light of the above statements, choose the correct answer from the option 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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Find the mutual inductance in the arrangement, when a small circular loop of wire of radius $R$ is placed inside
a large square loop of wire of side $L(L\gg R)$ . The loops are coplanar and their centres coincide :
a.
$M = \frac{\sqrt{2}\mu _{0}R^{2}}{L}$
b.
$M = \frac{2\sqrt{2}\mu _{0}R}{L^{2}}$
c.
$M = \frac{2\sqrt{2}\mu _{0}R^{2}}{L}$
d.
$M = \frac{\sqrt{2}\mu _{0}R}{L^{2}}$
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Two particles of equal mass $m$ move in a circle of radius $r$ under the action of their mutual gravitational
attraction. The speed of each particle will be :
a.
$\sqrt{\frac{Gm}{2r}}$
b.
$\sqrt{\frac{4Gm}{r}}$
c.
$\sqrt{\frac{Gm}{r}}$
d.
$\sqrt{\frac{Gm}{4r}}$
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A body cools from $60^{\circ}C$ to $40^{\circ}C$ in $6$ minutes. If, temperature of surroundings is $10^{\circ}C$ . Then, after the next $6$
minutes, its temperature will be ______ $^{\circ}C$ .
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Which of the following statement is not correct in the case of light emitting diodes?
A. It is a heavily doped p-n junction.
B. It emits light only when it is forward biased.
C. It emits light only when it is reverse biased.
D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.
Choose the correct answer from the options given below :
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As shown in figures, three identical polaroids $P_{1}$ , $P_{2}$ and $P_{3}$ are placed one after another. The pass axis of $P_{2}$ and $P_{3}$
are inclined at angle of $60$ ºand $90$ ºwith respect to axis of $P_{1}$ . The source $S$ has an intensity of $256\,W\,m^{-2}$ . The
intensity of light at point $O$ is _____ $W\,m^{-2}$ .
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A solid sphere of mass $2\, kg$ is making pure rolling on a horizontal surface with kinetic energy $2240\,J$ . The
velocity of centre of mass of the sphere will be ______ $m\,s^{-1}$ .
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A point charge $q_{1} = 4q_{0}$ is placed at origin. Another point charge $q_{2} = -q_{0}$ is placed at $x= 12\, cm$ . Charge of
proton is $q_{0}$ . The proton is placed on $x$ -axis so that the electrostatic force on the proton in zero. In this
situation, the position of the proton from the origin is ______ $cm$ .
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In a Young’s double slit experiment, two slits are illuminated with a light of wavelength $800\, nm$ . The line
joining $A_{1}P$ is perpendicular to $A_{1}A_{2}$ as shown in the figure. If the first minimum is detected at $P$ , the value
of slits separation $a$ will be :
The distance of screen from slits $D= 5\, cm$ .
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Surface tension of a soap bubble is $2. 0 \times 10^{-2}\,N\,m^{-1}$ . Work done to increase the radius of soap bubble from
$3. 5\, cm$ to $7\, cm$ will be : [Take $\pi$ $= \frac{22}{7}$ ]
a.
$0. 72 \times 10^{-4}\,J$
b.
$5. 76 \times 10^{-4}\,J$
c.
$18. 48 \times 10^{-4}\,J$
d.
$9. 24 \times 10^{-4}\,J$
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A $0. 4\, kg$ mass takes $8\,s$ to reach ground when dropped from a certain height $P$ above surface of earth. The loss
of potential energy in the last second of fall is ______ $J$ . [Take $g= 10\,m\,s^{-2}$ ]
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Match List I with List II :
List-I (Physical Quantity) List-II (Dimensional Formula)
A Pressure gradient I $[M^{0}L^{2}T^{-2}]$
B Energy density II $[M^{1}L^{-1}T^{-2}]$
C Electric Field III $[M^{1}L^{-2}T^{-2}]$
D Latent heat IV $[M^{1}L^{1}T^{-3}A^{-1}]$
Choose the correct answer from the options given below:
a.
A-III, B-II, C-I, D-IV
b.
A-II, B-III, C-IV, D-I
c.
A-III, B-II, C-IV, D-I
d.
A-II, B-III, C-I, D-IV
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In a cuboid of dimension $2L\times 2L\times L$ , a charge $q$ is placed at the centre of the surface $S$ having area of $4L^{2}$ .
The flux through the opposite surface to $S$ is given by
a.
$\frac{q}{12\in _{0}}$
b.
$\frac{q}{3\in _{0}}$
c.
$\frac{q}{2\in _{0}}$
d.
$\frac{q}{6\in _{0}}$
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If a radioactive element having half-life of $30$ min is undergoing beta decay, the fraction of radioactive element
remains undecayed after $90$ min will be :
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Consider the following reaction approaching equilibrium at $27^{\circ}C$ and $1$ atm pressure
$A + B_{K}^{K}_{r}\rightleftharpoons ^{t}_{=10}^{=0^{3}}_{2} C + D$
The standard Gibb's energy change $(\Delta _{r}G^{\circ})$ at $27^{\circ}C$ is $(-)$ _____ $kJmol^{-1}$ (Nearest integer).
(Given : $R = 8. 3J\,K^{-1}\, mol^{-1}$ and $ln 10 = 2. 3$ )
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Solid Lead nitrate is dissolved in $1$ litre of water. The solution was found to boil at $100. 15^{\circ}C$ . When $0. 2 mol$
of $NaCl$ is added to the resulting solution, it was observed that the solution froze at $-0. 8^{\circ}C$ . The solutbility
product of $PbCl_{2}$ formed is _____ $\times 10^{-6}$ at $298\,K$ . (Nearest integer)
Given : $K_{b} = 0. 5K\, kg \, mol^{-1}$ and $K_{f} = 1. 8 kg \, mol^{-1}$ . Assume molality to be equal to molarity in all cases.
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Millimoles of calcium hydroxyide required to produce $100 mL$ of the aqueous solution of $pH \,12$ is $x\times 10^{-1}$ .
The value of $x$ is _____ (Nearest integer).
Assume complete dissociation.
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Match List I with List II
List I List II
Antimicrobials Names
(A) Narrow Spectrum
(I) Furacin
Antibiotic
(B) Antiseptic (II) Sulphur dioxide
(C) Disinfectants (III) Penicillin-G
(D) Broad spectrum antibiotic (IV) Chloramphenicol
a.
(A) - III, (B) - I, (C) - II, (D) - IV
b.
(A) - I, (B) - II, (C) - IV, (D) - III
c.
(A) - II, (B) - I, (C) - IV, (D) - III
d.
(A) - III, (B) - I, (C) - IV, (D) - II
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“ $A$ ” obtained by Ostwald's method involving air oxidation of $NH_{3}$ , upon further air oxidation produces “ $B$ ”. “
$B$ ” on hydration forms an oxoacid of Nitrogen along with evolution of “ $A$ ”. The oxoacid also produces “ $A$ ”
and gives positive brown ring test
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The number of molecules or ions from the following, which do not have odd number of electrons are _____ .
(A) $NO_{2}$
(B) $ICl^{-}_{4}$
(C) $BrF_{3}$
(D) $ClO_{2}$
(E) $NO^{+}_{2}$
(F) $NO$
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Water decomposes at $2300\,K$
$H_{2}O(g) \rightarrow H\rightarrow_{2}(g) + \frac{1}{2}O_{2}(g)$
The percent of water decomposing at $2300\,K$ and 1 bar is _____ (Nearest integer). Equilibrium constant for the
reaction is $2 \times 10^{-3}$ at $2300\,K$
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Following chromatogram was developed by adsorption of compound ' $A$ ' on a $6 cm$ TLC glass plate.
Retardation factor of the compound ' $A$ ' is _____ $\times 10^{-1}$ .
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For certain chemical reaction $X \rightarrow Y\rightarrow$ , the rate of formation of product is plotted against the time as shown in
the figure. The number of Correctstatement/s from the following is _____
(A) Over all order of this reaction is one
(B) Order of this reaction can't be determined
(C) In region-I and III, the reaction is of first and zero order respectively
(D) In region-II, the reaction is of first order
(E) In region-II, the order of reaction is in the range of $0. 1$ to $0. 9$ .
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The increasing order of $pK_{a}$ for the following phenols is
(A) 2,4-Dinitrophenol
(B) 4-Nitrophenol
(C) 2,4,5-Trimethylphenol
(D) Phenol
(E) 3-Chlorophenol
Choose the correct answer from the option given below :
a.
(A), (E), (B), (D), (C)
b.
(A), (B), (E), (D), (C)
c.
(C), (D), (E), (B), (A)
d.
(C), (E), (D), (B), (A)
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Match List I with List II.
List I List II
Reaction Reagents
Hoffmann
(A) (I) Conc. $KOH, \Delta$
Degradation
Clemmensen
(B) (II) $CHCl_{3}, NaOH /H_{3}O^{+}$
reduction
(C) Cannizzaro reaction (III) $Br_{2}, NaOH$
Reimer-Tiemann
(D) (IV) $Zn -Hg / HCl$
reaction
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The correct order of hydration enthalpies is
(A) $K^{+}$
(B) $Rb^{+}$
(C) $Mg^{2+}$
(D) $Cs^{+}$
(E) $Ca^{2+}$
Choose the correct answer from the options given below:
a.
C > A > E > B > D
b.
E > C > A > B > D
c.
C > E > A > D > B
d.
C > E > A > B > D
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For $1\, mol$ of gas, the plot of $pV$ vs $p$ is shown below. p is the pressure and $V$ is the volume of the gas.
What is the value of compressibility factor at point $A$ ?
a.
$1 - \frac{a}{RTV}$
b.
$1 + \frac{b}{V}$
c.
$1 - \frac{b}{V}$
d.
$1 + \frac{a}{RTV}$
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The standard electrode potential ( $M^{3+}/M^{2+}$ ) for $V, \, Cr, \, Mn$ & $Co$ are $-0. 26\,V, \, -\,0. 41\,V, \, + 1. 57\,V$ and
$+1. 97\,V$ , respectively. The metal ions which can liberate $H_{2}$ from a dilute acid are
a.
$V^{2+}$ and $Mn^{2+}$
b.
$Cr^{2+}$ and $Co^{2+}$
c.
$V^{2+}$ and $Cr^{2+}$
d.
$Mn^{2+}$ and $Co^{2+}$
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Following figure shows dependence of molar conductance of two electrolytes on concentration. $\Lambda ^{0}_{m}$ is the
limiting molar conductivity.
The number of Incorrectstatement(s) from the following is _____
(A) $\Lambda ^{0}_{m}$ for electrolyte $A$ is obtained by extrapolation
(B) For electrolyte $B, \,\,\,\Lambda _{m}\, Vs \,\sqrt{c}$ graph is a straight line with intercept equal to $\Lambda ^{0}_{m}$
(C) At infinite dilution, the value of degree of dissociation approach zero for electrolyte $B$ .
(D) $\Lambda _{m}$ for any electrolyte $A$ or $B$ can be calculated using $\lambda$ $^{\circ}$ for individual ions.
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$17 mg$ of a hydrocarbon (M.F. $C_{10}H_{16}$ ) takes up $8. 40 mL$ of the $H_{2}$ gas measured at $0^{\circ}C$ and $760 mm$ of $Hg$ .
Ozonolysis of the same hydrocarbon yields
$CH_{3}-C_{||} -CH_{3}, H -C_{||} -H, H -C_{||} -CH_{2}-CH_{2}-C_{||} -C_{||} -H$
$O O O O O$
The number of double bond/s present in the hydrocarbon is
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Let $A={(x,y) \in R2 :y\geq 0, 2x\leq y\leq \sqrt{4 -(x-1)}^{2}}$ and
$B={(x,y) \in R\times R: 0 \leq y\leq min{2x, \sqrt{4 -(x-1)^{2}}}}$ . Then the ratio of the area of $A$ to the area of $B$
is
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A light ray emits from the origin making an angle $30^{\circ}$ with the positive $x$ -axis. After getting reflected by the
line $x + y = 1$ , if this ray intersects x-axis at Q, then the abscissa of Q is
a.
$\frac{2}{(\sqrt{3}-1)}$
b.
$\frac{2}{3+\sqrt{3}}$
c.
$\frac{2}{3-\sqrt{3}}$
d.
$\frac{\sqrt{3}}{2(\sqrt{3}+1)}$
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Let the tangents at the points $A(4, -11)$ and $B(8, -5)$ on the circle $x^{2} +y^{2} -3x+ 10y-15 = 0$ , intersect at
the point $C$ . Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal
to
a.
$\frac{3\sqrt{3}}{4}$
b.
$2\sqrt{13}$
c.
$\sqrt{13}$
d.
$\frac{2\sqrt{13}}{3}$
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Let $a_{1}, a_{2}, a_{3}, …$ . be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the
sum of fifth and seventh terms is $24$ , then $a_{1}a_{9}+ a_{2}a_{4}a_{9}+ a_{5}+ a_{7}$ is equal to
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If all the six digit numbers $x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}$ with $0 < x_{1} < x_{2} < x_{3} < x_{4} < x_{5} < x_{6}$ are arranged in the
increasing order, then the sum of the digits in the $72^{th\,}$ number is _______.
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For two non-zero complex number $z_{1}$ and $z_{2}$ , if $Re (z_{1}z_{2}) = 0$ and $Re (z_{1}+z_{2}) = 0$ , then which of the
following are possible?
(A) $Im (z_{1}) > 0$ and $Im (z_{2}) > 0$
(B) $Im (z_{1}) < 0$ and $Im (z_{2}) > 0$
(C) $Im (z_{1}) > 0$ and $Im (z_{2}) < 0$
(D) $Im (z_{1}) < 0$ and $Im (z_{2}) < 0$
Choose the correct answer from the options given below:
a.
$B$ and $D$
b.
$B$ and $C$
c.
$A$ and $B$
d.
$A$ and $C$
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If the co-efficient of $x^{9}$ in $(\alpha x^{3} + \frac{1}{\beta x} )^{11}$ and the co-efficient of $x^{-9}$ in $(\alpha x- \frac{1}{\beta x^{3}})^{11}$ are equal, then $(\alpha \beta )^{2}$ is
equal to
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Let $f(x) =x+ \frac{a}{\pi ^{2}-4} sinx+ \frac{b}{\pi ^{2}-4} cosx$ , $x\in R$ be a function which satisfies
$f(x) =x+\int _{0}^{\pi /2} sin(x+y)f(y)dy$ . Then $(a+b)$
is equal to
a.
$-\pi (\pi + 2)$
b.
$-2\pi (\pi + 2)$
c.
$-2\pi (\pi -2)$
d.
$-\pi (\pi -2)$
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Suppose f is a function satisfying $f(x + y) = f(x) + f(y)$ for all $x, y \in N$ and $f(1)= \frac{1}{5}$ . If
$\sum ^{m}_{n=1} \frac{f(n)}{n(n+1)(n+2)} = \frac{1}{12}$ then $m$ is equal to ______.
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$\{$
$\frac{1-cos(x^{2}-4px+q^{2}+8q+16)}{(x-2p)^{4}} , x\neq 2p$
Let $x= 2$ be a root of the equation $x^{2} +px+q= 0$ and $f(x) =$ . Then
$\{$
$0, x= 2p$
$_{x\rightarrow 2\rightarrow }lim_{p^{+}}[f(x)]$
where $[\cdot ]$ denotes greatest integer function, is
a.
$2$
b.
$1$
c.
$0$
d.
$-1$
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Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A, \,B, \,C$ and $D$
$---$
be $\vec{a}- \vec{b}+ \vec{c}, \,\,\lambda \vec{a}-3\vec{b}+ 4\vec{c}$ , $-\vec{a}+ 2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+ 6\vec{c}$ respectively. If $A\vec{B}$ , $A\vec{C}$ and $A\vec{D}$ are coplanar,
then $\lambda$ is :
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There rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without
replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma ^{2}$ represent mean and
variance of X, respectively, then $10(\mu ^{2} +\sigma ^{2})$ is equal to
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Let $y = f(x)$ be the solution of the differential equation $y(x+ 1)dx- x^{2}dy= 0, \,y(1)=e$ . Then $_{x}lim_{\rightarrow 0\rightarrow }_{+}f(x)$
is equal to
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Let $f :R\rightarrow \rightarrow R$ be a differentiable function that satisfies the relation $f(x+y) =f(x) +f(y) -1, \forall \,x$ , $y\in R$ .
If $f'(0) = 2$ , then $|f(-2)|$ is equal to
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Consider the following system of questions
$\alpha x+ 2y+z= 1$
$2\alpha x+ 3y+z= 1$
$3x+\alpha y+ 2z=\beta$
For some $\alpha ,\beta \in R$ . Then which of the following is NOT correct.
a.
It has no solution if $\alpha = -1$ and $\beta \neq 2$ It has no solution for $\alpha = -1$ and for all $\beta \in R$
c.
It has no solution for $\alpha = 3$ and for all $\beta \neq 2$
d.
It has a solution for all $\alpha \neq -1$ and $\beta = 2$
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If the vectors $\vec{a}=\lambda ˆi+\mu ˆj+ 4ˆk, \vec{b}= -2ˆi+ 4ˆj-2ˆk$ and $\vec{c}= 2ˆi+ 3ˆj+ˆk$ are coplanar and the projection of $\vec{a}$
on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda +\mu$ is equal to
a.
$0$
b.
$6$
c.
$24$
d.
$18$
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Let [x] denote the greatest integer $\leq x$ . Consider the function $f(x)= max{x^{2}, 1 +[x]}$ . Then the value of the
integral $\int _{0}^{2} f(x)dx$ is :
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Five digit numbers are formed using the digits $1, \,2, \,3, \,5, \,7$ with repetitions and are written in descending
order with serial numbers. For example, the number $77777$ has serial number $1$ . Then the serial number of
$35337$ is
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Let $\Delta$ be the area of the region ${(x,y) \in R2 :x^{2} +y^{2} \leq 21,y^{2} \leq 4x,x\geq 1}$ . Then $\frac{1}{2}(\Delta -21 sin^{-1} \frac{2}{\sqrt{7}})$ is
equal to
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Let $B$ and $C$ be the two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the
origin. Suppose $A$ is a point on $y -2x = 2$ such that $\Delta ABC$ is an equilateral triangle. Then, the area of the
$\Delta ABC$ is
a.
$3\sqrt{3}$
b.
$2\sqrt{3}$
c.
$\frac{8}{\sqrt{3}}$
d.
$\frac{10}{\sqrt{3}}$
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If $p,q$ and $r$ are three propositions, then which of the following combination of truth values of $p, \,q$ and $r$ makes
the logical expression ${(p\vee q) \wedge ((~p) \vee r)} \rightarrow ((~\rightarrow q) \vee r)$ false ?
a.
$p= T,q= F,r= T$
b.
$p= T,q= T,r= F$
c.
$p= F,q= T,r= F$
d.
$p= T,q= F,r= F$
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Let the co-ordinates of one vertex of $\Delta ABC$ be $A(0, 2,\alpha )$ and the other two vertices lie on the line
$\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$ . For $\alpha \in Z$ , if the area of $\Delta ABC$ is $21\, sq. \, units$ and the line segment $BC$ has length
$2\sqrt{21}$ units, then $\alpha ^{2}$ is equal to _______.
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Let $\lambda \neq 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14x^{2} -31x+ 3\lambda = 0$ and $\alpha ,\gamma$ be the roots of
the equation $35x^{2} -53x+ 4\lambda = 0$ . Then $\frac{3\alpha}{\beta}$ and $\frac{4\alpha}{\gamma}$ are the roots of the equation :
a.
$7x^{2} + 245x-250 = 0$
b.
$7x^{2} -245x+ 250 = 0$
c.
$49x^{2} -245x+ 250 = 0$
d.
$49x^{2} + 245x+ 250 = 0$
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Fifteen football players of a club-team are given $15$ T-shirts with their names written on the backside. If the
players pick up the T-shirts randomly, then the probability that at least $3$ players pick the correct T-shirt is
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Let the coefficients of three consecutive terms in the binomial expansion of $(1 + 2x)^{n}$ be in the ratio $2 : \,5 : \,8$ .
Then the coefficient of the term, which is in the middle of these three terms, is
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Let the equation of the plane $P$ containing the line $x+ 10 = \frac{8-y}{2} =z$ be $ax+by+ 3z= 2(a+b)$ and the
distance of the plane $P$ from the point $(1, 27, 7)$ be $c$ . Then $a^{2} + b^{2} + c^{2}$ is equal to
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