A wire of length $L$ is hanging from a fixed support. The length changes to $L_{1}$ and $L_{2}$ when masses $1\, kg$ and
$2\, kg$ are suspended respectively from its free end. Then the value of $L$ is equal to
a.
$\sqrt{LL}_{1}_{2}$
b.
$\frac{L_{1}+L_{2}}{2}$
c.
$2L_{1}- L_{2}$
Correct Answer
d.
$3L_{1}- L_{2}$
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Q2nat
The intensity of the light from a bulb incident on a surface is $0. 22\,W\,m^{-2}$ . The amplitude of the magnetic
field in this light-wave is _____ $\times 10^{-9}\,T$
(Given : Permittivity of vacuum $\epsilon _{0} = 8. 85 \times 10^{-12}\,C^{2}\,N^{-1}\,m^{-2}$ , speed of light in vacuum $c= 3 \times 10^{8}\,m\,s^{-1}$
)
✓ Accepted Answer:43
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Q3mcq
Using Young's double slit experiment, a monochromatic light of wavelength $5000$ Åproduces fringes of fringe
width $0. 5\, mm$ . If another monochromatic light of wavelength $6000$ Åis used and the separation between the
slits is doubled, then the new fringe width will be
a.
$0. 5\, mm$
b.
$1. 0\, mm$
c.
$0. 6\, mm$
d.
$0. 3\, mm$
Correct Answer
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Q4mcq
A body of mass $M$ at rest explodes into three pieces, in the ratio of masses $1 : 1 : 2$ . Two smaller pieces fly off
perpendicular to each other with velocities of $30\,m\,s^{-1}$ and $40\,m\,s^{-1}$ respectively. The velocity of the third
piece will be
a.
$35\,m\,s^{-1}$
b.
$50\,m\,s^{-1}$
c.
$25\,m\,s^{-1}$
Correct Answer
d.
$15\,m\,s^{-1}$
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Q5nat
A transistor is used in an amplifier circuit in common emitter mode. If the base current changes by $100\,\mu A$ , it
brings a change of $10\, mA$ in collector current. If the load resistance is $2\,k\Omega$ and input resistance is $1\,k\Omega$ , the
value of power gain is $x\times 10^{4}$ . The value of $x$ is _____ .
✓ Accepted Answer:2
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Q6mcq
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A :The photoelectric effect does not take place, if the energy of the incident radiation is less than
the work function of a metal.
Reason R :Kinetic energy of the photoelectrons is zero, if the energy of the incident radiation is equal to the
work function of a metal.
a.
Both A and R are correct and R is the correct
explanation of A
b.
Both A and R are correct but R is not the correct
explanation of A
Correct Answer
c.
A is correct but R is not correct
d.
A is not correct but R is correct
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Q7nat
As per the given figure, two plates $A$ and $B$ of thermal conductivity $K$ and $2K$ are joined together to form a
compound plate. The thickness of plates are $4. 0\, cm$ and $2. 5\, cm$ respectively and the area of cross-section is
$120\, cm^{2}$ for each plate. The equivalent thermal conductivity of the compound plate is $(1 + \frac{5}{\alpha})K$ , then the
value of $\alpha$ will be _____ .
✓ Accepted Answer:21
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Q8mcq
A particle of mass $500\,g$ is moving in a straight line with velocity $v= bx \frac{5}{2}$ . The work done by the net force
during its displacement from $x= 0$ to $x= 4\,m$ is (Take $b = 0. 25\,\,m\frac{-3}{2} s^{-1}$ ).
a.
$2\,J$
b.
$4\,J$
c.
$8\,J$
d.
$16\,J$
Correct Answer
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Q9mcq
The activity of a radioactive material is $2. 56 \times 10^{-3} Ci$ . If the half life of the material is $5$ days, after how
many days the activity will become $2 \times 10^{-5} Ci$ ?
a.
$30$ days
b.
$35$ days
Correct Answer
c.
$40$ days
d.
$25$ days
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Q10mcq
Two vectors and have equal magnitudes. If magnitude of $+$ is equal to two times the magnitude of
$\vec{A} \vec{B} \vec{A} \vec{B}$
$\vec{A}- \vec{B}$ , then the angle between $\vec{A}$ and $\vec{B}$ will be
a.
$cos^{-1}(\frac{3}{5} )$
Correct Answer
b.
$cos^{-1}(\frac{1}{3})$
c.
$sin^{-1}( \frac{1}{3})$
d.
$sin^{-1}(\frac{3}{5})$
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A charge particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of
the charge particle increases to $4$ times its initial value. What will be the ratio of new radius to the original
radius of circular path of the charge particle?
a.
$1 : 1$
b.
$1 : 2$
c.
$2 : 1$
d.
$1 : 4$
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A body is performing simple harmonic with an amplitude of $10\, cm$ . The velocity of the body was tripled by
air Jet when it is at $5\, cm$ from its mean position. The new amplitude of vibration is $\sqrt{x}\, cm$ . The value of $x$ is
_____ .
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$300\, calories$ of heat is given to a heat engine, and it rejects $225\, calories$ of heat. If source temperature is
$227^{\circ}C$ , then the temperature of sink will be _____ $^{\circ }C.$
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Two coils require $20$ minutes and $60$ minutes respectively to produce same amount of heat energy when
connected separately to the same source. If they are connected in parallel arrangement to the same source; the
time required to produce same amount of heat by the combination of coils, will be _____ $min$ .
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A longitudinal wave is represented by $y= 10 sin 2\pi (nt- \frac{x}{\lambda} )\, cm$ . The maximum particle velocity will be four
times the wave velocity if the determined value of wavelength is equal to
a.
$2\pi$
b.
$5\pi$
c.
$\pi$
d.
$\frac{5\pi}{2}$
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The variation of applied potential and current flowing through a given wire is shown in figure. The length of
wire is $31. 4\, cm$ . The diameter of wire is measured as $2. 4\, cm$ . The resistivity of the given wire is measured as
$x\times 10^{-3}\,\Omega \, cm$ . The value of $x$ is _____ .
[Take $\pi = 3. 14$ ]
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For a series $LCR$ circuit, $I$ vs $\omega$ curve is shown
(a) To the left of $\omega _{r}$ , the circuit is mainly capacitive.
(b) To the left of $\omega _{r}$ , the circuit is mainly inductive.
(c) At $\omega _{r}$ , impedance of the circuit is equal to the resistance of the circuit.
(d) At $\omega _{r}$ , impedance of the circuit is $0$ .
Choose the most appropriate answer from the options given below.
a.
(a) and (d) only
b.
(b) and (d) only
c.
(a) and (c) only
d.
(b) and (c) only
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A cylinder of fixed capacity of $44. 8$ litres contains helium gas at standard temperature and pressure. The
amount of heat needed to raise the temperature of gas in the cylinder by $20. 0^{\circ}C$ will be (Given gas constant
$R= 8. 3\,J\,K^{-1}\, mol^{-1}$ )
a.
$249\,J$
b.
$415\,J$
c.
$498\,J$
d.
$830\,J$
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Two balls $A$ and $B$ are placed at the top of $180\,m$ tall tower. Ball $A$ is released from the top at $t= 0\,s$ . Ball $B$ is
thrown vertically down with an initial velocity $u$ at $t= 2\,s$ . After a certain time, both balls meet $100\,m$ above
the ground. Find the value of $u$ in $m\,s^{-1}$ . [use $g= 10\,m\,s^{-2}$ ]
a.
$10$
b.
$15$
c.
$20$
d.
$30$
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A block of metal weighing $2\, kg$ is resting on a frictionless plane (as shown in figure). It is struck by a jet
releasing water at a rate of $1\, kg \,s^{-1}$ and at a speed of $10\,m\,s^{-1}$ . Then, the initial acceleration of the block, in
$m\,s^{-2}$ , will be
a.
$3$
b.
$6$
c.
$5$
d.
$4$
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A positive charge particle of $100\, mg$ is thrown in opposite direction to a uniform electric field of strength
$1 \times 10^{5}\,N\,C^{-1}$ . If the charge on the particle is $40\,\mu C$ and the initial velocity is $200\,m\,s^{-1}$ , how much distance
it will travel before coming to the rest momentarily
a.
$0. 5\,m$
b.
$1\,m$
c.
$5\,m$
d.
$10\,m$
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In van dar Wall equation $[P + \frac{a}{V^{2}}][V - b] =RT; \,P$ is pressure, $V$ is volume, $R$ is universal gas constant and
$T$ is temperature. The ratio of constants $\frac{a}{b}$ is dimensionally equal to :
a.
$\frac{P}{V}$
b.
$\frac{V}{P}$
c.
$PV$
d.
$PV^{3}$
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Only $2%$ of the optical source frequency is the available channel bandwidth for an optical communicating
system operating at $1000\, nm$ . If an audio signal requires a bandwidth of $8 kHz$ , how many channels can be
accommodated for transmission
a.
$375 \times 10^{7}$
b.
$75 \times 10^{7}$
c.
$375 \times 10^{8}$
d.
$75 \times 10^{9}$
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$\sqrt{d}_{1}$ and $\sqrt{d}_{2}$ are the impact parameters corresponding to scattering angles $60^{\circ}$ and $90^{\circ}$ respectively, when an
$\alpha$ particle is approaching a gold nucleus. For $d_{1} =xd_{2}$ , the value of $x$ will be _____ .
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A parallel beam of light is allowed to fall on a transparent spherical globe of diameter $30\, cm$ and refractive
index $1. 5$ . The distance from the centre of the globe at which beam of light can converge is _____ $mm$ .
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The escape velocity of a body on a planet $A$ is $12\, km \,s^{-1}$ . The escape velocity of the body on another planet $B$ ,
whose density is four times and radius is half of the planet $A$ , is
a.
$12\, km \,s^{-1}$
b.
$24\, km \,s^{-1}$
c.
$36\, km \,s^{-1}$
d.
$6\, km \,s^{-1}$
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A spherical shell of $1\, kg$ mass and radius $R$ is rolling with angular speed $\omega$ on horizontal plane (as shown in
figure). The magnitude of angular momentum of the shell about the origin $O$ is $\frac{a}{3}R^{2}\omega$ . The value of $a$ will be
a.
$2$
b.
$3$
c.
$5$
d.
$4$
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At a certain place the angle of dip is $30^{\circ}$ and the horizontal component of earth's magnetic field is $0. 5\,G$ . The
earth's total magnetic field (in $G$ ), at that certain place, is
a.
$\frac{1}{\sqrt{3}}$
b.
$\frac{1}{2}$
c.
$\sqrt{3}$
d.
$1$
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A parallel plate capacitor filled with a medium of dielectric constant $10$ , is connected across a battery and is
charged. The dielectric slab is replaced by another slab of dielectric constant $15$ . Then the energy of capacitor
will
a.
increase by $50%$
b.
decrease by $15%$
c.
increase by $25%$
d.
increase by $33%$
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$--C_{6}H_{12}O_{6}^{Zymase}\rightarrow \rightarrow A^{NaOI}\rightarrow \rightarrow B + CHI_{3}$
$\Delta$
The number of carbon atoms present in the product $B$ is
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Match List-I with Match List-II
List-I List-II
(Metal) (Application)
(A) $Cs$ (I) High temperature thermometer
(B) $Ga$ (II) Water repellent sprays
(C) $B$ (III) Photoelectric cells
(D) $Si$ (V) Bullet proof vest
Choose the most appropriate answer from the options given below
a.
$(A) -(III), (B) -(I), (C) -(IV), (D) -(II)$
b.
$(A) -(II), (B) -(III), (C) -(IV), (D) -(I)$
c.
$(A) -(IV), (B) -(III), (C) -(II), (D) -(I)$
d.
$(A) -(I), (B) -(IV), (C) -(II), (D) -(II)$
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Two isomers ' $A$ ' and ' $B$ ' with molecular formula $C_{4}H_{8}$ give different products on oxidation with $KMnO_{4}$ in
acidic medium. Isomer ' $A$ ' on reaction with $KMnO_{4}/H^{+}$ results in effervescence of a gas and gives ketone.
The compound ' $A$ ' is
a.
But- $1$ -ene.
b.
cis-But- $2$ -ene.
c.
trans-But- $2$ -ene.
d.
$2$ -methyl propene.
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An acidified manganate solution undergoes disproportionation reaction. The spin-only magnetic moment value
of the product having manganese in higher oxidation state is____B.M. (Nearest integer)
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A dilute solution of sulphuric acid is electrolysed using a current of $0. 10\,A$ for $2$ hours to produce hydrogen
and oxygen gas. The total volume of gases produced at $STP$ is $cm^{3}$ . (Nearest integer) [Given : Faraday
constant $F = 96500\, Cmol^{-1}$ at $STP$ , molar volume of an ideal gas is $22. 7\,L mol^{-1}$ ]
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Which of the following statements are correct?
(A) The electronic configuration of $Cr$ is $[Ar]\,3d^{5}4s^{1}$ .
(B) The magnetic quantum number may have a negative value.
(C) In the ground state of an atom, the orbitals are filled in order of their increasing energy order.
(D) The total number of nodes are given by $n -2$ .
Choose the most appropriate answer from the options given below.
a.
$(A), (C)\,&\,(D)$ only
b.
$(A)\,&\,(B)$ only
c.
$(A)\,&\,(C)$ only
d.
$(A), (B)\,&\,(C)$ only
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Geraniol, a volatile organic compound, is a component of rose oil. The density of the vapour is $0. 46 gL^{-1}$ at
$257\,^{\circ}C$ and $100\, mmHg$ . The molar mass of geraniol is (Nearest Integer)
[Given $R = 0. 082\,L$ atm $K^{-1} mol^{-1}$ ]
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Given below are two statements.
Statement I : Phenols are weakly acidic.
Statement II : Therefore they are freely soluble in $NaOH$ solution and are weaker acids than alcohols and
water.
Choose the most appropriate option
a.
Both Statement I and Statement II are correct.
b.
Statement I is correct but Statement II is
incorrect.
c.
Statement I is incorrect but Statement II is
correct.
d.
Both Statement I and Statement II are incorrect.
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The number of terminal oxygen atoms present in the product $B$ obtained from the following reaction is
$FeCr_{2}O_{4}+ Na_{2}CO_{3}+ O_{2} \rightarrow A + Fe\rightarrow_{2}O_{3}+ CO_{2}$
$A + H^{+} \rightarrow B + H\rightarrow_{2}O + Na^{+}$
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The activation energy of one of the reactions in a biochemical process is $532611\,J\, mol^{-1}$ . When the
temperature falls from $310\,K$ to $300\,K$ , the change in rate constant observed is $k_{300} = x \times 10^{-3}k_{310^{\circ}}$ . The
value of $x$ is
[Given: $ln 10 = 2. 3\,R = 8. 3\,J\,K^{-1}\, mol^{-1}$ ]
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$1. 2\, mL$ of acetic acid is dissolved in water to make $2. 0\,L$ of solution. The depression in freezing point
observed for this strength of acid is $0. 0198\,^{\circ}C$ . The percentage of dissociation of the acid is (Nearest integer)
[Given : Density of acetic acid is $1. 02\,g\, mL^{-1}$ Molar mass of acetic acid is $60\,g mol^{-1}$
$K_{f}(H_{2}O) = 1. 85\,K\, kg \, mol^{-1}$ ]
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Production of iron in blast furnace follows the following equation
$Fe_{3}O_{4}(s) + 4 CO(g) 3 Fe(l) + 4 CO_{2}(g)$
when $4. 640 kg$ of $Fe_{3}\vec{O}_{4}$ and $2. 520 kg$ of $CO$ are allowed to react then the amount of iror (in $g$ ) produced is :
[Given: Molar Atomic mass $(gmol^{-1}): Fe = 56$
Molar Atomic mass $(gmolm^{-1}): O = 16$
Molar Atomic mass $(gmolm^{-1}): C = 12$ ]
a.
$1400$
b.
$2200$
c.
$3360$
d.
$4200$
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$17. 0\,g$ of $NH_{3}$ completely vapourises at $-33. 42\,^{\circ}C$ and $1$ bar pressure and the enthalpy change in the process
is $23. 4\, kJ \, mol^{-1}$ . The enthalpy change for the vapourisation of $85\,g$ of $NH_{3}$ under the same conditions is $kJ$ .
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Arrange the following in the decreasing order of their covalent character :
(A) $LiCl$
(B) $NaCl$
(C) $KCl$
(D) $CsCl$
Choose the most appropriate answer from the options given below
a.
$(A) > (C) > (B) > (D)$
b.
$(B) > (A) > (C) > (D)$
c.
$(A) > (B) > (C) > (D)$
d.
$(A) > (B) > (D) > (C)$
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Kjeldahl's method was used for the estimation of nitrogen in an organic compound. The ammonia evolved
from $0. 55\,g$ of the compound neutralised $12. 5\, mL$ of $1M\,H_{2}SO_{4}$ solution. The percentage of nitrogen in the
compound is (Nearest integer)
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Match List-I with Match List-II
List-I List-II
(Metal) (Emitted light wavelength $(nm)$ )
(A) $Li$ (I) $670. 8$
(B) $Na$ (II) $589. 2$
(C) $Rb$ (III) $780. 0$
(D) $Cs$ (V) $455. 5$
Choose the most appropriate answer from the options given below
a.
$(A) -(I), (B) -(II), (C) -(III), (D) -(IV)$
b.
$(A) -(III), (B) -(II), (C) -(I), (D) -(IV)$
c.
$(A) -(III), (B) -(I), (C) -(II), (D) -(IV)$
d.
$(A) -(IV), (B) -(II), (C) -(I), (D) -(III)$
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Given below are two statements :
Statement I: The esterification of carboxylic acid with an alcohol is a nucleophilic acyl substitution.
Statement II: Electron withdrawing groups in the carboxylic acid will increase the rate of esterification
reaction.
Choose the most appropriate option
a.
Statement I is correct but Statement II is
incorrect.
b.
Both Statement I and Statement II are incorrect.
c.
Both Statement I and Statement II are correct.
d.
Statement I is incorrect but Statement II is
correct.
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Let $d$ be the distance between the foot of perpendiculars of the points $P(1, 2 -1)$ and $Q(2, -1, 3)$ on the plane
$-x+y+z= 1$ . Then $d^{2}$ is equal to ______.
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Let the tangent to the circle $C_{1} :x^{2} +y^{2} = 2$ at the point $M(-1, 1)$ intersect the circle $C_{2}$ :
$(x-3)^{2} + (y-2)^{2} = 5$ , at two distinct points $A$ and $B$ . If the tangents to $C_{2}$ at the points $A$ and $B$ intersect
at $N$ , then the area of the triangle $ANB$ is equal to
a.
$\frac{1}{2}$
b.
$\frac{2}{3}$
c.
$\frac{1}{6}$
d.
$\frac{5}{3}$
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Let $\vec{a}=\alpha ˆi+ 3ˆj- ˆk, \vec{b}= 3ˆi- \beta ˆj+ 4ˆk$ and $\vec{c}= ˆi+ 2ˆj-2ˆk$ where $\alpha ,\beta \in R$ be three vectors. If the projection
of $\vec{a}$ on $\vec{c}$ is $\frac{10}{3}$ and $\vec{b}\times \vec{c}= -6ˆi+ 10ˆj+ 7ˆk$ , then the value of $\alpha +\beta$ equal to
a.
$3$
b.
$4$
c.
$5$
d.
$6$
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Let the mean and the variance of $5$ observations $x_{1},x_{2},x_{3},x_{4},x_{5}$ be $\frac{24}{5}$ and $\frac{194}{25}$ respectively. If the mean and
variance of the first $4$ observation are $\frac{7}{2}$ and $a$ respectively, then $(4a+x_{5})$ is equal to
a.
$13$
b.
$15$
c.
$17$
d.
$18$
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Let $H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,a> 0, \,b> 0$ , be a hyperbola such that the sum of lengths of the transverse and the
conjugate axes is $4(2 + \sqrt{2} \sqrt{14})$ . If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$ , then value of $a^{2} +b^{2}$ is equal to ______.
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Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} + \frac{\sqrt{2}y}{2 cos^{4}x-cos 2x} =xe^{tan^{-1}(\sqrt{2}cot 2x)}, 0 <x< \frac{\pi}{2}$
with $y(\frac{\pi}{4})= \frac{\pi ^{2}}{32}$ . If $y( \frac{\pi}{3})= \frac{\pi ^{2}}{18} e^{-tan^{-1}(\alpha )}$ , then the value of $3\alpha ^{2}$ is equal to ______.
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$||$
Let a set $A=A_{1}\cup A_{2}\cup … \cup A_{k}$ , where $A_{i}\cap A_{j} =\phi$ for $i\neq j; \,1 \leq i,j\leq k.$ Define the relation $R$ from $A$
to $A$ by $R=$ { $(x,y) :y\in A_{i}$ if and only if $x\in A_{i}, 1 \leq i\leq k$ }. Then, $R$ is:
a.
reflexive, symmetric but not transitive
b.
reflexive, transitive but not symmetric
c.
reflexive but not symmetric and transitive
d.
an equivalence relation
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Let $S = {z\in C : |z-2| \leq 1,z(1 +i) +z(1 - i) \leq 2}$ . Let $|z-4\,i|$ attains minimum and maximum values,
respectively, at $z_{1} \in S$ and $z_{2} \in S$ . If $5(\bar{|}z_{1}|^{2} + |z_{2}|^{2})=\alpha +\beta \sqrt{5}$ , where $\alpha$ and $\beta$ are integers, then the value
of $\alpha +\beta$ is equal to ______.
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If the system of linear equations
$2x+y- z= 7$
$x-3y+ 2z= 1$
$x+ 4y+\delta z=k$ , where $\delta ,k\in R$
has infinitely many solutions, then $\delta +k$ is equal to
a.
$-3$
b.
$3$
c.
$6$
d.
$9$
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Let $c,k\in R$ . If $f(x) = (c+ 1)x^{2} +(1 - c^{2})x+ 2k$ and $f(x+y) =f(x) +f(y) - xy$ , for all $x,y\in R$ , then
the value of $|2(f(1) +f(2) +f(3) + … … +f(20))|$ is equal to ______.
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Let $PQ$ be a focal chord of the parabola $y^{2} = 4x$ such that it subtends an angle of $\frac{\pi}{2}$ at the point $(3, 0)$ . Let the
line segment $PQ$ be also a focal chord of the ellipse $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,a^{2} >b^{2}$ . If $e$ is the eccentricity of the
ellipse $E$ , then the value of $\frac{1}{e^{2}}$ is equal to
a.
$1 + \sqrt{2}$
b.
$3 + 2\sqrt{2}$
c.
$1 + 2\sqrt{3}$
d.
$4 + 5\sqrt{3}$
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Let $A= [a_{ij}]$ be a square matrix of order $3$ such that $a_{ij} = 2^{j-i}$ , for all $i, \,j= 1, 2, 3$ . Then, the matrix
$A^{2} +A^{3} + … +A^{10}$ is equal to
a.
$(\frac{3^{10}-1}{2})A$
b.
$(\frac{3^{10}+1}{2})A$
c.
$(\frac{3^{10}+3}{2})A$
d.
$(\frac{3^{10}-3}{2})A$
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A wire of length $22m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other
into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of
the square and the equilateral triangle is minimum, is
a.
$\frac{22}{9+4\sqrt{3}}$
b.
$\frac{66}{9+4\sqrt{3}}$
c.
$\frac{22}{4+9\sqrt{3}}$
d.
$\frac{66}{4+9\sqrt{3}}$
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Let $P_{1} : \vec{r}\cdot (2ˆi+ ˆj-3ˆk)= 4$ be a plane. Let $P_{2}$ be another plane which passes through the points
$(2, -\,3, 2)(2, -2, -3)$ and $(1, -4, 2)$ . If the direction ratios of the line of intersection of $P_{1}$ and $P_{2}$ be $16$ , $\alpha ,\beta$ ,
then the value of $\alpha +\beta$ is equal to ______.
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The distance between the two points $A$ and $A'$ which lie on $y= 2$ such that both the line segments $AB$ and
$A'B$ (where $B$ is the point $(2, 3)$ ) subtend angle $\frac{\pi}{4}$ at the origin, is equal to
a.
$10$
b.
$\frac{48}{5}$
c.
$\frac{52}{5}$
d.
$3$
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Let $\Delta \in {\wedge , \vee , \Rightarrow , \Leftrightarrow }$ be such that $(p\wedge q)\Delta ((p\vee q) \Rightarrow q)$ is a tautology. Then $\Delta$ is equal to
a.
$\wedge$
b.
$\vee$
c.
$\Rightarrow$
d.
$\Leftrightarrow$
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Let ${a_{n}}^{\infty }_{n=0}$ be a sequence such that $a_{0} =a_{1} = 0$ and $a_{n+2} = 2a_{n+1}- a_{n}+ 1$ for all $n\geq 0$ . Then, $\sum ^{\infty }_{n=2} \frac{a_{n}}{7^{n}}$
is equal to
a.
$\frac{6}{343}$
b.
$\frac{7}{216}$
c.
$\frac{8}{343}$
d.
$\frac{49}{216}$
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$\{max_{t\leq x} {t^{3}-3t}; x\leq 2$
Let be a function defined by : $x^{2} + 2x-6; 2 <x< 3$
$f :R\rightarrow \rightarrow R f(x) =$
$[x-3] + 9; 3 \leq x\leq 5$
$\{$
$2x+ 1; \, x> 5$
Where $[t]$ is the greatest integer less than or equal to $t$ . Let $m$ be the number of points where $f$ is not
differentiable and $I =\int _{-2}^{2} f(x)dx$ . Then the ordered pair $(m,I)$ is equal to
a.
$(3, \frac{27}{4})$
b.
$(3, \frac{23}{4})$
c.
$(4, \frac{27}{4})$
d.
$(4, \frac{23}{4})$
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If the constant term in the expansion of $(3x^{3} -2x^{2} + \frac{5}{x^{5}})^{10}$ is $2^{k}.l$ , where $l$ is an odd integer, then the value
of $k$ is equal to
a.
$6$
b.
$7$
c.
$8$
d.
$9$
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Let $b_{1}b_{2}b_{3}b_{4}$ be a $4$ -element permutation with $b_{i} \in {1, 2, 3, … … … , 100}$ for $1 \leq i\leq 4$ and $b_{i} \neq b_{j}$ for $i\neq j$
, such that either $b_{1}, \,b_{2}, \,b_{3}$ are consecutive integers or $b_{2},b_{3},b_{4}$ are consecutive integers. Then the number of
such permutations $b_{1}b_{2}b_{3}b_{4}$ is equal to ______.
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