The time taken for the magnetic energy to reach $25%$ of its maximum value, when a solenoid of resistance $R$ ,
inductance $L$ is connected to a battery, is :
a.
$\frac{L}{R} ln5$
b.
infinite
c.
$\frac{L}{R} ln2$
Correct Answer
d.
$\frac{L}{R}ln10$
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Q2nat
Consider a $72\, cm$ long wire $AB$ as shown in the figure. The galvanometer jockey is placed at $P$ on $AB$ at a
distance $x\, cm$ from $A$ . The galvanometer shows zero deflection.
The value of $x$ , to the nearest integer, is
✓ Accepted Answer:48
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Q3mcq
A plane electromagnetic wave propagating along y-direction can have the following pair of electric field $(\vec{E})$
and magnetic field $(\vec{B})$ components.
a.
$E_{y},B_{y}$ or $E_{z},B_{z}$
b.
$E_{y},B_{x}$ or $E_{x},B_{y}$
c.
$E_{x},B_{z}$ or $E_{z},B_{x}$
Correct Answer
d.
$E_{x},B_{y}$ or $E_{y},B_{x}$
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Q4nat
The projectile motion of a particle of mass $5\,g$ is shown in the figure.
The initial velocity of the particle is $5\sqrt{2}\, ms^{-1}$ and the air resistance is assumed to be negligible. The
magnitude of the change in momentum between the points $A$ and $B$ is $x\times 10^{-2}\, kgms^{-1}$ . The value of $x$ , to
the nearest integer, is ___ .
✓ Accepted Answer:5
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Q5nat
A galaxy is moving away from the earth at a speed of $286\, km \,s^{-1}$ . The shift in the wavelength of a red line at
$630\, nm$ is $x\times 10^{-10}\,m$ . The value of $x$ , to the nearest integer, is ___ .
[Take the value of the speed of the light $c$ , as $3 \times 10^{8}\,m\,s^{-1}$ ]
✓ Accepted Answer:6
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Q6mcq
An ideal gas in a cylinder is separated by a piston in such a way that the entropy of one part is $S_{1}$ and that of the
other part is $S_{2}$ . Given that $S_{1} >S_{2}$ . If the piston is removed then the total entropy of the system will be:
a.
$S_{1}\times S_{2}$
b.
$S_{1}- S_{2}$
c.
$\frac{S_{1}}{S_{2}}$
d.
$S_{1}+S_{2}$
Correct Answer
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Q7mcq
A particle of mass $m$ moves in a circular orbit under the central potential field, $U(r) = \frac{-C}{r}$ , where $C$ is a
positive constant. The correct radius - velocity graph of the particle's motion is :
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Q8mcq
A solid cylinder of mass $m$ is wrapped with an inextensible light string and, is placed on a rough inclined plane
as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:
[The coefficient of static friction, $\mu _{s}$ , is $0. 4$ ]
a.
$\frac{7}{2} mg$
b.
$5mg$
c.
$\frac{mg}{5}$
Correct Answer
d.
$0$
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Q9nat
The typical output characteristics curve for a transistor working in the common-emitter configuration is shown
in the figure.
The estimated current gain from the figure is
✓ Accepted Answer:200
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Q10mcq
A proton and an $\alpha$ -particle, having kinetic energies $K_{p}$ and $K_{\alpha }$ , respectively, enter into a magnetic field at
right angles. The ratio of the radii of the trajectory of proton to that of $\alpha$ -particle is $2 : 1$ . The ratio of $K_{p} :K_{\alpha }$
is:
a.
$1 : 8$
b.
$8 : 1$
c.
$1 : 4$
d.
$4 : 1$
Correct Answer
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Consider a water tank as shown in the figure. It's cross-sectional area is $0. 4\,m^{2}$ . The tank has an opening $B$
near the bottom whose cross-section area is $1\, cm^{2}$ . A load of $24\, kg$ is applied on the water at the top when the
height of the water level is $40\, cm$ above the bottom, the velocity of water coming out the opening $B$ is
$v\,m\,s^{-1}$ . The value of $v$ , to the nearest integer, is ___ .[Take the value of $g$ to be $10\,m\,s^{-2}$ ]
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Which of the following statements are correct?
(A) Electric monopoles do not exist whereas magnetic monopoles exist.
(B) Magnetic field lines due to a solenoid at its ends and outside cannot be completely straight and confined.
(C) Magnetic field lines are completely confined within a toroid.
(D) Magnetic field lines inside a bar magnet are not parallel.
(E) $χ= -1$ is the condition for a perfect diamagnetic material, where $χ$ is its magnetic susceptibility.
Choose the correct answer from the options given below :
a.
(C) and (E) only
b.
(B) and (D) only
c.
(A) and (B) only
d.
(B) and (C) only
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In a series LCR circuit, the inductive reactance $(X_{L})$ is $10\,\Omega$ and the capacitive reactance $(X_{C})$ is $4\,\Omega$ . The
resistance $(R)$ in the circuit is $6\,\Omega$ . The power factor of the circuit is :
a.
$\frac{1}{2}$
b.
$\frac{1}{2\sqrt{2}}$
c.
$\frac{1}{\sqrt{2}}$
d.
$\frac{\sqrt{3}}{2}$
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A TV transmission tower antenna is at a height of $20\,m$ . Suppose that the receiving antenna is at.
(i) ground level
(ii) a height of $5\,m$ .
The increase in antenna range in case (ii) relative to case (i) is $n%$ . The value of $n$ , to the nearest integer, is.
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Three rays of light, namely red $(R)$ , green $(G)$ and blue $(B)$ are incident on the face $PQ$ of a right angled
prism $PQR$ as shown in figure.
The refractive indices of the material of the prism for red, green and blue wavelength are $1. 27, 1. 42$ and $1. 49$
respectively. The colour of the ray(s) emerging out of the face $PR$ is :
a.
green
b.
red
c.
blue and green
d.
blue
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Consider a sample of oxygen behaving like an ideal gas. At $300\,K$ , the ratio of root-mean-square (RMS)
velocity to the average velocity of the gas molecule would be :
(Molecular weight of oxygen is $32\,g\, mol^{-1}; \,R= 8. 3\,J\,K^{-1}\, mol^{-1}$ )
a.
$\sqrt{\frac{3}{3}}$
b.
$\sqrt{\frac{8}{3}}$
c.
$\sqrt{\frac{3\pi}{8}}$
d.
$\sqrt{\frac{8\pi}{3}}$
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Two wires of same length and thickness having specific resistances $6\,\Omega \, cm$ and $3\,\Omega \, cm$ respectively are
connected in parallel. The effective resistivity is $\rho$ $\,\Omega cm$ . The value of $\rho$ to the nearest integer, is ___ .
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An infinite number of point charges, each carrying $1\,\mu C$ charge, are placed along the y-axis at
$y= 1\,m, 2\,m, 4\,m, 8\,m\,. . . \,.$
The total force on a $1\,C$ point charge, placed at the origin, is $x\times 10^{3}\,N$ . The value of $x$ , to the nearest integer,
is ___ .
[Take $\frac{1}{4\pi \epsilon _{0}} = 9 \times 10^{9}\,N\,m^{2}\,C^{-2}$ ]
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An object of mass $m_{1}$ collides with another object of mass $m_{2}$ , which is at rest. After the collision the objects
move with equal speeds in opposite direction. The ratio of the masses $m_{2} : m_{1}$ is :
a.
$3 : 1$
b.
$2 : 1$
c.
$1 : 2$
d.
$1 : 1$
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The radius of a sphere is measured to be $(7. 50 \pm 0. 85)\, cm$ . Suppose the percentage error in its volume is $x$ .
The value of $x$ , to the nearest $x$ , is ___ .
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A ball of mass $4\, kg$ , moving with a velocity of $10\,m\,s^{-1}$ , collides with a spring of length $8\,m$ and force
constant $100\,N\,m^{-1}$ . The length of the compressed spring is $x\,m$ . The value of $x$ , to the nearest integer, is ___
.
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Consider a uniform wire of mass $M$ and length $L$ . It is bent into a semicircle. Its moment of inertia about a line
perpendicular to the plane of the wire passing through the centre is :
a.
$\frac{1}{4} \frac{ML^{2}}{\pi ^{2}}$
b.
$\frac{2}{5} \frac{ML^{2}}{\pi ^{2}}$
c.
$\frac{ML^{2}}{\pi ^{2}}$
d.
$\frac{1}{2} \frac{ML^{2}}{\pi ^{2}}$
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The correct relation between $\alpha$ (ratio of collector current to emitter current) and $\beta$ (ratio of collector current to
base current) of a transistor is :
a.
$\beta = \frac{\alpha}{1+\alpha}$
b.
$\alpha = \frac{\beta}{1-\alpha}$
c.
$\beta = \frac{1}{1-\alpha}$
d.
$\alpha = \frac{\beta}{1+\beta}$
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The speed of electrons in a scanning electron microscope is $1 \times 10^{7}\, ms^{-1}$ . If the protons having the same
speed are used instead of electrons, then the resolving power of scanning proton microscope will be changed
by a factor of:
a.
$1837$
b.
$\frac{1}{1837}$
c.
$\sqrt{1837}$
d.
$\frac{1}{\sqrt{1837}}$
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If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration
of the day would be approximately :
(Take : $g= 10\, ms^{-2}$ , the radius of earth, $R= 6400 \times 10^{3}\,m$ , Take $\pi = 3. 14$ )
a.
$60$ minutes
b.
does not change
c.
$1200$ minutes
d.
$84$ minutes
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The angular momentum of a planet of mass $M$ moving around the sun in an elliptical orbit is $\vec{L}$ . The magnitude
of the areal velocity of the planet is :
a.
$\frac{4L}{M}$
b.
$\frac{L}{M}$
c.
$\frac{2L}{M}$
d.
$\frac{L}{2M}$
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Given below are two statements :
Statement I : $C_{2}H_{5}OH$ and $AgCN$ both can generate nucleophile.
Statement II : $KCN$ and $AgCN$ both will generate nitrile nucleophile with all reaction conditions.
Choose the most appropriate option :
a.
Statement I is true but statement II is false
b.
Both statement I and statement II are true
c.
Statement I is false but statement II is true
d.
Both statement I and statement II are false
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In basic medium, $H_{2}O_{2}$ exhibits which of the following reactions?
(A) $Mn^{2+} \rightarrow Mn\rightarrow^{4+}$
(B) $I_{2} I^{-}$
(C) $Pb\vec{S}\rightarrow PbSO\rightarrow_{4}$
Choose the most appropriate answer from the options given below :
a.
(A),(C) only
b.
(A) only
c.
(B) only
d.
(A), (B) only
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Given below are two statements:
Statement I : Bohr's theory accounts for the stability and line spectrum of $Li^{+}$ ion.
Statement II : Bohr's theory was unable to explain the splitting of spectral lines in the presence of a magnetic
field.
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 true.
b.
Statement $I$ is false but statement $II$ is true.
c.
Both statement $I$ and statement $II$ are false.
d.
Statement $I$ is true but statement $II$ is false.
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Match list-I with list-II :
(a) $Be$ (i) Treatment of cancer
(b) $Mg$ (ii) Extraction of metals
(c) $Ca$ (iii) Incendiary bombs and signals
(d) $Ra$ (iv) Windows of X-ray tubes
(v) Bearings for motor engines.
Choose the most appropriate answer, the option given below :
a.
$a -iv, \,b -iii, \,c -i, \,d -ii$
b.
$a -iv, b -iii, c -ii, d -i$
c.
$a -iii, \,b -iv, \,c -v, \,d -ii$
d.
$a -iii, \,b -iv, \,c -ii, \,d -v$
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The number of species below that have two lone pairs of electrons in their central atom is ___ (Round off to
the Nearest integer) $SF_{4}, BF^{-}_{4} , ClF_{3}, AsF_{3}, PCl_{5}, BrF_{5}, XeF_{4}, SF_{6}$
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The solubility of $CdSO_{4}$ in water is $8. 0 \times 10^{-4}\, mol \,L^{-1}$ . Its solubility in $0. 01M\,H_{2}SO_{4}$ solution is ___
$\times 10^{-6}\, mol \,L^{-1}$ (Round off to the Nearest integer) (Assume that solubility is much less than $0. 01\,M$ )
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The first ionization energy of magnesium is smaller, as compared to that of elements $X$ and $Y$ , but higher than
that of $Z$ . The elements $X, \,Y$ and $Z$ , respectively, are :
a.
chlorine, lithium and sodium
b.
argon, lithium and sodium
c.
argon, chlorine and sodium
d.
neon, sodium and chlorine
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Consider the above reaction where $6. 1\,g$ of benzoic acid is used to get $7. 8\,g$ of m-bromo benzoic acid. The
percentage yield of the product is ___ .
(Round off to the Nearest integer)
[Given : Atomic masses : $C = 12. 0\,u, \,H : 1. 0\,u, \,O : 16. 0\,u, \,Br = 80. 0\,u$ ]
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A xenon compound $A$ upon partial hydrolysis gives $XeO_{2}F_{2}$ . The number of lone pair of electrons present in
compound $A$ is ___ (Round off to the Nearest integer)
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Match list-I with list-II :
List-I List-II
(a) Mercury (i) Vapour phase refining
(b) Copper (ii) Distillation refining
(c) Silicon (iii) Electrolytic refining
(d) Nickel (iv) Zone refining
Choose the most appropriate answer from the option given below:
a.
$a -i, \,b -iv, \,c -ii, \,d -iii$
b.
$a -ii, \,b -iii, \,c -i, \,d -iv$
c.
$a -ii, b -iii, c -iv, d -i$
d.
$a -ii, \,b -iv, \,c -iii, \,d -i$
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$10. 0\, ml$ of $Na_{2}CO_{3}$ solution is titrated against $0. 2\,M\, HCl$ solution. The following values were obtained in $5$
readings. $4. 8\, ml, 4. 9\, ml, 5. 0\, ml, 5. 0\, ml$ and $5. 0\, ml$
Based on these readings, and convention of titrimetric estimation of concentration of $Na_{2}CO_{3}$ solution is ___
$mM$ .
(Round off to the Nearest integer)
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The molar conductivity at infinite dilution of barium chloride, sulphuric arid and hydrochloric acid are
$280, \,860 426\, Scm^{2} mol^{-1}$ respectively. The molar conductivity at infinite dilution of barium sulphate is ___
$Scm^{2} \, mol^{-1}$ ( Round off to the Nearest Integer).
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Match List - I with List - II :
List-I (Class of Chemicals) List-II (Example)
(a) Antifertility drug (i) Meprobamate
(b) Antibiotic (ii) Alitame
(c) Tranquilizer (iii) Norethindrone
(d) Artificial Sweetener (iv) Salvarsan
Given below are two statements :
Statement I : Non-biodegradable wastes are generated by the thermal power plants.
Statement II : Bio-degradable detergents leads to eutrophication.
In the light of the above statements, choose the most appropriate answer from the option given below:
a.
Both statement I and statement II are false
b.
Statement I is true but statement II is false
c.
Statement I is false but statement II is true
d.
Both statement I and statement II are true.
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In Tollen's test for aldehyde, the overall number of electron(s) transferred to the Tollen's reagent formula
$[Ag (NH_{3})_{2}]^{+}$ per aldehyde group to form silver mirror is ___.(Round off to the Nearest integer)
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A reaction has a half life of $1\, min$ . The time required for $99. 9%$ completion of the reaction is ___ min.
(Round off to the Nearest integer)
[Use : $ln 2 = 0. 69, ln 10 = 2. 3$ ]
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A solute $A$ dimerizes in water. The boiling point of a $2$ molal solution of $A$ is $100. 52^{\circ }C$ . The percentage
association of $A$ is ___ .
(Round off to the Nearest integer)
Use : $K_{b}$ for water $= 0. 52\,K\, kg \, mol^{-1}$
Boiling point of water $= 100^{\circ}C$
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Let $f\, : \,R R$ be a function defined as
$\{\frac{si\vec{n}(a+1)x+sin 2x}{2x} ,if x< 0$
$f(x) = b , if \,x= 0$
$\{$
$\frac{\sqrt{x+bx^{3}}-\sqrt{x}}{bx^{5/2}} , if \,x> 0$
If $f$ is continuous at $x= 0$ , then the value of $a+b$ is equal to :
a.
$-\frac{5}{2}$
b.
$-2$
c.
$-3$
d.
$- \frac{3}{2}$
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Let $S_{1}$ be the sum of first $2n$ terms of an arithmetic progression. Let $S_{2}$ be the sum of first $4n$ terms of the
same arithmetic progression. If $(S_{2}- S_{1})$ is $1000$ , then the sum of the first $6n$ terms of the arithmetic
progression is equal to:
a.
$1000$
b.
$7000$
c.
$5000$
d.
$3000$
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Let the mirror image of the point $(1, 3,a)$ with respect to the plane $\vec{r}\cdot (2ˆi-ˆj+ˆk)- b= 0$ be $(-3, 5, 2)$ .
Then the value of $|a+ b|$ is equal to ___ .
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Let the centroid of an equilateral triangle $ABC$ be at the origin. Let one of the sides of the equilateral triangle
be along the straight line $x+y= 3$ . If $R$ and $r$ be the radius of circumcircle and incircle respectively of
$\Delta ABC$ , then $(R+r)$ is equal to :
a.
$\frac{9}{\sqrt{2}}$
b.
$7\sqrt{2}$
c.
$2\sqrt{2}$
d.
$3\sqrt{2}$
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Consider a hyperbola $H\, : \,x^{2} -2y^{2} = 4$ . Let the tangent at a point $P(4, \sqrt{6})$ meet the $x$ -axis at $Q$ and latus
rectum at $R(x_{1},y_{1}),x_{1} > 0$ . If $F$ is a focus of $H$ which is nearer to the point $P$ , then the area of $\Delta QFR$ (in
sq. units) is equal to
a.
$4\sqrt{6}$
b.
$\sqrt{6}-1$
c.
$\frac{7}{\sqrt{6}} -2$
d.
$4 -1\sqrt{6}$
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Let $^{n}C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1 +x)^{n}$ . If
$\sum ^{10}_{k=0}(2^{2}+ 3k)^{n}C_{k} =\alpha . 3^{10}+\beta \cdot 2^{10}, \,\alpha , \,\beta \in R$ , then $\alpha +\beta$ is equal to ___ .
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$2 -1$
Let $I$ be an identity matrix of order $2 \times 2$ and $P =[ ]$ . Then the value of $n\in N$ for which
$5 -3$
$P^{n} = 5I-8P$ is equal to ___ .
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Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx} = (y+ 1)((y+ 1)e^{x^{2}/2}- x), 0 <x< 2. 1$ , with
$y(2) = 0$ . Then the value of $\frac{dy}{dx}$ at $x= 1$ is equal to
a.
$\frac{-e^{3/2}}{(e^{2}+1)^{2}}$
b.
$- \frac{2e^{2}}{(1+e^{2})^{2}}$
c.
$\frac{e^{5/2}}{(1+e^{2})^{2}}$
d.
$\frac{5e^{1/2}}{(e^{2}+1)^{2}}$
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Let $f :R-{3} \rightarrow \rightarrow R-{1}$ be defined by $f(x) = \frac{x-2}{x-3}$ . Let $g\, : \,R\rightarrow \rightarrow R$ be given as $g(x) = 2x-3$ . Then, the
sum of all the values of $x$ for which $f^{-1}(x) +g^{-1}(x) = \frac{13}{2}$ is equal to
a.
$7$
b.
$2$
c.
$5$
d.
$3$
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Let $f\, : \,R \rightarrow R\rightarrow$ satisfy the equation $f(x+y) =f(x) \cdot f(y)$ for all $x,y\in R$ and $f(x) \neq 0$ for any $x\in R$ . If
the function $f$ is differentiable at $x= 0$ and $f'(0) = 3$ , then $_{h}lim_{0} \frac{1}{h} (f(h) -1)$ is equal to ___ .
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Let a complex number be $w= 1 - \sqrt{3}i$ . Let another complex number $z$ be such that $|zw| = 1$ and
$arg(z) -arg(w) = \frac{\pi}{2}$ . Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to
a.
$4$
b.
$\frac{1}{2}$
c.
$\frac{1}{4}$
d.
$2$
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A pole stands vertically inside a triangular park $ABC$ . Let the angle of elevation of the top of the pole from
each corner of the park be $\frac{\pi}{3}$ . If the radius of the circumcircle of $\Delta ABC$ is $2$ , then the height of the pole is
equal to :
a.
$\frac{2\sqrt{3}}{3}$
b.
$2\sqrt{3}$
c.
$\sqrt{3}$
d.
$\frac{1}{\sqrt{3}}$
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Let $P$ be a plane containing the line $\frac{x-1}{3} = \frac{y+6}{4} = \frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4} = \frac{y-2}{-3} = \frac{z+5}{7}$ . If the point
$(1, -1,\alpha )$ lies on the plane $P$ , then the value of $|5\alpha |$ is equal to ___ .
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$| || || |$
$|| |||$
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $\vec{a} = \vec{b}$ , If $\vec{a}\times \vec{b} = \vec{a}$ , then the angle
between the vectors
$(\vec{a}+ \vec{b}+(\vec{a}\times \vec{b}))$ and $\vec{a}$ is equal to :
a.
$sin^{-1}(\frac{1}{\sqrt{3}})$
b.
$cos^{-1}(\frac{1}{\sqrt{3}} )$
c.
$cos^{-1}( \frac{1}{\sqrt{2}})$
d.
$sin^{-1}( \frac{1}{\sqrt{6}})$
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Let in a Binomial distribution, consisting of $5$ independent trials, probabilities of exactly $1$ and $2$ successes be
$0. 4096$ and $0. 2048$ respectively. Then the probability of getting exactly $3$ successes is equal to :
a.
$\frac{32}{625}$
b.
$\frac{80}{243}$
c.
$\frac{40}{243}$
d.
$\frac{128}{625}$
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Let $P(x)$ be a real polynomial of degree $3$ which vanishes at $\vec{x}= -3$ . Let $P(x)$ have local minima at $x= 1$ ,
local maxima at $x= -1$ and $\int _{-1}^{1} P(x)dx= 18$ , then the sum of all the coefficients of the polynomial $P(x)$ is
equal to ___ .
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Let $y=y(x)$ be the solution of the differential equation $xdy- ydx=\sqrt{(x^{2}-y^{2})}dx, \,x\geq 1$ , with $y(1) = 0$ .
If the area bounded by the line $x= 1,x=e^{\pi },y= 0$ and $y=y(x)$ is $\alpha e^{2\pi } +\beta ,$ then the value of $10(\alpha +\beta )$ is
equal to ___ .
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Let in a series of $2n$ observations, half of them are equal to $a$ and remaining half are equal to $-a$ . Also by
adding a constant $b$ in each of these observations, the mean and standard deviation of new set become $5$ and $20$
, respectively. Then the value of $a^{2} +b^{2}$ is equal to :
a.
$425$
b.
$650$
c.
$250$
d.
$925$
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Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as " $ARB$ iff there exists a non-singular matrix
$P$ such that $PAP^{-1} =B$ ". Then which of the following is true ?
a.
$R$ is symmetric, transitive but not reflexive
b.
$R$ is reflexive, symmetric but not transitive
c.
$R$ is an equivalence relation
d.
$R$ is reflexive, transitive but not symmetric
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In a triangle $ABC$ , if $B\vec{C} = 8, C\vec{A} = 7, A\vec{B} = 10$ , then the projection of the vector $A\vec{B}$ on $A\vec{C}$ is equal to
:
a.
$\frac{25}{4}$
b.
$\frac{85}{14}$
c.
$\frac{127}{20}$
d.
$\frac{115}{16}$
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Let a tangent be drawn to the ellipse $\frac{x^{2}}{27} +y^{2} = 1$ at $(3\sqrt{3}cos\theta , sin\theta )$ where $\theta \in (0, \frac{\pi}{2} )$ . Then the value of $\theta$
such that the sum of intercepts on axes made by this tangent is minimum is equal to :
a.
$\frac{\pi}{8}$
b.
$\frac{\pi}{4}$
c.
$\frac{\pi}{6}$
d.
$\frac{\pi}{3}$
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Let $S_{1}\, : \,x^{2} +y^{2} = 9$ and $S_{2}\, : \,(x-2)^{2} +y^{2} = 1$ .
Then the locus of center of a variable circle $S$ which touches $S_{1}$ internally and $S_{2}$ externally always passes
through the points :
a.
$(0, \pm \sqrt{3})$
b.
$(\frac{1}{2}, \pm \frac{\sqrt{5}}{2})$
c.
$(2, \pm \frac{3}{2})$
d.
$(1, \pm 2)$
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Let $g(x) =\int _{0}^{x} f(t)dt$ , where $f$ is continuous function in $[0, 3]$ such that $\frac{1}{3} \leq f(t) \leq 1$ for all $t\in [0, 1]$ and
$0 \leq f(t) \leq \frac{1}{2}$ for all $t\in (1, 3]$ .
The largest possible interval in which $g(3)$ lies is :
a.
$[-1, - \frac{1}{2}]$
b.
$[- \frac{3}{2}, -1]$
c.
$[ \frac{1}{3}, 2]$
d.
$[1, 3]$
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If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f(x^{3})+xg(x^{3})$ is divisible by
$x^{2} +x+ 1$ , then $P(1)$ is equal to ___ .
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Let the system of linear equations
$4x+\lambda y+ 2z= 0$
$2x- y+z= 0$
$\mu x+ 2y+ 3z= 0, \,\lambda , \,\mu \in R$
has a non-trivial solution. Then which of the following is true?
a.
$\mu = 6, \,\lambda \in R$
b.
$\lambda = 2, \,\mu \in R$
c.
$\lambda = 3, \,\mu \in R$
d.
$\mu = -6, \,\lambda \in R$
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