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JEE Main Full Mock Test: Practice Test 13

Question paper with answer key. Selection Pro

Q1. A particle executes SHM as $x=A\cos(\omega t+\phi)$. At $t=0$, $x=A/2$ and velocity is negative. Then $\phi$ is:
  1. $\pi/3$
  2. $2\pi/3$
  3. $4\pi/3$
  4. $5\pi/3$

Answer: (A) $\pi/3$

$\cos\phi=1/2$ and $v=-A\omega\sin\phi<0$, so $\sin\phi>0$. Hence $\phi=\pi/3$.

Q2. A uniform rod of mass $M$ and length $L$ is hinged at one end and released from rest from horizontal. Its angular speed on reaching the vertical position is:
  1. $\sqrt{3g/L}$
  2. $\sqrt{2g/L}$
  3. $\sqrt{6g/L}$
  4. $\sqrt{g/L}$

Answer: (A) $\sqrt{3g/L}$

$MgL/2=(1/2)(ML^2/3)\omega^2$, giving $\omega=\sqrt{3g/L}$.

Q3. A mass $m$ is attached between two identical springs of force constant $k$ on a frictionless surface. The angular frequency of small oscillations is:
  1. $\sqrt{k/m}$
  2. $\sqrt{2k/m}$
  3. $\sqrt{m/2k}$
  4. $2\sqrt{k/m}$

Answer: (B) $\sqrt{2k/m}$

The restoring force is $-2kx$, so $m\ddot x=-2kx$ and $\omega=\sqrt{2k/m}$.

Q4. A projectile is fired with speed $u$. At its highest point its speed is $u/3$. The projection angle is:
  1. $30^\circ$
  2. $45^\circ$
  3. $60^\circ$
  4. $\cos^{-1}(1/3)$

Answer: (D) $\cos^{-1}(1/3)$

At the highest point the speed is the horizontal component $u\cos\theta$. Thus $\cos\theta=1/3$.

Q5. A particle of mass $m$ moving at speed $v$ collides elastically head-on with a particle of mass $3m$ at rest. The first particle moves after collision with speed:
  1. $v/2$
  2. $v/4$
  3. $-v/2$
  4. $-v$

Answer: (C) $-v/2$

For an elastic one-dimensional collision, $v_1^\prime=(m-3m)/(m+3m)v=-v/2$.

Q6. A satellite moves from a circular orbit of radius $R$ to one of radius $9R$. The ratio of its kinetic energies (old:new) is:
  1. $1:3$
  2. $9:1$
  3. $3:1$
  4. $1:9$

Answer: (B) $9:1$

For circular orbit $K\propto1/r$, hence $K_R/K_{9R}=9$.

Q7. A conducting spherical shell of radius $R$ carries charge $Q$. Its potential at any point inside the shell is:
  1. $0$
  2. $kQ/R$
  3. $2kQ/R$
  4. $kQ/(2R)$

Answer: (B) $kQ/R$

Potential inside a charged conducting shell is constant and equal to its surface value $kQ/R$.

Q8. Capacitors $C$ and $2C$ are charged separately to $V$ and $2V$, then connected positive-to-positive. Their final common potential is:
  1. $V$
  2. $5V/3$
  3. $4V/3$
  4. $2V$

Answer: (B) $5V/3$

Total initial charge is $CV+2C(2V)=5CV$ and total capacitance is $3C$, so $V_f=5V/3$.

Q9. A cell of emf $E$ and internal resistance $r$ supplies a variable resistance $R$. Power delivered to $R$ is maximum when:
  1. $R=r/2$
  2. $R=r$
  3. $R=2r$
  4. $R\to\infty$

Answer: (B) $R=r$

$P=E^2R/(R+r)^2$ is maximum at $R=r$.

Q10. A long wire carries current $I$. If both current and distance from the wire are doubled, its magnetic field becomes:
  1. $B/4$
  2. $B/2$
  3. $B$
  4. $2B$

Answer: (C) $B$

$B=\mu_0I/(2\pi r)$. Doubling both $I$ and $r$ leaves $I/r$ unchanged.

Q11. A charged particle enters a uniform magnetic field at $60^\circ$. The ratio of helical pitch to orbit radius is:
  1. $\pi/\sqrt3$
  2. $2\pi/\sqrt3$
  3. $\sqrt3\pi$
  4. $2\pi\sqrt3$

Answer: (B) $2\pi/\sqrt3$

$p=v\cos60^\circ(2\pi m/qB)$ and $r=v\sin60^\circ m/qB$, giving $p/r=2\pi/\sqrt3$.

Q12. A coil of $N$ turns and resistance $R$ has flux per turn changed uniformly from $\Phi$ to zero. Total charge passing through it is:
  1. $N\Phi/R$
  2. $N\Phi/(Rt)$
  3. $N\Phi t/R$
  4. $\Phi/(NR)$

Answer: (A) $N\Phi/R$

$q=(1/R)\int|\mathcal E|dt=N|\Delta\Phi|/R=N\Phi/R$.

Q13. In a series LCR circuit at resonance, the voltage across the inductor is $100$ V. The voltage across the capacitor is:
  1. $0$
  2. $50$ V
  3. $100$ V
  4. $200$ V

Answer: (C) $100$ V

At resonance $X_L=X_C$, so $V_L$ and $V_C$ are equal in magnitude and opposite in phase.

Q14. A convex lens forms an image at the same size when the object is at $2f$. If the object is moved to $3f$, the magnitude of magnification is:
  1. $1/2$
  2. $1/3$
  3. $2$
  4. $3$

Answer: (A) $1/2$

For $u=3f$, lens formula gives $v=3f/2$, hence $|m|=v/u=1/2$.

Q15. In YDSE, individual slit intensities are $I$ and $4I$. At phase difference $\pi$, resultant intensity is:
  1. $I$
  2. $3I$
  3. $5I$
  4. $9I$

Answer: (A) $I$

$I_R=I+4I+2\sqrt{4I^2}\cos\pi=I$.

Q16. A metal has work function $2$ eV. Light of wavelength $248$ nm falls on it. Taking $hc=1240$ eV nm, stopping potential is:
  1. $2$ V
  2. $3$ V
  3. $4$ V
  4. $5$ V

Answer: (B) $3$ V

Photon energy is $1240/248=5$ eV. Thus $K_{max}=3$ eV and stopping potential is $3$ V.

Q17. A radioactive sample has decay constant $\lambda$. Time for activity to become $A_0/16$ is:
  1. $\ln2/\lambda$
  2. $2\ln2/\lambda$
  3. $3\ln2/\lambda$
  4. $4\ln2/\lambda$

Answer: (D) $4\ln2/\lambda$

$A=A_0e^{-\lambda t}$; $1/16=e^{-4\ln2}$, so $t=4\ln2/\lambda$.

Q18. An ideal gas expands isobarically from $V$ to $2V$. If initial temperature is $T$, final temperature is:
  1. $T/2$
  2. $T$
  3. $2T$
  4. $4T$

Answer: (C) $2T$

At constant pressure, $V/T$ is constant, so doubling $V$ doubles $T$.

Q19. A reversible adiabatic process satisfies:
  1. $PV=\text{constant}$
  2. $PV^\gamma=\text{constant}$
  3. $P/T=\text{constant}$
  4. $PT=\text{constant}$

Answer: (B) $PV^\gamma=\text{constant}$

For a reversible adiabatic process, $PV^\gamma$ is constant.

Q20. Capillary rise is $h$ in a tube of radius $r$. If radius becomes $r/3$, the rise becomes:
  1. $h/3$
  2. $h$
  3. $3h$
  4. $9h$

Answer: (C) $3h$

$h\propto1/r$, so reducing radius to $r/3$ makes the rise $3h$.

Q21. A wire has extension $\Delta L=FL/(AY)$. If both $L$ and $A$ are doubled for the same force and material, the new extension is:
  1. $\Delta L/2$
  2. $\Delta L$
  3. $2\Delta L$
  4. $4\Delta L$

Answer: (B) $\Delta L$

$\Delta L^\prime=F(2L)/(2AY)=\Delta L$.

Q22. A body is projected vertically upward with speed $u$. Ratio of time to reach maximum height to total flight time is:
  1. $1:1$
  2. $1:2$
  3. $2:1$
  4. $1:4$

Answer: (B) $1:2$

Upward time is $u/g$ and total time is $2u/g$.

Q23. A block slides down a rough incline of angle $\theta$ with coefficient $\mu$. Its acceleration is zero when:
  1. $\tan\theta=\mu$
  2. $\tan\theta=1/\mu$
  3. $\sin\theta=\mu$
  4. $\cos\theta=\mu$

Answer: (A) $\tan\theta=\mu$

$mg\sin\theta=\mu mg\cos\theta$, hence $\tan\theta=\mu$.

Q24. A solid sphere rolls without slipping down from height $h$. Its speed at the bottom is:
  1. $\sqrt{2gh}$
  2. $\sqrt{10gh/7}$
  3. $\sqrt{7gh/10}$
  4. $\sqrt{gh}$

Answer: (B) $\sqrt{10gh/7}$

$mgh=\frac12mv^2+\frac12(2mR^2/5)(v^2/R^2)=7mv^2/10$, so $v=\sqrt{10gh/7}$.

Q25. A pendulum clock is taken to a planet where $g$ is four times Earth value. Its period becomes:
  1. $T/4$
  2. $T/2$
  3. $2T$
  4. $4T$

Answer: (B) $T/2$

$T\propto1/\sqrt g$; with $g^\prime=4g$, $T^\prime=T/2$.

Q26. For a buffer containing equal concentrations of a weak acid $HA$ and its salt, the pH is:
  1. $pK_a$
  2. $2pK_a$
  3. $pK_a/2$
  4. $14-pK_a$

Answer: (A) $pK_a$

$pH=pK_a+\log([A^-]/[HA])=pK_a$.

Q27. A $0.1$ M weak acid has $K_a=10^{-5}$. Its approximate degree of ionisation is:
  1. $0.001$
  2. $0.01$
  3. $0.1$
  4. $0.1\%$

Answer: (B) $0.01$

$\alpha\approx\sqrt{K_a/C}=\sqrt{10^{-4}}=0.01$.

Q28. For $A(g)\rightleftharpoons2B(g)$, the relation between $K_p$ and $K_c$ is:
  1. $K_p=K_cRT$
  2. $K_p=K_c/(RT)$
  3. $K_p=K_c(RT)^2$
  4. $K_p=K_c$

Answer: (A) $K_p=K_cRT$

$K_p=K_c(RT)^{\Delta n}$ and $\Delta n=1$.

Q29. For a first-order reaction, time for $99\%$ completion is approximately:
  1. $2t_{1/2}$
  2. $4t_{1/2}$
  3. $6.64t_{1/2}$
  4. $10t_{1/2}$

Answer: (C) $6.64t_{1/2}$

Remaining fraction is $0.01$. Thus $t/t_{1/2}=\log_2(100)=6.64$.

Q30. If a first-order reaction has $k=0.0231\,min^{-1}$, its half-life is:
  1. $15$ min
  2. $20$ min
  3. $30$ min
  4. $60$ min

Answer: (C) $30$ min

$t_{1/2}=0.693/k\approx30$ min.

Q31. At equilibrium, if $Q_c
  1. Backward
  2. Forward
  3. Not at all
  4. Randomly

Answer: (B) Forward

The system must form more products until $Q_c$ reaches $K_c$.

Q32. Which free ion has the largest number of unpaired electrons?
  1. $Fe^{2+}$
  2. $Fe^{3+}$
  3. $Co^{2+}$
  4. $Ni^{2+}$

Answer: (B) $Fe^{3+}$

$Fe^{3+}$ is $3d^5$ and has five unpaired electrons.

Q33. The spin-only magnetic moment of a species with three unpaired electrons is:
  1. $\sqrt3$ BM
  2. $2\sqrt2$ BM
  3. $\sqrt{15}$ BM
  4. $3$ BM

Answer: (C) $\sqrt{15}$ BM

$\mu=\sqrt{n(n+2)}$ BM; for $n=3$, $\mu=\sqrt{15}$ BM.

Q34. The geometry and hybridisation of $[Ni(CN)_4]^{2-}$ are:
  1. Tetrahedral, $sp^3$
  2. Square planar, $dsp^2$
  3. Octahedral, $d^2sp^3$
  4. Linear, $sp$

Answer: (B) Square planar, $dsp^2$

$CN^-$ is strong field; $Ni^{2+}$ forms a square-planar $dsp^2$ complex.

Q35. If $E^\circ_{cell}>0$, then under standard conditions:
  1. $\Delta G^\circ>0$
  2. $\Delta G^\circ=0$
  3. $\Delta G^\circ<0$
  4. $K<1$

Answer: (C) $\Delta G^\circ<0$

$\Delta G^\circ=-nFE^\circ_{cell}$, so positive $E^\circ$ means negative $\Delta G^\circ$.

Q36. Increasing the concentration of products relative to reactants generally makes a galvanic-cell potential:
  1. Increase
  2. Decrease
  3. Unchanged
  4. Infinite

Answer: (B) Decrease

Nernst equation: $E=E^\circ-(RT/nF)\ln Q$; increasing $Q$ lowers $E$.

Q37. The oxidation state of Cr in $Cr_2O_7^{2-}$ is:
  1. $+3$
  2. $+4$
  3. $+6$
  4. $+7$

Answer: (C) $+6$

$2x-14=-2$ gives $x=+6$.

Q38. Which molecule has zero dipole moment despite polar bonds?
  1. $H_2O$
  2. $NH_3$
  3. $CO_2$
  4. $SO_2$

Answer: (C) $CO_2$

Linear symmetric $CO_2$ has equal and opposite bond dipoles.

Q39. The bond order of $O_2^+$ is:
  1. $1.5$
  2. $2$
  3. $2.5$
  4. $3$

Answer: (C) $2.5$

Removing one electron from antibonding $\pi^*$ raises the $O_2$ bond order from 2 to 2.5.

Q40. Among LiF, NaF, KF and RbF, the highest lattice enthalpy is expected for:
  1. LiF
  2. NaF
  3. KF
  4. RbF

Answer: (A) LiF

With the same charges, smaller ionic size gives stronger electrostatic attraction; $LiF$ is highest.

Q41. The major product of propene with HBr in presence of peroxide is:
  1. 1-bromopropane
  2. 2-bromopropane
  3. 1,2-dibromopropane
  4. Propane

Answer: (A) 1-bromopropane

Peroxide effect gives anti-Markovnikov radical addition of HBr.

Q42. A primary alcohol is best converted selectively to an aldehyde using:
  1. Excess $KMnO_4$
  2. PCC
  3. Excess $K_2Cr_2O_7/H^+$
  4. $O_3$

Answer: (B) PCC

PCC oxidises a primary alcohol to an aldehyde under controlled conditions.

Q43. Phenol is more acidic than ethanol because:
  1. Phenol has lower mass
  2. Phenoxide is resonance-stabilised
  3. Ethanol is aromatic
  4. Phenol has no O-H bond

Answer: (B) Phenoxide is resonance-stabilised

Resonance delocalisation stabilises the phenoxide ion.

Q44. Which compound gives a positive Tollens test?
  1. Acetone
  2. Propanone
  3. Ethanal
  4. Diethyl ether

Answer: (C) Ethanal

Aldehydes reduce Tollens reagent; ethanal is an aldehyde.

Q45. Aniline is less basic than ethylamine because:
  1. It has no nitrogen
  2. Its N lone pair is delocalised into the benzene ring
  3. It is aromatic
  4. It has no lone pair

Answer: (B) Its N lone pair is delocalised into the benzene ring

Resonance makes the nitrogen lone pair in aniline less available for protonation.

Q46. The number of geometrical isomers of square-planar $[Pt(NH_3)_2Cl_2]$ is:
  1. $1$
  2. $2$
  3. $3$
  4. $4$

Answer: (B) $2$

There are cis and trans forms, so two geometrical isomers.

Q47. Which is a condensation polymer?
  1. Polyethene
  2. PVC
  3. Nylon-6,6
  4. Polystyrene

Answer: (C) Nylon-6,6

Nylon-6,6 forms by condensation of a diamine and a dicarboxylic acid.

Q48. For an ideal solution, the activity coefficient of each component is:
  1. $0$
  2. $1$
  3. Always greater than 1
  4. Always less than 1

Answer: (B) $1$

Ideal solutions obey Raoult law and have activity coefficients equal to one.

Q49. For a very dilute solution of a macromolecule, the most suitable colligative property for molar-mass determination is:
  1. Elevation of boiling point
  2. Depression of freezing point
  3. Osmotic pressure
  4. Relative lowering of vapour pressure

Answer: (C) Osmotic pressure

Osmotic pressure is especially sensitive and measurable for dilute macromolecular solutions.

Q50. The standard enthalpy of formation of an element in its most stable standard state is:
  1. $+1$ kJ mol$^{-1}$
  2. $0$
  3. $-1$ kJ mol$^{-1}$
  4. Always positive

Answer: (B) $0$

By definition it is zero for an element in its standard state.

Q51. If $|z-1|=|z+1|$, the locus of $z$ is:
  1. Real axis
  2. Imaginary axis
  3. Unit circle
  4. $y=x$

Answer: (B) Imaginary axis

Writing $z=x+iy$ and squaring gives $(x-1)^2+y^2=(x+1)^2+y^2$, hence $x=0$.

Q52. How many roots of $z^4=16$ have positive real part?
  1. $1$
  2. $2$
  3. $3$
  4. $4$

Answer: (A) $1$

The roots are $2,2i,-2,-2i$; only $2$ has positive real part.

Q53. The coefficient of $x^5$ in $(1+x)^3(1+x^2)^4$ is:
  1. $18$
  2. $20$
  3. $22$
  4. $24$

Answer: (C) $22$

For $i+2j=5$, $(i,j)=(1,2),(3,1)$. Coefficient $=\binom31\binom42+\binom33\binom41=18+4=22$.

Q54. If $x^2-2ax+a^2-b^2=0$ has equal roots, then:
  1. $b=0$
  2. $a=0$
  3. $a=b$
  4. $a=-b$

Answer: (A) $b=0$

Discriminant is $4b^2$. Equal roots require $b=0$.

Q55. For a GP with first term $a$ and ratio $r\ne1$, which are equivalent forms of the sum $S_n$?
  1. $a(r^n-1)/(r-1)$
  2. $a(1-r^n)/(1-r)$
  3. Both A and B
  4. Neither

Answer: (C) Both A and B

The first two expressions are algebraically identical.

Q56. If $A$ is a $3\times3$ matrix with $|A|=3$, then $|A^{-1}|$ is:
  1. $-3$
  2. $1/3$
  3. $3$
  4. $9$

Answer: (B) $1/3$

$|A^{-1}|=1/|A|=1/3$.

Q57. For non-zero vectors $\vec a,\vec b$, if $\vec a\times\vec b=0$, they are:
  1. Perpendicular
  2. Parallel or antiparallel
  3. Equal only
  4. Always identical

Answer: (B) Parallel or antiparallel

$ab\sin\theta=0$ with non-zero magnitudes gives $\theta=0$ or $\pi$.

Q58. The angle between $(1,1,0)$ and $(1,0,1)$ is:
  1. $30^\circ$
  2. $45^\circ$
  3. $60^\circ$
  4. $90^\circ$

Answer: (C) $60^\circ$

Dot product is 1 and magnitudes are $\sqrt2$ each, so $\cos\theta=1/2$.

Q59. The tangent to $y=x^3-3x$ at $x=1$ is:
  1. $y=-2$
  2. $y=2x-4$
  3. $y=-2x$
  4. $y=2x$

Answer: (A) $y=-2$

At $(1,-2)$ the derivative is $3(1)^2-3=0$, hence the tangent is $y=-2$.

Q60. The maximum value of $x^2(4-x)$ on $[0,4]$ is:
  1. $256/27$
  2. $128/27$
  3. $64/9$
  4. $32/9$

Answer: (A) $256/27$

Using f^\prime=x(8-3x), the interior maximum is at x=8/3, giving f=256/27.

Q61. $\lim_{x\to0}(\sin x-x\cos x)/x^3$ equals:
  1. $1/3$
  2. $1/2$
  3. $1$
  4. $2/3$

Answer: (A) $1/3$

Using expansions, numerator $=x^3/3+O(x^5)$, so the limit is $1/3$.

Q62. If $y=x^x$, then $y^{\prime\prime}$ is:
  1. $x^x[(\ln x+1)^2+1/x]$
  2. $x^x(\ln x+1)$
  3. $x^{x-1}$
  4. $x^x(\ln x)^2$

Answer: (A) $x^x[(\ln x+1)^2+1/x]$

$y^\prime=x^x(\ln x+1)$; differentiating gives $y^{\prime\prime}=x^x[(\ln x+1)^2+1/x]$.

Q63. $\int dx/(x^2+a^2)$ for $a>0$ equals:
  1. $\frac1a\tan^{-1}(x/a)+C$
  2. $a\tan^{-1}(x/a)+C$
  3. $\ln(x^2+a^2)+C$
  4. $1/(x^2+a^2)+C$

Answer: (A) $\frac1a\tan^{-1}(x/a)+C$

This is the standard integral $\frac1a\tan^{-1}(x/a)+C$.

Q64. $\int_0^{\pi/2}\sin^3x\,dx$ equals:
  1. $1/3$
  2. $2/3$
  3. $1$
  4. $4/3$

Answer: (B) $2/3$

Write $\sin^3x=\sin x(1-\cos^2x)$ and put $u=\cos x$; the result is $2/3$.

Q65. The area enclosed between $y=x^2$ and $y=2x$ is:
  1. $2/3$
  2. $4/3$
  3. $8/3$
  4. $4$

Answer: (B) $4/3$

Intersections are 0 and 2. Area $=\int_0^2(2x-x^2)dx=4/3$.

Q66. The distance of $(1,2)$ from $3x+4y-12=0$ is:
  1. $1/5$
  2. $1$
  3. $2$
  4. $3$

Answer: (A) $1/5$

Distance $=|3+8-12|/5=1/5$.

Q67. The circle with diameter endpoints $(1,2)$ and $(5,6)$ is:
  1. $(x-3)^2+(y-4)^2=8$
  2. $(x-3)^2+(y-4)^2=4$
  3. $(x-2)^2+(y-3)^2=8$
  4. $(x-3)^2+(y-4)^2=16$

Answer: (A) $(x-3)^2+(y-4)^2=8$

Centre is $(3,4)$ and radius squared is $[(4)^2+(4)^2]/4=8$.

Q68. The eccentricity of $x^2/25+y^2/9=1$ is:
  1. $3/5$
  2. $4/5$
  3. $5/3$
  4. $2/5$

Answer: (B) $4/5$

$e=\sqrt{1-b^2/a^2}=\sqrt{1-9/25}=4/5$.

Q69. The parabola with focus $(2,0)$ and directrix $x=-2$ is:
  1. $y^2=8x$
  2. $y^2=4x$
  3. $x^2=8y$
  4. $y^2=16x$

Answer: (A) $y^2=8x$

Its vertex is origin and focal length is 2. Thus $y^2=4ax=8x$.

Q70. If $\tan^{-1}x+\tan^{-1}y=\pi/4$ for $x,y>0$, then:
  1. $x+y=1$
  2. $xy=1$
  3. $x+y+xy=1$
  4. $x-y=1$

Answer: (C) $x+y+xy=1$

Taking tangent gives $(x+y)/(1-xy)=1$, hence $x+y+xy=1$.

Q71. The probability that an integer from 1 to 100 is divisible by 2 or 5 is:
  1. $3/5$
  2. $1/2$
  3. $7/10$
  4. $2/5$

Answer: (A) $3/5$

By inclusion-exclusion: $50+20-10=60$ favourable integers, so probability $=3/5$.

Q72. Two fair dice are thrown. Probability that the sum is prime is:
  1. $5/12$
  2. $1/2$
  3. $7/18$
  4. $1/3$

Answer: (A) $5/12$

Prime sums 2,3,5,7,11 have $1+2+4+6+2=15$ outcomes, so $15/36=5/12$.

Q73. The variance of the first $n$ natural numbers is:
  1. $(n^2-1)/12$
  2. $(n^2+1)/12$
  3. $n(n+1)/12$
  4. $n^2/12$

Answer: (A) $(n^2-1)/12$

The standard result for $1,2,\ldots,n$ is $(n^2-1)/12$.

Q74. The mean of five observations is 12. If 8 is replaced by 18, the new mean is:
  1. $12$
  2. $13$
  3. $14$
  4. $15$

Answer: (C) $14$

Old sum is 60; new sum is $60-8+18=70$, so new mean is 14.

Q75. The number of distinct arrangements of the letters of LEVEL is:
  1. $20$
  2. $30$
  3. $60$
  4. $120$

Answer: (B) $30$

Two Ls and two Es repeat: $5!/(2!2!)=30$.