JEE Main Full Mock Test: Practice Test 16

Question paper with answer key. Selection Pro

Q1. A particle executes SHM with amplitude $A$ and angular frequency $\omega$. At displacement $x=A/2$, the ratio of kinetic energy to potential energy is:

$3:1$
$1:1$
$1:3$
$3:4$

Answer: (A) $3:1$

$U/E=x^2/A^2=1/4$, so $K=3E/4$. Hence $K:U=3:1$.

Q2. A projectile is launched with speed $u$ at an angle $\theta$ such that its speed at the highest point is $u/3$. Its horizontal range is:

$\frac{2u^2}{3g}$
$\frac{8u^2}{9g}$
$\frac{4u^2}{9g}$
$\frac{4\sqrt2u^2}{9g}$

Answer: (D) $\frac{4\sqrt2u^2}{9g}$

At the highest point $u\cos\theta=u/3$, so $\cos\theta=1/3$. Therefore $\sin2\theta=4\sqrt2/9$, giving $R=4\sqrt2u^2/(9g)$.

Q3. A block of mass $m$ is attached between two identical springs, each of force constant $k$, on a frictionless surface. The angular frequency of small oscillations is:

$\sqrt{k/m}$
$\sqrt{2k/m}$
$\sqrt{m/2k}$
$2\sqrt{k/m}$

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

Both springs provide restoring force, so $F=-2kx$. Thus $m\ddot x+2kx=0$ and $\omega=\sqrt{2k/m}$.

Q4. A uniform rod of length $L$ and mass $M$ is pivoted at one end. A point mass $M$ is attached at the other end. The moment of inertia about the pivot is:

$\frac43ML^2$
$\frac12ML^2$
$ML^2$
$\frac13ML^2$

Answer: (A) $\frac43ML^2$

For the rod $I=ML^2/3$ and for the end mass $I=ML^2$. Total $I=4ML^2/3$.

Q5. A solid sphere rolls without slipping from rest through a vertical height $h$. Its speed at the bottom is:

$\sqrt{7gh/10}$
$\sqrt{10gh/7}$
$\sqrt{gh}$
$\sqrt{2gh}$

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

Using $I=2MR^2/5$ and $v=\omega R$, energy conservation gives $Mgh=7Mv^2/10$. Hence $v=\sqrt{10gh/7}$.

Q6. A satellite is shifted from a circular orbit of radius $R$ to another circular orbit of radius $9R$. The ratio of its orbital periods $T_1:T_2$ is:

$1:27$
$1:9$
$3:1$
$1:3$

Answer: (A) $1:27$

Kepler's law gives $T\propto r^{3/2}$, so $T_2/T_1=9^{3/2}=27$.

Q7. A capacitor of capacitance $C$ charged to voltage $V$ is disconnected from the battery and its plate separation is doubled. The new electrostatic energy is:

$2CV^2$
$\frac14CV^2$
$CV^2$
$\frac12CV^2$

Answer: (C) $CV^2$

Charge remains constant and capacitance becomes $C/2$. Since $U=Q^2/(2C)$, energy doubles to $CV^2$.

Q8. Two point charges $+q$ and $+4q$ are separated by distance $d$. The electric field is zero between them at distance $x$ from $+q$. Then $x$ is:

$d/4$
$d/2$
$2d/3$
$d/3$

Answer: (D) $d/3$

Equating field magnitudes: $q/x^2=4q/(d-x)^2$, hence $d-x=2x$ and $x=d/3$.

Q9. A wire of resistance $R$ is stretched uniformly to twice its original length, with volume constant. Its new resistance is:

$2R$
$R/2$
$4R$
$R$

Answer: (C) $4R$

Constant volume gives $A' = A/2$. Since $R=\rho L/A$, doubling length and halving area gives $R'=4R$.

Q10. A battery of emf $E$ and internal resistance $r$ delivers maximum power to a load. The load resistance and maximum power are:

$r/2,\ E^2/(2r)$
$r,\ E^2/r$
$r,\ E^2/(4r)$
$2r,\ E^2/(2r)$

Answer: (C) $r,\ E^2/(4r)$

Maximum power transfer requires $R=r$. Then $P_{max}=E^2/(4r)$.

Q11. A charged particle enters a uniform magnetic field with speed $v$ making angle $\theta$ with the field. The pitch of its helical path is:

$\frac{2\pi qB}{mv\cos\theta}$
$\frac{2\pi mv\cos\theta}{qB}$
$\frac{mv}{qB}$
$\frac{2\pi mv\sin\theta}{qB}$

Answer: (B) $\frac{2\pi mv\cos\theta}{qB}$

Parallel speed is $v\cos\theta$ and cyclotron period is $2\pi m/(qB)$. Their product is the pitch.

Q12. A conducting loop of resistance $R$ and area $A$ is in a uniform perpendicular magnetic field. If the field decreases uniformly from $B$ to zero in time $t$, the induced current magnitude is:

$B/(ARt)$
$BAt/R$
$BA/(Rt)$
$BR/(At)$

Answer: (C) $BA/(Rt)$

Induced emf is $BA/t$, so $I=BA/(Rt)$.

Q13. In a series LCR circuit at resonance, the impedance and phase difference between current and applied voltage are:

$\sqrt{R^2+(X_L+X_C)^2}$ and $\pi/2$
$R$ and $\pi/2$
$X_L-X_C$ and $0$
$R$ and $0$

Answer: (D) $R$ and $0$

At resonance $X_L=X_C$, hence net reactance is zero, $Z=R$, and current is in phase with voltage.

Q14. An ideal transformer has $N_p=1000$ turns and $N_s=100$ turns. If the primary voltage is $220$ V, the secondary voltage is:

$110$ V
$22$ V
$2.2$ V
$2200$ V

Answer: (B) $22$ V

$V_s/V_p=N_s/N_p=1/10$, hence $V_s=22$ V.

Q15. A convex lens of focal length $20$ cm forms a real image twice the size of the object. The object distance is:

$10$ cm
$30$ cm
$40$ cm
$20$ cm

Answer: (B) $30$ cm

For a real image formed by a convex lens, magnification $m = -2$. Using $m = \frac{f}{f+u}$ with $f = 20\text{ cm}$, we get $-2 = \frac{20}{20+u} \implies -40 - 2u = 20 \implies u = -30\text{ cm}$. Hence, $|u| = 30\text{ cm}$.

Q16. In Young's double-slit experiment, the original central intensity is $I_0$. If one slit amplitude is reduced to half while the other is unchanged, the new central intensity is:

$I_0$
$9I_0/16$
$I_0/2$
$I_0/4$

Answer: (B) $9I_0/16$

Original amplitude sum is $2a$; new sum is $3a/2$. Intensities are proportional to squared amplitudes, so ratio is $9/16$.

Q17. An electron and a proton are accelerated through the same potential difference. The ratio of their de Broglie wavelengths $\lambda_e/\lambda_p$ is:

$m_p/m_e$
$1$
$\sqrt{m_p/m_e}$
$\sqrt{m_e/m_p}$

Answer: (C) $\sqrt{m_p/m_e}$

For the same potential, $p=\sqrt{2mqV}$ and $\lambda\propto1/\sqrt m$. Thus $\lambda_e/\lambda_p=\sqrt{m_p/m_e}$.

Q18. A radioactive nucleus has decay constant $\lambda$. The ratio of mean life to half-life is:

$2$
$1/\ln2$
$\ln2$
$1/2$

Answer: (B) $1/\ln2$

$\tau=1/\lambda$ and $T_{1/2}=\ln2/\lambda$, so the ratio is $1/\ln2$.

Q19. For a reversible adiabatic process of an ideal gas, which relation is correct?

$V/T=\text{constant}$
$P/T=\text{constant}$
$PV^\gamma=\text{constant}$
$PV=\text{constant}$

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

The reversible adiabatic equation is $PV^\gamma=constant$.

Q20. One mole of an ideal monatomic gas is heated at constant pressure from temperature $T$ to $3T$. The work done is:

$3RT$
$2RT$
$\frac32RT$
$RT$

Answer: (B) $2RT$

At constant pressure, $W=nR\Delta T=R(3T-T)=2RT$.

Q21. A fluid flows through a horizontal pipe whose area changes from $A$ to $A/4$. If speed in the wider section is $v$, speed in the narrow section is:

$v$
$4v$
$2v$
$v/4$

Answer: (B) $4v$

Continuity gives $Av=(A/4)v_2$, hence $v_2=4v$.

Q22. A liquid rises to height $h$ in a capillary of radius $r$. If radius becomes $r/3$, all else unchanged, the new rise is:

$h/3$
$h$
$9h$
$3h$

Answer: (D) $3h$

Capillary rise varies inversely with radius, so reducing radius by 3 increases rise by 3.

Q23. A particle moves in a circle of radius $R$ with constant angular acceleration $\alpha$ from rest. Its tangential speed after time $t$ is:

$R/(\alpha t)$
$\alpha Rt$
$\alpha t^2/2$
$\alpha t/R$

Answer: (B) $\alpha Rt$

Starting from rest, $\omega=\alpha t$. Therefore $v=R\omega=\alpha Rt$.

Q24. A body of mass $m$ is attached to a vertical spring of constant $k$. Its equilibrium extension is:

$mgk$
$mg/k$
$g/k$
$k/mg$

Answer: (B) $mg/k$

At equilibrium $kx=mg$, so $x=mg/k$.

Q25. Two identical resistors are connected first in series and then in parallel across the same battery. The ratio of power in series to power in parallel is:

$1:4$
$1:2$
$2:1$
$4:1$

Answer: (A) $1:4$

Equivalent resistances are $2R$ and $R/2$. Since $P=V^2/R$, the ratio is $1:4$.

Q26. For a buffer containing equal concentrations of a weak acid HA and its salt NaA, the pH is:

$pK_b$
$pK_a$
$7$
$2pK_a$

Answer: (B) $pK_a$

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

Q27. For $Ag_2CrO_4\rightleftharpoons2Ag^++CrO_4^{2-}$, if molar solubility is $s$, then:

$K_{sp}=4s^3$
$K_{sp}=s^2$
$K_{sp}=2s^2$
$K_{sp}=s^3$

Answer: (A) $K_{sp}=4s^3$

$K_{sp}=(2s)^2(s)=4s^3$.

Q28. A first-order reaction has $k=0.0693\,min^{-1}$. Its half-life is:

$5$ min
$20$ min
$100$ min
$10$ min

Answer: (D) $10$ min

$t_{1/2}=0.693/k=10$ min.

Q29. For rate law $r=k[A]^2[B]$, the overall order is:

Answer: (C) 3

Overall order is $2+1=3$.

Q30. A galvanic cell has positive $E^\circ_{cell}$ when:

$\Delta G^\circ<0$
$K=0$
$\Delta G^\circ=0$
$\Delta G^\circ>0$

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

$\Delta G^\circ=-nFE^\circ_{cell}$, so positive cell potential corresponds to negative standard Gibbs energy.

Q31. At 298 K, for a one-electron cell with $E^\circ_{cell}=0.0591$ V, the equilibrium constant is:

$10$
$1000$
$100$
$1$

Answer: (A) $10$

$E^\circ=(0.0591/n)\log K$. With $n=1$, $\log K=1$, hence $K=10$.

Q32. The oxidation state of chromium in $K_2Cr_2O_7$ is:

$+7$
$+5$
$+6$
$+3$

Answer: (C) $+6$

$2(+1)+2x+7(-2)=0$ gives $x=+6$.

Q33. Which molecule has the largest bond angle?

$NH_3$
$BF_3$
$H_2O$
$CH_4$

Answer: (B) $BF_3$

$BF_3$ is trigonal planar with $120^\circ$ bond angles.

Q34. The molecular shape of $XeF_4$ is:

Square planar
See-saw
Tetrahedral
Trigonal bipyramidal

Answer: (A) Square planar

Six electron domains occur around Xe; two lone pairs occupy opposite positions, giving square-planar molecular geometry.

Q35. Which species has the highest bond order?

$O_2^-$
$O_2^{2-}$
$O_2^+$
$O_2$

Answer: (C) $O_2^+$

Removing one antibonding electron from O2 raises bond order from 2 to 2.5 in $O_2^+$.

Q36. Among these, the strongest acid in water is:

$Cl_3CCOOH$
$ClCH_2COOH$
$CH_3COOH$
$Cl_2CHCOOH$

Answer: (A) $Cl_3CCOOH$

The -I effect of chlorine stabilizes the conjugate base; three Cl atoms give the strongest acid.

Q37. The major product of dehydration of 3-methyl-2-butanol is:

2-methylbutane
3-methyl-1-butene
2-methyl-2-butene
2-methyl-1-butene

Answer: (C) 2-methyl-2-butene

Acid-catalyzed dehydration predominantly gives the Zaitsev alkene, 2-methyl-2-butene.

Q38. Which reagent converts a primary alcohol to an aldehyde without significant over-oxidation?

$K_2Cr_2O_7/H^+$
PCC
$KMnO_4$
$LiAlH_4$

Answer: (B) PCC

PCC is a mild oxidant that stops primary alcohol oxidation at the aldehyde stage.

Q39. The major product of phenol with bromine water is:

p-bromophenol only
No reaction
Bromobenzene
2,4,6-tribromophenol

Answer: (D) 2,4,6-tribromophenol

The strongly activating -OH group directs bromination to both ortho and para positions, giving 2,4,6-tribromophenol.

Q40. Aniline is less basic than cyclohexylamine mainly because:

Cyclohexylamine has no lone pair
Nitrogen in aniline is sp3
Aniline is ionic
Its lone pair is delocalized into the benzene ring

Answer: (D) Its lone pair is delocalized into the benzene ring

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

Q41. The number of stereoisomers of 2,3-dibromopentane is:

Answer: (D) 4

There are two stereogenic centers and no internal plane of symmetry, giving $2^2=4$ stereoisomers.

Q42. Which reagent enables anti-Markovnikov addition of HBr to an alkene?

$Br_2/CCl_4$
Peroxide ($ROOR$)
$Hg(OAc)_2$
$H_2SO_4$

Answer: (B) Peroxide ($ROOR$)

Peroxide initiates a radical chain mechanism for anti-Markovnikov addition of HBr.

Q43. Ozonolysis of 2-methylpropene followed by reductive work-up gives:

Acetic acid only
Acetone + formaldehyde
Propanone only
Methanol + CO

Answer: (B) Acetone + formaldehyde

Cleavage of $CH_2=C(CH_3)_2$ produces $HCHO$ and acetone.

Q44. In $[Fe(CN)_6]^{4-}$, the oxidation state of Fe and number of unpaired electrons are:

+4, 0
+3, 1
+2, 2
+2, 0

Answer: (D) +2, 0

Fe is +2. It is d6 and CN- is strong-field, giving low-spin $t_{2g}^6$ with zero unpaired electrons.

Q45. The geometry of $[PtCl_4]^{2-}$ is generally:

Trigonal planar
Square planar
Tetrahedral
Octahedral

Answer: (B) Square planar

Pt(II) is d8 and commonly adopts square-planar geometry.

Q46. Which has the greatest lattice enthalpy magnitude?

NaCl
CaO
MgO
KCl

Answer: (C) MgO

The 2+/2- ionic charges in MgO produce much stronger electrostatic attraction than the 1+/1- salts.

Q47. For an ideal solution, the enthalpy of mixing is:

Infinite
Zero
Positive
Negative

Answer: (B) Zero

By definition, an ideal solution has $\Delta H_{mix}=0$.

Q48. A non-volatile solute raises the boiling point because it:

Increases vapour pressure
Decreases molar mass
Makes surface tension zero
Decreases vapour pressure

Answer: (D) Decreases vapour pressure

The solute lowers solvent vapour pressure, requiring a higher temperature to reach external pressure.

Q49. The van't Hoff factor for complete dissociation of $Al_2(SO_4)_3$ is:

Answer: (B) 5

Complete dissociation gives 2 Al3+ and 3 sulfate ions, i.e. 5 particles.

Q50. Among Na, Mg, Al and Si, the highest first ionization enthalpy is for:

Answer: (A) Si

Across Period 3, Si has the highest first ionization enthalpy among these four.

Q51. If $z$ satisfies $|z-2|=|z+2|$, the locus of $z=x+iy$ is:

$x=0$
$y=0$
$y=2$
$x=2$

Answer: (A) $x=0$

The locus is the perpendicular bisector of $(2,0)$ and $(-2,0)$, hence $x=0$.

Q52. The number of roots of $z^4=16$ lying on the real axis is:

Answer: (B) 2

The four roots have modulus 2; the real ones are $2$ and $-2$.

Q53. The coefficient of $x^5$ in $(1+x)^9(1-x)^5$ is:

$-30$
$-36$
$20$
$-20$

Answer: (C) $20$

Rewrite as $(1+x)^4(1-x^2)^5$. Expanding $(1+x)^4 = 1+4x+6x^2+4x^3+x^4$ and $(1-x^2)^5 = 1-5x^2+10x^4-\dots$. The $x^5$ term comes from $(4x)(10x^4) + (4x^3)(-5x^2) = 40x^5 - 20x^5 = 20x^5$. So the coefficient is $20$.

Q54. If the roots of $x^2-2(a+1)x+a^2+1=0$ are equal, then $a$ is:

$0$
$1$
$-1$
$2$

Answer: (A) $0$

Discriminant $=4[(a+1)^2-(a^2+1)]=8a$. Equal roots require it to be zero, so $a=0$.

Q55. The determinant of a square matrix with two identical rows is:

$1$
$-1$
$0$
Cannot be determined

Answer: (C) $0$

A determinant is zero whenever two rows or two columns are identical.

Q56. If $A^2=A$, the possible eigenvalues of $A$ are:

Only 1
Only 0
$-1$ and $1$
$0$ and $1$

Answer: (D) $0$ and $1$

For an eigenvalue $\lambda$, $\lambda^2=\lambda$, hence $\lambda=0$ or 1.

Q57. The maximum value of $x/(1+x^2)$ for real $x$ is:

$1/4$
$1$
$2$
$1/2$

Answer: (D) $1/2$

For $x>0$, $1+x^2\ge2x$, so the ratio is at most $1/2$, attained at $x=1$.

Q58. For $f(x)=x^3-3x$, the number of local extrema is:

Answer: (A) 2

$f'(x)=3(x^2-1)$ has critical points $x=\pm1$, and the derivative changes sign at both.

Q59. The area under $y=\sqrt{x}$ from $0$ to $4$ is:

$32/3$
$8/3$
$4$
$16/3$

Answer: (D) $16/3$

$\int_0^4x^{1/2}dx=\frac23(4^{3/2})=16/3$.

Q60. The value of $\int_0^{\pi/2}\sin^3x\,dx$ is:

$1$
$1/3$
$2/3$
$4/3$

Answer: (C) $2/3$

Using $\sin^3x=\sin x(1-\cos^2x)$ gives $\int_0^1(1-u^2)du=2/3$.

Q61. The limit $\lim_{x\to0}(\sin x-x\cos x)/x^3$ equals:

$1/6$
$1/3$
$1/2$
$2/3$

Answer: (B) $1/3$

Expansions give $\sin x=x-x^3/6+\cdots$ and $x\cos x=x-x^3/2+\cdots$, so the difference divided by $x^3$ tends to $1/3$.

Q62. The general solution of $dy/dx=y/x$ for $x\ne0$ is:

$y=x+C$
$y=Cx$
$y=Ce^x$
$y=C/x$

Answer: (B) $y=Cx$

Separating gives $dy/y=dx/x$, hence $\ln|y|=\ln|x|+C$ and $y=Cx$.

Q63. The solution of $dy/dx+y=e^x$ satisfying $y(0)=0$ is:

$\frac12(e^x+e^{-x})$
$e^x-1$
$xe^x$
$\frac12(e^x-e^{-x})$

Answer: (D) $\frac12(e^x-e^{-x})$

The integrating factor is $e^x$. Thus $(ye^x)'=e^{2x}$, giving $y=\frac12(e^x-e^{-x})$ after applying $y(0)=0$.

Q64. The distance of $(2,-3)$ from the line $3x-4y+5=0$ is:

$4$
$5$
$3$
$23/5$

Answer: (D) $23/5$

Distance is $|3(2)-4(-3)+5|/\sqrt{9+16}=23/5$.

Q65. The circle passing through the origin and having centre $(3,4)$ is:

$x^2+y^2+6x+8y=0$
$x^2+y^2=25$
$x^2+y^2-6x-8y=0$
$x^2+y^2-3x-4y=0$

Answer: (C) $x^2+y^2-6x-8y=0$

Radius squared is $25$. Expanding $(x-3)^2+(y-4)^2=25$ gives the stated equation.

Q66. The length of the tangent from $(5,0)$ to the circle $x^2+y^2=9$ is:

$3$
$4$
$\sqrt{34}$
$2$

Answer: (B) $4$

Tangent length is $\sqrt{5^2-3^2}=4$.

Q67. The eccentricity of $x^2/25+y^2/9=1$ is:

$9/25$
$5/3$
$3/5$
$4/5$

Answer: (D) $4/5$

Here $a=5,b=3,c=4$, so $e=c/a=4/5$.

Q68. The tangent to $x^2+y^2=25$ at $(3,4)$ is:

$x+y=7$
$4x+3y=25$
$3x+4y=25$
$3x-4y=25$

Answer: (C) $3x+4y=25$

The tangent formula is $xx_1+yy_1=r^2$, hence $3x+4y=25$.

Q69. If $\vec a=(1,2,2)$ and $\vec b=(2,-1,1)$, then $|\vec a\times\vec b|$ is:

$5\sqrt2$
$5$
$7$
$\sqrt{35}$

Answer: (A) $5\sqrt2$

$\vec a\times\vec b=(4,3,-5)$, whose magnitude is $\sqrt{16+9+25}=5\sqrt2$.

Q70. The angle between vectors $(1,1,0)$ and $(1,0,1)$ is:

$45^\circ$
$30^\circ$
$60^\circ$
$90^\circ$

Answer: (C) $60^\circ$

Dot product is 1 and both magnitudes are $\sqrt2$, so $\cos\theta=1/2$ and $\theta=60^\circ$.

Q71. A fair die is thrown twice. The probability that the sum is a prime number is:

$5/12$
$7/18$
$13/36$
$1/2$

Answer: (A) $5/12$

Prime sums 2,3,5,7,11 have respectively 1,2,4,6,2 outcomes: 15 out of 36, giving $5/12$.

Q72. From five distinct letter types $A,B,C,D,E$, the number of ways of selecting exactly three distinct letters is:

$15$
$10$
$5$
$20$

Answer: (B) $10$

The number of selections is $\binom53=10$.

Q73. The coefficient of variation for a dataset with mean 20 and standard deviation 4 is:

$5\%$
$25\%$
$20\%$
$80\%$

Answer: (C) $20\%$

CV $=(4/20)\times100=20\%$.

Q74. If $Var(X)=9$, then $Var(5X-7)$ is:

$25$
$18$
$225$
$9$

Answer: (C) $225$

$Var(aX+b)=a^2Var(X)$, so $25\times9=225$.

Q75. The number of onto functions from a 3-element set to a 2-element set is:

Answer: (C) $6$

There are $2^3=8$ total functions and 2 constant non-onto functions, so $8-2=6$.