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JEE Main Full Mock Test: Practice Test 05

Question paper with answer key. Selection Pro

Q1. A block of mass $2\,\text{kg}$ is attached to a spring of force constant $200\,\text{N/m}$. If the block is displaced by $0.10\,\text{m}$ from its equilibrium position and released, its maximum speed is:
  1. $0.5\,\text{m/s}$
  2. $1\,\text{m/s}$
  3. $\sqrt{2}\,\text{m/s}$
  4. $2\,\text{m/s}$

Answer: (B) $1\,\text{m/s}$

For SHM, $\frac{1}{2}kA^2=\frac{1}{2}mv_{\max}^2$. Hence $v_{\max}=A\sqrt{\frac{k}{m}}=0.10\sqrt{100}=1\,\text{m/s}$.

Q2. A projectile is fired with speed $20\,\text{m/s}$ at an angle $30^\circ$ to the horizontal. Taking $g=10\,\text{m/s}^2$, its horizontal range is:
Diagram for question 2
  1. $10\sqrt{3}\,\text{m}$
  2. $20\sqrt{3}\,\text{m}$
  3. $30\,\text{m}$
  4. $40\,\text{m}$

Answer: (B) $20\sqrt{3}\,\text{m}$

$R=\frac{u^2\sin2\theta}{g}=\frac{400\sin60^\circ}{10}=20\sqrt{3}\,\text{m}$.

Q3. A body of mass $m$ moving with speed $u$ collides perfectly inelastically with an identical body at rest. The loss of kinetic energy is:
  1. $\frac{1}{8}mu^2$
  2. $\frac{1}{4}mu^2$
  3. $\frac{1}{2}mu^2$
  4. $\frac{3}{4}mu^2$

Answer: (B) $\frac{1}{4}mu^2$

Common speed is $u/2$. Initial KE is $\frac12mu^2$ and final KE is $\frac12(2m)(u/2)^2=\frac14mu^2$. Loss is $\frac14mu^2$.

Q4. A satellite moves in a circular orbit of radius $R$ around Earth with speed $v$. If its orbital radius becomes $4R$, its new speed is:
  1. $v/2$
  2. $v/4$
  3. $2v$
  4. $4v$

Answer: (A) $v/2$

Orbital speed varies as $v\propto 1/\sqrt{r}$. Thus $v'=v/\sqrt4=v/2$.

Q5. A conducting sphere of radius $R$ is charged to potential $V$. Its electrostatic energy is:
  1. $4\pi\varepsilon_0RV^2$
  2. $2\pi\varepsilon_0RV^2$
  3. $\frac12(4\pi\varepsilon_0R)V^2$
  4. $\frac{V^2}{4\pi\varepsilon_0R}$

Answer: (C) $\frac12(4\pi\varepsilon_0R)V^2$

Capacitance of an isolated sphere is $C=4\pi\varepsilon_0R$. Hence $U=\frac12CV^2=2\pi\varepsilon_0RV^2$.

Q6. Two capacitors of capacitances $C$ and $2C$ are connected in series across a battery of voltage $V$. The potential difference across the capacitor $C$ is:
  1. $V/3$
  2. $2V/3$
  3. $V/2$
  4. $V$

Answer: (B) $2V/3$

Series capacitors carry equal charge. $C_{eq}=\frac{2C}{3}$, so $Q=\frac{2CV}{3}$. Voltage across $C$ is $Q/C=2V/3$.

Q7. A cell of emf $E$ and internal resistance $r$ supplies current $I$ to an external resistance $R$. The terminal voltage is:
  1. $E+Ir$
  2. $E-Ir$
  3. $IR+Ir+E$
  4. $E/R$

Answer: (B) $E-Ir$

For a discharging cell, terminal voltage is $V=E-Ir=IR$.

Q8. A wire carries current $I$ in a magnetic field $B$ perpendicular to its length. If its length is doubled while current and field remain unchanged, the magnetic force becomes:
  1. Half
  2. Unchanged
  3. Double
  4. Four times

Answer: (C) Double

$F=BIL$ for perpendicular geometry, so doubling $L$ doubles the force.

Q9. An electron enters a uniform magnetic field perpendicular to the field with speed $v$. If its speed is doubled, its circular orbit radius becomes:
  1. $r/2$
  2. $r$
  3. $2r$
  4. $4r$

Answer: (C) $2r$

The radius is $r=mv/(eB)$, hence it is directly proportional to $v$.

Q10. A coil of $N$ turns and area $A$ rotates with angular speed $\omega$ in a uniform magnetic field $B$. The maximum induced emf is:
  1. $NBA$
  2. $NBA\omega$
  3. $NBA/\omega$
  4. $B/NA\omega$

Answer: (B) $NBA\omega$

For rotating coil, $\mathcal{E}_{\max}=NBA\omega$.

Q11. In an AC circuit, $V_{rms}=220\,\text{V}$. The peak voltage is:
  1. $110\,\text{V}$
  2. $220\,\text{V}$
  3. $220\sqrt2\,\text{V}$
  4. $440\,\text{V}$

Answer: (C) $220\sqrt2\,\text{V}$

$V_0=\sqrt2V_{rms}=220\sqrt2\,\text{V}$.

Q12. A convex lens of focal length $15\,\text{cm}$ is placed in contact with a concave lens of focal length $-30\,\text{cm}$. The equivalent focal length is:
Diagram for question 12
  1. $10\,\text{cm}$
  2. $30\,\text{cm}$
  3. $-30\,\text{cm}$
  4. $-10\,\text{cm}$

Answer: (B) $30\,\text{cm}$

$\frac1F=\frac1{15}-\frac1{30}=\frac1{30}$, hence $F=30\,\text{cm}$.

Q13. In Young's double-slit experiment, if the wavelength is doubled while slit separation and screen distance remain unchanged, fringe width becomes:
Diagram for question 13
  1. Half
  2. Unchanged
  3. Double
  4. Four times

Answer: (C) Double

Fringe width is $\beta=\lambda D/d$, so it doubles with wavelength.

Q14. A photon has energy $6.2\,\text{eV}$. Its approximate wavelength is:
  1. $100\,\text{nm}$
  2. $200\,\text{nm}$
  3. $400\,\text{nm}$
  4. $620\,\text{nm}$

Answer: (B) $200\,\text{nm}$

Using $\lambda(\text{nm})\approx1240/E(\text{eV})$, $\lambda\approx1240/6.2=200\,\text{nm}$.

Q15. The work function of a metal is $2\,\text{eV}$. Light of photon energy $5\,\text{eV}$ falls on it. The maximum kinetic energy of emitted electrons is:
  1. $2\,\text{eV}$
  2. $3\,\text{eV}$
  3. $5\,\text{eV}$
  4. $7\,\text{eV}$

Answer: (B) $3\,\text{eV}$

Einstein's equation gives $K_{\max}=E_\gamma-\phi=5-2=3\,\text{eV}$.

Q16. A radioactive sample has a half-life of $4$ hours. The fraction remaining after $12$ hours is:
  1. $1/2$
  2. $1/4$
  3. $1/8$
  4. $1/16$

Answer: (C) $1/8$

$12$ hours corresponds to 3 half-lives, so remaining fraction is $(1/2)^3=1/8$.

Q17. The de Broglie wavelength of a particle with momentum $p$ is $\lambda$. If its momentum becomes $3p$, the new wavelength is:
  1. $3\lambda$
  2. $\lambda$
  3. $\lambda/3$
  4. $\lambda/9$

Answer: (C) $\lambda/3$

Since $\lambda=h/p$, tripling momentum makes wavelength one-third.

Q18. A gas absorbs $500\,\text{J}$ of heat and does $200\,\text{J}$ of work. The change in internal energy is:
  1. $300\,\text{J}$
  2. $700\,\text{J}$
  3. $-300\,\text{J}$
  4. $-700\,\text{J}$

Answer: (A) $300\,\text{J}$

First law: $\Delta U=Q-W=500-200=300\,\text{J}$.

Q19. One mole of an ideal gas expands isothermally from $V$ to $3V$ at temperature $T$. Work done is:
  1. $RT\ln3$
  2. $3RT\ln3$
  3. $RT/\ln3$
  4. $2RT$

Answer: (A) $RT\ln3$

For one mole in isothermal expansion, $W=RT\ln(V_2/V_1)=RT\ln3$.

Q20. A metal wire of length $L$ and area $A$ has Young's modulus $Y$. Its spring constant for longitudinal deformation is:
  1. $YA/L$
  2. $YL/A$
  3. $Y/(AL)$
  4. $A/(YL)$

Answer: (A) $YA/L$

Stress $=Y\times$ strain gives $F/\Delta L=YA/L$.

Q21. A liquid drop of radius $R$ has surface tension $T$. Its excess pressure over atmosphere is:
  1. $T/R$
  2. $2T/R$
  3. $3T/R$
  4. $4T/R$

Answer: (B) $2T/R$

For a liquid drop with one surface, $\Delta P=2T/R$.

Q22. A block is projected up a rough incline with initial speed $u$. If friction acts opposite motion, the acceleration while moving upward is:
  1. $g\sin\theta-\mu g\cos\theta$
  2. $-g(\sin\theta+\mu\cos\theta)$
  3. $g(\sin\theta+\mu\cos\theta)$
  4. $-g(\sin\theta-\mu\cos\theta)$

Answer: (B) $-g(\sin\theta+\mu\cos\theta)$

Both gravity component and friction act down the incline, so acceleration is $-g(\sin\theta+\mu\cos\theta)$.

Q23. A rod of length $L$ rotates about one end with angular speed $\omega$. The speed of its midpoint is:
  1. $\omega L$
  2. $\omega L/2$
  3. $2\omega L$
  4. $\omega/L$

Answer: (B) $\omega L/2$

Linear speed is $v=\omega r$; at midpoint $r=L/2$.

Q24. A uniform disc and a uniform ring have the same mass and radius and roll without slipping with the same speed. Which has greater total kinetic energy?
  1. Disc
  2. Ring
  3. Both equal
  4. Cannot be determined

Answer: (B) Ring

$K=\frac12Mv^2+\frac12I\omega^2$. The ring has $I=MR^2$, larger than the disc's $MR^2/2$.

Q25. For a simple pendulum of length $L$, the time period is $T$. If its length is changed to $9L$, the new period is:
Diagram for question 25
  1. $T/9$
  2. $T/3$
  3. $3T$
  4. $9T$

Answer: (C) $3T$

$T\propto\sqrt L$, so $T'=\sqrt9T=3T$.

Q26. The pH of a $0.001\,\text{M}$ strong monoprotic acid is approximately:
  1. $1$
  2. $2$
  3. $3$
  4. $11$

Answer: (C) $3$

For a strong acid, $[H^+]=10^{-3}\,\text{M}$, so pH $=3$.

Q27. For a first-order reaction, the half-life is $20$ minutes. The time required for $87.5\%$ completion is:
  1. $20$ min
  2. $40$ min
  3. $60$ min
  4. $80$ min

Answer: (C) $60$ min

Remaining fraction is $12.5\%=1/8=(1/2)^3$, so three half-lives are required: $60$ min.

Q28. For the reaction $N_2+3H_2\rightleftharpoons2NH_3$, increasing pressure at constant temperature shifts equilibrium toward:
  1. Reactants
  2. Products
  3. No change
  4. Both equally

Answer: (B) Products

There are 4 moles of gaseous reactants and 2 moles of gaseous products, so higher pressure favors products.

Q29. The oxidation state of sulfur in $H_2SO_4$ is:
  1. $+4$
  2. $+5$
  3. $+6$
  4. $-2$

Answer: (C) $+6$

$2(+1)+x+4(-2)=0$, giving $x=+6$.

Q30. Which species is paramagnetic?
  1. $N_2$
  2. $O_2$
  3. $CO$
  4. $CN^-$

Answer: (B) $O_2$

Molecular orbital theory shows $O_2$ has two unpaired electrons in $\pi^*$ orbitals.

Q31. The hybridisation of the central atom in $SF_6$ is:
  1. $sp^2$
  2. $sp^3$
  3. $sp^3d$
  4. $sp^3d^2$

Answer: (D) $sp^3d^2$

Six sigma bonds around sulfur correspond to octahedral arrangement and $sp^3d^2$ hybridisation.

Q32. Among the following, the strongest oxidising agent under standard conditions is the species with the:
  1. Most negative $E^\circ$
  2. Most positive $E^\circ$
  3. Zero $E^\circ$
  4. Largest atomic mass

Answer: (B) Most positive $E^\circ$

A more positive standard reduction potential indicates greater tendency to undergo reduction and hence stronger oxidising character.

Q33. The molarity of a solution containing $4\,\text{g}$ of $NaOH$ in $500\,\text{mL}$ solution is:
  1. $0.1\,M$
  2. $0.2\,M$
  3. $0.4\,M$
  4. $0.8\,M$

Answer: (C) $0.4\,M$

Moles of NaOH $=4/40=0.1$. Volume $=0.5$ L, so molarity $=0.2\,M$.

Q34. Which molecule has the highest bond angle?
  1. $NH_3$
  2. $H_2O$
  3. $CH_4$
  4. $BF_3$

Answer: (D) $BF_3$

$BF_3$ is trigonal planar with bond angle $120^\circ$, larger than the others listed.

Q35. The major product of addition of HBr to propene in the absence of peroxide is:
  1. 1-bromopropane
  2. 2-bromopropane
  3. 1,2-dibromopropane
  4. Propane

Answer: (B) 2-bromopropane

In the absence of peroxide, HBr adds by Markovnikov's rule to give 2-bromopropane.

Q36. Which compound gives a positive iodoform test?
  1. Methanol
  2. Ethanol
  3. Propanol
  4. Methanoic acid

Answer: (B) Ethanol

Ethanol is oxidised to ethanal, which contains the required $CH_3CO-$ group for the iodoform reaction.

Q37. The number of stereoisomers possible for 2,3-dichlorobutane is:
  1. $2$
  2. $3$
  3. $4$
  4. $6$

Answer: (B) $3$

It has two stereocentres and a meso form, giving three stereoisomers in total.

Q38. Which reagent converts an alkene into a vicinal dihalide?
  1. $H_2/Ni$
  2. $Br_2/CCl_4$
  3. $KMnO_4$
  4. $O_3/Zn$

Answer: (B) $Br_2/CCl_4$

Bromine adds across the double bond to form a vicinal dibromide.

Q39. The major product of dehydration of 2-methyl-2-butanol is:
  1. 2-methyl-1-butene
  2. 2-methyl-2-butene
  3. 1-pentene
  4. 2-pentene

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

Dehydration follows Zaitsev's rule; the more substituted alkene, 2-methyl-2-butene, predominates.

Q40. Which of the following is a strong field ligand?
  1. $I^-$
  2. $Br^-$
  3. $F^-$
  4. $CN^-$

Answer: (D) $CN^-$

$CN^-$ is a strong-field ligand and causes significant splitting in coordination complexes.

Q41. The coordination number of the central metal ion in $[Co(NH_3)_5Cl]^{2+}$ is:
  1. $4$
  2. $5$
  3. $6$
  4. $7$

Answer: (C) $6$

There are five $NH_3$ ligands and one $Cl^-$ ligand directly attached to cobalt, giving coordination number $6$.

Q42. The geometry of $[Ni(CN)_4]^{2-}$ is generally:
  1. Tetrahedral
  2. Square planar
  3. Octahedral
  4. Trigonal planar

Answer: (B) Square planar

$CN^-$ is strong field and $Ni^{2+}$ in this complex forms a square-planar $dsp^2$ arrangement.

Q43. The major product of nitration of benzene is:
  1. Nitrobenzene
  2. Aniline
  3. Phenol
  4. Benzoic acid

Answer: (A) Nitrobenzene

Electrophilic substitution of benzene with nitrating mixture gives nitrobenzene.

Q44. Which group is meta-directing in electrophilic aromatic substitution?
  1. $-OH$
  2. $-NH_2$
  3. $-CH_3$
  4. $-NO_2$

Answer: (D) $-NO_2$

The strongly electron-withdrawing $-NO_2$ group is deactivating and meta-directing.

Q45. The product of reduction of nitrobenzene with $Sn/HCl$ is:
  1. Phenol
  2. Aniline
  3. Azobenzene
  4. Benzene

Answer: (B) Aniline

$Sn/HCl$ reduces the nitro group to an amino group, giving aniline.

Q46. The monomer used to prepare PVC is:
  1. Ethene
  2. Vinyl chloride
  3. Styrene
  4. Tetrafluoroethene

Answer: (B) Vinyl chloride

PVC is polyvinyl chloride, formed by polymerisation of $CH_2=CHCl$.

Q47. The number of $\sigma$ bonds in ethyne, $C_2H_2$, is:
  1. $2$
  2. $3$
  3. $4$
  4. $5$

Answer: (C) $4$

There are three sigma bonds: one C-C and two C-H bonds.

Q48. For an ideal gas, the compressibility factor $Z$ is:
  1. $0$
  2. $1$
  3. $RT$
  4. $P/V$

Answer: (B) $1$

For an ideal gas, $PV=nRT$, so $Z=PV/(nRT)=1$.

Q49. The enthalpy change for formation of one mole of a compound from its elements in their standard states is called:
  1. Bond enthalpy
  2. Standard enthalpy of formation
  3. Lattice enthalpy
  4. Ionisation enthalpy

Answer: (B) Standard enthalpy of formation

This is the definition of standard enthalpy of formation.

Q50. Which of the following has the highest boiling point due to strong hydrogen bonding?
  1. $CH_4$
  2. $NH_3$
  3. $H_2O$
  4. $H_2S$

Answer: (C) $H_2O$

Water forms an extensive three-dimensional hydrogen-bond network, giving it an unusually high boiling point.

Q51. If $z=3+4i$, then $|z|^2$ is:
  1. $5$
  2. $7$
  3. $25$
  4. $49$

Answer: (C) $25$

$|z|^2=3^2+4^2=25$.

Q52. The coefficient of $x^3$ in $(1+x)^8$ is:
  1. $28$
  2. $56$
  3. $70$
  4. $84$

Answer: (B) $56$

Coefficient is ${}^{8}C_3=56$.

Q53. If the roots of $x^2-6x+k=0$ are equal, then $k$ is:
  1. $6$
  2. $8$
  3. $9$
  4. $12$

Answer: (C) $9$

Equal roots require discriminant zero: $36-4k=0$, hence $k=9$.

Q54. The value of $\sum_{r=1}^{10}r$ is:
  1. $45$
  2. $50$
  3. $55$
  4. $60$

Answer: (C) $55$

$\sum r=n(n+1)/2=10\times11/2=55$.

Q55. The determinant $\begin{vmatrix}2&1\\3&4\end{vmatrix}$ is:
  1. $3$
  2. $5$
  3. $8$
  4. $11$

Answer: (B) $5$

Determinant $=2(4)-1(3)=5$.

Q56. If $\vec a\cdot\vec b=0$ for two non-zero vectors, the angle between them is:
  1. $0^\circ$
  2. $45^\circ$
  3. $90^\circ$
  4. $180^\circ$

Answer: (C) $90^\circ$

$\vec a\cdot\vec b=|a||b|\cos\theta=0$ and both vectors are non-zero, so $\theta=90^\circ$.

Q57. The equation of the tangent to $y=x^2$ at $x=1$ is:
  1. $y=x+1$
  2. $y=2x-1$
  3. $y=2x+1$
  4. $y=x-1$

Answer: (B) $y=2x-1$

At $(1,1)$, slope is $2$. Hence $y-1=2(x-1)$, or $y=2x-1$.

Q58. The maximum value of $x(4-x)$ for real $x$ is:
  1. $2$
  2. $4$
  3. $6$
  4. $8$

Answer: (B) $4$

$x(4-x)=4-(x-2)^2$, so maximum value is $4$.

Q59. The integral $\int_0^1 2x\,dx$ equals:
  1. $1/2$
  2. $1$
  3. $2$
  4. $4$

Answer: (B) $1$

$\int_0^1 2x\,dx=[x^2]_0^1=1$.

Q60. The derivative of $x^x$ for $x>0$ is:
  1. $x^x$
  2. $x^x\ln x$
  3. $x^x(\ln x+1)$
  4. $x^{x-1}$

Answer: (C) $x^x(\ln x+1)$

Taking logarithms, $\ln y=x\ln x$ gives $y'/y=\ln x+1$, hence $y'=x^x(\ln x+1)$.

Q61. The limit $\lim_{x\to0}\frac{e^x-1}{x}$ is:
  1. $0$
  2. $1$
  3. $e$
  4. $\infty$

Answer: (B) $1$

This is the standard exponential limit and equals $1$.

Q62. The distance between points $(1,2)$ and $(4,6)$ is:
  1. $4$
  2. $5$
  3. $6$
  4. $7$

Answer: (B) $5$

Distance $=\sqrt{(4-1)^2+(6-2)^2}=\sqrt{25}=5$.

Q63. The equation of the circle with centre $(2,-1)$ and radius $3$ is:
  1. $(x-2)^2+(y+1)^2=9$
  2. $(x+2)^2+(y-1)^2=9$
  3. $(x-2)^2+(y-1)^2=3$
  4. $(x+2)^2+(y+1)^2=3$

Answer: (A) $(x-2)^2+(y+1)^2=9$

Circle equation is $(x-h)^2+(y-k)^2=r^2$.

Q64. The eccentricity of a parabola is:
  1. $0$
  2. $1$
  3. $1/2$
  4. $\sqrt2$

Answer: (B) $1$

Every parabola has eccentricity $e=1$.

Q65. The eccentricity of a hyperbola is always:
  1. Less than $1$
  2. Equal to $1$
  3. Greater than $1$
  4. Zero

Answer: (C) Greater than $1$

A hyperbola has $e>1$.

Q66. If $\tan\theta=1$ and $0<\theta<\pi/2$, then $\theta$ is:
  1. $\pi/6$
  2. $\pi/4$
  3. $\pi/3$
  4. $\pi/2$

Answer: (B) $\pi/4$

In the first quadrant, $\tan\theta=1$ gives $\theta=\pi/4$.

Q67. The principal value of $\cos^{-1}(-1)$ is:
  1. $0$
  2. $\pi/4$
  3. $\pi/2$
  4. $\pi$

Answer: (D) $\pi$

The principal range of $\cos^{-1}x$ is $[0,\pi]$, so $\cos^{-1}(-1)=\pi$.

Q68. The probability of getting exactly one head when two fair coins are tossed is:
  1. $1/4$
  2. $1/2$
  3. $3/4$
  4. $1$

Answer: (B) $1/2$

Favourable outcomes are HT and TH, so probability is $2/4=1/2$.

Q69. The mean of $3,5,7,9,11$ is:
  1. $6$
  2. $7$
  3. $8$
  4. $9$

Answer: (B) $7$

Mean $=(3+5+7+9+11)/5=35/5=7$.

Q70. The variance of the numbers $1,1,1,1$ is:
  1. $0$
  2. $1$
  3. $2$
  4. $4$

Answer: (A) $0$

All observations are equal, so every deviation from the mean is zero and variance is $0$.

Q71. If $A$ is a $3\times3$ matrix and $|A|=2$, then $|2A|$ is:
  1. $4$
  2. $6$
  3. $8$
  4. $16$

Answer: (C) $8$

For an $n\times n$ matrix, $|kA|=k^n|A|$. Thus $|2A|=2^3\times2=16$.

Q72. The general solution of $\frac{dy}{dx}=3x^2$ is:
  1. $y=x^3+C$
  2. $y=3x^3+C$
  3. $y=x^2+C$
  4. $y=6x+C$

Answer: (A) $y=x^3+C$

Integrating gives $y=\int3x^2dx=x^3+C$.

Q73. The area enclosed between $y=x$ and the $x$-axis from $x=0$ to $x=2$ is:
  1. $1$
  2. $2$
  3. $3$
  4. $4$

Answer: (B) $2$

Area $=\int_0^2x\,dx=[x^2/2]_0^2=2$.

Q74. If $\alpha,\beta$ are roots of $x^2-5x+6=0$, then $\alpha^2+\beta^2$ is:
  1. $7$
  2. $11$
  3. $13$
  4. $25$

Answer: (C) $13$

$\alpha+\beta=5$ and $\alpha\beta=6$. Hence $\alpha^2+\beta^2=25-12=13$.

Q75. The number of ways of choosing $3$ students from $8$ students is:
  1. $24$
  2. $48$
  3. $56$
  4. $64$

Answer: (C) $56$

Number of choices is ${}^{8}C_3=56$.