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JEE Main Full Mock Test: Practice Test 03

Question paper with answer key. Selection Pro

Q1. A particle of mass $m$ is projected with velocity $v_0$ at an angle of $45^\circ$ with the horizontal. The magnitude of angular momentum of the particle about the point of projection when it is at the highest point of its trajectory is:
  1. $\frac{m v_0^3}{2\sqrt{2} g}$
  2. $\frac{m v_0^3}{4\sqrt{2} g}$
  3. $\frac{m v_0^3}{\sqrt{2} g}$
  4. $\frac{m v_0^3}{2 g}$

Answer: (B) $\frac{m v_0^3}{4\sqrt{2} g}$

At the highest point, velocity is horizontal: $v_x = v_0 \cos 45^\circ = \frac{v_0}{\sqrt{2}}$. Maximum height $H = \frac{v_0^2 \sin^2 45^\circ}{2g} = \frac{v_0^2}{4g}$. Angular momentum about point of projection $L = m v_x H = m \left(\frac{v_0}{\sqrt{2}}\right) \left(\frac{v_0^2}{4g}\right) = \frac{m v_0^3}{4\sqrt{2} g}$.

Q2. An ideal gas undergoes an isothermal expansion from volume $V$ to $2V$ at temperature $T$. The work done by the gas is $W_1$. If the same gas undergoes an adiabatic expansion from $V$ to $2V$ starting from the same initial state, the work done is $W_2$. Which relation is correct?
  1. $W_1 < W_2$
  2. $W_1 = W_2$
  3. $W_1 > W_2$
  4. $W_1 = 2 W_2$

Answer: (C) $W_1 > W_2$

In a $P-V$ diagram, the isothermal curve lies above the adiabatic curve during expansion. Area under $P-V$ curve represents work done. Thus $W_{\text{isothermal}} > W_{\text{adiabatic}}$, i.e., $W_1 > W_2$.

Q3. A parallel plate capacitor with plate area $A$ and separation $d$ is charged to a potential difference $V$. The battery is then disconnected and a dielectric slab of thickness $d$ and dielectric constant $K$ is inserted. The work done in inserting the dielectric slab is:
  1. $\frac{1}{2} \frac{\varepsilon_0 A V^2}{d} (K - 1)$
  2. $\frac{1}{2} \frac{\varepsilon_0 A V^2}{d} \left(1 - \frac{1}{K}\right)$
  3. $\frac{\varepsilon_0 A V^2}{d} \left(1 - \frac{1}{K}\right)$
  4. $\frac{1}{2} \frac{\varepsilon_0 A V^2}{d} \left(1 + \frac{1}{K}\right)$

Answer: (B) $\frac{1}{2} \frac{\varepsilon_0 A V^2}{d} \left(1 - \frac{1}{K}\right)$

Initial energy $U_i = \frac{1}{2} C_0 V^2$. When battery is disconnected, charge $Q$ is constant. Final capacitance $C_f = K C_0$. Final energy $U_f = \frac{Q^2}{2 C_f} = \frac{U_i}{K}$. Work done magnitude $= U_i - U_f = \frac{1}{2} \frac{\varepsilon_0 A V^2}{d} \left(1 - \frac{1}{K}\right)$.

Q4. A particle performs simple harmonic motion with amplitude $A$ and time period $T$. The time taken by the particle to travel from $x = 0$ to $x = \frac{A}{\sqrt{2}}$ is:
  1. $\frac{T}{4}$
  2. $\frac{T}{6}$
  3. $\frac{T}{12}$
  4. $\frac{T}{8}$

Answer: (D) $\frac{T}{8}$

Equation of motion: $x = A \sin(\omega t)$. For $x = \frac{A}{\sqrt{2}}$, $\sin(\omega t) = \frac{1}{\sqrt{2}} \implies \omega t = \frac{\pi}{4} \implies \frac{2\pi}{T} t = \frac{\pi}{4} \implies t = \frac{T}{8}$.

Q5. A circular ring of radius $R$ carries a uniform charge $Q$. A point charge $q$ is placed on the axis of the ring at a distance $x$ from its center. The force on $q$ is maximum when $x$ is equal to:
  1. $R$
  2. $\frac{R}{\sqrt{2}}$
  3. $\sqrt{2} R$
  4. $\frac{R}{2}$

Answer: (B) $\frac{R}{\sqrt{2}}$

Electric field along axis of ring $E = \frac{k Q x}{(R^2 + x^2)^{3/2}}$. Differentiating $E$ w.r.t $x$ and equating to $0$ gives $x = \frac{R}{\sqrt{2}}$.

Q6. In an LCR series circuit, $R = 10\;\Omega$, $L = 100\text{ mH}$, and $C = 100\;\mu\text{F}$. The resonance frequency of the circuit in $\text{rad/s}$ is approximately:
  1. $100\text{ rad/s}$
  2. $1000\text{ rad/s}$
  3. $316\text{ rad/s}$
  4. $500\text{ rad/s}$

Answer: (C) $316\text{ rad/s}$

Resonant angular frequency $\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{100 \times 10^{-3} \times 100 \times 10^{-6}}} = \sqrt{10^5} \approx 316.2\text{ rad/s}$.

Q7. A planet revolves around the Sun in an elliptical orbit. If $v_1$ and $v_2$ are the velocities at perihelion and aphelion respectively, and $r_1$ and $r_2$ are the corresponding distances, then $v_1 / v_2$ equals:
  1. $\frac{r_1}{r_2}$
  2. $\frac{r_2}{r_1}$
  3. $\left(\frac{r_2}{r_1}\right)^2$
  4. $\sqrt{\frac{r_2}{r_1}}$

Answer: (B) $\frac{r_2}{r_1}$

By conservation of angular momentum: $m v_1 r_1 = m v_2 r_2 \implies \frac{v_1}{v_2} = \frac{r_2}{r_1}$.

Q8. The de Broglie wavelength of an electron accelerated from rest through a potential difference of $100\text{ V}$ is approximately:
  1. $0.123\text{ \AA}$
  2. $12.27\text{ \AA}$
  3. $1.227\text{ \AA}$
  4. $2.45\text{ \AA}$

Answer: (C) $1.227\text{ \AA}$

$\lambda = \frac{12.27}{\sqrt{V}}\text{ \AA} = \frac{12.27}{\sqrt{100}}\text{ \AA} = 1.227\text{ \AA}$.

Q9. A convex lens of focal length $20\text{ cm}$ forms a real image three times the size of the object. The magnitude of distance of the object from the lens is:
  1. $20\text{ cm}$
  2. $\frac{40}{3}\text{ cm}$
  3. $40\text{ cm}$
  4. $\frac{80}{3}\text{ cm}$

Answer: (D) $\frac{80}{3}\text{ cm}$

Magnification $m = -3 = \frac{v}{u} \implies v = -3u$. Lens formula: $\frac{1}{20} = \frac{1}{-3u} - \frac{1}{u} = -\frac{4}{3u} \implies 3u = -80 \implies |u| = \frac{80}{3}\text{ cm}$.

Q10. A long straight wire carries a current of $10\text{ A}$. The magnetic field at a distance of $5\text{ cm}$ from the wire is:
  1. $2 \times 10^{-5}\text{ T}$
  2. $4 \times 10^{-5}\text{ T}$
  3. $1 \times 10^{-5}\text{ T}$
  4. $8 \times 10^{-5}\text{ T}$

Answer: (B) $4 \times 10^{-5}\text{ T}$

$B = \frac{\mu_0 I}{2\pi r} = \frac{(4\pi \times 10^{-7})(10)}{2\pi (0.05)} = 4 \times 10^{-5}\text{ T}$.

Q11. The efficiency of a Carnot engine operating between $27^\circ\text{C}$ and $327^\circ\text{C}$ is:
  1. $25\%$
  2. $75\%$
  3. $50\%$
  4. $90\%$

Answer: (C) $50\%$

$T_1 = 600\text{ K}, T_2 = 300\text{ K}$. Efficiency $\eta = 1 - \frac{300}{600} = 0.5 = 50\%$.

Q12. A wire of length $L$ and resistance $R$ is stretched uniformly to twice its length. The new resistance of the wire will be:
  1. $2 R$
  2. $4 R$
  3. $R / 2$
  4. $R / 4$

Answer: (B) $4 R$

Since volume is constant, stretching to length $2L$ reduces area to $A/2$. New resistance $R' = \rho \frac{2L}{A/2} = 4R$.

Q13. Two sound waves of wavelengths $1.0\text{ m}$ and $1.01\text{ m}$ produce $4\text{ beats/s}$. The velocity of sound in the medium is:
  1. $330\text{ m/s}$
  2. $400\text{ m/s}$
  3. $404\text{ m/s}$
  4. $360\text{ m/s}$

Answer: (C) $404\text{ m/s}$

Beat frequency $= v \left(\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right) = v \left(1 - \frac{1}{1.01}\right) = 4 \implies v = 404\text{ m/s}$.

Q14. A force $\vec{F} = 3\hat{i} + 4\hat{j}\text{ N}$ acts on a body and displaces it by $\vec{d} = 3\hat{i} + 4\hat{k}\text{ m}$. The work done by the force is:
  1. $25\text{ J}$
  2. $12\text{ J}$
  3. $16\text{ J}$
  4. $9\text{ J}$

Answer: (D) $9\text{ J}$

$W = \vec{F} \cdot \vec{d} = (3)(3) + (4)(0) + (0)(4) = 9\text{ J}$.

Q15. The dimensional formula for magnetic flux is:
  1. $[M^1 L^2 T^{-2} A^{-1}]$
  2. $[M^1 L^1 T^{-2} A^{-1}]$
  3. $[M^1 L^2 T^{-1} A^{-2}]$
  4. $[M^1 L^0 T^{-2} A^{-1}]$

Answer: (A) $[M^1 L^2 T^{-2} A^{-1}]$

$[\Phi] = [B][A] = [M T^{-2} A^{-1}][L^2] = [M^1 L^2 T^{-2} A^{-1}]$.

Q16. In a p-n junction diode, the reverse saturation current is mainly due to:
  1. Diffusion of majority charge carriers
  2. Drift of minority charge carriers
  3. Drift of majority charge carriers
  4. Diffusion of minority charge carriers

Answer: (B) Drift of minority charge carriers

Reverse current in p-n junction is caused by the drift of minority charge carriers across the depletion layer under reverse bias.

Q17. A disc of mass $2\text{ kg}$ and radius $0.5\text{ m}$ rolls without slipping on a horizontal plane with velocity $4\text{ m/s}$. Total kinetic energy of the disc is:
  1. $16\text{ J}$
  2. $12\text{ J}$
  3. $24\text{ J}$
  4. $8\text{ J}$

Answer: (C) $24\text{ J}$

Total $\text{KE} = \frac{1}{2} m v^2 \left(1 + \frac{k^2}{R^2}\right) = \frac{1}{2}(2)(16)(1 + 0.5) = 24\text{ J}$.

Q18. An unpolarized light of intensity $I_0$ falls on a single ideal polarizer. The intensity of transmitted light is:
  1. $I_0$
  2. $I_0 / 2$
  3. $I_0 / 4$
  4. $I_0 / \sqrt{2}$

Answer: (B) $I_0 / 2$

When unpolarized light passes through a linear polarizer, the transmitted intensity is half of the initial intensity: $I = I_0 / 2$.

Q19. A body is thrown vertically upwards with velocity $u$. The ratio of times taken to reach half the maximum height during upward motion and downward motion from top is:
  1. $\sqrt{2} + 1$
  2. $1$
  3. $\sqrt{2} - 1$
  4. $\frac{1}{\sqrt{2}}$

Answer: (C) $\sqrt{2} - 1$

Time to top $t = u/g$. Time from top to $H/2$ is $t_2 = u/(\sqrt{2}g)$. Upward time to $H/2$ is $t_1 = t - t_2$. Ratio $t_1 / t_2 = \sqrt{2} - 1$.

Q20. The binding energy per nucleon is maximum for the nucleus:
  1. $\text{Fe}^{56}$
  2. $\text{U}^{235}$
  3. $\text{He}^4$
  4. $\text{O}^{16}$

Answer: (A) $\text{Fe}^{56}$

$\text{Fe}^{56}$ has maximum binding energy per nucleon (about $8.8\text{ MeV}$).

Q21. A block of mass $m$ slides down an inclined plane of inclination $30^\circ$ with constant velocity. The coefficient of kinetic friction between block and plane is:
  1. $\sqrt{3}$
  2. $\frac{1}{\sqrt{3}}$
  3. $\frac{1}{2}$
  4. $\frac{\sqrt{3}}{2}$

Answer: (B) $\frac{1}{\sqrt{3}}$

Constant velocity $\implies mg \sin 30^\circ = \mu_k mg \cos 30^\circ \implies \mu_k = \tan 30^\circ = \frac{1}{\sqrt{3}}$.

Q22. A soap bubble of radius $r$ has surface tension $T$. The excess pressure inside the bubble is:
  1. $\frac{2T}{r}$
  2. $\frac{T}{r}$
  3. $\frac{4T}{r}$
  4. $\frac{8T}{r}$

Answer: (C) $\frac{4T}{r}$

A soap bubble has two surfaces, so $\Delta P = \frac{4T}{r}$.

Q23. An electric dipole of dipole moment $\vec{p}$ is placed in a uniform electric field $\vec{E}$. The potential energy of dipole is minimum when angle between $\vec{p}$ and $\vec{E}$ is:
  1. $180^\circ$
  2. $90^\circ$
  3. $45^\circ$
  4. $0^\circ$

Answer: (D) $0^\circ$

$U = -p E \cos\theta$. Minimum value occurs at $\theta = 0^\circ$ where $U = -pE$.

Q24. In an electromagnetic wave, the amplitude of electric field is $60\text{ V/m}$. The amplitude of magnetic field is:
  1. $2 \times 10^{-7}\text{ T}$
  2. $1.8 \times 10^{10}\text{ T}$
  3. $2 \times 10^{-6}\text{ T}$
  4. $6 \times 10^{-7}\text{ T}$

Answer: (A) $2 \times 10^{-7}\text{ T}$

$B_0 = \frac{E_0}{c} = \frac{60}{3 \times 10^8} = 2 \times 10^{-7}\text{ T}$.

Q25. A copper wire of length $2\text{ m}$ is stretched by $1\text{ mm}$. If Young's modulus of copper is $1.1 \times 10^{11}\text{ N/m}^2$, energy stored per unit volume in the wire is:
  1. $2.75 \times 10^4\text{ J/m}^3$
  2. $1.375 \times 10^4\text{ J/m}^3$
  3. $5.5 \times 10^4\text{ J/m}^3$
  4. $1.1 \times 10^4\text{ J/m}^3$

Answer: (B) $1.375 \times 10^4\text{ J/m}^3$

Energy density $= \frac{1}{2} Y (\text{strain})^2 = \frac{1}{2} (1.1 \times 10^{11}) (0.5 \times 10^{-3})^2 = 1.375 \times 10^4\text{ J/m}^3$.

Q26. Which of the following molecules has zero dipole moment?
  1. $\text{NH}_3$
  2. $\text{H}_2\text{O}$
  3. $\text{BF}_3$
  4. $\text{SO}_2$

Answer: (C) $\text{BF}_3$

$\text{BF}_3$ has symmetrical trigonal planar geometry ($sp^2$), cancelling bond dipoles to give zero overall dipole moment.

Q27. The coordination number and oxidation state of cobalt in $[\text{Co(en)}_2\text{Cl}_2]^+$ are respectively:
  1. $6$ and $+3$
  2. $4$ and $+2$
  3. $6$ and $+2$
  4. $4$ and $+3$

Answer: (A) $6$ and $+3$

'en' is bidentate $\implies \text{Coordination Number} = 2(2) + 2 = 6$. Oxidation state: $x + 2(0) - 2 = +1 \implies x = +3$.

Q28. Among the following, the most basic compound in aqueous medium is:
  1. Methylamine
  2. Dimethylamine
  3. Trimethylamine
  4. Aniline

Answer: (B) Dimethylamine

For methyl amines in aqueous medium, order of basicity is $2^\circ > 1^\circ > 3^\circ > \text{Aniline}$. Dimethylamine is most basic.

Q29. The number of unpaired electrons in $[\text{Fe(CN)}_6]^{3-}$ is:
  1. $5$
  2. $3$
  3. $1$
  4. $0$

Answer: (C) $1$

$\text{Fe}^{3+} = 3d^5$. $\text{CN}^-$ is strong field, pairs electrons to $t_{2g}^5 e_g^0$, leaving 1 unpaired electron.

Q30. Which alkali metal carbonate is thermally least stable?
  1. $\text{Na}_2\text{CO}_3$
  2. $\text{K}_2\text{CO}_3$
  3. $\text{Cs}_2\text{CO}_3$
  4. $\text{Li}_2\text{CO}_3$

Answer: (D) $\text{Li}_2\text{CO}_3$

$\text{Li}^+$ has high polarising power due to small size, decomposing $\text{Li}_2\text{CO}_3$ easily upon heating.

Q31. The spin-only magnetic moment of $\text{Cr}^{3+}$ ion ($Z = 24$) is:
  1. $3.87\text{ BM}$
  2. $4.90\text{ BM}$
  3. $2.84\text{ BM}$
  4. $1.73\text{ BM}$

Answer: (A) $3.87\text{ BM}$

$\text{Cr}^{3+} = [\text{Ar}] 3d^3 \implies n = 3$. $\mu = \sqrt{3(5)} = \sqrt{15} \approx 3.87\text{ BM}$.

Q32. Which of the following compounds gives a positive Tollens' test?
  1. Acetone
  2. Glucose
  3. Acetophenone
  4. Acetic acid

Answer: (B) Glucose

Glucose is a reducing sugar containing an aldehyde group in open-chain form, giving positive Tollens' test.

Q33. The correct order of acidic strength of halogen hydracids is:
  1. $\text{HF} > \text{HCl} > \text{HBr} > \text{HI}$
  2. $\text{HCl} > \text{HBr} > \text{HI} > \text{HF}$
  3. $\text{HI} > \text{HBr} > \text{HCl} > \text{HF}$
  4. $\text{HBr} > \text{HI} > \text{HCl} > \text{HF}$

Answer: (C) $\text{HI} > \text{HBr} > \text{HCl} > \text{HF}$

Bond strength decreases down halogen group: $\text{HI} > \text{HBr} > \text{HCl} > \text{HF}$.

Q34. The half-life period for a zero order reaction $A \rightarrow B$ with initial concentration $[A]_0$ and rate constant $k$ is:
  1. $\frac{0.693}{k}$
  2. $\frac{[A]_0}{k}$
  3. $\frac{1}{k [A]_0}$
  4. $\frac{[A]_0}{2k}$

Answer: (D) $\frac{[A]_0}{2k}$

For zero order reaction $t_{1/2} = \frac{[A]_0}{2k}$.

Q35. Ozonolysis of 2-methylbut-2-ene followed by reduction with $\text{Zn/H}_2\text{O}$ gives:
  1. Propanone and Ethanal
  2. Propanal and Ethanal
  3. Methanal and Propanone
  4. Butanone and Methanal

Answer: (A) Propanone and Ethanal

$(\text{CH}_3)_2\text{C}=\text{CH}-\text{CH}_3 \xrightarrow{\text{O}_3, \text{Zn/H}_2\text{O}} (\text{CH}_3)_2\text{C=O} + \text{CH}_3\text{CHO}$.

Q36. Which of the following is an aromatic compound according to Huckel's rule?
  1. Cyclooctatetraene
  2. Cyclopentadienyl anion
  3. Cyclopentadienyl cation
  4. Cyclobutadiene

Answer: (B) Cyclopentadienyl anion

Cyclopentadienyl anion is planar and conjugated with $6\,\pi$ electrons ($4n+2$ for $n=1$).

Q37. The elevation in boiling point of a $0.1\text{ m}$ solution of $\text{BaCl}_2$ assuming complete dissociation is ($K_b = 0.52\text{ K}\cdot\text{kg/mol}$):
  1. $0.052\text{ K}$
  2. $0.104\text{ K}$
  3. $0.156\text{ K}$
  4. $0.208\text{ K}$

Answer: (C) $0.156\text{ K}$

$i = 3$ for $\text{BaCl}_2$. $\Delta T_b = i K_b m = 3 \times 0.52 \times 0.1 = 0.156\text{ K}$.

Q38. The oxidation number of phosphorus in orthophosphoric acid ($\text{H}_3\text{PO}_4$) is:
  1. $+3$
  2. $+1$
  3. $+4$
  4. $+5$

Answer: (D) $+5$

$3(+1) + x + 4(-2) = 0 \implies x = +5$.

Q39. Which of the following carbocations is most stable?
  1. Tropylium cation
  2. Allyl cation
  3. tert-Butyl cation
  4. Benzyl cation

Answer: (A) Tropylium cation

Tropylium cation is aromatic with $6\,\pi$ electrons delocalized over 7 ring carbons, making it exceptionally stable.

Q40. Lucas reagent is a mixture of:
  1. Hydrated $\text{ZnCl}_2$ and dilute $\text{HCl}$
  2. Anhydrous $\text{ZnCl}_2$ and concentrated $\text{HCl}$
  3. Anhydrous $\text{AlCl}_3$ and concentrated $\text{HCl}$
  4. Concentrated $\text{HNO}_3$ and concentrated $\text{H}_2\text{SO}_4$

Answer: (B) Anhydrous $\text{ZnCl}_2$ and concentrated $\text{HCl}$

Lucas reagent is equimolar anhydrous $\text{ZnCl}_2$ in concentrated $\text{HCl}$.

Q41. The IUPAC name of $\text{CH}_3-\text{CH}(\text{OH})-\text{CH}_2-\text{COOH}$ is:
  1. 2-Hydroxybutanoic acid
  2. 3-Hydroxybutyric acid
  3. 3-Hydroxybutanoic acid
  4. beta-Hydroxybutyric acid

Answer: (C) 3-Hydroxybutanoic acid

Chain is 4 carbons long with -COOH at position 1 and -OH at position 3: 3-hydroxybutanoic acid.

Q42. Which pair of orbitals has electronic density along the axes?
  1. $d_{xy}$ and $d_{yz}$
  2. $d_{xz}$ and $d_{z^2}$
  3. $d_{xy}$ and $d_{x^2-y^2}$
  4. $d_{z^2}$ and $d_{x^2-y^2}$

Answer: (D) $d_{z^2}$ and $d_{x^2-y^2}$

The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals are axial orbitals with electron lobes lying along coordinate axes.

Q43. The conjugate base of $\text{HSO}_4^-$ is:
  1. $\text{SO}_4^{2-}$
  2. $\text{H}_2\text{SO}_4$
  3. $\text{SO}_3$
  4. $\text{HSO}_3^-$

Answer: (A) $\text{SO}_4^{2-}$

Loss of $H^+$ from $\text{HSO}_4^-$ forms its conjugate base $\text{SO}_4^{2-}$.

Q44. Bakers' yeast converts glucose into ethyl alcohol by the enzyme:
  1. Invertase
  2. Zymase
  3. Maltase
  4. Diastase

Answer: (B) Zymase

Zymase enzyme present in yeast ferment glucose into ethyl alcohol.

Q45. Which of the following compounds reacts fastest via $S_{\text{N}}2$ mechanism?
  1. $\text{CH}_3\text{CH}_2\text{Br}$
  2. $(\text{CH}_3)_2\text{CHBr}$
  3. $\text{CH}_3\text{Br}$
  4. $(\text{CH}_3)_3\text{CBr}$

Answer: (C) $\text{CH}_3\text{Br}$

Least sterically hindered methyl halide reacts fastest in $S_{\text{N}}2$.

Q46. Which gas is evolved when sodium metal reacts with ethanol?
  1. Oxygen
  2. Methane
  3. Carbon dioxide
  4. Hydrogen

Answer: (D) Hydrogen

$2\text{C}_2\text{H}_5\text{OH} + 2\text{Na} \rightarrow 2\text{C}_2\text{H}_5\text{ONa} + \text{H}_2\uparrow$.

Q47. Which polymer is prepared from monomer tetrafluoroethene?
  1. Teflon
  2. Nylon-6
  3. Buna-S
  4. Bakelite

Answer: (A) Teflon

Tetrafluoroethene polymerizes to form Polytetrafluoroethylene (Teflon).

Q48. The shape of $\text{XeF}_4$ molecule according to VSEPR theory is:
  1. Tetrahedral
  2. Square planar
  3. See-saw
  4. Trigonal bipyramidal

Answer: (B) Square planar

$\text{XeF}_4$ has 4 bond pairs and 2 lone pairs $\implies$ square planar geometry.

Q49. Standard reduction potentials of metals $A, B, C$ are $+0.5\text{ V}, -3.0\text{ V}, -1.2\text{ V}$ respectively. The reducing power order is:
  1. $A > C > B$
  2. $C > B > A$
  3. $B > C > A$
  4. $A > B > C$

Answer: (C) $B > C > A$

More negative $E^\circ$ value corresponds to stronger reducing power: $B (-3.0) > C (-1.2) > A (+0.5)$.

Q50. In metallurgy, calcination is carried out in the:
  1. Presence of excess air
  2. Presence of oxygen and sulfur dioxide
  3. Presence of carbon monoxide
  4. Absence or limited supply of air

Answer: (D) Absence or limited supply of air

Calcination is thermal treatment of ore in absence/limited supply of air.

Q51. If $A$ is a square matrix of order $3$ such that $|A| = 5$, then $|\text{adj}(A)|$ is equal to:
  1. $25$
  2. $125$
  3. $5$
  4. $625$

Answer: (A) $25$

$|\text{adj}(A)| = |A|^{n-1} = 5^{3-1} = 25$.

Q52. The value of integral $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ is:
  1. $\frac{\pi}{2}$
  2. $\frac{\pi}{4}$
  3. $\pi$
  4. $0$

Answer: (B) $\frac{\pi}{4}$

Applying King's property $I + I = \int_0^{\pi/2} 1 dx = \frac{\pi}{2} \implies I = \frac{\pi}{4}$.

Q53. The distance of point $(1, 2, 3)$ from the plane $x - y + z = 5$ measured parallel to line $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ is:
  1. $\sqrt{14}$
  2. $2\sqrt{14}$
  3. $\frac{3}{2}\sqrt{14}$
  4. $\frac{\sqrt{14}}{4}$

Answer: (C) $\frac{3}{2}\sqrt{14}$

Point on line: $(1+r, 2+2r, 3+3r)$. Lies on plane $\implies 1+r - 2-2r + 3+3r = 5 \implies 2r = 3 \implies r = 3/2$. Distance $= |r|\sqrt{1^2+2^2+3^2} = \frac{3}{2}\sqrt{14}$.

Q54. The sum of the infinite geometric series $1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \dots$ is:
  1. $2$
  2. $\frac{3}{2}$
  3. $4$
  4. $3$

Answer: (D) $3$

$S_\infty = \frac{a}{1 - r} = \frac{1}{1 - 2/3} = 3$.

Q55. The number of ways to arrange the letters of the word 'MATHEMATICS' is:
  1. $\frac{11!}{2! 2! 2!}$
  2. $\frac{11!}{2! 2!}$
  3. $11!$
  4. $\frac{11!}{4!}$

Answer: (A) $\frac{11!}{2! 2! 2!}$

11 letters with 2 M's, 2 A's, and 2 T's $\implies \frac{11!}{2! 2! 2!}$.

Q56. The solution of differential equation $\frac{dy}{dx} + y = e^{-x}$ is:
  1. $y e^{-x} = x + C$
  2. $y e^x = x + C$
  3. $y = e^x + C$
  4. $y = x e^x + C$

Answer: (B) $y e^x = x + C$

$\text{IF} = e^x$. Integrating $y e^x = \int e^x e^{-x} dx = x + C$.

Q57. If roots of quadratic equation $x^2 - p x + q = 0$ differ by $1$, then:
  1. $p^2 = 4q - 1$
  2. $p^2 = 4q + 1$
  3. $q^2 = 4p + 1$
  4. $p^2 = q + 4$

Answer: (B) $p^2 = 4q + 1$

$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta = 1 \implies p^2 - 4q = 1 \implies p^2 = 4q + 1$.

Q58. If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$, then magnitude of projection of $\vec{a}$ on $\vec{b}$ is:
  1. $\sqrt{3}$
  2. $\frac{2}{\sqrt{3}}$
  3. $\frac{1}{\sqrt{3}}$
  4. $1$

Answer: (C) $\frac{1}{\sqrt{3}}$

Projection $= \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{1 - 1 + 1}{\sqrt{3}} = \frac{1}{\sqrt{3}}$.

Q59. The general solution of $\sin 2x + \cos x = 0$ is:
  1. $x = 2n\pi \pm \frac{\pi}{3}$
  2. $x = n\pi + \frac{\pi}{4}$
  3. $x = n\pi \pm \frac{\pi}{6}$
  4. $x = n\pi + \frac{\pi}{2}$ or $x = n\pi + (-1)^n \left(-\frac{\pi}{6}\right)$

Answer: (D) $x = n\pi + \frac{\pi}{2}$ or $x = n\pi + (-1)^n \left(-\frac{\pi}{6}\right)$

$\cos x (2\sin x + 1) = 0 \implies \cos x = 0$ or $\sin x = -1/2$. Solutions are $n\pi + \pi/2$ and $n\pi + (-1)^n(-\pi/6)$.

Q60. If two dice are thrown simultaneously, the probability of getting a sum greater than $9$ is:
  1. $\frac{1}{6}$
  2. $\frac{1}{12}$
  3. $\frac{5}{36}$
  4. $\frac{1}{4}$

Answer: (A) $\frac{1}{6}$

Sums > 9 are 10, 11, 12 (6 pairs total: (4,6),(5,5),(6,4),(5,6),(6,5),(6,6)). $P = 6/36 = 1/6$.

Q61. The derivative of $\tan^{-1}\left(\frac{2x}{1 - x^2}\right)$ with respect to $\sin^{-1}\left(\frac{2x}{1 + x^2}\right)$ is:
  1. $2$
  2. $1$
  3. $1/2$
  4. $0$

Answer: (B) $1$

Both expressions reduce to $2 \tan^{-1} x$. Thus derivative of $u$ w.r.t $u$ is $1$.

Q62. If $z$ is a complex number such that $|z| = 1$, then $\frac{1 + z}{1 + \bar{z}}$ is equal to:
  1. $\bar{z}$
  2. $1$
  3. $z$
  4. $-z$

Answer: (C) $z$

$\bar{z} = 1/z \implies \frac{1 + z}{1 + 1/z} = z$.

Q63. The eccentricity of the ellipse $9x^2 + 16y^2 = 144$ is:
  1. $\frac{7}{16}$
  2. $\frac{3}{4}$
  3. $\frac{\sqrt{5}}{4}$
  4. $\frac{\sqrt{7}}{4}$

Answer: (D) $\frac{\sqrt{7}}{4}$

$a^2 = 16, b^2 = 9 \implies e = \sqrt{1 - 9/16} = \frac{\sqrt{7}}{4}$.

Q64. The focus of the parabola $y^2 = -8x$ is:
  1. $(-2, 0)$
  2. $(2, 0)$
  3. $(0, -2)$
  4. $(0, 2)$

Answer: (A) $(-2, 0)$

For $y^2 = -4ax$, $4a = 8 \implies a = 2$. Focus is $(-2, 0)$.

Q65. The term independent of $x$ in the expansion of $\left(x + \frac{1}{x}\right)^{10}$ is:
  1. $210$
  2. $252$
  3. $120$
  4. $45$

Answer: (B) $252$

$T_{r+1} = {}^{10}C_r x^{10-2r}$. For constant term $10 - 2r = 0 \implies r = 5$. Value $={}^{10}C_5 = 252$.

Q66. The limit $\lim_{x \to 0} \frac{\sin x - x}{x^3}$ is equal to:
  1. $\frac{1}{6}$
  2. $0$
  3. $-\frac{1}{6}$
  4. $-\frac{1}{3}$

Answer: (C) $-\frac{1}{6}$

Using series expansion: $\lim_{x \to 0} \frac{(x - x^3/6 + \dots) - x}{x^3} = -\frac{1}{6}$.

Q67. The radius of the circle $x^2 + y^2 - 4x + 6y - 12 = 0$ is:
  1. $25$
  2. $\sqrt{13}$
  3. $7$
  4. $5$

Answer: (D) $5$

$r = \sqrt{g^2 + f^2 - c} = \sqrt{4 + 9 - (-12)} = \sqrt{25} = 5$.

Q68. The slope of normal to the curve $y = x^3 - x$ at $x = 2$ is:
  1. $-\frac{1}{11}$
  2. $11$
  3. $-11$
  4. $\frac{1}{11}$

Answer: (A) $-\frac{1}{11}$

$\frac{dy}{dx} = 3x^2 - 1$. At $x = 2$, slope of tangent $= 11 \implies$ slope of normal $= -1/11$.

Q69. If matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, then $A^2 - 5A$ is equal to:
  1. $-2 I$
  2. $2 I$
  3. $I$
  4. $0$

Answer: (B) $2 I$

By Cayley-Hamilton Theorem: $\lambda^2 - 5\lambda - 2 = 0 \implies A^2 - 5A = 2I$.

Q70. The angle between two lines whose direction cosines are proportional to $(1, 1, 0)$ and $(1, 0, 1)$ is:
  1. $\frac{\pi}{4}$
  2. $\frac{\pi}{6}$
  3. $\frac{\pi}{3}$
  4. $\frac{\pi}{2}$

Answer: (C) $\frac{\pi}{3}$

$\cos\theta = \frac{1(1) + 1(0) + 0(1)}{\sqrt{2}\sqrt{2}} = \frac{1}{2} \implies \theta = \frac{\pi}{3}$.

Q71. The value of $\int e^x (\tan x + \sec^2 x) dx$ is:
  1. $e^x \sec x + C$
  2. $e^x \sec^2 x + C$
  3. $e^x \ln(\sec x) + C$
  4. $e^x \tan x + C$

Answer: (D) $e^x \tan x + C$

Form $\int e^x [f(x) + f'(x)]dx = e^x f(x) + C \implies e^x \tan x + C$.

Q72. The mean deviation about mean for observations $2, 4, 6, 8, 10$ is:
  1. $2.4$
  2. $2.0$
  3. $1.8$
  4. $3.0$

Answer: (A) $2.4$

Mean $= 6$. Deviations: $4, 2, 0, 2, 4$. Mean deviation $= 12/5 = 2.4$.

Q73. If $f(x) = |x - 1| + |x - 2|$, then $f(x)$ is non-differentiable at:
  1. $x = 1$ only
  2. $x = 1$ and $x = 2$
  3. $x = 2$ only
  4. Everywhere differentiable

Answer: (B) $x = 1$ and $x = 2$

Modulus functions have corners where expressions inside vanish, i.e., $x = 1$ and $x = 2$.

Q74. If $A$ and $B$ are two events such that $P(A) = 0.4$, $P(B) = 0.8$, and $P(B|A) = 0.6$, then $P(A \cup B)$ is:
  1. $0.84$
  2. $0.76$
  3. $0.96$
  4. $0.60$

Answer: (C) $0.96$

$P(A \cap B) = 0.4 \times 0.6 = 0.24 \implies P(A \cup B) = 0.4 + 0.8 - 0.24 = 0.96$.

Q75. The principal value of $\sin^{-1}\left(\sin\frac{2\pi}{3}\right)$ is:
  1. $\frac{2\pi}{3}$
  2. $-\frac{\pi}{3}$
  3. $\frac{4\pi}{3}$
  4. $\frac{\pi}{3}$

Answer: (D) $\frac{\pi}{3}$

$\sin(2\pi/3) = \sin(\pi/3)$. Since $\pi/3 \in [-\pi/2, \pi/2]$, principal value is $\frac{\pi}{3}$.