JEE Main Full Mock Test: Practice Test 06

Question paper with answer key. Selection Pro

Q1. A projectile is fired with speed $u$ at an angle $\theta$ with the horizontal. If its maximum height is equal to one-fourth of its horizontal range, then $\tan\theta$ is:

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

Answer: (C) $4$

Using $H=u^2\sin^2\theta/(2g)$ and $R=u^2\sin2\theta/g$, the condition $H=R/4$ gives $\tan\theta=2$.

Q2. A block of mass $m$ is connected between two identical springs of force constant $k$ on a frictionless surface. The angular frequency of small oscillations is:

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

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

Both springs provide restoring forces in the same direction, so $F=-2kx$. Hence $\omega=\sqrt{2k/m}$.

Q3. A uniform rod of mass $M$ and length $L$ is hinged at one end and released from the horizontal position. Its angular speed when it reaches the vertical position is:

$\sqrt{g/L}$
$\sqrt{2g/L}$
$\sqrt{3g/L}$
$\sqrt{6g/L}$

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

The centre of mass falls by $L/2$. Thus $MgL/2=(1/2)(ML^2/3)\omega^2$, giving $\omega=\sqrt{3g/L}$.

Q4. A particle moves in a potential $U(x)=\frac12kx^2+\alpha x^4$, where $\alpha>0$. For small oscillations of amplitude $A$, the angular frequency to first order in $A^2$ is:

$\sqrt{k/m}\left(1+\frac{3\alpha A^2}{2k}\right)$
$\sqrt{k/m}\left(1-\frac{3\alpha A^2}{2k}\right)$
$\sqrt{k/m}\left(1+\frac{3\alpha A^2}{k}\right)$
$\sqrt{k/m}$

Answer: (A) $\sqrt{k/m}\left(1+\frac{3\alpha A^2}{2k}\right)$

The equation is $\ddot x+(k/m)x+(4\alpha/m)x^3=0$. The first-order Duffing correction gives $\omega\approx\sqrt{k/m}[1+3\alpha A^2/(2k)]$.

Q5. A satellite changes from a circular orbit of radius $R$ to another circular orbit of radius $4R$. The ratio of its total mechanical energy in the final orbit to that in the initial orbit is:

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

Answer: (A) $1/4$

For a circular orbit $E=-GMm/(2r)$. Hence $E_f/E_i=R/(4R)=1/4$.

Q6. A point charge $q$ is placed at the centre of a cube. The electric flux through any one face is:

$q/\varepsilon_0$
$q/(2\varepsilon_0)$
$q/(6\varepsilon_0)$
$q/(8\varepsilon_0)$

Answer: (C) $q/(6\varepsilon_0)$

Gauss's law gives total flux $q/\varepsilon_0$. By symmetry it is equally divided among six faces.

Q7. Two capacitors $C$ and $2C$ are connected in parallel, charged to potential $V$, and disconnected from the battery. A dielectric of constant $2$ is completely inserted into capacitor $C$. The final potential difference across the combination is:

$V/2$
$3V/4$
$3V/2$
$V$

Answer: (B) $3V/4$

Initially, the total charge is $(C+2C)V=3CV$. After the dielectric is inserted, the capacitance $C$ becomes $2C$, so the total capacitance is $4C$. Since the system is isolated, charge remains constant. Therefore $V_f=3CV/(4C)=3V/4$.

Q8. A cell of emf $E$ and internal resistance $r$ is connected to a variable resistance $R$. The power delivered to $R$ is maximum when:

$R=0$
$R=r$
$R=2r$
$R\to\infty$

Answer: (B) $R=r$

The external power is $P=E^2R/(R+r)^2$. Differentiation gives its maximum at $R=r$.

Q9. Two resistors $R$ and $2R$ are connected in parallel. The fraction of the total current flowing through the resistor $2R$ is:

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

Answer: (A) $1/3$

Current divides inversely to resistance. Thus $I_{2R}:I_R=R:2R=1:2$, so $I_{2R}=I_{total}/3$.

Q10. A charged particle enters a uniform magnetic field with velocity making an angle $\theta$ with the field. The ratio of the radius of its helical path to its pitch is:

$\frac{\sin\theta}{2\pi\cos\theta}$
$\frac{\cos\theta}{2\pi\sin\theta}$
$\frac{\sin\theta}{\cos\theta}$
$2\pi\tan\theta$

Answer: (A) $\frac{\sin\theta}{2\pi\cos\theta}$

The radius of the helical path is $r=mv\sin\theta/(qB)$, while the pitch is $p=v\cos\theta(2\pi m/qB)$. Hence $r/p=\tan\theta/(2\pi)=\sin\theta/(2\pi\cos\theta)$, which is option 1.

Q11. A rectangular coil of $N$ turns and area $A$ rotates in a uniform magnetic field $B$ with angular speed $\omega$. The maximum induced emf is:

$NBA$
$NBA\omega$
$NBA/\omega$
$B/(NA\omega)$

Answer: (B) $NBA\omega$

Magnetic flux is $NBA\cos\omega t$, so the maximum magnitude of $-d\Phi/dt$ is $NBA\omega$.

Q12. In a series LCR circuit at resonance, the rms current is $I$. If the resistance is doubled while $L,C$ and source voltage remain unchanged, the new rms current is:

$I/2$
$I$
$2I$
$I/\sqrt2$

Answer: (A) $I/2$

At resonance impedance equals $R$. Doubling $R$ halves the current.

Q13. A convex lens of focal length $20$ cm and a concave lens of focal length $-30$ cm are kept in contact. The power of the combination is:

$+5/3$ D
$+1/60$ D
$-1/60$ D
$+1/5$ D

Answer: (A) $+5/3$ D

The powers are $+5$ D and $-10/3$ D. Hence $P=5-10/3=5/3$ D.

Q14. In Young's double-slit experiment, the slit separation is doubled and the wavelength is halved while the screen distance remains fixed. The fringe width becomes:

$\beta/2$
$\beta/4$
$2\beta$
$4\beta$

Answer: (B) $\beta/4$

Since $\beta=\lambda D/d$, replacing $\lambda$ by $\lambda/2$ and $d$ by $2d$ makes $\beta'=\beta/4$.

Q15. A photon of energy $8$ eV strikes a metal of work function $3$ eV. The stopping potential is:

$3$ V
$5$ V
$8$ V
$11$ V

Answer: (B) $5$ V

Einstein's equation gives $K_{max}=8-3=5$ eV. Therefore stopping potential is $5$ V.

Q16. An electron and a proton are accelerated through the same potential difference. The ratio of their de Broglie wavelengths $\lambda_e/\lambda_p$ is approximately:

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

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

For the same accelerating potential, $\lambda=h/\sqrt{2mqV}$, so $\lambda_e/\lambda_p=\sqrt{m_p/m_e}$.

Q17. A radioactive nucleus has a half-life of 6 h. The fraction of the original sample left after 24 h is:

$1/2$
$1/4$
$1/8$
$1/16$

Answer: (D) $1/16$

Twenty-four hours correspond to four half-lives, so the fraction left is $(1/2)^4=1/16$.

Q18. For a gas undergoing an adiabatic process, which relation is correct for an ideal gas?

$PV=constant$
$PV^\gamma=constant$
$P/V^\gamma=constant$
$P+V^\gamma=constant$

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

For a reversible adiabatic process of an ideal gas, $PV^\gamma=constant$.

Q19. One mole of an ideal gas expands isothermally from volume $V$ to $4V$ at temperature $T$. The work done by the gas is:

$RT\ln2$
$RT\ln4$
$4RT\ln4$
$3RT$

Answer: (B) $RT\ln4$

For one mole in an isothermal process, $W=RT\ln(V_2/V_1)=RT\ln4$.

Q20. A body is projected up a rough incline of angle $\theta$ with coefficient of kinetic friction $\mu$. Its retardation while moving upward is:

$g(\sin\theta-\mu\cos\theta)$
$g(\sin\theta+\mu\cos\theta)$
$g(\cos\theta+\mu\sin\theta)$
$g\sin\theta$

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

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

Q21. A solid cylinder and a hollow ring have the same mass and radius and roll without slipping with the same translational speed. The ratio of their total kinetic energies, cylinder:ring, is:

$3:4$
$4:3$
$1:2$
$2:1$

Answer: (A) $3:4$

For the cylinder $I=MR^2/2$, giving $K=3Mv^2/4$. For the ring $I=MR^2$, giving $K=Mv^2$. Ratio is $3:4$.

Q22. A liquid drop of radius $R$ is split into 8 identical smaller drops. If surface tension is unchanged, the total surface energy after splitting compared with before splitting is:

$1:1$
$2:1$
$4:1$
$8:1$

Answer: (B) $2:1$

The new radius is $R/2$. Total area becomes $8\times4\pi(R/2)^2=2(4\pi R^2)$, so surface energy doubles.

Q23. A wire has Young's modulus $Y$, length $L$ and cross-sectional area $A$. If both its length and radius are doubled, its longitudinal spring constant changes by a factor:

$1/2$
$2$
$4$
$8$

Answer: (B) $2$

Doubling the radius gives four times the area, while doubling the length doubles the denominator in $k=YA/L$. Hence the new spring constant is $2k$.

Q24. A simple pendulum has time period $T$ at the Earth's surface. If it is taken to a height equal to the Earth's radius, neglecting rotation, its new time period is:

$T/2$
$T\sqrt2$
$2T$
$4T$

Answer: (C) $2T$

At height $h=R$, $g'=GM/(2R)^2=g/4$. Since $T\propto1/\sqrt g$, the new period is $2T$.

Q25. An electric dipole of moment $p$ is placed in a uniform electric field $E$ at angle $\theta$. The torque magnitude is:

$pE\cos\theta$
$pE\sin\theta$
$p/E$
$E/p$

Answer: (B) $pE\sin\theta$

The torque on a dipole is $\tau=pE\sin\theta$.

Q26. For a first-order reaction, the half-life is 20 min. The time required for 93.75% completion is:

40 min
60 min
80 min
100 min

Answer: (C) 80 min

6.25% remains, which is $1/16=(1/2)^4$. Therefore four half-lives are required: $80$ min.

Q27. For a reaction $A\rightleftharpoons B$ with $K_c=4$, initially only $A$ is present at concentration $1$ M. The equilibrium concentration of $B$ is:

$0.20$ M
$0.50$ M
$0.80$ M
$1.20$ M

Answer: (C) $0.80$ M

Let $B=x$. Then $A=1-x$ and $K=x/(1-x)=4$. Hence $x=0.8$ M.

Q28. The pH of a $10^{-3}$ M weak acid with $K_a=10^{-5}$ is approximately:

Answer: (B) $4$

For a weak acid, $[H^+]\approx\sqrt{K_aC}=\sqrt{10^{-8}}=10^{-4}$ M, so pH $\approx4$.

Q29. A buffer contains equal concentrations of $CH_3COOH$ and $CH_3COO^-$. If $pK_a=4.76$, its pH is:

$2.38$
$4.76$
$7.00$
$9.52$

Answer: (B) $4.76$

Henderson-Hasselbalch equation gives $pH=pK_a+\log([salt]/[acid])=pK_a$.

Q30. For a galvanic cell, $E^\circ_{cell}=1.10$ V and $n=2$. The standard Gibbs energy change is approximately:

$-106.1$ kJ mol$^{-1}$
$-212.3$ kJ mol$^{-1}$
$+212.3$ kJ mol$^{-1}$
$+106.1$ kJ mol$^{-1}$

Answer: (B) $-212.3$ kJ mol$^{-1}$

Using $\Delta G^\circ=-nFE^\circ$, $=-2\times96485\times1.10\approx-212.3$ kJ mol$^{-1}$.

Q31. For the cell reaction $Zn+Cu^{2+}\rightarrow Zn^{2+}+Cu$, if $[Zn^{2+}]$ is increased while $[Cu^{2+}]$ is kept fixed, the cell potential:

increases
decreases
remains unchanged
becomes zero

Answer: (B) decreases

The reaction quotient increases, so from the Nernst equation $E=E^\circ-(0.0591/2)\log Q$, the cell potential decreases.

Q32. The major product of acid-catalysed dehydration of 2-methyl-2-butanol is:

2-methyl-1-butene
2-methyl-2-butene
1-pentene
2-pentanol

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

Dehydration proceeds mainly by an E1 mechanism and gives the more substituted alkene, 2-methyl-2-butene.

Q33. Which compound gives the fastest SN1 reaction among the following?

CH3Cl
CH3CH2Cl
(CH3)2CHCl
(CH3)3CCl

Answer: (D) (CH3)3CCl

SN1 rate depends strongly on carbocation stability. The tertiary carbocation is the most stable.

Q34. The major product of propene reacting with HBr in the presence of peroxide is:

1-bromopropane
2-bromopropane
1,2-dibromopropane
propanol

Answer: (A) 1-bromopropane

In the presence of peroxide, HBr undergoes radical anti-Markovnikov addition, producing 1-bromopropane.

Q35. Which species is aromatic according to Hückel's rule?

Cyclobutadiene
Cyclopentadienyl anion
Cyclopentadienyl cation
Cycloheptatrienyl anion

Answer: (B) Cyclopentadienyl anion

The cyclopentadienyl anion has 6 $\pi$ electrons and is planar/conjugated, satisfying the $4n+2$ rule with $n=1$.

Q36. The number of stereoisomers possible for 2,3-dichlorobutane is:

Answer: (B) 3

It has two stereocentres and a meso form, giving two enantiomers plus one meso form: three stereoisomers.

Q37. Which compound is most acidic?

Phenol
p-Cresol
p-Nitrophenol
Cyclohexanol

Answer: (C) p-Nitrophenol

The nitro group withdraws electron density and stabilizes the phenoxide ion, making p-nitrophenol the strongest acid here.

Q38. The major product of nitration of anisole is predominantly:

m-Nitroanisole
p-Nitroanisole
Nitrobenzene
Benzoic acid

Answer: (B) p-Nitroanisole

The methoxy group is strongly activating and ortho/para directing; the para product is favored because of lower steric hindrance.

Q39. Aniline treated with $NaNO_2/HCl$ at $0-5^\circ C$ forms:

Nitrobenzene
Benzenediazonium chloride
Phenol
Benzene

Answer: (B) Benzenediazonium chloride

Primary aromatic amines undergo diazotisation with nitrous acid in cold acidic medium.

Q40. Which reagent converts an aldehyde selectively into a primary alcohol?

$KMnO_4$
$LiAlH_4$
$K_2Cr_2O_7$
$Br_2/H_2O$

Answer: (B) $LiAlH_4$

$LiAlH_4$ reduces aldehydes to primary alcohols.

Q41. The strongest acid among the following is:

CH3COOH
ClCH2COOH
Cl2CHCOOH
Cl3CCOOH

Answer: (D) Cl3CCOOH

The electron-withdrawing $-I$ effect of chlorine stabilizes the carboxylate ion. More chlorine atoms increase acidity.

Q42. The coordination number of Co in $[Co(NH_3)_4Cl_2]^+$ is:

Answer: (C) 6

Four ammonia ligands and two chloride ligands are directly coordinated, so the coordination number is 6.

Q43. The geometry of $[Ni(CN)_4]^{2-}$ is:

Tetrahedral
Square planar
Octahedral
Trigonal planar

Answer: (B) Square planar

$CN^-$ is a strong-field ligand. For $Ni^{2+}$, pairing leads to a square-planar $dsp^2$ complex.

Q44. For an ideal gas, the compressibility factor $Z=PV/(nRT)$ is:

0
1
$RT$
$P/V$

Answer: (B) 1

The ideal-gas equation is $PV=nRT$, so $Z=1$.

Q45. For an ideal solution, the enthalpy change of mixing is:

positive
negative
zero
infinite

Answer: (C) zero

In an ideal solution, intermolecular interactions are effectively unchanged on mixing, so $\Delta H_{mix}=0$.

Q46. The number of unpaired electrons in the ground-state $Fe^{3+}$ ion is:

Answer: (C) 5

$Fe$ is $[Ar]3d^64s^2$; $Fe^{3+}$ is $[Ar]3d^5$, with five unpaired electrons by Hund's rule.

Q47. Which molecule has the largest bond angle?

Answer: (C) BF3

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

Q48. For the reaction $N_2+3H_2\rightleftharpoons2NH_3$, increasing pressure at constant temperature shifts equilibrium toward:

Reactants
Products
Neither side
Both equally

Answer: (B) Products

There are four gaseous moles on the reactant side and two on the product side, so increased pressure favors products.

Q49. The oxidation state of chromium in $K_2Cr_2O_7$ is:

Answer: (C) +6

Let chromium be $x$: $2(+1)+2x+7(-2)=0$, giving $x=+6$.

Q50. Which of the following is a strong-field ligand?

$I^-$
$F^-$
$H_2O$
$CN^-$

Answer: (D) $CN^-$

$CN^-$ lies toward the strong-field end of the spectrochemical series and produces large crystal-field splitting.

Q51. If the roots of $x^2-2ax+(a^2-1)=0$ are both positive, then the range of $a$ is:

$a>1$
$a<-1$
$|a|<1$
$a\ge1$

Answer: (A) $a>1$

The roots are $a\pm1$. Both are positive iff $a-1>0$, hence $a>1$.

Q52. The number of real solutions of $|x-1|+|x+1|=4$ is:

Answer: (C) 2

For $x\ge1$, the left side is $2x=4$, giving $x=2$. For $x\le-1$, it is $-2x=4$, giving $x=-2$. Thus there are two solutions.

Q53. If $z$ satisfies $|z-2|=|z+2|$, then the locus of $z=x+iy$ is:

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

Answer: (A) $x=0$

Squaring gives $(x-2)^2+y^2=(x+2)^2+y^2$, hence $x=0$.

Q54. The value of $\lim_{x\to0}\frac{\tan x-x}{x^3}$ is:

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

Answer: (A) $1/3$

Using $\tan x=x+x^3/3+O(x^5)$, the limit is $1/3$.

Q55. If $f(x)=x^x$ for $x>0$, then $f'(e)$ equals:

$e^e$
$2e^e$
$e^{e+1}$
$e$

Answer: (B) $2e^e$

Since $f'(x)=x^x(\ln x+1)$, at $x=e$ it is $e^e(1+1)=2e^e$.

Q56. The maximum value of $x^2(4-x)^2$ for $0\le x\le4$ is:

Answer: (C) 16

Since $x(4-x)=4-(x-2)^2\le4$, its square is at most $16$, attained at $x=2$.

Q57. Evaluate $\int_0^{\pi/2}\ln(\sin x)\,dx$.

$-\pi\ln2$
$-\frac{\pi}{2}\ln2$
$0$
$\frac{\pi}{2}\ln2$

Answer: (B) $-\frac{\pi}{2}\ln2$

The standard integral is $\int_0^{\pi/2}\ln(\sin x)dx=-\frac{\pi}{2}\ln2$.

Q58. If $A$ is a $3\times3$ matrix with $|A|=2$, then $|\operatorname{adj}A|$ is:

Answer: (B) 4

For an $n\times n$ matrix, $|\operatorname{adj}A|=|A|^{n-1}$. Hence it is $2^2=4$.

Q59. The number of solutions of $\sin x+\sin2x=0$ in $[0,2\pi]$ is:

Answer: (C) 5

$\sin x(1+2\cos x)=0$. Thus $x=0,\pi,2\pi$ and $\cos x=-1/2$ gives $2\pi/3,4\pi/3$, totaling 5.

Q60. The coefficient of $x^5$ in $(1+x)^8(1-x)^4$ is:

Answer: (D) $5$

$\sin x(1+2\cos x)=0$. The solutions in $[0,2\pi]$ are $0,\pi,2\pi,2\pi/3,4\pi/3$, giving 5 solutions.

Q61. The distance between the parallel lines $3x-4y+7=0$ and $6x-8y-5=0$ is:

$-14$
$-8$
$8$
$14$

Answer: (A) $-14$

The coefficient of $x^5$ in $(1+x)^8(1-x)^4$ is $\sum_{j=0}^4(-1)^j\binom4j\binom8{5-j}=-14$.

Q62. The equation of the circle passing through $(0,0),(4,0),(0,6)$ is:

$x^2+y^2-4x-6y=0$
$x^2+y^2+4x+6y=0$
$x^2+y^2-6x-4y=0$
$x^2+y^2+6x-4y=0$

Answer: (A) $x^2+y^2-4x-6y=0$

For $x^2+y^2+Dx+Ey=0$, substituting $(4,0)$ gives $D=-4$ and substituting $(0,6)$ gives $E=-6$.

Q63. The eccentricity of the ellipse $9x^2+16y^2=144$ is:

$3/4$
$\sqrt7/4$
$1/4$
$\sqrt7/3$

Answer: (B) $\sqrt7/4$

The ellipse is $x^2/16+y^2/9=1$, so $a=4,b=3$. Hence $e=\sqrt{1-b^2/a^2}=\sqrt7/4$.

Q64. The number of terms in the expansion of $(1+x+x^2)^5$ after collecting like powers is:

Answer: (B) 11

The minimum power is 0 and maximum is 10, and every integer power from 0 to 10 occurs. Hence there are 11 terms.

Q65. The sum of the infinite series $\frac12+\frac{1}{4}+\frac18+\cdots$ is:

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

Answer: (B) $1$

This is a geometric series with first term $1/2$ and ratio $1/2$, so sum $=(1/2)/(1-1/2)=1$.

Q66. If $\alpha,\beta$ are roots of $x^2-3x+1=0$, then $\alpha^3+\beta^3$ is:

Answer: (B) 18

$\alpha+\beta=3$, $\alpha\beta=1$. Thus $\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=27-9=18$.

Q67. The probability that a randomly selected integer from $1$ to $100$ is divisible by neither 2 nor 3 is:

$33/100$
$1/3$
$1/2$
$1/6$

Answer: (A) $33/100$

Multiples of 2 or 3 among 1 to 100 are $50+33-16=67$. Hence 33 integers are divisible by neither, so probability is $33/100$.

Q68. For the differential equation $\frac{dy}{dx}=\frac{y}{x}$, $x\ne0$, the general solution is:

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

Answer: (B) $y=Cx$

Separating variables gives $dy/y=dx/x$. Hence $\ln|y|=\ln|x|+C$, so $y=Cx$.

Q69. The area bounded by $y=x$ and $y=x^2$ is:

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

Answer: (B) $1/6$

The curves intersect at $x=0,1$. The area is $\int_0^1(x-x^2)dx=1/2-1/3=1/6$.

Q70. If $\vec a=(1,2,3)$ and $\vec b=(2,-1,1)$, then $|\vec a\times\vec b|$ is:

$5\sqrt3$
$\sqrt{35}$
$5$
$7$

Answer: (A) $5\sqrt3$

$\vec a\times\vec b=(5,5,-5)$, so its magnitude is $\sqrt{25+25+25}=5\sqrt3$.

Q71. The shortest distance between the skew lines $\vec r=\hat i+\lambda(\hat j+\hat k)$ and $\vec r=\hat j+\mu(\hat i+\hat k)$ is:

$1/\sqrt2$
$1/\sqrt3$
$\sqrt2$
$1$

Answer: (A) $1/\sqrt2$

Using $d=| (\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)|/|\vec b_1\times\vec b_2|$ gives $1/\sqrt2$.

Q72. The mean of the first $n$ positive integers is 15. Then $n$ is:

Answer: (A) 29

Mean is $(n+1)/2=15$, so $n+1=30$ and $n=29$.

Q73. If the variance of the numbers $2,4,6,8,10$ is $8$, then their standard deviation is:

$2$
$2\sqrt2$
$4$
$8$

Answer: (B) $2\sqrt2$

Standard deviation is the square root of variance, so $\sigma=\sqrt8=2\sqrt2$.

Q74. The number of ways to arrange 5 distinct books on a shelf if two particular books must not be together is:

Answer: (B) 72

Total arrangements are $5!=120$. If the two specified books stay together, there are $2\times4!=48$ arrangements. Hence not together: $120-48=72$.

Q75. If $\tan^{-1}x+\tan^{-1}2x=\pi/4$, then the positive value of $x$ is:

$\frac{\sqrt{17}-3}{4}$
$1/2$
$1$
$2$

Answer: (A) $\frac{\sqrt{17}-3}{4}$

Taking tangent gives $3x/(1-2x^2)=1$, so $2x^2+3x-1=0$. The positive root is $(\sqrt{17}-3)/4$.