Mathematics
Q1Sets and relations
A set $P$ has 3 elements and a set $Q$ has 4 elements. The number of relations from $P$ to $Q$ is
- a4096
- b144
- c12
- d128
Answer (a): A relation is any subset of $P\times Q$, which has 12 elements, so there are $2^{12}=4096$ relations.
Revise: Algebra (Class 11 notes)
Q2Logic
The negation of the statement $p \wedge \neg q$ is
- a$\neg p \vee q$
- b$\neg p \wedge q$
- c$p \vee \neg q$
- d$p \wedge q$
Answer (a): By De Morgan's law, $\neg(p\wedge\neg q)=\neg p\vee\neg(\neg q)=\neg p\vee q$.
Revise: Algebra (Class 11 notes)
Q3Functions
The domain of $f(x)=\sqrt{x^2-5x+6}$ is
- a$[2,3]$
- b$(-\infty,2]\cup[3,\infty)$
- c$(2,3)$
- d$(-\infty,2)\cup(3,\infty)$
Answer (b): $x^2-5x+6=(x-2)(x-3)\ge 0$ holds for $x\le 2$ or $x\ge 3$, endpoints included since the root of zero is defined.
Revise: Algebra (Class 11 notes)
Q4Determinants
If $A$ is a $3\times 3$ matrix with $|A|=4$, then $|2A|$ equals
- a8
- b16
- c64
- d32
Answer (d): For an $n\times n$ matrix $|kA|=k^n|A|$, so $|2A|=2^3\cdot 4=32$.
Revise: Algebra (Class 12 notes)
Q5Complex numbers
The value of $(1+i)^8$ is
- a-16
- b16
- c256
- d16i
Answer (b): $(1+i)^2=2i$, so $(1+i)^8=(2i)^4=16,i^4=16$.
Revise: Algebra (Class 12 notes)
Q6Sequences and series
The sum of the first 20 terms of the AP $3,7,11,\dots$ is
- a800
- b840
- c780
- d820
Answer (d): $a=3,\ d=4$: $S_{20}=\frac{20}{2}[2(3)+19(4)]=10(82)=820$.
Revise: Algebra (Class 11 notes)
Q7Permutations and combinations
The number of distinct arrangements of the letters of the word BANANA is
- a120
- b720
- c60
- d30
Answer (c): Six letters with A repeated 3 times and N twice: $\frac{6!}{3!,2!}=\frac{720}{12}=60$.
Revise: Algebra (Class 11 notes)
Q8Binomial theorem
The sum of the coefficients in the expansion of $(2x-1)^5$ is
- a32
- b243
- c1
- d-1
Answer (c): Put $x=1$: $(2-1)^5=1$.
Revise: Algebra (Class 11 notes)
Q9Trigonometry
The value of $\tan 15^\circ$ is
- a$2-\sqrt3$
- b$2+\sqrt3$
- c$\sqrt3-1$
- d$\frac{1}{\sqrt3}$
Answer (a): $\tan(45^\circ-30^\circ)=\frac{1-\frac1{\sqrt3}}{1+\frac1{\sqrt3}}=\frac{\sqrt3-1}{\sqrt3+1}=2-\sqrt3$.
Revise: Trigonometry (Class 11 notes)
Q10Inverse trigonometric functions
$\tan^{-1}1+\tan^{-1}2+\tan^{-1}3$ equals
- a$\frac{\pi}{2}$
- b$\frac{3\pi}{4}$
- c2π
- dπ
Answer (d): $\tan^{-1}2+\tan^{-1}3=\pi+\tan^{-1}\frac{5}{1-6}=\pi-\frac{\pi}{4}=\frac{3\pi}{4}$ (product greater than 1). Adding $\frac{\pi}{4}$ gives $\pi$.
Revise: Trigonometry (Class 12 notes)
Q11Properties of triangles
In triangle $ABC$, $a=7,\ b=8,\ c=9$. Then $\cos A$ equals
- a$\frac34$
- b$\frac23$
- c$\frac13$
- d$\frac56$
Answer (b): $\cos A=\frac{b^2+c^2-a^2}{2bc}=\frac{64+81-49}{144}=\frac{96}{144}=\frac23$.
Revise: Trigonometry (Class 11 notes)
Q12Straight lines
The distance between the parallel lines $3x+4y=5$ and $3x+4y=15$ is
- a2
- b4
- c5
- d$\frac{10}{7}$
Answer (a): Distance $=\frac{|15-5|}{\sqrt{3^2+4^2}}=\frac{10}{5}=2$.
Revise: Analytic geometry (Class 11 notes)
Q13Pair of lines
The acute angle between the lines given by $x^2-5xy+6y^2=0$ is
- a$\frac{\pi}{4}$
- b$\tan^{-1}\frac17$
- c$\tan^{-1}\frac15$
- d$\tan^{-1}\frac57$
Answer (b): The lines are $x=2y$ and $x=3y$, with slopes $\frac12$ and $\frac13$: $\tan\theta=\frac{\frac12-\frac13}{1+\frac16}=\frac{1/6}{7/6}=\frac17$.
Revise: Analytic geometry (Class 11 notes)
Q14Circle
The radius of the circle $x^2+y^2-6x+8y-11=0$ is
- a6
- b5
- c7
- d$\sqrt{11}$
Answer (a): Centre $(3,-4)$; $r=\sqrt{9+16+11}=\sqrt{36}=6$.
Revise: Analytic geometry (Class 11 notes)
Q15Conics
The eccentricity of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is
- a$\frac45$
- b$\frac{9}{25}$
- c$\frac53$
- d$\frac35$
Answer (d): $e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{16}{25}}=\frac35$.
Revise: Analytic geometry (Class 12 notes)
Q16Three-dimensional geometry
The distance between the points $(1,2,3)$ and $(4,6,15)$ is
- a12
- b13
- c19
- d5
Answer (b): $\sqrt{3^2+4^2+12^2}=\sqrt{169}=13$.
Revise: Analytic geometry (Class 12 notes)
Q17Limits
$\lim_{x\to0}\dfrac{\sin 5x}{\tan 2x}$ equals
- a$\frac25$
- b3
- c1
- d$\frac52$
Answer (d): $\frac{\sin5x}{5x}\cdot\frac{2x}{\tan2x}\cdot\frac{5x}{2x}\to \frac52$.
Revise: Calculus (Class 11 notes)
Q18Derivatives
The derivative of $x\ln x$ at $x=e$ is
- a2
- b1
- ce
- d$e+1$
Answer (a): $\frac{d}{dx}(x\ln x)=\ln x+1$, which is $1+1=2$ at $x=e$.
Revise: Calculus (Class 11 notes)
Q19Integration
$\int_0^{\pi/2}\sin^2x,dx$ equals
- a$\frac{\pi}{2}$
- b$\frac{\pi}{4}$
- c$\frac12$
- d$\frac{\pi}{8}$
Answer (b): $\sin^2x=\frac{1-\cos2x}{2}$, so the integral is $\frac12\left[x-\frac{\sin2x}{2}\right]_0^{\pi/2}=\frac{\pi}{4}$.
Revise: Calculus (Class 12 notes)
Q20Probability
Two fair dice are thrown. The probability that the sum is 9 is
- a$\frac1{12}$
- b$\frac19$
- c$\frac5{36}$
- d$\frac16$
Answer (b): Favourable: $(3,6),(4,5),(5,4),(6,3)$, i.e. 4 of 36, so $\frac{4}{36}=\frac19$.
Revise: Statistics and Probability (Class 11 notes)
Q21 (2 marks)Matrices
If $A=\begin{pmatrix}2&1\1&1\end{pmatrix}$, then $A^2-3A+I$ equals
- aI
- bA
- c2A
- dthe zero matrix
Answer (d): $A^2=\begin{pmatrix}5&3\3&2\end{pmatrix}$ and $3A=\begin{pmatrix}6&3\3&3\end{pmatrix}$, so $A^2-3A=-I$ and $A^2-3A+I=O$ (Cayley-Hamilton).
Revise: Algebra (Class 12 notes)
Q22 (2 marks)Determinants
If $\omega$ is a complex cube root of unity, the value of $\begin{vmatrix}1&1&1\1&\omega&\omega^2\1&\omega^2&\omega\end{vmatrix}$ is
- a$3\sqrt3,i$
- b$-3\sqrt3,i$
- c0
- d$3\sqrt3$
Answer (b): Expanding along row 1 gives $3(\omega^2-\omega)$. Since $\omega^2-\omega=-i\sqrt3$, the value is $-3\sqrt3,i$.
Revise: Algebra (Class 12 notes)
Q23 (2 marks)Complex numbers
The principal argument of $\dfrac{1+i}{\sqrt3-i}$ is
- a$\frac{\pi}{12}$
- b$\frac{5\pi}{12}$
- c$\frac{7\pi}{12}$
- d$\frac{\pi}{4}$
Answer (b): $\arg(1+i)=45^\circ$ and $\arg(\sqrt3-i)=-30^\circ$. The argument of the quotient is $45^\circ+30^\circ=75^\circ=\frac{5\pi}{12}$.
Revise: Algebra (Class 12 notes)
Q24 (2 marks)Binomial theorem
The term independent of $x$ in the expansion of $\left(x^2-\dfrac1x\right)^9$ is
- a-84
- b84
- c126
- d36
Answer (b): $T_{r+1}=\binom9r(-1)^r x^{18-3r}$. Independence needs $r=6$, so the term is $\binom96(+1)=84$.
Revise: Algebra (Class 11 notes)
Q25 (2 marks)Sequences and series
The sum to infinity of the series $1+\dfrac23+\dfrac39+\dfrac4{27}+\cdots$ is
- a$\frac32$
- b$\frac43$
- c$\frac92$
- d$\frac94$
Answer (d): The series is $\sum n x^{n-1}$ with $x=\frac13$, whose sum is $\frac1{(1-x)^2}=\frac{1}{(2/3)^2}=\frac94$.
Revise: Algebra (Class 11 notes)
Q26 (2 marks)Permutations and combinations
A committee of 4 is chosen from 6 men and 4 women. The number of committees with at least 2 women is
- a95
- b90
- c115
- d125
Answer (c): Total $\binom{10}{4}=210$. With 0 women: $\binom64=15$; with 1 woman: $4\binom63=80$. So $210-95=115$.
Revise: Algebra (Class 11 notes)
Q27 (2 marks)Inverse trigonometric functions
The value of $\sin!\left(2\cos^{-1}\frac35\right)$ is
- a$\frac{24}{25}$
- b$\frac{12}{25}$
- c$\frac{7}{25}$
- d$\frac{24}{7}$
Answer (a): Let $\theta=\cos^{-1}\frac35$, so $\sin\theta=\frac45$. Then $\sin2\theta=2\cdot\frac45\cdot\frac35=\frac{24}{25}$.
Revise: Trigonometry (Class 12 notes)
Q28 (2 marks)Conics
The length of the latus rectum of the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{16}=1$ is
- a$\frac{32}{3}$
- b$\frac{16}{3}$
- c8
- d6
Answer (a): Here $a^2=9,\ b^2=16$. Latus rectum $=\frac{2b^2}{a}=\frac{32}{3}$.
Revise: Analytic geometry (Class 12 notes)
Q29 (2 marks)Three-dimensional geometry
The cosine of the angle between two lines with direction ratios $(1,2,2)$ and $(2,3,6)$ is
- a$\frac{19}{21}$
- b$\frac57$
- c$\frac{20}{21}$
- d$\frac23$
Answer (c): Dot product $=2+6+12=20$; magnitudes $3$ and $7$. So $\cos\theta=\frac{20}{21}$.
Revise: Analytic geometry (Class 12 notes)
Q30 (2 marks)Differential equations
The general solution of $\dfrac{dy}{dx}+\dfrac{2y}{x}=x^2$ is
- a$y=\frac{x^3}{3}+\frac{C}{x^2}$
- b$y=\frac{x^3}{5}+Cx^2$
- c$y=\frac{x^3}{5}+\frac{C}{x^2}$
- d$y=\frac{x^2}{4}+\frac{C}{x}$
Answer (c): Integrating factor $=e^{\int 2/x,dx}=x^2$. Then $\frac{d}{dx}(x^2y)=x^4$, so $x^2y=\frac{x^5}{5}+C$.
Revise: Calculus (Class 12 notes)
Q31 (2 marks)Integration
$\int_1^e x\ln x,dx$ equals
- a$\frac{e^2+1}{4}$
- b$\frac{e^2-1}{4}$
- c$\frac{e^2}{4}$
- d$\frac{e^2+1}{2}$
Answer (a): By parts: $\left[\frac{x^2}{2}\ln x-\frac{x^2}{4}\right]_1^e=\left(\frac{e^2}{2}-\frac{e^2}{4}\right)-\left(0-\frac14\right)=\frac{e^2+1}{4}$.
Revise: Calculus (Class 12 notes)
Q32 (2 marks)Applications of derivatives
The local maximum value of $f(x)=x^3-12x+5$ is
- a21
- b-11
- c5
- d14
Answer (a): $f'(x)=3x^2-12=0$ gives $x=\pm2$; $f''(-2)=-12<0$ so the maximum is at $x=-2$: $f(-2)=-8+24+5=21$.
Revise: Calculus (Class 12 notes)
Q33 (2 marks)Vectors
The area of a triangle whose adjacent sides are $\vec a=\hat i+2\hat j+3\hat k$ and $\vec b=2\hat i-\hat j+\hat k$ is
- a$5\sqrt3$
- b$\frac{5\sqrt2}{2}$
- c$\frac{5\sqrt3}{2}$
- d$\frac{15}{2}$
Answer (c): $\vec a\times\vec b=5\hat i+5\hat j-5\hat k$, magnitude $5\sqrt3$. Triangle area $=\frac12\cdot5\sqrt3=\frac{5\sqrt3}{2}$.
Revise: Vectors (Class 11 notes)
Q34 (2 marks)Statistics
The mean of 5 observations is 8 and their variance is 4. If a sixth observation equal to 8 is added, the new variance is
- a$\frac{10}{3}$
- b4
- c3
- d$\frac{20}{3}$
Answer (a): $\sum(x-8)^2=5\times4=20$. The new mean is still 8 and the added term contributes 0, so variance $=\frac{20}{6}=\frac{10}{3}$.
Revise: Statistics and Probability (Class 11 notes)
Q35 (2 marks)Probability (Bayes' theorem)
Box I has 3 white and 2 black balls, Box II has 1 white and 4 black balls. A box is chosen at random and a ball drawn is white. The probability that it came from Box I is
- a$\frac34$
- b$\frac35$
- c$\frac12$
- d$\frac{3}{10}$
Answer (a): $P(W)=\frac12\cdot\frac35+\frac12\cdot\frac15=\frac25$. $P(\text{I}|W)=\frac{3/10}{2/5}=\frac34$.
Revise: Statistics and Probability (Class 11 notes)
Physics
Q36Kinematics
A ball is thrown vertically upward with a speed of 20 m/s. Taking $g=10\ \mathrm{m/s^2}$, the maximum height reached is
- a40 m
- b20 m
- c10 m
- d25 m
Answer (b): $H=\frac{u^2}{2g}=\frac{400}{20}=20$ m.
Revise: Kinematics (Class 11 notes)
Q37Rotational dynamics
The moment of inertia of a uniform solid sphere of mass $M$ and radius $R$ about its diameter is
- a$\frac23MR^2$
- b$\frac12MR^2$
- c$\frac75MR^2$
- d$\frac25MR^2$
Answer (d): Standard result for a solid sphere: $I=\frac25MR^2$ (the hollow sphere has $\frac23MR^2$).
Revise: Rotational Dynamic (Class 12 notes)
Q38Fluid statics
The gauge pressure at a depth of 5 m in water ($\rho=1000\ \mathrm{kg/m^3},\ g=10\ \mathrm{m/s^2}$) is
- a$5\times10^3$ Pa
- b$5\times10^5$ Pa
- c$1.5\times10^5$ Pa
- d$5\times10^4$ Pa
Answer (d): $P=\rho gh=1000\times10\times5=5\times10^4$ Pa.
Revise: Fluid Statics (Class 12 notes)
Q39Elasticity
A wire of length $L$ and radius $r$ is stretched by a force $F$. If both its length and radius are doubled, the extension for the same force becomes
- athe same
- bdouble
- chalf
- done quarter
Answer (c): $\Delta L=\frac{FL}{\pi r^2Y}$. Doubling $L$ and $r$ multiplies it by $\frac{2}{4}=\frac12$.
Revise: Elasticity (Class 11 notes)
Q40Gravitation
At a height equal to the Earth's radius above the surface, the acceleration due to gravity is
- a$\frac g4$
- b$\frac g2$
- c$\frac g9$
- d$\frac g8$
Answer (a): Distance from the centre is $2R$, and $g\propto\frac1{r^2}$, so $g'=\frac g4$.
Revise: Gravitation (Class 11 notes)
Q41Quantity of heat
The heat needed to raise the temperature of 200 g of water from 20 degC to 50 degC is ($c=4200\ \mathrm{J,kg^{-1}K^{-1}}$)
- a8400 J
- b25200 J
- c42000 J
- d12600 J
Answer (b): $Q=mc\Delta T=0.2\times4200\times30=25200$ J.
Revise: Quantity of Heat (Class 11 notes)
Q42Second law of thermodynamics
The efficiency of a Carnot engine working between 500 K and 400 K is
- a25%
- b20%
- c80%
- d10%
Answer (b): $\eta=1-\frac{T_2}{T_1}=1-\frac{400}{500}=0.2=20%$.
Revise: Second Law of Thermodynamics (Class 12 notes)
Q43Reflection at curved mirrors
An object is placed 30 cm from a concave mirror of focal length 20 cm. The image distance is
- a12 cm
- b10 cm
- c50 cm
- d60 cm
Answer (d): $\frac1v=\frac1f-\frac1u=\frac1{20}-\frac1{30}=\frac1{60}$, so $v=60$ cm (real image).
Revise: Reflection at Curved Mirrors (Class 11 notes)
Q44Waves in pipes
The fundamental frequency of a pipe closed at one end of length 0.5 m is (speed of sound 340 m/s)
- a340 Hz
- b170 Hz
- c680 Hz
- d85 Hz
Answer (b): $f=\frac{v}{4L}=\frac{340}{2}=170$ Hz.
Revise: Wave in Pipes and Strings (Class 12 notes)
Q45Capacitors
Capacitors of $6\ \mu$F and $3\ \mu$F are connected in series. The equivalent capacitance is
- a$9\ \mu$F
- b$4.5\ \mu$F
- c$2\ \mu$F
- d$0.5\ \mu$F
Answer (c): $C=\frac{6\times3}{6+3}=2\ \mu$F.
Revise: Capacitor (Class 11 notes)
Q46Current electricity
A wire of resistance 10 ohm is stretched uniformly to double its length. Its new resistance is
- a20 ohm
- b40 ohm
- c5 ohm
- d80 ohm
Answer (b): Volume is constant, so area halves; $R\propto\frac{l}{A}$ becomes $2\times2=4$ times, i.e. 40 ohm.
Revise: DC Circuit (Class 11 notes)
Q47Magnetic force on a conductor
A straight wire 0.5 m long carrying 4 A is perpendicular to a field of 0.2 T. The force on it is
- a0.8 N
- b0.04 N
- c0.4 N
- d4 N
Answer (c): $F=BIl=0.2\times4\times0.5=0.4$ N.
Revise: Magnetic Field (Class 12 notes)
Q48Electromagnetic induction
The magnetic flux through a coil falls from 0.04 Wb to 0.01 Wb in 0.1 s. The magnitude of the induced emf is
- a0.03 V
- b0.3 V
- c0.5 V
- d3 V
Answer (b): $\varepsilon=\frac{\Delta\Phi}{\Delta t}=\frac{0.03}{0.1}=0.3$ V.
Revise: Electromagnetic Induction (Class 12 notes)
Q49Alternating current
The rms value of an alternating voltage with a peak value of 100 V is about
- a70.7 V
- b100 V
- c141 V
- d50 V
Answer (a): $V_{rms}=\frac{V_0}{\sqrt2}=\frac{100}{1.414}\approx70.7$ V.
Revise: Alternating Currents (Class 12 notes)
Q50Radioactivity
A radioactive sample has a half-life of 10 days. The fraction remaining after 30 days is
- a$\frac16$
- b$\frac14$
- c$\frac18$
- d$\frac13$
Answer (c): 30 days is 3 half-lives, so the fraction is $\left(\frac12\right)^3=\frac18$.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Q51Photoelectric effect
Light of photon energy 5 eV falls on a metal of work function 2 eV. The maximum kinetic energy of the emitted electrons is
- a7 eV
- b2.5 eV
- c3 eV
- d5 eV
Answer (c): $K_{max}=h\nu-\phi=5-2=3$ eV.
Revise: Photons (Class 12 notes)
Q52 (2 marks)Dynamics on an incline
A block starts from rest on a smooth incline of $30^\circ$. Taking $g=10\ \mathrm{m/s^2}$, the distance it slides in 4 s is
- a40 m
- b80 m
- c20 m
- d10 m
Answer (a): $a=g\sin30^\circ=5\ \mathrm{m/s^2}$. $s=\frac12at^2=\frac12\times5\times16=40$ m.
Revise: Dynamics (Class 11 notes)
Q53 (2 marks)Conservation of angular momentum
A skater of moment of inertia $3\ \mathrm{kg,m^2}$ spins at 4 rad/s. On pulling in the arms the moment of inertia becomes $1.5\ \mathrm{kg,m^2}$. The final rotational kinetic energy is
- a48 J
- b24 J
- c96 J
- d12 J
Answer (a): $I_1\omega_1=I_2\omega_2$ gives $\omega_2=8$ rad/s. $K=\frac12I_2\omega_2^2=\frac12(1.5)(64)=48$ J.
Revise: Rotational Dynamic (Class 12 notes)
Q54 (2 marks)Fluid flow, Bernoulli's principle
Water flows at 2 m/s in a pipe of diameter 6 cm which narrows to a diameter of 3 cm (horizontal pipe). The pressure drop in the narrow section is ($\rho=1000\ \mathrm{kg/m^3}$)
- a$3.2\times10^4$ Pa
- b$6\times10^4$ Pa
- c$1.5\times10^4$ Pa
- d$3\times10^4$ Pa
Answer (d): Continuity: $v_2=2\times4=8$ m/s. $\Delta P=\frac12\rho(v_2^2-v_1^2)=500(64-4)=3\times10^4$ Pa.
Revise: Fluid Statics (Class 12 notes)
Q55 (2 marks)Gravitation
A planet has 8 times the mass and 2 times the radius of the Earth. If the Earth's escape velocity is 11.2 km/s, the planet's escape velocity is
- a44.8 km/s
- b11.2 km/s
- c15.8 km/s
- d22.4 km/s
Answer (d): $v_e=\sqrt{\frac{2GM}{R}}\propto\sqrt{\frac MR}=\sqrt{\frac82}=2$, so $v=22.4$ km/s.
Revise: Gravitation (Class 11 notes)
Q56 (2 marks)Thermal expansion
A steel rod 2 m long at 20 degC is heated to 120 degC ($\alpha=1.2\times10^{-5}\ \mathrm{K^{-1}}$). The increase in length is
- a1.2 mm
- b2.4 mm
- c24 mm
- d4.8 mm
Answer (b): $\Delta L=L\alpha\Delta T=2\times1.2\times10^{-5}\times100=2.4\times10^{-3}$ m $=2.4$ mm.
Revise: Thermal Expansion (Class 11 notes)
Q57 (2 marks)First law of thermodynamics
Two moles of a monatomic ideal gas are heated at constant volume from 300 K to 400 K ($R=8.3\ \mathrm{J,mol^{-1}K^{-1}}$). The change in internal energy is
- a4150 J
- b1660 J
- c830 J
- d2490 J
Answer (d): $\Delta U=nC_V\Delta T=2\times\frac32(8.3)\times100=2490$ J.
Revise: First Law of Thermodynamics (Class 12 notes)
Q58 (2 marks)Interference
In Young's experiment the slit separation is 0.5 mm, the screen is 1 m away, and the wavelength is 500 nm. The distance of the 3rd bright fringe from the central maximum is
- a1 mm
- b1.5 mm
- c3 mm
- d6 mm
Answer (c): Fringe width $\beta=\frac{\lambda D}{d}=\frac{500\times10^{-9}\times1}{0.5\times10^{-3}}=1$ mm. The 3rd bright fringe is at $3\beta=3$ mm.
Revise: Interference (Class 12 notes)
Q59 (2 marks)Waves on strings
A string 0.8 m long fixed at both ends carries waves of speed 200 m/s. The frequency of its third harmonic is
- a125 Hz
- b250 Hz
- c500 Hz
- d375 Hz
Answer (d): $f_n=\frac{nv}{2L}$, so $f_3=\frac{3\times200}{1.6}=375$ Hz.
Revise: Wave in Pipes and Strings (Class 12 notes)
Q60 (2 marks)Electric potential
Two point charges, each $+4\ \mu$C, are 0.6 m apart in air. The electric potential at the midpoint is ($k=9\times10^9\ \mathrm{N,m^2C^{-2}}$)
- a$2.4\times10^5$ V
- b$1.2\times10^5$ V
- c0
- d$4.8\times10^5$ V
Answer (a): Each charge is 0.3 m away: $V=2\times\frac{9\times10^9\times4\times10^{-6}}{0.3}=2.4\times10^5$ V.
Revise: Potential Difference and Potential Energy (Class 11 notes)
Q61 (2 marks)AC circuits
A series LCR circuit has $R=30\ \Omega$, $X_L=60\ \Omega$, $X_C=20\ \Omega$ and is connected to a 100 V rms supply. The rms current is
- a3.3 A
- b1.4 A
- c2 A
- d5 A
Answer (c): $Z=\sqrt{R^2+(X_L-X_C)^2}=\sqrt{900+1600}=50\ \Omega$. $I=\frac{100}{50}=2$ A.
Revise: Alternating Currents (Class 12 notes)
Q62 (2 marks)Motional emf
A rod 0.5 m long moves at 4 m/s perpendicular to a field of 0.3 T, sliding on rails that complete a circuit of resistance 2 ohm. The induced current is
- a0.6 A
- b0.3 A
- c1.2 A
- d0.15 A
Answer (b): $\varepsilon=Blv=0.3\times0.5\times4=0.6$ V. $I=\frac{0.6}{2}=0.3$ A.
Revise: Electromagnetic Induction (Class 12 notes)
Q63 (2 marks)Nuclear physics, binding energy
The mass defect of a helium-4 nucleus is 0.0304 u (1 u = 931.5 MeV). The binding energy per nucleon is about
- a28.3 MeV
- b3.5 MeV
- c7.1 MeV
- d14.2 MeV
Answer (c): Total $B=0.0304\times931.5\approx28.3$ MeV. Dividing by 4 nucleons gives about 7.1 MeV.
Revise: Nuclear Physics (Class 11 notes)
Chemistry
Q64Mole concept
The number of moles in 11 g of carbon dioxide (molar mass 44 g/mol) is
- a0.5
- b2.5
- c0.25
- d4
Answer (c): $n=\frac{11}{44}=0.25$ mol.
Revise: Stoichiometry (Class 11 notes)
Q65Ionic equilibrium
The pH of a $0.005$ M solution of Ba(OH)$_2$ (fully dissociated) is
- a2
- b12
- c11.7
- d13
Answer (b): $[\mathrm{OH^-}]=2\times0.005=0.01$ M, so pOH $=2$ and pH $=12$.
Revise: Ionic Equilibrium (Class 12 notes)
Q66Chemical kinetics
The unit of the rate constant of a first order reaction is
- a$\mathrm{mol,L^{-1}s^{-1}}$
- b$\mathrm{L,mol^{-1}s^{-1}}$
- c$\mathrm{s^{-1}}$
- d$\mathrm{mol,s^{-1}}$
Answer (c): Rate $=k[A]$, so $k=\frac{\mathrm{mol,L^{-1}s^{-1}}}{\mathrm{mol,L^{-1}}}=\mathrm{s^{-1}}$.
Revise: Chemical Kinetics (Class 12 notes)
Q67Oxidation and reduction
The oxidation number of chromium in K$_2$Cr$_2$O$_7$ is
- a+7
- b+3
- c+6
- d+12
Answer (c): $2(+1)+2x+7(-2)=0$ gives $x=+6$.
Revise: Oxidation and Reduction (Class 11 notes)
Q68d-block elements
The number of unpaired electrons in the Fe$^{3+}$ ion (atomic number 26) is
- a4
- b3
- c6
- d5
Answer (d): Fe$^{3+}$ is $3d^5$, with all five d orbitals singly occupied, so 5 unpaired electrons.
Revise: Transition Metals (Class 12 notes)
Q69Chemical bonding, VSEPR
The shape of the XeF$_4$ molecule is
- atetrahedral
- bsee-saw
- coctahedral
- dsquare planar
Answer (d): Xe has 4 bond pairs and 2 lone pairs (octahedral arrangement); the lone pairs sit opposite each other, giving a square planar shape.
Revise: Chemical Bonding (Class 11 notes)
Q70s-block elements
The flame colour imparted by strontium salts is
- agolden yellow
- bbrick red
- ccrimson red
- dviolet
Answer (c): Sr gives crimson red; Na golden yellow, Ca brick red, K violet.
Revise: Chemistry of Metals (Class 11 notes)
Q71Metallurgy
Froth flotation is mainly used for the concentration of
- asulphide ores
- boxide ores
- ccarbonate ores
- dhalide ores
Answer (a): Sulphide ores are wetted by oil and float in the froth, while the gangue is wetted by water and sinks.
Revise: Chemistry of Metals (Class 11 notes)
Q72IUPAC nomenclature
The IUPAC name of $\mathrm{CH_3CH(CH_3)CH_2CH_2OH}$ is
- a3-methylbutan-1-ol
- b2-methylbutan-1-ol
- c3-methylbutan-2-ol
- d2-methylbutan-4-ol
Answer (a): The chain of four carbons is numbered from the OH end, giving OH at C1 and the methyl at C3.
Revise: Fundamental Principles (Class 11 notes)
Q73Isomerism
The number of structural isomers of pentane, $\mathrm{C_5H_{12}}$, is
- a2
- b4
- c5
- d3
Answer (d): n-pentane, isopentane (2-methylbutane) and neopentane (2,2-dimethylpropane).
Revise: Fundamental Principles (Class 11 notes)
Q74Alkynes
Hydration of propyne in the presence of dilute H$_2$SO$_4$ and HgSO$_4$ gives
- apropanone
- bpropanal
- cpropan-2-ol
- dpropan-1-ol
Answer (a): Markovnikov addition of water gives the enol $\mathrm{CH_3C(OH)=CH_2}$, which tautomerises to propanone.
Revise: Hydrocarbons (Class 11 notes)
Q75Carboxylic acids
Which of the following is the strongest acid?
- a$\mathrm{CH_3COOH}$
- b$\mathrm{ClCH_2COOH}$
- c$\mathrm{Cl_2CHCOOH}$
- d$\mathrm{Cl_3CCOOH}$
Answer (d): Each chlorine withdraws electrons (-I effect) and stabilises the carboxylate, so acidity increases with more Cl: trichloroacetic acid is strongest.
Revise: Carboxylic Acids (Class 12 notes)
Q76Amines
The carbylamine test (CHCl$_3$ with alcoholic KOH) is given by
- asecondary amines
- btertiary amines
- cquaternary ammonium salts
- dprimary amines
Answer (d): Only primary amines form the foul-smelling isocyanide (carbylamine).
Revise: Amines (Class 12 notes)
Q77Haloalkanes
The IUPAC name of the compound $\mathrm{CH_3CH_2CH_2CH_2Br}$ is
- a2-bromobutane
- bbutyl bromine
- c1-bromobutane
- d4-bromobutane
Answer (c): A four-carbon chain with Br on the first carbon, numbered from the end nearer the halogen: 1-bromobutane.
Revise: Haloalkanes (Class 12 notes)
Q78 (2 marks)Chemical equilibrium
For $\mathrm{N_2+3H_2\rightleftharpoons 2NH_3}$, the equilibrium concentrations are $[\mathrm{N_2}]=0.25$ M, $[\mathrm{H_2}]=2$ M and $[\mathrm{NH_3}]=2$ M. The value of $K_c$ is
- a2
- b4
- c0.5
- d8
Answer (a): $K_c=\frac{[\mathrm{NH_3}]^2}{[\mathrm{N_2}][\mathrm{H_2}]^3}=\frac{4}{0.25\times8}=2$.
Revise: Chemical Equilibrium (Class 11 notes)
Q79 (2 marks)Chemical kinetics
A first order reaction has a half-life of 20 minutes. The time required for 87.5% of it to complete is
- a60 minutes
- b40 minutes
- c80 minutes
- d100 minutes
Answer (a): 87.5% complete leaves $\frac18=\left(\frac12\right)^3$, which is 3 half-lives: $3\times20=60$ min.
Revise: Chemical Kinetics (Class 12 notes)
Q80 (2 marks)Electrochemistry, Faraday's laws
A charge of 9650 C is passed through aqueous CuSO$_4$. The mass of copper deposited is (Cu = 63.5, F = 96500 C/mol)
- a6.35 g
- b3.175 g
- c1.59 g
- d0.635 g
Answer (b): Moles of electrons $=\frac{9650}{96500}=0.1$. Cu$^{2+}+2e^-\to$ Cu gives 0.05 mol Cu $=0.05\times63.5=3.175$ g.
Revise: Electrochemistry (Class 12 notes)
Q81 (2 marks)Stoichiometry
The volume of hydrogen at STP liberated when 6.5 g of zinc (Zn = 65) reacts with excess dilute acid is
- a4.48 L
- b1.12 L
- c22.4 L
- d2.24 L
Answer (d): $\mathrm{Zn+2HCl\to ZnCl_2+H_2}$: 0.1 mol Zn gives 0.1 mol H$_2$, i.e. $0.1\times22.4=2.24$ L.
Revise: Stoichiometry (Class 11 notes)
Q82 (2 marks)Haloalkanes and carboxylic acids
Ethyl bromide is treated with KCN to form X, which on acid hydrolysis gives Y. Y is
- aethanoic acid
- bpropanoic acid
- cpropan-1-amine
- dpropanal
Answer (b): $\mathrm{CH_3CH_2Br\to CH_3CH_2CN}$ (X, chain grows by one carbon), and hydrolysis of the nitrile gives $\mathrm{CH_3CH_2COOH}$.
Revise: Haloalkanes (Class 12 notes)
Q83 (2 marks)Aldehydes and alcohols
Acetaldehyde reacts with methylmagnesium bromide, and the adduct is hydrolysed. The product is
- apropan-1-ol
- bbutan-2-ol
- cpropan-2-ol
- dethanol
Answer (c): The Grignard reagent adds to the carbonyl carbon to give $\mathrm{CH_3CH(OH)CH_3}$, a secondary alcohol.
Revise: Aldehydes and Ketones (Class 12 notes)
Q84 (2 marks)Amines
Acetamide on heating with bromine and aqueous NaOH (Hofmann bromamide reaction) gives
- aethylamine
- bmethylamine
- cmethanol
- dacetic acid
Answer (b): The amide loses its carbonyl carbon as carbonate, so the amine has one carbon fewer: $\mathrm{CH_3NH_2}$.
Revise: Amines (Class 12 notes)
Q85 (2 marks)d-block elements
The spin-only magnetic moment of Fe$^{2+}$ (high spin, atomic number 26) is about
- a5.9 BM
- b3.87 BM
- c2.83 BM
- d4.9 BM
Answer (d): Fe$^{2+}$ is $3d^6$ with 4 unpaired electrons: $\mu=\sqrt{n(n+2)}=\sqrt{24}\approx4.9$ BM.
Revise: Transition Metals (Class 12 notes)
English
Q86Subject-verb agreement
Choose the correct option: Neither of the boys ___ present at the meeting.
- awas
- bwere
- care
- dhave been
Answer (a): 'Neither of' takes a singular verb, so 'was' is correct.
Revise: Grammar and Usage (Class 11 notes)
Q87Past perfect tense
By the time we reached the cinema, the film ___.
- aalready started
- bhas already started
- chad already started
- dis already starting
Answer (c): An earlier past action before 'we reached' needs the past perfect: 'had already started'.
Revise: Grammar and Usage (Class 11 notes)
Q88Prepositions
She is allergic ___ dust.
- ato
- bwith
- cfrom
- dof
Answer (a): The adjective 'allergic' takes the preposition 'to'.
Revise: Grammar and Usage (Class 11 notes)
Q89Conditionals
If I ___ you, I would apologise at once.
- abe
- bam
- cwere
- dhad been
Answer (c): The second conditional of an unreal present uses 'were' for all persons.
Revise: Grammar and Usage (Class 11 notes)
Q90Vocabulary
Choose the word closest in meaning to 'meticulous'.
- acareless
- bhasty
- cgenerous
- dcareful and precise
Answer (d): 'Meticulous' means showing great attention to detail.
Revise: Vocabulary and Word Formation (Class 11 notes)
Q91Vocabulary, antonyms
Choose the antonym of 'scarce'.
- arare
- babundant
- climited
- dcostly
Answer (b): 'Scarce' means in short supply, so its opposite is 'abundant'.
Revise: Vocabulary and Word Formation (Class 11 notes)
Q92Phonetics, word stress
In which word does the main stress fall on the second syllable?
- aphotography
- bphotograph
- cpolitics
- dcinema
Answer (a): Photography is pho-TOG-ra-phy, while photograph, politics and cinema are stressed on the first syllable.
Q93Phonetics, silent letters
Which word begins with a silent letter?
- akitchen
- bmarket
- ckettle
- dknowledge
Answer (d): In 'knowledge' the initial k is silent: it is pronounced /ˈnɒlɪdʒ/.
Q94Idioms
What does the idiom 'to bite the bullet' mean?
- ato eat very quickly
- bto face a hard situation bravely
- cto refuse to cooperate
- dto make a costly mistake
Answer (b): It means to endure something unpleasant with courage.
Revise: Vocabulary and Word Formation (Class 11 notes)
Q95Active and passive voice
Choose the passive form of: 'They are building a new bridge.'
- aA new bridge was built by them.
- bA new bridge is built by them.
- cA new bridge is being built by them.
- dA new bridge has been built by them.
Answer (c): The present continuous passive is 'is/are being + past participle'.
Revise: Grammar and Usage (Class 11 notes)
Q96 (2 marks)Reading comprehension
Read: 'The village library opened in 1952 with only two hundred books. Today it holds ten thousand because every family donates one book each year.' According to the passage, the collection grew mainly because
- athe government buys new books
- beach family donates a book every year
- cvisitors pay a fee
- dthe library opened in 1952
Answer (b): The passage states the growth is due to every family donating one book each year.
Revise: Reading Comprehension (Class 11 notes)
Q97 (2 marks)Reading comprehension, inference
Read: 'Mina rose before dawn, packed her bag and walked eight kilometres to school without complaint. She had missed only one day all year.' Which conclusion is best supported?
- aMina lives near the school.
- bMina dislikes school.
- cMina is committed to her education.
- dMina travels by bus.
Answer (c): Her early start, long walk and near-perfect attendance show commitment; the other options contradict the text.
Revise: Reading Comprehension (Class 11 notes)
Q98 (2 marks)Error spotting, agreement
Identify the part with an error: 'Each of the students (A) have submitted (B) their assignments (C) on time (D).'
- aEach of the students
- btheir assignments
- chave submitted
- don time
Answer (c): 'Each' is singular, so the verb must be 'has submitted'.
Revise: Grammar and Usage (Class 11 notes)
Q99 (2 marks)Reported speech
Choose the correct indirect speech: She said, 'I will call you tomorrow.'
- aShe said that she will call you tomorrow.
- bShe said that she would call you tomorrow.
- cShe said that she called me the next day.
- dShe said that she would call me the next day.
Answer (d): 'Will' becomes 'would', 'I' becomes 'she', 'you' becomes 'me' and 'tomorrow' becomes 'the next day'.
Revise: Grammar and Usage (Class 11 notes)
Q100 (2 marks)Phonetics, homophones
Which pair of words are homophones?
- aadvice - advise
- bquiet - quite
- caccess - excess
- dprincipal - principle
Answer (d): 'Principal' and 'principle' are both pronounced /ˈprɪnsəpəl/ but differ in meaning and spelling.