- a
- b
- c
- d
Answer (d): The values fit $f(x)=x^2+1$, so $f(-1)=1+1=2$.
Revise: Algebra (Class 11 notes)
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Answer (d): The values fit $f(x)=x^2+1$, so $f(-1)=1+1=2$.
Revise: Algebra (Class 11 notes)
Answer (c): Taking complements reverses inclusion, so $A\subseteq B$ gives $B^c\subseteq A^c$.
Revise: Algebra (Class 11 notes)
Answer (c): $\log_a(ab)=1+\log_a b=x$, so $\log_a b=x-1$ and $\log_b a=\frac{1}{x-1}$. Then $\log_b(ab)=1+\frac{1}{x-1}=\frac{x}{x-1}$.
Revise: Algebra (Class 11 notes)
Answer (b): $AB=\begin{pmatrix}1&0\0&-(2x-3)\end{pmatrix}$. For this to be $I$, $-(2x-3)=1$, so $x=1$.
Revise: Algebra (Class 11 notes)
Answer (d): Multiply by $A^{-1}$: $A-I+A^{-1}=0$, so $A^{-1}=I-A$.
Revise: Algebra (Class 11 notes)
Answer (a): For $a/r, a, ar$ the product is $a^3=1728$, so $a=12$.
Revise: Algebra (Class 11 notes)
Answer (c): $x^4=1$ gives $(x^2-1)(x^2+1)=0$, so $x=\pm1,\pm i$.
Revise: Algebra (Class 11 notes)
Answer (d): The inverse is $\frac{3-i}{2+3i}=\frac{(3-i)(2-3i)}{13}=\frac{3-11i}{13}$.
Revise: Algebra (Class 11 notes)
Answer (d): Clearing fractions gives $x^2-2(a+b)x+3ab=0$. Roots equal and opposite means the sum is zero, so $a+b=0$.
Revise: Algebra (Class 11 notes)
Answer (d): By the remainder theorem put $x=-2k$: $k(4k^2)-2k-1=4k^3-2k-1$.
Revise: Algebra (Class 11 notes)
Answer (d): $\sin^{-1}(4/5)=\cos^{-1}(3/5)$, so the right side is $0$ and $x=\cos 0=1$.
Revise: Trigonometry (Class 11 notes)
Answer (c): $2-2\cos^2\theta=3\cos\theta$ gives $(2\cos\theta-1)(\cos\theta+2)=0$, so $\cos\theta=\frac12$ and $\theta=\frac{\pi}{3},\frac{5\pi}{3}$.
Revise: Trigonometry (Class 11 notes)
Answer (a): The pair condition $abc+2fgh-af^2-bg^2-ch^2=0$ with $a=1,b=2,c=2,f=\frac52,g=\frac32,h=\frac k2$ gives $2k^2-15k+27=0$, so $k=3$ or $\frac92$. Only $3$ is listed.
Revise: Analytic geometry (Class 11 notes)
Answer (c): Externally touching circles have two direct tangents and one common tangent at the point of contact, so three.
Revise: Analytic geometry (Class 11 notes)
Answer (a): Put $y=2x$: $5x^2-10x=0$, so $x=0$ or $x=2$, giving $(0,0)$ and $(2,4)$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): $\tan\theta=\frac{2\sqrt{h^2-ab}}{a+b}=\frac{2\sqrt{49/4-6}}{5}=1$, so $\theta=\frac{\pi}{4}$.
Revise: Analytic geometry (Class 11 notes)
Answer (b): Divide by $x^2$: the lower powers vanish and the ratio of leading coefficients $\frac{a}{d}$ remains.
Revise: Calculus (Class 12 notes)
Answer (c): The limit equals $n\cdot3^{n-1}$. For $n=3$ it is $3\cdot9=27$.
Revise: Calculus (Class 12 notes)
Answer (a): The standard result is $\frac{d}{dx}\operatorname{cosech}^{-1}x=\frac{-1}{|x|\sqrt{1+x^2}}$.
Revise: Calculus (Class 12 notes)
Answer (d): $y=\frac{\ln x}{\frac12\ln x}=2$, a constant, so $\frac{dy}{dx}=0$.
Revise: Calculus (Class 12 notes)
Answer (b): This is the standard integral giving $\sin^{-1}x+c$.
Revise: Calculus (Class 12 notes)
Answer (b): $\cos2x=2\cos^2x-1$, so the integrand is $2\cos x-\sec x$, giving $2\sin x-\log(\sec x+\tan x)$.
Revise: Calculus (Class 12 notes)
Answer (b): Put $u=\sin x$: $\int a^u du=\frac{a^u}{\log a}+c$.
Revise: Calculus (Class 12 notes)
Answer (d): $y''=6x-24=0$ gives $x=4$, and $y''$ changes sign there.
Revise: Calculus (Class 12 notes)
Answer (d): Area $=2\int_0^{\pi}\sin x,dx=2\cdot2=4$ (the negative half is taken as positive).
Revise: Calculus (Class 12 notes)
Answer (d): $v=\frac{dx}{dt}=b+2ct$, and $a=\frac{dv}{dt}=2c$.
Revise: Kinematics (Class 11 notes)
Answer (b): $p=\sqrt{2mK}\propto\sqrt m$, so $p_1:p_2=\sqrt1:\sqrt4=1:2$.
Revise: Work, Energy and Power (Class 11 notes)
Answer (a): Earth bulges at the equator and its rotation reduces effective gravity there, so $g$ is least at the equator.
Revise: Gravitation (Class 11 notes)
Answer (b): Surface energy $=T\times4\pi r^2\propto r^2$.
Revise: Fluid Statics (Class 12 notes)
Answer (b): For floating, $\rho_{body}V=\rho_{water}\cdot\frac34V$, so $\rho_{body}=0.75$ g/cc.
Revise: Fluid Statics (Class 12 notes)
Answer (c): $P_2=P_1(V_1/V_2)^\gamma=4^{1.5}=8$ atm.
Revise: First Law of Thermodynamics (Class 12 notes)
Answer (d): $727^\circ\text{C}=1000$ K and the Stefan law gives rate $\propto T^4=(1000)^4$.
Revise: Rate of Heat Flow (Class 11 notes)
Answer (a): Boiling starts when saturated vapour pressure equals the external pressure on the liquid surface, usually atmospheric.
Revise: Heat and Temperature (Class 11 notes)
Answer (a): Red has the smallest refractive index, so by $\frac1f\propto(\mu-1)$ its focal length is largest.
Revise: Lenses (Class 11 notes)
Answer (a): $\sqrt2=\frac{\sin\frac{60+\delta}{2}}{\sin30^\circ}$ gives $\sin\frac{60+\delta}{2}=\frac{1}{\sqrt2}$, so $\frac{60+\delta}{2}=45^\circ$ and $\delta=30^\circ$.
Revise: Refraction Through Prisms (Class 11 notes)
Answer (b): Fringe width $\beta=\frac{\lambda D}{d}$, so it decreases as $d$ increases.
Revise: Interference (Class 12 notes)
Answer (d): Total flux is $4\pi I=2000\pi$ lm, 2% of it is $40\pi$ lm, and the disc area is $\pi(0.1)^2=0.01\pi$ m$^2$, so $E=40\pi/0.01\pi=4000$ lux.
Revise: Nature and Propagation of Light (Class 12 notes)
Answer (b): Inside a conducting sphere the field is zero, so the potential is constant and equals the surface value, 80 V.
Revise: Potential Difference and Potential Energy (Class 11 notes)
Answer (b): In parallel the voltage is the same, so $Q=CV$ gives $Q_1:Q_2=2:3$.
Revise: Capacitor (Class 11 notes)
Answer (b): By definition this angle is the magnetic declination.
Revise: Magnetic Field (Class 12 notes)
Answer (d): The field inside is along the axis, parallel to the velocity, so $F=qvB\sin0=0$.
Revise: Magnetic Field (Class 12 notes)
Answer (a): $E=I^2Rt$ so $I=\sqrt{\frac{E}{Rt}}$.
Revise: DC Circuit (Class 11 notes)
Answer (c): The wave number is $k=\pi$, so $\lambda=\frac{2\pi}{k}=2$ cm.
Revise: Wave Motion (Class 12 notes)
Answer (a): $f'=f\frac{v}{v-v_s}=150\times\frac{330}{220}=225$ Hz.
Revise: Acoustic Phenomena (Class 12 notes)
Answer (a): At nodes the medium is displaced least but the neighbouring particles squeeze or stretch it most, so strain (pressure variation) is maximum at nodes.
Revise: Wave in Pipes and Strings (Class 12 notes)
Answer (c): The orbit holds a whole number of wavelengths: $2\pi R=n\lambda$.
Revise: Quantization of Energy (Class 12 notes)
Answer (a): Absorption means going up in n. The 1 to 2 jump has the largest energy gap (10.2 eV), hence highest frequency.
Revise: Quantization of Energy (Class 12 notes)
Answer (d): 15 years is 3 half-lives, so $\frac18$ remains and $1-\frac18=\frac78$ has decayed.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (c): Beta rays are fast electrons (or positrons) ejected from the nucleus, so they are charged particles emitted by the nucleus.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (a): The soldiers are the object of 'helped' and are not the subject, so the plain pronoun 'them' is used.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): The subject is 'He', so the reflexive pronoun is 'himself'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): The verb agrees with 'The leader', not with the phrase 'with his helpers', so 'has'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Two subjects joined by 'and' are plural, so 'have'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): The main clause is negative, so the tag is positive, and 'one' is repeated as 'should one?'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): A negative statement takes a positive tag, and 'Those books' becomes 'they': 'are they?'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'For' is used with a period of time such as ten years.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Beside' means next to; 'besides' means in addition to.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'Unless' already carries the negative sense, and a present condition takes 'will' in the main clause: 'It will not improve'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Third conditional: 'If + past perfect' goes with 'would have + past participle'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): No article is used before the name of a game.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'The' is used for a specific known place such as the theatre he went to.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): To sow is to plant seeds; the opposite action is to reap (harvest).
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): 'Faulty' means having defects; 'flawless' means without defects.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): 'Check' (to examine) means the same as 'test'.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): As a noun, 'present' means 'gift'.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Present simple question passive: 'Is + object + past participle'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Present simple passive: 'is/are + past participle', so 'Fish is sold here.'
Revise: Grammar and Usage (Class 11 notes)
Answer (c): In indirect speech 'said to' becomes 'told', 'I' becomes 'she', 'will' becomes 'would' and 'myself' becomes 'herself'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): An imperative becomes 'told + object + to-infinitive', and 'your' changes to 'my' for the listener.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'Every year' signals a habit, so the present simple 'goes'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): Acrophobia is the fear of heights.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Of two past actions, the earlier one takes the past perfect: 'had died'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): Mis- gives 'misfire' (to fail to fire properly).
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): For verbs ending in -ie, change ie to y before -ing: die gives dying.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): COOH is the principal functional group, so the secondary suffix is 'oic acid' (2-hydroxy-4-oxobutanoic acid).
Revise: Fundamental Principles (Class 11 notes)
Answer (a): Electrophilic addition to an alkene goes through a carbocation; Markovnikov's rule follows the more stable one.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): HC$\equiv$CH has two C-H $\sigma$ bonds and one C-C $\sigma$ bond, plus two $\pi$ bonds in the triple bond.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): Lewis acids accept an electron pair and oxidising agents accept electrons, so both are electron-seeking electrophiles.
Revise: Fundamental Principles (Class 11 notes)
Answer (a): Wurtz coupling joins two alkyl halides ($2R\text{-}X\to R\text{-}R$), so it cannot give methane (one carbon).
Revise: Hydrocarbons (Class 11 notes)
Answer (c): The $\ce{H+}$ from strong HCl suppresses ionisation of the weak acid $\ce{H2S}$. The other pairs are both strong electrolytes.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (c): Saturated monocarboxylic acids (e.g. $\ce{CH3COOH}$, $\ce{C2H4O2}$) follow $\ce{C_nH_{2n}O2}$.
Revise: Carboxylic Acids (Class 12 notes)
Answer (d): Glucose, fructose and galactose are all isomers with the formula $\ce{C6H12O6}$.
Answer (c): This is the statement of the law of conservation of mass.
Revise: Stoichiometry (Class 11 notes)
Answer (a): By the $(n+l)$ rule: 4d is 6, 3d is 5, 5s is 5, 4p is 5. So 4d is highest.
Revise: Atomic Structure (Class 11 notes)
Answer (b): In peroxides ($\ce{O2^{2-}}$) each oxygen has oxidation state $-1$.
Revise: Oxidation and Reduction (Class 11 notes)
Answer (a): The standard hydrogen electrode is the reference and is assigned exactly zero potential.
Revise: Electrochemistry (Class 12 notes)
Answer (a): Na has 11 protons and 11 electrons; $\ce{Mg+}$ has 12 protons for 11 electrons and $\ce{Al^{2+}}$ is smaller still, so Na is largest.
Revise: Classification of Elements (Class 11 notes)
Answer (d): 1000 g of water is $\frac{1000}{18}=55.55$ mol in 1 L, so molarity is 55.55 M.
Revise: Stoichiometry (Class 11 notes)
Answer (c): Only tritium ($^3$H) is radioactive; protium and deuterium are stable.
Revise: Nuclear Chemistry (Class 12 notes)
Answer (a): The contact process oxidises $\ce{SO2}$ to $\ce{SO3}$ using vanadium pentoxide.
Revise: Modern Chemical Manufactures (Class 11 notes)
Answer (b): White vitriol is hydrated zinc sulphate, $\ce{ZnSO4.7H2O}$ (green vitriol is the iron salt).
Revise: Chemistry of Metals (Class 11 notes)
Answer (d): Proteins are held by peptide bonds (a covalent bond) in the chain and by hydrogen bonds in the folded structure, so all of the above apply.
Revise: Biomolecules (Class 11 notes)
Answer (a): A chromophore is the group with multiple bonds that gives colour; an auxochrome only deepens it.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (b): Calomel is mercurous chloride, $\ce{Hg2Cl2}$.
Revise: Chemistry of Metals (Class 11 notes)
Answer (c): In the brown ring test $\ce{FeSO4}$ combines with NO to give $[\ce{Fe(H2O)5NO}]^{2+}$.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (c): Chalcopyrite ($\ce{CuFeS2}$) is a brassy yellow ore often mistaken for gold, hence 'fool's gold'.
Revise: Copper (Class 12 notes)
Answer (c): Hexane has 5 isomers: hexane, 2-methylpentane, 3-methylpentane, 2,2-dimethylbutane and 2,3-dimethylbutane.
Revise: Fundamental Principles (Class 11 notes)
Answer (c): For a first order reaction the rate is $k[A]$, so it depends on the concentration of the reactant; time is not a direct factor, so 'all of above' is wrong.
Revise: Chemical Kinetics (Class 12 notes)
Answer (b): Water is neutral (pH 7). $\ce{CO2}$ and $\ce{N2O3}$ are acidic and ZnO is amphoteric.
Revise: Chemistry of Non-Metals (Class 11 notes)