- a$a\neq0,\ b=0$
- b$a=0,\ b=0$
- c$a=0,\ b\neq0$
- d$a\neq0,\ b\neq0$
Answer (c): A fraction is zero only when the numerator is zero, and the denominator must be non-zero for the fraction to exist.
Revise: Algebra (Class 11 notes)
TU IOST | Entrance paper
Answer (c): A fraction is zero only when the numerator is zero, and the denominator must be non-zero for the fraction to exist.
Revise: Algebra (Class 11 notes)
Answer (b): The inequality $x\ge5$ is satisfied by the single number 6, while $[5,\infty)$ is the whole set of solutions rather than a value of $x$.
Revise: Algebra (Class 11 notes)
Answer (d): $x-2=\pm5$ gives $x=7$ or $x=-3$.
Revise: Algebra (Class 11 notes)
Answer (c): The range is the set of second elements: ${2,3,4}$.
Revise: Algebra (Class 11 notes)
Answer (b): Put $x=4,y=81$: $a^4=81=3^4$, so $a=3$.
Revise: Algebra (Class 11 notes)
Answer (a): $\tan^2x=\tan^2(\pi/4)$ gives the general solution $x=n\pi\pm\pi/4$.
Revise: Trigonometry (Class 11 notes)
Answer (c): Let $\theta=\sin^{-1}x$. Then $\sin2\theta=2\sin\theta\cos\theta=2x\sqrt{1-x^2}$.
Revise: Trigonometry (Class 11 notes)
Answer (d): $a=\frac14$, $r=\frac14$, so $S_\infty=\frac{a}{1-r}=\frac{1/4}{3/4}=\frac13$.
Revise: Algebra (Class 11 notes)
Answer (a): For an A.P. the middle term is the mean of the others: $2(6a-2)=(8a+4)+(3a+7)$, so $12a-4=11a+11$ and $a=15$.
Revise: Algebra (Class 11 notes)
Answer (c): Row 3 is twice row 2, so two rows are proportional and the determinant is 0.
Revise: Algebra (Class 12 notes)
Answer (b): $AB=1\cdot1+2\cdot2+3\cdot3=14$, a $1\times1$ matrix.
Revise: Algebra (Class 12 notes)
Answer (b): Determinant $=3-1=2$. Swap the diagonal, negate the off-diagonal: $\frac12\begin{bmatrix}-3&-1\-1&-1\end{bmatrix}$.
Revise: Algebra (Class 12 notes)
Answer (b): $1+\omega^2=-\omega$, so $1-\omega+\omega^2=-2\omega$ and $(-2\omega)^3=-8\omega^3=-8$.
Revise: Algebra (Class 12 notes)
Answer (c): $|1/(1+i)|=1/|1+i|=1/\sqrt2$.
Revise: Algebra (Class 12 notes)
Answer (b): The equation is $x^2-(3+4)x+3\cdot4=0$, that is $x^2-7x+12=0$.
Revise: Algebra (Class 11 notes)
Answer (b): $d=\frac{|3(-1)+4(-2)+12|}{\sqrt{9+16}}=\frac{1}{5}$.
Revise: Analytic geometry (Class 11 notes)
Answer (b): Perpendicular lines need coefficient sum $a+b=0$: $k-3=0$, so $k=3$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): Tangent at $(x_1,y_1)$ is $xx_1+yy_1=5$, i.e. $2x+y=5$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): $\frac{x^2+x}{x^2}=1+\frac1x\to1$.
Revise: Calculus (Class 11 notes)
Answer (c): $\sec^2x-2=\tan^2x-1=(\tan x-1)(\tan x+1)$, so the limit is $\tan\frac\pi4+1=2$.
Revise: Calculus (Class 11 notes)
Answer (b): $y=1/t=x^{-2}$, so $dy/dx=-2x^{-3}=-2/x^3$.
Revise: Calculus (Class 11 notes)
Answer (d): $\frac{d(x^2)}{d(-x^2)}=\frac{2x}{-2x}=-1$.
Revise: Calculus (Class 11 notes)
Answer (a): The integrand is an odd function and the limits are symmetric, so the integral is 0.
Revise: Calculus (Class 12 notes)
Answer (a): Put $u=1+\ln x$, $du=dx/x$: $\int u^{-2}du=-1/u+c$.
Revise: Calculus (Class 12 notes)
Answer (b): $\int_1^4 4x^2dx=\frac43(64-1)=84$.
Revise: Calculus (Class 12 notes)
Answer (b): Tension is a force with dimension $[MLT^{-2}]$ and surface tension is $[MT^{-2}]$, so their ratio has dimension $[L]$.
Revise: Physical Quantities (Class 11 notes)
Answer (c): Mass $=10/10=1$ kg (taking $g=10$), so $a=F/m=5$ m/s$^2$.
Revise: Dynamics (Class 11 notes)
Answer (c): $g'=g(1-d/R)=g/2$ gives $d=R/2$.
Revise: Gravitation (Class 11 notes)
Answer (c): Spokes keep the mass near the rim, so the wheel has a large moment of inertia and runs steadily.
Revise: Rotational Dynamic (Class 12 notes)
Answer (c): Young's modulus is a property of the material, independent of length and diameter.
Revise: Elasticity (Class 11 notes)
Answer (a): $\eta=18\times10^{-6}$ Pa s. $F=6\pi\eta rv=6\pi(18\times10^{-6})(0.3\times10^{-3})(1)\approx1.01\times10^{-7}$ N.
Revise: Fluid Statics (Class 12 notes)
Answer (b): Invar (an iron-nickel alloy) has a very small coefficient of linear expansion.
Revise: Thermal Expansion (Class 11 notes)
Answer (d): Three-fourths of the air remains, so $T'=300\times\frac43=400$ K $=127^\circ$C.
Revise: Ideal Gas (Class 11 notes)
Answer (a): $\eta=1-T_2/T_1=1$ only when $T_2=0$ K.
Revise: Second Law of Thermodynamics (Class 12 notes)
Answer (c): Area $=2\pi rl=2\pi(6\times10^{-3})(0.6)=2.26\times10^{-2}$ m$^2$. $T^4=\frac{1500}{6\times10^{-8}\times2.26\times10^{-2}}\approx1.1\times10^{12}$, so $T\approx1025$ K.
Revise: Rate of Heat Flow (Class 11 notes)
Answer (b): $a\sin\theta=2\lambda$: $\sin\theta=\frac{2\times550}{2200}=0.5$, so $\theta=30^\circ$.
Revise: Diffraction of Light (Class 12 notes)
Answer (a): $P=5-7.5=-2.5$ D, so $f=1/P=-0.4$ m $=-40$ cm.
Revise: Lenses (Class 11 notes)
Answer (d): Relaxed eye: $v_0=L-f_e=9$ cm. $M=\frac{v_0}{u_0}\cdot\frac{D}{f_e}=25$ gives $\frac{v_0}{u_0}=5$, so $u_0=9/5=1.8$ cm.
Revise: Lenses (Class 11 notes)
Answer (d): $\lambda=v/f=360/500=0.72$ m. A phase difference of $60^\circ$ is $\lambda/6$, so the distance is $0.72/6=0.12$ m $=12$ cm.
Revise: Wave Motion (Class 12 notes)
Answer (b): Like charges repel, so work is done against the force and the potential energy rises.
Revise: Potential Difference and Potential Energy (Class 11 notes)
Answer (a): Parallel: $nC$ (maximum). Series: $C/n$ (minimum). Ratio $=nC\div(C/n)=n^2$.
Revise: Capacitor (Class 11 notes)
Answer (d): Each semicircle is $6\ \Omega$, and the two are in parallel: $\frac{6\times6}{12}=3\ \Omega$.
Revise: DC Circuit (Class 11 notes)
Answer (a): An ideal ammeter is connected in series and must not change the current, so its resistance is zero.
Revise: DC Circuit (Class 11 notes)
Answer (c): $v=\sqrt{2E/m}=\sqrt{\frac{2\times10\times1.6\times10^{-19}}{9.1\times10^{-31}}}\approx1.88\times10^{6}$ m/s. $r=\frac{mv}{eB}\approx0.107$ m $\approx11$ cm.
Revise: Magnetic Field (Class 12 notes)
Answer (d): By Faraday's law the emf exists only while the flux is changing.
Revise: Electromagnetic Induction (Class 12 notes)
Answer (a): High retentivity means the material keeps its magnetism, which is what a permanent magnet needs.
Revise: Magnetic Properties of Materials (Class 12 notes)
Answer (c): The glow comes from excited gas atoms, so its colour is set by the gas.
Revise: Electrons (Class 12 notes)
Answer (a): $K_{max}=hf-\phi$ depends on frequency (and on work function), not on intensity.
Revise: Photons (Class 12 notes)
Answer (d): Energy per fission $=200\times1.6\times10^{-13}=3.2\times10^{-11}$ J. $N=\frac{16000}{3.2\times10^{-11}}=5\times10^{14}$.
Revise: Nuclear Physics (Class 11 notes)
Answer (a): A trivalent impurity (aluminium) creates holes and gives p-type silicon.
Revise: Semiconductor Devices (Class 12 notes)
Answer (c): 'Let on' means to reveal a secret, so 'I won't let on to anyone' fits the promise.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'Tie down' means to restrict someone, so 'feel tied down by my presence' matches 'go out whenever you like'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Show up' means appear or arrive.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Call by' means to visit briefly and needs no object, so it completes 'Mahesh is calling by this evening'; the other options need an object.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): Magniloquent means using high-flown, pompous language, so 'inflated' is closest.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Vivacious means lively and high-spirited; the opposite is depressed.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Mendacious means lying; its opposite is veracious (truthful).
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Taciturn means habitually silent and reserved.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): The fixed phrases are 'cause for anxiety' and 'take care of your health'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'Sympathy with' (fellow feeling) pairs with 'taste for music' (liking), giving 'with, for'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): The idioms are 'contrary to' and 'turn a deaf ear to', so both gaps take 'to'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): We say 'addicted to drugs' and 'anxious for (about) their future'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Touch upon' a subject and 'turn into' something.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): Claustrophobia is the fear of confined spaces.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Latin for 'great work': the finest work of an artist or writer.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Uxoricide is wife-killing (homicide = any person, filicide = child).
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): An apiary houses bees; an aquarium houses fish.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): 'To swear like a trooper' means to use swear words very frequently.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): The idiom means perfectly healthy and well.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): To 'do time' or 'serve time' means to be in prison.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): It means to stop what you are doing and go to bed.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): 'To play it by ear' means to improvise and act according to the situation.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Being 'in the red' means being overdrawn or in debt.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): A greeting such as 'Congratulations' is reported with the verb 'congratulated'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): $\frac{1.008\ \mathrm{g}}{6.022\times10^{23}}\approx1.67\times10^{-24}$ g.
Revise: Stoichiometry (Class 11 notes)
Answer (d): Atomic number is the count of protons, so it is always an integer.
Revise: Atomic Structure (Class 11 notes)
Answer (b): The azimuthal quantum number $l$ gives the shape (s, p, d, f) of the orbital.
Revise: Atomic Structure (Class 11 notes)
Answer (d): $K_w$ rises with temperature, so $[H^+]=[OH^-]$ both rise and both pH and pOH fall, while water stays neutral.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (a): Noble gas neon has a complete octet and small size, giving the highest first ionization energy here.
Revise: Classification of Elements (Class 11 notes)
Answer (a): $\ce{Na2SO4}$ comes from a strong acid and strong base, so there is no hydrolysis.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (b): Half-life is 50 min. $75%$ completion needs two half-lives: $2\times50=100$ min.
Revise: Chemical Kinetics (Class 12 notes)
Answer (c): Malachite is $\ce{CuCO3.Cu(OH)2}$, an ore of copper.
Revise: Chemistry of Metals (Class 11 notes)
Answer (d): Carbon monoxide formed in the furnace is the actual reducing agent: $\ce{Fe2O3 + 3CO -> 2Fe + 3CO2}$.
Revise: Chemistry of Metals (Class 11 notes)
Answer (b): Equal volumes of 1.0 M solutions make 0.5 M of each reactant and the same amount of NaCl, so the volume doubles and $[\text{NaCl}]=1.0/2=0.5$ M.
Revise: Volumetric Analysis (Class 12 notes)
Answer (d): The contact process uses vanadium pentoxide (formerly platinum).
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (d): Molecularity is the number of reacting molecules: $2+1=3$.
Revise: Chemical Kinetics (Class 12 notes)
Answer (a): Heavy water is water made from deuterium, $\ce{D2O}$.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (d): Zinc coating of iron is galvanizing.
Revise: Electrochemistry (Class 12 notes)
Answer (a): The acid salt and base react: $\ce{NaHCO3 + NaOH -> Na2CO3 + H2O}$.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (d): Delocalised pi electrons (resonance) make all C-C bonds equal.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (d): Picric acid is 2,4,6-trinitrophenol, a phenol with no carboxyl group.
Revise: Phenols (Class 12 notes)
Answer (a): Primary aliphatic amines give diazonium salts that decompose to alcohols with $\ce{N2}$.
Revise: Amines (Class 12 notes)
Answer (c): In the Cannizzaro reaction one aldehyde molecule is oxidised to acid and another reduced to alcohol.
Revise: Aldehydes and Ketones (Class 12 notes)
Answer (b): $\ce{NO2+}$ attacks the ring and replaces H.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (d): Dettol contains chloroxylenol as the active antiseptic.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (a): DDT is a chlorinated organic insecticide.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (d): A zwitter ion carries both a positive and a negative charge, and amino acids such as glycine are the standard example. Glycine hydrochloride is the only amino acid option; glucose, DNA and soap are not zwitter ions.
Revise: Amines (Class 12 notes)
Answer (b): Fructose has its carbonyl group in the middle of the chain, so it is a ketose (it is also a hexose, but ketose is the distinguishing class).
Answer (a): $k=\text{rate}/[A]$ has units $\mathrm{mol,L^{-1}s^{-1}}/\mathrm{mol,L^{-1}}=\mathrm{s^{-1}}$.
Revise: Chemical Kinetics (Class 12 notes)