- a
- b
- c
- d
Answer (a): Put $x=2$: $f(2)=\sqrt{4-4}=0$.
Revise: Algebra (Class 11 notes)
TU IOST | Entrance paper
Answer (a): Put $x=2$: $f(2)=\sqrt{4-4}=0$.
Revise: Algebra (Class 11 notes)
Answer (b): $|x|=2$ gives $x=2$ or $x=-2$, so the solution set is ${-2,2}$. Option a prints ${2,2}$, which is just ${2}$, and the interval options are not solution sets of an equation.
Revise: Algebra (Class 11 notes)
Answer (b): Every element of $A$ is already in $B$, so the common part is $A$.
Revise: Algebra (Class 11 notes)
Answer (c): 1 is not prime and 0 and -1 are not natural primes; the smallest prime is 2.
Revise: Algebra (Class 11 notes)
Answer (b): Replacing $y$ by $-y$ leaves the equation unchanged, so it is symmetric about the X axis.
Revise: Analytic geometry (Class 11 notes)
Answer (b): Sum $=(x-4)+(2x-4)=3x-8=13$, so $x=7$.
Revise: Algebra (Class 11 notes)
Answer (a): $A^T=A$ since the off-diagonal entries are both 3, so it is symmetric.
Revise: Algebra (Class 11 notes)
Answer (a): Two rows (R1 and R2) are identical, so the determinant is 0.
Revise: Algebra (Class 11 notes)
Answer (d): Expand: $1-\omega-\omega^2+\omega^3=1-(\omega+\omega^2)+1=1+1+1=3$.
Revise: Algebra (Class 11 notes)
Answer (a): $|2-3i|=\sqrt{2^2+3^2}=\sqrt{13}$.
Revise: Algebra (Class 11 notes)
Answer (a): Sum of roots 3 and product 2 give $x^2-3x+2=0$.
Revise: Algebra (Class 11 notes)
Answer (b): $\tan(\tan^{-1}x)=x$, so the value is 1.
Revise: Trigonometry (Class 11 notes)
Answer (c): $\cos x=\cos\frac{\pi}{3}$, and cosine is even, so $x=2n\pi\pm\frac{\pi}{3}$.
Revise: Trigonometry (Class 11 notes)
Answer (a): Distance $=\frac{|0+0+1|}{\sqrt{1+1}}=\frac{1}{\sqrt2}$.
Revise: Analytic geometry (Class 11 notes)
Answer (d): Here $a+b=2-2=0$, so the lines are perpendicular and the angle is $90^0$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): Centre is $(-g,-f)$ with $2g=-2$, $2f=-4$, giving $(1,2)$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): Tangent at $(x_1,y_1)$ is $xx_1+yy_1=r^2$: $2x+3y=13$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): This is the derivative of $x^3$ at 2: $3\cdot2^2=12$.
Revise: Calculus (Class 11 notes)
Answer (b): Split: $\frac{\tan x}{x}-5\to1-5=-4$.
Revise: Calculus (Class 11 notes)
Answer (b): At $x=2$: left limit $2(2)=4$, right limit and value $2+2=4$, so it is continuous there.
Revise: Calculus (Class 11 notes)
Answer (b): $2x+3=1$ gives $x=-1$.
Revise: Calculus (Class 11 notes)
Answer (c): Differentiate $y=t^2$ directly with respect to $t$: $2t$.
Revise: Calculus (Class 11 notes)
Answer (a): $f''(x)=6x=0$ at $x=0$, and $f''$ changes sign there.
Revise: Calculus (Class 11 notes)
Answer (b): $2\cdot\left[\frac{x^2}{2}\right]_0^1=2\cdot\frac12=1$.
Revise: Calculus (Class 11 notes)
Answer (a): The integral of $\cos x$ is $\sin x$ (+ C).
Revise: Calculus (Class 11 notes)
Answer (a): A clause follows, so a conjunction is needed: 'Although'. 'In spite of' and 'despite' take a noun phrase.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Both ... and' is the correlative pair; 'friendly' follows 'and'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'People' is plural and the present perfect fits, so 'have come'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'Both' pairs with 'and'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): The list is of gerunds (reading, observing, teaching), so parallel form 'discussing'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): Names of towns and places take no article.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): The adjective 'concerned' takes 'about' (concerned about something).
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'By' marks the agent (robber); 'with' marks the instrument (gun).
Revise: Grammar and Usage (Class 11 notes)
Answer (a): The subject is the singular 'fear', so the present perfect 'has caused'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'Need' followed by a gerund has a passive sense: 'needs cutting'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): Here 'from this evidence' means 'judging from' it, the usual collocation for drawing a conclusion.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): The verb 'object' takes the preposition 'to'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Work out' means to solve or calculate, so 'worked out the answer' is correct. 'Worked over' means revised or beaten up.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): We form historic, athletic and artistic, but 'success' takes -ful or -ive, not -ic.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): 'Marvel' (to be amazed) is a verb; the others are adjectives.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): We have unprovided, unprotected and undetected, but 'poisonous' does not take a negative prefix.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): In a time clause with 'before' the simple present is used for future: 'comes'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Memoranda' is the plural of memorandum, so 'have been'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): Anemia is the condition of too little blood or haemoglobin.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): A veteran is a person with long experience in a field. A cosmopolitan is a person at home in many countries.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): 'Doubt if' expresses uncertainty about whether she will come.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): In 'They walk very fast', fast modifies the verb walk and is itself modified by very, so it is an adverb.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): An exclamation of congratulation is reported with the verb 'congratulate'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): From the mean position $y=A\sin\omega t=A/2$ gives $\omega t=\pi/6$, so $t=T/12=1$ s.
Revise: Periodic Motion (Class 12 notes)
Answer (b): Distance $=1.6$ km, so $t=1.6/64=0.025$ h $=1.5$ min.
Revise: Kinematics (Class 11 notes)
Answer (c): Both work and energy have dimension $[ML^2T^{-2}]$.
Revise: Physical Quantities (Class 11 notes)
Answer (d): $F=\frac{mv^2}{r}=\frac{1\cdot v^2}{0.5}=2v^2$, so $v=\sqrt{F/2}$.
Revise: Circular Motion (Class 11 notes)
Answer (b): $v\propto\sqrt{T/M}$: $v=2\sqrt{\frac{1083}{273}\cdot\frac{2}{32}}\approx2\times0.5=1$ km/s.
Revise: Ideal Gas (Class 11 notes)
Answer (d): A bolometer detects the intensity of heat (thermal) radiation.
Revise: Rate of Heat Flow (Class 11 notes)
Answer (b): $C=\frac{5}{9}(14-32)=-10^0$C, which is $263$ K.
Revise: Heat and Temperature (Class 11 notes)
Answer (d): A diverging lens of $f=-1.5$ m is needed: $P=1/f=-0.66$ D.
Revise: Lenses (Class 11 notes)
Answer (a): $\sin C=\frac{n_w}{n_g}=\frac{4/3}{3/2}=\frac89$.
Revise: Refraction at Plane Surface (Class 11 notes)
Answer (b): $d=\frac{10^{-2}}{5000}=2\times10^{-6}$ m; $n_{max}\le\frac{d}{\lambda}=3.3$, so $n=3$.
Revise: Diffraction of Light (Class 12 notes)
Answer (c): Speed in glass is $c/n$, so $time=\frac{t}{c/n}=\frac{nt}{c}$.
Revise: Refraction at Plane Surface (Class 11 notes)
Answer (c): $\frac{2}{x^2}=\frac{4}{(1-x)^2}$ gives $\frac{1-x}{x}=\sqrt2$, so $x=\frac{1}{1+\sqrt2}\approx0.41$ m.
Revise: Electric Field (Class 11 notes)
Answer (c): An open pipe has antinodes at both ends and supports all harmonics.
Revise: Wave in Pipes and Strings (Class 12 notes)
Answer (a): $P=VI$, so $\frac{W}{A}=V$, the unit of potential.
Revise: DC Circuit (Class 11 notes)
Answer (b): Resistivity depends only on the material, not on length, so it is equal.
Revise: DC Circuit (Class 11 notes)
Answer (d): Adding a capacitor in series lowers the net capacitance and raises the capacitive reactance. $Z=\sqrt{R^2+(X_L-X_C)^2}$ then falls if $X_L>X_C$ and rises if $X_C>X_L$, so it depends on the frequency.
Revise: Alternating Currents (Class 12 notes)
Answer (d): $Y=IR$ and $X=I(R+r)$, so $r=\frac{(X-Y)R}{Y}$.
Revise: DC Circuit (Class 11 notes)
Answer (a): $\frac1{\lambda_L}=R\left(1-\frac14\right)=\frac{3R}{4}$, $\frac1{\lambda_B}=R\left(\frac14-\frac19\right)=\frac{5R}{36}$; ratio $=\frac{5/36}{3/4}=\frac{5}{27}$.
Revise: Quantization of Energy (Class 12 notes)
Answer (b): $\nu_0=\frac{\phi}{h}=\frac{3.3\times1.6\times10^{-19}}{6.63\times10^{-34}}\approx8\times10^{14}$ Hz.
Revise: Photons (Class 12 notes)
Answer (d): 20 days is 4 half lives, so remaining $=\frac{1000}{16}=62.5$ g and decayed $=937.5$ g.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (a): $T=\frac{2\pi m}{qB}$ has no speed term.
Revise: Magnetic Field (Class 12 notes)
Answer (c): $v\propto\sqrt T$: $v=330\sqrt{300/273}\approx346$, nearest option is 360.
Revise: Mechanical Waves (Class 12 notes)
Answer (c): $T=\frac{2u\sin\theta}{g}$, $R=\frac{u^2\sin2\theta}{g}$; $gT^2=2R$ gives $\tan\theta=1$, so $45^0$.
Revise: Kinematics (Class 11 notes)
Answer (a): $a=\frac{eE}{m}=\frac{1.6\times10^{-19}}{9.1\times10^{-31}}\approx1.7\times10^{11}$.
Revise: Electric Field (Class 11 notes)
Answer (c): Heat produced equals kinetic energy $\frac12mv^2$, which becomes 4 times when $v$ doubles.
Revise: Quantity of Heat (Class 11 notes)
Answer (a): Saturated hydrocarbons with little affinity for reagents are called paraffins, i.e. alkanes.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): Nitrogen gives NaCN, then sodium ferrocyanide, which with Fe(III) forms ferric ferrocyanide (Prussian blue), seen as a blue-green colour.
Revise: Fundamental Principles (Class 11 notes)
Answer (d): $\ce{Al4C3 + 12H2O -> 4Al(OH)3 + 3CH4}$.
Revise: Hydrocarbons (Class 11 notes)
Answer (c): An amine condenses with an aldehyde to form an imine, called a Schiff's base.
Revise: Amines (Class 12 notes)
Answer (a): Tetraethyl lead (TEL) was the classic antiknock additive in petrol.
Revise: Hydrocarbons (Class 11 notes)
Answer (b): $\ce{CHCl3 + HNO3 -> CCl3NO2 + H2O}$, giving chloropicrin (tear gas).
Revise: Haloalkanes (Class 12 notes)
Answer (b): Aldehydes reduce Tollen's reagent to give a silver mirror.
Revise: Aldehydes and Ketones (Class 12 notes)
Answer (b): Terylene is a polyester formed from ethylene glycol and terephthalic acid with loss of water.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (b): Zinc blende is the sulphide ore ZnS.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): Mohr's salt is $\ce{FeSO4.(NH4)2SO4.6H2O}$, which contains Fe(II).
Revise: Oxidation and Reduction (Class 11 notes)
Answer (b): $\ce{CO2}$ is a non-metal oxide that gives carbonic acid with water; CaO and FeO are basic and CO is neutral.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (d): $\ce{NH3}$ has three bond pairs and one lone pair on N, giving a trigonal pyramidal shape.
Revise: Chemical Bonding (Class 11 notes)
Answer (c): $\ce{H3PO4}$ has three replaceable hydrogens, so it is tribasic.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (a): Epsom salt is hydrated magnesium sulphate, MgSO4.7H2O.
Revise: Chemistry of Metals (Class 11 notes)
Answer (b): Ostwald's process oxidises ammonia over Pt catalyst to make $\ce{HNO3}$.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (a): Chlorine has the highest electron affinity (more than fluorine, which has a small, crowded 2p).
Revise: Classification of Elements (Class 11 notes)
Answer (a): Molar mass of $\ce{CO2}$ is 44 g, so 44 g is one mole, i.e. $6.023\times10^{23}$ molecules.
Revise: Stoichiometry (Class 11 notes)
Answer (b): $\ce{Cr2O7^2-}$ gains 6 electrons (Cr from +6 to +3 in two atoms), so $E=M/6$.
Revise: Volumetric Analysis (Class 12 notes)
Answer (a): Equivalent weight of $Na_2CO_3$ is $106/2=53$. Mass $=0.1\times53\times0.1\text{ L}=0.53$ g.
Revise: Volumetric Analysis (Class 12 notes)
Answer (b): Surface tension is force per unit length: dyne per cm.
Revise: States of Matter (Class 11 notes)
Answer (a): Filling degenerate orbitals singly with parallel spins is Hund's rule of maximum multiplicity.
Revise: Atomic Structure (Class 11 notes)
Answer (b): A filled 3d shell is extra stable, so one 4s electron moves to 3d: $[Ar]4s^13d^{10}$.
Revise: Atomic Structure (Class 11 notes)
Answer (b): $[H^+]=10^{-5.2}=10^{0.8}\times10^{-6}=6.3\times10^{-6}$ M.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (a): Rate $=k[A]$ so $k=\frac{mol,L^{-1}s^{-1}}{mol,L^{-1}}=s^{-1}$.
Revise: Chemical Kinetics (Class 12 notes)
Answer (c): Formation can be exothermic (negative, e.g. water) or endothermic (positive, e.g. NO).
Revise: Chemical Thermodynamic (Class 12 notes)