- a
- b
- c
- d
Answer (a): A singleton subset holds exactly one element, so there is one for each of the 4 elements.
Revise: Algebra (Class 11 notes)
TU IOST | Model Questions
Answer (a): A singleton subset holds exactly one element, so there is one for each of the 4 elements.
Revise: Algebra (Class 11 notes)
Answer (c): A function has an inverse only when it is a bijection, that is, both one-one and onto.
Revise: Algebra (Class 11 notes)
Answer (a): Put $y=3x-4$, so $x=\frac{y+4}{3}$. Swapping variables gives $f^{-1}(x)=\frac{x+4}{3}$.
Revise: Algebra (Class 11 notes)
Answer (a): Using $\lim\left(1+\frac kx\right)^x=e^k$ with $k=3$, the limit is $e^3$.
Revise: Calculus (Class 11 notes)
Answer (d): $\sin x$ stays between $-1$ and $1$ while $x\to\infty$, so the ratio goes to $0$ (squeeze theorem).
Revise: Calculus (Class 11 notes)
Answer (c): Left limit $=4$, right limit $=4$, so $\lim_{x\to2}f(x)=4$, but $f(2)=2$. The limit exists and differs from the value, so the discontinuity is removable.
Revise: Calculus (Class 11 notes)
Answer (c): For $x>0$ the derivative is $\frac1x$, and for $x<0$, $\log(-x)$ also gives $\frac1x$. So it is $\frac1x$ for all $x\ne0$.
Revise: Calculus (Class 11 notes)
Answer (b): $\frac{d(\sin^3x)}{d(\cos^3x)}=\frac{3\sin^2x\cos x}{-3\cos^2x\sin x}=-\tan x$.
Revise: Calculus (Class 11 notes)
Answer (b): $f'(x)=\frac{1}{x\log x}$. At $x=e$, $\log e=1$, so $f'(e)=\frac1e$.
Revise: Calculus (Class 11 notes)
Answer (a): $\sin x\cos x=\frac12\sin2x$, and $\sin2x$ has minimum $-1$, so the minimum is $-\frac12$.
Revise: Calculus (Class 11 notes)
Answer (d): By parts with $u=\log x$, $dv=dx$: $x\log x-\int1,dx=x\log x-x+C=x(\log x-1)+C$.
Revise: Calculus (Class 11 notes)
Answer (a): The integral is $\sin^{-1}x$ evaluated from $0$ to $\frac{\sqrt3}{2}$, which is $\frac\pi3-0$.
Revise: Calculus (Class 11 notes)
Answer (c): Area $=\int_0^4y^2,dy=\frac{4^3}{3}=\frac{64}{3}$.
Revise: Calculus (Class 11 notes)
Answer (d): $(1-i)^2=-2i$, so $(1-i)^3=-2i(1-i)=-2-2i=-2(1+i)$. Also $1-i^3=1+i$. The ratio is $-2$. The paper prints 'modulus', a misprint for 'value': the modulus would be $2$, which is not an option, while $-2$ is.
Revise: Algebra (Class 11 notes)
Answer (a): $D=9-8=1>0$ and a perfect square, so the roots ($1$ and $\frac12$) are real, unequal and rational.
Revise: Algebra (Class 11 notes)
Answer (b): $AB=\begin{pmatrix}19&22\43&50\end{pmatrix}$. Its transpose is $\begin{pmatrix}19&43\22&50\end{pmatrix}$.
Revise: Algebra (Class 11 notes)
Answer (c): The first equation is twice the second, so both represent the same line.
Revise: Analytic geometry (Class 11 notes)
Answer (d): $G^2=A\cdot H$, so $H=\frac{18^2}{81}=\frac{324}{81}=4$.
Revise: Algebra (Class 11 notes)
Answer (b): $\tan^2\theta=\frac13=\tan^2\frac\pi6$, so $\theta=n\pi\pm\frac\pi6$.
Revise: Trigonometry (Class 11 notes)
Answer (b): For $x>0$, $\tan^{-1}x+\cot^{-1}x=\frac\pi2$.
Revise: Trigonometry (Class 11 notes)
Answer (b): The inradius is $r=(s-a)\tan\frac A2=(s-b)\tan\frac B2=(s-c)\tan\frac C2$, so $(s-b)\tan\frac B2$ fits.
Revise: Trigonometry (Class 11 notes)
Answer (d): Rewrite the first as $6x+8y=12$. Distance $=\frac{|15-12|}{\sqrt{36+64}}=\frac{3}{10}$.
Revise: Analytic geometry (Class 11 notes)
Answer (b): Use $abc+2fgh-af^2-bg^2-ch^2=0$ with $a=12,b=2,h=-5,g=\frac{11}2,f=-\frac52,c=\lambda$: $24\lambda+137.5-75-60.5-25\lambda=0$, so $\lambda=2$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): Diameter form: $(x-1)(x-0)+(y+1)(y-2)=0$, which gives $x^2-x+y^2-y-2=0$.
Revise: Analytic geometry (Class 11 notes)
Answer (d): Equal direction cosines $l=m=n$ with $l^2+m^2+n^2=1$ give $3l^2=1$, so $l=\pm\frac1{\sqrt3}$.
Revise: Vectors (Class 11 notes)
Answer (a): Both $L/R$ and $CR$ are time constants, so both have the dimension of time.
Revise: Physical Quantities (Class 11 notes)
Answer (d): In the first 3 cm, $u^2-\left(\frac u2\right)^2=2a(3)$, so $\frac{3u^2}{4}=6a$ and $2a=\frac{u^2}{4}\text{ per cm}$. Then $\left(\frac u2\right)^2=2as'$ gives $s'=\frac{u^2/4}{u^2/4}=1$ cm.
Revise: Kinematics (Class 11 notes)
Answer (c): $\frac RH=4\cot\theta=4\sqrt3$, so $\cot\theta=\sqrt3$ and $\theta=30^\circ$.
Revise: Kinematics (Class 11 notes)
Answer (b): Zero net force means zero acceleration (Newton's first law). The body may be at rest or move uniformly.
Revise: Dynamics (Class 11 notes)
Answer (c): Young's modulus is a property of the material, not of the dimensions of the wire.
Revise: Elasticity (Class 11 notes)
Answer (b): $W=8\pi TR^2$ (two surfaces). New work $=8\pi T(9R^2-R^2)=8\times8\pi TR^2=8W$.
Revise: Elasticity (Class 11 notes)
Answer (a): Alcohol expands more, so its density falls faster than the ball's volume rises. The upthrust drops and the apparent weight increases.
Revise: Thermal Expansion (Class 11 notes)
Answer (b): At the critical temperature the liquid and vapour phases become identical, so no latent heat is needed.
Revise: Quantity of Heat (Class 11 notes)
Answer (c): By the equipartition theorem each degree of freedom carries $\frac12kT$.
Revise: Ideal Gas (Class 11 notes)
Answer (c): Radiated heat $\propto$ surface area $\propto r^2$, so the ratio is $9:4$.
Revise: Rate of Heat Flow (Class 11 notes)
Answer (b): $a\sin\theta=n\lambda$: $\sin\theta=\frac{2\times550\times10^{-9}}{22\times10^{-7}}=0.5$, so $\theta=30^\circ$.
Revise: Diffraction of Light (Class 12 notes)
Answer (b): Three mutually perpendicular mirrors act like a corner reflector, giving $2^3-1=7$ images.
Revise: Reflection at Curved Mirrors (Class 11 notes)
Answer (a): $\mu=\frac{\sin\frac{A+\delta}2}{\sin\frac A2}$: $\sin\frac{A+\delta}{2}=1.414\times0.5=0.707$, so $\frac{A+\delta}2=45^\circ$ and $\delta=30^\circ$.
Revise: Refraction Through Prisms (Class 11 notes)
Answer (d): $\lambda=\frac{360}{500}=0.72$ m. Path difference $=\frac{\lambda}{360^\circ}\times60^\circ=\frac{0.72}{6}=0.12$ m $=12$ cm.
Revise: Mechanical Waves (Class 12 notes)
Answer (a): Let the point be $x$ from $q$: $\frac{q}{x^2}=\frac{4q}{(r-x)^2}$ so $r-x=2x$ and $x=\frac r3$. It is $\frac{2r}{3}$ from $4q$.
Revise: Electric Charges (Class 11 notes)
Answer (a): Series gives $\frac Cn$ (minimum), parallel gives $nC$ (maximum). Ratio $=\frac{nC}{C/n}=n^2$.
Revise: Capacitor (Class 11 notes)
Answer (b): Half the length gives half the resistance. Heat $=\frac{V^2t}{R}$, so it doubles.
Revise: DC Circuit (Class 11 notes)
Answer (d): $E=I(R+r)$: $50=4.5(10+r)$, so $r=\frac{50}{4.5}-10\approx1.1\ \Omega$.
Revise: DC Circuit (Class 11 notes)
Answer (b): In a non-uniform field the two poles feel unequal forces, so there is a net force as well as a torque.
Revise: Magnetic Field (Class 12 notes)
Answer (c): $Q=\frac{X_L}{R}=\frac{100}{5}=20$. The paper prints the reactance as 100 Hz, a misprint for 100 ohm, since a reactance is measured in ohms.
Revise: Alternating Currents (Class 12 notes)
Answer (c): $L=\frac{1}{4\pi^2f^2C}=\frac{1}{39.48\times(1.02\times10^6)^2\times420\times10^{-12}}\approx5.8\times10^{-5}$ H.
Revise: Alternating Currents (Class 12 notes)
Answer (d): $v=\sqrt{\frac{2eV}{m}}=\sqrt{\frac{2\times1.6\times10^{-19}\times100}{9.1\times10^{-31}}}\approx5.9\times10^6$ m/s, about $6\times10^6$.
Revise: Electrons (Class 12 notes)
Answer (c): Mean life $=\frac{T_{1/2}}{0.693}=1.443\times10=14.43$ years.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (d): In an NPN transistor the majority carriers are electrons, which move faster than holes, giving faster response.
Revise: Semiconductor Devices (Class 12 notes)
Answer (c): The azimuthal quantum number $l$ gives the subshell and so the shape of the orbital (s, p, d, f).
Revise: Atomic Structure (Class 11 notes)
Answer (c): Gamma rays have no mass or charge and penetrate the most; alpha rays the least.
Revise: Nuclear Chemistry (Class 12 notes)
Answer (a): $\ce{MgCl2}$ is formed from a metal and a non-metal by transfer of electrons. The others are covalent.
Revise: Chemical Bonding (Class 11 notes)
Answer (b): Electronegativity increases across a period and up a group, so fluorine is highest.
Revise: Classification of Elements (Class 11 notes)
Answer (a): $2x+7(-2)=-2$, so $2x=12$ and $x=+6$.
Revise: Oxidation and Reduction (Class 11 notes)
Answer (d): ZnO reacts with both acids and bases, so it is amphoteric. CaO and MgO are basic.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): $\ce{C60}$ is a truncated icosahedron with 20 hexagons and 12 pentagons.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (c): Down's process electrolyses molten NaCl to get sodium metal.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): Iso-butane has a three carbon chain with a methyl on carbon 2.
Revise: Hydrocarbons (Class 11 notes)
Answer (b): Cold dilute alkaline $\ce{KMnO4}$ is Baeyer's reagent, used to test for unsaturation.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): $2R-X+2Na\to R-R+2NaX$ is the Wurtz reaction.
Revise: Haloalkanes (Class 12 notes)
Answer (b): Alkanes are saturated, with formula $C_nH_{2n+2}$.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): Lassaigne's sodium fusion test detects N, S and halogens.
Revise: Fundamental Principles (Class 11 notes)
Answer (c): Carbon in methane is $sp^3$ hybridised with bond angle $109.5^\circ$.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): Each $\ce{Cr2O7^2-}$ gains 6 electrons in acid, so equivalent weight $=\frac M6$.
Revise: Volumetric Analysis (Class 12 notes)
Answer (d): A Lewis base donates an electron pair. $\ce{NH3}$ has a lone pair, while the others are electron deficient acids.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (d): Galvanic and voltaic cells are the same thing, and both are electrochemical cells.
Revise: Electrochemistry (Class 12 notes)
Answer (a): The thermochemical calorie is defined as J.
Revise: Chemical Thermodynamic (Class 12 notes)
Answer (d): Concentration, temperature and surface area all change the rate.
Revise: Chemical Kinetics (Class 12 notes)
Answer (b): Each carbon forms three sigma bonds in a plane, so it is $sp^2$ hybridised.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (c): Tertiary alcohols form the chloride fastest, so turbidity appears immediately.
Revise: Alcohols (Class 12 notes)
Answer (a): Fats are esters of glycerol and fatty acids, which are lipids.
Revise: Bio-Inorganic Chemistry (Class 11 notes)
Answer (a): Bakelite is a condensation polymer of phenol and formaldehyde.
Revise: Phenols (Class 12 notes)
Answer (b): Mercuric chloride $\ce{HgCl2}$ is corrosive sublimate. $\ce{Hg2Cl2}$ is calomel.
Revise: Study of Heavy Metals (Class 12 notes)
Answer (a): Haematite is $\ce{Fe2O3}$, an ore of iron.
Revise: Chemistry of Metals (Class 11 notes)
Answer (b): The word is stressed per-MIS-si-ble, on the second syllable.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): PER-ish has first syllable stress. In-CRED-i-ble, in-CUR and te-MER-i-ty stress later syllables.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): The adjective glad takes of with the noun: glad of your help.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): Slave to is the fixed collocation.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): A vendetta is a long-running quarrel, a feud.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Acerbity means sharpness or bitterness, close to acrimony.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Replete means full, so the opposite is empty.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Quell means to put down or suppress, so encourage is the opposite. Promote is a weaker fit.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): Contagious means spread by direct or indirect contact.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Ad nauseam means to the point of sickness, that is, to a disgusting degree.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): The main clause has will, so the tag is won't she?
Revise: Grammar and Usage (Class 11 notes)
Answer (c): To stand on ceremony means to be too formal or to insist on strict rules of etiquette.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): Double Dutch means something that is hard or impossible to understand.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): Analog computers process continuously varying electrical signals that represent physical quantities.
Revise: Computer System (Class 11 notes)
Answer (c): The wild cards are ? (one character) and * (any number of characters).
Revise: Computer Software and Operating System (Class 11 notes)
Answer (b): The order from large to small is terabyte, gigabyte, megabyte, byte, so gigabyte is second largest.
Revise: Computer System (Class 11 notes)
Answer (c): Cache is a fast buffer between the CPU and the main memory (RAM).
Revise: Computer System (Class 11 notes)
Answer (b): Nano means $10^{-9}$, so a nanosecond is one billionth of a second.
Revise: Computer System (Class 11 notes)
Answer (b): RAM loses its contents when power is off, so it is volatile memory.
Revise: Computer System (Class 11 notes)
Answer (d): CPU means Central Processing Unit.
Revise: Computer System (Class 11 notes)
Answer (a): Ctrl + C copies the selected text on Windows. (Command + C is the Mac equivalent.)
Revise: Computer Software and Operating System (Class 11 notes)
Answer (d): LaTeX is a document preparation system. The others are operating systems.
Revise: Computer Software and Operating System (Class 11 notes)