TU IOST | Entrance paper
Answer (b): $n(A\cap B)=n(A)+n(B)-n(A\cup B)=35+60-80=15$.
Revise: Algebra (Class 11 notes)
Answer (c): $A-B$ keeps the points of $A$ that are not in $B$: those below $-2$, and $-2$ itself is in $B$, so the result is $[-3,-2)$.
Revise: Algebra (Class 11 notes)
Answer (c): A statement is a sentence that is definitely true or false. Questions and requests are not statements, $2x+3=5$ is an open sentence (true only for $x=1$), and only the Everest sentence is true.
Revise: Algebra (Class 11 notes)
Answer (c): $2x_1=2x_2$ gives $x_1=x_2$, so $f$ is one to one. Odd numbers are never images, so it is not onto.
Revise: Algebra (Class 11 notes)
Answer (b): The derivative is $f'(x)=-3<0$ everywhere, so the function is decreasing.
Revise: Calculus (Class 11 notes)
Answer (b): $\operatorname{cosec}x=-1$ means $\sin x=-1$, so $x=\frac{3\pi}{2}+2n\pi$, which is an odd multiple of $\frac{\pi}{2}$. Only option b has that form.
Revise: Trigonometry (Class 11 notes)
Answer (a): From $2b=a+c$, $c^2=bd$ and $d=\frac{2ce}{c+e}$ we get $c^2=\frac{(a+c)}{2}\cdot\frac{2ce}{c+e}$, so $c(c+e)=(a+c)e$ and $c^2=ae$. Hence $a,c,e$ are in G.P.
Revise: Algebra (Class 11 notes)
Answer (d): The determinant of a $1\times1$ matrix is its only entry, so $\det[-7]=-7$.
Revise: Algebra (Class 11 notes)
Answer (b): The second equation is twice the first, so the lines coincide and there are infinitely many solutions: consistent and dependent.
Revise: Algebra (Class 11 notes)
Answer (a): $1+\omega+\omega^2=0$.
Revise: Algebra (Class 11 notes)
Answer (c): $x^{\circ}=\frac{\pi x}{180}$ radians, so the limit is $\frac{\pi}{180}\lim\frac{\sin(\pi x/180)}{\pi x/180}=\frac{\pi}{180}$.
Revise: Calculus (Class 11 notes)
Answer (c): By the chain rule, $\frac{d}{dx}e^{\sqrt{x}}=e^{\sqrt{x}}\cdot\frac{1}{2\sqrt{x}}$.
Revise: Calculus (Class 11 notes)
Answer (c): Write $\frac{x}{1+x}=1-\frac{1}{1+x}$ and integrate: $x-\ln(1+x)$.
Revise: Calculus (Class 11 notes)
Answer (a): $\int_0^2 3(1+x)^2dx=\left[(1+x)^3\right]_0^2=27-1=26$.
Revise: Calculus (Class 11 notes)
Answer (b): De Morgan's law: the complement of a union is the intersection of the complements.
Revise: Algebra (Class 11 notes)
Answer (b): For $0\le x<1$, take $\theta=\sin^{-1}x$. Then $\cos\theta=\sqrt{1-x^2}$, so $\theta=\cos^{-1}\sqrt{1-x^2}$. No other option matches this identity.
Revise: Trigonometry (Class 11 notes)
Answer (d): The determinant is $10-9=1$. Swap the diagonal entries and negate the off-diagonal entries: $\begin{pmatrix}5&-3\-3&2\end{pmatrix}$.
Revise: Algebra (Class 11 notes)
Answer (b): By the remainder theorem, put $x=-1$: $-1-2-5+10=2$.
Revise: Algebra (Class 11 notes)
Answer (c): The slopes are $\frac13$ and $2$. $\tan\theta=\left|\frac{2-\frac13}{1+\frac23}\right|=1$, so $\theta=\frac{\pi}{4}$.
Revise: Analytic geometry (Class 11 notes)
Answer (d): The general term is $\binom3r x^{6-4r}a^{2r}$. For $x^2$ we need $r=1$, giving $3a^2$.
Revise: Algebra (Class 11 notes)
Answer (a): The hundreds digit can be $1,2,3$ or $4$ (4 ways). The other two places use the remaining 4 digits in $4\times3=12$ ways. Total $4\times12=48$.
Revise: Algebra (Class 11 notes)
Answer (a): $y^2=4ax$ with $4a=8$ gives $a=2$, so the focus is $(a,0)=(2,0)$.
Revise: Analytic geometry (Class 11 notes)
Answer (c): Direction cosines $l,m,n$ always satisfy $l^2+m^2+n^2=1$.
Revise: Vectors (Class 11 notes)
Answer (d): A unit vector is the vector divided by its magnitude. Here $|\vec i+\vec j-2\vec k|=\sqrt{1+1+4}=\sqrt6$, so the unit vector is $\frac{\vec i+\vec j-2\vec k}{\sqrt6}$. The first two options are scalars.
Revise: Vectors (Class 11 notes)
Answer (b): Comparing with $x^2+y^2+2gx+2fy+c=0$ gives $g=-a,f=-a$, so the centre $(-g,-f)=(a,a)$.
Revise: Analytic geometry (Class 11 notes)
Answer (b): Glissando is pronounced gli-SSAN-do, with primary stress on the second syllable.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Investigate is in-VES-ti-gate. The others (INtroverted, INtimate, INtricate) are stressed on the first syllable.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): Magnitude means size or importance, and of the choices 'weight' (as in the weight of a matter) is the closest in sense. The other options are unrelated.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): To outpace is to go faster than or exceed, which is to surpass.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Impulsive means acting on sudden urge without thinking, and its opposite is 'caution' (careful thought).
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Reputation here means good name or standing; notoriety (being known for something bad) is the nearest opposite among the options.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): An apostate is a person who gives up their religious belief.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): A sinecure is a position that pays well but demands little or no work.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): We warn someone of a danger and warn someone against making errors (collocations of warn).
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Sympathetic takes to, and satisfied takes with.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): The clause shows contrast (had the book but did not give it), so the concessive conjunction Although fits.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): The feminine of gander is goose.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): To go through the roof means to rise very sharply, as prices do.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Hellenism refers to ancient Greek culture and the imitation of it.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Set on means to attack, which matches dogs suddenly attacking him.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Clear out' means to empty and tidy a place, which fits clearing out a writing desk. 'Clear off' as a phrasal verb means to go away, so it does not fit with a desk as its object here.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): A stick in the mud is an old-fashioned person who resists change.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Otherwise means "if not", which links the condition to the result.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): While is used to contrast two situations happening at the same time.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Nature' meaning the natural world is used without an article, so no article is needed.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): The statement is positive with had, so the tag is negative: hadn't they?
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Bear no resemblance to is the standard collocation.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Respectable means decent and of good appearance, which fits clothes.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Site means a place chosen for building; sight means vision.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Honorary describes a position or work done without payment.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): $ct^2$ has the dimension of velocity, so $[c]=\frac{[LT^{-1}]}{[T^2]}=[LT^{-3}]$.
Revise: Physical Quantities (Class 11 notes)
Answer (c): $AB\cos\theta=6$ gives $12\cos\theta=6$, so $\cos\theta=\frac12$ and $\theta=60^\circ$.
Revise: Vectors (Class 11 notes)
Answer (b): The earth spins from west to east, so the sky appears to turn the opposite way, east to west.
Revise: Kinematics (Class 11 notes)
Answer (b): Ice has very small friction, so the available grip is limited. Taking smaller steps keeps the required horizontal force, and so the needed friction, small. The cause is the smaller friction of ice.
Revise: Dynamics (Class 11 notes)
Answer (a): $108,\text{km/h}=30,\text{m/s}$. $F=\frac{m\Delta v}{t}=\frac{120\times30}{10}=360,\text{N}$.
Revise: Dynamics (Class 11 notes)
Answer (c): $y_n=\frac{n\lambda D}{d}$, so $d=\frac{10\times589\times10^{-9}\times2}{3.44\times10^{-2}}\approx3.4\times10^{-4},\text{m}=0.34,\text{mm}$.
Revise: Interference (Class 12 notes)
Answer (c): By Archimedes' principle, upthrust equals the weight of the liquid displaced, which depends on the volume displaced.
Revise: Fluid Statics (Class 12 notes)
Answer (c): $\Delta L=L\alpha\Delta T$ does not depend on the diameter, so both expand equally.
Revise: Thermal Expansion (Class 11 notes)
Answer (b): Mean free path $\lambda=\frac{kT}{\sqrt2\pi d^2p}\propto\frac1p$.
Revise: Ideal Gas (Class 11 notes)
Answer (c): Higher pressure raises the boiling point, so water boils at about $120^\circ$C in a pressure cooker.
Revise: Quantity of Heat (Class 11 notes)
Answer (b): A correct path bends toward the normal on entering and bends back on leaving, so the emergent ray is parallel to the incident ray and only shifted sideways. This is the path shown in B.
Revise: Refraction at Plane Surface (Class 11 notes)
Answer (b): A far-sighted eye is too short or too weak, so parallel rays from a distant object would focus behind the retina without extra accommodation.
Revise: Lenses (Class 11 notes)
Answer (c): A charge always has an electric field, and a moving charge (a current) also produces a magnetic field.
Revise: Magnetic Field (Class 12 notes)
Answer (d): A dielectric raises capacitance by the factor $K$. Mica is a dielectric, while metals fill the gap and short it.
Revise: Capacitor (Class 11 notes)
Answer (c): $\Phi=LI=5\times10^{-3}\times2=0.010,\text{Wb}$.
Revise: Electromagnetic Induction (Class 12 notes)
Answer (d): $R=\frac{V^2}{P}=\frac{220^2}{100}=484,\Omega$. At $440$ V the power is $\frac{440^2}{484}=400,\text{W}$ (ignoring the bulb burning out).
Revise: DC Circuit (Class 11 notes)
Answer (d): Charges at rest on a conductor mean no tangential field, so the field at the surface is perpendicular to it.
Revise: Electric Field (Class 11 notes)
Answer (c): Above resonance $X_L=\omega L$ exceeds $X_C=\frac1{\omega C}$, so the circuit behaves inductively.
Revise: Alternating Currents (Class 12 notes)
Answer (d): A permanent magnet must keep its magnetism (high retentivity) and resist demagnetisation (high coercivity).
Revise: Magnetic Properties of Materials (Class 12 notes)
Answer (c): Angle of dip is the angle the total magnetic field of the earth makes with the horizontal.
Revise: Magnetic Field (Class 12 notes)
Answer (b): The field gives a constant acceleration perpendicular to the initial velocity, as in projectile motion, so the path is a parabola.
Revise: Electrons (Class 12 notes)
Answer (d): The photoelectric effect gives $h$ from the stopping potential versus frequency graph, with slope $h/e$. That is the standard experimental method.
Revise: Photons (Class 12 notes)
Answer (b): Heat is $99.6%$ of input power, so $P=\frac{600}{0.996}\approx602,\text{W}$ and $I=\frac{P}{V}=\frac{602}{50000}\approx0.012,\text{A}$.
Revise: Electrons (Class 12 notes)
Answer (c): Both have 92 protons and 92 electrons; the mass numbers differ by 3, so $U^{238}$ has 3 more neutrons.
Revise: Nuclear Physics (Class 11 notes)
Answer (d): Mobile carriers diffuse across and recombine, leaving only immobile ionised donor and acceptor atoms in the depletion layer.
Revise: Semiconductor Devices (Class 12 notes)
Answer (b): $32,\text{g}$ of $O_2$ is $\frac{32}{32}=1$ mole, which holds $6.02\times10^{23}$ molecules.
Revise: Stoichiometry (Class 11 notes)
Answer (a): This is the statement of Pauli's exclusion principle.
Revise: Atomic Structure (Class 11 notes)
Answer (c): Sulfur has 6 bond pairs and no lone pair, giving an octahedral shape.
Revise: Chemical Bonding (Class 11 notes)
Answer (d): Protium, deuterium and tritium are the isotopes of hydrogen; fullerene is an allotrope of carbon.
Revise: Atomic Structure (Class 11 notes)
Answer (a): Nitric oxide formed in the test combines with $\ce{FeSO4}$ to give the brown complex $[\ce{Fe(H2O)5NO}]\ce{SO4}$, written $\ce{FeSO4.NO}$.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (a): Hypo is sodium thiosulphate, $\ce{Na2S2O3.5H2O}$.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (c): Hydrated magnesium sulfate is called Epsom salt.
Revise: Chemistry of Metals (Class 11 notes)
Answer (d): Minamata disease is mercury (methylmercury) poisoning.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): Lassaigne's test detects N, S and halogens as sodium salts, but fluorine is not detected as fluoride precipitates with silver nitrate are soluble.
Revise: Haloalkanes (Class 12 notes)
Answer (c): Alkynes decolourise cold alkaline $\ce{KMnO4}$ (Bayer's reagent) because of unsaturation, but alkanes do not.
Revise: Hydrocarbons (Class 11 notes)
Answer (c): Friedel-Crafts alkylation uses anhydrous $\ce{AlCl3}$ as the Lewis acid catalyst.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (a): Urea, $\ce{CO(NH2)2}$, is about $46%$ nitrogen and is a nitrogenous fertiliser.
Revise: Modern Chemical Manufactures (Class 11 notes)
Answer (c): Acetone has three carbons with the carbonyl on carbon 2: propan-2-one.
Revise: Aldehydes and Ketones (Class 12 notes)
Answer (d): $\ce{H2SO4}$ is dibasic (2 replaceable H), so equivalent weight $=\frac{98}{2}=49$.
Revise: Volumetric Analysis (Class 12 notes)
Answer (a): Rate depends on temperature, concentration and surface area; molecularity is a theoretical property of an elementary step.
Revise: Chemical Kinetics (Class 12 notes)
Answer (b): The first law is conservation of energy.
Revise: Chemical Thermodynamic (Class 12 notes)
Answer (c): Malathion is an organophosphate; DDT, BHC and dieldrin are organochlorines.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (d): Aniline does couple with diazonium salts, so the statement that it does not is false.
Revise: Amines (Class 12 notes)
Answer (a): Ethyl alcohol that is 100 percent pure is called absolute alcohol; rectified spirit is about 95 percent.
Revise: Alcohols (Class 12 notes)
Answer (b): $2R\text{-}X+2Na\to R\text{-}R+2NaX$ in dry ether is the Wurtz reaction.
Revise: Haloalkanes (Class 12 notes)
Answer (c): Ketones are not oxidised easily, so they do not reduce Tollens', Fehling's or Benedict's reagents.
Revise: Aldehydes and Ketones (Class 12 notes)
Answer (d): $\ce{CO2}$ has no proton to donate but accepts an electron pair, so it is a Lewis acid.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (a): Galvanising coats iron with zinc.
Revise: Chemistry of Metals (Class 11 notes)
Answer (b): Silver nitrate is called lunar caustic.
Revise: Chemistry of Metals (Class 11 notes)
Answer (c): Three electron-withdrawing chlorine atoms stabilise the anion most, so trichloroacetic acid is strongest.
Revise: Carboxylic Acids (Class 12 notes)