TU IOST | Entrance paper
Answer (b): $0/0$ has no definite value, so $a/b$ is indeterminate only when both $a=0$ and $b=0$.
Revise: Algebra (Class 11 notes)
Answer (c): An open interval $(a,b)$ means $a<x<b$, so $(-\infty,\infty)$ is ${x: -\infty<x<\infty}$ (infinity is never included).
Revise: Algebra (Class 11 notes)
Answer (a): A statement is a sentence that is definitely true or false. 'Bagmati is a River' is a declarative sentence with a fixed truth value, while a question and an open expression like $x>0$ are not statements.
Revise: Algebra (Class 11 notes)
Answer (c): Equal ordered pairs have equal components: $2x-1=3$ gives $x=2$, and $y+3=-3$ gives $y=-6$.
Revise: Algebra (Class 11 notes)
Answer (a): $(f\circ g)(x)=f(g(x))=f(x)=x$.
Revise: Algebra (Class 11 notes)
Answer (d): $\sin 2x=1$ gives $2x=\frac{\pi}{2}+2n\pi$, so $x=n\pi+\frac{\pi}{4}$. Option d is the closest printed form (a general-solution pattern with $\pi/4$).
Revise: Trigonometry (Class 11 notes)
Answer (b): Let $\theta=\tan^{-1}x$, so the opposite side is $x$, adjacent is 1 and hypotenuse $\sqrt{1+x^2}$. Then $\sin\theta=x/\sqrt{1+x^2}$.
Revise: Trigonometry (Class 11 notes)
Answer (a): In an A.P. the middle term is the mean: $2(p+8)=2p+3p+1$, so $16+2p=5p+1$ and $p=5$.
Revise: Algebra (Class 11 notes)
Answer (d): For two positive numbers $a,b$: $AH=\frac{a+b}{2}\cdot\frac{2ab}{a+b}=ab=G^2$.
Revise: Algebra (Class 11 notes)
Answer (a): A $1\times3$ row times a $3\times1$ column gives $1\times1$: $(-1)(2)+0\cdot3+2\cdot4=6$.
Revise: Algebra (Class 11 notes)
Answer (c): Column 2 is twice column 1, so the determinant is 0. (Check: $1(8-6)-2(4-3)+0=0$.)
Revise: Algebra (Class 11 notes)
Answer (d): For a $2\times2$ matrix, swap the diagonal entries and change the sign of the off-diagonal ones.
Revise: Algebra (Class 11 notes)
Answer (a): $(1+i)^2=2i$. Then $\frac{2i(2+i)}{(2-i)(2+i)}=\frac{4i-2}{5}$, so the real part is $-2/5$.
Revise: Algebra (Class 11 notes)
Answer (a): Since $1+\omega+\omega^2=0$, the product is $0$.
Revise: Algebra (Class 11 notes)
Answer (a): Equal roots need $D=0$: $4(k+2)^2-36k=0$, so $k^2-5k+4=0$ and $k=1,4$.
Revise: Algebra (Class 11 notes)
Answer (c): $d=\frac{|3(1)+4(2)-7|}{\sqrt{3^2+4^2}}=\frac{4}{5}$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): The lines are real and distinct if $h^2>ab$, coincident if $h^2=ab$, imaginary if $h^2<ab$.
Revise: Analytic geometry (Class 11 notes)
Answer (c): Tangent at $(x_1,y_1)$ is $xx_1+yy_1=r^2$: $2x+y=5$.
Revise: Analytic geometry (Class 11 notes)
Answer (b): Write $\sqrt{x^2-4}=\sqrt{(x-2)(x+2)}$, so the expression is $\sqrt{\frac{x-2}{x+2}}\to0$.
Revise: Calculus (Class 11 notes)
Answer (b): $1-\cos ax=2\sin^2\frac{ax}{2}$, so the limit is $2\cdot\frac{a^2}{4}=\frac{a^2}{2}$.
Revise: Calculus (Class 11 notes)
Answer (a): $\frac{dy}{dx}=3x^2-2x=x(3x-2)=0$ gives $x=0$ or $x=2/3$.
Revise: Calculus (Class 11 notes)
Answer (b): $y''=2$ is never zero and never changes sign, so the parabola has no point of inflection.
Revise: Calculus (Class 11 notes)
Answer (d): Put $u=1-x^2$, $du=-2x,dx$: the integral is $\left[-2\sqrt{1-x^2}\right]_0^1=0-(-2)=2$.
Revise: Calculus (Class 11 notes)
Answer (a): By parts: $-x\cos x+\int\cos x,dx=-x\cos x+\sin x+C$.
Revise: Calculus (Class 11 notes)
Answer (a): $\int_3^4 3x^2dx=[x^3]_3^4=64-27=37$.
Revise: Calculus (Class 11 notes)
Answer (b): Both 'hut' and 'but' have the short vowel /ʌ/. 'Seat, sit' differ in length and quality and 'salt, mark' differ too; 'pair, tear' is only arguable depending on the meaning of 'tear'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): A plain statement of fact normally takes a falling tone.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'joy' begins with /dʒ/, as does 'gesture' (soft g). 'gig' and 'game' begin with /g/.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): plea-sur-a-ble: four syllables.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Omniscient' means all-knowing; omnipresent is everywhere, omnipotent is all-powerful.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): Aspiration is a strong desire or ambition, so 'yearning' is the synonym. Apathy is lack of interest.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Hypocrisy is pretending to have virtues one does not have; its opposite is sincerity.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): We say 'died of' a cause such as neglect or disease and 'died in' an accident.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Go with' someone is the natural collocation: 'Who did you go with?'.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): The words are definite, poignant and existence, so the missing vowels are e, i, e in that order.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): The plane passed above my head, so 'over' is correct.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): The compound subject 'You and I' is plural, so the verb is 'are'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): To 'turn the tables' means to reverse a situation so that a likely loser wins.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): A conservative person is averse to change; revolutionary means sudden big change, evolutionary means gradual change.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Greed is classically called a vice, a moral failing or bad habit. 'Sin' is also plausible, but 'vice' is the closest general term here.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): 'Go off' means to fire or explode, so the gun went off by mistake.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): After 'has' the past participle is needed, and the participle of 'lie' (recline) is 'lain': she has lain down for a rest. 'Laid' would need an object.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Brother + ly gives 'brotherly' (like a brother).
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): The opposite of desirable is 'undesirable'.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): 'Interesting' completes the meaning of the object 'her class', so it is an object complement.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'This evening' is correct. 'Today afternoon' should be 'this afternoon', and 'a worth seeing sight' should be 'a sight worth seeing'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): To beckon is to make a gesture inviting someone to come, i.e. to signal.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): Hemato- refers to blood, so the fear of blood is hematophobia.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): A short temper drives friends away, so the negative sense 'few' (almost none) fits, not 'a few'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'Having my hair style criticized' is a passive construction (hair style is criticized by others).
Revise: Grammar and Usage (Class 11 notes)
Answer (a): $W=\vec F\cdot\vec s=6+6+8=20$ J. Power $=W/t=20/5=4$ W.
Revise: Work, Energy and Power (Class 11 notes)
Answer (c): $\cos\omega t=\sin(\omega t+\pi/2)$, so the phase difference is $\pi/2$.
Revise: Periodic Motion (Class 12 notes)
Answer (d): For the shortest path the drift must be cancelled: $10\sin\theta=5$, so $\theta=30^\circ$ from North, pointing upstream, i.e. towards the West.
Revise: Kinematics (Class 11 notes)
Answer (a): Surface energy is energy per unit area: $\frac{ML^2T^{-2}}{L^2}=ML^0T^{-2}$.
Revise: Physical Quantities (Class 11 notes)
Answer (d): Stokes' law gives $v=\frac{2r^2(\rho-\sigma)g}{9\eta}$, so $v\propto r^2$.
Revise: Elasticity (Class 11 notes)
Answer (a): $\tau=\frac{\Delta L}{t}=\frac{4L_0-L_0}{4}=\frac{3}{4}L_0$.
Revise: Rotational Dynamic (Class 12 notes)
Answer (c): $v_{rms}\propto\sqrt{T/M}$, so $T_H/2=320/32$ giving $T_H=20$ K.
Revise: Ideal Gas (Class 11 notes)
Answer (a): $\Delta S=Q/T$, so the dimension is $ML^2T^{-2}K^{-1}$.
Revise: Second Law of Thermodynamics (Class 12 notes)
Answer (b): $\Delta A=\beta A\Delta\theta=2x\cdot4\cdot10=80x$.
Revise: Thermal Expansion (Class 11 notes)
Answer (c): $E\propto T^4$; ratio $=16$, so $T$ doubles: $T_2=2\times400=800$ K $=527^\circ$C.
Revise: Rate of Heat Flow (Class 11 notes)
Answer (a): Frequency stays the same; speed falls in glass, so $\lambda=v/f$ decreases.
Revise: Refraction at Plane Surface (Class 11 notes)
Answer (d): Fringe width $\beta=\lambda D/d$ is smallest for the shortest wavelength, so violet fringes lie nearest the centre.
Revise: Interference (Class 12 notes)
Answer (b): Power $\propto(\mu_g/\mu_w-1)$; the relative index drops in water, so power decreases for every colour.
Revise: Lenses (Class 11 notes)
Answer (c): $f'=f\frac{v+v_o}{v-v_s}=1000\times\frac{340}{320}=1062.5$ Hz.
Revise: Acoustic Phenomena (Class 12 notes)
Answer (b): Fundamental: open $L=\lambda/2$, closed $L=\lambda/4$, so $L_{open}:L_{closed}=2:1$.
Revise: Wave in Pipes and Strings (Class 12 notes)
Answer (c): $v_d=\frac{I}{nAe}$; doubling the radius makes $A$ four times larger, so $v_d$ falls to one fourth.
Revise: DC Circuit (Class 11 notes)
Answer (a): $R=V^2/P$, so the 25 W lamp has four times the resistance. In series the current is the same, and $P=I^2R$ is larger for the 25 W lamp.
Revise: DC Circuit (Class 11 notes)
Answer (d): $C=4\pi\epsilon_0R$, so $R=C/(4\pi\epsilon_0)=9\times10^{9}$ m, far larger than Earth's radius.
Revise: Capacitor (Class 11 notes)
Answer (c): $\varepsilon\propto Nf$: $\frac{50\times50}{100\times150}=\frac{1}{6}$.
Revise: Electromagnetic Induction (Class 12 notes)
Answer (c): $S=\frac{I_gG}{I-I_g}=\frac{0.1\times1}{0.9}=0.111\ \Omega$.
Revise: Electrical Circuit One (Class 12 notes)
Answer (d): Proton: $e/m$. Alpha: $2e/4m=e/2m$. Ratio $=2:1$.
Revise: Electrons (Class 12 notes)
Answer (a): Bohr: $L=\frac{nh}{2\pi}$ with $n=1$.
Revise: Quantization of Energy (Class 12 notes)
Answer (c): $R=R_0A^{1/3}$ with $A=27$ gives $3R_0$; with $R_0\approx1.2$ to $1.5$ fm the radius is about 3.6 to 4.5 fm, so 4.5 fermi is the nearest.
Revise: Nuclear Physics (Class 11 notes)
Answer (d): $\lambda=\frac{h}{\sqrt{2mK}}\propto K^{-1/2}$, so doubling $K$ gives $\lambda/\sqrt2$.
Revise: Quantization of Energy (Class 12 notes)
Answer (a): $T_{1/2}=0.693,\tau$, so half-life is smaller than mean life.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (c): Molar mass of water is 18 g/mol: $90/18=5$ mol.
Revise: Stoichiometry (Class 11 notes)
Answer (b): $CaCO_3+2HCl\to CaCl_2+H_2O+CO_2$. $CaCO_3$ is $0.01$ mol and needs $0.02$ mol HCl, but HCl is $1/36.5=0.027$ mol, so $CaCO_3$ is limiting. $CO_2=0.01$ mol $=0.224$ L.
Revise: Stoichiometry (Class 11 notes)
Answer (a): $m_l$ gives the spatial orientation of an orbital; $n$ gives size and $l$ gives shape.
Revise: Atomic Structure (Class 11 notes)
Answer (b): At constant volume $P/T$ is constant: $10/300=20/T$ gives $T=600$ K $=327^\circ$C.
Revise: States of Matter (Class 11 notes)
Answer (a): Molality uses mass of solvent, which does not change with temperature; the others depend on volume.
Revise: Stoichiometry (Class 11 notes)
Answer (b): It is the salt of a strong base and a weak acid; carbonate hydrolyses to give $OH^-$, so the solution is basic.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (d): $BF_3$ has an empty p orbital and accepts an electron pair, so it is a Lewis acid.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (a): Mn goes from $+7$ to $+6$, a change of 1 electron, so equivalent weight $=M/1$.
Revise: Oxidation and Reduction (Class 11 notes)
Answer (d): VSEPR applies to covalent molecules. $MgCl_2$ is essentially ionic, so VSEPR does not apply.
Revise: Chemical Bonding (Class 11 notes)
Answer (a): ZnO reacts with both acids and bases, forming zincates with alkalis.
Revise: Classification of Elements (Class 11 notes)
Answer (c): Haber's process: $N_2+3H_2\rightleftharpoons2NH_3$.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (c): Dry ice is solid carbon dioxide, which sublimes without melting.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (a): Rhombic and monoclinic sulfur exist as puckered $S_8$ rings.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (d): He, Ne and Ar occur in air, whereas Rn is radioactive and short-lived, so it is not a normal constituent of the atmosphere.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (b): Chalcopyrite is $CuFeS_2$, the main ore of copper.
Revise: Chemistry of Metals (Class 11 notes)
Answer (c): Limestone decomposes to CaO, which combines with silica gangue to form fusible $CaSiO_3$ slag.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): Bleaching powder is calcium chlorohypochlorite, $Ca(OCl)Cl$. The other formulas are gypsum, plaster of Paris and Epsom salt.
Revise: Chemistry of Metals (Class 11 notes)
Answer (b): Successive members differ by $CH_2$, so molecular weights differ, while the general formula and chemical properties are the same.
Revise: Fundamental Principles (Class 11 notes)
Answer (d): Aldehydes (not ketones) respond to Tollens', Fehling's and Schiff's tests. $Br_2$ in $CCl_4$ tests for unsaturation and gives no distinction here.
Revise: Aldehydes and Ketones (Class 12 notes)
Answer (a): In acidic medium (Sn/HCl) $C_6H_5NO_2$ is fully reduced to aniline; the other products form in neutral or alkaline media.
Revise: Nitro Compounds (Class 12 notes)
Answer (c): $6HCHO+4NH_3\to(CH_2)_6N_4+6H_2O$; hexamethylenetetramine is urotropine.
Revise: Aldehydes and Ketones (Class 12 notes)
Answer (d): Dettol (chloroxylenol) is applied on living tissue to kill germs, so it is an antiseptic.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (d): BHC (benzene hexachloride, gammexane) is an insecticide.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (c): Amino acids join through peptide (amide) bonds to form proteins.
Revise: Biomolecules (Class 11 notes)
Answer (c): DNA contains the pentose sugar deoxyribose (RNA has ribose).
Revise: Biomolecules (Class 11 notes)