- a
- b
- c$\frac{22}{7}$
- d
Answer (b): $\pi$ is non-terminating and non-repeating, so it is irrational. The others are rational numbers.
Revise: Algebra (Class 11 notes)
TU IOST | Entrance paper
Answer (b): $\pi$ is non-terminating and non-repeating, so it is irrational. The others are rational numbers.
Revise: Algebra (Class 11 notes)
Answer (a): If $P\subset Q$, every element of $P$ is also in $Q$, so $P\cap Q=P$.
Revise: Algebra (Class 11 notes)
Answer (d): A statement must be definitely true or false. $x>2$ depends on the value of $x$, so it is an open sentence, not a statement.
Revise: Algebra (Class 11 notes)
Answer (b): $x-3=5$ gives $x=8$ and $x-3=-5$ gives $x=-2$.
Revise: Algebra (Class 11 notes)
Answer (a): Put $x=3$, $y=64$: $a^3=64=4^3$, so $a=4$.
Revise: Algebra (Class 11 notes)
Answer (d): Let $y=x-5$, so $x=y+5$. Hence $f^{-1}(x)=x+5$.
Revise: Algebra (Class 11 notes)
Answer (a): Period of $\cos(bx+c)$ is $2\pi/b = 2\pi/2=\pi$.
Revise: Trigonometry (Class 11 notes)
Answer (c): Equal angles with the same sine and cosine means the angle is $\pi/4$, so $x=\sin(\pi/4)=\frac{1}{\sqrt2}$.
Revise: Trigonometry (Class 11 notes)
Answer (b): $\sin^2x=\sin^2(\pi/4)$, and $\sin^2x=\sin^2\alpha$ gives $x=n\pi\pm\alpha$, so $x=n\pi\pm\frac{\pi}{4}$.
Revise: Trigonometry (Class 11 notes)
Answer (c): Terms $ar^2, ar^4, ar^6$: $pr=a^2r^8=(ar^4)^2=q^2$.
Revise: Algebra (Class 11 notes)
Answer (a): Middle term twice: $2(6p-2)=(3p+4)+(2p+7)$, so $12p-4=5p+11$ and $p=\frac{15}{7}$.
Revise: Algebra (Class 11 notes)
Answer (a): $a_{11}=2$, $a_{12}=a_{21}=3$, $a_{22}=4$.
Revise: Algebra (Class 11 notes)
Answer (c): The determinant is $(x+1)(1)-0=x+1=0$, so $x=-1$.
Revise: Algebra (Class 11 notes)
Answer (d): Same left side with different right sides is contradictory, so the lines are parallel and there is no solution.
Revise: Algebra (Class 11 notes)
Answer (c): Expanding: $1-\omega-\omega^2+\omega^3=1-(\omega+\omega^2)+1=1+1+1=3$.
Revise: Algebra (Class 11 notes)
Answer (a): $\sqrt{-3}\sqrt{-5}=(i\sqrt3)(i\sqrt5)=i^2\sqrt{15}=-\sqrt{15}$.
Revise: Algebra (Class 11 notes)
Answer (c): Sum of roots $=-1$, product $=-6$, so $x^2-(\text{sum})x+\text{product}=x^2+x-6=0$.
Revise: Algebra (Class 11 notes)
Answer (a): For $ax^2+2hxy+by^2=0$ the lines are perpendicular when $a+b=0$: $3+k=0$, so $k=-3$.
Revise: Analytic geometry (Class 11 notes)
Answer (b): The tangent to $x^2+y^2=r^2$ at $(x_1,y_1)$ is $xx_1+yy_1=r^2$, giving $2x+y=5$.
Revise: Analytic geometry (Class 11 notes)
Answer (c): The limit equals $3a^2=27$, so $a^2=9$ and $a=\pm3$.
Revise: Calculus (Class 11 notes)
Answer (d): $\frac{dy}{dx}=\frac{2x}{p^2}-\frac{2p^2}{x^3}$. At $x=p$: $\frac2p-\frac2p=0$.
Revise: Calculus (Class 11 notes)
Answer (a): $\int3\sqrt x,dx=2x^{3/2}$, $\int\sin x,dx=-\cos x$ and $\int2e^{2x}dx=e^{2x}$.
Revise: Calculus (Class 11 notes)
Answer (b): Area $=\int_2^4 3x^2dx=[x^3]_2^4=64-8=56$.
Revise: Calculus (Class 11 notes)
Answer (a): $f'(x)=4x^3-12x+8=4(x-1)^2(x+2)$, which is zero at $x=1$ and $x=-2$.
Revise: Calculus (Class 11 notes)
Answer (b): 'Permissible' (per-MIS-si-ble) is stressed on its second syllable.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): PER-ish-es is stressed on the first syllable; in-CRED-i-ble, in-CUR and te-MER-i-ty are not.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): "Glad of" something is the fixed collocation used here (glad of your help).
Revise: Grammar and Usage (Class 11 notes)
Answer (a): The stem is garbled, but 'addicted to' is the only natural preposition for a habit of addiction.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): A vendetta is a long-running quarrel or blood 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 well filled or stocked, so its opposite is empty.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): 'Quell' means to suppress, so its opposite is 'encourage'.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): A contagious disease spreads by contact or touch.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Pork, mutton and venison are names of meat, while lamb is the name of the animal.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): "Ad nauseam" means to a sickening or excessive 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 (b): "Autumnal" describes a noun, so it is an adjective; width and advantage are nouns.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'To stand on ceremony' means to be too formal or to pay too much attention to correct rules of behaviour.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): "Double Dutch" means unintelligible gibberish.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (d): "Nature" in the general sense takes no article.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): 'See about' means to arrange or attend to something, as in 'see about getting a room ready'.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Past perfect passive needs the participle, so "worn away" (gradually eroded).
Revise: Grammar and Usage (Class 11 notes)
Answer (c): "So + auxiliary + subject" agrees with a positive statement: "so am I".
Revise: Grammar and Usage (Class 11 notes)
Answer (d): Palatable means pleasant to taste.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): A sentence beginning with 'Never' needs inversion, so 'Never did I harm anyone' is correct.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): "Nowadays" signals a present continuous action: "is becoming".
Revise: Grammar and Usage (Class 11 notes)
Answer (c): "Very interesting" describes the object "presentation", so it is an object complement.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Of the options only 'perform' sits naturally with 'skills'; the intended verb may have been 'acquire', which is not offered.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): A dietician specialises in food and nutrition.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): $L/R$ and $CR$ both have dimension of time.
Revise: Physical Quantities (Class 11 notes)
Answer (d): From $v^2=u^2-2as$: $(u/2)^2=u^2-6a$ gives $a=u^2/8$. Then to stop from $u/2$: $s=\frac{(u/2)^2}{2a}=1$ cm.
Revise: Kinematics (Class 11 notes)
Answer (c): $R/H=\frac{2\sin\theta\cos\theta\cdot2}{\sin^2\theta}=4\cot\theta=4\sqrt3$, so $\cot\theta=\sqrt3$ and $\theta=30^0$.
Revise: Kinematics (Class 11 notes)
Answer (b): Zero net force means zero acceleration (rest or uniform motion), so saying acceleration is not zero is wrong.
Revise: Dynamics (Class 11 notes)
Answer (c): Young's modulus is a property of the material, so it does not change with the length of the wire.
Revise: Elasticity (Class 11 notes)
Answer (b): Work is surface tension times the area increase with two surfaces: $W=8\pi R^2T$. Going from $R$ to $3R$ gives $8\pi T(9R^2-R^2)=64\pi R^2T=8W$.
Revise: Elasticity (Class 11 notes)
Answer (a): The alcohol expands more than the ball, so its density falls faster than the ball's volume rises and the buoyant force decreases; the apparent weight increases.
Revise: Thermal Expansion (Class 11 notes)
Answer (b): At the critical temperature the liquid and vapour phases become identical, so latent heat is zero.
Revise: Quantity of Heat (Class 11 notes)
Answer (c): By equipartition each degree of freedom carries $\frac12kT$.
Revise: Ideal Gas (Class 11 notes)
Answer (c): Radiated heat $\propto$ surface area $\propto r^2$, so $9:4$.
Revise: Rate of Heat Flow (Class 11 notes)
Answer (b): For minima $a\sin\theta=2\lambda$. With $a=2.2\times10^{-6}$ m: $\sin\theta=\frac{1.1\times10^{-6}}{2.2\times10^{-6}}=0.5$, so $\theta=30^0$.
Revise: Diffraction of Light (Class 12 notes)
Answer (a): $n=\frac{\sin\frac{A+\delta}{2}}{\sin\frac A2}$ gives $\sin\frac{60+\delta}{2}=1.414\times0.5=0.707$, so $\frac{60+\delta}{2}=45^0$ and $\delta=30^0$.
Revise: Refraction Through Prisms (Class 11 notes)
Answer (d): $\lambda=v/f=0.72$ m. Path difference $=\frac{\lambda}{360}\times60=0.12$ m $=12$ cm.
Revise: Wave Motion (Class 12 notes)
Answer (a): At distance $x$ from $q$: $\frac{q}{x^2}=\frac{4q}{(r-x)^2}$ gives $r-x=2x$, so $x=r/3$ from $q$, which is $2r/3$ from $4q$.
Revise: Electric Charges (Class 11 notes)
Answer (a): Series gives $C/n$ (minimum), parallel gives $nC$ (maximum). Ratio $=nC\big/(C/n)=n^2$.
Revise: Capacitor (Class 11 notes)
Answer (b): Resistance halves, and $H=\frac{V^2}{R}t$ doubles for the same voltage.
Revise: DC Circuit (Class 11 notes)
Answer (d): $E=I(R+r)$: $50=4.5(10+r)$, so $10+r=11.1$ and $r\approx1.1,\Omega$.
Revise: DC Circuit (Class 11 notes)
Answer (a): In a uniform field the forces on the two poles are equal and opposite (net force zero) but form a couple, giving a torque.
Revise: Magnetic Field (Class 12 notes)
Answer (c): $Q=\frac{X_L}{R}=\frac{100}{5}=20$.
Revise: Alternating Currents (Class 12 notes)
Answer (c): $L=\frac{1}{4\pi^2f^2C}=\frac{1}{4\pi^2(1.02\times10^6)^2(420\times10^{-12})}\approx5.8\times10^{-5}$ H. The digits match option c; its exponent is likely misprinted.
Revise: Alternating Currents (Class 12 notes)
Answer (d): $\frac12mv^2=eV$: $v=\sqrt{\frac{2\times1.6\times10^{-19}\times100}{9.1\times10^{-31}}}\approx5.9\times10^6$ m/s.
Revise: Electrons (Class 12 notes)
Answer (c): Mean life $=\frac{T_{1/2}}{0.693}=\frac{10}{0.693}=14.43$ years.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (d): In NPN the majority carriers are electrons, which are more mobile than holes, giving better frequency response.
Revise: Semiconductor Devices (Class 12 notes)
Answer (b): The magnetic quantum number $m_l$ gives the spatial orientation of an orbital.
Revise: Atomic Structure (Class 11 notes)
Answer (c): Gamma rays are uncharged high-energy photons and penetrate most.
Revise: Nuclear Chemistry (Class 12 notes)
Answer (a): Mg is a metal that transfers electrons to Cl, forming ionic $MgCl_2$; the others are covalent.
Revise: Chemical Bonding (Class 11 notes)
Answer (b): Fluorine is the most electronegative element (4.0).
Revise: Classification of Elements (Class 11 notes)
Answer (a): $2x+7(-2)=-2$ gives $2x=12$, so $x=+6$.
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 and CuO is basic.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): $C_{60}$ looks like a football: 20 hexagons and 12 pentagons.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (c): Down's cell electrolyses molten NaCl to obtain sodium.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): Iso-butane is $(CH_3)_3CH$, a propane chain with a methyl on C-2.
Revise: Hydrocarbons (Class 11 notes)
Answer (b): Cold dilute alkaline $KMnO_4$ is Baeyer's reagent, used to test unsaturation.
Revise: Hydrocarbons (Class 11 notes)
Answer (a): $2R\text{-}X+2Na\to R\text{-}R+2NaX$ in dry ether is the Wurtz reaction.
Revise: Haloalkanes (Class 12 notes)
Answer (b): Saturated hydrocarbons follow $C_nH_{2n+2}$.
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 is $sp^3$ hybridised with four bonds, giving a tetrahedral shape.
Revise: Chemical Bonding (Class 11 notes)
Answer (a): $Mn^{7+}\to Mn^{2+}$ transfers 5 electrons, so $E=M/5$.
Revise: Volumetric Analysis (Class 12 notes)
Answer (d): $NH_3$ has a lone pair to donate, so it is a Lewis base; the other three are electron-deficient Lewis acids.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (d): An electrochemical, galvanic or voltaic cell all mean a cell that produces electricity from a redox reaction.
Revise: Electrochemistry (Class 12 notes)
Answer (a): The thermochemical calorie is defined as 4.184 J.
Revise: Chemical Thermodynamic (Class 12 notes)
Answer (d): Concentration, temperature and surface area of reactants all affect the reaction rate.
Revise: Chemical Kinetics (Class 12 notes)
Answer (a): Each C has three sigma bonds in a planar ring, so it is $sp^2$.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (c): Tertiary alcohols form the chloride fastest via a stable carbocation, so turbidity appears at once.
Revise: Alcohols (Class 12 notes)
Answer (a): Fats are triglyceride esters, classed as lipids.
Revise: Fundamental Principles (Class 11 notes)
Answer (a): Bakelite is the phenol-formaldehyde condensation polymer.
Revise: Phenols (Class 12 notes)
Answer (b): Mercuric chloride $HgCl_2$ is called corrosive sublimate; $Hg_2Cl_2$ is calomel.
Revise: Study of Heavy Metals (Class 12 notes)
Answer (a): Hematite is $Fe_2O_3$, an iron ore.
Revise: Chemistry of Metals (Class 11 notes)