- a$7$ and $-1$
- b$7$ and $1$
- c
- d
Answer (a): $x-3=4$ gives $x=7$, and $x-3=-4$ gives $x=-1$.
Revise: Algebra (Class 11 notes)
TU IOST | Entrance paper
Answer (a): $x-3=4$ gives $x=7$, and $x-3=-4$ gives $x=-1$.
Revise: Algebra (Class 11 notes)
Answer (a): $x-x^2=9$ means $x^2-x+9=0$, whose discriminant $1-36<0$ gives no real root, and $2x=6$ gives $x=3$ which does not satisfy it. So the solution set is empty.
Revise: Algebra (Class 11 notes)
Answer (b): $f^{-1}(4)$ is the set of $x$ with $x^2+3=4$, so $x^2=1$ and $x=\pm1$.
Revise: Algebra (Class 11 notes)
Answer (b): $\log_2 2^{1/2}=\frac12$, $\log_3 3^{1/2}=\frac12$, and $\log_4 2=\frac12$, so the sum is $\frac32$.
Revise: Algebra (Class 11 notes)
Answer (a): $\cos\theta=\pm\frac{1}{\sqrt2}$, so $\cos^2\theta=\cos^2\frac\pi4$ and $\theta=n\pi\pm\frac\pi4$.
Revise: Trigonometry (Class 11 notes)
Answer (b): Let $\sin\alpha=\frac35$; then $\cos\alpha=\sqrt{1-\frac{9}{25}}=\frac45$.
Revise: Trigonometry (Class 11 notes)
Answer (a): Reciprocals are in A.P.: $\frac{2}{b^2}=\frac1{a^2}+\frac1{c^2}=\frac{a^2+c^2}{a^2c^2}$, so $2a^2c^2=b^2(a^2+c^2)$.
Revise: Algebra (Class 11 notes)
Answer (b): $A^2=\begin{pmatrix}5&8\8&13\end{pmatrix}$. Comparing the $(1,2)$ entry: $8-2n=0$ so $n=4$ (and $5-n-1=0$ agrees).
Revise: Algebra (Class 11 notes)
Answer (d): Expanding along the first row, $|A|=1(0-0)-0+1(0-1)=-1\neq0$, so $A$ is nonsingular. It is not symmetric since $a_{23}\neq a_{32}$.
Revise: Algebra (Class 11 notes)
Answer (b): The second equation reduces to $2x+y=4$, which contradicts $2x+y=5$, so the system has no solution (inconsistent). The equations are not multiples of each other, so they are independent.
Revise: Algebra (Class 11 notes)
Answer (d): $i^7=i^4\cdot i^3=-i$ and $i^4=1$, so the sum is $1-i$.
Revise: Algebra (Class 11 notes)
Answer (a): Multiply the denominator by $w$: $w(y+wz+w^2x)=wy+w^2z+x$ since $w^3=1$. That is the numerator, so the ratio is $w$.
Revise: Algebra (Class 11 notes)
Answer (b): Equal roots need $16-4k=0$, so $k=4$.
Revise: Algebra (Class 11 notes)
Answer (a): Two lines coincide exactly when all three coefficient ratios are equal.
Revise: Analytic geometry (Class 11 notes)
Answer (b): A pair of lines $ax^2+2hxy+by^2=0$ is perpendicular when $a+b=0$. Here $2+k=0$, so $k=-2$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): The centre is $(2,2)$, so $(x-2)^2+(y-2)^2=4$, which expands to $x^2+y^2-4x-4y+4=0$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): The point $(3,4)$ lies on the line, so $k=3(3)+4(4)=25$.
Revise: Analytic geometry (Class 11 notes)
Answer (a): Using $\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}$: $5\cdot2^4=80$.
Revise: Calculus (Class 11 notes)
Answer (c): $3+4\cdot\frac{\tan2x}{x}\to3+4\cdot2=11$.
Revise: Calculus (Class 11 notes)
Answer (d): $\frac{d}{dx}b^x=b^x\ln b$.
Revise: Calculus (Class 11 notes)
Answer (a): Chain rule: $-\sin(\cos\sqrt x)\cdot(-\sin\sqrt x)\cdot\frac{1}{2\sqrt x}=\frac{\sin(\cos\sqrt x)\sin\sqrt x}{2\sqrt x}$.
Revise: Calculus (Class 11 notes)
Answer (d): $\frac{d}{dx}(x^2+2x+2)=2x+2=0$ gives $x=-1$.
Revise: Calculus (Class 11 notes)
Answer (a): $y''=6x+6=0$ gives $x=-1$, and $y''$ changes sign there.
Revise: Calculus (Class 11 notes)
Answer (b): $1-\cos x=2\sin^2\frac x2$, so the integral is $\frac12\int\csc^2\frac x2,dx=-\cot\frac x2$.
Revise: Calculus (Class 11 notes)
Answer (b): Put $u=1+\sin x$: the integral is $\ln(1+\sin x)\Big|_0^{\pi/2}=\ln2-\ln1=\ln2$.
Revise: Calculus (Class 11 notes)
Answer (c): In edu-CA-tion the primary stress falls on the third syllable (-ca-).
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Arduous means requiring great effort, so its synonym is difficult.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Obscure means unclear or hard to understand; explicit means stated clearly, the best opposite here.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Chef begins with the /sh/ sound, the same as chute. Chess, child and cheap start with /ch/.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Pair' and 'tear' (to rip) share the vowel /eə/. Hut/hot, sit/seat and salt/mark have different vowels.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): The verb 'differ' takes 'from' for the person and 'on' for the point of disagreement: 'differ from you on this point'.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): The noun 'sympathy' is followed by 'for' when it is directed at a group: 'sympathy for the poor'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): The verb insist takes the preposition on.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): A hole and corner method means done in a secret, underhand way.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Palatable means pleasant to the taste.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): Displeased is the correct word formed with the prefix dis-.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Omnipresent means present everywhere; omniscient means all-knowing and omnipotent means all-powerful.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (a): Cibophobia (cibus, food) is the fear of food. Ergophobia is fear of work.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (b): A knell is the sound of a bell rung slowly at a funeral.
Revise: Vocabulary and Word Formation (Class 11 notes)
Answer (c): The time word now and the ongoing trend need a present continuous with a modal of deduction: must be buying.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Chipmunks are like most other ground squirrels' is a complete, correct predicate. 'Alike' cannot take an object and 'like as'/'like to' are wrong.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): A duration phrase (the whole day long) with an action continuing up to now takes the present perfect continuous.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): 'Too busy to take care' is the 'too ... to' pattern, meaning so busy that he cannot. 'So' would need 'that'.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): 'Demand to know' is the correct construction, an infinitive after demand.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'But' here means 'except': the mail comes every day except Saturday.
Revise: Grammar and Usage (Class 11 notes)
Answer (a): Paid by the month is the fixed phrase with the definite article.
Revise: Grammar and Usage (Class 11 notes)
Answer (b): Subjects joined by and but each preceded by every take a singular verb: is.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'None of the squares was properly drawn' is acceptable. 'Some of rice' needs 'the', and 'wait for none' is not the idiom ('no man').
Revise: Grammar and Usage (Class 11 notes)
Answer (a): A complex sentence has one main clause and a dependent clause. 'However hard she worked' is a dependent clause, so a is complex.
Revise: Grammar and Usage (Class 11 notes)
Answer (d): 'Moral' is an adjective. Sympathize and lighten are verbs and perception is a noun.
Revise: Grammar and Usage (Class 11 notes)
Answer (c): $[h]=ML^2T^{-1}$ and $[e]=AT$, so $[h/e]=ML^2T^{-2}A^{-1}$, which is the dimension of magnetic flux (weber).
Revise: Physical Quantities (Class 11 notes)
Answer (d): Leading zeros do not count; the digits 3 and the trailing 0 are significant, giving 2.
Revise: Physical Quantities (Class 11 notes)
Answer (a): Limiting static friction is at least as large as kinetic friction, so mu_s is greater than or equal to mu_k.
Revise: Dynamics (Class 11 notes)
Answer (d): Poisson's ratio for ordinary materials lies between -1 and 0.5, so 0.65 is impossible.
Revise: Elasticity (Class 11 notes)
Answer (c): Equilibrium needs $h\rho g=\frac{2T}{r}$, so $h=\frac{2(72\times10^{-3})}{0.4\times10^{-3}\times1000\times9.8}=0.03674$ m $=3.674$ cm.
Answer (d): Expansion is uniform, so area expansion is twice the linear one: $2\times2%=4%$.
Revise: Thermal Expansion (Class 11 notes)
Answer (c): A domestic pressure cooker works at roughly twice atmospheric pressure, raising the boiling point to about 120 C.
Revise: Heat and Temperature (Class 11 notes)
Answer (d): $\lambda=\frac{kT}{\sqrt2\pi d^2P}$, so at constant pressure $\lambda\propto T$.
Revise: Ideal Gas (Class 11 notes)
Answer (b): $\eta=1-\frac{T_2}{T_1}=1-\frac{273}{373}=0.2681=26.81%$.
Revise: Second Law of Thermodynamics (Class 12 notes)
Answer (d): Bending of waves around corners or obstacles is diffraction.
Revise: Diffraction of Light (Class 12 notes)
Answer (c): The two prongs move in opposite directions at each instant, so they are out of phase by pi.
Revise: Wave in Pipes and Strings (Class 12 notes)
Answer (d): For a convex mirror $f>0$ with the usual sign rule, $u=-f$: $1/v=1/f+1/f=2/f$, so $v=f/2$ behind the mirror.
Revise: Reflection at Curved Mirrors (Class 11 notes)
Answer (a): The Moon has no atmosphere, so there is no scattering of sunlight and the sky looks black.
Revise: Dispersion (Class 11 notes)
Answer (c): A lens of lower refractive index than its surroundings acts opposite to normal: a convex air bubble in glass is divergent.
Revise: Lenses (Class 11 notes)
Answer (d): Two prisms combined can give dispersion without deviation or deviation without dispersion, depending on the arrangement.
Revise: Refraction Through Prisms (Class 11 notes)
Answer (b): The short de Broglie wavelength of fast electrons gives high resolving power.
Revise: Electrons (Class 12 notes)
Answer (a): Three free charges can balance only when collinear, with the middle one of opposite sign and smaller magnitude.
Revise: Electric Charges (Class 11 notes)
Answer (d): $C=4\pi\varepsilon_0R=\frac{6.4\times10^6}{9\times10^9}=7.11\times10^{-4}$ F $=711,\mu$F.
Revise: Capacitor (Class 11 notes)
Answer (a): Specific resistance is a property of the material, so for the same material the ratio is 1:1.
Revise: DC Circuit (Class 11 notes)
Answer (c): Inside a long solenoid the field is uniform with $B=\mu_0 n i$.
Revise: Magnetic Field (Class 12 notes)
Answer (b): Electrons are bound to the nucleus by the electrostatic (Coulomb) attraction. Nuclear forces act only between nucleons and gravity is negligible.
Revise: Quantization of Energy (Class 12 notes)
Answer (c): The Balmer series (transitions to n=2) lies in the visible region.
Revise: Quantization of Energy (Class 12 notes)
Answer (d): $\tau=\frac{T_{1/2}}{0.693}=\frac{1600}{0.693}\approx2309$ years.
Revise: Radioactivity and Nuclear Reaction (Class 12 notes)
Answer (d): The Rydberg constant depends slightly on the nuclear mass through the reduced mass, $R_M=R_\infty/(1+m/M)$. The effect is noticeable for light nuclei and negligible for heavy ones.
Revise: Quantization of Energy (Class 12 notes)
Answer (d): The number of orbitals in a shell is $n^2=9$ (1 s, 3 p and 5 d).
Revise: Atomic Structure (Class 11 notes)
Answer (b): Moles $=\frac{18}{18}=1$ since the molar mass of water is 18 g/mol.
Revise: Stoichiometry (Class 11 notes)
Answer (c): One quarter means two half-lives: 2 x 30 = 60 minutes.
Revise: Nuclear Chemistry (Class 12 notes)
Answer (a): H is +1 and O is -2, so Cl is +1 to make the molecule neutral.
Revise: Oxidation and Reduction (Class 11 notes)
Answer (b): Mass of Fe per molecule $=0.0033\times67200=221.8$; atoms $=\frac{221.8}{55.85}\approx4$.
Revise: Stoichiometry (Class 11 notes)
Answer (a): At constant temperature, pressure and volume of a fixed gas are inversely proportional: Boyle's law.
Revise: States of Matter (Class 11 notes)
Answer (c): C has a single outer electron which it loses most easily, making it the most metallic.
Revise: Classification of Elements (Class 11 notes)
Answer (a): VSEPR gives the shape around a single central atom; ethane has two carbon centres, so no single molecular geometry is assigned.
Revise: Chemical Bonding (Class 11 notes)
Answer (d): Taking the rate law from the stoichiometry, rate $=k[A]^2[B]$, so tripling both gives $3^3=27$ times.
Revise: Chemical Kinetics (Class 12 notes)
Answer (c): NaCl is a salt of a strong acid and strong base, so it does not hydrolyse and the solution is neutral.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (c): $K_w$ rises with temperature, so $[H^+]>10^{-7}$ and pH falls below 7 while water stays neutral.
Revise: Ionic Equilibrium (Class 12 notes)
Answer (b): Only very active metals such as Mg and Mn release hydrogen with very dilute nitric acid; Zn, Cu and Pb give nitrogen oxides.
Revise: Chemistry of Metals (Class 11 notes)
Answer (b): $N=M\times$ basicity $=1\times2=2N$.
Revise: Volumetric Analysis (Class 12 notes)
Answer (a): Ortho hydrogen has nuclear spins in the same direction; para hydrogen has them opposite.
Revise: Chemistry of Non-Metals (Class 11 notes)
Answer (b): Haber's process combines $\ce{N2}$ and $\ce{H2}$ over an iron catalyst.
Revise: Modern Chemical Manufactures (Class 11 notes)
Answer (a): Haematite is $\ce{Fe2O3}$, an ore of iron.
Revise: Chemistry of Metals (Class 11 notes)
Answer (a): $\ce{BF3}$ has an empty orbital on boron and is an electron acceptor (electrophile); the others have lone pairs.
Revise: Haloalkanes (Class 12 notes)
Answer (a): Anti-Markovnikov addition of HBr (peroxide effect) needs an unsymmetrical alkene, and propene is the only one listed; 2-butene is symmetrical, and 1-butane is an alkane, not an alkene.
Revise: Hydrocarbons (Class 11 notes)
Answer (d): Glycerol with oxalic acid at 110 C gives formic acid (and CO2).
Revise: Carboxylic Acids (Class 12 notes)
Answer (c): Electron-withdrawing groups such as -COOH direct the incoming group to the meta position.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (b): Dichlorobenzene exists as ortho, meta and para isomers: 3.
Revise: Aromatic Hydrocarbons (Class 11 notes)
Answer (b): Terylene (polyester) forms by condensation of a diacid and a diol with loss of water; the others are addition polymers.
Revise: Modern Chemical Manufactures (Class 11 notes)
Answer (c): Penicillin is an antibiotic.
Revise: Chemistry in the Service to Mankind (Class 12 notes)
Answer (b): Enzymes are biological catalysts made of protein.
Revise: Biomolecules (Class 11 notes)
Answer (c): Uracil replaces thymine in RNA and is absent from DNA.
Revise: Biomolecules (Class 11 notes)