JEE Main 28 Jan 2026, Shift 1

75 questions. Answers are hidden — tap “Show answer” under each one.

Physics

Q1 · MCQ · Thermal Properties of MatterOpen
10 kg of ice at $-10^\circ$ C is added to 100 kg of water to lower its temperature from $25^\circ$ C. Consider no heat exchange to surroundings. The decrement to the temperature of water is $^\circ$ C.
(specific heat of ice $= 2100$ J/kg. $^\circ$ C, specific heat of water $= 4200$ J/kg. $^\circ$ C, latent heat of fusion of ice $= 3.36\times10^{5}$ J/kg)
1. (1)10
2. (2)15
3. (3)6.67
4. (4)11.6
We corrected a printing error in the original paper
Source printed the unit as “J/Kg” (three times).
Q2 · MCQ · Alternating CurrentOpen
The electric current in the circuit is given as $i = i_0(t/T)$ . The r.m.s current for the period $t = 0$ to $t = T$ is
1. (1) $\dfrac{i_0}{\sqrt2}$
2. (2) $i_0$
3. (3) $\dfrac{i_0}{\sqrt6}$
4. (4) $\dfrac{i_0}{\sqrt3}$
Q3 · MCQ · Current ElectricityOpen
In the potentiometer, when the cell in the secondary circuit is shunted with $4\ \Omega$ resistance, the balance is obtained at the length 120 cm of wire. Now when the same cell is shunted with $12\ \Omega$ resistance, the balance is shifted to a length of 180 cm. The internal resistance of cell is $\Omega$ .
1. (1)3
2. (2)4
3. (3)12
4. (4)6
We corrected a printing error in the original paper
Source printed “the same cell in shunted”.
Q4 · MCQ · Rotational MotionOpen
Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure.
If the mass of each disc is 600 gm and applied torque between two discs is $43\times10^{5}$ dyne cm, the angular acceleration of the discs about the given axis $AB$ is $\mathrm{rad/s^2}$ .
1. (1)22
2. (2)11
3. (3)100
4. (4)27
Q5 · MCQ · Motion in a Straight LineOpen
Water drops fall from a tap on the floor, 5 m below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is m.
( $g = 10\ \mathrm{m/s^2}$ )
1. (1)2.5
2. (2)4.0
3. (3)4.2
4. (4)3.8
Q6 · MCQ · NucleiOpen
An atom ${}_{3}X$ is bombarded by shower of fundamental particles and in 10 s this atom absorbed 10 electrons, 10 protons and 9 neutrons. The percentage growth in the surface area of the nucleons is recorded by :
1. (1)250%
2. (2)150%
3. (3)225%
4. (4)900%
Flawed question. The mass number of ${}_{3}X$ is not printed, and with the evident $A = 8 \to 27$ the surface area grows by 125%, which is not an option (the key 225% is the final area as a percentage of the initial); Allen gives the question as BONUS. It is shown for completeness and is left out of practice tests.
Q7 · MCQ · Electromagnetic WavesOpen
The electric field of an electromagnetic wave travelling through a medium is given by $\vec E(x,t) = 25\sin(2.0\times10^{15}t - 10^{7}x)\,\hat n$ then the refractive index of the medium is .
(All given measurements are in SI units)
1. (1)1.2
2. (2)2
3. (3)1.5
4. (4)1.7
We corrected a printing error in the original paper
Source printed “All given measurement are”.
Q8 · MCQ · Thermal Properties of MatterOpen
Which of the following best represents the temperature versus heat supplied graph for water, in the range of $-20^\circ$ C to $120^\circ$ C ?
1. (1)
2. (2)
3. (3)
4. (4)
Q9 · MCQ · Moving Charges and MagnetismOpen
Three long straight wires carrying current are arranged mutually parallel as shown in the figure. The force experienced by 15 cm length of wire $Q$ is .
( $\mu_0 = 4\pi\times10^{-7}$ T.m/A)
1. (1) $6\times10^{-7}$ N towards $P$
2. (2) $6\times10^{-6}$ N towards $R$
3. (3) $6\times10^{-7}$ N towards $R$
4. (4) $6\times10^{-6}$ N towards $P$
Q10 · MCQ · Semiconductor ElectronicsOpen
Assuming in forward bias condition there is a voltage drop of 0.7 V across a silicon diode, the current through diode $\mathrm{D_1}$ in the circuit is mA. (Assume all diodes in the given circuit are identical)
1. (1)20.15
2. (2)11.7
3. (3)17.6
4. (4)18.8
Q11 · MCQ · Ray OpticsOpen
The magnitudes of power of a biconvex lens (refractive index 1.5) and that of a plano-concave lens (refractive index $= 1.7$ ) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is .
1. (1) $5:2$
2. (2) $5:12$
3. (3) $12:5$
4. (4) $2:5$
Q12 · MCQ · ThermodynamicsOpen
In the following $p$ - $V$ diagram the equation of state along the curved path is given by $(V-2)^2 = 4ap$ where $a$ is a constant. The total work done in the closed path is
1. (1) $-\dfrac{1}{a}$
2. (2) $+\dfrac{1}{3a}$
3. (3) $\dfrac{1}{2a}$
4. (4) $-\dfrac{1}{3a}$
Q13 · MCQ · Current ElectricityOpen
For the two cells having same EMF E and internal resistance $r$ , the current passing through the external resistor $6\ \Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is $\Omega$ .
1. (1)3
2. (2)4
3. (3)9
4. (4)6
Q14 · MCQ · Mechanical Properties of SolidsOpen
Two wires $A$ and $B$ made of different materials of length 6.0 cm and 5.4 cm, respectively and area of cross sections $3.0\times10^{-5}\ \mathrm{m^2}$ and $4.5\times10^{-5}\ \mathrm{m^2}$ , respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $x:3$ . The value of $x$ is .
1. (1)1
2. (2)4
3. (3)2
4. (4)5
Q15 · MCQ · Newton's Laws of MotionOpen
A block of mass 5 kg is moving on an inclined plane which makes an angle of $30^\circ$ with the horizontal. Friction coefficient between the block and inclined plane surface is $\dfrac{\sqrt3}{2}$ . The force to be applied on the block so that the block will move down without acceleration is N.
1. (1)25
2. (2)12.5
3. (3)7.5
4. (4)15
Q16 · MCQ · Wave OpticsOpen
Given below are two statements :
Statement-I : A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave.
Statement-II : The curvature of a spherical wave emerging from a slit will increase for increasing slit width.
In the light of the above statements, choose the correct answer from the options given below.
1. (1)Both Statement-I and Statement-II are false.
2. (2)Both Statement-I and Statement-II are true.
3. (3)Statement-I is true but Statement-II is false.
4. (4)Statement-I is false but Statement-II is true.
Q17 · MCQ · Units and MeasurementsOpen
When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text{th}}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes 15 divisions on main scale and $5^{\text{th}}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is mm. (Least count of Vernier calliper $= 0.1$ mm)
1. (1)15.4
2. (2)15.1
3. (3)15.5
4. (4)15.9
We corrected a printing error in the original paper
Source printed “coincides with a min scale division”.
Q18 · MCQ · Moving Charges and MagnetismOpen
The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16\ \mu$ T. The magnetic field at a distance $x = \sqrt3 R$ on its axis from the centre is $\mu$ T.
1. (1) $2\sqrt2$
2. (2)4
3. (3)2
4. (4)8
Q19 · MCQ · Electric Potential and CapacitanceOpen
Two point charges of 1 nC and 2 nC are placed at the two corners of equilateral triangle of side 3 cm. The work done in bringing a charge of 3 nC from infinity to the third corner of the triangle is $\mu$ J. $\frac{1}{4\pi\epsilon_0} = 9\times10^{9}\ \mathrm{N.m^2/C^2}$
1. (1)2.7
2. (2)5.4
3. (3)3.3
4. (4)27
Q20 · MCQ · Newton's Laws of MotionOpen
A particle of mass m falls from rest through a resistive medium having resistive force, $F = -kv$ , where v is the velocity of the particle and k is a constant. Which of the following graphs represents velocity (v) versus time(t) ?
1. (1)
2. (2)
3. (3)
4. (4)
Q21 · Numerical · OscillationsOpen
The displacement of a particle, executing simple harmonic motion with time period $T$ , is expressed as $x(t) = \mathrm{A}\sin\omega t$ , where $A$ is the amplitude. The maximum value of potential energy of this oscillator is found at $t = T/2\beta$ . The value of $\beta$ is .
Q22 · Numerical · Dual Nature of Radiation and MatterOpen
The ratio of de Broglie wavelength of a deuteron with kinetic energy $E$ to that of an alpha particle with kinetic energy $2E$ , is $n:1$ . The value of $n$ is .
(Assume mass of proton $=$ mass of neutron) :
We corrected a printing error in the original paper
Source printed “deutron”.
Q23 · Numerical · Rotational MotionOpen
A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is $\sqrt{n}$ cm. The value of $n$ is
Q24 · Numerical · Ray OpticsOpen
A convex lens of refractive index 1.5 and focal length $f = 18$ cm is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha\times f$ . The value of $\alpha$ is .
(refractive index of water $= 4/3$ )
Q25 · Numerical · Current ElectricityOpen
The equivalent resistance between the points $A$ and $B$ in the following circuit is $\dfrac{x}{5}\,\Omega$ . The value of $x$ is .

Chemistry

Q1 · MCQ · ThermodynamicsOpen
20.0 dm $^3$ of an ideal gas ‘X' at 600 K and 0.5 MPa undergoes isothermal reversible expansion until pressure of the gas is 0.2 MPa. Which of the following option is correct ?
(Given: $\log 2 = 0.3010$ and $\log 5 = 0.6989$ )
1. (1) $w = -9.1$ kJ, $\Delta U = 0$ , $\Delta H = 0$ , $q = 9.1$ kJ
2. (2) $w = 9.1$ J, $\Delta U = 9.1$ J, $\Delta H = 0$ ; $q = 0$
3. (3) $w = +4.1$ kJ, $\Delta U = 0$ , $\Delta H = 0$ ; $q = -4.1$ kJ
4. (4) $w = -3.9$ kJ, $\Delta U = 0$ , $\Delta H = 0$ ; $q = 3.9$ kJ
Q2 · MCQ · Organic Basics (GOC)Open
CORRECT order of stability for the following is
$\ce{CH2=CH^-}$ , $\ce{CH3-CH2^-}$ , $\ce{CH\bond{3}C^-}$
1. (1) $\ce{CH3-CH2^-}$ $>$ $\ce{CH2=CH^-}$ $>$ $\ce{CH\bond{3}C^-}$
2. (2) $\ce{CH2=CH^-}$ $>$ $\ce{CH\bond{3}C^-}$ $>$ $\ce{CH3-CH2^-}$
3. (3) $\ce{CH\bond{3}C^-}$ $>$ $\ce{CH2=CH^-}$ $>$ $\ce{CH3-CH2^-}$
4. (4) $\ce{CH\bond{3}C^-}$ $>$ $\ce{CH3-CH2^-}$ $>$ $\ce{CH2=CH^-}$
Q3 · MCQ · SolutionsOpen
At T(K), 2 moles of liquid A and 3 moles of liquid B are mixed. The vapour pressure of ideal solution formed is 320 mm Hg. At this stage, one mole of A and one mole of B are added to the solution. The vapour pressure is now measured as 328.6 mm Hg. The vapour pressures (in mm Hg) of A and B are respectively:
1. (1)300, 200
2. (2)600, 400
3. (3)400, 300
4. (4)500, 200
We corrected a printing error in the original paper
Source printed “The vapour pressure (in mm Hg) of A and B are”.
Q4 · MCQ · Alcohols, Phenols & EthersOpen
Consider the following reaction sequence
Compound (y) develops characteristic colour with neutral $\ce{FeCl3}$ solution.
Identify the INCORRECT statement from the following for the above sequence.
1. (1)Both compounds x and y will dissolve in NaOH.
2. (2)Compound y will dissolve in $\ce{NaHCO3}$ and evolve a gas.
3. (3)Compound x is more acidic than compound y.
4. (4)Both compounds x and y will burn with sooty flame.
Q5 · MCQ · Purification & CharacterisationOpen
Method used for separation of mixture of products (B and C) obtained in the following reaction is:
1. (1)Simple distillation
2. (2)sublimation
3. (3)steam distillation
4. (4)fractional distillation
Q6 · MCQ · EquilibriumOpen
Consider a weak base ‘B' of $\mathrm{p}K_\mathrm{b} = 5.699$ . ‘x' mL of 0.02 M HCl and ‘y' mL of 0.02 M weak base ‘B' are mixed to make 100 mL of a buffer of pH 9 at 25 $^\circ$ C. The values of ‘x' and ‘y' respectively are:
[Given: $\log 2 = 0.3010$ , $\log 3 = 0.4771$ , $\log 5 = 0.699$ ]
1. (1)
  x y
  11.1 88.9
2. (2)
  x y
  42.7 57.3
3. (3)
  x y
  14.3 85.7
4. (4)
  x y
  85.7 14.3
Q7 · MCQ · Chemical KineticsOpen
An organic compound undergoes first order decomposition. The times taken for decomposition to $\left(\dfrac{1}{8}\right)^{\mathrm{th}}$ and $\left(\dfrac{1}{10}\right)^{\mathrm{th}}$ of its initial concentration are $t_{1/8}$ and $t_{1/10}$ respectively.
What is the value of $\dfrac{t_{1/8}}{t_{1/10}}\times 10$ ?
( $\log 2 = 0.3$ )
1. (1)9
2. (2)0.9
3. (3)3
4. (4)30
We corrected a printing error in the original paper
Source printed “The time taken for decomposition to ... are”.
Q8 · MCQ · HydrocarbonsOpen
Given below are two statements for the following reaction sequence.
Statement I: Compound ‘Z' will give yellow precipitate with NaOI.
Statement II: Compound ‘Q' has two different types of ‘H' atoms (aromatic : aliphatic) in the ratio 1 : 3.
In the light of the above statements, choose the correct answer from the option given below:
1. (1)Statement I is true but Statement II is false
2. (2)Both Statement I and Statement II are true
3. (3)Statement I is false but Statement II is true
4. (4)Both Statement I and Statement II are false
Q9 · MCQ · Aldehydes, Ketones & AcidsOpen
Given below are the four isomeric compounds (P, Q, R, S)
Identify correct statements from below.
- (A)Q, R and S will give precipitate with 2, 4 - DNP.
- (B)P and Q will give positive Baeyer's test.
- (C)Q and R will give sooty flame.
- (D)R and S will give yellow precipitate with $\ce{I2}$ /NaOH.
- (E)Q alone will deposit silver with Tollen's reagent
Choose the correct option.
1. (1)A, C and E only
2. (2)A and E only
3. (3)C and E only
4. (4)A, B, D and E only
We corrected a printing error in the original paper
Source printed “Bayer's test”.
Q10 · MCQ · Chemical BondingOpen
Given below are two statements:
Statement I: The number of species among $\ce{BF4^-}$ , $\ce{SiF4}$ , $\ce{XeF4}$ and $\ce{SF4}$ , that have unequal E–F bond lengths is two. Here, E is the central atom.
Statement II: Among $\ce{O2^-}$ , $\ce{O2^{2-}}$ , $\ce{F2}$ and $\ce{O2^+}$ , $\ce{O2^-}$ has the highest bond order.
In the light of the above statements, choose the correct answer from the options given below
1. (1)Both Statement I and Statement II are false
2. (2)Both Statement I and Statement II are true
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
Q11 · MCQ · p-Block (15–18)Open
Regarding the hydrides of group 15 elements $\ce{EH3}$ (E = N, P, As, Sb), select the correct statement from the following:
- (A)The stability of hydrides decreases down the group.
- (B)The basicity of hydrides decreases down the group.
- (C)The reducing character increases down the group.
- (D)The boiling point increases down the group.
Choose the correct answer from the options given below:
1. (1)A, B & C only
2. (2)A & D only
3. (3)A, B, C & D
4. (4)B & C only
Q12 · MCQ · Practical / Qualitative AnalysisOpen
Given below are two statements:
Statement I: Griess–Ilosvay test is used for the detection of nitrite ion, which involves the use of sulphanilic acid and $\alpha$ -naphthylamine reagent.
Statement II: In the above test, sulphanilic acid is diazotized by the acidified nitrite ion, which on further coupling with $\alpha$ -naphthylamine forms an azo-dye.
In the light of the above statements, choose the correct answer from the options given below
1. (1)Statement I is false but Statement II is true
2. (2)Both Statement I and Statement II are true
3. (3)Statement I is true but Statement II is false
4. (4)Both Statement I and Statement II are false
We corrected a printing error in the original paper
Source printed “Griss–Ilosvay test”.
Q13 · MCQ · AminesOpen
Consider the following reactions giving major product. Identify the correct reaction.
1. (1)
2. (2)
3. (3)
4. (4)
Q14 · MCQ · BiomoleculesOpen
In the given pentapeptide, find out an essential amino acid (Y) and the sequence present in the pentapeptide:
Choose the correct answer from the options given below:
1. (1)
  (Y) (Sequence)
  Threonine Ser – Thr – Asp – Gly – Ala
2. (2)
  (Y) (Sequence)
  Serine Thr – Ser – Asp – Ala – Gly
3. (3)
  (Y) (Sequence)
  Threonine Thr – Ser – Asp – Gly – Ala
4. (4)
  (Y) (Sequence)
  Serine Ser – Asp – Thr – Ala – Gly
Q15 · MCQ · HydrocarbonsOpen
$\ce{Ph-CH=CH2 ->[(PhCOO)2][HBr] Product}$
Consider the above reaction
- (A)The reaction proceeds through a more stable radical intermediate.
- (B)The role of peroxide is to generate $\dot{\mathrm{H}}$ (Hydrogen radical).
- (C)During this reaction, benzene is formed as a by-product.
- (D)1-Bromo-2-phenylethane is formed as the minor product.
- (E)The same reaction in absence of peroxide proceeds via carbocation intermediate.
Identify the correct statements. Choose the correct answer from the options given below:
1. (1)A & E Only
2. (2)A, B & D Only
3. (3)C, D & E Only
4. (4)A, C & E Only
We corrected a printing error in the original paper
Source printed “benzene is formed as a biproduct” and “1-Bromo-2-phenylethane is fanned as the minor product”.
Q16 · MCQ · Structure of AtomOpen
The wave numbers of three spectral lines of H atom are considered. Identify the set of spectral lines belonging to Balmer series.
( $R$ = Rydberg constant)
1. (1) $\dfrac{5R}{36}, \dfrac{3R}{16}, \dfrac{21R}{100}$
2. (2) $\dfrac{5R}{36}, \dfrac{8R}{9}, \dfrac{15R}{16}$
3. (3) $\dfrac{7R}{144}, \dfrac{3R}{16}, \dfrac{16R}{255}$
4. (4) $\dfrac{3R}{4}, \dfrac{3R}{16}, \dfrac{7R}{144}$
Q17 · MCQ · Structure of AtomOpen
Figure 1. electron probability density for 2s orbital
Figure 2. wave function for 2s orbital
Which of the following point in Figure 2 most accurately represents the nodal surface as shown in Figure 1 ?
1. (1)B
2. (2)D
3. (3)C
4. (4)A
Q18 · MCQ · d- and f-BlockOpen
Given below are two statements :
Statement I : The number of pairs, from the following, in which both the ions are coloured in aqueous solution is 3.
[ $\ce{Sc^{3+}}$ , $\ce{Ti^{3+}}$ ], [ $\ce{Mn^{2+}}$ , $\ce{Cr^{2+}}$ ], [ $\ce{Cu^{2+}}$ , $\ce{Zn^{2+}}$ ] and [ $\ce{Ni^{2+}}$ , $\ce{Ti^{4+}}$ ]
Statement II : $\ce{Th^{4+}}$ is the strongest reducing agent among $\ce{Th^{4+}}$ , $\ce{Ce^{4+}}$ , $\ce{Gd^{3+}}$ and $\ce{Eu^{2+}}$ .
In the light of the above statements, choose the correct answer from the options given below
1. (1)Statement I is true but Statement II is false
2. (2)Statement I is false but Statement II is true
3. (3)Both Statement I and Statement II are false
4. (4)Both Statement I and Statement II are true
Q19 · MCQ · Classification & PeriodicityOpen
In period 4 of the periodic table, the elements with highest and lowest atomic radii are respectively.
1. (1)Na & Cl
2. (2)K & Se
3. (3)K & Br
4. (4)Rb & Br
Q20 · MCQ · Coordination CompoundsOpen
The correct statement among the following is :
1. (1) $\ce{[Ni(CN)4]^{2-}}$ and $\ce{[NiCl4]^{2-}}$ are diamagnetic and $\ce{Ni(CO)4}$ is paramagnetic.
2. (2) $\ce{Ni(CO)4}$ and $\ce{[NiCl4]^{2-}}$ are diamagnetic and $\ce{[Ni(CN)4]^{2-}}$ is paramagnetic.
3. (3) $\ce{Ni(CO)4}$ and $\ce{[Ni(CN)4]^{2-}}$ are diamagnetic and $\ce{[NiCl4]^{2-}}$ is paramagnetic.
4. (4) $\ce{Ni(CO)4}$ is diamagnetic and $\ce{[NiCl4]^{2-}}$ and $\ce{[Ni(CN)4]^{2-}}$ are paramagnetic.
Q21 · Numerical · Redox ReactionsOpen
500 mL of 1.2 M KI solution is mixed with 500 mL of 0.2 M $\ce{KMnO4}$ solution in basic medium. The liberated iodine was titrated with standard 0.1 M $\ce{Na2S2O3}$ solution in the presence of starch indicator till the blue color disappeared. The volume (in L) of $\ce{Na2S2O3}$ consumed is . (Nearest integer)
Q22 · Numerical · Redox ReactionsOpen
Consider the following redox reaction taking place in acidic medium
$\ce{BH4^-(aq) + ClO3^-(aq) -> H2BO3^-(aq) + Cl^-(aq)}$
If the Nernst equation for the above balanced reaction is $E_\mathrm{cell} = E^\circ_\mathrm{cell} - \frac{RT}{nF}\ln Q,$ Then the value of $n$ is . (Nearest integer)
Q23 · Numerical · Coordination CompoundsOpen
X is the number of geometrical isomers exhibited by $\ce{[Pt(NH3)(H2O)BrCl]}$ .
Y is the number of optically inactive isomer(s) exhibited by $\ce{[CrCl2(ox)2]^{3-}}$
Z is the number of geometrical isomers exhibited by $\ce{[Co(NH3)3(NO2)3]}$
The value of X + Y + Z is .
Q24 · Numerical · Purification & CharacterisationOpen
0.53 g of an organic compound (x) when heated with excess of nitric acid (concentrated) and then with silver nitrate gave 0.75 g of silver bromide precipitate. 1.0 g of (x) gave 1.32 g of $\ce{CO2}$ gas on combustion. The percentage of hydrogen in the compound (x) is %. [Nearest Integer]
[Given : Molar mass in g mol $^{-1}$ H : 1, C : 12, Br : 80, Ag : 108, O : 16; Compound (x) : $\ce{C_{x}H_{y}Br_{z}}$ ]
Q25 · Numerical · EquilibriumOpen
Consider the dissociation equilibrium of the following weak acid $\ce{HA <=> H^+(aq) + A^-(aq)}$
If the $\mathrm{p}K_\mathrm{a}$ of the acid is 4, then the pH of 10 mM HA solution is . (Nearest integer)
[Given : The degree of dissociation can be neglected with respect to unity]

Maths

Q1 · MCQ · Sets, Relations & FunctionsOpen
If $g(x) = 3x^2 + 2x - 3$ , $f(0) = -3$ and $4g(f(x)) = 3x^2 - 32x + 72$ , where $f$ is a differentiable function, then $f(g(2))$ is equal to:
1. (1) $\dfrac{25}{6}$
2. (2) $-\dfrac{25}{6}$
3. (3) $\dfrac{7}{2}$
4. (4) $-\dfrac{7}{2}$
We corrected a printing error in the original paper
Assumption “where $f$ is a differentiable function” added; the source leaves it unstated, and without it $f$ may change at $x = 16/3$ to the other root $(14 - 3x)/6$ of the equation, which gives $f(13) = -25/6$ (option 2).
Q2 · MCQ · Sequences & SeriesOpen
The value of $\displaystyle\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is:
1. (1) $2/e$
2. (2) $1/e$
3. (3) $\sqrt{e}$
4. (4) $e/2$
Q3 · MCQ · CirclesOpen
Let $y = x$ be the equation of a chord of the circle $C_1$ (in the closed half-plane $x \ge 0$ ) of diameter 10 passing through the origin. Let $C_2$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_2$ , which passes through the point $(2, 3)$ and is farthest from the center of $C_2$ , is $x + ay + b = 0$ , then $a - b$ is equal to:
1. (1)10
2. (2) $-6$
3. (3) $-2$
4. (4)6
Q4 · MCQ · TrigonometryOpen
If $\dfrac{\tan(A - B)}{\tan A} + \dfrac{\sin^2 C}{\sin^2 A} = 1$ , $A, B, C \in \left(0, \dfrac{\pi}{2}\right)$ , then
1. (1) $\tan A$ , $\tan C$ , $\tan B$ are in G.P.
2. (2) $\tan A$ , $\tan B$ , $\tan C$ are in G.P.
3. (3) $\tan A$ , $\tan C$ , $\tan B$ are in A.P.
4. (4) $\tan A$ , $\tan B$ , $\tan C$ are in A.P.
Q5 · MCQ · Complex NumbersOpen
Let $z$ be a complex number such that $\lvert z - 6\rvert = 5$ and $\lvert z + 2 - 6i\rvert = 5$ . Then the value of $z^3 + 3z^2 - 15z + 141$ is equal to
1. (1)42
2. (2)37
3. (3)50
4. (4)61
Q6 · MCQ · Straight LinesOpen
Let $ABC$ be an equilateral triangle with orthocenter at the origin and the side $BC$ on the line $x + 2\sqrt2\,y = 4$ . If the co-ordinates of the vertex $A$ are $(\alpha, \beta)$ , then the greatest integer less than or equal to $\lvert\alpha + \sqrt2\,\beta\rvert$ is
1. (1)2
2. (2)3
3. (3)5
4. (4)4
Q7 · MCQ · Permutations & CombinationsOpen
Let $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ . Let $x$ be the number of 9-digit numbers formed using the digits of the set $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of the set $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then,
1. (1) $29x = 5y$
2. (2) $45x = 7y$
3. (3) $21x = 4y$
4. (4) $56x = 9y$
Q8 · MCQ · Quadratic EquationsOpen
Let $S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\}$ be a set of polynomials. Then the number of polynomials in $S$ , which are divisible by $x^2 + 2$ , is
1. (1)20
2. (2)6
3. (3)120
4. (4)10
Q9 · MCQ · ProbabilityOpen
A bag contains 10 balls out of which $k$ are red and $(10 - k)$ are black, where $0 \le k \le 10$ . If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:
1. (1) $\dfrac{7}{11}$
2. (2) $\dfrac{7}{55}$
3. (3) $\dfrac{7}{110}$
4. (4) $\dfrac{14}{55}$
We corrected a printing error in the original paper
Source printed “the bag contains l red and 9 black balls” (letter l for the digit 1).
Q10 · MCQ · Quadratic EquationsOpen
If $\alpha, \beta$ , where $\alpha < \beta$ , are the roots of the equation $\lambda x^2 - (\lambda + 3)x + 3 = 0$ such that $\dfrac{1}{\alpha} - \dfrac{1}{\beta} = \dfrac{1}{3}$ , then the sum of all possible values of $\lambda$ is:
1. (1)6
2. (2)2
3. (3)4
4. (4)8
Q11 · MCQ · Indefinite IntegrationOpen
If $\displaystyle\int\left(\frac{1 - 5\cos^2 x}{\sin^5 x\cos^2 x}\right)dx = f(x) + C$ where $C$ is the constant of integration, then $f\left(\dfrac{\pi}{6}\right) - f\left(\dfrac{\pi}{4}\right)$ is equal to
1. (1) $\dfrac{1}{\sqrt3}\left(26 + \sqrt3\right)$
2. (2) $\dfrac{4}{\sqrt3}\left(8 - \sqrt6\right)$
3. (3) $\dfrac{1}{\sqrt3}\left(26 - \sqrt3\right)$
4. (4) $\dfrac{2}{\sqrt3}\left(4 + \sqrt6\right)$
Q12 · MCQ · Definite IntegrationOpen
Let $f$ be a polynomial function such that $f(x^2 + 1) = x^4 + 5x^2 + 2$ , for all $x \in \mathbb{R}$ . Then $\displaystyle\int_0^3 f(x)\,dx$ is equal to
1. (1) $\dfrac{41}{3}$
2. (2) $\dfrac{33}{2}$
3. (3) $\dfrac{27}{2}$
4. (4) $\dfrac{5}{3}$
Q13 · MCQ · Area under CurvesOpen
The area of the region $R = \{(x, y) : xy \le 8,\ 1 \le y \le x^2,\ x \ge 0\}$ is
1. (1) $\dfrac{1}{3}\left(49\log_e(2) - 15\right)$
2. (2) $\dfrac{2}{3}\left(20\log_e(2) + 9\right)$
3. (3) $\dfrac{2}{3}\left(24\log_e(2) - 7\right)$
4. (4) $\dfrac{1}{3}\left(40\log_e(2) + 27\right)$
Q14 · MCQ · Limits, Continuity & DifferentiabilityOpen
The value of $\lim_{x\to0}\frac{\log_e\left(\sec(ex)\cdot\sec\left(e^2x\right)\cdot\ldots\cdot\sec\left(e^{10}x\right)\right)}{e^2 - e^{2\cos x}}$ is equal to
1. (1) $\dfrac{(e^{10} - 1)}{2e^2(e^2 - 1)}$
2. (2) $\dfrac{(e^{20} - 1)}{2e^2(e^2 - 1)}$
3. (3) $\dfrac{(e^{20} - 1)}{2(e^2 - 1)}$
4. (4) $\dfrac{(e^{10} - 1)}{2(e^2 - 1)}$
Q15 · MCQ · StatisticsOpen
The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations is
1. (1)5
2. (2)4
3. (3)6
4. (4)7
Q16 · MCQ · Matrices & DeterminantsOpen
Let $A$ , $B$ and $C$ be three $2\times2$ matrices with real entries such that $B = (I + A)^{-1}$ and $A + C = I$ . If $BC = \begin{bmatrix} 1 & -5\\ -1 & 2\end{bmatrix} \quad\text{and}\quad CB\begin{bmatrix} x_1\\ x_2\end{bmatrix} = \begin{bmatrix} 12\\ -6\end{bmatrix},$ then $x_1 + x_2$ is
1. (1)2
2. (2)0
3. (3) $-2$
4. (4)4
Q17 · MCQ · Sequences & SeriesOpen
The common difference of the A.P.: $a_1, a_2, \ldots, a_m$ is 13 more than the common difference of the A.P.: $b_1, b_2, \ldots, b_n$ . If $b_{31} = -277$ , $b_{43} = -385$ and $a_{78} = 327$ , then $a_1$ is equal to
1. (1)21
2. (2)24
3. (3)19
4. (4)16
Q18 · MCQ · 3D GeometryOpen
If the distances of the point $(1, 2, a)$ from the line $\dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1}$ along the lines $L_1 : \frac{x-1}{3} = \frac{y-2}{4} = \frac{z-a}{b} \quad\text{and}\quad L_2 : \frac{x-1}{1} = \frac{y-2}{4} = \frac{z-a}{c}$ are equal, then $a + b + c$ is equal to
1. (1)7
2. (2)5
3. (3)6
4. (4)4
Q19 · MCQ · VectorsOpen
For three unit vectors $\vec a, \vec b, \vec c$ satisfying $\lvert\vec a - \vec b\rvert^2 + \lvert\vec b - \vec c\rvert^2 + \lvert\vec c - \vec a\rvert^2 = 9$ and $\lvert 2\vec a + k\vec b + k\vec c\rvert = 3$ , the positive value of $k$ is:
1. (1)3
2. (2)6
3. (3)4
4. (4)5
Q20 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation $x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\quad x \ne 0.$ If $y(2) = 0$ , then $\tan(y(1))$ is equal to
1. (1) $\dfrac{3}{4}$
2. (2) $\dfrac{7}{4}$
3. (3) $-\dfrac{7}{4}$
4. (4) $-\dfrac{3}{4}$
We corrected a printing error in the original paper
Source printed “If $y(2) = x$ ”; corrected to the intended “ $y(2) = 0$ ” (the source's own solution uses $y(2) = 0$ , and the key (2) follows from it).
Q21 · Numerical · Sequences & SeriesOpen
In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R} - (a, b)$ , then $a^2 + b^2$ is equal to .
Q22 · Numerical · Conic SectionsOpen
For some $\theta \in \left(0, \dfrac{\pi}{2}\right)$ , let the eccentricity and the length of the latus rectum of the hyperbola $x^2 - y^2\sec^2\theta = 8$ be $e_1$ and $\ell_1$ , respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta + y^2 = 6$ be $e_2$ and $\ell_2$ , respectively. If $e_1^2 = e_2^2\left(\sec^2\theta + 1\right)$ , then $\left(\dfrac{\ell_1\ell_2}{e_1e_2}\right)\tan^2\theta$ is equal to .
Q23 · Numerical · Inverse TrigonometryOpen
If $k = \tan\left(\frac{\pi}{4} + \frac12\cos^{-1}\left(\frac23\right)\right) + \tan\left(\frac12\sin^{-1}\left(\frac23\right)\right)$ then the number of solutions of the equation $\sin^{-1}(kx - 1) = \sin^{-1}x - \cos^{-1}x$ is .
Q24 · Numerical · Definite IntegrationOpen
The value of $\displaystyle\sum_{r=1}^{20}\left(\left\lvert\sqrt{\pi\left(\int_0^r x\,\lvert\sin\pi x\rvert\,dx\right)}\right\rvert\right)$ is .
Q25 · Numerical · VectorsOpen
Let $PQR$ be a triangle such that $\overrightarrow{PQ} = -2\hat i - \hat j + 2\hat k$ and $\overrightarrow{PR} = a\hat i + b\hat j - 4\hat k$ , $a, b \in \mathbb{Z}$ . Let $S$ be the point on $QR$ , which is equidistant from the lines $PQ$ and $PR$ . If $\lvert\overrightarrow{PR}\rvert = 9$ and $\overrightarrow{PS}$ is parallel to $\hat i - 7\hat j + 2\hat k$ , then the value of $3a - 4b$ is .
We corrected a printing error in the original paper
Source printed “ $\overrightarrow{PS} = \hat i - 7\hat j + 2\hat k$ ”; corrected to the intended “ $\overrightarrow{PS}$ is parallel to $\hat i - 7\hat j + 2\hat k$ ”: the printed vector has length $3\sqrt6$ and puts $S$ off the line $QR$ (no such triangle; the source marks the question Bonus), while its direction is the internal bisector for $a = 7$ , $b = -4$ , which the key 37 uses.