JEE Main 4 Apr 2026, Shift 1

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

Physics

Q1 · MCQ · Units and MeasurementsOpen
In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm. If the circular scale has 100 divisions, the least count of screw gauge is mm.
1. (1) $1\times10^{-2}$
2. (2) $1\times10^{-3}$
3. (3) $5\times10^{-2}$
4. (4) $5\times10^{-3}$
Q2 · MCQ · Mechanical Properties of SolidsOpen
The increase in the pressure required to decrease the volume ( $\Delta V$ ) of water is $6.3\times10^{7}\ \mathrm{N/m^2}$ . The percentage decrease in the volume is . (Bulk modulus of water $= 2.1\times10^{9}\ \mathrm{N/m^2}$ .)
1. (1)2 %
2. (2)3 %
3. (3)6 %
4. (4)4 %
Q3 · MCQ · Newton's Laws of MotionOpen
The time taken by a block of mass m to slide down from the highest point to the lowest point on a rough inclined plane is 50 % more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at $45^\circ$ with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is .
1. (1)3/4
2. (2)2/3
3. (3)5/9
4. (4)4/9
Q4 · MCQ · NucleiOpen
Two nuclei of mass number 3 combine with another nucleus of mass number 4 to yield a nucleus of mass number 10. If the binding energy per nucleon for the mass number 3, 4 and 10 are 5.6 MeV, 7.4 MeV and 6.1 MeV, respectively, then in the process, $\Delta Mc^2 =$ MeV.
1. (1)6.9
2. (2)7.9
3. (3)2.2
4. (4)4.3
Q5 · MCQ · Rotational MotionOpen
A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$ . If $I_1$ is moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1 =$ .
1. (1)35
2. (2)70
3. (3)140
4. (4)210
We corrected a printing error in the original paper
Source printed “mas”.
Q6 · MCQ · Motion in a PlaneOpen
The two projectiles are projected with the same initial velocities at the $15^\circ$ and $30^\circ$ with respect to the horizontal. The ratio of their ranges is $1 : x$ . The value of $x$ is
1. (1) $\sqrt2$
2. (2) $\sqrt3$
3. (3) $2\sqrt3$
4. (4) $\dfrac{1}{\sqrt2}$
Q7 · MCQ · Dual Nature of Radiation and MatterOpen
The graph shows variation of stopping potential $V_o$ with the frequency $\nu$ of the incident radiation for three photosensitive metals $X_1$ , $X_2$ and $X_3$ . Which metal will give out electrons with greater kinetic energy, for the same wavelength of incident radiation?
1. (1) $X_1$
2. (2) $X_2$
3. (3) $X_3$
4. (4)All the metals will give out photo electrons with same kinetic energies.
Q8 · MCQ · Wave OpticsOpen
A slit of width $a$ is illuminated by light of wavelength $\lambda$ . The linear separation between $1^{\text{st}}$ and $3^{\text{rd}}$ minima in the diffraction pattern produced on a screen placed at a distance $D$ from the slit system is .
1. (1) $\dfrac{D\lambda}{a}$
2. (2) $1.5\dfrac{D\lambda}{a}$
3. (3) $2\dfrac{D\lambda}{a}$
4. (4) $3\dfrac{D\lambda}{a}$
Q9 · MCQ · Mechanical Properties of SolidsOpen
A string A of length 0.314 m of Young's modulus $2\times10^{10}\ \mathrm{N/m^2}$ is connected to another string B of length and Young's modulus both twice of those of A. This series combination of strings is then suspended from a rigid support and its free end is fixed to a load of mass 0.8 kg. The net change in length of the combination is mm.
(radius of both the strings is 0.2 mm and acceleration due to gravity $= 10\ \mathrm{m/s^2}$ )
(Mass of both strings is to be neglected as compared to the mass of load)
1. (1)3
2. (2)2
3. (3)1.9
4. (4)1
Q10 · MCQ · Kinetic Theory of GasesOpen
One gas of $n_1$ mole of molecules at temperature $T_1$ , volume $V_1$ , and pressure $P_1$ , and another gas of $n_2$ mole of molecules at temperature $T_2$ , volume $V_2$ , and pressure $P_2$ , are mixed resulting in pressure $P$ and volume $V$ of the mixture. The temperature of the mixture is .
1. (1) $(T_1 + T_2)/2$
2. (2) $T_1T_2PV/(T_2P_1V_1 + T_1P_2V_2)$
3. (3) $(T_2P_1V_1 + T_1P_2V_2)/(T_1T_2PV)$
4. (4) $|T_1 - T_2|/2$
We corrected a printing error in the original paper
Source printed option (3) as “ $(T_2P_1V_1 + T_1P_2V_2)\,(T_1T_2PV)$ ” (the “/” missing).
Q11 · MCQ · Kinetic Theory of GasesOpen
An ideal gas undergoes a process maintaining relation between pressure ( $P$ ) and volume ( $V$ ) as $P = P_o\left(1 + \left(\dfrac{V_o}{V}\right)^{2}\right)^{-1}$ , where $P_o$ and $V_o$ are constants. If two samples A and B (two moles each) with initial volume $V_o$ and $3V_o$ respectively undergo above mentioned process, then the difference of the initial temperatures of these samples, $T_B - T_A$ is
( $R$ = gas constant)
1. (1) $\dfrac{9P_oV_o}{8R}$
2. (2) $\dfrac{11P_oV_o}{10R}$
3. (3) $\dfrac{7P_oV_o}{6R}$
4. (4) $\dfrac{13P_oV_o}{11R}$
We corrected a printing error in the original paper
Source printed “undergo above mentioned process and attain same pressure, then the difference at the temperatures”; corrected to the intended “undergo above mentioned process, then the difference of the initial temperatures” (two samples on the same $P(V)$ curve at the same pressure would have equal temperatures; the key is $T_B - T_A$ at $V_o$ and $3V_o$ ).
Q12 · MCQ · Moving Charges and MagnetismOpen
A voltmeter with internal resistance of $x\ \Omega$ can be used to measure upto 20 V. In order to increase its measuring range to 30 V, the required modification is to .
1. (1)connect resistor of $\dfrac{x}{2}\,\Omega$ , in series with voltmeter.
2. (2)connect resistor of $\dfrac{x}{2}\,\Omega$ , in parallel to voltmeter.
3. (3)connect resistor of $x\ \Omega$ , in series with voltmeter.
4. (4)connect resistor of $2x\ \Omega$ , in parallel to voltmeter.
Q13 · MCQ · Semiconductor ElectronicsOpen
Two 4 bits binary numbers, $A = 1101$ and $B = 1010$ are given in the inputs of a logic circuit shown in figure below. The output ( $Y$ ) will be :
1. (1) $Y = 1101$
2. (2) $Y = 0010$
3. (3) $Y = 0111$
4. (4) $Y = 1000$
Q14 · MCQ · Ray OpticsOpen
A rod of length 10 cm lies along the principal axis of a concave mirror of focal length 10 cm as shown in figure. The length of the image is cm.
1. (1)2.5
2. (2)5
3. (3)7.5
4. (4)7
We corrected a printing error in the original paper
Source printed “principle axis”.
Q15 · MCQ · Electric Potential and CapacitanceOpen
A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed $v$ . If $x$ is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to $x^{\alpha}$ , where $\alpha$ is .
1. (1) $-2$
2. (2)1
3. (3) $-1$
4. (4)2
Q16 · MCQ · Moving Charges and MagnetismOpen
An insulated wire is wound so that it forms a flat coil with $N = 200$ turns. The radius of the innermost turn is $r_1 = 3$ cm, and of the outermost turn $r_2 = 6$ cm. If 20 mA current flows in it then the magnetic moment will be $\alpha\times10^{-2}\ \mathrm{A\,m^2}$ . The value of $\alpha$ is .
1. (1)4.4
2. (2)2.64
3. (3)3.25
4. (4)1.2
We corrected a printing error in the original paper
Source printed “If 20 mA current flow in it”.
Q17 · MCQ · Semiconductor ElectronicsOpen
Consider a circuit consisting of a capacitor ( $20\ \mu$ F), resistor ( $100\ \Omega$ ) and two identical diodes as shown in figure. The resistance of diode under forward biasing condition is $10\ \Omega$ . The time constant of the circuit is $\alpha\times10^{-3}$ s. The value of $\alpha$ is .
1. (1)2.2
2. (2)2.0
3. (3)2.1
4. (4)2.4
Q18 · MCQ · Current ElectricityOpen
The voltage and the current between A and B points shown in the circuit are .
1. (1)24 V, 12 A
2. (2)24 V, 4 A
3. (3)18 V, 12 A
4. (4)27 V, 4 A
Q19 · MCQ · Ray OpticsOpen
A telescope with objective diameter $R$ is used to observe a distant star emitting light of wavelength 500 nm, at a resolution of $5\times10^{-7}$ radian. The value of $R$ is cm.
1. (1)61
2. (2)122
3. (3)244
4. (4)305
Q20 · MCQ · Wave OpticsOpen
An unpolarized light is incident on the plane interface of air-dielectric medium shown in figure. If the incident angle is equal to Brewster angle, identify the expression representing reflected wave.
1. (1) $E_y\hat j\sin(kx - kz - \omega t)$
2. (2) $(E_x\hat i + E_z\hat k)\sin(kx + ky - \omega t)$
3. (3) $(E_y\hat j + E_z\hat k)\sin(ky + kz - \omega t)$
4. (4) $(E_x\hat i + E_y\hat j + E_z\hat k)\sin(kx + ky - kz - \omega t)$
We corrected a printing error in the original paper
Source printed option (1) as “ $(E_x\hat i + E_y\hat j)\sin(kx - kz - \omega t)$ ”; corrected to the intended “ $E_y\hat j\sin(kx - kz - \omega t)$ ” (at the Brewster angle the reflected light is polarised perpendicular to the plane of incidence $xz$ , and an $E_x\hat i$ component is not transverse to the reflected direction); source printed option (3) as “ $(E_x\hat j + E_y\hat k)$ ”.
Q21 · Numerical · Work, Energy and PowerOpen
A 1 kg block subjected to two simultaneous forces $(2\hat i + 3\hat j + 4\hat k)$ N and $(3\hat i - \hat j - 2\hat k)$ N is moved a distance of 25 m along $(3\hat i - 4\hat j)$ direction. The work done in this process is J.
Q22 · Numerical · Mechanical Properties of FluidsOpen
The surface tension of a soap solution is $3.5\times10^{-2}$ N/m. The work required to increase the radius of a soap bubble from 1 cm to 2 cm is $\alpha\times10^{-6}$ J. The value of $\alpha$ is . ( $\pi = 22/7$ )
Q23 · Numerical · OscillationsOpen
The velocity of a particle executing simple harmonic motion along x-axis is described as $v^2 = 50 - x^2$ , where $x$ represents displacement. If the time period of motion is $\dfrac{\alpha}{7}$ s, the value of $\alpha$ is .
We corrected a printing error in the original paper
Source printed the period as “ $\dfrac{x}{7}$ s, the value of $x$ ” ( $x$ already denotes the displacement).
Q24 · Numerical · Work, Energy and PowerOpen
A body of mass 2 kg begins to move under the influence of time dependent force $\vec F = (2t\hat i + 6t^2\hat j)$ N, where $\hat i$ and $\hat j$ are unit vectors along x and y-axis respectively. The power produced by the force at $t = 2$ s is W.
Q25 · Numerical · Alternating CurrentOpen
An inductor, a capacitor and a resistor of $100\ \Omega$ are connected in series across an a.c power supply 220 V, 70 Hz. The power factor of the given circuit is 0.5. The difference in the inductive reactance and capacitance reactance is $\sqrt3\,\alpha\ \Omega$ . The value of $\alpha$ is .
We corrected a printing error in the original paper
Source printed “An inductor of 10 mH, capacitor of $0.1\ \mu$ F and a resistor of $100\ \Omega$ are connected in series aross”; corrected to the intended “An inductor, a capacitor and a resistor of $100\ \Omega$ are connected in series across” (10 mH and $0.1\ \mu$ F at 70 Hz give a power factor of about 0.004, contradicting the stated 0.5 on which the key rests).

Chemistry

Q1 · MCQ · Some Basic ConceptsOpen
Number of moles and number of molecules in 1.4187 L of $\ce{SO2}$ at STP respectively are
1. (1)0.1266; $3.812\times10^{22}$
2. (2)0.0633; $3.812\times10^{22}$
3. (3)0.1266; $7.6238\times10^{22}$
4. (4)0.0633; $7.6238\times10^{22}$
Q2 · MCQ · Structure of AtomOpen
What is the ratio of wave number of first line (lowest energy line) of Balmer series of H atomic spectrum to first line of its Brackett series?
1. (1)5:1
2. (2)5:0.81
3. (3)5:1.75
4. (4)5:27
Q3 · MCQ · Structure of AtomOpen
Which of the following is correct set of 4 quantum number of 19 $^\text{th}$ electron in Chromium (Atomic number $=$ 24) in accordance with Aufbau principle?
1. (1) $n = 3$ , $\ell = 2$ , $m = +2$ , $s = +\dfrac{1}{2}$
2. (2) $n = 3$ , $\ell = 2$ , $m = -2$ , $s = +\dfrac{1}{2}$
3. (3) $n = 4$ , $\ell = 1$ , $m = 0$ , $s = +\dfrac{1}{2}$
4. (4) $n = 4$ , $\ell = 0$ , $m = 0$ , $s = +\dfrac{1}{2}$
Q4 · MCQ · ThermodynamicsOpen
Given below are two statements:
Statement I : For an ideal gas, heat capacity at constant volume is always greater than the heat capacity at constant pressure.
Statement II : In a constant volume process, no work is produced and all the heat withdrawn goes into the chaotic motion and is reflected by a temperature increase of the ideal gas.
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 true
2. (2)Both Statement I and Statement II are false
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
Q5 · MCQ · EquilibriumOpen
At T(K), the equilibrium constant of
$\ce{A2(g) + B2(g) <=> C(g)}$ is $2.7\times10^{-5}$ .
What is the equilibrium constant for
$\ce{1/3 A2(g) + 1/3 B2(g) <=> 1/3 C(g)}$ at the same temperature?
1. (1) $(2.7\times10^{-5})^{3}$
2. (2) $6\times10^{-2}$
3. (3) $\sqrt{2.7\times10^{-5}}$
4. (4) $3\times10^{-2}$
Q6 · MCQ · Redox ReactionsOpen
In order to oxidise a mixture of 1 mole each of $\ce{FeC2O4}$ , $\ce{Fe2(C2O4)3}$ , $\ce{FeSO4}$ and $\ce{Fe2(SO4)3}$ in acidic medium, the number of moles of $\ce{KMnO4}$ required is :
1. (1)3
2. (2)2
3. (3)5
4. (4)7
Q7 · MCQ · Chemical KineticsOpen
Consider the first order reaction $\ce{R -> P}$ .
The fraction of molecules decomposed in the given first order reaction can be expressed as
1. (1) $1-e^{k_1t}$
2. (2) $1+e^{k_1t}$
3. (3) $1+e^{-k_1t}$
4. (4) $1-e^{-k_1t}$
Q8 · MCQ · Structure of AtomOpen
A monoatomic anion ( $\ce{A^-}$ ) has 45 neutrons and 36 electrons. Atomic mass, group in the periodic table and physical state at room temperature of the element (A) respectively are :
1. (1)80, 17, liquid
2. (2)81, 16, solid
3. (3)80, 16, gas
4. (4)81, 15, gas
Q9 · MCQ · p-Block (15–18)Open
Given below are two statements :
Statement I : The covalency of oxygen is generally two but it can exceed upto four. The oxidation state of oxygen in $\ce{SO2}$ is $-2$ and in $\ce{OF2}$ it is $+2$ .
Statement II : The anomalous behaviour of oxygen when compared to the other elements of group 16 is due to its small size and high electronegativity.
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 true
2. (2)Both Statement I and Statement II are false
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
Q10 · MCQ · d- and f-BlockOpen
The correct statements among the following are,
- (A)Mo(VI) and W(VI) are less stable than Cr(VI).
- (B) $\ce{Ce^{4+}}$ and $\ce{Tb^{4+}}$ are oxidant while $\ce{Eu^{2+}}$ and $\ce{Yb^{2+}}$ are reductant.
- (C)Cm and Am have seven unpaired electrons.
- (D)Actinoid contraction is greater from element to element than lanthanoid contraction.
Choose the correct answer from the options given below :
1. (1)A and B Only
2. (2)C and D Only
3. (3)B and D Only
4. (4)A and C Only
Q11 · MCQ · d- and f-BlockOpen
Correct statements from the following are
- (A)Potassium dichromate is an oxidising agent and it oxidises $\ce{FeSO4}$ to $\ce{Fe2(SO4)3}$ in acidic medium.
- (B)Sodium dichromate can be used as primary standard in volumetric estimation.
- (C) $\ce{CrO4^2-}$ and $\ce{Cr2O7^2-}$ are interconvertible in aqueous solution by varying the pH of the solution.
- (D)Cr–O–Cr bond angle in $\ce{Cr2O7^2-}$ is 126 $^\circ$ .
Choose the correct answer from the options given below :
1. (1)A, B and C only
2. (2)A, C and D only
3. (3)A and C only
4. (4)B and D only

Match the List-I with List-II

List-I		List-II
Complex ion		Calculated spin only
		magnetic moment (BM)
A.	$\ce{[Cr(H2O)6]^2+}$	I.	3.87
B.	$\ce{[Co(H2O)6]^2+}$	II.	5.92
C.	$\ce{[Cu(H2O)6]^2+}$	III.	4.90
D.	$\ce{[Mn(H2O)6]^2+}$	IV.	1.73

Choose the correct answer from the options given below:

(1)A-I, B-III, C-IV, D-II
(2)A-II, B-I, C-III, D-IV
(3)A-IV, B-II, C-I, D-III
(4)A-III, B-I, C-IV, D-II

Q13 · MCQ · Organic Basics (GOC)Open
Increasing order of electron withdrawing power of following functional groups is :
- (a) $\ce{-CN}$
- (b) $\ce{-COOH}$
- (c) $\ce{-NO2}$
- (d) $\ce{-I}$
1. (1)c $<$ b $<$ d $<$ a
2. (2)c $<$ a $<$ b $<$ d
3. (3)d $<$ b $<$ a $<$ c
4. (4)a $<$ b $<$ c $<$ d
Q14 · MCQ · HydrocarbonsOpen
An alkene (X) on ozonolysis followed by reduction gives following products.
The alkene (X) is :
1. (1)
2. (2)
3. (3)
4. (4)

Q15 · MCQ · Haloalkanes & HaloarenesOpen

Match the List-I with List-II

List_I		List-II
Name of reaction		Reagent or catalyst used
A.	Finkelstein reaction	I.	$\ce{SbF3}$
B.	Swarts reaction	II.	Na, dry ether
C.	Sandmeyer's reaction	III.	NaI
D.	Fittig reaction	IV.	$\ce{Cu2Cl2}$

Choose the correct answer from the options given below :

(1)A-I, B-IV, C-III, D-III
(2)A-III, B-I, C-IV, D-II
(3)A-IV, B-II, C-I, D-III
(4)A-I, B-III, C-II D-IV

We corrected a printing error in the original paper

Source printed “Fitting reaction” in List-I (D).

Q16 · MCQ · Alcohols, Phenols & EthersOpen
Amongst the following, the total number of compounds soluble in aqueous NaOH at room temperature is:
1. (1)5
2. (2)4
3. (3)6
4. (4)3
Q17 · MCQ · AminesOpen
Product C of the following reaction sequence will be
1. (1)1-Bromo-4-nitrobenzene
2. (2)1,3,5-Tribromo-2-nitrobenzene
3. (3)4-Bromo-1-nitrobenzene
4. (4)1,3,5-Tribromobenzene
Q18 · MCQ · BiomoleculesOpen
Statement-I : The structure of Maltose is given below :
Maltose is a non reducing sugar
Statement-II : The structure of Lactose is given below
Lactose is a reducing sugar
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 true
2. (2)Both Statement I and Statement II are false
3. (3)Statement I is true but Statement II is false
4. (4)Statement I is false but Statement II is true
Q19 · MCQ · BiomoleculesOpen
Match the LIST-I with LIST-II :
List-I List-II
Name of One letter
amino acid symbol/type)
A Arginine I. D / Non-essential
B Aspartic acid II. R / Essential
C Lysine III. E / Non-essential
D Glutamic acid IV. K/Essential
Choose the correct answer from the options given below :
1. (1)A-II, B-I, C-IV, D-III
2. (2)A-IV, B-III, C-II, D-I
3. (3)A-III, B-IV, C-I, D-II
4. (4)A-II, B-IV, C-I, D-III
Q20 · MCQ · EquilibriumOpen
Identify the colour of compound ‘X' in the sequence of the reaction.
1. (1)Violet
2. (2)Green
3. (3)Red
4. (4)Colourless
Q21 · Numerical · Chemical BondingOpen
According to Lewis theory, the total number of $\sigma$ bond-pairs and lone pair of electrons around the central atom of $\ce{XeO6^4-}$ ion is .
Q22 · Numerical · Aldehydes, Ketones & AcidsOpen
Consider the following sequence of reactions to give the major product (X)
P g of the major product (X) formed is reacted with $\ce{NaHCO3}$ solution to liberate a gas which occupied 11.2 dm $^3$ at STP.
P $=$ g.
(Given molar mass in g mol $^{-1}$ H : 1, C : 12, O : 16, Cl : 35.5)
We corrected a printing error in the original paper
Source printed “ $\ce{NaHCO3}$ soluion”.
Q23 · Numerical · Purification & CharacterisationOpen
2.0 g of a bromo hydrocarbon (X) was subjected to Carius analysis, gave 3.36 g of AgBr. The percentage of carbon in the compound (X) is 26.7%. Total number of carbon atoms in the empirical formula for compound (X) is .
(Given molar mass in g mol $^{-1}$ H : 1, C : 12, Br : 80, Ag : 108)
Q24 · Numerical · EquilibriumOpen
The pH of a solution obtained by mixing 5 mL of 0.1 M $\ce{NH4OH}$ solution with 250 mL of 0.1 M $\ce{NH4Cl}$ solution is $\times\,10^{-2}$ . (Nearest integer)
Given : p $K_\mathrm{b}$ ( $\ce{NH4OH}$ ) $= 4.74$
$\log 2 = 0.30$
$\log 3 = 0.48$
$\log 5 = 0.70$
We corrected a printing error in the original paper
Source printed “The pH a solution”.
Q25 · Numerical · SolutionsOpen
A non-volatile, non-electrolyte solid solute when dissolved in 40 g of a solvent, the vapour pressure of the solvent decreased from 760 mm Hg to 750 mm Hg. If the same solution boils at 320 K, then the number of moles of the solvent present in the solution is . (Nearest integer)
[Given : boiling point of the pure solvent $= 319.5$ K, $K_\mathrm{b}$ of the solvent $= 0.3$ K kg mol $^{-1}$ ]

Maths

Q1 · MCQ · Inverse TrigonometryOpen
Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x) = \cos^{-1}\left(\dfrac{4x + 2\left[x\right]}{3}\right)$ is $[\alpha, \beta]$ , then $12(\alpha + \beta)$ is equal to:
1. (1)6
2. (2)8
3. (3)9
4. (4)4
Q2 · MCQ · Quadratic EquationsOpen
If the set of all solutions of $\lvert x^2 + x - 9\rvert = \lvert x\rvert + \lvert x^2 - 9\rvert$ is $[\alpha, \beta] \cup [\gamma, \infty)$ , then $(\alpha^2 + \beta^2 + \gamma^2)$ is equal to:
1. (1)9
2. (2)18
3. (3)36
4. (4)72
Q3 · MCQ · Complex NumbersOpen
Let $z$ be a complex number such that $\lvert z + 2\rvert = \lvert z - 2\rvert$ and $\arg\left(\dfrac{z + 3}{z - i}\right) = \dfrac{\pi}{4}$ . Then $\lvert z\rvert^2$ is equal to:
1. (1)9
2. (2)4
3. (3)5
4. (4)1
Q4 · MCQ · Sets, Relations & FunctionsOpen
The number of functions $f : \{1, 2, 3, 4\} \to \{a, b, c\}$ , which are not onto, is:
1. (1)48
2. (2)45
3. (3)51
4. (4)35
Q5 · MCQ · Matrices & DeterminantsOpen
Let $S = \left\{A = \begin{bmatrix} a & b\\ c & d\end{bmatrix} : a, b, c, d \in \{0, 1, 2, 3, 4\} \text{ and } A^2 - 4A + 3I = 0\right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in $S$ , for which the sum of the diagonal elements is equal to 4, is:
1. (1)20
2. (2)17
3. (3)21
4. (4)19
Q6 · MCQ · Matrices & DeterminantsOpen
Let $A = \begin{bmatrix} 1 & 1 & 2\\ -2 & 0 & 1\\ 1 & 3 & 5\end{bmatrix}.$ Then the sum of all elements of the matrix $\operatorname{adj}(\operatorname{adj}(2(\operatorname{adj}A)^{-1}))$ is equal to:
1. (1)3
2. (2)4
3. (3) $-4$
4. (4) $-3$
Q7 · MCQ · Sequences & SeriesOpen
The first term of an A.P. of 30 non-negative terms is $\dfrac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is:
1. (1) $\dfrac{5}{87}$
2. (2) $\dfrac{25}{83}$
3. (3) $\dfrac{15}{29}$
4. (4) $\dfrac{5}{29}$
Q8 · MCQ · Permutations & CombinationsOpen
The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is:
1. (1)5040
2. (2)3050
3. (3)3410
4. (4)4320
Q9 · MCQ · Binomial TheoremOpen
Let the smallest value of $k \in \mathbb{N}$ , for which the coefficient of $x^3$ in $(1 + x)^3 + (1 + x)^4 + (1 + x)^5 + \ldots + (1 + x)^{99} + (1 + kx)^{100}$ , $x \neq 0$ , is $\left(43n + \dfrac{101}{4}\right)\left({}^{100}C_3\right)$ for some $n \in \mathbb{N}$ , be $p$ . Then the value of $p + n$ is:
1. (1)10
2. (2)11
3. (3)12
4. (4)13
Q10 · MCQ · StatisticsOpen
Suppose that the mean and median of the non-negative numbers 21, 8, 17, $a$ , 51, 103, $b$ , 13, 67, $(a > b)$ , are 40 and 21, respectively. If the mean deviation about the median is 26, then $2a$ is equal to:
1. (1)109
2. (2)117
3. (3)161
4. (4)131
Q11 · MCQ · Straight LinesOpen
Let the line $L_1 : x + 3 = 0$ intersect the lines $L_2 : x - y = 0$ and $L_3 : 3x + y = 0$ at the points $A$ and $B$ , respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ at the point $C$ . Then $BC^2 : AC^2$ is equal to:
1. (1) $5 : 1$
2. (2) $1 : 5$
3. (3) $2 : 3$
4. (4) $3 : 2$
Q12 · MCQ · Straight LinesOpen
Let the vertex $A$ of a triangle $ABC$ be $(1, 2)$ , and the mid-point of the side $AB$ be $(5, -1)$ . If the centroid of this triangle is $(3, 4)$ and its circumcenter is $(\alpha, \beta)$ , then $21(\alpha + \beta)$ is equal to:
1. (1)309
2. (2)403
3. (3)497
4. (4)524
Q13 · MCQ · CirclesOpen
Suppose the two chords, drawn from the point $(1, 2)$ on the circle $x^2 + y^2 + x - 3y = 0$ are bisected by the $y$ -axis. If the other ends of these chords are $R$ and $S$ , and the mid point of the line segment $RS$ is $(\alpha, \beta)$ , then $6(\alpha + \beta)$ is equal to:
1. (1)1
2. (2)3
3. (3)4
4. (4)6
Q14 · MCQ · 3D GeometryOpen
A line with direction ratios $1, -1, 2$ intersects the lines $\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z + 1}{3}$ and $\dfrac{x + 1}{-1} = \dfrac{y - 2}{1} = \dfrac{z}{4}$ at the points $P$ and $Q$ , respectively. If the length of the line segment $PQ$ is $\alpha$ , then $225\alpha^2$ is equal to:
1. (1)1024
2. (2)1014
3. (3)1104
4. (4)1204
Q15 · MCQ · 3D GeometryOpen
The square of the distance of the point $(-2, -8, 6)$ from the line $\dfrac{x - 1}{1} = \dfrac{y - 1}{2} = \dfrac{z}{-1}$ along the line $\dfrac{x + 5}{1} = \dfrac{y + 5}{-1} = \dfrac{z}{2}$ is equal to:
1. (1)3
2. (2)6
3. (3)8
4. (4)12
Q16 · MCQ · Limits, Continuity & DifferentiabilityOpen
If $y = \tan^{-1}\left(\dfrac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\dfrac{x}{1 + \sqrt{1 - x^2}}\right)$ , then $\dfrac{dy}{dx}$ at $x = \dfrac{\sqrt3}{2}$ is equal to:
1. (1)3
2. (2) $-1$
3. (3)1
4. (4)2
Q17 · MCQ · Limits, Continuity & DifferentiabilityOpen
Let $f$ be a real polynomial of degree $n$ such that $f(x) = f'(x)\,f''(x)$ , for all $x \in \mathbb{R}$ . If $f(0) = 0$ , then $36\left(f'(2) + f''(2) + \displaystyle\int_0^2 f(x)\,dx\right)$ is equal to:
1. (1)42
2. (2)46
3. (3)56
4. (4)66
Q18 · MCQ · Area under CurvesOpen
The area of the region $\{(x, y) : y \le \pi - \lvert x\rvert,\ y \le \lvert x\sin x\rvert,\ y \ge 0\}$ is:
1. (1) $1 + \dfrac{\pi^2}{8}$
2. (2) $2 + \dfrac{\pi^2}{4}$
3. (3) $\dfrac{\pi^2}{8} - 1$
4. (4) $4 + \dfrac{\pi^2}{2}$
Q19 · MCQ · Definite IntegrationOpen
Let $\displaystyle\int_{-2}^{2}\left(\lvert\sin x\rvert + [x\sin x]\right)dx = 2(3 - \cos 2) + \beta$ , where $[\cdot]$ is the greatest integer function. Then $\beta\sin\left(\dfrac{\beta}{2}\right)$ equals.
1. (1)1
2. (2)2
3. (3)4
4. (4)8
Q20 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation $\dfrac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)$ , $y(0) = \dfrac12$ . Then $(2y(1) - 1)$ is equal to
1. (1) $\sqrt3\tan\left(\dfrac{11\sqrt3}{6}\right)$
2. (2) $\dfrac{\sqrt3}{2}\tan\left(\dfrac{11\sqrt3}{12}\right)$
3. (3) $\sqrt3\tan\left(\dfrac{11\sqrt3}{12}\right)$
4. (4) $\dfrac{\sqrt3}{2}\tan\left(\dfrac{11\sqrt3}{6}\right)$
Flawed question. Ill-posed: the solution with $y(0) = 1/2$ blows up near $x = 0.99$ (its arctangent would have to reach $11\sqrt3/12 > \pi/2$ ), so $y(1)$ does not exist; the keyed (3) is only a formal value (about $-102$ , impossible for an increasing $y$ starting at $1/2$ ), and no minimal change gives a valid question with these options. It is shown for completeness and is left out of practice tests.
Q21 · Numerical · ProbabilityOpen
A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $p$ , then $96p$ is equal to .
Q22 · Numerical · Conic SectionsOpen
Consider the parabola $P : y^2 = 4kx$ and the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ . Let the line segment joining the points of intersection of $P$ and $E$ , be their latus rectums. If the eccentricity of $E$ is $e$ , then $e^2 + 2\sqrt2$ is equal to .
Q23 · Numerical · TrigonometryOpen
If $A = \dfrac{\sin 3^\circ}{\cos 9^\circ} + \dfrac{\sin 9^\circ}{\cos 27^\circ} + \dfrac{\sin 27^\circ}{\cos 81^\circ}$ and $B = \tan 81^\circ - \tan 3^\circ$ , then $\dfrac{B}{A}$ is equal to .
Q24 · Numerical · TrigonometryOpen
Let $\vec a_k = (\tan\theta_k)\hat i + \hat j$ and $\vec b_k = \hat i - (\cot\theta_k)\hat j$ , where $\theta_k = \dfrac{2^{k-1}\pi}{2^n + 1}$ , for some $n \in \mathbb{N}$ , $n > 5$ . Then the value of $\dfrac{\displaystyle\sum_{k=1}^{n}\lvert\vec a_k\rvert^2}{\displaystyle\sum_{k=1}^{n}\lvert\vec b_k\rvert^2}$ is .
We corrected a printing error in the original paper
Source printed “ $\vec b_k = \hat i - (\cos\theta_k)\hat j$ ”; corrected to the intended “ $(\cot\theta_k)\hat j$ ” (with $\cos$ the ratio depends on $n$ , e.g. 166.5 for $n = 6$ , and is never 3; with $\cot$ it is 3 for every $n$ , as the key and the source's solution use).
Q25 · Numerical · Limits, Continuity & DifferentiabilityOpen
The number of points, at which the function $f(x) = \max\{6x, 2 + 3x^2\} + \lvert x - 1\rvert\cos\left\lvert x^2 - \dfrac14\right\rvert$ , $x \in (-\pi, \pi)$ , is not differentiable, is .
We corrected a printing error in the original paper
Source printed “fucntion”.

	List-I		List-II
	Name of		One letter
	amino acid		symbol/type)
A	Arginine	I.	D / Non-essential
B	Aspartic acid	II.	R / Essential
C	Lysine	III.	E / Non-essential
D	Glutamic acid	IV.	K/Essential