JEE Main 23 Jan 2026, Shift 1

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

Physics

Q1 · MCQ · Ray OpticsOpen
A thin prism with angle $5^\circ$ of refractive index 1.72 is combined with another prism of refractive index 1.9 to produce dispersion without deviation. The angle of second prism is .
1. (1) $4.5^\circ$
2. (2) $6^\circ$
3. (3) $4^\circ$
4. (4) $5^\circ$
Q2 · MCQ · Work, Energy and PowerOpen
A small bob A of mass $m$ is attached to a massless rigid rod of length 1m pivoted at point P and kept at an angle of $60^\circ$ with vertical as shown in figure. At distance of 1m below point P, an identical bob B is kept at rest on a smooth horizontal surface that extends to a circular track of radius R as shown in figure. If bob B just manages to complete the circular path of radius R upto a point Q after being hit elastically by bob A, then radius R is m.
1. (1) $\dfrac35$
2. (2) $\dfrac15$
3. (3) $\dfrac{2+\sqrt3}{5}$
4. (4) $\dfrac{2-\sqrt3}{5}$
We corrected a printing error in the original paper
Source printed “A small both A”.

Match List-I with List-II.

List-I (Relation)		List-II (Law)
A.	$\displaystyle\oint\vec E\cdot d\vec l = -\frac{d}{dt}\oint\vec B\cdot d\vec a$	I.	Ampere's circuital law.
B.	$\displaystyle\oint\vec B\cdot d\vec l = \mu_0\left(I + \epsilon_0\frac{d\phi_E}{dt}\right)$	II.	Faraday's laws of electromagnetic induction.
(C)	$\displaystyle\oint\vec E\cdot d\vec a = \frac{1}{\epsilon_0}\int\rho\,dv$	III.	Ampere-Maxwell law
(D)	$\displaystyle\oint\vec B\cdot d\vec l = \mu_0 I$	IV.	Gauss's law of electrostatics

Choose the correct answer from the options given below:

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

We corrected a printing error in the original paper

Source printed row B as “ $\oint B\cdot d\vec l = \mu_0\left(1 + \epsilon_0\frac{d\phi_E}{dt}\right)$ ” (digit 1 for the current $I$ , no arrow on $B$ ).

Q4 · MCQ · Units and MeasurementsOpen
Four persons measure the length of a rod as 20.00 cm, 19.75 cm, 17.01 cm and 18.25 cm. The relative error in the measurement of average length of the rod is:
1. (1)0.24
2. (2)0.18
3. (3)0.06
4. (4)0.08
Q5 · MCQ · Dual Nature of Radiation and MatterOpen
The de Broglie wavelength of an oxygen molecule at $27^\circ$ C is $x\times10^{-12}$ m. The value of $x$ is (take Planck's constant $= 6.63\times10^{-34}$ J.s, Boltzmann constant $= 1.38\times10^{-23}$ J/K, mass of oxygen molecule $= 5.31\times10^{-26}$ kg).
1. (1)26
2. (2)24
3. (3)30
4. (4)20
We corrected a printing error in the original paper
Source printed “mass of oxygen. Molecule”.
Q6 · MCQ · OscillationsOpen
A simple pendulum of string length 30 cm performs 20 oscillations in 10s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is cm. [Assume that the mass of the pendulum remains same.]
1. (1)120
2. (2)0.75
3. (3)7.5
4. (4)15
Q7 · MCQ · Work, Energy and PowerOpen
Two blocks with masses 100g and 200g are attached to the ends of springs A and B as shown in figure. The energy stored in A is E. The energy stored in B, when spring constants $k_A$ , $k_B$ of A and B, respectively satisfy the relation $4k_A = 3k_B$ , is:
1. (1)4E
2. (2)2E
3. (3)3E
4. (4) $\dfrac43$ E
Q8 · MCQ · Rotational MotionOpen
The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L $(R < L)$ about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M):
1. (1) $\dfrac38 MR^2 + \dfrac{7}{12}ML^2$
2. (2) $\dfrac34 MR^2 + \dfrac16 ML^2$
3. (3) $\dfrac34 MR^2 + \dfrac{7}{12}ML^2$
4. (4) $\dfrac38 MR^2 + \dfrac16 ML^2$
Q9 · MCQ · Current ElectricityOpen
A wire of uniform resistance $\lambda\ \Omega$ /m is bent into a circle of radius r and another piece of wire with length 2r is connected between points A and B (AOB) as shown in figure. The equivalent resistance between points A and B is $\Omega$ .
1. (1) $\dfrac{3\pi\lambda r}{8}$
2. (2) $(\pi + 1)\,2r\lambda$
3. (3) $\dfrac{6\pi\lambda r}{3\pi + 16}$
4. (4) $2\pi\lambda r$
Q10 · MCQ · Mechanical Properties of SolidsOpen
The strain-stress plot for materials A,B,C and D is shown in the figure. Which material has the largest Young's modulus?
1. (1)C
2. (2)D
3. (3)A
4. (4)B
Q11 · MCQ · Ray OpticsOpen
Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are $2.4\times10^{8}$ m/s and $2.7\times10^{8}$ m/s respectively, then the value of critical angle is:
1. (1) $\cot^{-1}\left(\dfrac{3}{\sqrt{13}}\right)$
2. (2) $\sin^{-1}\left(\dfrac98\right)$
3. (3) $\tan^{-1}\left(\dfrac{8}{\sqrt{17}}\right)$
4. (4) $\cos^{-1}\left(\dfrac89\right)$
Q12 · MCQ · Semiconductor ElectronicsOpen
The following diagram shows a Zener diode as a voltage regulator. The Zener diode is rated at $V_Z = 5$ V and the desired current in load is 5 mA. The unregulated voltage source can supply upto 25V. Considering the Zener diode can withstand four times of the load current, the value of resistor $R_S$ (shown in circuit) should be $\Omega$ .
1. (1)4000
2. (2)10
3. (3)100
4. (4)1000
Flawed question. Dropped in the source; as printed $R_S = (25-5)\,\mathrm{V}/(20+5)\,\mathrm{mA} = 800\ \Omega$ , not among the options, and no evident intended version. It is shown for completeness and is left out of practice tests.

Q13 · MCQ · AtomsOpen

In hydrogen atom spectrum, (

R \rightarrow

Rydberg's constant)

A.	the maximum wavelength of the radiation of Lyman series is $\dfrac{4}{3R}$
B.	the Balmer series lies in the visible region of the spectrum
C.	the minimum wavelength of the radiation of Paschen series is $\dfrac{9}{R}$
D.	the minimum wavelength of Lyman series is $\dfrac{5}{4R}$

Choose the correct answer from the options given below:

(1)B, D Only
(2)A, B and C Only
(3)A, B and D Only
(4)A, B Only

Q14 · MCQ · Motion in a PlaneOpen
An object is projected with kinetic energy K from a point A at an angle $60^\circ$ with the horizontal. The ratio of the difference in kinetic energies at points B and C to that at point A (see figure), in the absence of air friction is:
1. (1)1 : 2
2. (2)2 : 3
3. (3)1 : 4
4. (4)3 : 4
We corrected a printing error in the original paper
Source printed “the difference in kinetic energies points B and C” (“at” missing).
Q15 · MCQ · Electric Charges and FieldsOpen
Two point charges 2q and q are placed at vertex A and centre of face CDEF of the cube as shown in figure. The electric flux passing through the cube is:
1. (1) $\dfrac{3q}{\varepsilon_0}$
2. (2) $\dfrac{q}{\varepsilon_0}$
3. (3) $\dfrac{3q}{2\varepsilon_0}$
4. (4) $\dfrac{3q}{4\varepsilon_0}$
Q16 · MCQ · Electromagnetic InductionOpen
A 20 m long uniform copper wire held horizontally is allowed to fall under the gravity ( $g = 10\ \mathrm{m/s^2}$ ) through a uniform horizontal magnetic field of 0.5 Gauss perpendicular to the length of the wire. The induced EMF across the wire when it travels a vertical distance of 200 m is mV.
1. (1) $0.2\sqrt{10}$
2. (2) $20\sqrt{10}$
3. (3) $2\sqrt{10}$
4. (4) $200\sqrt{10}$
We corrected a printing error in the original paper
Source printed “across the wire it travells”.
Q17 · MCQ · Rotational MotionOpen
Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is:
1. (1) $\dfrac32\dfrac{L}{md^2}$
2. (2) $\dfrac{2L}{md^2}$
3. (3) $\dfrac43\dfrac{L}{md^2}$
4. (4) $\dfrac{2L}{5md^2}$
Q18 · MCQ · Centre of Mass, Momentum and CollisionOpen
In a perfectly inelastic collision, two spheres made of the same material with masses 15 kg and 25 kg, moving in opposite directions with speeds of 10 m/s and 30 m/s, respectively, strike each other and stick together. The rise in temperature (in $^\circ$ C), if all the heat produced during the collision is retained by these spheres, is:
(specific heat of sphere material 31 cal/kg. $^\circ$ C and 1 cal $= 4.2$ J)
1. (1)1.75
2. (2)1.44
3. (3)1.15
4. (4)1.95
Q19 · MCQ · Units and MeasurementsOpen
In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the correct thickness of the sheet is mm.
[Assume least count is 0.01 mm]
1. (1)1.50
2. (2)1.48
3. (3)1.54
4. (4)1.51
Q20 · MCQ · Magnetism and MatterOpen
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Consider a ferromagnetic material:
Assertion (A): The individual atoms in a ferromagnetic material possess a magnetic dipole moment and interact with one another in such a way that they spontaneously align themselves forming domains.
Reason (R): At high enough temperature, the domain structure of ferromagnetic material disintegrates. Thus, magnetization will disappear at high enough temperature known as Curie temperature.
In the light of the above statements, choose the correct answer from the options given below:
1. (1)(A) is true but (R) is false
2. (2)Both (A) and (R) are true but (R) is not the correct explanation of (A)
3. (3)Both (A) and (R) are true and (R) is the correct explanation of (A)
4. (4)(A) is false but (R) is true
Q21 · Numerical · Electromagnetic InductionOpen
A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T. The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of $60^\circ$ with vertical, then maximum induced EMF between the point of suspension and point of oscillation is mV. (Take $g = 10\ \mathrm{m/s^2}$ )
Q22 · Numerical · Electromagnetic WavesOpen
The equation of the electric field of an electromagnetic wave propagating through free space is given by: $E = \sqrt{377}\,\sin(6.27\times10^{3}t - 2.09\times10^{-5}x)\ \mathrm{N/C}$ The average power of the electromagnetic wave is $\left(\dfrac{1}{\alpha}\right)\mathrm{W/m^2}$ . The value of $\alpha$ is
$\left(\text{Take } \sqrt{\dfrac{\mu_0}{\varepsilon_0}} = 377 \text{ in SI units}\right)$
We corrected a printing error in the original paper
Source printed “he average power”.
Q23 · Numerical · Alternating CurrentOpen
Using a variable-frequency a.c. voltage source, the maximum current measured in the given LCR circuit is 50 mA for $V = 5\sin(100t)$ . The values of L and R are shown in the figure. The capacitance of the capacitor (C) used is $\mu$ F.
Q24 · Numerical · Wave OpticsOpen
In two separate Young's double-slit experimental set-ups, two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slit separations and that of the wavelengths of light used are 2 : 1 and 1 : 2 respectively. The corresponding ratio of the distances between the slits and the respective screens $(D_1/D_2)$ is .
Q25 · Numerical · Electric Potential and CapacitanceOpen
The space between the plates of a parallel-plate capacitor of capacitance C (without any dielectric) is now filled with three dielectric slabs of dielectric constants $K_1 = 2$ , $K_2 = 3$ , and $K_3 = 5$ (as shown in the figure). If new capacitance is $\dfrac{n}{3}C$ then the value of n is .

Chemistry

Q1 · MCQ · Structure of AtomOpen
Which of the following statements regarding the energy of the stationary state is true in the following one-electron system ?
1. (1) $-1.09\times10^{-18}$ J for second orbit of H atom.
2. (2) $+2.18\times10^{-18}$ J for second orbit of $\ce{He+}$ ion
3. (3) $+8.72\times10^{-18}$ J for first orbit of $\ce{He+}$ ion
4. (4) $-2.18\times10^{-18}$ J for third orbit of $\ce{Li^{2+}}$ ion
Q2 · MCQ · SolutionsOpen
Which one of the following graphs accurately represents the plot of partial pressure of $\ce{CS2}$ vs its mole fraction in a mixture of acetone and $\ce{CS2}$ at constant temperature ?
1. (1)
2. (2)
3. (3)
4. (4)
Q3 · MCQ · Classification & PeriodicityOpen
The correct trend in the first ionization enthalpies of the elements in the 3 $^\text{rd}$ period of periodic table is:
1. (1)Al $<$ Si $<$ S $<$ P $<$ Cl
2. (2)Al $<$ S $<$ P $<$ Si $<$ Cl
3. (3)Si $<$ S $<$ Al $<$ P $<$ Cl
4. (4)S $<$ Si $<$ Al $<$ P $<$ Cl
Q4 · MCQ · Alcohols, Phenols & EthersOpen
The correct sequence of reagents for the above conversion of X to Y is :
1. (1)(i) NaOH (aq) (ii) Jones reagent (iii) $\ce{H3O+}$
2. (2)(i) $\ce{B2H6}$ / $\ce{H2O2}$ (ii) NaOEt (iii) Jones reagent
3. (3)(i) Jones reagent (ii) NaOEt (iii) Hot $\ce{KMnO4}$ /KOH
4. (4)(i) NaOEt (ii) $\ce{B2H6}$ / $\ce{H2O2}$ (iii) Jones reagent
We corrected a printing error in the original paper
Option (2): source printed “ $\ce{B2H6}$ / $\ce{H2O2}$ (ii) NaOEt (iii) Jones reagent” without the step label “(i)”.
Q5 · MCQ · ElectrochemistryOpen
In the given electrochemical cell, Ag(s) $|$ AgCl(s) $|$ $\ce{FeCl2}$ (aq), $\ce{FeCl3}$ (aq) $|$ Pt(s) at 298 K, the cell potential ( $E_\mathrm{cell}$ ) will increase when :
- (A)Concentration of $\ce{Fe^{2+}}$ is increased.
- (B)Concentration of $\ce{Fe^{3+}}$ is decreased
- (C)Concentration of $\ce{Fe^{2+}}$ is decreased
- (D)Concentration of $\ce{Fe^{3+}}$ is increased
- (E)Concentration of $\ce{Cl-}$ is increased
Choose the correct answer from the options given below :
1. (1)A and B only
2. (2)A and E only
3. (3)B only
4. (4)C, D and E only
We corrected a printing error in the original paper
Source labelled the fifth statement “(D)” (two statements (D), none (E), while the options refer to E).
Q6 · MCQ · Aldehydes, Ketones & AcidsOpen
‘x' is the product which is obtained from propanenitrile and stannous chloride in the presence of hydrochloric acid followed by hydrolysis. ‘y' is the product which is obtained from the but-2-ene by the ozonolysis followed by hydrolysis. From the following, which product is not obtained when one mole of ‘x' and one mole of ‘y' react with each other in the presence of alkali followed by heating ?
1. (1)2-Methylbut-2-enal
2. (2)Pent-2-enal
3. (3)2-Methylpent-2-enal
4. (4)3-Methylbut-2-enal
Q7 · MCQ · AminesOpen
Consider the following sequence of reactions.
4-Nitrotoluene
Assuming that the reaction proceeds to completion, then 137 mg of 4-nitrotoluene will produce mg of B.
(Given molar mass in g mol $^{-1}$ H : 1, C : 12, N : 14, O : 16, Br : 80)
1. (1)301
2. (2)146
3. (3)228
4. (4)208
Q8 · MCQ · ThermodynamicsOpen
A cup of water at 5 $^\circ$ C (system) is placed in a microwave oven and the oven is turned on for one minute during which, the water begins to boil. Which of the following option is true ?
1. (1) $q = +$ ve, $w = 0$ , $\Delta U = -$ ve
2. (2) $q = +$ ve, $w = -$ ve, $\Delta U = +$ ve
3. (3) $q = -$ ve, $w = -$ ve, $\Delta U = -$ ve
4. (4) $q = +$ ve, $w = -$ ve, $\Delta U = -$ ve
Q9 · MCQ · Coordination CompoundsOpen
Given below are two statements :
Statement I : $\ce{[CoBr4]^{2-}}$ ion will absorb light of lower energy than $\ce{[CoCl4]^{2-}}$ ion.
Statement II : In $\ce{[CoI4]^{2-}}$ ion, the energy separation between the two set of d-orbitals is more than $\ce{[CoCl4]^{2-}}$ ion.
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)Statement I is true but Statement II is false
3. (3)Statement I is false but Statement II is true
4. (4)Both Statement I and Statement II are true
Q10 · MCQ · HydrocarbonsOpen
But-2-yne and hydrogen (one mole each) are separately treated with (i) Pd/C and (ii) Na/liq. $\ce{NH3}$ to give the products X and Y respectively.
Identify the incorrect statements.
- (A)X and Y are stereoisomers.
- (B)Dipole moment of X is zero
- (C)Boiling point of X is higher than Y.
- (D)X and Y react with $\ce{O3}$ /Zn + $\ce{H2O}$ to give different products.
Choose the correct answer from the options given below :
1. (1)B and C only
2. (2)B and D only
3. (3)A and B only
4. (4)A and C only
Q11 · MCQ · Structure of AtomOpen
Given,
(A) $n = 5$ , $m_l = -1$
(B) $n = 3$ , $l = 2$ , $m_l = -1$ , $m_s = +\dfrac{1}{2}$
The maximum number of electron(s) in an atom that can have the quantum numbers as given in (A) and (B) respectively are :
1. (1)26 and 1
2. (2)4 and 1
3. (3)2 and 4
4. (4)8 and 1
We corrected a printing error in the original paper
Source printed the magnetic quantum number as “m $_1$ ” and the azimuthal quantum number in (B) as “1 = 2” (digit 1 for l).
Q12 · MCQ · HydrocarbonsOpen
Consider the following compounds
Arrange these compounds in the increasing order of reactivity with nitrating mixture.
1. (1)c $<$ a $<$ b
2. (2)b $<$ c $<$ a
3. (3)c $<$ b $<$ a
4. (4)b $<$ a $<$ c
Q13 · MCQ · Coordination CompoundsOpen
The statements that are incorrect about the nickel (II) complex of dimethylglyoxime are :
- (A)It is red in colour
- (B)It has a high solubility in water at pH = 9
- (C)The Ni ion has two unpaired d-electrons
- (D)The N – Ni – N bond angle is almost close to 90 $^\circ$
- (E)The complex contains four five-membered metallacycles (metal containing rings)
Choose the correct answer from the options given below :
1. (1)C and E only
2. (2)A, D and B only
3. (3)B, C and E only
4. (4)C and D only
Q14 · MCQ · BiomoleculesOpen
From the given following (A to D) cyclic structures, those which will not react with Tollen's reagent are :
1. (1)B and D
2. (2)A and D
3. (3)A and B
4. (4)B and C
Q15 · MCQ · Chemical BondingOpen
Identify the molecule(X) with maximum number of lone pairs of electrons (obtained using Lewis dot structure) among $\ce{HNO3}$ , $\ce{H2SO4}$ , $\ce{NF3}$ and $\ce{O3}$ . Choose the correct bond angle made by the central atom of the molecule (X).
1. (1)120 $^\circ$
2. (2)107 $^\circ$
3. (3)102 $^\circ$
4. (4)116 $^\circ$

Q16 · MCQ · Practical / Qualitative AnalysisOpen

Match List-I with List-II.

	List-I		List-II
	Functional group (detection)		Change observed during detection
(A)	Unsaturation (Baeyer's test)	(I)	Red colour Appears
(B)	Alcoholic group (Ceric ammonium nitrate test)	(II)	Silver mirror appears
(C)	Aldehyde group (Tollen's reagent)	(III)	Violet colour appears
(D)	Phenolic group ( $\ce{FeCl3}$ test)	(IV)	Discharge of pink colour

Choose the correct answer from the options given below :

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

Q17 · MCQ · Purification & CharacterisationOpen
Given below are two statements :
Statement-I : Sublimation is used for the separation and purification of compounds with low melting point.
Statement-II : The boiling point of a liquid increases as the external pressure is reduced.
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)Statement-I is true but Statement-II is false.
3. (3)Both Statement-I and Statement-II are true.
4. (4)Both Statement-I and Statement-II are false.
Q18 · MCQ · AminesOpen
Compound ‘P' undergoes the following sequence of reactions :
‘P' is :
1. (1)
2. (2)
3. (3)
4. (4)
Q19 · MCQ · p-Block (13–14)Open
The correct statements from the following are :
- (A)Ionic radii of trivalent cations of group 13 elements decrease down the group.
- (B)Electronegativity of group 13 elements decreases down the group.
- (C)Among the group 13 elements, Boron has highest first ionisation enthalpy.
- (D)The trichloride and triiodide of group 13 elements are covalent in nature.
Choose the correct answer from the options given below :
1. (1)A and C only
2. (2)A and D only
3. (3)C and D only
4. (4)B and D only
We corrected a printing error in the original paper
Source printed “Ionic radii … decreases down the group” and “from the otpions given below”.
Q20 · MCQ · EquilibriumOpen
Consider the general reaction given below at 400 K
$\ce{xA(g) <=> yB(g)}$
The values of $K_\mathrm{p}$ and $K_\mathrm{c}$ are studied under the same condition of temperature but variation in x and y.
(i) $K_\mathrm{p} = 85.87$ and $K_\mathrm{c} = 2.586$ appropriate units
(ii) $K_\mathrm{p} = 0.862$ and $K_\mathrm{c} = 28.62$ appropriate units
The value of x and y in (i) and (ii) respectively are:
1. (1)(i) 3, 1 ; (ii) 3, 1
2. (2)(i) 4, 1 ; (ii) 4, 1
3. (3)(i) 1, 3 ; (ii) 2, 1
4. (4)(i) 1, 2 ; (ii) 2, 1
Q21 · Numerical · Chemical KineticsOpen
For the thermal decomposition of reaction AB(g), the following is constructed.
The half life of the reaction is ‘x' min.
x = min. (Nearest integer)
Q22 · Numerical · EquilibriumOpen
For the following gas phase equilibrium reaction at constant temperature,
$\ce{NH3(g) <=> 1/2 N2(g) + 3/2 H2(g)}$
If the total pressure is $\sqrt{3}$ atm and the pressure equilibrium constant ( $K_\mathrm{P}$ ) is 9 atm, then the degree of dissociation is given as $(x\times10^{-2})^{-1/2}$ .
The value of x is (Nearest integer)
Q23 · Numerical · Some Basic ConceptsOpen
x mg of pure HCl was used to make an aqueous solution. 25.0 mL of 0.1 M $\ce{Ba(OH)2}$ solution is used when the HCl solution was titrated against it. The numerical value of x is $\times\,10^{-1}$ . (Nearest integer)
Given : Molar mass of HCl and $\ce{Ba(OH)2}$ are 36.5 and 171.0 g mol $^{-1}$ respectively.
Q24 · Numerical · Haloalkanes & HaloarenesOpen
Consider all the structural isomers with molecular formula $\ce{C5H11Br}$ that are separately treated with KOH (aq) to give respective substitution products, without any rearrangement. The number of products which can exhibit optical isomerism from these is .
We corrected a printing error in the original paper
Source printed “Consider all the structural isomers with molecular formula $\ce{C5H11Br}$ are separately treated” (“that” missing).
Q25 · Numerical · Coordination CompoundsOpen
The crystal field stabilisation energy of $\ce{[Co(oxalate)3]^{3-}}$ complex is ‘n' times that of the $\ce{[Cr(oxalate)3]^{3-}}$ complex. Here ‘n' is . [Assume $\Delta_\mathrm{o} \gg P$ ]
We corrected a printing error in the original paper
Source printed “The crystal field splitting energy”; corrected to the intended “crystal field stabilisation energy” (the splitting $\Delta_\mathrm{o}$ is about the same for both oxalato complexes; the key 2 is the CFSE ratio $-2.4\Delta_\mathrm{o}/-1.2\Delta_\mathrm{o}$ ).

Maths

Q1 · MCQ · Sets, Relations & FunctionsOpen
Let the domain of the function $f(x) = \log_3\log_5\log_7(9x - x^2 - 13)$ be the interval $(m, n)$ . Let the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ have eccentricity $\dfrac{n}{3}$ and the length of the latus rectum $\dfrac{8m}{3}$ . Then $b^2 - a^2$ is equal to:
1. (1)5
2. (2)11
3. (3)9
4. (4)7
Q2 · MCQ · Indefinite IntegrationOpen
Let $f(x) = \displaystyle\int\frac{(2 - x^2)\cdot e^x}{\left(\sqrt{1+x}\right)(1-x)^{3/2}}\,dx$ . If $f(0) = 0$ , then $f\left(\dfrac12\right)$ is equal to:
1. (1) $\sqrt{3e} - 1$
2. (2) $\sqrt{2e} + 1$
3. (3) $\sqrt{2e} - 1$
4. (4) $\sqrt{3e} + 1$
We corrected a printing error in the original paper
Source printed “If $f(0) - 0$ ” (minus sign instead of the equals sign).
Q3 · MCQ · VectorsOpen
Let $\vec a = -\hat i + \hat j + 2\hat k$ , $\vec b = \hat i - \hat j - 3\hat k$ , $\vec c = \vec a\times\vec b$ and $\vec d = \vec c\times\vec a$ . Then $\left(\vec a - \vec b\right)\cdot\vec d$ is equal to:
1. (1)4
2. (2) $-4$
3. (3) $-2$
4. (4)2
Q4 · MCQ · Straight LinesOpen
A rectangle is formed by the lines $x = 0$ , $y = 0$ , $x = 3$ and $y = 4$ . Let the line $L$ be perpendicular to $3x + y + 6 = 0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $\left(\dfrac12, -5\right)$ from the line $L$ is equal to:
1. (1) $2\sqrt5$
2. (2) $3\sqrt{10}$
3. (3) $\sqrt{10}$
4. (4) $2\sqrt{10}$
Q5 · MCQ · Conic SectionsOpen
Let the line $y - x = 1$ intersect the ellipse $\dfrac{x^2}{2} + \dfrac{y^2}{1} = 1$ at the points $A$ and $B$ . Then the angle made by the line segment $AB$ at the center of the ellipse is:
1. (1) $\pi - \tan^{-1}\left(\dfrac14\right)$
2. (2) $\dfrac{\pi}{2} + \tan^{-1}\left(\dfrac14\right)$
3. (3) $\dfrac{\pi}{2} + 2\tan^{-1}\left(\dfrac14\right)$
4. (4) $\dfrac{\pi}{2} - \tan^{-1}\left(\dfrac14\right)$
Q6 · MCQ · Sets, Relations & FunctionsOpen
Let $A = \{-2, -1, 0, 1, 2, 3, 4\}$ . Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x + y \le 2$ . Let $l$ be the number of elements in $R$ . Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations respectively. Then $l + m + n$ is equal to:
1. (1)32
2. (2)34
3. (3)33
4. (4)35
Q7 · MCQ · Differential EquationsOpen
Let $y = y(x)$ be the solution of the differential equation $x^4\,dy + (4x^3y + 2\sin x)\,dx = 0,\quad x > 0,\quad y\left(\frac{\pi}{2}\right) = 0.$ Then $\pi^4 y\left(\dfrac{\pi}{3}\right)$ is equal to:
1. (1)81
2. (2)92
3. (3)64
4. (4)72
Q8 · MCQ · Quadratic EquationsOpen
If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation $\left(-2 + \sqrt3\right)\left(\left\lvert\sqrt x - 3\right\rvert\right) + \left(x - 6\sqrt x\right) + \left(9 - 2\sqrt3\right) = 0,$ $x \ge 0$ , then $\sqrt{\dfrac{\beta}{\alpha}} + \sqrt{\alpha\beta}$ is equal to:
1. (1)8
2. (2)9
3. (3)10
4. (4)11
Q9 · MCQ · Limits, Continuity & DifferentiabilityOpen
Let $f(x) = \begin{cases} \dfrac{ax^2 + 2ax + 3}{4x^2 + 4x - 3}, & x \neq -\dfrac32, \dfrac12\\[2ex] b, & x = -\dfrac32, \dfrac12\end{cases}$ be continuous at $x = -\dfrac32$ . If $f\circ f(x) = \dfrac75$ , then $x$ is equal to:
1. (1)2
2. (2)1
3. (3)0
4. (4)1.4
Q10 · MCQ · Definite IntegrationOpen
The value of the integral $\displaystyle\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}}\frac{dx}{1 + \sqrt[3]{\tan 2x}}$ is:
1. (1) $\dfrac{\pi}{12}$
2. (2) $\dfrac{\pi}{18}$
3. (3) $\dfrac{\pi}{6}$
4. (4) $\dfrac{\pi}{3}$
Q11 · MCQ · Matrices & DeterminantsOpen
Among the statements:
I: If $\begin{vmatrix} 1 & \cos\alpha & \cos\beta\\ \cos\alpha & 1 & \cos\gamma\\ \cos\beta & \cos\gamma & 1\end{vmatrix} = \begin{vmatrix} 0 & \cos\alpha & \cos\beta\\ \cos\alpha & 0 & \cos\gamma\\ \cos\beta & \cos\gamma & 0\end{vmatrix}$ , then $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = \dfrac32$ , and
II: If $\begin{vmatrix} x^2 + x & x + 1 & x - 2\\ 2x^2 + 3x - 1 & 3x & 3x - 3\\ x^2 + 2x + 3 & 2x - 1 & 2x - 1\end{vmatrix} = px + q$ , then $p^2 = 196q^2$ ,
1. (1)both are false
2. (2)only II is true
3. (3)both are true
4. (4)only I is true
Q12 · MCQ · Binomial TheoremOpen
The value of $\dfrac{{}^{100}C_{50}}{51} + \dfrac{{}^{100}C_{51}}{52} + \ldots + \dfrac{{}^{100}C_{100}}{101}$ is:
1. (1) $\dfrac{2^{101}}{100}$
2. (2) $\dfrac{2^{100}}{100}$
3. (3) $\dfrac{2^{101}}{101}$
4. (4) $\dfrac{2^{100}}{101}$
Q13 · MCQ · StatisticsOpen
Let the mean and variance of 8 numbers $-10$ , $-7$ , $-1$ , $x$ , $y$ , 9, 2, 16 be $\dfrac72$ and $\dfrac{293}{4}$ , respectively. Then the mean of 4 numbers $x$ , $y$ , $x + y + 1$ , $\lvert x - y\rvert$ is:
1. (1)11
2. (2)9
3. (3)10
4. (4)12
Q14 · MCQ · Binomial TheoremOpen
The sum of all possible values of $n \in \mathbb{N}$ , so that the coefficients of $x$ , $x^2$ and $x^3$ in the expansion of $(1 + x^2)^2(1 + x)^n$ , are in arithmetic progression is:
1. (1)3
2. (2)7
3. (3)12
4. (4)9
Q15 · MCQ · 3D GeometryOpen
The vertices $B$ and $C$ of a triangle $ABC$ lie on the line $\dfrac{x}{1} = \dfrac{1 - y}{-2} = \dfrac{z - 2}{3}$ . The coordinates of $A$ and $B$ are $(1, 6, 3)$ and $(4, 9, \alpha)$ respectively and $C$ is at a distance of 10 units from $B$ . The area (in sq. units) of $\Delta ABC$ is:
1. (1) $5\sqrt{13}$
2. (2) $15\sqrt{13}$
3. (3) $20\sqrt{13}$
4. (4) $10\sqrt{13}$
Q16 · MCQ · TrigonometryOpen
Number of solutions of $\sqrt3\cos2\theta + 8\cos\theta + 3\sqrt3 = 0$ , $\theta \in [-3\pi, 2\pi]$ is:
1. (1)0
2. (2)5
3. (3)3
4. (4)4
Q17 · MCQ · Complex NumbersOpen
Let $S = \{z : 3 \le \lvert 2z - 3(1 + i)\rvert \le 7\}$ be a set of complex numbers. Then $\displaystyle\min_{z\in S}\left\lvert\left(z + \frac12(5 + 3i)\right)\right\rvert$ is equal to:
1. (1) $\dfrac12$
2. (2) $\dfrac32$
3. (3)2
4. (4) $\dfrac52$
Q18 · MCQ · TrigonometryOpen
Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right),\quad \theta \in \mathbb{R}.$ Then $\alpha + 2\beta$ is equal to:
1. (1)4
2. (2)5
3. (3)3
4. (4)6
Q19 · MCQ · Quadratic EquationsOpen
A building construction work can be completed by two masons $A$ and $B$ together in 22.5 days. Mason $A$ alone can complete the construction work in 24 days less than mason $B$ alone. Then mason $A$ alone will complete the construction work in:
1. (1)24 days
2. (2)42 days
3. (3)30 days
4. (4)36 days
Q20 · MCQ · 3D GeometryOpen
Let the direction cosines of two lines satisfy the equations: $4\ell + m - n = 0$ and $2mn + 10n\ell + 3\ell m = 0$ . Then the cosine of the acute angle between these lines is:
1. (1) $\dfrac{10}{\sqrt{38}}$
2. (2) $\dfrac{20}{3\sqrt{38}}$
3. (3) $\dfrac{10}{7\sqrt{38}}$
4. (4) $\dfrac{10}{3\sqrt{38}}$
Q21 · Numerical · Matrices & DeterminantsOpen
Let $\lvert A\rvert = 6$ , where $A$ is a $3\times3$ matrix. If $\lvert\operatorname{adj}(3\operatorname{adj}(A^2\cdot\operatorname{adj}(2A)))\rvert = 2^m\cdot3^n$ , $m, n \in \mathbb{N}$ , then $m + n$ is equal to .
Q22 · Numerical · Area under CurvesOpen
Let the area of the region bounded by the curve $y = \max\{\sin x, \cos x\}$ , lines $x = 0$ , $x = \dfrac{3\pi}{2}$ , and the $x$ -axis be $A$ . Then, $A + A^2$ is equal to .
Q23 · Numerical · Differential EquationsOpen
Let $f$ be a twice differentiable non-negative function such that $(f(x))^2 = 25 + \int_0^x\left((f(t))^2 + (f'(t))^2\right)dt.$ Then the mean of $f(\log_e(1)), f(\log_e(2)), \ldots, f(\log_e(625))$ is equal to .
Q24 · Numerical · ProbabilityOpen
From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a - b \ge 10$ is $\dfrac{m}{n}$ , $\gcd(m, n) = 1$ , then $m + n$ is equal to .
Q25 · Numerical · Permutations & CombinationsOpen
The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is .