JEE Main 2023 Jan 24 Shift 1 Question Paper with Solution PDF- Download Here

Two long straight wires P and Q carrying equal currents of 10A each were kept parallel to each other at a distance of 5cm. The magnitude of the magnetic force experienced by a 10cm length of wire P is F₁. If the distance between the wires is halved and the currents in them are doubled, the force F₂ on a 10cm length of wire P will be:

• (1) 8F₁
• (2) 10F₁
• (3) F₁/8
• (4) F₁/10

Given below are two statements: Statement-I : An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable. Statement-II : Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight when the elevator goes down with increasing speed. In the light of the above statements, choose the correct answer from the options given below:

• (1) Both statement I and statement II are false
• (2) Statement I is true but Statement II is false
• (3) Both Statement I and Statement II are true
• (4) Statement I is false but Statement II is true

From the photoelectric effect experiment, the following observations are made. Identify which of these are correct: A. The stopping potential depends only on the work function of the metal. B. The saturation current increases as the intensity of incident light increases. C. The maximum kinetic energy of a photo electron depends on the intensity of the incident light. D. Photoelectric effect can be explained using wave theory of light. Choose the correct answer from the options given below:

• (1) B, C only
• (2) A, C, D only
• (3) B only
• (4) A, B, D only

The weight of a body at the surface of Earth is 18 N. The weight of the body at an altitude of 3200 km above the Earth's surface is (given, radius of Earth Re = 6400 km):

• (1) 9.8N
• (2) 4.9N
• (3) 19.6N
• (4) 8N

A 100 m long wire having cross-sectional area 6.25 × 10⁻⁴ m² and Young's modulus is 10¹⁰ Nm⁻² is subjected to a load of 250 N, then the elongation in the wire will be:

• (1) 6.25 × 10⁻³ m
• (2) 4 × 10⁻⁴ m
• (3) 6.25 × 10⁻⁶ m
• (4) 4 × 10⁻³ m

1 g of a liquid is converted to vapour at 3 × 10⁵ Pa pressure. If 10% of the heat supplied is used for increasing the volume by 1600 cm³ during this phase change, then the increase in internal energy in the process will be :

• (1) 4320 J
• (2) 432000 J
• (3) 4800 J
• (4) 4.32 × 10⁸ J

A modulating signal is a square wave, as shown in the figure. If the carrier wave is given as c(t) = 2sin(8πt) volts, the modulation index is :

• (1) 1/4
• (2) 1
• (3) 1/1
• (4) 1/2

As per given figure, a weightless pulley P is attached on a double inclined frictionless surface. The tension in the string (massless) will be (if g = 10 m/s²): (Figure shows a pulley with a 4kg mass at 60 degrees incline and 1kg at 30 degree incline)

• (1) (4√3 + 1) N
• (2) 4(√3 + 1) N
• (3) 4(√3 – 1) N
• (4) (4√3 + 1) N

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R: Assertion (A): Photodiodes are preferably operated in reverse bias condition for light intensity measurement. Reason (R): The current in the forward bias is more than the current in the reverse bias for a p-n junction diode. In the light of the above statements, choose the correct answer from the options given below:

• (1) A is false but R is true
• (2) Both A and R are true but R is NOT the correct explanation of A
• (3) A is true but R is false
• (4) Both A and R are true and R is the correct explanation of A

If E represents the electric field vector and K the propagation vector of electromagnetic waves in vacuum, then the magnetic field vector is given by (w - angular frequency):

• (1) B = (K X E)/w
• (2) B = w(E x K)
• (3) B = w(K × E)
• (4) (K × Ể)

A circular loop of radius r is carrying current I A. The ratio of magnetic field at the center of the circular loop and at a distance r from the center of the loop on its axis is:

• (1) 1:3√2
• (2) 3√2:2
• (3) 2√2:1
• (4) 1: √2

A travelling wave is described by the equation: y(x, t) = 0.05 sin(8x – 4t) m. The velocity of the wave is: (All quantities are in SI units.)

• (1) 4 ms⁻¹
• (2) 2 ms⁻¹
• (3) 0.5 ms⁻¹
• (4) 8 ms⁻¹

As shown in the figure, a network of resistors is connected to a battery of 24 V with an internal resistance of 3Ω. The currents through the resistors R₄ and R₅ are I₄ and I₅ respectively. The values of I₄ and I₅ are: (Figure shows a circuit diagram with various resistors)

• (1) I₄ = 2/5 A and I₅ = 8/5 A
• (2) I₄ = 8/5 A and I₅ = 2/5 A
• (3) I₄ = 3/4 A and I₅ = 3/4 A
• (4) I₄ = 2/5 A and I₅ = 2/5 A

Given below are two statements: Statement I: If the Brewster's angle for the light propagating from air to glass is θ_b, then Brewster's angle for the light propagating from glass to air is: π/2 - θ_b Statement II: The Brewster's angle for the light propagating from glass to air is tan⁻¹(μ_g), where μ_g is the refractive index of glass. In the light of the above statements, choose the correct answer from the options given below:

• (1) Both Statement I and Statement II are true.
• (2) Statement I is true but Statement II is false.
• (3) Statement I is false but Statement II is true.
• (4) Both Statement I and Statement II are false.

If two charges q₁ and q₂ are separated with distance d and placed in a medium of dielectric constant K, what will be the equivalent distance between charges in air for the same electrostatic force?

• (1) d√K
• (2) k√d
• (3) 1.5d√ K
• (4) 2d/K

Consider the following radioactive decay process: 218/84A → 214/82A₁→ 214/83A₂ → 214/83A₃ → 210/81A₄ → 210/80A₅→ 210/80A₆ The mass number and the atomic number of A₆ are given by:

• (1) 210 and 82
• (2) 210 and 84
• (3) 210 and 80
• (4) 211 and 80

Given below are two statements: Statement I: The temperature of a gas is −73°C. When the gas is heated to 527°C, the root mean square speed of the molecules is doubled. Statement II: The product of pressure and volume of an ideal gas will be equal to the translational kinetic energy of the molecules. In the light of the above statements, choose the correct answer from the options given below:

• (1) Both Statement I and Statement II are true.
• (2) Statement I is true but Statement II is false.
• (3) Both Statement I and Statement II are false.
• (4) Statement I is false but Statement II is true.

The maximum vertical height to which a man can throw a ball is 136 m. The maximum horizontal distance up to which he can throw the same ball is:

• (1) 192 m
• (2) 136 m
• (3) 272 m
• (4) 68 m

A conducting loop of radius π¹/² * (10) cm is placed perpendicular to a uniform magnetic field of 0.5 T. The magnetic field is decreased to zero in 0.5s at a steady rate. The induced emf in the circular loop at 0.25 s is:

• (1) 1 mV
• (2) 10 mV
• (3) 100 mV
• (4) 5 mV

Match List I with List II:

List I
A. Planck's constant (h)
B. Stopping potential (Vₛ)
C. Work function (Φ)
D. Momentum (p)

List II
I. M¹L²T⁻²
II. M¹L¹T⁻¹
III. M¹L²T⁻¹
IV. M¹L²T⁻³A⁻¹

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

Section – B

Question 21:

A spherical body of mass 2 kg starting from rest acquires a kinetic energy of 10000 J at the end of 5th second. The force acted on the body is _________ Ν.

A block of mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface, and the ends of the springs are attached to rigid supports. When the mass is displaced from its equilibrium position, it executes simple harmonic motion. The time period of oscillation is given by π/√x in SI units. The value of x is:

A hole is drilled in a metal sheet. At 27°C, the diameter of the hole is 5 cm. When the sheet is heated to 177°C, the change in the diameter of the hole is Δd. The value of Δd will be, if the coefficient of linear expansion of the metal is 1.6 × 10⁻⁵/°C:

In the circuit shown in the figure, the ratio of the quality factor and the bandwidth is __________ s. (Figure shows a circuit diagram with various components)

A hollow cylindrical conductor has a length of 3.14m, while its inner and outer diameters are 4mm and 8mm, respectively. The resistance of the conductor is 2 × 10⁻³ Ω. If the resistivity of the material is 2.4 × 10⁻⁸ Ωm, the value of n is: 2 × 10⁻³ Ω.

As shown in the figure, a combination of a thin plano-concave lens and a thin plano-convex lens is used to image an object placed at infinity. The radius of curvature of both lenses is 30 cm, and the refractive index of the material for both lenses is 1.75. Both lenses are placed at a distance of 40 cm from each other. Due to the combination, the image of the object is formed at distance x cm, from the concave lens.

A solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm, then its radius of gyration about PQ will be √x cm. The value of x is:

Vectors a = aì + bj + k and b = 2ì – 3ĵ + 4k are perpendicular to each other when 3a + 2b = 7. The ratio of a to b is x/2. The value of x is:

Assume that protons and neutrons have equal masses. The mass of a nucleon is 1.6×10⁻²⁷ kg, and the radius of a nucleus is 1.5×10⁻¹⁵ A¹/³ m. The approximate ratio of nuclear density to water density is n × 10¹³. The value of n is:

A stream of positively charged particles having q/m = 2 × 10¹¹ C/kg and velocity v₀ = 3 × 10⁷ m/s is deflected by an electric field Ē = 1.8 ĵ kV/m. The electric field exists in a region of 10 cm along the x-direction. Due to the electric field, the deflection of the charged particles in the y-direction is ____ mm.

Compound (X) undergoes the following sequence of reactions to give the lactone (Y):

What is compound X?

• (1) CH₃CH(CH₃)CHO
• (2) HOCH₂C(CH₃)₂CHO
• (3) H₂C-CH₂-CH₂-CH₂-CHO
• (4) H₂C-CH₂-CHO

Assertion (A): Hydrolysis of an alkyl chloride is a slow reaction, but in the presence of NaI, the rate of hydrolysis increases.

Reason (R): I^- is a good nucleophile as well as a good leaving group.

• (1) A is false but R is true.
• (2) A is true but R is false.
• (3) Both A and R are true, and R is the correct explanation of A.
• (4) Both A and R are true, but R is NOT the correct explanation of A.

Order of covalent bond:

A. KF > KI; LiF > KF
B. KF < KI; LiF > KF
C. SnCl₂ > SnCl₄; CuCl > NaCl
D. LiF > KF; CuCl < NaCl
E. KF < KI; CuCl > NaCl

Choose the correct combination of options

• (1) C, E only
• (2) B, C only
• (3) B, C, E only
• (4) A, B only

Increasing order of stability of the resonance structures is:[Image of resonance structures is present]

• (1) C, D, B, A
• (2) C, D, A, B
• (3) D, C, A, B
• (4) D, C, B, A

The magnetic moment of a transition metal compound has been calculated to be 3.87 B.M. The metal ion is:

• (1) Cr²⁺
• (2) Mn²⁺
• (3) V²⁺
• (4) Ti²⁺

Match List I with List II:

List I: A. Reverberatory furnace, B. Electrolytic cell, C. Blast furnace, D. Zone Refining furnace

List II: I. Pig Iron, II. Aluminum, III. Silicon, IV. Copper

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

It is observed that characteristic X-ray spectra of elements show regularity. When frequency to the power n, i.e., νⁿ of X-rays emitted, is plotted against atomic number Z, the following graph is obtained. What is the value of n?

• (1) 1
• (2) 2
• (3) ½
• (4) 3

Which of the phosphorus oxoacids can create a silver mirror from AgNO₃ solution?

• (1) (HPO₃)_n
• (2) H₄P₂O₅
• (3) H₄P₂O₆
• (4) H₄P₂O₇

The primary and secondary valencies of cobalt, respectively, in [Co(NH₃)₅Cl]Cl₂ are:

• (1) 3 and 5
• (2) 2 and 6
• (3) 2 and 8
• (4) 3 and 6

An ammoniacal metal salt solution gives a brilliant red precipitate on the addition of dimethylglyoxime. The metal ion is:

• (1) Cu²⁺
• (2) Co²⁺
• (3) Fe²⁺
• (4) Ni²⁺

'R' formed in the following sequence of reactions is: [Image of a reaction scheme is present]

Match List I with List II:

List I: A. Chlorophyll, B. Soda ash, C. Dentistry, Ornamental work, D. Used in white washing

List II: I. Na₂CO₃, II. CaSO₄, III. Mg²⁺, IV. Ca(OH)₂

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

Statement I: For colloidal particles, the values of colligative properties are of small order as compared to values shown by true solutions at the same concentration.

Statement II: For colloidal particles, the potential difference between the fixed layer and the diffused layer of same charges is called the electrokinetic potential or zeta potential.

• (1) Statement I is true but Statement II is false.
• (2) Statement I is false but Statement II is true.
• (3) Both Statement I and Statement II are true.
• (4) Both Statement I and Statement II are false.

Reaction of BeO with ammonia and hydrogen fluoride gives 'A' which on thermal decomposition gives BeF₂ and NH₄F. What is 'A'?

• (1) (NH₄)₂BeF₄
• (2) H₃NBeF₃
• (3) (NH₃)BeF₃
• (4) (NH₄)Be₂F₅

'A' and 'B' formed in the following set of reactions are:[Image of a reaction scheme is present]

In the following reaction, ‘A' is:[Image of a reaction scheme is present]

The decreasing order of the hydrogen bonding in the following forms of water is correctly represented by:

A. Liquid water, B. Ice, C. Impure water

• (1) A = B > C
• (2) B > A > C
• (3) C > B > A
• (4) A > B > C

Given below are two statements:

Statement I: Noradrenaline is a neurotransmitter.

Statement II: Low level of noradrenaline is not the cause of depression in humans.

• (1) Statement I is correct but Statement II is incorrect.
• (2) Statement I is incorrect but Statement II is correct.
• (3) Both Statement I and Statement II are correct.
• (4) Both Statement I and Statement II are incorrect.

In the depression of freezing point experiment:

A. Vapour pressure of the solution is less than that of pure solvent.
B. Vapour pressure of the solution is more than that of pure solvent.
C. Only solute molecules solidify at the freezing point.
D. Only solvent molecules solidify at the freezing point.

• (1) A and D only
• (2) B and C only
• (3) A and C only
• (4) A only

Which of the following is true about freons?

• (1) These are chlorofluorocarbon compounds.
• (2) These are chemicals causing skin cancer.
• (3) These are radicals of chlorine and chlorine monoxide.
• (4) All radicals are called freons.

Section B

Question 21:

The dissociation constant of acetic acid is x × 10^-5. When 25 mL of 0.2 M CH₃COONa solution is mixed with 25 mL of 0.02 M CH₃COOH solution, the pH of the resultant solution is found to be equal to 5. The value of x is ________

5 g of NaOH was dissolved in deionized water to prepare a 450 mL stock solution. What volume (in mL) of this solution would be required to prepare 500 mL of 0.1 M solution?

If the wavelength of the first line of the Paschen series of the hydrogen atom is 720 nm, then the wavelength of the second line of this series is _______ nm (nearest integer).

The number of correct statements from the following is _______

A. Larger the activation energy, smaller is the value of the rate constant.
B. The higher is the activation energy, higher is the value of the temperature coefficient.
C. At lower temperatures, increase in temperature causes more change in the value of k than at higher temperatures.
D. A plot of ln k vs 1/T is a straight line with slope equal to -Ea/R.

At 298 K, a 1-liter solution containing 10 mmol of Cr₂O₇^2- and 100 mmol of Cr³⁺ shows a pH of 3.0. The potential for the half-cell reaction is x × 10^-3 V. The value of x is ________

When Fe_0.93O is heated in the presence of oxygen, it converts to Fe₂O₃. The number of correct statements from the following is _______.

A. The equivalent weight of Fe_0.93O is Molecular weight / 0.79
B. The number of moles of Fe²⁺ and Fe³⁺ in 1 mole of Fe_0.93O is 0.79 and 0.14, respectively.
C. Fe_0.93O is metal deficient with a lattice comprising cubic close-packed arrangement of O^2- ions.
D. The percentage composition of Fe²⁺ and Fe³⁺ in Fe_0.93O is 85 and 15, respectively.

The d-electronic configuration of [CoCl₄]^2- in tetrahedral crystal field is e^mt_n². Sum of 'm' and the number of unpaired electrons is _______.

For independent processes at 300 K, the number of non-spontaneous processes from the following is ________

Process A: ∆H = -25 kJ mol^-1, ∆S = -80 J K^-1mol^-1
Process B: ∆H = -22 kJ mol^-1, ∆S = 40 J K^-1mol^-1
Process C: ∆H = 25 kJ mol^-1, ∆S = -50 J K^-1mol^-1
Process D: ∆H = 22 kJ mol^-1, ∆S = 20 J K^-1mol^-1

Uracil is a base present in RNA with the following structure. The percentage of nitrogen in uracil is _______.

Number of moles of AgCl formed in the following reaction is _______

Mathematics
Section – A

Question 1:

The distance of the point (7, -3, -4) from the plane passing through the points (2, -3, 1), (-1, 1, -2), and (3, -4, 2) is:

• (1) \( 4 \)
• (2) \( 5 \)
• (3) \( 5\sqrt{2} \)
• (4) \( 4\sqrt{2} \)

The value of the following limit is: \[ \lim_{t \to 0} \left( \frac{1}{1^{2n^2 t} + 2^{\frac{1}{2n^2 t}} + \ldots + n^{\frac{1}{2n^2 t}}} \right)^{2n^2 t} \]

• (1) \( n^2 + n \)
• (2) \( n \)
• (3) \( \frac{n(n+1)}{2} \)
• (4) \( n^2 \)

Let \( \mathbf{u} = \hat{i} - \hat{j} - 2\hat{k}, \mathbf{v} = 2\hat{i} + \hat{j} - \hat{k}, \mathbf{w} = 2 \) and \( \mathbf{v} \times \mathbf{w} = \mathbf{u} + \lambda \mathbf{v}. \) Then \( \mathbf{u} \cdot \mathbf{w} \) is equal to:

• (1) \( 1 \)
• (2) \( \frac{3}{2} \)
• (3) \( 2 \)
• (4) \( \frac{2}{3} \)

The value of \( \sum_{r=0}^{22} 22C_r 23C_{r} \) is:

• (1) \( 45 C_{23} \)
• (2) \( 44 C_{23} \)
• (3) \( 45 C_{24} \)
• (4) \( 44 C_{22} \)

Let a tangent to the curve \( y^2 = 24x \) meet the curve \( xy = 2 \) at the points A and B. Then the midpoints of such line segments AB lie on a parabola with the:

• (1) Directrix \( 4x = 3 \)
• (2) Directrix \( 4x = -3 \)
• (3) Length of latus rectum \( \frac{3}{2} \)
• (4) Length of latus rectum 2

Let \( N \) denote the number that turns up when a fair die is rolled. If the probability that the system of equations \[ x + y + z = 1 \] \[ 2x + Ny + 2z = 2 \] \[ 3x + 3y + Nz = 3 \]
has unique solution is \( \frac{k}{6} \), then the sum of value of \( k \) and all possible values of \( N \) is:

• (1) 18
• (2) 19
• (3) 20
• (4) 21

tan^{-1 \left( \frac{1 + \sqrt{3{3 + \sqrt{3 \right) + \sec^{-1 \sqrt{\left( \frac{8 + 4 \sqrt{3{6 + 3 \sqrt{3 \right) is equal to:

• (1) \( \frac{\pi}{4} \)
• (2) \( \frac{\pi}{2} \)
• (3) \( \frac{\pi}{3} \)
• (4) \( \frac{\pi}{6} \)

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that \[ \frac{QA}{AR} = \frac{RB}{BP} = \frac{PC}{CQ} = \frac{1}{2} \]
Then \[ \frac{Area of \, \Delta PQR}{Area of \, \Delta ABC} is equal to: \]

• (1) 4
• (2) 3
• (3) 2
• (4) \( \frac{5}{2} \)

If A and B are two non-zero n \(\times\) n matrices such that \(A^2 + B = A^2 B\), then:

• (1) \( AB = I \)
• (2) \( A^2 B = I \)
• (3) \( A^2 = I \) or \( B = I \)
• (4) \( A^2 B = BA^2 \)

Let \( y = y(x) \) be the solution of the differential equation \( x^3 \frac{dy}{dx} + (xy - 1) dx = 0, x > 0, y(1) = e^{-3} \). Then \( y(1) \) is equal to:

• (1) 1
• (2) \( e \)
• (3) \( 2 - e \)
• (4) \( 3 - e \)

The area enclosed by the curves \( y^2 + 4x = 4 \) and \( y - 2x = 2 \) is:

• (1) \( \frac{25}{3} \)
• (2) \( \frac{22}{3} \)
• (3) \( 9 \)
• (4) \( \frac{23}{3} \)

Let \( \alpha \) be a root of the equation \( (a - c) x^2 + (b - a) x + (c - b) = 0 \), where \( a, b, c \) are distinct real numbers such that the matrix \[ \begin{pmatrix} \alpha^2 & \alpha & 1
1 & 1 & 1
a & b & c \end{pmatrix} \]
is singular. Then the value of \( (a - c)^2 \) is:

• (1) 6
• (2) 9
• (3) 12
• (4) 5

The distance of the point \( (-1, 9, -16) \) from the plane \( 2x + 3y - z = 5 \) measured parallel to the line \( x+4 = 2y - z = 3 \) is:

• (1) \( 13 \sqrt{2} \)
• (2) \( 31 \)
• (3) \( 26 \)
• (4) \( 20 \sqrt{2} \)

For three positive integers \( p, q, r \), such that \( x^n = y^m = z^p \) and \( r = pq + 1 \), and \( 3, 3 \log x, 3 \log y, 7 \log z \) are in A.P. with common difference \( \frac{1}{2} \). Then \( r - p - q \) is equal to:

• (1) 2
• (2) 6
• (3) 12
• (4) -6

Let \( p, q \in \mathbb{R} \) and \( (1 - \sqrt{3}i)^{200} = 2^{199} (p + iq) \), then \( p \) and \( q \) are equal to:

• (1) \( \sqrt{3} \) and \( -\sqrt{3} \)
• (2) \( 2 \) and \( -3 \)
• (3) \( 4 \) and \( 5 \)
• (4) \( 3 \) and \( 2 \)

The relation \( R = \{(a, b): gcd(a, b) = 1, 2a = b, a, b \in \mathbb{Z}\} \) is:

• (1) transitive but not reflexive
• (2) symmetric but not transitive
• (3) reflexive but not symmetric
• (4) neither symmetric nor transitive

The compound statement \( (\neg (P \land Q)) \lor ((\neg P) \land Q) \) is equivalent to:

• (1) \( (\neg P) \lor Q \)
• (2) \( \neg Q \lor P \)
• (3) \( (\neg Q) \land (\neg P) \)
• (4) \( (\neg P) \lor Q \)

If \( f(x) = \frac{\sin(\frac{1}{x})}{x} \), then at \( x = 0 \):

• (1) \( f(x) \) is continuous but not differentiable
• (2) \( f(x) \) is continuous but \( f' \) is not continuous
• (3) \( f' \) is continuous but not differentiable
• (4) \( f' \) is continuous and differentiable

The equation \( x^2 - 4x + [x] + 3 = [x] \), where \( [x] \) denotes the greatest integer function, has:

• (1) exactly two solutions in \( (-\infty, \infty) \)
• (2) no solution
• (3) a unique solution in \( (-\infty, 1) \)
• (4) a unique solution in \( (-\infty, 0) \)

Let \( \Omega \) be the sample space and \( A \subset \Omega \) be an event. Given below are two statements:
(S1): If \( P(A) = 0 \), then \( A = \Omega \)
(S2): If \( P(A) = 1 \), then \( A = \Omega \).
Which of the following is correct?

• (1) only (S1) is true
• (2) only (S2) is true
• (3) both (S1) and (S2) are true
• (4) both (S1) and (S2) are false

Section – B

Question 21:

Let C be the largest circle centered at (2, 0) and inscribed in the ellipse \( \frac{x^2}{36} + \frac{y^2}{16} = 1 \). If the point \( (1, \alpha) \) lies on C, then \( 10 \alpha^2 \) is equal to:

Suppose \( \sum_{r=0}^{2023} r^2 \binom{2023}{r} = 2023 \times \alpha \times 2022^{2022} \). Then the value of \( \alpha \) is:

The value of \( \int_0^{12} |x^2 - 3x + 2| dx \) is:

The number of 9-digit numbers that can be formed using all the digits of the number 123412341 such that the even digits occupy only even places, is:

Let \( \lambda \in \mathbb{R} \) and let the equation \( E = |x^2 - 2|x| + |x - 3| = 0 \). Then the largest element in the set \( S = \{ x + \lambda : x is an integer solution of E \} \) is:

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is:

Let a tangent to the curve \( 9x^2 + 16y^2 = 144 \) intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is:

The value of \( \int_0^{\frac{\pi}{2}} \frac{( \cos x )^{2023}}{( \sin x )^{2023} + ( \cos x )^{2023}} dx \) is:

The shortest distance between the lines \( \frac{x-2}{3} = \frac{y+1}{2} = \frac{z-6}{2} \) and \( \frac{x-6}{1} = \frac{y-1}{2} = \frac{z+8}{3} \) is equal to:

The 4^th term of a GP is 500 and its common ratio is \( \frac{1}{m} \), where \( m \in \mathbb{N} \). Let \( S_n \) denote the sum of the first \( n \) terms of this GP. If \( S_6 > S_5 + 1 \) and \( S_7 < S_6 + \frac{1}{2} \), then the number of possible values of \( m \) is: