Practice Questions

Q19.Let the curve z(1 + i) + ¯z(1 −i) = 4, z ∈C, divide the region |z −3| ≤1 into two parts of areas α and β . Then |α −β| equals : (1) 1 + π2 (2) 1 + π3 (3) 1 + π6 (4) 1 + π4

Q19.If in the expansion of (1 + x)p(1 −x)q , the coefficients of x and x2 are 1 and -2 , respectively, then p2 + q2 is equal to : (1) 18 (2) 13 (3) 8 (4) 20 a

Q20.Two equal sides of an isosceles triangle are along −x + 2y = 4 and x + y = 4. If m is the slope of its third side, then the sum, of all possible distinct values of m, is : (1) −2√10 (2) 12 (3) 6 (4) −6

Q20.If α > β > γ > 0, then the expression cot−1 {β (α−β) } + cot−1 {γ (β−γ) } + cot−1 {α (γ−α) } equal to : (1) π (2) 0 (3) π 2 −(α + β + γ) (4) 3π L.

Q20.If sin x + sin2 x = 1, x ∈(0, π2 ), then (cos12 x + tan12 x) + 3 (cos10 x + tan10 x + cos8 x + tan8 x) + (cos6 x + tan6 x) is equal to : (1) 4 (2) 1 (3) 3 (4) 2 π

Q20.If the area of the region {(x, y) : −1 ≤x ≤1, 0 ≤y ≤a + e|x| −e−x, a > 0} is e2+8e+1e , then the value of is : (1) 8 (2) 7 (3) 5 (4) 6

Q20.Let →a = ^i + 2^j + 3^k,→b = 3^i + ^j −^k and →c be three vectors such that →c is coplanar with →a and →b. If the vector →C is perpendicular to →b and →a ⋅→c = 5, then |→c| is equal to (1) √116 (2) 3√21 (3) 16 (4) 18

Q20.Let the area of the region {(x, y) : 2y ≤x2 + 3, y + |x| ≤3, y ⩾|x −1|} be A. Then 6 A is equal to : (1) 16 (2) 12 (3) 14 (4) 18

Q20.Let z1, z2 and z3 be three complex numbers on the circle |z| = 1 with arg (z1) = −π4 , arg (z2) = 0 and arg (z3) = π4 . If |z1¯z2 + z2¯z3 + z3¯z1|2 = α + β√2, α, β ∈Z, then the value of α2 + β2 is : (1) 24 (2) 29 (3) 41 (4) 31

Q20.Let E : x2 + y2 = 1, a > b and H : x2 − y2 = 1. Let the distance between the foci of E and the foci of H a2 b2 A2 B2 be 2√3. If a −A = 2, and the ratio of the eccentricities of E and H is 13 , then the sum of the lengths of their latus rectums is equal to: (1) 10 (2) 9 (3) 8 (4) 7 = α × 229 , then α is equal to ______

Q20.If π 2 ≤x ≤3π4 , then cos−1 ( 1213 cos x + 135 sin x) is equal to (1) x −tan−1 43 (2) x + tan−1 45 (3) x −tan−1 125 (4) x + tan−1 125

Q21.Let S = {p1, p2 … . , p10} be the set of first ten prime numbers. Let A = S ∪P , where P is the set of all possible products of distinct elements of S . Then the number of all ordered pairs ( x, y ), x ∈S , y ∈A , such that x divides y, is ______.

Q21.The variance of the numbers 8, 21, 34, 47, … , 320 is

Q26.During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is 2000Å and it becomes 6000Å when the electron jumps from state B to state C. Then the wavelength of the radiation emitted during the transition of electrons from state A to state B is (1) 4000Å (2) 2000Å (3) 3000Å (4) 6000Å

Q26.Three infinitely long wires with linear charge density λ are placed along the x −axis, y-axis and z− axis respectively. Which of the following denotes an equipotential surface? (1) xyz = constant (2) xy + yz + zx = constant (3) (x2 + y2) (y2 + z2) (z2 + x2) = constant (4) (x + y)(y + z)(z + x) = constant

Q26. A B Y 0 0 1 0 1 1 1 0 0 1 1 1 To obtain the given truth table, following logic gate should be placed at G: 2025 (22 Jan Shift 2) JEE Main Previous Year Paper (1) OR Gate (2) AND Gate (3) NOR Gate (4) NAND Gate

Q26. A bar magnet has total length 2l = 20 units and the field point P is at a distance d = 10 units from the centre of the magnet. If the relative uncertainty of length measurement is 1% , then uncertainty of the magnetic field at point P is : (1) 4% (2) 15% (3) 5% (4) 10%

Q26.A galvanometer having a coil of resistance 30Ω need 20 mA of current for full-scale deflection. If a maximum current of 3 A is to be measured using this galvanometer, the resistance of the shunt to be added to the galvanometer should be 30 X Ω, where X is Options (1) 596 (2) 149 (3) 298 (4) 447

Q26.A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r . The dipoles are ∣∣ 2025 (23 Jan Shift 1) JEE Main Previous Year Paper made of opposite charge q separated by a distance 2a . For the charge particle at P not to experience any net force, which of the following correctly describes the situation? (1) a r ∼10 (2) ar ∼20 (3) a r ∼0.5 (4) ar ∼3

Q26.An electric dipole of mass m, charge q, and length l is placed in a uniform electric field E = E0^i. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be: (1) 1 ml (2) ml 2π √ 2qE0 2π√ qE0 (3) 1 ml (4) ml 2π √2qE0 2π√ 2qE0

Q26.An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component of velocity 106 m/s. If the magnitude of the electric field between the plates is 9.1 V/cm , then the vertical component of velocity of electron is (mass of electron = 9.1 × 10−31 kg and charge of electron = 1.6 × 10−19C ) (1) 0 (2) 1 × 106 m/s (3) 16 × 106 m/s (4) 16 × 104 m/s

Q27.Two identical symmetric double convex lenses of focal length f are cut into two equal parts L1, L2 by AB plane and L3, L4 by XY plane as shown in figure respectively. The ratio of focal lengths of lenses L1 and L3 is (1) 1:1 (2) 1:2 (3) 1:4 (4) 2:1

Q27.A small rigid spherical ball of mass M is dropped in a long vertical tube containing glycerine. The velocity of the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the viscous force acting on the ball will be (consider g as acceleration due to gravity) (1) 2 Mg (2) Mg (3) 3 2 Mg (4) Mg2

Q27.A concave mirror produces an image of an object such that the distance between the object and image is 20 cm . If the magnification of the image is ' -3 ', then the magnitude of the radius of curvature of the mirror is : (1) 30 cm (2) 3.75 cm (3) 15 cm (4) 7.5 cm 2025 (28 Jan Shift 2) JEE Main Previous Year Paper

Q27.The position vector of a moving body at any instant of time is given as →r = . The magnitude (5t2^i −5t^j)m and direction of velocity at t = 2 s is, (1) 5√15 m/s, making an angle of tan−1 4 with - ve (2) 5√15 m/s, making an angle of tan−1 4 with + ve Y axis X axis (3) 5√17 m/s, making an angle of tan−1 4 with + ve (4) 5√17 m/s, making an angle of tan−1 4 with - ve X axis Y axis