Practice Questions

Q77.Let 𝑦= 𝑦( 𝑥) be the solution of the differential equation 𝑑𝑦 1 + 𝑥𝑒𝑦- 𝑥, - √2 < 𝑥< √2, 𝑦0 = 0 𝑑𝑥= , then the minimum value of 𝑦𝑥, 𝑥∈-√2, √2 is equal to : (1) 2 - √3 - loge2 (2) 2 + √3 + loge2 (3) 1 + √3 - loge√3 - 1 (4) 1 - √3 - loge√3 - 1

Q77.Let y(x) be the solution of the differential equation 2x2 dy + (ey −2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to: (1) loge(2e) (2) loge 2 (3) 2 (4) 0

Q77.If the curve y = y(x) is the solution of the differential equation 2(x2 + x5/4)dy −y(x + x1/4)dx = 2x9/4dx, x > 0 which passes through the point (1, 1 −43 loge 2), then the value of y(16) is equal to (1) 4( 313 + 38 loge 3) (2) ( 313 + 38 loge 3) (3) 4( 313 −83 loge 3) (4) ( 313 −83 loge 3) −−

Q77.Let y = y(x) be the solution of the differential equation dxdy = (y + 1)((y + 1)ex2/2 −x), y(2) = 0. Then the value of dxdy at x = 1 is equal to (1) −e3/2 (2) − 2e2 (e2+1)2 (1+e2)2 (3) e5/2 (4) 5e1/2 (1+e2)2 (e2+1)2 −−−−−

Q77.Let y = y(x) be solution of the differential equation loge( dxdy ) y(−23 loge 2) = α loge 2 , then the value of α is equal to: JEE Main 2021 (27 Jul Shift 1) JEE Main Previous Year Paper (1) −14 (2) 41 (3) 2 (4) −12 →

Q77.Let y = y(x) be the solution of the differential equation (x −x3)dy = (y + yx2 −3x4)dx, x > 2 If y(3) = 3, then y(4) is equal to: (1) 4 (2) 12 (3) 8 (4) 16 b If magnitudes of the vectors →a, b and →care √2, 1 and

Q77.Let three vectors →a, b and →cbe such that →a× b =→c, b ×→c=→a and →a = 2. Then which one of the following is not true? b b b × is 2 (1) →a× ((→ → → (2) → +→c) ( −→c)) = 0 Projection of →a on ( ×→c) + = 8 (4) 3→a+→b −2→c 2 = 51 (3) [→a →b →c] [→c →a →b ] JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper = 2. If P(α, β, γ) is the

Q77.Let →a = ˆi + 2ˆj −3ˆk and b = 2ˆi −3ˆj + 5ˆk. If →r×→a = b ×→r,→r⋅(αˆi + 2ˆj + ˆk) 2 is equal to : = −1, α ∈R, then the value of α + →r →r⋅(2ˆi + 5ˆj −αˆk) (1) 9 (2) 15 (3) 13 (4) 11

Q78.Let L be the line of intersection of planes →r⋅(ˆi −ˆj + 2ˆk) = 2 and →r⋅(2ˆi + ˆj −ˆk) foot of perpendicular on L from the point (1, 2, 0), then the value of 35(α + β + γ) is equal to: (1) 101 (2) 119 (3) 143 (4) 134

Q78.If dy dx = 2y , y(0) = 1, then y(1) is equal to : (1) log2(1 + e2) (2) log2(2e) (3) log2(2 + e) (4) log2(1 + e) → → → → 1 is a unit

Q78.Let y = y(x) be the solution of the differential equation ex√1 −y2 dx + ( xy )dy = 0, y(1) = −1 Then the value of (y(3))2 is equal to: (1) 1 −4e3 (2) 1 −4e6 (3) 1 + 4e3 (4) 1 + 4e6 →

Q78.Let the position vectors of two points P and Q be 3ˆi −ˆj + 2ˆk and ˆi + 2ˆj −4ˆk, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, −1, 2) and (−2, 1, −2), respectively. Let −−−→ → → lines PR and QS intersect at T . If the vector TA is perpendicular to both PR and QS and the length of vector −→ TA is √5 units, then the modulus of a position vector of A is : (1) √482 (2) √171 (3) √5 (4) √227 P divides the line

Q78.A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, −1, 1) −→ , then the projection of OP on this plane is of length: (1) √25 (2) √27 (3) √23 (4) √211

Q79.If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line x+1 2 = y−31 = z+2−1 and containing the line x−23 = 1−y2 = z+11 is αx + βy + γz = 24 then α + β + γ is equal to: (1) 20 (2) 19 (3) 18 (4) 21

Q79.Let the foot of perpendicular from a point 𝑃( 1, 2, - 1 ) to the straight line 𝐿: 𝑥 = 𝑦 = 𝑧 be 𝑁. Let a line be 1 0 -1 drawn from 𝑃 parallel to the plane 𝑥+ 𝑦+ 2𝑧= 0 which meets 𝐿 at point 𝑄. If 𝛼 is the acute angle between the lines 𝑃𝑁 and 𝑃𝑄, then cos𝛼 is equal to . 1 √3 (1) (2) √5 2 1 1 (3) (4) √3 2√3

Q79.The distance of the point ( - 1, 2, - 2 ) from the line of intersection of the planes 2𝑥+ 3𝑦+ 2𝑧= 0 and 𝑥- 2𝑦+ 𝑧= 0 is : 1 √42 (1) (2) √2 2 5 √34 (3) (4) 2 2

Q79.If the foot of the perpendicular from point (4, 3, 8) on the line L1 : x−al = y−23 = z−b4 , l ≠0 is (3, 5, 7), then the shortest distance between the line L1 and line L2 : x−23 = y−44 = z−55 is equal to (1) 1 (2) 1 2 √6 (3) √23 (4) √31 JEE Main 2021 (16 Mar Shift 2) JEE Main Previous Year Paper

Q79.Let a, b ∈R. If the mirror image of the point P(a, 6, 9) with respect to the line x−37 = y−25 = z−1−9 is (20, b, −a −9), then |a + b| is equal to: (1) 86 (2) 90 (3) 84 (4) 88

Q79.Consider the line L given by the equation x−3 2 = y−11 = z−21 . Let Q be the mirror image of the point (2, 3, −1) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P? (1) (−1, 1, 2) (2) (1, 1, 1) (3) (1, 1, 2) (4) (1, 2, 2)

Q80.The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is: (1) 65 (2) 65 28 27 (3) 35 (4) 135 27 29

Q80.Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 × 2 matrices. The probability that such formed matrices have all different entries and are non-singular, is: (1) 45 (2) 23 162 81 (3) 22 (4) 43 81 162

Q80.Let 𝑆= {1, 2, 3, 4, 5, 6} . Then the probability that a randomly chosen onto function 𝑔 from 𝑆 to 𝑆 satisfies 𝑔3 = 2 𝑔1 is : 1 1 (1) (2) 15 5 (3) 1 (4) 1 30 10

Q80.Let A be a set of all 4 -digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is: (1) 1 (2) 122 5 297 (3) 97 (4) 2 297 9

Q80.Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is: JEE Main 2021 (20 Jul Shift 1) JEE Main Previous Year Paper (1) 1 (2) 1 66 11 (3) 1 (4) 2 9 11

Q81.If the least and the largest real values of 𝛼, for which the equation 𝑧+ 𝛼𝑧- 1 + 2𝑖= 0 𝑧∈𝐶 and 𝑖= √-1 has a solution, are 𝑝 and 𝑞 respectively; then 4𝑝2 + 𝑞2 is equal to_______.