Practice Questions

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.If (1, 5, 35), (7, 5, 5), (1, λ, 7) and (2λ, 1, 2) are coplanar, then the sum of all possible values of λ is: (1) 445 (2) −445 (3) 395 (4) −395 JEE Main 2021 (26 Feb Shift 1) JEE Main Previous Year Paper

Q79.The angle between the straight lines, whose direction cosines l, m, n are given by the equations 2l + 2 m −n = 0 and mn + nl+ lm= 0, is: (1) π (2) π 3 2 (3) cos−1( 89 ) (4) π −cos−1( 94 )

Q79.Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes = 6. Then which of the following points does NOT lie on P ? →r⋅(ˆi + ˆj + 4ˆk) = 16 & →r⋅(−ˆi + ˆj + ˆk) JEE Main 2021 (26 Aug Shift 2) JEE Main Previous Year Paper (1) (4, 2, 2) (2) (6, −6, 2) (3) (−8, 8, 6) (4) (3, 3, 2)

Q79.Let →a and b be two non-zero vectors perpendicular to each other and →a = b , If →a× b = →a , then the angle between the vectors and →a is equal to : + b + × (→a → → (→a b)) JEE Main 2021 (18 Mar Shift 2) JEE Main Previous Year Paper (1) sin−1( √31 ) (2) cos−1( √31 ) (3) cos−1( √21 ) (4) sin−1( √61 )

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)

Q79.The equation of the plane passing through the point 1, 2, - 3 and perpendicular to the planes 3𝑥+ 𝑦- 2𝑧= 5 and 2𝑥- 5𝑦- 𝑧= 7, is (1) 11𝑥+ 𝑦+ 17𝑧+ 38 = 0 (2) 3𝑥- 10𝑦- 2𝑧+ 11 = 0 (3) 6𝑥- 5𝑦+ 2𝑧+ 10 = 0 (4) 6𝑥- 5𝑦- 2𝑧- 2 = 0

Q79.If the shortest distance between the straight lines 3(x −1) = 6(y −2) = 2(z −1) and 4(x −2) = 2(y −λ) = (z −3), λ ∈R is 1 , then the integral value of λ is equal to: √38 (1) 3 (2) 2 (3) 5 (4) −1

Q79.Consider the three planes P1 : 3x + 15y + 21z = 9 P2 : x −3y −z = 5, and P3 : 2x + 10y + 14z = 5 Then, which one of the following is true? (1) P2 and P3 are parallel. (2) P1, P2 and P3 all are parallel. (3) P1 and P2 are parallel. (4) P1 and P3 are parallel.

Q79.The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is (1) dy 2 dy (2) dy 2 dy −y = 0 + y = 0 y( dx ) −2x( dx ) y( dx ) −2x( dx ) + −y = 0 + −y = 0 (4) y( dxdy ) 2x( dxdy ) (3) y( dxdy ) 2 2x( dxdy )

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 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.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.Let →a and b be two vectors such that 2→a+ 3b = 3→a+ b and the angle between →a and b is 60°. If 8→a → vector, then b is equal to : (1) 8 (2) 4 (3) 6 (4) 5

Q79.The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is: (1) x + 3z = 10 (2) x + 3z = 0 (3) 3x + z = 6 (4) 3x −z = 0

Q79.If the mirror image of the point (1, 3, 5) with respect to the plane 4x −5y + 2z = 8 is (α, β, γ), then 5(α + β + γ) equals : (1) 43 (2) 47 (3) 41 (4) 39

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 = 2ˆi + ˆj −2ˆk and b = ˆi + ˆj. If →cis a vector such that →a⋅→c= →c, →c−→a = 2√2 and the angle between π , then the value of is: and →cis × × 6 (→a → → b) (→a b) ×→c (1) 2 (2) 4 3 (3) 3 (4) 32

Q79.The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is: (1) 1 (2) 1 72 36 (3) 1 (4) 5 54 216

Q79.A plane P contains the line x + 2y + 3 z + 1 = 0 = x −y −z −6, and is perpendicular to the plane −2x + y + z + 8 = 0. Then which of the following points lies on P? (1) (2, −1, 1) (2) (0, 1, 1) (3) (−1, 1, 2) (4) (1, 0, 1)

Q79.Let P be a plane lx + my + nz = 0 containing the line, 1−x1 = y+42 = z+23 . If plane segment AB joining points A(−3, −6, 1) and B(2, 4, −3) in ratio k : 1 then the value of k is equal to : (1) 1. 5 (2) 3 (3) 2 (4) 4

Q79.Let the plane passing through the point (−1, 0, −2) and perpendicular to each of the planes 2x + y −z = 2 and x −y −z = 3 be ax + by + cz + 8 = 0. Then the value of a + b + c is equal to: (1) 3 (2) 8 (3) 5 (4) 4

Q79.Equation of a plane at a distance √221 planes x −y −z −1 = 0 and 2x + y −3 z + 4 = 0, is (1) −x + 2y + 2z −3 = 0 (2) 3x −4z + 3 = 0 (3) 3x −1y −5z + 2 = 0 (4) 4x −y −5z + 2 = 0

Q79.Let the acute angle bisector of the two planes 𝑥- 2𝑦- 2𝑧+ 1 = 0 and 2𝑥- 3𝑦- 6𝑧+ 1 = 0 be the plane 𝑃. Then which of the following points lies on 𝑃 ? 1 (1) ( 0, 2, - 4 ) (2) -2, 0, - 2 (3) ( 4, 0, - 2 ) (4) 3, 1, - 1 2

Q79.In a group of 400 people, 160 are smokers and non-vegetarian; 100 are smokers and vegetarian and the remaining 140 are non-smokers and vegetarian. Their chances of getting a particular chest disorder are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is : (1) 14 (2) 7 45 45 (3) 8 (4) 28 45 45