Practice Questions

Q77.If →a and→b are perpendicular, then →a× (→a (→a (→a →b))) 4→ (1) →a b (2) →0 → 4→ 1 (3) →a× b (4) 2 →a b

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 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 f(x) = {ax2 + b ; |x| < 1 respectively: (1) 1 2 , 12 (2) 12 , −32 (3) 2 5 , −32 (4) −12 , 32

Q77.A differential equation representing the family of parabolas with axis parallel to y−axis and whose length of latus rectum is the distance of the point (2, −3) from the line 3x + 4y = 5, is given by: (1) 11 d2x dy2 = 10 (2) 11 dx2d2y = 10 d2y (3) 10 = 11 (4) 10 d2xdy2 = 11 dx2 = 1 and

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 →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

Q77.Let y = y(x) be the solution of the differential equation dydx = 2(y + 2 sin x −5)x −2 cos x such that y(0) = 7. Then y(π) is equal to (1) 7eπ2 + 5 (2) eπ2 + 5 (3) 2eπ2 + 5 (4) 3eπ2 + 5

Q77.Which of the following is true for y(x) that satisfies the differential equation dy = xy −1 + x −y; y(0) = 0 dx (1) y(1) = e−12 −1 (2) y(1) = e 12 −e−12 (3) y(1) = 1 (4) y(1) = e 21 −1 → → + 2ˆj + = −3, then →r⋅(2ˆi −3ˆj + ˆk) is

Q77.Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If dt, 0 ≤x ≤1 and f(0) = 0, then : lim x21 ∫x0 x→0 ∫x0 √1 −(f ′(t))2 dt = ∫x0 f(t) f(t)dt (1) does not exist (2) equals 0 (3) equals 1 (4) equals 21 2x+y−2x

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 (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.If 𝑦= 𝑦( 𝑥) is the solution curve of the differential equation 𝑥2 d𝑦+ 𝑦- 1 0; 𝑥> 0 and 𝑦( 1 ) = 1, 𝑥d𝑥= 1 then 𝑦 is equal to : 2 (1) 3 + e (2) 3 - e 3 1 1 (3) - (4) 3 + 2 √e √e

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.If for a > 0, the feet of perpendiculars from the points A(a, −2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, −a, −1) and D respectively, then the length of line segment CD is equal to : (1) √31 (2) √41 (3) √55 (4) √66

Q77.If 𝑦0 = 0, then for 𝑦= 1, the value of 𝑥 lies in the interval : 𝑑𝑥= 2𝑥+ 2𝑥+ 𝑦log𝑒2, 1 (1) 1, 2 (2) 2, 1 (3) 2, 3 (4) 0, 1 2

Q78.Let →a = ˆi + ˆj + ˆk and→b = ˆj −ˆk. If →cis a vector such that →a×→c=→b and →a⋅→c= 3, then →a⋅(→ ×→c) to: (1) 6 (2) −2 (3) 2 (4) −6

Q78.The distance of the point (1, −2, 3) from the plane x −y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, −6 , is (1) 2 (2) 5 (3) 3 (4) 1 units from the origin, which contains the line of intersection of the

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 a, b and c be distinct positive numbers. If the vectors aˆi + aˆj + cˆk,ˆi + ˆk and cˆi + cˆj + bˆk are co-planar, then c is equal to: 2 (1) (2) a+b 1 2 1 + a b (3) a 1 + 1b (4) √ab

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

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

Q78.The vector equation of the plane passing through the intersection of the planes →r⋅(ˆi +ˆj + ˆk) = −2, and the point (1, 0, 2) is: →r⋅(ˆi −2ˆj) = 73 (1) →r⋅(ˆi + 7ˆj + 3ˆk) = 7 (2) →r⋅(ˆi −7ˆj + 3ˆk) = 7 = 37 (4) →r⋅(3ˆi + 7ˆj + 3ˆk) (3) →r⋅(ˆi + 7ˆj + 3ˆk)

Q78.Let O be the origin. Let OP→ = xˆi + yˆj −ˆk and OQ→ = −ˆi + 2ˆj + 3xˆk, x, y ∈R, x > 0, be such that −−−−−→ → → → → PQ = √20 and the vector OP is perpendicular to OQ. If OR = 3ˆi + zˆj −7ˆk, z ∈R, is coplanar with OP −→ and OQ, then the value of x2 + y2 + z2 is equal to (1) 7 (2) 9 (3) 2 (4) 1