Practice Questions

Q89.Let a curve y = y(x) be given by the solution of the differential equation y-axis at y = −1, and the intersection point of the cos( 12 cos−1(e−x))dx = (√e2x −1)dy. If it intersects curve with x− axis is (α, 0), then eα is equal to

Q89.If the projection of the vector ˆi + 2ˆj + ˆk on the sum of the two vectors 2ˆi + 4ˆj −5ˆk and −λˆi + 2ˆj + 3ˆk is 1, then λ is equal to _______.

Q89.Let →a = ˆi + αˆj + 3ˆk and →b = 3ˆi −αˆj + ˆk. If the area of the parallelogram whose adjacent sides are represented → → by the vectors →a and b is 8√3 square units, then →a⋅ b is equal to ___ .

Q89.Let the mirror image of the point (1, 3, a) with respect to the plane →r⋅(2ˆi −ˆj + ˆk) Then the value of |a + b| is equal to ___ . y+6

Q89.Let 𝑦= 𝑦( 𝑥) be solution of the following differential equation 𝑒𝑦𝑑𝑦 2𝑒𝑦sin𝑥+ sin𝑥cos2𝑥= 0, 𝑦 𝜋 = 0. 𝑑𝑥- 2 If 𝑦0 = loge𝛼+ 𝛽e-2, then 4 ( 𝛼+ 𝛽) is equal to .

Q89.Let P be a plane passing through the points (1, 0, 1), (1, −2, 1) and (0, 1, −2). Let a vector →a = αˆi + βˆj + γˆk = 2 , then be such that →a is parallel to the plane P , perpendicular to (ˆi + 2ˆj + 3ˆk) and →a⋅(ˆi + ˆj + 2ˆk) (α −β + γ)2 equals______. → + λ ∈R, α > 0 and

Q89.If ∫ b( x2+x+12x+1 ) (x2+x+1)2 dx = a tan−1( 2x+1√3 ) + value of 9(√3a b) is equal to _________. → →

Q89.If the lines x−k k is _______. 1 = 2 = z−33 and x+13 = y+22 = z+31 are co-planar, then the value of

Q89.If the area of the triangle formed by the x-axis, the normal and the tangent to the circle (x −2)2 + (y −3)2 = 25 at the point (5, 7) is A, then 24A is equal to _________.

Q89.Let →cbe a vector perpendicular to the vectors →a = ˆi + ˆj −ˆk and b = ˆi + 2ˆj + ˆk. If →c⋅(ˆi + ˆj + 3ˆk) → is equal to × the value of →c⋅(→a b)

Q89.Let f : R →R be a continuous function such that f(x) + f(x + 1) = 2 for all x ∈R . If I1 = ∫80 f(x)dx and I2 = ∫3−1 f(x)dx , then the value of I1 + 2I2 is equal to ________.

Q89.Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equal to ______. dx = αIm,n, α ∈R, then α equals

Q89.If the line 𝑦= 𝑚𝑥 bisects the area enclosed by the lines 𝑥= 0, 𝑦= 0, 𝑥= and the curve 2 𝑦= 1 + 4𝑥- 𝑥2, then 12𝑚 is equal to .

Q89.Let three vectors →𝑎, →𝑏 and →𝑐 be such that →𝑐 is coplanar with →𝑎 and →𝑏, →𝑎· →𝑐= 7 and →𝑏 is perpendicular to →𝑐, 2 where →𝑎= - ^𝑖+ ^𝑗+ ^𝑘 and →𝑏= 2 ^𝑖+ ^𝑘, then the value of 2 →𝑎+ →𝑏+ →𝑐 is

Q89.The square of the distance of the point of intersection of the line x−1 2 = y−23 = z+16 and the plane 2 x −y + z = 6 from the point (−1, −1, 2) is

Q89.The area of the region S = {(x, y) : 3x2 ≤4y ≤6x + 24} is______.

Q89.If the equation of the plane passing through the line of intersection of the planes 2x −7y + 4z −3 = 0, 3x −5y + 4z + 11 = 0 and the point (−2, 1, 3) is ax + by + cz −7 = 0, then the value of 2a + b + c −7 is _________.

Q89.Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, −3, 1) and (2, 3, −5) at the right angles. If a, b, c, d are integers, then the minimum value of (a2 + b2 + c2 + d2) is

Q90.Let →a = ˆi + 2ˆj −ˆk, b = ˆi −ˆj and →c= ˆi −ˆj −ˆk be three given vectors. If →ris a vector such that →r×→a =→c×→a → and →r⋅ b = 0, then →r⋅→a is equal to JEE Main 2021 (25 Feb Shift 1) JEE Main Previous Year Paper

Q90.Suppose the line 𝑥- 2 = 𝑦- 2 = 𝑧+ 2 lies on the plane 𝑥+ 3𝑦- 2𝑧+ 𝛽= 0 . Then ( 𝛼+ 𝛽) is equal to 𝛼 -5 2 . JEE Main 2021 (31 Aug Shift 2) JEE Main Previous Year Paper

Q90.Let y = y(x) be the solution of the differential equation x+2 ) + (y + = (x + 2)dy, y(1) = 1. If the domain of y = y(x) is an open interval (α, β), + 2)e( 1))dx ((x y+1 then |α + β| is equal to ___________. JEE Main 2021 (22 Jul Shift 1) JEE Main Previous Year Paper

Q90.Let there be three independent events E1, E2 and E3. The probability that only E1 occurs is α only E2 occurs is β and only E3 occurs is γ. Let ′p′ denote the probability of none of events occurs that satisfies the equations (α −2β)p = αβ and (β −3γ)p = 2βγ. All the given probabilities are assumed to lie in the interval (0, 1). Then, Probability of occurrence of E1 is equal to ________. Probability of occurrence of E3 JEE Main 2021 (17 Mar Shift 1) JEE Main Previous Year Paper

Q90.An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0. 9 and that of the second unit is 0. 8. The instrument is switched on and it fails to operate. If the probability that only the first unit failed and second unit is functioning is p, then 98p is equal to JEE Main 2021 (31 Aug Shift 1) JEE Main Previous Year Paper

Q90.If Im,n = ∫10 xm−1(1 −x)n−1dx, for m, n ⩾1, and ∫10 xm−1+xn−1(1+x)m+n ________. JEE Main 2021 (26 Feb Shift 2) JEE Main Previous Year Paper

Q90.Let the curve y = y(x) be the solution of the differential equation, dxdy = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is 4√83 , then the value of y(1) is equal to ________. JEE Main 2021 (16 Mar Shift 1) JEE Main Previous Year Paper