Practice Questions

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 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.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 S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x −y + z + 3 = 0 and let R(3, 5, γ) be a point of this plane. Then the square of the length of the line segment SR is

Q89.Let →a = ˆi −αˆj + βˆk, b = 3ˆi + βˆj −αˆk and →c= −αˆi −2ˆj + ˆk, where α and β are integers. If →a⋅ b = −1 and → → is equal to ______. × b ⋅→c= 10, then (→a b) ⋅→c

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.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.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.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 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= 2 ^𝑖- ^𝑗+ 2 ^𝑘 and →𝑏= ^𝑖+ 2 ^𝑗- ^𝑘. Let a vector →𝑣 be in the plane containing →𝑎 and →𝑏. If →𝑣 is 2 is equal to _____. perpendicular to the vector 3 ^𝑖+ 2 ^𝑗- ^𝑘 and its projection on →𝑎 is 19 units, then |2→𝑣|

Q89.The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A4 is equal to →

Q89.Let →a = ˆi + ˆj + ˆk, b and →c= ˆj −ˆk be three vectors such that →a× b =→cand →a⋅ b = 1. If the length of → projection vector of the vector b on the vector →a×→cis l, then the value of 3l2 is equal to _____.

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

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 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 y = y(x) is the solution of the equation esin y cos y dxdy + esin y cos x = cos x, y(0) = 0; then 1 + y( π6 ) + √32 y( π3 ) + √21 y( π4 ) 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.The area (in sq. units) of the region bounded by the curves x2 + 2y −1 = 0, y2 + 4x −4 = 0 and y2 −4x −4 = 0 in the upper half plane is _________.

Q90.Let a plane P pass through the point (3, 7, −7) and contain the line, x−2−3 = y−32 = z+21 . If distance of the plane P from the origin is d, then d2 is equal to JEE Main 2021 (27 Jul Shift 1) 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

Q90.If the shortest distance between the lines r1 = αˆi + 2ˆj + 2ˆk + λ(ˆi −2ˆj 2ˆk), → μ ∈R is 9, then α is equal to_____. r2 = −4ˆi −ˆk + μ(3ˆi −2ˆj −2ˆk), JEE Main 2021 (20 Jul Shift 1) JEE Main Previous Year Paper

Q90.The probability distribution of random variable X is given by: X 1 2 3 4 5 P(X) K 2K 2K 3K K Let p = P(1 < X < 4 ∣X < 3). If 5p = λK , then λ is equal to JEE Main 2021 (27 Aug Shift 2) JEE Main Previous Year Paper

Q90.Let →𝑝= 2 ^i + 3 ^j + ^k and →𝑞= ^i + 2 ^j + ^k be two vectors. If a vector →𝑟= 𝛼 ^i + 𝛽 ^j + 𝛾 ^k is perpendicular to each of the vectors ( →𝑝+ →𝑞) and ( →𝑝- →𝑞), and | →𝑟| = √3, then |𝛼| + | 𝛽| + | 𝛾| is equal to JEE Main 2021 (25 Jul Shift 1) JEE Main Previous Year Paper

Q90.The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points Q(3, −4, −5) and R(2, −3, 1) and the plane 2x + y + z = 7, is equal to _____. JEE Main 2021 (27 Jul Shift 2) JEE Main Previous Year Paper