Practice Questions

Q88.A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then ( π4 + 1)k is equal to

Q88.If a + α = 1, b + β = 2 and af(x) + αf( x1 ) = bx + xβ , x ≠0, then the value of the expression f(x)+f(x+ x ___________.

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.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.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 x be a vector in the plane containing vectors →a = 2ˆi −ˆj + ˆk and b = ˆi + 2ˆj −ˆk. If the vector x is 17√6 → 2 is 2 , then the value of x is equal to _______. perpendicular to (3ˆi + 2ˆj −ˆk) and its projection on →a

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.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.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 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 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.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 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 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

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 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= 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.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.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 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.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 _________.

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 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 = 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.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 _____.