Maths

If ∫₀^(π/2) (sin³x)e⁻sin²ˣdx = α - (β/e)∫₀¹ √t eᵗdt, then α + β is equal to ………

Let A = {n∈N|n² ≤ n + 10,000}, B = {3k + 1|k ∈ N} and C = {2k|k∈ N}, then the sum of all the elements of the set A ∩ (B-C) is equal to……

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……….

Let n be a non-negative integer. Then the number of divisors of the form “4n+1” of the number (10)¹⁰ . (11)¹¹ . (13)¹³ is equal to……….

If the real part of the complex number z = (3 + 2icosθ)/(1-3icosθ), θ ∈ (0, π/2) is zero, then the value of sin²3θ + cos²θ is equal to……….

. The number of real roots of the equation e⁴ˣ + e³ˣ – 4e²ˣ – eˣ + 1 = 0 is equal to………….

Let f:(a,b) →R be twice differentiable such that f(x) = ∫ₐˣ g(t)dt for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

A student appeared in an examination consisting of 8 true – false type questions. The student guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least 'n' correct answers is less than 1/2, is :

Which of the following is the negation of the statement “for all M > 0, there exists x ∈ S such that x ≥ M”?

Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point: