Set, Relation and Function Questions

If X = {4n - 3n - 1 : n ∈ N} and Y = {9(n - 1) : n ∈ N}, where N is the set of natural numbers, then X ∪ Y is equal to

If g is the inverse of a function f and f′(x) = , then g′(x) is equal to

If f and g are differentiable functions in [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 and f(1) = 6, then for some c ∈ ]0, 1[

Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1. Let F(x) = f(√t)dt for x ∈ [0, 2]. If F′(x) = f′(x) for all ...

Two sets A and B are as under A = {(a, b) ∈R × R : |a - 5| < 1 and |b - 5| < 1}; B = {(a, b)∈R × R : 4(a - 6)2 + 9(b - 5)2 ≤ 36}. Then :-...

Let S = {t ∈R : f(x) = |x - π|.(e|π|-1) sin|x| is not differentiable at t}. Then the set S is equal to:...

Let S = {x ∈ R : x ≥ 0 and a |x - 3| + x(x - 6) + 6 = 0}. Then S :...

Let f : [a, b] → [1, ∞) be a continuous function and let g : R → R be defined as g(x) = . Then...

Let ƒ, g : [−1, 2] → be continuous function which are twice differentiable on the interval (−1, 2). Let the values of ƒ and g at the points −1, 0 and 2 be as given in...

In , consider the planes P1 : y = 0 and P2 : x + z = 1. Let P3 be a plane, different from P1 and P2, which passes through the intersection of P1 and P2. If the distance of the point (0,1,0) from P3 is...

Let g : → be a differentiable functions with g(0) = 0, g′(0) = 0 and g′(1) ≠ 0. Let and h(x) = e|x| for all x ∈ . Let (ƒoh)(x) denote ƒ(h(x)) and (hoƒ)(x) ...

Let ƒ(x) = for all for x ∈ and and g(x) = sin x for all x ∈ Let (ƒog)(x) denotes ƒ(g(x)) and (goƒ)(x) denotes g(ƒ(x)). Then which of the following is (are) true?...

For every twice differentiable function ƒ : → [–2, 2] with (ƒ(0))2 + (ƒ'(0))2 = 85, which of the following statement(s) is (are) TRUE ?...

Let ƒ : ℝ→ℝand g :ℝ→ℝ be two non-constant differentiable functions. If ƒ'(x) = (e(ƒ(x) – (x)))g'(x) for all x ∈ℝ, and &fnof...

Let X = {1, 2, 3, 4, 5}. The number of differentordered pairs (Y, Z) that can be formed suchthat and is empty, is :...

Let {θ : sin θ - cosθ = √2 cos θ} and Q = {θ : sin θ + cosθ = √2 sinθ} be two sets. Then

The function f : [0, 3] → [1, 29], defined by f(x) = 2x3 - 15x2 + 36x + 1, is

The domain of the function f (x) = is

The value of p and q for which the function f(x) = is continuous for all x in R, is

Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number x ∈ S such that x > 0. Which of the following statements is the negation of the statement P ?...