List of top Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

Let S be the set of all continuous functions f: [-1,1]→\(\R\) satisfying the following three conditions:
(i) f is infinitely differentiable on the open interval (-1,1),
(ii) the Taylor series
\(f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+...\) of f at 0 converges to f(x) for each x ∈ (-1,1),
(iii) \(f(\frac{1}{n})=0\ \text{for all}\ n\isin\N\)
Then which of the following is/are true?

Let S be the set of all functions f: \(\R\rightarrow\R\) satisfying \(|f(x)-f(y) |^2 \le |x - y|^3\ for\ all\ x, y \isin \R.\) Then which of the following is/are true?

Let V be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 5, together with the zero polynomial. Let T: V→ \(\R\) be the linear map defined by 7(1) = 1 and T(x(x-1)... (x-k+1)) = 1 for 1 ≤ k ≤ 5.
Then which of the following is/are true?

Let (x_n) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?