# Sample Multiple Choice Questions Sample Questions for JGEEBILS Part 1

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1. Let f: R ⇾ R be a continuous bounded function. Then

(a) f has to be uniformly continuous

(b) there exists an x ∈ R such that f (x) = x

(c) f cannot be increasing

(d) lim (x ⇾ ∞) of f (x) exists.

2. For the function

f (x) = (x + x^{2}) ⚹ cos (πx)

Consider the statements:

I. f is differentiable at x = 0 and f (0) = 1.

II. f is differentiable everywhere and f (x) is continuous at x = 0.

III. f is increasing in a neighbourhood around x = 0.

IV. f is not increasing in any neighbourhood of x = 0.

Which one of the following combinations of the above statements is true.

(a) I. and II.

(b) I. and III.

(c) II. and IV.

(d) I. and IV.

Sample true/false questions

1. If A and B are 3 × 3 matrices and A is invertible, then there exists an integer n such that A + nB is invertible.

2. Let P be a degree 3 polynomial with complex coefficients such that the constant term is 2010. Then P has a root α with | α|> 10.

3. The symmetric group S5 consisting of permutations on 5 symbols has an element of order 6.

4. Suppose fn (x) is a sequence of continuous functions on the closed interval [0; 1] converging to 0 pointwise. Then the integral fn (x) dx converges to 0.

5. There are n homomorphisms from the group Z/nZ to the additive group of rationals Q.

6. A bounded continuous function on R is uniformly continuous.