Ques 1.

Are the following statements true or false? Give reasons for your answer.
 a) Complement of the open interval ]1,0] is an open set.

 b) Every bounded sequences is not convergent.

 c) The function [:f − 2,2 ] → R defined by  is uniformly continuous.

 d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.

 
 e) The function  defined by  is not integrable. 

Ques 2.

Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:

Also check whether the function f is derivable at x = 1.

Ques 3.

Find the following limit.

Ques 4.

Check whether the intervals ]9,5] and [6,12[ are equivalent or not.

Ques 5.

Prove that a strictly decreasing function is always one-one.

Ques 6.

Write the inequality 4 ≤ 2x + 3 ≤ 6 in the modulus form.

Ques 7.

Verify Bozano–Weierstrass Theorem for the following sets:
 i) Set of non-negative integers.
 ii) Interval [− ,1 ∞] 

Ques 8.

Check whether the  exists or not? 

Ques 9.

Test the following series for convergence,

(i) 

(ii) 

Ques 10.

Show that     is conditionally convergent.

Ques 11.

Determine the local minimum and local maximum values of the function f defined by 

Ques 12.

Show that  the lagrange’s form of remainder in the Maclaurin series expansion of  , tends to zero as n → ∞.Hence obtain the Maclaurin’s infinite expansion for 

Ques 13.

If the partition  is a refinement of the partition  of  then  and  Verify this result for the function  defined over the interval  and the partitions  and

Ques 14.

Evaluate:

Ques 15.

Use Cauchy’s mean value theorem to prove that:

Ques 16.

Find a and b such that  

Ques 17.

Show that  is an algebraic number.

Ques 18.

Using the principle of mathematical induction, show that 

Ques 19.

Show that the equation  has a real root other than x = − .1

Ques 20.

Check whether the set of integers is countable or not.

Ques 21.

Using Weiestrass M-test, show that the following series converges uniformly. 

Ques 22.

Use the Fundamental Theorem of Integral Calculus to evaluate the integral 

Ques 23.

Apply Bonnet Mean Value Theorem for integrals to show that 

Ques 24.

Show that the function  defined by  has an inverse by applying the inverse function theorem. Find its inverse also.

Ques 25.

Verify the second mean value theorem for the function  and   in the interval