11.

12.

13.

14.

15.

16.

By Lagranges Mean value theorem the value of 2/65 lies in the interval :

1 1
(A) (+ 43) (8) (+ ன்‌

0 1 1
(© (+ 45) (D) (+ 43)

Let {f,} bea sequence of functions defined by :

0 ^) = nxe™ , x>0 is not uniformly convergent on [0, 1]

൬ f,(x) = 2 zy, xé€R is uniformly convergent on [0, 1]
l+nx
(A) _ both (i) and (ii) are true (B) (i) is false and (ii) are true
(0 @) is true and (ii) is false (09) both (i) and (ii) are false

Let g be a bounded function on [a, b]. Then which of the following is correct?
(^) Ifgisa constant function on [a, b], then it is Riemann Integrable on [a, }]
(B) If gis continuous on [a, b], then it is Riemann Integrable on [a, ಗಿ]
(0) Ifgis monotonic on [a, b], then g is Riemann Integrable on [a, b]
(D) allof the above

Which of the following improper integrals are convergent?

മ്മ + cosx
(A) Jeu (B) dx
J ॐ +] | 1+
= 2
© [Sra (D) all of these
x

0

Let K be a positive integer and let J, ={xe [0,1]: decimal expansion of x contains a prime

digit at K® place}. Then the Lebesgue measure of the set J is:

4
(A) 1 ® 16
o (4)
(©) (>) (D) 0
ey?

vy.
Consider f(x, 9) =4 ‏قوب قب‎ if (x, 9) #O, 0)
0; ४, 9) =, 9)
(A) fis continuous at (0, 0)
(B) fis bounded in a neighbourhood of (0, 0)
(C) fis not bounded in a neighbourhood of (0, 0)
(D) fhas all directional derivatives at (0, 0)

. Then which of the following is true?

5 083/2023
[P.T.0.]