RA Grand Quiz 2024.pdf by ali raza ppsc mcqs

real analysis mcqs

1. Real Analysis
for lecturer mathematics
Grand Quiz (2024)
0301-4889903 Mathematics by aliraza
Ali Raza(Lecturer Mathematics)
M.PhilMathematics
M.sc(Gold Medalist)
[email protected]

2. 0301-4889903 ALI RAZA 2
Q No.1 :
For the function ��,�=�
4
+�
4
−4��+1,which of the following is a critical point?
a) (0,0)b) (1,1) c) (−1,−1) d) All of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

3. 0301-4889903 ALI RAZA 3
Q No.2 :
Minimum value of the function ��,�=�
2
−��+�
2
+6�is
a) −8 b) −10 c) −12 d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

4. 0301-4889903 ALI RAZA 4
Q No.3 :
Young’s theorem states that, “If �
�and �
�exist some neighbourhoodof (�,�)and both are differentiable at
(�,�)then
a) �
&#3627408485;&#3627408486;&#3627408462;,&#3627408463;<&#3627408467;
&#3627408486;&#3627408485;(&#3627408462;,&#3627408463;)b)&#3627408467;
&#3627408485;&#3627408486;&#3627408462;,&#3627408463;>&#3627408467;
&#3627408486;&#3627408485;(&#3627408462;,&#3627408463;) c)&#3627408467;
&#3627408485;&#3627408486;&#3627408462;,&#3627408463;=&#3627408467;
&#3627408486;&#3627408485;(&#3627408462;,&#3627408463;) d)&#3627408467;
&#3627408485;&#3627408486;&#3627408462;,&#3627408463;≠&#3627408467;
&#3627408486;&#3627408485;(&#3627408462;,&#3627408463;)
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

5. 0301-4889903 ALI RAZA 5
Q No.4 :
If &#3627408467;(&#3627408485;,&#3627408486;)is a Homogeneous function of degree &#3627408475;then
a) &#3627408485;
????????????
??????&#3627408485;
+&#3627408486;
????????????
??????&#3627408485;
=&#3627408467;(&#3627408485;,&#3627408486;) b)&#3627408485;
????????????
??????&#3627408485;
+&#3627408486;
????????????
??????&#3627408485;
=&#3627408475;&#3627408467;(&#3627408485;,&#3627408486;)
c)&#3627408485;
????????????
??????&#3627408485;
+&#3627408486;
????????????
??????&#3627408485;
=(&#3627408475;−1)&#3627408467;(&#3627408485;,&#3627408486;) d)&#3627408485;
????????????
??????&#3627408485;
+&#3627408486;
????????????
??????&#3627408485;
=
1
??????
&#3627408467;(&#3627408485;,&#3627408486;)
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

6. 0301-4889903 ALI RAZA 6
Q No.5 :
For the function &#3627408467;&#3627408485;=&#3627408485;
3
,Which of the following derivative does not exists at &#3627408485;=0
a) &#3627408467;′(&#3627408485;) b)&#3627408467;′′(&#3627408485;) c)&#3627408467;′′′(&#3627408485;) d) All of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

7. 0301-4889903 ALI RAZA 7
Q No.6 :
If &#3627408467;is differentiable at &#3627408485;=&#3627408464;then
a) lim
&#3627408485;→&#3627408464;
&#3627408485;??????&#3627408485;−&#3627408464;??????(&#3627408464;)
&#3627408485;−&#3627408464;
=&#3627408467;&#3627408464;+&#3627408464;&#3627408467;′(&#3627408464;) b)lim
&#3627408485;→&#3627408464;
&#3627408485;??????&#3627408485;−&#3627408464;??????(&#3627408464;)
&#3627408485;−&#3627408464;
=&#3627408464;&#3627408467;&#3627408464;+&#3627408467;′(&#3627408464;)
c)lim
&#3627408485;→&#3627408464;
&#3627408485;??????&#3627408485;−&#3627408464;??????(&#3627408464;)
&#3627408485;−&#3627408464;
=&#3627408464;&#3627408467;&#3627408464;−&#3627408467;′(&#3627408464;) d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

8. 0301-4889903 ALI RAZA 8
Q No.7 :
Function &#3627408467;&#3627408485;=
&#3627408485;
3
??????&#3627408467;&#3627408485;≤1
&#3627408485;
2
??????&#3627408467;&#3627408485;>1
is
a) Both continuous and differentiable at &#3627408485;=1
b) Continuous but not differentiable at &#3627408485;=1
c) Neither continuous nor differentiable at &#3627408485;=1
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

9. 0301-4889903 ALI RAZA 9
Q No.8:
Function &#3627408467;&#3627408485;=
1
&#3627408485;
is
a) Both continuous and uniformly continuous on 0<&#3627408485;≤1
b) Continuous but not uniformly continuous on 0<&#3627408485;≤1
c) Neither continuous nor uniformly continuous on 0<&#3627408485;≤1
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

10. 0301-4889903 ALI RAZA 10
Q No.9:
Let &#3627408467;is continuous at &#3627408485;=&#3627408464;then &#3627408467;is also continuous at &#3627408485;=&#3627408464;. Converse of this statement
a) is always true
b) is never true
c) May not be true
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

11. 0301-4889903 ALI RAZA 11
Q No.10:
Consider the sequence &#3627408485;
??????+1=41−
1
&#3627408485;??????
, &#3627408485;
1>0then limit of sequence is
a) 1 b) 2 c) 3 d) Does not exist
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

12. 0301-4889903 ALI RAZA 12
Q No.11:
lim
&#3627408485;→∞
1+3+5+⋯+(2??????−1)
??????
2
=___________.
a) 1 b) 2 c) 3 d) Does not exist
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

13. 0301-4889903 ALI RAZA 13
Q No.12:
Let &#3627408485;be a real number such that &#3627408485;<1and let &#3627408477;>&#3627408478;>0be two rational numbers than
a)
&#3627408485;
&#3627408477;
−1
&#3627408477;
<
&#3627408485;
&#3627408478;
−1
&#3627408478;
b)
&#3627408485;
&#3627408477;
−1
&#3627408477;
=
&#3627408485;
&#3627408478;
−1
&#3627408478;
c)
&#3627408485;
&#3627408477;
−1
&#3627408477;
>
&#3627408485;
&#3627408478;
−1
&#3627408478;
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

14. 0301-4889903 ALI RAZA 14
Q No.13:
Which of the following is true?
a) ??????&#3627408475;&#3627408467;&#3627408436;=&#3627408454;&#3627408482;&#3627408477;(−&#3627408436;) b) ??????&#3627408475;&#3627408467;&#3627408436;=−&#3627408454;&#3627408482;&#3627408477;(−&#3627408436;) c) ??????&#3627408475;&#3627408467;&#3627408436;=−&#3627408454;&#3627408482;&#3627408477;(&#3627408436;) d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

15. 0301-4889903 ALI RAZA 15
Q No.14:
Which of the following is not true?
a) &#3627408452;is field b) &#3627408452;is an ordered field c) &#3627408452;is complete ordered field d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

16. 0301-4889903 ALI RAZA 16
Q No.15 :
For &#3627408485;,&#3627408486;∈&#3627408453;equality &#3627408485;+&#3627408486;=|&#3627408485;−&#3627408486;|holds if and only if
a) &#3627408485;&#3627408486;>0 b) &#3627408485;&#3627408486;<0 c) &#3627408485;&#3627408486;=0 d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

17. 0301-4889903 ALI RAZA 17
Q No.16 :
Which of the following function is one-one?
a) &#3627408467;&#3627408485;=&#3627408480;&#3627408468;&#3627408475;(&#3627408485;) b) &#3627408467;&#3627408485;=[&#3627408485;] c) &#3627408467;&#3627408485;=|&#3627408485;| d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

18. 0301-4889903 ALI RAZA 18
Q No.17 :
If &#3627408467;&#3627408485;=
2&#3627408485;−1??????&#3627408467;&#3627408485;??????&#3627408480;&#3627408479;&#3627408462;&#3627408481;??????&#3627408476;&#3627408475;&#3627408462;??????
5−&#3627408485;??????&#3627408467;&#3627408485;??????&#3627408479;&#3627408479;&#3627408462;&#3627408481;??????&#3627408476;&#3627408475;&#3627408462;??????
then lim
&#3627408485;→2
&#3627408467;(&#3627408485;)
a) 2 b) 3 c) −2 d) does not exist
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

19. 0301-4889903 ALI RAZA 19
Q No.18 :
lim
&#3627408485;→0
&#3627408462;
??????
−&#3627408463;
??????
??????????????????&#3627408485;
=________________where &#3627408462;,&#3627408463;>0
a) &#3627408462;−&#3627408463;b) &#3627408462;+&#3627408463;c)
&#3627408462;
&#3627408463;
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

20. 0301-4889903 ALI RAZA 20
Q No.19 :
lim
ℎ→0
??????&#3627408462;+
ℎ
2
−??????&#3627408462;−
ℎ
2
ℎ
=______________.
a) &#3627408467;′(&#3627408462;) b) 2&#3627408467;′(&#3627408462;) c) &#3627408467;′(2&#3627408462;) d) Zero
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

21. 0301-4889903 ALI RAZA 21
Q No.20 :
&#3627408467;&#3627408485;=
1
&#3627408485;
on [1,∞)is
a) Monotonically increasing b) Strictly increasing c) Monotonically decreasing d) Strictly decreasing
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

22. 0301-4889903 ALI RAZA 22
Q No.21 :
&#3627408467;&#3627408485;=&#3627408480;&#3627408468;&#3627408475;|&#3627408485;|is continuous on
a) &#3627408485;=0b) −∞,0∪(0,∞)c) (−∞,∞)d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

23. 0301-4889903 ALI RAZA 23
Q No.22 :
&#3627408467;&#3627408485;=
&#3627408485;&#3627408485;
1+&#3627408485;
2
then which of the following is not true?
a) &#3627408467;(&#3627408485;)is continuous on &#3627408453;b) &#3627408467;(&#3627408485;)is bounded on &#3627408453;c) &#3627408467;(&#3627408485;)is odd d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

24. 0301-4889903 ALI RAZA 24
Q No.23 :
For what values of &#3627408462;and &#3627408463;is &#3627408467;(&#3627408485;)differentiable at &#3627408485;=1if &#3627408467;&#3627408485;=
&#3627408485;
3
??????&#3627408467;&#3627408485;<1
&#3627408462;&#3627408485;+&#3627408463;??????&#3627408467;&#3627408485;≥1
a) &#3627408462;=2,&#3627408463;=3 b) &#3627408462;=−3,&#3627408463;=−2 c) &#3627408462;=3,&#3627408463;=−2 d) &#3627408462;=2,&#3627408463;=−3
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

25. 0301-4889903 ALI RAZA 25
Q No.24:
If &#3627408485;<1then lim
??????→∞
&#3627408485;
??????
is
a) 0 b) 1 c) ∞ d) Does not exist
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

26. 0301-4889903 ALI RAZA 26
Q No.25:
Area under the curve &#3627408467;&#3627408485;=cos&#3627408485;from &#3627408485;=−??????to &#3627408485;=??????bounded by &#3627408485;−axis is
a) 0 b) 1 c) 2 d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

27. 0301-4889903 ALI RAZA 27
Q No.26: If |&#3627408436;&#3627408437;|=1and &#3627408437;&#3627408438;=1then what will be the value of |&#3627408436;&#3627408438;|for which the area of triangle &#3627408436;&#3627408437;&#3627408438;is
minimum?
a)
1
2
b) 1 c) 2 d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

28. 0301-4889903 ALI RAZA 28
Q No.27 :
Let &#3627408467;&#3627408485;=
&#3627408485;
2
??????&#3627408467;&#3627408485;≤1
&#3627408485;??????&#3627408467;&#3627408485;>1
then
a) &#3627408467;(&#3627408485;)is continuous on
1
2
,2 b) &#3627408467;(&#3627408485;)is differentiable
1
2
,2
c) &#3627408467;(&#3627408485;)Satisfy Lagrange’s Mean value theorem d) All of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

29. 0301-4889903 ALI RAZA 29
Q No.28 :
If 0<&#3627408462;<&#3627408463;then lim
??????→∞
&#3627408462;
??????+1
+&#3627408463;
??????+1
&#3627408462;
??????
+&#3627408463;
??????
is equal to
a) &#3627408462;b) &#3627408463; c)
&#3627408462;+&#3627408463;
2
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

30. 0301-4889903 ALI RAZA 30
Q No.29:
Which of the following is true?
a) Every Riemann integrablefunction is continuous
b) Every continuous function is Riemann integrable
c) Every bounded function is Riemann integrable
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

31. 0301-4889903 ALI RAZA 31
Q No.30:
Bernoulli’s inequality 1+&#3627408485;
??????
≥1+&#3627408475;&#3627408485;,for all &#3627408475;∈??????is true for
a) &#3627408485;>0 b) &#3627408485;≥0 c) &#3627408485;>1 d) &#3627408485;>−1
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

32. 0301-4889903 ALI RAZA 32
Q No.31:
Which of the following is not an indeterminate form?
a) ∞+∞ b) ∞−∞ c)
∞
∞
d) None of these
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

33. 0301-4889903 ALI RAZA 33
Q No.32:
Radius of convergence of the series σ
??????=1
∞
2
??????
??????
2
&#3627408487;
??????
is
a) 2 b)
1
2
c) 2 d) ∞
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

34. 0301-4889903 ALI RAZA 34
Q No.33:
Suppose &#3627408462;,&#3627408464;∈&#3627408453;,c>0and &#3627408467;is defined on −1,1by &#3627408467;&#3627408485;=
&#3627408485;
&#3627408462;
sin
1
&#3627408485;
??????
??????&#3627408467;&#3627408485;≠0
0 ??????&#3627408467;&#3627408485;=0
then &#3627408467;′(0)exists iff
a) &#3627408462;>0 b) &#3627408462;>1 c) &#3627408462;>1+&#3627408464;d) &#3627408462;≥1+&#3627408464;
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

35. 0301-4889903 ALI RAZA 35
Q No.34:
׬
0
∞sin&#3627408485;
&#3627408485;
2
=___________.
a) ?????? b) ?????? c)
??????
2
d)
??????
2
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation

36. 0301-4889903 ALI RAZA 36
Q No.35:
෍
??????=1
∞
1
&#3627408475;
2
=_________
a)
??????
2
2
b)
??????
2
4
c)
??????
2
6
d)
??????
2
8
Real AnalysisGrand Quiz(2024) | Lecturer Mathematics preparation