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**topic**Riemann Integral of the

**Chapter Real Analysis.**These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve

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**IIT JAM Mathematics Test Series 2023 : Real Analysis – Riemann Integral**

**Q1. Let ****a**** be a positive real number. Which of the following Integrals are convergent ?**

(a) _{a}∫^{0} ^{1}/x^{4} dx

(b) _{a}∫^{0} ^{1}/√x dx

(c) _{4}∫^{0} ^{1}/xlog_{e}x dx

(d) _{5}∫^{0} ^{1}/x(logx)^{2} dx

**Q2. Let k be a positive integer. The radius of convergence of the series ^{∞}Σ_{n=0} (n!)^{k}x^{n}/_{(kn)!} is**

(a) k

(b) k^{-k}

(c) k^{k}

(d) ∞

**Q3. Let {a _{n} : n≥1} be a sequence of real numbers such that ^{∞}Σ_{n=1} an is convergent and ^{∞}Σ_{n=1} |a_{n}| is divergent. Let R be the radius of convergence of the power series Σn=1∞ a_{n}x^{n} then we can conclude that**

(a) 0 < R < 1

(b) R = 1

(c) 1 < R < ∞

(d) R = ∞

**Q4. Let f be a monotonically increasing function from [0,1] into [0,1]. Which of the following statements is/are true ?**

(a) f must be continuous at all but finitely many points in [0,1]

(b) f must be continuous at all but countably many points in [0,1]

(c) f must be Riemann integrable

(d) f must be Lebesgue integrable

**Q5. Let a, p be real numbers and a > 1**

(a) If p > 1 then _{-∞}∫^{∞} ^{1}/|x|^{pa} dx < ∞

(b) If p > 1/a then _{-∞}∫^{∞} ^{1}/|x|^{pa} dx < ∞

(c) If p < 1/a then _{-∞}∫^{∞} ^{1}/|x|^{pa} dx < ∞

(d) For any p ∈ R we have _{-∞}∫^{∞} ^{1}/|x|^{pa} dx = ∞

**Q6. Let L = _{0}∫^{1} ^{dx}/(1+x^{8}). Then**

(a) L < 1

(b) L > 1

(c) L < ^{π}/_{4}

(d) L >^{ π}/_{4}

**Q7. Let f, g and h be bounded functions on the closed interval [a,b], such that f(x)≤g(x)≤h(x) for all x ∈ [a,b]. Let P = {a = a_{o}= a_{1}< a_{2} < … < a_{n} = b} be a partition of [a,b]. We denote by U(f,P) and L(f,P), the upper and lower Riemann sums of f with respect to the partition P and similarly for g and h. Which of the following statements is necessarily true ?**

(a) If U(h,P) – U(f,P) < 1 then U(g,P) – L(g,P) < 1

(b) If L(h,P) – L(f,P) < 1 then U(g,P) – L(g,P) < 1

(c) If U(h,P) – L(f,P) < 1 then U(g,P) – L(g,P) < 1

(d) If L(h,P) – U(f,P) < 1 then U(g,P) – L(g,P) < 1

**Q8. Let f:R → R be a differentiable function such that f’ is bounded. Given a closed and bounded interval [a,b], and a partition P = {a = a_{o}= a_{1}< a_{2} < … < a_{n} = b} of [a,b], let M(f,P) and m(f,P) denote respectively, the upper Riemann sum and the lower Riemann sum of with respect to P. Then**

(a) |M(f,P) – _{a}∫^{b} f(x)dx| ≤ (b – a) sup{|f(x)| : x∈[a,b]}

(b) |m(f,P) – _{a}∫^{b} f(x)dx| ≤ (b – a) inf{|f(x)| : x∈[a,b]}

(c) |M(f,P) – _{a}∫^{b} f(x)dx| ≤ (b – a)^{2} sup{|f'(x)| : x∈[a,b]}

(d) |m(f,P) – _{a}∫^{b} f(x)dx| ≤ (b – a)^{2} imf{|f'(x)| : x∈[a,b]}

**Q9. Suppose f is an increasing real valued function on [0,∞] with f(x) > 0 ∀x and let g(x) = ^{1}/_{x} _{0}∫^{x}f(u)du ; 0< x < ∞. Then which of the following are true :**

(a) g(x) ≤ f(x) for all x ∈ (0,∞)

(b) xg(x) ≤ f(x) for all x ∈ (0,∞)

(c) xg(x) ≥ f(0) x ∈ (0,∞)

(d) yg(y) – xg(x) ≤ (y-x) f(y) for all x < y

**Q10. Let f:[0,1] → R be defined by f(x) = xcos[π/(2x)] if x ≠ 0, and f(x) = 0 if x = 0 then**

(a) f is continuous on [0,1]

(b) f is of bounded variation on [0,1]

(c) f is differentiable [0,1] and its derivative f’ is bounded on (0,1)

(d) f is Riemann integrable on [0,1]

**Q11. Let f:[0,1] → R be continuous such that f(t) ≥ 0 for all t in [0,1]. Define g(x) = _{0}∫^{x} f(t) dt then**

(a) g is monotone and bounded

(b) g is monotone, but not bounded

(c) g is bounded, but not monotone

(d) g is neither monotone nor bounded

**Q12. Let f be a continuous function on [0,-1] with f(0) = 1. Let G(a) = ^{1}/_{a} _{0}∫^{a} f(x) dx**

(a) lim_{a→0} G(a) = ^{1}/_{2}

(b) lim_{a→0} G(a) = 1

(c) lim_{a→0} G(a) = 0

(d) The Limit lim_{a→0} G(a) does not exist

**Q13. The improper Integral _{0}∫^{4} ^{dx}/_{x(4-x)} is**

(a) Convergent & value in 6

(b) Divergent & diverges to +∞

(c) Convergent & value is 4

(d) None of these

**Q14. Let f(x) = x ^{3} – sinx, x ∈ [-^{π}/_{2}, ^{π}/_{2}]. Then f is**

(a) Bounded and of bounded variation

(b) Unbounded and of bounded variation

(c) Unbounded but not of bounded variation

(d) Bounded but not of bounded variation

**Q15. Match the Following**

**A. _{1}∫^{2} ^{x}/_{√(x-1)} dx B. _{0}∫^{π} ^{dx}/_{(1+cosx)}**

**(i) Divergent & diverges to +∞ (ii) Convergent & value is ^{4}/_{3}**

**(iii) Divergent & diverges to -∞ (iv) Convergent & value is ^{8}/_{3}**

(a) A-i, B-ii

(b) A-iii, B-iv

(c) A-iv, B-i

(d) A-ii, B-iii

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*Answer Key*

*Answer Key*

01. | (b), (d) | 06. | (a), (d) | 11. | (b) |

02. | (c) | 07. | (c) | 12. | (c) |

03. | (b) | 08. | (a) | 13. | (b) |

04. | (b), (c), (d) | 09. | (a), (d) | 14. | (c) |

05. | (d) | 10. | (b) | 15. | (a), (d) |

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