# IIT JAM Mathematics Test Series 2023 : Real Analysis – Differentiability

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Differentiability 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 IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Share with Your Friends.

## IIT JAM Mathematics Test Series 2023 : Real Analysis – Differentiability

Q1. Let X = (0,1)∪(2,3) be an open set in R. Let f be a continuous function on X such that the derivative f'(x) = 0 for all x. Then the range of f has

(a) Uncountable number of points
(b) Count ably infinite number of points
(c) At most 2 points
(d) At most 1 point

Q2. If  f(x) is real valued function defined on [0,∞] such that f(0) = 0 and f(x) > 0 for all x, then the function h(x) = f(x)/x is

(a) Increasing in [0,∞]
(b) Decreasing in [0,1]
(c) Increasing in [0,1] and decreasing in [1,∞]
(d) Decreasing in [0,1] and increasing in [1,∞]

Q3. Let f:R R be a continuous function and f(x+1) = f(x) for all xR. Then

(a) f is bounded above, but not bounded below.
(b) f is bounded above and below but may not attain its bounds
(c) f is bounded above and below and attains its bounds
(d) f is uniformly comtituous

Q4. Decide which of the following functions are Uniformly continuous on (0,1).

(a) f(x) = ex
(b) f(x) = x
(c) f(x) = tan(πx/2)
(d) f(x) = sin(x)

Q5. Which of the following statements is (are) on the interval (0,π/2) ?

(a) cosx < cos(sin x)
(b) tanx < x
(c) √(1+x) < 1 + x/2 – x2/8
(d) (1-x2)/2 < In(2+x)

Q6. Let f:R → R be a differentiable function with f(0)=0 If for all x∈R, l < f1(x) < 2, then Which one of the following statements is true on (0,∞) ?

(a) f is unbounded
(b) f is increasing and bounded
(c) f has at least one zero
(d) f is periodic

Q7. Let f(x) = 1/(x+|x|) + 1/(x+|x-1|) for all x∈[-1,1]. Then which one of the following is TRUE ?

(a) Maximum value of f(x) is 3/2
(b) Minimum value of f(x) is 1/2
(c) Maximum of f(x) occurs at x=1/2
(d) Minimum of f(x) occurs at x=1

Q8. Consider the function f(x) = |cosx|+|sin(2-x)| . At which of the following points is f not differentiable?

(a) {(2n+1)π/2 : n∈Z}
(b) {nπ : n∈Z}
(c) {nπ+2 : n∈Z}
(d) {/2 : n∈Z}

Q9. Let F = {f:R → R: |f(x)-f(y)| ≤ K|(x-y)|a} For all x, y∈R and for some a > 0 and some K > 0.Which of the following is/are true?

(a) Every f∈F is continuous
(b) Every f∈F is uniformly continuous
(c) Every differentiable function f is in F
(d) Every f∈F is differentiable

Q10. Let f:R → R be defined by f(x) = x2sin(1/x2), x ≠ 0, f(0) = 0

(a) f is differentiable everywhere with continuous derivative
(b) f is differentiable everywhere other than the point 1
(c) f is nowhere differentiable
(d) f is differentiable but its derivative is not continuous

Q11. Let f:(0,1) → R be continuous. Suppose that |f(x) – f(y)| ≤ |cosx – cosy| for all x,y ∈ (0,1). Then

(a) f is discontinuous at least at one point in (0,1).
(b) f is continuous everywhere on (0,1), but not uniformly continuous on (0,1).
(c) f is uniformly continuous on (0,1).
(d) lim f(x) exists, where x → 0

Q12. Let f(x) = |sinπx| , xR.Then

(a) f is contimuous nowhere
(b) f is continuous everywhere and differentiable nowhere
(c) f is continuous everywhere and differentiable everywhere except at integral value of x
(d) f is differentiable everywhere

Q13. The function f(x) = 1-|1-x| on R is

(a) Not continuous
(b) Continuous but not differentiable
(c) Differentiable but not continuous
(d) Differentiable at only one point

Q14. f:[0,1] R is a function. Which of the following is possible?

(a) f is differentiable but not continuous
(b) S is unbounded and continuous
(c)S is differentiable but its derivative is not continuous
(d) f is continuous but not integrable

Q15. Let f = R R be differentiable. Which of these will follow from mean value theorem

(a) For all a, b in R, if a<c<b then [f(b) – f(a)]/(b-a) = f ‘(c)
(b) For some a, b in R and for all c in (a, b) , [f(b) – f(a)]/(b-a) = f ‘(c)
(c) For all a, b in R there is some c in (a, b) such that [f(b) – f(a)]/(b-a) = f ‘(c)
(d) For all a, b in R there is a unique element  c in (a, b) then  the [f(b) – f(a)]/(b-a) = f ‘(c)

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