Maths · Calculus · chapter 5 of 13
Class 12 Maths Chapter 5: Continuity and Differentiability MCQs with answers
10 multiple-choice questions on Continuity and Differentiability, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter.
Question 1
The function f(x) = x3 − 8x − 2 for x ≠ 2, and f(2) = k, is continuous at x = 2. The value of k is
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Answer: (c) 12
For x ≠ 2, (x3 − 8)/(x − 2) = x2 + 2x + 4, so lim_(x→2) f(x) = 4 + 4 + 4 = 12. Continuity at 2 needs f(2) = 12, so k = 12.
Question 2
Let f(x) = 2x + 1 for x < 3, f(3) = 4, and f(x) = x + 4 for x > 3. At x = 3, f is
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Answer: (d) discontinuous, since the limit at 3 exists but is not equal to f(3)
Left hand limit = 2(3) + 1 = 7; right hand limit = 3 + 4 = 7; so lim_(x→3) f(x) = 7, but f(3) = 4 ≠ 7. The limit exists and differs from the value, so f is discontinuous at 3.
Question 3
A real function f has domain {5} and f(5) = 7. Then f is
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Answer: (c) a continuous function
If the domain of a real function is a singleton, the function is continuous: there is no other point at which it could fail.
Question 4
The function f(x) = x2 + 1x2 − 5x + 6 is continuous on
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Answer: (a) R − {2, 3}
A rational function p(x)/q(x) is continuous on its domain, which is R minus the zeros of q. x2 − 5x + 6 = (x − 2)(x − 3), so the domain is R − {2, 3}.
Question 5
Which statement about f(x) = 1x, x ≠ 0, is correct?
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Answer: (a) f is a continuous function, since it is continuous at every point of its domain
0 is not in the domain, so continuity at 0 is not asked. At every c ≠ 0, lim_(x→c) 1/x = 1/c = f(c), so f is a continuous function. +∞ is not a real number, so (d) is false.
Question 6
For a real function f and a point c in its domain, which statement is always true?
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Answer: (b) If f is differentiable at c, then f is continuous at c
Theorem 3: differentiability at c implies continuity at c. The converse fails: |x| is continuous at 0 but not differentiable there, which rules out (a) and (c).
Question 7
ddx(x3 tan x) is
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Answer: (d) 3x2 tan x + x3 sec2 x
Product rule: (uv)′ = u′v + uv′ with u = x3, v = tan x: 3x2 tan x + x3 sec2 x.
Question 8
ddx cos(x3) is
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Answer: (a) −3x2 sin(x3)
With t = x3: d/dx cos t = −sin t · dt/dx = −sin(x3) · 3x2.
Question 9
The derivative of 5x at x = 0 is
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Answer: (d) log 5
d/dx (ax) = ax log a, so the derivative is 5x log 5, which at x = 0 is 50 log 5 = log 5.
Question 10
The value of log₂₇ 9 is
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Answer: (a) 23
Change of base: log₂₇ 9 = log₃ 9 / log₃ 27 = 2/3. (Equivalently 272/3 = 9.)
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