{"id":19248,"date":"2022-06-03T13:00:48","date_gmt":"2022-06-03T07:30:48","guid":{"rendered":"https:\/\/mcq-questions.com\/?p=19248"},"modified":"2022-05-21T12:26:55","modified_gmt":"2022-05-21T06:56:55","slug":"mcq-questions-for-class-12-maths-chapter-5","status":"publish","type":"post","link":"https:\/\/mcq-questions.com\/mcq-questions-for-class-12-maths-chapter-5\/","title":{"rendered":"MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers"},"content":{"rendered":"

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers<\/a> during preparation and score maximum marks in the exam. Students can download the Continuity and Differentiability Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 5 Continuity and Differentiability Objective Questions.<\/p>\n

Continuity and Differentiability Class 12 MCQs Questions with Answers<\/h2>\n

Students are advised to solve the Continuity and Differentiability Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Continuity and Differentiability Class 12 with answers will boost your confidence thereby helping you score well in the exam.<\/p>\n

Explore numerous MCQ Questions of Continuity and Differentiability Class 12 with answers provided with detailed solutions by looking below.<\/p>\n

Question 1.
\nThe function
\nf(x) = \"MCQ
\nis continuous at x = 0, then the value of ‘k\u2019 is:
\n(a) 3
\n(b) 2
\n(c) 1
\n(d) 1.5.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) 2<\/p>\n<\/details>\n

\n

Question 2.
\nThe function f(x) = [x], where [x] denotes the greatest integer function, is continuous at:
\n(a) 4
\n(b)-2
\n(c) 1
\n(d) 1.5.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) 1.5.<\/p>\n<\/details>\n

\n

Question 3.
\nThe value of \u2018k\u2019 which makes the function defined by
\n\"MCQ
\ncontinuous at x = 0 is
\n(a) -8
\n(b) 1
\n(c) -1
\n(d) None of these.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) None of these.<\/p>\n<\/details>\n

\n

Question 4.
\nDifferential coefficient of sec (tan-1<\/sup> x) w.r.t. x is
\n(a) \\(\\frac { x }{\\sqrt{1+x^2}}\\)
\n(b) \\(\\frac { x}{1+x^2}\\)
\n(c) x\\(\\sqrt { 1+x^2}\\)
\n(d) \\(\\frac { 1 }{\\sqrt{1+x^2}}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) \\(\\frac { x }{\\sqrt{1+x^2}}\\)<\/p>\n<\/details>\n

\n

Question 5.
\nIf y = log (\\(\\frac { 1-x^2 }{1+x^2}\\)) then \\(\\frac { dy }{dx}\\) is equal to:
\n(a) \\(\\frac { 4x^3 }{1-x^4}\\)
\n(b) \\(\\frac { -4x}{1-x^4}\\)
\n(c) \\(\\frac {1}{ 4-x^4}\\)
\n(d) \\(\\frac { -4x^3 }{1-x^4}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) \\(\\frac { -4x}{1-x^4}\\)<\/p>\n<\/details>\n

\n

Question 6.
\nIf y = \\(\\sqrt { sin x+ y}\\), then \\(\\frac { dy }{dx}\\) is equal to
\n(a) \\(\\frac { cos x }{2y-1}\\)
\n(b) \\(\\frac { cos x}{1-2y}\\)
\n(c) \\(\\frac {sin x}{1-2y}\\)
\n(d) \\(\\frac { sin x }{2y-1}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) \\(\\frac { cos x }{2y-1}\\)<\/p>\n<\/details>\n

\n

Question 7.
\nIf u = sin-1<\/sup> (\\(\\frac { 2x }{1+x^2}\\)) and u = tan-1<\/sup> (\\(\\frac { 2x }{1-x^2}\\)) then \\(\\frac { dy }{dx}\\) is
\n(a) \\(\\frac { 1 }{2}\\)
\n(b) x
\n(c) \\(\\frac {1-x^2}{1+x^2}\\)
\n(d) 1<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) 1<\/p>\n<\/details>\n

\n

Question 8.
\nIf x = t\u00b2, y = t\u00b3, then \\(\\frac { d^2y }{dx^2}\\) is
\n(a) \\(\\frac { 3 }{2}\\)
\n(b) \\(\\frac { 3 }{4t}\\)
\n(c) \\(\\frac {3}{2t}\\)
\n(d) \\(\\frac { 3t }{2}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) \\(\\frac { 3 }{4t}\\)<\/p>\n<\/details>\n

\n

Question 9.
\nThe value of \u2018c\u2019 in Rolle\u2019s Theorem for the function f(x) = x\u00b3 – 3x in the interval [0, \u221a3] is
\n(a) 1
\n(b) -1
\n(c) \\(\\frac {3}{2}\\)
\n(d) \\(\\frac {1}{3}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) 1<\/p>\n<\/details>\n

\n

Question 10.
\nThe value of \u2018c\u2019 in Mean Value Theorem for the function f(x) = x(x – 2), x \u2208 [1, 2] is
\n(a) \\(\\frac {3}{2}\\)
\n(b) \\(\\frac {2}{3}\\)
\n(c) \\(\\frac {1}{2}\\)
\n(d) \\(\\frac {3}{4}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) \\(\\frac {3}{2}\\)<\/p>\n<\/details>\n

\n

Question 11.
\nLet f : (- 1, 1) \u2192 R be a differentiable function with f(0) = – 1 and f'(0) = 1.
\nLet g(x) = [f (2f(x) + 2)]\u00b2. Then g'(0) =
\n(a) 4
\n(b) -4
\n(c) log 2
\n(d) -log 2.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) -4
\nHint:
\nHere g (x)= [f (2 f(x) + 2)]\u00b2
\ng'(x) = 2[f(2f(x) + 2)] \\(\\frac { d }{dx}\\) [2f(x) + 2 ]
\n= 2f(2f(x) + 2) . [2 f'(x)]
\n\u2234 g'(0) = 2f(2f(0) + 2) . [2f'(0)]
\n= 2f(2 (-1) +2). 2f’\/(0)
\n= 2f(0) . 2f'(0) = 4f(0) f'(0)
\n= 4 (-1) (1) = -4.<\/p>\n<\/details>\n

\n

Question 12.
\n\\(\\frac { d^2x }{dy^2}\\) equals
\n\"MCQ<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: d
\n\"MCQ
\nHint:
\n\"MCQ<\/p>\n<\/details>\n

\n

Question 13.
\nIf function f(x) is differentiable at x = a, then
\n\\(\\lim _{x \\rightarrow a}\\) \\(\\frac { x^2 f(a) – a^2 f(x) }{x-a}\\) is
\n(a) a\u00b2 f(a)
\n(b) af(a) – a\u00b2 f'(a)
\n(c) 2a f(a) – a\u00b2 f'(a)
\n(d) 2a f (a) + a\u00b2 f'(a).<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (c) 2a f(a) – a\u00b2 f'(a)
\nHint:
\n\"MCQ<\/p>\n<\/details>\n

\n

Question 14.
\nIf f: R \u2192 R is a function defined by
\nf(x) = [x] cos (\\(\\frac { 2x-1 }{2}\\))\u03c0, where [x] denotes the greatest integer function, then \u2018f’ is
\n(a) continuous for every real x
\n(b) discontinuous only at x = 0
\n(c) discontinuous only at non-zero integral values of x
\n(d) continuous only at x = 0.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) continuous for every real x
\nHint:
\nContinuous for every real x.<\/p>\n<\/details>\n

\n

Question 15.
\nIf y = sec (tan-1<\/sup> x), then \\(\\frac { dy }{dx}\\) at x = 1 is equal to
\n(a) \\(\\frac {1}{2}\\)
\n(b) 1
\n(c) \u221a2
\n(d) \\(\\frac {1}{\u221a2}\\)<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (d) \\(\\frac {1}{\u221a2}\\)
\nHint:
\nHere y = sec (tan-1<\/sup> x).
\n\u2234 \\(\\frac { dy }{dx}\\) = sec (tan-1<\/sup> x) tan (tan-1<\/sup> x). \\(\\frac { 1 }{1+x^2}\\)
\n= sec (tan-1<\/sup> x). x . (tan-1<\/sup> x). \\(\\frac { 1 }{1+x}\\)
\n\\(\\left.\\frac{d y}{d x}\\right]_{x=1}\\) = sec (tan-1<\/sup> 1). \\(\\frac { 1 }{1+1}\\)
\n= sec (\\(\\frac { \u03c0 }{4}\\)) \\(\\frac { 1 }{2}\\) = \\(\\frac { \u221a2 }{2}\\) = \\(\\frac { 1 }{\u221a2}\\)<\/p>\n<\/details>\n

\n

Question 16.
\nIf g is the inverse of a function f and f'(x) = \\(\\frac { 1 }{1+x^5}\\), then g'(x) is equal to
\n(a) 5x4<\/sup>
\n(b) \\(\\frac {1}{1+{g(x)}^5}\\)
\n(c) 1 + {g(x)}5<\/sup>
\n(d) 1 + x5<\/sup><\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) \\(\\frac {1}{1+{g(x)}^5}\\)
\nHint:
\nHere f(g(x)) = x. [\u2235 g is the inverse of f]
\nf'(g(x)) g'(x) = 1
\n\u21d2 g'(x) = \\(\\frac { 1 }{f'{g(x)}}\\) = \\(\\frac { 1 }{1+{g(x)}^5}\\)<\/p>\n<\/details>\n

\n

Question 17.
\nIf the function
\ng(x) = \\(\\left\\{\\begin{array}{ll}
\nk \\sqrt{x+1} & ; 0 \\leq x \\leq 3 \\\\
\nm x+2 & ; 3 \\end{array}\\right.\\)
\nis differentiable, then the value of k + m is
\n(a) 2
\n(b) \\(\\frac {16}{5}\\)
\n(c) \\(\\frac {10}{3}\\)
\n(d) 4<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (a) 2
\nHint:
\nWe have
\ng(x) = \\(\\left\\{\\begin{array}{ll}
\nk \\sqrt{x+1} & ; 0 \\leq x \\leq 3 \\\\
\nm x+2 & ; 3 \\end{array}\\right.\\)
\nWhen this function is differentiable, then it is continuous
\n\u21d2 \\(\\lim _{x \\rightarrow 3^{-}}\\) g(x) = \\(\\lim _{x \\rightarrow 3^{+}}\\) g(x) = g(3)
\n\u21d2 2k = 3m + 2 = 2k
\n\u21d2 2k = 3m + 2 ………… (1)
\nAlso, LHD = \\(\\lim _{x \\rightarrow 3^{-}}\\) g(x) = \\(\\frac {k}{4}\\)
\nRHD = \\(\\lim _{x \\rightarrow 3^{+}}\\) g(x) = m
\n\u2234 LHD = RHD k
\n\u21d2 \\(\\frac {k}{4}\\) = m
\nSolving (1) and (2),
\nk = \\(\\frac {8}{5}\\) and m = \\(\\frac {2}{5}\\)
\nHence, k + m = \\(\\frac {8}{5}\\) + \\(\\frac {2}{5}\\) = \\(\\frac {10}{2}\\) = 2.<\/p>\n<\/details>\n

\n

Question 18.
\nFor x \u2208 R, f(x) = |log 2 – sin x| and g(x) =f(f(x)), then
\n(a) g is not differentiable at x = 0
\n(b) g'(0) = cos (log 2)
\n(c) g'(0) = -cos (log 2)
\n(d) g is differentiable at x = 0 and g'(0) = – sin (log 2).<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: (b) g'(0) = cos (log 2)
\nHint:
\nWe have : f(x) = log 2 – sin x
\nand g(x) = f(f cos x)
\n= log 2 – sin (log 2 – sin x).
\nSince \u2018g\u2019 is the sum of two differentiable functions,
\n\u2234 g is differentiable.
\ng'(x) = 0 – cos (log 2 – sin x) (0 – cos x)
\n= cos (log 2 – sin x) cos x.
\nHence, g’ (x) = cos (log 2).<\/p>\n<\/details>\n

\n

Fill in the blanks<\/span><\/p>\n

Question 1.
\nIf f(x) = \\(\\left\\{\\begin{array}{c}
\n\\frac{x^{2}-1}{x-1}, \\text { when } x \\neq 1 \\\\
\nk, \\text { when } x=1
\n\\end{array}\\right.\\) is continuous then the value of k = …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: 2.<\/p>\n<\/details>\n

\n

Question 2.
\nIf f(x) = x + 7, and g(x) = x – 7, x \u2208R, then \\(\\frac { d }{dx}\\) (fog) (x) = ……………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: 1.<\/p>\n<\/details>\n

\n

Question 3.
\nIf 2x + 3y = sin x, then \\(\\frac { dy }{dx}\\) = …………………..<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: \\(\\frac { cos x-2 }{3}\\)<\/p>\n<\/details>\n

\n

Question 4.
\n\\(\\frac { d }{dx}\\) (cosec-1<\/sup> x) = …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: \\(\\frac { -1 }{|x|\\sqrt{x^2-1}}\\)<\/p>\n<\/details>\n

\n

Question 5.
\n\\(\\frac { d }{dx}\\) (\\(\\sqrt { e^{ \\sqrt{x}} }\\)) = …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: \\(\\frac { 1 }{4\u221ax}\\) \\(\\sqrt { e ^{\\sqrt{x}} }\\)<\/p>\n<\/details>\n

\n

Question 6.
\nIf x = at\u00b2, y = 2at, then \\(\\frac { dy }{dx}\\) = ……………….<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: \\(\\frac { 1 }{t}\\)<\/p>\n<\/details>\n

\n

Question 7.
\nThe derivative of xx<\/sup> w.r.t. x is.<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: xx<\/sup> (1 + log x).<\/p>\n<\/details>\n

\n

Question 8.
\nIf y = x\u00b2 + 3x + 2, then \\(\\frac { d^2y }{dx^2}\\) = ………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: 2.<\/p>\n<\/details>\n

\n

Question 9.
\nValue of ‘c’ in Rolle\u2019s Theorem for the function f(x) = x\u00b3 – 3x in [-\u221a3, 0] is ……………..<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: c = -1.<\/p>\n<\/details>\n

\n

Question 10.
\nValue of \u2018c\u2019 in LMV Theorem for f(x) = x\u00b2 in [2, 4] is …………………<\/p>\n

\nAnswer<\/span><\/summary>\n

Answer: c = 3.<\/p>\n<\/details>\n

\n

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Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students …<\/p>\n

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