VBQs Continuity and Differentiability Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Continuity and Differentiability Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

**Continuity and Differentiability VBQs Class 12 Mathematics**

**Question. For x ∈ R, f(x) = |log2 – sinx| and g(x) = f(f(x)), then :**

(a) g'(0) = – cos(log2)

(b) g is differentiable at x = 0 and g'(0) = – sin(log2)

(c) g is not differentiable at x = 0

(d) g'(0) = cos(log2)

## Answer

D

**Question. (image 32)**

(a) continuous on R – {1} and differentiable on R – {–1, 1}.

(b) both continuous and differentiable on R – {1}.

(c) continuous on R – {–1} and differentiable on R – {–1, 1}.

(d) both continuous and differentiable on R – {–1}.

## Answer

A

**Question. Let S be the set of all functions f : [0,1] → R, which are continuous on [0, 1] and differentiable on (0,1). Then for every f in S, there exists a c ∈ (0,1), depending on f, such that:**

(image 35)

## Answer

None

**Question. Let f(x) = x|x|, g(x) = sin x and h(x) = (gof) (x). Then**

(a) h(x) is not differentiable at x = 0.

(b) h(x) is differentiable at x = 0, but h'(x) is not continuous at x = 0

(c) h'(x) is continuous at x = 0 but it is not differentiable at x = 0

(d) h'(x) is differentiable at x = 0

## Answer

C

**Question. Let for i = 1, 2, 3, p _{i}(x) be a polynomial of degree 2 in x, p’_{i}(x) and p”_{i}(x) be the first and second order derivatives of p_{i}(x) respectively. Let, (image 107) and B(x) = [A(x)]^{T} A(x). Then determinant of B(x):**

(a) is a polynomial of degree 6 in x.

(b) is a polynomial of degree 3 in x.

(c) is a polynomial of degree 2 in x.

(d) does not depend on x.

## Answer

A

**Question. Let the function, f: [–7, 0] → R be continuous on [ –7, 0] and differentiable on (–7, 0). If f(–7) = –3 and f'(x) d” 2, for all x∈(–7, 0), then for all such functions f, f'(–1) + f(0) lies in the interval:**

(a) (– ∞ , 20]

(b) [–3, 11]

(c) (– ∞, 11]

(d) [–6, 20]

## Answer

A

**Question. Let a, b ∈ R, (a ≠ 0) . if the function f defined as (image 13) is continuous in the interval [0,∞) , then an ordered pair (a, b) is :**

(a) (-√2, 1- √3)

(b) ( √2, -1+ √3)

(c) ( √2, 1 – √3)

(d) (-√2, 1+ √3)

## Answer

C

**Question. Let k be a non– ero real number.**

(image 14)**is a continuous function then the value of k is:**

(a) 4

(b) 1

(c) 3

(d) 2

## Answer

C

**Question. Let f(x) = log _{e}(sinx), (0 < x < π) and g(x) = sin^{–1}(e^{–x}), (x ≥ 0). If a is a positive real number such that a = (fog)'(α) and b = (fog)(α), then:**

(a) aα

^{2}+ bα + a = 0

(b) aα

^{2}– bα – a =1

(c) aα

^{2}– bα – a = 0

(d) aα

^{2}+ bα – a = – 2a

^{2}

## Answer

B

**Question. Let f : R → R be differentiable at c ∈ R and f(c) = 0. If g(x) = | f(x) |, then at x = c, g is :**

(a) not differentiable if f'(c) = 0

(b) differentiable if f”(c) ≠ 0

(c) differentiable if f’ (c) = 0

(d) not differentiable

## Answer

C

**Question. Let f : R → R be a function defined as (image 9)Then, f is :**

(a) continuous if a = 5 and b = 5

(b) continuous if a = – 5 and b = 10

(c) continous if a = 0 and b = 5

(d) not continuous for any values of a and b

## Answer

D

**Question. If the function f defined as (image 10) x ≠ 0, is continuous at x = 0, then the ordered pair (k, f (0)) is equal to?**

(a) (3, 1)

(b) (3, 2)

(c) (1/3, 2)

(d) (2, 1)

## Answer

A

**Question. Let f be a differentiable function such that f(1) = 2 and f'(x) = f(x) for all x ∈ R. If h(x) = f(f(x)), then h'(1) is equal to :**

(a) 2e^{2}

(b) 4e

(c) 2e

(d) 4e^{2}

## Answer

B

**Question. Let (image 44) and g(x) = |f(x)| + f(|x|). Then, in the interval (–2, 2), g is :**

(a) differentiable at all points

(b) not continuous

(c) not differentiable at two points

(d) not differentiable at one point

## Answer

D

**Question. (image 97)**

(a) y ”(0) = 0

(b) | y ‘(0) | + | y ”(0) |=1

(c) | y ”(0) |= 2

(d) | y ‘(0) | + | y ”(0) |= 3

## Answer

C

**Question. If c is a point at which Rolle’s theorem holds for the function, f(x) = log _{e}(x^{2}+a / 7x) in the interval [3, 4], where a ∈ R, then f”(c) is equal to:**

(a) − 1/12

(b) 1/12

(c) − 1/24

(d) √3/7

## Answer

B

**Question. Let (image 47) Let S be the set of points in the interval (– 4, 4) at which ƒ is not differentiable. Then S:**

(a) is an empty set

(b) equals {– 2, – 1, 0, 1, 2}

(c) equals {– 2, – 1, 1, 2}

(d) equals {– 2, 2}

## Answer

B

**Question. Let ƒ : (– 1, 1) → R be a function defined by ƒ(x) = max { − | x |, − √(1 – x ^{2}) }. If K be the set of all points at which ƒ is not differentiable, then K has exactly:**

(a) five elements

(b) one element

(c) three elements

(d) two elements

## Answer

C

** Question. If the equation an x a_{n}x^{n} + a_{n-1}x^{n-1} + …………. + a_{1}x = 0, a_{1} ≠ 0, n ≥ 2, has a positive root x = α , then the equation na_{n}x^{n-1} + (n – 1)a_{n-1}x^{n-2} + ……… + a_{1} = 0 has a positive root, which is**(a) greater than α

(b) smaller than α

(c) greater than or equal to α

(d) equal to α

## Answer

B

Question. If 2a + 3b + 6c = 0, (a, b, c ∈ R) then the quadratic equation ax^{2} + bx + c = 0 has

(a) at least one root in [0, 1]

(b) at least one root in [2, 3]

(c) at least one root in [4, 5]

(d) None of these

## Answer

A

** Question. A value of c for which conclusion of Mean Value Theorem holds for the function f (x) = loge x on the interval [1, 3] is**(a) log

_{3}e

(b) log

_{e}3

(c) 2 loBg

_{3}e

(d) 1/2 log

_{3}e

## Answer

C

** Question.** (image 119)

(a) n

^{2}y

(b) – n

^{2}y

(c) –y

(d) 2x

^{2}y

## Answer

A

**Question. If (image 69) then dy/da at a = 5π/6 is:**

(a) 4

(b) 4/3

(c) –4

(d) – 1/4

## Answer

A

**Question. Let y = y(x) be a function of x satisfying y√(1−x ^{2}) = k − x√(1−y^{2}) where k is a constant and y(1/2) = −(1/4). Then dy/dx at x = 1/2, is equal to:**

(a) – √5/4

(b) – √5/2

(c) 2/√5

(d) √5/2

## Answer

B

**Question. If the function (image 51) is differentiable at x = 1, then a/b is equal to :**

(a) (π+2)/2

(b) (π – 2)/2

(c) (–π – 2)/2

(d) –1 – cos^{–1}(2)

## Answer

A

**Question. If the function.**

(image 52)**is differentiable, then the value of k + m is :**

(a) 10/3

(b) 4

(c) 2

(d) 16/5

## Answer

C

**Question. Let f : [1, 3] → R be a function satisfying x/[x] ≤ f[x] ≤ √(6 – x), for all x ≠ 2 and f (2) = 1, where R is the set of all real numbers and [x] denotes the largest integer less than or equal to x.****Statement 1:** (image 21)**Statement 2:** f is continuous at x = 2.

(a) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

(b) Statement 1 is false, Statement 2 is true.

(c) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

(d) Statement 1 is true, Statement 2 is false.

## Answer

D

**Question.Statement 1:** A function f : R → R is continuous at x_{0} if and only if (image 22)**Statement 2:** A function f : R → R is discontinuous at x_{0} if and only if, (image 22)

(a) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

(b) Statement 1 is false, Statement 2 is true.

(c) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.

(d) Statement 1 is true, Statement 2 is false.

## Answer

A

**Question. If x + | y | = 2y, then y as a function of x, at x = 0 is**

(a) differentiable but not continuous

(b) continuous but not differentiable

(c) continuous as well as differentiable

(d) neither continuous nor differentiable

## Answer

B

**Question. If function f(x) is differentiable at x = a, then (image 58)**

(a) – a^{2}f'(a)

(b) af(a) – a^{2}f'(a)

(c) 2af(a) – a^{2}f'(a)

(d) 2af(a) + a^{2}f'(a)

## Answer

C

**Question. Let f (x) be a polynomial function of second degree. If f(1) = f(-1) and a,b,c are in A. P , then f'(a), f'(b), f'(c) are in**

(a) Arithmetic -Geometric Progression

(b) A.P

(c) G..P

(d) H.P.

## Answer

B

**Question. If f(x + y) = f(x). f(y)∀x.y and f(5) = 2, f'(0) = 3, then f'(5) is**

(a) 0

(b) 1

(c) 6

(d) 2

## Answer

C

**Question. The set of points where f(x) = x / (1 + |x|) is differentiable is**

(a) (-∞,0)∪(0,∞)

(b) (-∞,-1)∪(-1,∞)

(c) (-∞,∞)

(d) (0,∞)

## Answer

C

**Question. If f is a real valued differentiable function satisfying | f(x) – f(y) | ≤ (x – y) ^{2} , x, y ∈ R and f(0) = 0, then f(1) equals**

(a) – 1

(b) 0

(c) 2

(d) 1

## Answer

B

**Question. The derivative of tan ^{-1}( √(1+x^{2}−1) / x ) with respect to tan^{-1}( 2x√(1−x^{2}) / (1−2x^{2}) ) at x = 1/2 is :**

(a) 2√3/5

(b) √3/12

(c) 2√3/3

(d) √3/10

## Answer

D

**Question. Let f: R → R be a function such that | f(x) | ≤ x ^{2} , for all x ∈ R . Then, at x = 0, f is:**

(a) continuous but not differentiable.

(b) continuous as well as differentiable.

(c) neither continuous nor differentiable.

(d) differentiable but not continuous.

## Answer

B

**Question. Let f, g: R → R be two functions defined by (image 54) ****Statement I:** f is a continuous function at x = 0.**Statement II:** g is a differentiable function at x = 0.

(a) Both statement I and II are false.

(b) Both statement I and II are true.

(c) Statement I is true, statement II is false.

(d) Statement I is false, statement II is true.

## Answer

B

**Question. If (a + √2bcos x)(a – √2bcos y) = a ^{2} – b^{2}, where a > b > 0, then dx/dy at (π/4, π/4) is :**

(a) (a−2b)/(a+2b)

(b) (a−b)/(a+b)

(c) (a+b)/(a−b)

(d) (2a+b)/(2a−b)

## Answer

C

**Question. If (image 67) then dy/dx at x = 0 is ___________.**

## Answer

91

**Question. If f(x) is continuous and f(9/2) = 2/9, then (image 16) is equal to:**

(a) 9/2

(b) 2/9

(c) 0

(d) 8/9

## Answer

B

**Question. Consider the function :****f (x) = [ x] + | 1 – x |, -1 ≤ x ≤ 3 where [x] is the greatest integer function.****Statement 1 :** f is not continuous at x = 0, 1, 2 and 3.**Statement 2 : (image 17)**

(a) Statement 1 is true ; Statement 2 is false,

(b) Statement 1 is true; Statement 2 is true; Statement 2 is not correct explanation for Statement 1.

(c) Statement 1 is true; Statement 2 is true; Statement It is a correct explanation for Statement 1.

(d) Statement 1 is false; Statement 2 is true.

## Answer

A

**Question. If x = 2sinθ – sin2θ and y = 2cosθ – cos2θ, θ ∈ [0, 2π], then d ^{2}y/dx^{2} at θ = π is :**

(a) 3/4

(b) – 3/8

(c) 3/2

(d) – 3/4

## Answer

None

**Question. Let (image 88) (wherever it is defined) is equal to :**

(a) –1/(1 – x)^{2}

(b) 3/(1 – x)^{2}(c) 1/(1 – x)^{2}

(d) –3/(1 – x)^{2}

## Answer

B

**Question. If ƒ'(x) = sin (log x) and y = ƒ( (2x+3)/(3–2x) ), then dy/dx equals**

(image 89)

## Answer

C

**Question. If e ^{y} + xy = e, the ordered pair (dy/dx, d^{2}y/dx^{2}) at x = 0 is equal to :**

(a) (1/e, − 1/e

^{2})

(b) (− 1/e, 1/e

^{2})

(c) (1/e, 1/e

^{2})

(d) (− 1/e, − 1/e

^{2})

## Answer

B

**Question. The derivative of tan ^{-1}( (sinx − cosx) / (sinx + cosx) ), with respect to x/2 , where ( x ∈ (0, π/2) ) is :**

(a) 1

(b) 2/3

(c) 1/2

(d) 2

## Answer

D

**Question. Let S be the set of points where the function, f(x) = |2 – |x – 3||, x∈R, is not differentiable. Then (image 37) is equal to _______.**

## Answer

3

**Question. If (image 38) is continuous at x = 0, then the ordered pair (p, q) is equal to:**

(a) (− 3/2, − 1/2)

(b) (− 1/2, 3/2)

(c) (− 3/2, 1/2)

(d) (5/2, − 1/2)

## Answer

C

**Question. Let (image 59) Then which one of the following is true?**

(a) ƒ is neither differentiable at x = 0 nor at x =1

(b) ƒ is differentiable at x = 0 and at x =1

(c) ƒ is differentiable at x = 0 but not at x = 1

(d) ƒ is differentiable at x = 1 but not at x = 0

## Answer

C

**Question. Let f : R → R be a function defined by f(x) = min {x +1, |x| + 1},Then which of the following is true?**

(a) f (x) is differentiable everywhere

(b) f (x) is not differentiable at x = 0

(c) f (x) ≥ 1 for all x ∈ R

(d) f (x) is not differentiable at x = 1

## Answer

A

**Question. If (image 73) then dy/dx is equal to :**

(a) π/6 − x

(b) x − π/6

(c) π/3 − x

(d) 2x − π/3

## Answer

None

**Question. Let f : R → R be a function such that f(x) = x ^{3} + x^{2}f'(1) + xf”(2) + f”'(3), x ∈ R. Then f(2) equals:**

(a) – 4

(b) 30

(c) – 2

(d) 8

## Answer

C

**Question. If x = 3 tan t and y = 3 sec t, then the value of d ^{2}y/dx^{2} at t = π/4, is :**

(a) 1/3√2

(b) 1/6√2

(c) 3/2√2

(d) 1/6

## Answer

B

**Question. If the function (image 29) is twice differentiable, then the ordered pair (k _{1}, k_{2}) is equal to:**

(a) (1/2, 1)

(b) (1, 0)

(c) (1/2, −1)

(d) (1, 1)

## Answer

A

**Question. Let f be a twice differentiable function on (1, 6). If f (2) = 8, f'(2) = 5, f'(x) ≥ 1 and f”(x) ≥ 4, for all x ∈ (1, 6), then :**

(a) f(5) + f ‘(5) ≤ 26

(b) f(5) + f ‘(5) ≥ 28

(c) f'(5) + f ”(5) ≤ 20

(d) f(5) ≤ 10

## Answer

B

**Question. If f(x) = x ^{n} , then the value of**

(image 117)

(a) 1

(b) 2

^{n}

(c) 2

^{n}-1

(d) 0

## Answer

D

**Question. Let f be differentiable for all x. If f (1) = – 2 and f ‘(x) ≥ 2 for x ∈ [1, 6], then**

(a) f (6) ≥ 8

(b) f (6) < 8

(c) f (6) < 5

(d) f (6) = 5

## Answer

A

**Question. Let f (a) = g(a) = k and their nth derivatives f ^{n}(a) , g^{n}(a) exist and are not equal for some n. Further if**

(image 118)

**then the value of k is**

(a) 0

(b) 4

(c) 2

(d) 1

## Answer

B

**Question. If 2a + 3b + 6c = 0, then at least one root of the equation ax ^{2} + bx + c = 0 lies in the interval**

(a) (1, 3)

(b) (1, 2)

(c) (2, 3)

(d) (0, 1)

## Answer

D

**Question. If f (x) = sin ^{-1}((2×3^{x})/(1+9^{x})), then f'(– 1/2) equals.**

(a) √3log

_{e}√3

(b) – √3log

_{e}√3

(c) – √3log

_{e}3

(d) √3log

_{e}3

## Answer

A

**Question. If x ^{2} + y^{2} + sin y = 4, then the value of d^{2}y/dx^{2} at the point (– 2, 0) is**

(a) – 34

(b) – 32

(c) – 2

(d) 4

## Answer

A

**Question. If f(x) = x ^{2} – x + 5, x > 1/2, and g(x) is its inverse function, then g'(7) equals:**

(a) – 1/3

(b) 1/13

(c) 1/3

(d) – 1/13

## Answer

C

**Question. Suppose f(x) is differentiable at x = 1 and (image 63) then f'(1) equals**

(a) 3

(b) 4

(c) 5

(d) 6

## Answer

C

**Question. If (image 64) then f(x) is**

(a) discontinuous every where

(b) continuous as well as differentiable for all x

(c) continuous for all x but not differentiable at x = 0

(d) neither differentiable nor continuous at x = 0

## Answer

C

**Question. If y = sec(tan ^{–1}x), then dy/dx at x = 1 is equal to :**

(a) 1/√2

(b) 1/2

(c) 1

(d) √2

## Answer

A

**Question. Let [t] denote the greatest integer ≤ t and (image 3) Then the function, f(x) = [x ^{2}] sin(πx) is discontinuous, when x is equal to :**

(a) √(A+1)

(b) √(A + 5)

(c) √(A+ 21)

(d) √A

## Answer

A

**Question. If the function f defined on (- 1/3, 1/3) (image 4) is continuous, then k is equal to __________.**

## Answer

5

**Question. If the curves x ^{2}/α + y^{2}/4 = 1 and y^{3} = 16x intersect at right angles, then a value of α is :**

(a) 2

(b) 4/3

(c) 1/2

(d) 3/4

## Answer

B

**Question. (image 103)**

(a) 12y

(b) 224y^{2}

(c) 225y^{2}**(d) 225 y**

## Answer

D

**Question. If Rolle’s theorem holds for the function f(x) 2x ^{3} + bx^{2} + cx, x ∈ [–1, 1], at the point x = 1/2 , then 2b + c equals :**

(a) –3

(b) –1

(c) 2

(d) 1

## Answer

B

**Question. For (image 87) Then, 1 + (dy/dx) ^{2} equals :**

(a) x

^{2}/y

^{2}

(b) y

^{2}/x

^{2}

(c) (x

^{2}+y

^{2})/y

^{2}

(d) (x

^{2}+y

^{2})/x

^{2}

## Answer

D

**Question. Let f : (–1, 1) → R be a differentiable function with f(0) = – 1 and f'(0) = 1. Let g(x) = [f(2f(x) + 2)] ^{2}. Then g'(0) =**

(a) –4

(b) 0

(c) –2

(d) 4

## Answer

A

**Question. d ^{2}x/dy^{2} equals :**

(image 111)

## Answer

C

**Question. Let f(x) = x | x | and g(x) = sin x.****Statement-1 :** gof is differentiable at x = 0 and its derivative is continuous at that point.**Statement-2 :** gof is twice differentiable at x = 0.

(a) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(b) Statement-1 is true, Statement-2 is false.

(c) Statement-1 is false, Statement-2 is true.

(d) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

## Answer

B

**Question. If (image 93) x > 0, then dy/dx is**

(a) (1 + x)/x

(b) 1/x

(c) (1 − x)/x

(d) x/(1 + x)

## Answer

C

**Question. For all twice differentiable functios f : R → R, with f(0) = f(1) = f’(0) = 0**

(a) f “(x) ¹ 0 at every point x∈(0,1)

(b) f “(x) = 0, for some x∈(0,1)

(c) f “(0) = 0

(d) f “(x) = 0, at every point x∈(0,1)

## Answer

B

**Question. Let x ^{k} + y^{k} = a^{k}, (a, k > 0) and dy/dx + (y/x)^{1/3} = 0, then k is:**

(a) 3/2

(b) 4/3

(c) 2/3

(d) 1/3

## Answer

C

**Question. Let f be a polynomial function such that f (3x) = f'(x) , f”(x), for all x ∈ R. Then :**

(a) f(b) + f'(b) = 28

(b) f”(b) – f'(b) = 0

(c) f “(b) – f'(b) = 4

(d) f(b) – f'(b) + f”(b) = 10

## Answer

B

**Question. If f(x) = [x] – [x/4], x ∈ R, where [x] denotes the greatest integer function, then:**

(image 6)

## Answer

A

**Question. If the function (image 7) is continuous at x = 5, then the value of a – b is:**

(a) 2/(π+5)

(b) -2/(π+5)

(c) 2/(π – 5)

(d) 2/(5 – π)

## Answer

D

**Question. If f and g are differentiable functions in [0, 1] satisfying f(0) = 2 = g(1), g(0) = 0 and f(1) = 6, then for some c ∈ ]0,1[**

(a) f'(c) = g'(c)

(b) f'(c) = 2g'(c)

(c) 2f'(c) = g'(c)

(d) 2f'(c) = 3g'(c)

## Answer

B

**Question. If the Rolle’s theorem holds for the function f(x) = 2x ^{3} + ax^{2} + bx in the interval [– 1, 1] for the point c = 1/2, then the value of 2a + b is:**

(a) 1

(b) – 1

(c) 2

(d) – 2

## Answer

B

**Question. The value of c in the Lagrange’s mean value theorem for the function f(x) = x ^{3} – 4x^{2} + 8x + 11, when x ∈ [0,1] is:**

(a) (4 − √5)/3

(b) (4 − √7)/3

(c) 2/3

(d) (√7 − 2)/3

## Answer

B

**Question. If (image 101) then λ + k is equal to :**

(a) – 23

(b) – 24

(c) 26

(d) – 26

## Answer

B

**Question. Let f(x) = 15 – |x – 10|; x ∈ R. Then the set of all values of x, at which the function, g(x) = f (f(x)) is not differentiable, is:**

(a) {5, 10, 15}

(b) {10, 15}

(c) {5, 10, 15, 20}

(d) {10}

## Answer

A

**Question. If f (1) = 1, f'(1) = 3, then the derivative of f(f(f(x))) + (f(x)) ^{2} at x = 1 is :**

(a) 33

(b) 12

(c) 15

(d) 9

## Answer

A

**Question. Let (image 26) If f(x) is continuous in [0, π/2], then f(π/4) is**

(a) –1

(b) 1/2

(c) – 1/2

(d) 1

## Answer

C

**Question. f is defined in [-5, 5] as f(x) = x if x is rational = – x if x is irrational. Then**

(a) f(x) is continuous at every x, except x = 0

(b) f(x) is discontinuous at every x, except x = 0

(c) f(x) is continuous everywhere

(d) f(x) is discontinuous everywhere

## Answer

B

**Question. Let y be an implicit function of x defined by x ^{2x} – 2x^{x} cot y – 1= 0. Then y'(1) equals**

(a) 1

(b) log 2

(c) –log 2

(d) –1

## Answer

D

**Question. If x ^{m}.y^{n} = (x+y)^{m+n}, then dy/dx is**

(a) y/x

(b) x+y/xy

(c) xy

(d) x/y

## Answer

A