CBSE Class 12 Mathematics Continuity and Differentiability MCQs Set D

Question: f (x) = 1/1+tan x
(a) is a continuous, real-valued function for all x ∈ (– ∞, ∞)
(b) is discontinuous only at x = 3π/4
(c) has only finitely many discontinuities on (– ∞, ∞)
(d) has infinitely many discontinuities on (– ∞, ∞)
Answer: d

Question: If f (x+Y/3) = 2+f(x) f(y)/3 for all real x and y and f′(2)=2,
(a) 2x+1
(b) 2x
(c) 2x+2
(d) Constant
Answer: c

Question: Let f Q :[1 ,10] → Q be a continuous function and f(1) = 10, then f(10) is equal to
(a) 1/10
(b) 10
(c) 1
(d) Cannot be obtained
Answer: b

Question: If (x + y)= f(x) f(y) for all x ,y ∈ R, f(5)=2 and (0) =3. Then, f ′(5) equals
(a) 6
(b) 5
(c) 4
(d) 3
Answer: a

Question: ABC is an isosceles triangle inscribed in a circle of radius r. If AB AC = and h is the altitude from
(a) 1/r
(b) 1/64r
(c) 1/128r
(d) 1/2r
Answer: c 

Question: If f is strictly increasing function, then lim f(x2)-f(x)/f(x)-f(0) is equal to
x→0
(a) 0
(b) 1
(c) −1
(d) 2
Answer: c 

Question: lim |x| , ^{[cos x]} ,where [.] is the greatest integer
x →o
function, is
(a) 1
(b) does not exist
(c) 0
(d) None of these
Answer: a

Question: If , y = e^3x+7 , then the value of dy/dx|x=0 is
(a) 1
(b) 0
(c) – 1
(d) 3e7
Answer: d

Question: If y = sec x°, then dy/dx is equal to :
(a) sec x tan x
(b) sec x° tan x°
(c) π/180 sec x° tan x°
(d) None of these
Answer: c 

Question: Let f(x) = sinx, g(x) = x² and h(x) = log_ex.
If F(x) = (hogof) (x), then F'(x) is equal to
(a) a cosec³x
(b) 2 cot x² – 4x² cosec²x²
(c) 2x cot x²
(d) – 2 cosec²x
Answer: d 

Question: The value of the derivative of |x – 1| + |x – 3| at x = 2 is :
(a) –2
(b) 0
(c) 2
(d) not defined
Answer: b

Question: If f (x) = (x + 1)^{cot x} be continuous at x = 0 then f (0) is equal to:
(a) 0
(b) – e
(c) e
(d) None
Answer: c 

Question: The number of discontinuous functions y(x) on [– 2, 2] satisfying x² + y² = 4 is
(a) 0
(b) 1
(c) 2
(d) > 2
Answer: a

Question: A value of c for which the Mean Value Theorem holds for the function f(x) = logex on the interval [1, 3] is
(a) 2 log₃e
(b) 1/2 log₃e
(c) l log₃e
(d)  log₃
Answer: a 

Question: If y = log_ax + log_xa + log_xx + log_aa, then dy/dx is equal to
(a) 1/x + x log a
(b) log a/x + x/log a
(c) 1/x log a + x log a
(d) 1/x log a – log a/x (log x)²
Answer: d

Question: If x = 1–t²/1+t² and 2t/1+t² , then dy/dx is equal to :
(a) – y/x
(b) y/x
(c) – x/y
(d) x/y
Answer: c

Question: If y = log tan √x then the value of dy/dx is :
(a) 1/2√x
(b) sec²√x/√x tan x
(c) 2sec²√x
(d) sec²√x/2√xtan√x
Answer: d 

Question: In the interval [7, 9] the function f(x) = [x] is discontinuous at _______, where [x] denotes the greatest integer function
(a) 2
(b) 4
(c) 6
(d) 8
Answer: d

Question: If 2f (sin x) + f (cos x) = x , then d/dx f (x) is
(a) sin x + cos x
(b) 2
(c) 1/√1 – x²
(d) None of these
Answer: c

Question: The number of points at which the function f(x) = 1/x-[x], [.] denotes the greatest integer function is not continuous is
(a) 1
(b) 2
(c) 3
(d) None of these
Answer: d

Question: The function f(x) = cot x is discontinuous on the set
(a) {x = nπ,n ∈ Z}
(b) {x = 2nπ,n ∈ Z}
(c) {x = (2n+1)π/2 ;n ∈ Z}
(d) {x = nπ/2,n ∈ Z}
Answer: d

Question: If y = (tanx)^{sin x}, then dy/dx is equal to
(a) sec x + cos x
(b) sec x + log tan x
(c) (tan x)sin x
(d) None of these
Answer: d

Question: If y = (cos x²)², then dy/dx is equal to :
(a) – 4x sin 2 x²
(b) – x sin x²
(c) – 2x sin 2 x²
(d) – x cos 2 x²
Answer: c 

Question: The point of discontinuity of f (x) = tan (πx/x+1) other than x = –1 are :
(a) x = 0
(b) x = π
(c) x = 2m+1/1– 2m
(d) x = 2m–1/2m+1
Answer: c

Question: If f(x) = (log_cot xtan x)(log_tanxcot x)^–1 + tan^–14x/4–x² , then f'(2) is equal to
(a) 1/2
(b) –1/2
(c) 1
(d) – 1
Answer: a 

Question: If y = x – x² , then the derivative of y² with respect to x² is
(a) 1– 2x
(b) 2 – 4x
(c) 3x – 2x²
(d) 1– 3x + 2x²
Answer: a

Question: Let f (x) = 1 – tan x/4x – π , x ≠ π/4 , x ∈ (0,π/2). If f(x) is continuous in (0,π/2) , then f(π/4) =
(a) 1
(b) 1/2
(c) –1/2
(d) – 1
Answer: c

Question: Let 3f(x) – 2f(1/x) = x, then f ‘(2) is equal to
(a) 2/7
(b) 1/2
(c) 2
(d) 7/2
Answer: b

Question: If f (x) = x² sin1/x, where x ≠ 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is
(a) 0
(b) – 1
(c) 1
(d) None of these
Answer: a

Question: If y = log (1–x²/1+x²), then dy/dx , is equal to 
(a) 4x³/1– x⁴
(b) –4x/1– x⁴
(c) 1/4 – x⁴
(d) –4x³/1– x⁴
Answer: b

Question: If sin y + e^{–x cos y} = e , then dy/dx at (1, π) is equal to
(a) sin y
(b) – x cos y
(c) e
(d) sin y – x cos y
Answer: c

ASSERTION – REASON TYPE QUESTIONS

(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Question:
Assertion : If y = log₁₀x + log_ey, then
dy/dx = log₁₀e/x (y/y–1)
Reason : d/dx( log₁₀x) = log x/ log₁₀
and d/dx (log_ex) = log x/loge
Answer: c

Question:
Assertion : f (x) = xⁿ sin(1/x) is differentiable for all real values of x (n ≥ 2).
Reason : For n ≥ 2,limx→0 f(x) = 0
Answer: d

Question: Consider the function
f(x) = [sin x], x ∈ [0, π]
Assertion: f(x) is not continuous at x = π/2
Reason : lim f(x)_x→π/2 does not exist
Answer: c

Question:
Assertion : The function f(x) = |x|/x is continuous at x = 0.
Reason : The left hand limit and right hand limit of the function f(x) = |x|/x = are not equal at x = 0.
Answer: d

Question: Assertion : If a function f is discontinuous at c, then c is called a point of discontinuity.
Reason : A function is continuous at x = c, if the function is defined at x = c and the value of the function at x = c equals the limit of the function at x = c.
Answer: b

Question:
Assertion : d/dx e^{cos x} = e^{cos x} (– sin x)
Reason : d/dx e^x = e^x
Answer: b

Question:
Assertion : For x < 0, d/dx(ln|x|) = –1/x
Reason : For x < 0, |x| = – x
Answer: d

ASSERTION – REASON TYPE QUESTIONS

(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Question: Assertion : The function defined by f(x) = cos(x²) is a continuous function.
Reason : The cosine function is continuous in its domain i.e., x ∈ R.
Answer: b

Question: Assertion : f (x) = | [x] x | in x ∈ [–1, 2], where [ . ] represents greatest integer function, is non-differentiable at x = 2. 
Reason : Discontinuous function is always non differentiable.
Answer: a

Question: Assertion : If x = at² and y = 2at, then d²y/dx²|_t=2 = –1/6a
Reason : d²y/dx² = (dy/dt)² × (dt/dx)²
Answer: c

Question: Assertion : f (x) = | x | sin x, is differentiable at x = 0.
Reason : If f (x) is not differentiable and g (x) is differentiable at x = a, then f (x) . g (x) can still be differentiable at x = a.
Answer: a

Question: Assertion : Rolle’s theorem can not be verified for the function f (x) = |x| in the interval [–1, 1].
Reason : The function f (x) = |x| is differentiable in the interval (–1, 1) everywhere.
Answer: c

Question: Assertion : If u = f(tanx), v = g(secx) and f ‘(1) = 2,g (√2)= 4, then (du/dv)_{x = π / 4} = 1/√2
Reason : If u = f(x), v = g(x), then the derivative of f with respect to g is du/dv = du/dx / dv/dx
Answer: a

Question: Assertion : The function f (x) = |sin x| is not differentiable at points x = nπ.
Reason : The left hand derivative and right hand derivative of the function f (x) = |sin x| are not equal at points x = nπ.
Answer: d

Question: Assertion : Every differentiable function is continuous but converse is not true.
Reason : Function f(x) = |x| is continuous.
Answer: b