Calculus – MCQs & Full Test (50 Questions)

This test covers the entire Calculus syllabus of NCERT Class 12 and follows the CBSE objective-question pattern.

1. 1. A function is continuous at x = a if:

      (A) LHL = RHL only

      (B) Limit exists only

      (C) f(a) exists

      (D) LHL = RHL = f(a)

   2. If a function is differentiable at x = a, then it is:

      (A) discontinuous

      (B) continuous

      (C) undefined

      (D) constant

   3. The function |x| is not differentiable at:

      (A) x = 1

      (B) x = −1

      (C) x = 0

      (D) x = 2

   4. d/dx (sin x) equals:

      (A) −cos x

      (B) cos x

      (C) sin x

      (D) −sin x

   5. d/dx (cos x) equals:

      (A) sin x

      (B) cos x

      (C) −sin x

      (D) −cos x

   6. The derivative of tan x is:

      (A) sec x

      (B) sec²x

      (C) cosec²x

      (D) tan²x

   7. If y = sin(3x), then dy/dx equals:

      (A) cos(3x)

      (B) 3 sin(3x)

      (C) 3 cos(3x)

      (D) −3 cos(3x)

   8. Implicit differentiation is used when:

      (A) y is isolated

      (B) y is constant

      (C) y is not isolated

      (D) function is continuous

   9. If f′(x) > 0, then f(x) is:

      (A) decreasing

      (B) constant

      (C) increasing

      (D) maximum

   10. A point where f′(x) = 0 is called:

       (A) critical point

       (B) inflection point

       (C) stationary point

       (D) turning point

1. 1. For maximum value, f′(x) changes from:

      (A) − to +

      (B) + to −

      (C) 0 to +

      (D) 0 to −

   2. For minimum value, f′(x) changes from:

      (A) + to −

      (B) − to +

      (C) + to 0

      (D) 0 to +

   3. Which test is compulsory in NCERT for maxima–minima?

      (A) Second derivative test

      (B) First derivative test

      (C) Graph test

      (D) Value test

1. 1. ∫ x² dx equals:

      (A) x³

      (B) x³/3 + C

      (C) 2x

      (D) x² + C

   2. ∫ cos x dx equals:

      (A) sin x + C

      (B) −sin x + C

      (C) cos x + C

      (D) −cos x + C

   3. The constant C appears because:

      (A) integral is unique

      (B) derivative of constant is zero

      (C) integral is zero

      (D) derivative is constant

   4. ∫ 1/x dx equals:

      (A) x

      (B) 1/x

      (C) ln|x| + C

      (D) eˣ

   5. ∫ 0 dx equals:

      (A) 0

      (B) 1

      (C) C

      (D) x

1. 1. ∫_a^a f(x) dx equals:

      (A) 1

      (B) −1

      (C) 0

      (D) f(a)

   2. ∫_a^b f(x) dx equals:

      (A) F(a) − F(b)

      (B) F(b) − F(a)

      (C) F(a) + F(b)

      (D) 0

   3. If f(x) is odd, then ∫_−a^a f(x) dx equals:

      (A) a

      (B) −a

      (C) 0

      (D) 1

   4. Constant of integration is used in:

      (A) definite integral

      (B) indefinite integral

      (C) both

      (D) none

1. 1. Area under curve y = f(x) is given by:

      (A) f(x)

      (B) d/dx f(x)

      (C) ∫ f(x) dx

      (D) f′(x)

   2. If curve lies below x-axis, area is:

      (A) negative

      (B) zero

      (C) positive

      (D) undefined

   3. Area between two curves is:

      (A) ∫ (lower − upper) dx

      (B) ∫ (upper − lower) dx

      (C) ∫ upper dx

      (D) ∫ lower dx

1. 1. A differential equation involves:

      (A) integrals

      (B) limits

      (C) derivatives

      (D) constants only

   2. The order of dy/dx = x² is:

      (A) 1

      (B) 2

      (C) 0

      (D) undefined

   3. The degree of (dy/dx)³ = y is:

      (A) 1

      (B) 2

      (C) 3

      (D) not defined

   4. The solution of dy/dx = 2x is:

      (A) x²

      (B) 2x²

      (C) x² + C

      (D) 2x + C

   5. General solution contains:

      (A) no constants

      (B) fixed constants

      (C) arbitrary constants

      (D) variables only

1. If f′(x) = 0 for all x, then f(x) is:

   (A) linear

   (B) quadratic

   (C) constant

   (D) exponential

2. The derivative of eˣ is:

   (A) eˣ

   (B) xeˣ

   (C) ln x

   (D) 1

3. ∫ eˣ dx equals:

   (A) xeˣ

   (B) eˣ + C

   (C) ln x

   (D) 1

4. ∫ sin x dx equals:

   (A) cos x

   (B) −cos x + C

   (C) sin x + C

   (D) −sin x

5. Which chapter has maximum board weightage?

   (A) Limits

   (B) Continuity

   (C) Integrals

   (D) Differential equations