MCQ Questions Chapter 6 Application of Derivatives Class 12 Mathematics

Please refer to MCQ Questions Chapter 6 Application of Derivatives Class 12 Mathematics with answers provided below. These multiple-choice questions have been developed based on the latest NCERT book for class 12 Mathematics issued for the current academic year. We have provided MCQ Questions for Class 12 Mathematics for all chapters on our website. Students should learn the objective based questions for Chapter 6 Application of Derivatives in Class 12 Mathematics provided below to get more marks in exams.

Chapter 6 Application of Derivatives MCQ Questions

Question. The given function f(x) = -3x +15 is
(a) Increasing on R
(b) Decreasing on R
(c) Strictly increasing on R
(d) Strictly decreasing on R

Question. The tangent to a given curve f(x) = y is perpendicular to x-axis if
(a) 𝑑𝑦/dx = 0
(b) 𝑑𝑦/dx = 1
(c) 𝑑𝑥/dy = 0
(d) 𝑑𝑥/dy = 1

Question. If a function f(x) has ƒ′ (a) = 0 and ƒ′′(a) = 0, then
(a) x = a is a maximum for f(x)
(b) x = a is a minimum for f(x)
(c) It is difficult to say (a) and (b)
(d) f(x) is necessarily a constant function

Question. If the curves y=2𝑒𝑥 and y = a𝑒−𝑥 intersect orthogonally then the value of a is
(a) -1/2
(b) -2
(c) 1/2
(d) 2

Question. The equation of tangent to the curve y (1 + 𝑥2) = 2 –x, where it crosses x-axis is:
(a) x + 5y = 2
(b) x – 5y = 2
(c) 5x – y = 2
(d) 5x + y = 2

Question. The function f (x ) = a cos x + b tan x + x has extreme values at x = 0 and x = π/6 , then
(a) a = – 2/3 , b = -1
(b) a = -2/3 , b = -1
(c) a = -2/3 , b = 1
(d) a = 2/3 , b = 1

Question. If a differential function f (x ) has a relative minimum at x = 0, then the function f(x ) = f (x ) + ax + b has a relative minimum at x = 0 for
(a) all a and all b
(b) all b, if a = 0
(c) all b > 0
(d) all a > 0

Question. The minimum value of 9x + 4y, where xy =16 is
(a) 48
(b) 28
(c) 38
(d) 18

Question. The absolute maximum and minimum values of the function f given by f (x ) = cos² x + sin x, x ∈ [0, π]
(a) 2.25 and 2
(b) 1.25 and 1
(c) 1.75 and 1.5
(d) None of these

Question. The function f (x ) = 4x² – 18x² + 27x – 7 has
(a) one local maxima 
(b) one local minima
(c) one local maxima and two local minima
(d) neither maxima nor minima

Question. The function f(x) = x² – 2/x² – 4 has
(a) no point of local minima
(b) no point of local maxima
(c) exactly one point of local minima
(d) exactly one point of local maxima

Question. Let f : R → R be defined by f(x) =

If f has a local minimum at x = -1, then a possible value of k is
(a) 1
(b) 0
(c) – 1/2
(d) -1

Question. Let f (x ) be a polynomial of degree four having extreme

equal to 
(a) -8
(b) -4
(c) 0
(d) 4

Question. Let f be a function defined by f(x) =

Statement I x = 0 is point of minima of f . 
Statement II f ‘ (0) = 0. 
(a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I
(b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I
(c) Statement I is true; Statement II is false
(d) Statement I is false; Statement II is true

Question. If f is defined as f(x) = x+1/x , then which of following is true?
(a) Local maximum value of f (x) is – 2
(b) Local minimum value of f (x) is 2
(c) Local maximum value of f (x) is less than local minimum value of f (x)
(d) All the above are true

Question. If f (x ) = x² + 2bx + 2c² and g(x ) = – x² – 2cx + b² such that minimumf (x ) > maximumg(x ), then the relation between b and c is
(a) 0 < c < b √2
(b) | c | < | b | √2
(c) | c | > | b | √2
(d) No real values of b and c

Question. If the tangent to the curve y = x/(x² – 3), x ∈ R, (x ≠ ± 3), at a point (α, β) (0, 0) on it is parallel to the line 2x + 6y – 11 = 0, then :
(a) |6α + 2β| = 19
(b) |6α + 2β| = 9
(c) |2α + 6β| = 19
(d) |2α + 6β| = 11

Question. If the tangent to the curve, y = x³ + ax – b at the point (1, –5) is perpendicular to the line, – x + y + 4 = 0, then which one of the following points lies on the curve?
(a) (–2, 1)
(b) (–2, 2)
(c) (2, –1)
(d) (2, –2)

Question. Let AD and BC be two vertical poles at A and B respectively on a hori ontal ground. If AD = 8 m, BC = 11 m and AB = 10 m; then the distance (in meters) of a point M on AB from the point A such that MD² + MC² is minimum is ______.
Ans : 5

Question. The set of all real values of l for which the function f(x) = (1- cos² x) · (λ + sin x), x ∈ (− π/2, π/2), has exactly one maxima and exactly minima, is:
(a) (− 1/2, 1/2) − {0}
(b) (− 3/2, 3/2)
(c) (− 1/2, 1/2)
(d) (− 3/2, 3/2) − {0}

Question. Let S be the set of all values of x for which the tangent to the curve y = f(x) = x³ – x² – 2x at (x, y) is parallel to the line segment oining the points (1, f(1)) and (– 1, f(– 1)), then S is equal to:
(a) {1/3, 1}
(b) {− 1/3, −1}
(c) {1/3, −1}
(d) {− 1/3, 1}

Question. The tangent and the normal lines at the point ( √3 , 1) to the circle x² + y² = 4 and the x-axis form a triangle. The area of this triangle (in square units) is :
(a) 4/√3
(b) 1/3
(c) 2/√3
(d) 1/√3

Question. The shortest distance between the point (3/2, 0) and the curve y = √x, (x > 0), is:
(a) √5/2
(b) √3/2
(c) 3/2
(d) 5/4

Question. The tangent to the curve, y = xe^_x2 passing through the point (1, e) also passes through the point:
(a) (2, 3e)
(b) (4/3, 2e)
(c) (5/3, 2e)
(d) (3, 6e)

Question. Suppose f(x) is a polynomial of degree four, having critical points at –1, 0, 1. If T = {x ∈ R | f (x) = f(0)}, then the sum of squares of all the elements of T is :
(a) 4
(b) 6
(c) 2
(d) 8

Question. Let f(x) be a polynomial of degree 3 such that f(–1) = 10, f(1)= –6, f(x) has a critical point at x = –1 and f'(x) has a critical point at x = 1. Then f(x) has a local minima at x = ________.
Ans : 3

Question. The minimum distance of a point on the curve y = x² – 4 from the origin is :
(a) √15/2
(b) √19/2
(c) √15/2
(d) √19/2

Question. Let k and K be the minimum and the maximum values of the function f(x) = ( (1+x)^0.6 ) / ( (1+x)^0.6 ) in [0, 1] respectively, then the ordered pair (k, K) is equal to :
(a) (2^–0.4, 1)
(b) (2^–0.4, 2^0.6)
(c) (2^–0.6, 1)
(d) (1, 2^0.6)

Question. A helicopter is flying along the curve given by y – x^3/2 = 7, (x ≥ 0). A soldier positioned at the point (1/2, 7) wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is:
(a) √5/6
(b) 1/7 √(3/3)
(c) 1/7 √(6/3)
(d) 1/2

Question. Let P be a point on the parabola,x² = 4y. If the distance of P from the centre of the circle, x² + y² + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is
(a) x + 4y – 2 = 0
(b) x + 2y = 0
(c) x + y + 1 = 0
(d) x – y + 3 = 0

Question. The maximum area of a right angled triangle with hypotenuse h is :
(a) h²/(2√2)
(b) h²/2
(c) h²/√2
(d) h²/4

Question. Let a, b ∈ R be such that the function f given by f(x) = ln | x | + bx² + ax, x ≠ 0 has extreme values at x = –1 and x = 2
Statement-1 : f has local maximum at x = –1 and at x = 2.
Statement-2 : a = 1/2 and b = -1/4
(a) Statement-1 is false, Statement-2 is true.
(b) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
(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.

Question. If the tangents drawn to the hyperbola 4y² = x² + 1 intersect the co-ordinate axes at the distinct points A and B, then the locus of the mid point of AB is
(a) x² – 4y² + 16 x²y² = 0
(b) 4x² – y² + 16 x²y² = 0
(c) 4x² – y² – 16 x²y² = 0
(d) x² – 4y² – 16 x²y² = 0

Question. The normal to the curve y(x – 2)(x – 3) = x + 6 at the point where the curve intersects the y-axis passes through the point:
(a) (1/2, 1/3)
(b) (– 1/2, 1/2)
(c) (1/2, 1/2)
(d) (1/2, – 1/3)

Question. The eccentricity of an ellipse whose centre is at the origin is 1/2. If one of its directices is x = – 4, then the equation of the normal to it at (1, 3/2) is :
(a) x + 2y = 4
(b) 2y – x = 2
(c) 4x – 2y = 1
(d) 4x + 2y = 7

Question. Let f : R → R be defined by
(image 117)
If f has a local minimum at x = – 1 , then a possible value of k is
(a) 0
(b) – 1/2
(c) –1
(d) 1

Question. Given P(x) = x⁴ + ax³ + bx² + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(–1) < P(1), then in the interval [ –1, 1] :
(a) P(–1) is not minimum but P(1) is the maximum of P
(b) P(–1) is the minimum but P(1) is not the maximum of P
(c) Neither P(–1) is the minimum nor P(1) is the maximum of P
(d) P(–1) is the minimum and P(1) is the maximum of P

Question. A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(a) = 1 , then the curve also passes through the point :
(a) (1/3, 24)
(b) (1/2, 4)
(c) (2, 1/8)
(d) (3, 1/28)

Question. Let C be a curve given by y(x) = 1 + √(4x – 3), x > 3/4. If P is a point on C, such that the tangent at P has slope 2/3, then a point through which the normal at P passes, is :
(a) (1, 7)
(b) (3, –4)
(c) (4, –3)
(d) (2, 3)

Question. A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of ice is 5 cm,then the rate at which the thickness of ice decreases is
(a) 1/36π cm/min.
(b) 1/18π cm/min.
(c) 1/54π cm/min.
(d) 5/6π cm/min.

Question. A point on the parabola y² = 18x at which the ordinate increases at twice the rate of the abscissa is
(a) (9/8, 9/2)
(b) (2, -4)
(c) (-9/8, 9/2)
(d) (2, 4)

Question. The equation of a normal to the curve, sin y = x sin(π/3 + y) at x = 0, is :
(a) 2x – √3y = 0
(b) 2x + √3y = 0
(c) 2y – √3x = 0
(d) 2y +√ 3x = 0

Question. The maximum value of f(x) = x/4+x+x2 on [-1,1] is
(a) -1/4
(b) -1/3
(c) 1/6
(d) 1/5

Question. The minimum radius vector of the curve 4/x2 + 9/y2 = 1 is of length
(a) 1
(b) 5
(c) 7
(d) None of these

Question. The denominator of a fraction is greater than 16 of the square of numerator, then least value of fraction is
(a) -1/4
(b) – 1/ 8
(c) 1/12
(d) 1/16

Question. The function f(x) = ax +b/x , b , x > 0 takes the least value at x equal to
(a) b
(b) √a
(c) √b
(d) √b/a

Question. If the sum of two numbers is 3, then the maximum value of the product of the first and the square of second is
(a) 4
(b) 1
(c) 3
(d) 0

Question. If y = a logx + bx² + x has its extremum value at x =1 and x = 2, then (a,b) is equal to
(a) (1,1/2)
(b) (1/2 , 2)
(c) (2,-1/2)
(d) (-2/3 , -1/6)

Question. If the function f (x ) =2x³ – 9ax² + 12a² x+1 , where a > 0 attains its maximum and minimum at p and q respectively such that p2 = q , then a is equal to
(a) 3
(b) 1
(c) 2
(d) 1/2

Question. The interval in which the given function f(x) = 𝑡𝑎𝑛−1(𝑠i𝑛𝑥 + 𝑐𝑜𝑠𝑥) is increasing is
(a) (0, 𝜋/2)
(b) (0, 𝜋/6)
(c) (0, 𝜋/4)
(d) (2 , 𝜋)

Question. The two curves 𝑥3 − 3𝑥𝑦2 + 2 = 0 𝑎𝑛𝑑 3𝑥2𝑦 − 𝑦3 − 2 = 0 intersect at an angle of
(a) 𝜋/4
(b) 𝜋/3
(c) 𝜋/2
(d) 𝜋/6

Question. The interval in which y= x²e-x is increasing is
(a) (-∞, ∞)
(b) (- 2, 0)
(c) (2, ∞)
(d) (0, 2)

Question. The equation of normal to the curve 3𝑥2 –𝑦2 = 8 which is parallel to the line x + 3y = 8 is
(a) 3x – y = 8
(b) 3x + y + 8 = 0
(c) x + 3y ± 8 = 0
(d) x + 3y = 0

Question. Consider a rectangle whose length is increasing at the uniform rate of 2 m/sec, breadth is decreasing at the uniform rate of 3 m/sec and the area is decreasing at the uniform rate of 5 m²/sec. If after some time the breadth of the rectangle is 2 m then the length of the rectangle is
(a) 2 m
(b) 4 m
(c) 1 m
(d) 3 m

Question. If a circular iron sheet of radius 30 cm is heated such that its area increases at the uniform rate of 6π cm²/hr, then the rate (in mm/hr) at which the radius of the circular sheet increases is
(a) 1.0
(b) 0.1
(c) 1.1
(d) 2.0

Question. For the curve y = 3 sinθ cosθ, x = e^θ sin θ, 0 ≤ θ ≤ π, the tangent is parallel to x-axis when q is:
(a) 3π/4
(b) π/2
(c) π/4
(d) π/6

Question. If an equation of a tangent to the curve, y – cos(x + f), – 1 -1 ≤ x ≤ 1+ π, is x + 2y = k then k is equal to :
(a) 1
(b) 2
(c) π/4
(d) π/2

Question. Let f be any function continuous on [a, b] and twice differentiable on (a, b). If for all x ∈ (a, b), f’2 (x) > 0 and f”(x) < 0, then for any c ∈ (a, b), ( f(c) − f(a) ) / ( f(b) − f(c) ) is greater than:
(a) b+a / b−a
(b) 1
(c) b−c / c−a
(d) c−a / b−c

Question. Let f(x) = x cos^–1 (–sin |x|), x ∈ [− π/2, π/π ], then which of the following is true?
(a) f’ is increasing in (− π/2, 0) and decreasing in (0, π/2)
(b) f’(0) = – /2
(c) f ’ is not differentiable at x = 0
(d) f ’ is decreasing in (− π/2, 0) and increasing in (0, π/2)

Question. If x = –1 and x = 2 are extreme points of f(x) = αlog |x| + βx² + x then
(a) α = 2, β = − 1/2
(b) α = 2, β = 1/2
(c) α = −6, β = − 1/2
(d) α = −6, β = − 1/2

Question. If the lines x + y = a and x – y = b touch the curve y = x² – 3x + 2 at the points where the curve intersects the x-axis, then a/b is equal to _____.
Ans : 0.50

Question. If the tangent to the curve, y = e^x at a point (c, e^c) and the normal to the parabola, y² = 4x at the point (1, 2) intersect at the same point on the x-axis, then the value of c is ____________.
Ans : 4

Question. The minimum area of a triangle formed by any tangent to the ellipse x²/16 + y²/81 = 1 and the co-ordinate axes is:
(a) 12
(b) 18
(c) 26
(d) 36

Question. The normal to the curve x = a(1+ cosθ), y = a sinθ at ‘θ’ always passes through the fixed point
(a) (a, a)
(b) (0, a)
(c) (0, 0)
(d) (a, 0)

Question. Let f: (-∞,∞) → (-∞,∞) be defined by f(x) = x³ + 1.
Statement 1: The function f has a local extremum at x = 0
Statement 2: The function f is continuous and differentiable on (-∞,∞) and f'(0) = 0
(a) Statement 1 is true, Statement 2 is false.
(b) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(c) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation for Statement 1.
(d) Statement 1 is false, Statement 2 is true.

Question. A function y = f(x) has a second order derivative f”(x) = 6(x -1). If its graph passes through the point (2,1) and at that point the tangent to the graph is y = 3x – 5, then the function is
(a) (x +1)²
(b) (x -1)³
(c) (x +1)³
(d) (x -1)²

Question. Let m and M be respectively the minimum and maximum values of (image 81) Then the ordered pair (m, M) is equal to :
(a) (– 3, 3)
(b) (– 3, – 1)
(c) (– 4, – 1)
(d) (1, 3)

Question. If x = 1 is a critical point of the function f(x) = (3x² + ax − 2 − a)e^x, then :
(a) x = 1 and x = (− 2/3) are local minima of f.
(b) x = 1 and x = (− 2/3) are local maxima of f.
(c) x = 1 is a local maxima and x = (− 2/3) is a local minima of f.
(d) x = 1 is a local minima and x = (− 2/3) is a local maxima of f.

Question. Let the normal at a point P on the curve y2 – 3×2 + y + 10 = 0 intersect the y-axis at (0, 3/2). If m is the slope of the tangent at P to the curve, then |m| is equal to ______.
Ans : 4.0

Question. The length of the perpendicular from the origin, on the normal to the curve, x² + 2xy – 3y² = 0 at the point (2, 2) is:
(a) √2
(b) 4√2
(c) 2
(d) 2√2

Question. The slope of tangent to the curve x = 𝑡2 + 3t – 8, y = 2𝑡2 – 2t – 5 at the point (2, –1) is
(a) 22/7
(b) 6/7
(c) −6/7
(d) -6

Question. The points at which the tangents to the curve y = 𝑥3 – 12x + 18 are parallel to x-axis are:
(a) (2, –2), (–2, –34)
(b) (2, 34), (–2, 0)
(c) (0, 34), (–2, 0)
(d) (2, 2), (–2, 34)

Question. The function f(x) = 2 𝑥3- 9 𝑥2 + 12𝑥 + 29 i𝑠 strictly 𝑑𝑒𝑐𝑟𝑒𝑎𝑠i𝑛𝑔 when
(a) x < 2
(b) x > 2
(c) x > 3
(d) 1 < x < 2

Question. The given function f(x) = 2x+5 is
(a) Increasing on R
(b) Decreasing on R
(c) Strictly increasing on R
(d) Strictly decreasing on R

Assertion Reasoning Questions :

a. Both A and R are correct; R is the correct explanation of A.
b. Both A and R are correct; R is not the correct explanation of A.
c. A is correct; R is incorrect.
d. R is correct; A is incorrect.

Question. Assertion (A): From the graph, x = 2 is a point of local maxima

Reason (R): The slope of the curve changes from positive to negative.

Question. Assertion (A): The equation of tangent at (3, 2) is y = 2.

Reason (R): The slope of tangent of the curve at (3, 2) is zero.

Case Based Questions :

Suppose a car is travelling around the hill, and the path car follows along the hill is represented as y = 𝑥³– 3𝑥²+2.

Question. When the car is at the instant (0, 2), suddenly landslide starts and on applying brakes the car slides tangentially. What is the equation of the path in which the car slides?
(a) x = 2
(b) y = 2
(c) x + y =2
(d) x – y = 2

Question. What is the maximum value the path of the curve reaches in the domain [-1, 2]
(a) 0
(b) 1
(c) 2
(d) 3

Question. Determine the continuity of the function at x = 3?
(a) Continuous and value is 3
(b) Continuous and value is 2
(c) not continuous
(d) data insufficient

Question. When the car is at the instant (2,-2), the driver realized some slippery material like oil is spilled on the road. The driver applied brakes but the car slides normal at the instant. Find the equation of the normal path car took.
(a) x = 2
(b) y = 2
(c) x + y =2
(d) x – y = 2

Question. When the car follows the path from x= 2 to x = 1, then what is the change in the slope of the path of the hill?
(a) Increases
(b) decreases
(c) no change
(d) cannot be determined

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