Application of Derivatives VBQs Class 12 Mathematics

VBQs Application of Derivatives 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 Application of Derivatives 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.

Application of Derivatives VBQs Class 12 Mathematics

Question. The function 𝑓(𝑥) = tan 𝑥 − 𝑥 is:
(a) always increasing
(b) always decreasing
(c) not always decreasing
(d) sometimes increasing and sometimes decreasing

Question. The point on the curve x2 = 2y which is nearest to the point ( 0 , 5 ) is
(a) (2√2 , 4)
(b) (2√2 , 0 )
(c) ( 0 , 0 )
(d) ( 2 , 2 )

Question. The maximum value of [x(x-1) + 1] 1/3, 0 ≤ x ≤1 is
(a) (1/3)^1/3
(b) ½
(c) 1
(d) 0

Question. The function 𝑓(𝑥) = 𝑥³ − 6𝑥² + 15 𝑥 − 12 is:
(a) strictly decreasing on R
(b) strictly increasing on R
(c) increasing on (−∞, 2] and decreasing on (2, ∞)
(d) none of these

Question. Is the function 𝑓(𝑥) = cos(2𝑥 + 𝜋/4); is increasing or decreasing in the interval (3𝜋/8 , 7𝜋/8)
(a) increasing
(b) decreasing
(c) neither increasing nor decreasing
(d) none of these

Question. The equation of normal x = acos3θ , y=a sin3θ at the point θ= 𝜋/4 is
(a) x = 0
(b) y = 0
(c) x = y
(d) x + y = a

Question.The angle of intersection of the parabolas y2 =4ax and x2 = 4ay at the origin is
(a)𝜋/6
(b) 𝜋/3
(c) 𝜋/2
(d) 𝜋/4

Question. The line y = x+1 touches y2=4x at the point
(a) ( 1 , 2)
(b) (2 , 1)
(c) (1 ,-2 )
(d) ( -1 , 2)

Question. The function 𝑓(𝑥) = 𝑥𝑥 is decreasing in the interval:
(a) (0, 𝑒)
(b) [0, 1)
(c)) (0, 1/e)
(d) none of these

Question. The function 𝑓(𝑥) = [𝑥(𝑥 − 3)]2 is increasing in :
(a) (0, ∞)
(b) (− ∞, 0)
(c) (1, 3)
(d) [0, 1.5] ∪ (3, ∞)

Question. The function 𝑓(𝑥) = 𝑥/𝑥 +1 is increasing in :
(a) (−1, 1)
(b) (−1, ∞)
(c) (− ∞, −1) ∪ (1, ∞)
(d) none of these

Question. If the function f(x) = x3+ax2+bx+1 is maximum at x=0 and x=1 then :
(a) a = 2/3 , b = 0
(b) a=- 3/2 , b=0
(c) a = 0 , b = 3/2
(d) None of These

Question. The smallest value of polynomial 3x⁴−8x³+12x²− 48x +1 in [1, 4]is:
(a)-49
(b) 59
(c) -59
(d) 257

Question. The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x² – 1 below the x-axis, is :
(a) 2/3√3
(b) 1/3√3
(c) 4/3
(d) 4/3√3

Question. Let f be a function defined by – 
(image 114)
Statement – 1 : x = 0 is point of minima of f
Statement – 2 : f'(0) = 0.
(a) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for statement-1.
(b) Statement-1 is true, statement-2 is true; statement-2 is NOT a correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-2 is true.

Question. For x ∈ (0, 5π/2), define (image 115) Then f has
(a) local minimum at π and 2π
(b) local minimum at π and local maximum at 2π
(c) local maximum at π and local minimum at 2π
(d) local maximum at π and 2π

Question. Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If = 4, then which one of the following is not true ?
(a) f is an odd function.
(b) f(1) – 4f(–1) = 4.
(c) x = 1 is a point of maxima and x = –1 is a point of minima of f.
(d) x = 1 is a point of minima and x = –1 is a point of maxima of f.

Question. If m is the minimum value of k for which the function f(x) = x√(kx − x²) is increasing in the interval [0,3] and M is the maximum value of f in [0,3] when k = m, then the ordered pair (m, M) is equal to :
(a) (4,3 2)
(b) (4,3 3)
(c) (3,3 3)
(d) (5,3 6)

Question. Let a₁, a₂, a₃, …. be an A. P. with a₆ = 2. Then the common difference of this A.P., which maximises the product a₁ a₄ a₅, is :
(a) 3/2
(b) 8/5
(c) 6/5
(d) 2/3

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 equals
(a) 2/1
(b) 3
(c) 1
(d) 2

Question. The maximum distance from origin of a point on the curve x = a sin t–b sin(at/b), y = a cos t–b cos(at/b), both a, b > 0 is
(a) a – b
(b) a + b
(c) √(a² + b²)
(d) √(a² – b²)

Question. The maximum value of 3cos θ + 5 sin(θ − π/6) for any real value of θ is:
(a) √19
(b) √79/2
(c) √34
(d) √31

Question. If the tangent at a point P, with parameter t, on the curve x = 4t² + 3, y = 8t³ – 1, t ∈ R, meets the curve again at a point Q, then the coordinates of Q are :
(a) (16t² + 3, –64t³ – 1)
(b) (4t² + 3, –8t³ – 2)
(c) (t² + 3, t³ – 1)
(d) (t² + 3, – t³ – 1)

Question. The normal to the curve, x² + 2xy – 3y² = 0, at (1, 1)
(a) meets the curve again in the third quadrant.
(b) meets the curve again in the fourth quadrant.
(c) does not meet the curve again.
(d) meets the curve again in the second quadrant.

Question. Let P(4, –4) and Q(9, 6) be two points on the parabola, y² = 4x and let this X be any point arc POQ of this parabola, where O is vertex of the parabola, such that the area of ΔPXQ is maximum. Then this minimum area (in sq. units) is:
(a) 75/2
(b) 125/4
(c) 625/4
(d) 125/2

Question. The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:
(a) 6π
(b) 3√3π
(c) 4/3 π
(d) 2√3 π

Question. Let f(x) = x² + 1/x² and g(x) = x − 1/x, x ∈ R − {−1, 0, 1}. If h(x) = f(x)/g(x) , then the local minimum value of h(x) is :
(a) – 3
(b) -2√2
(c) 2√2
(d) 3

Question. The equation of the normal to the parabola, x² = 8y at x = 4 is
(a) x + 2y = 0
(b) x + y = 2
(c) x – 2y = 0
(d) x + y = 6

Question. The equation of the tangent to the curve y = x + 4/x² , that is parallel to the x-axis, is
(a) y = 1
(b) y = 2
(c) y = 3
(d) y = 0

Question. If a function f (x) defined by (image 3) be continuous for some a, b, c ∈ R and f'(0) + f'(2) = e, then the value of a is :
(a) 1/(e² – 3e + 13)
(b) e/(e² – 3e – 13)
(c) e/(e² + 3e + 13)
(d) e/(e² – 3e + 13)

Question. A spherical iron ball of 10 cm 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 (in cm/min.) at which of the thickness of ice decreases, is:
(a) 5/6π
(b) 1/54π
(c) 1/36π
(d) 1/18π

Question. Twenty metres of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is :
(a) 30
(b) 12.5
(c) 10
(d) 25

Question. A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:
(a) x = 2r
(b) 2x = r
(c) 2x = (π + 4)r
(d) (4 – π) x = πr

Question. If the tangent to the conic, y – 6 = x² at (2, 10) touches the circle, x² + y² + 8x – 2y = k (for some fixed k) at a point (α, β) ; then (α, β) is :
(a) (− 7/17, 6/17)
(b) (− 4/17, 1/17)
(c) (− 6/17, 10/17)
(d) (− 8/17, 2/17)

Question. The distance, from the origin, of the normal to the curve, x = 2 cost + 2t sint, y = 2 sint – 2t cost at t = π/4, is :
(a) 2
(b) 4
(c) 2
(d) 2√2

Question. From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of 48 m/s. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration g = 32 m s², is:
(a) 128
(b) 88
(c) 112
(d) 100

Question. The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = √3 is:
(a) 4/3 √3π
(b) 8/3 √3π
(c) 4π
(d) 2π

Question. The cost of running a bus from A to B, is ₹(av + b/v), where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be ₹ 75 while at 40 km/h, it is ₹ 65. Then the most economical speed (in km/ h) of the bus is :
(a) 45
(b) 50
(c) 60
(d) 40

Question. A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :
(a) 1/18π
(b) 1/36π
(c) 5/6π
(d) 1/9π

Question. A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is tan^–1 . Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is:
a) 1/15 π
(b) 1/10 π
(c) 2/π
(d) 1/5 π

Question. A line is drawn through the point (1,2) to meet the coordinate axes at P and Q such that it forms a triangle OPQ, where O is the origin. If the area of the triangle OPQ is least, then the slope of the line PQ is :
(a) – 1/4
(b) – 4
(c) – 2
(d) – 1/2

Question. Let f : R → R be a continuous function defined by f(x) = 1/(e^x + 2e^-x)
Statement -1 : f (c) = 1/3, for some c ∈ R.
Statement -2 : 0 < f(x) ≤ 1/(2√2), for all x ∈ R
(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.

Question. Suppose the cubic x³ – px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds?
(a) The cubic has minima at √(p /3) and maxima at – √(p /3)
(b) The cubic has minima at – √(p /3) and maxima at √(p /3)
(c) The cubic has minima at both √(p /3) and – √(p /3)
(d) The cubic has maxima at both √(p /3) and – √(p /3)

Question. The real number k for which the equation, 2x³ + 3x + k = 0 has two distinct real roots in [0, 1]
(a) lies between 1 and 2
(b) lies between 2 and 3
(c) lies between 1 and 0
(d) does not exist

Question. Which of the following functions are strictly decreasing on (0 , π/2)
(a) Cos x
(b) tan 2x
(c) Cos 3x
(d) tan x

Question. The least value of a such that f(x) =𝑥² + ax +1 is strictly increasing on ( 1 , 2) is
(a) – 2
(b) -4
(c) 2
(d) 4

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

Question. The point on the curve y² = x, where the tangent makes an angle of π/4 with x-axis is
(a) (½, ¼)
(b) ( ¼ , ½ )
(c) (4, 2)
(d) (1, 1)

Question. The tangent to the curve y = e²x at the point (0, 1) meets x-axis at
(a) (0,1)
(b) (−1/2 ,0)
(c) (2,0)
(d) (0,2)

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

Question. The point on the curve y² = x where tangent makes 450 angle with x-axis is
(a) (0,0)
(b) (2,16)
(c) (3, 9)
(d) none of these

Question. The angle between the curves y² = x and x2 = y at (1,1)is:
(a) tan^-14/3
(b) tan^-13/4
(c ) 900
(d) 450

Question. The abscissaof the point on the curve 3y = 6x − 5x³, the normal at which passes through the origin is
(a) 1
(b) 2
(,(c) -1
(d) -2

Question. The Equation normal to the curve y=x + Sinx + Cosx at x = is 𝜋/2
(a) x = 2
(b) x = 𝜋
(c) x + 𝜋 = 0
(d) 2x =𝜋

Question. The Point on the curve y = x²– 3x + 2 where tangent is perpendicular to y=x is
(a) ( ½ , ¼)
(b) (¼ , ½)
(c) ( 4 , 2 )
(d) ( 1 ,1 )

Question. At what point the slope of the tangent to the curve x²+y²− 2x −3 is zero?
(a) (3, 0), (−1, 0)
(b) (3,0),(1,2)
(c) (−1, 0),(1,2)
(d) (1, 2),(1, − 2)

Question. If the curve ay + x² = 7 and x³ = y cut each other at 900 at ( 1 , 1) , then value of a is :
(a) 1
(b) -6
(c) 6
(d) 0

Question. The tangent to the curve y = e^2x at the point ( 0 , 1) meets x-axix at
(a) ( 0 , 1)
(b) [-1/2 , 0]
(c) (2 , 0 )
(d) (0 ,2)

Question. The maximum value of x²+ 250/𝑥 is 
(a) 0
(b) 25
(c) 50
(d) 75

Question. The equation of tangent at those points where the curve y = x² – 3x + 2 meets x- axis are
(a) x−y+2=0, x−y−1=0
(b) x −y −1 = 0 , x −y = 0
(c) x +y −1 = 0 , x −y − 2 = 0
(d) x −y = 0 , x +y = 0

Question. The two curves 𝑥3 – 3x𝑦² + 2 = 0 and 3𝑥²𝑦²– 𝑦³ = 2
(a) Touch each other
(b) Cut at right angle
(c) Cut at an angle π/3
(d) Cut at an angle π/4

Question. The tangent to the curve given by x = et.cos t, y =et.sin t at t = π/4 makes with x-axis an angle
(a) 0
(b) π/4
(c) π/3
(d) π/2

Question. The value of f(x) = ( x-2 )(x – 3)² is
(a) 7/3
(b) 3
(c) 4/27
(d) 0

Question. The Least value f(x) = ex + e-x
(a) -2
(b) 0
(c) 2
(d) can’t be determine

Question. The maximum value of y = sinx. cosx is
(a) 1/4
(b) 1/2
(c) √2
(d) 2√2

Question. The function f (x)=2×3−3x²−12x + 4, has:
(a) Two points of local maximum
(b) Two points of local minimum
(c) one maxima and one minima
(d) no maxima or minima

Question. The sum of two non-zero numbers is 8, the minimum value of the sum of their reciprocals is:
(a) ¼
(b) ½
(c) 1/8
(d) None of these

Question. The function 𝑓(𝑥) = 4 − 3𝑥 + 3𝑥² − 𝑥³ is:
(a) decreasing on R
(b) increasing on R
(c) strictly decreasing on R
(d) strictly increasing on R

Question. The function 𝑓(𝑥) = 𝑥/sin 𝑥 is:
(a) increasing in (0, 1)
(b) decreasing in (0, 1)
(c) increasing in (0, 1/2) and decreasing in (1/2, 1)
(d) none of these

Question. The Curve y = 4x²+ 2x -8 and y = x³ – x + 13 touch each other at the point
(a) ( 3 , 23)
(b) (23 , -3 )
(c) ( 34 , 3)
(d) ( 3 , 34)

Question. The Maximum value of f(x) = is logx/𝑥
(a) 1/e
(b) 2/e
(c) e
(d) 1  

Question. The slope of normal to the curve y = 2x² + 3 sin x at x = 0 is
(a) -1/3
(b) ½
(c) 1/3
(d) 3

Question. The line y = x + 1 is a tangent to the curve y² = 4x at the point
(a) (1, 2)
(b) ( 2 , 1)
(c) ( -1, 2 )
(d) ( -1 , -2 )

Question. The equation of the normal to the curve 3x² − y² = 8 which is parallel to the line x + 3y = 8 is
(a) x + 3y = 8
(b) x + 3y + 8 = 0
(c) x + 3y ± 8 = 0
(d) None of These

Question.Statement-1: The function x² (e^x + e^–x) is increasing for all x > 0.
Statement-2: The functions x²e^x and x²e^–x are increasing for all x > 0 and the sum of two increasing functions in any interval (a, b) is an increasing function in (a, b).
(a) Statement-1 is false; Statement-2 is true.
(b) Statement-1is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true; Statement-2 is false.
(d) Statement-1is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1.

Question. If S₁ and S₂ are respectively the sets of local minimum and local maximum points of the function, f(x) = 9x⁴ + 12x³ – 36x² + 25, x∈R, then :
(a) S₁ = {–2}; S₂ = {0, 1}
(b) S₁ = {–2, 0}; S₂ = {1}
(c) S₁ = {–2, 1}; S₂ = {0}
(d) S₁ = {–1}; S₂ = {0, 2}

Question. The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is : 
(a) 6
(b) 2/3 √3
(c) 2√3
(d) √3

Question. A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is:
(a) 9/7
(b) 7/9
(c) 2/9
(d) 9/2

Question. If a metallic circular plate of radius 50 cm is heated so that its radius increases at the rate of 1 mm per hour, then the rate at which, the area of the plate increases (in cm²/hour) is
(a) 5 π
(b) 10 π
(c) 100 π
(d) 50 π

Question. The function f (x) = tan^–1(sin x + cos x) is an increasing function in
(a) (0, π/2)
(b) (− π/2, π/2)
(c) (π/2, π/2)
(d) (− π/2, π/4)

Question. A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
(image 39)

Question. Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x³ – 9x² + 12x + 5 in the interval [0, 3]. Then M – m is equal to
(a) 1
(b) 5
(c) 4
(d) 9

Question. If a right circularcone having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm²) of this cone is
(a) 8√3π
(b) 6√2π
(c) 6√3π
(d) 8√2π