CBSE Class 12 Mathematics Application Of Derivatives MCQs Set 07

Question. The abscissa of the point on the curve \( 3y = 6x - 5x^3 \), at which the normal passes through origin is :
(a) 1
(b) \( \frac{1}{3} \)
(c) 2
(d) \( \frac{1}{2} \)
Answer: (a) 1

Question. The two curves \( x^3 - 3xy^2 + 2 = 0 \) and \( 3x^2y - y^3 = 2 \) :
(a) touch each other
(b) cut at right angle
(c) cut at an angle \( \frac{\pi}{3} \)
(d) cut at an angle \( \frac{\pi}{4} \)
Answer: (b) cut at right angle

Question. The tangent to the curve given by \( x = e^t \cdot \cos t, y = e^t \cdot \sin t \) at \( t = \frac{\pi}{4} \) makes with x-axis an angle:
(a) \( \pi \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{3} \)
(d) \( \frac{\pi}{2} \)
Answer: (d) \( \frac{\pi}{2} \)

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) \( x - y = 0 \)
Answer: (c) \( x + y = 0 \)

Question. The point on the curve \( y^2 = x \), where the tangent makes an angle of \( \frac{\pi}{4} \) with x-axis is
(a) \( (\frac{1}{2}, \frac{1}{4}) \)
(b) \( (\frac{1}{4}, \frac{1}{2}) \)
(c) (4, 2)
(d) (1, 1)
Answer: (b) \( (\frac{1}{4}, \frac{1}{2}) \)

Question. The angle of intersection of the curves \( y^2 = x \) and \( x^2 = y \) at (1, 1) is:
(a) \( \tan^{-1} (\frac{4}{3}) \)
(b) \( \tan^{-1} (\frac{3}{4}) \)
(c) Positive
(d) None of the options
Answer: (b) \( \tan^{-1} (\frac{3}{4}) \)

Question. The interval on which the function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is decreasing is :
(a) [– 1, \( \infty \))
(b) [– 2, – 1]
(c) (– \( \infty \), – 2]
(d) [– 1, 1]
Answer: (b) [– 2, – 1]

Question. If \( y = x(x - 3)^2 \) decreases for the values of x given by : 
(a) 1 < x < 3
(b) x < 0
(c) x > 0
(d) 0 < x < \( \frac{3}{2} \)
Answer: (a) 1 < x < 3

Question. The function \( f(x) = \tan x - x \) : 
(a) always increases
(b) always decreases
(c) never increases
(d) sometime increase and sometimes decreases
Answer: (a) always increases

Question. The function \( f(x) = 4 \sin^3 x - 6 \sin^2 x + 12 \sin x + 100 \) is strictly : 
(a) increasing in \( (\pi, \frac{3\pi}{2}) \)
(b) decreasing in \( (\frac{\pi}{2}, \pi) \)
(c) decreasing in \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
(d) decreasing in \( [0, \frac{\pi}{2}] \)
Answer: (b) decreasing in \( (\frac{\pi}{2}, \pi) \)

Question. Which of the following function is decreasing on \( (0, \frac{\pi}{2}) \)? 
(a) sin 2x
(b) tan x
(c) cos x
(d) cos 3x
Answer: (c) cos x

Question. The curve \( y = x^{1/5} \) has at (0, 0) :
(a) a vertical tangent (parallel to Y-axis)
(b) a horizontal tangent (parallel to X-axis)
(c) an oblique tangent
(d) not tangent
Answer: (a) a vertical tangent (parallel to Y-axis)

Question. If \( x + y = K \) is normal to \( y^2 = 12x \), then K is :
(a) 3
(b) 9
(c) – 9
(d) – 3
Answer: (a) 9

Question. If the curve \( ay + x^2 = 7 \) and \( x^3 = y \), cut orthogonally at (1, 1), the the value of a is :
(a) 1
(b) 0
(c) – 6
(d) 6
Answer: (d) 6

Question. The points at which the tangents to the curve \( y = x^3 - 12x + 18 \) 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)
Answer: (d) (2, 2), (– 2, 34)

Question. If x is real, then the minimum value of \( x^2 - 8x + 17 \) is :
(a) – 1
(b) 0
(c) 1
(d) 2
Answer: (c) 1

Question. The function \( f(x) = 2x^3 - 3x^2 - 12x + 4 \), has :
(a) two points of local maximum
(b) two points of local minimum
(c) one maxima and one minima
(d) no maxima or minima
Answer: (c) one maxima and one minima

Question. The maximum slope of curve : \( y = - x^3 + 3x^2 + 9x - 27 \) is :
(a) 0
(b) 12
(c) 16
(d) 32
Answer: (b) 12

Question. The function \( f(x) = x^x \) has a statinoary point at :
(a) x = e
(b) x = \( \frac{1}{e} \)
(c) x = 1
(d) x = \( \sqrt{e} \)
Answer: (b) x = \( \frac{1}{e} \)

Question. A right circular cylinder which is open at the top and has a given surface area, will have the greatest volume, if its height h and radius r are related by :
(a) 2h = r
(b) h = 4r
(c) h = 2r
(d) h = r
Answer: (d) h = r

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

Question. The tangent to the curve \( y = e^{2x} \) at the point (0, 1) meets x-axis at :
(a) (0, 1)
(b) \( (-\frac{1}{2}, 0) \)
(c) (2, 0)
(d) (0, 2)
Answer: (b) \( (-\frac{1}{2}, 0) \)

Question. What is the slope of the tangent to the curve \( y = \frac{2x}{(x^2 + 1)} \) at (0, 0) ?
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (c) 2

Question. What will be the differential function of \( \sqrt{x^2 + 2} \) ?
(a) \( x \sqrt{x^2 + 2} \)
(b) \( \frac{x}{\sqrt{x^2 + 2}} \)
(c) \( \frac{x}{\sqrt{x^2 - 2}} \)
(d) \( \frac{-x}{\sqrt{x^2 + 2}} \)
Answer: (b) \( \frac{x}{\sqrt{x^2 + 2}} \)

Question. What is the derivative of \( \log (x^2 + 4) \) ?
(a) \( \frac{2x}{x^2 + 4} dx \)
(b) \( \frac{2x}{(x^2 - 4)} dx \)
(c) \( \frac{-2x}{(x^2 + 4)} dx \)
(d) \( \frac{-2x}{(x^2 - 4)} dx \)
Answer: (a) \( \frac{2x}{x^2 + 4} dx \)

Question. What is the nature of function of \( f(x) = 7x - 4 \) on R ?
(a) increasing
(b) decreasing
(c) strictly increasing
(d) increasing and decreasing
Answer: (c) strictly increasing

Question. Find the interval in which function \( f(x) = x^2 - 4x + 5 \) is increasing :
(a) (2, \( \infty \))
(b) (– \( \infty \), 2)
(c) (3, \( \infty \))
(d) (– \( \infty \), \( \infty \))
Answer: (a) (2, \( \infty \))

Question. Find the interval in which function, \( f(x) = \sin x + \cos x, 0 \le x \le 2\pi \) is decreasing :
(a) \( (\frac{\pi}{4}, \frac{5\pi}{4}) \)
(b) \( (-\frac{\pi}{4}, \frac{5\pi}{4}) \)
(c) \( (\frac{\pi}{4}, -\frac{5\pi}{4}) \)
(d) \( (-\frac{\pi}{4}, \frac{\pi}{4}) \)
Answer: (a) \( (\frac{\pi}{4}, \frac{5\pi}{4}) \)

Question. The minimum value of the function, \( y = 2x^3 - 21x^2 + 36x - 20 \) is :
(a) – 120
(b) – 126
(c) – 128
(d) None of the options
Answer: (c) – 128

Question. Let l be the length and b be the breadth of a rectangle such that \( l + b = k \). What is the minimum area of rectangle ?
(a) \( 2k^2 \)
(b) \( k^2 \)
(c) \( \frac{k^2}{2} \)
(d) \( \frac{k^2}{4} \)
Answer: (d) \( \frac{k^2}{4} \)

Question. The critical point and nature for the function \( f(x, y) = x^2 - 2x + y^2 + 2y - 2 \) is :
(a) (1, 1) Max.
(b) (1, - 1) Max.
(c) (1, 1) Min.
(d) (1, – 1) Min.
Answer: (d) (1, – 1) Min.

Question. What is the maximum area of a triangle that can be inscribed in a circle of radius ‘a’ ?
(a) \( \frac{3}{4} a^2 \)
(b) \( \frac{a^2}{2} \)
(c) \( \frac{3\sqrt{3}}{4} a^2 \)
(d) \( \frac{\sqrt{3}}{4} a^2 \)
Answer: (c) \( \frac{3\sqrt{3}}{4} a^2 \)

Question. The greatest value of \( 5 \sin^2 x + 7 \cos^2 x - 4 \sin x \cos x \) will be :
(a) \( 6 - \sqrt{5} \)
(b) \( 6 + \sqrt{5} \)
(c) \( -6 + \sqrt{5} \)
(d) \( -6 - \sqrt{5} \)
Answer: (b) \( 6 + \sqrt{5} \)

Question. Which one of following is corect in respect of the function, \( f(x) = x^3 \sin x \) ?
(a) It has local maximum at x = 0
(b) It has local minimum at x = 0
(c) It has neither maximum nor minimum at x = 0
(d) It has maximum value as 1
Answer: (c) It has neither maximum nor minimum at x = 0