JEE Mathematics Application of Derivatives MCQs Set E

Question. If the rate of increase of area of a circle is not constant but the rate of increase of perimeter is constant, then the rate of increase of area varies
(a) as the radius
(b) inversely as the perimeter
(c) as the square of the perimeter
(d) inversely as the radius

Answer: A

Question. If the path of a moving point is the curve x = at, y = b sin at, then its acceleration at any instant
(a) varies as the distance from the axis of y
(b) varies as the distance of the point from the origin
(c) varies as the distance from the axis of x
(d) is constant

Answer: A

Question.  At which point the line x/a + y/b = touches the curve y = be^-x/a
(a) (0,b)
(b) (0,0)
(c) (0,a)
(d) (b,0)

Answer: A

Question. The sum of intercepts on co-ordinate axes made by tangent to the curve √x + √y = √a is
(a) a
(b) 2√a
(c) 2a
(d) None of these

Answer: A

Question. The length of the normal at point 't ' of the curve x = a(t + sin t), y = a(1- cos t) is
(a) 2asin(t / 2) tan(t / 2)
(b) a sin t
(c) 2asin(t / 2)
(d) None of these

Answer: A

Question. The curve y – e^xy + x = 0 has a vertical tangent at the point:
(a) (1, 0)
(b) at no point
(c) (1, 1)
(d) (0, 1)

Answer: A

Question. If y = (4x – 5) is a tangent to the curve y2 = px3 + q at (2, 3), then:
(a) p = 2, q = – 7
(b) p = – 2, q = – 7
(c) p = 2, q = 7
(d) p = – 2, q = 7

Answer: A

Question. The equation of its tangent to the curve y = 1 – ex/2 at its point of intersection with the y-axis is:
(a) x + 2y = 0
(b) x – y = 0
(c) 2x + y = 0
(d) none of these

Answer: A

Question. If x denotes displacement in time t and x = a cos t, then acceleration is given by:
(a) – a cos t
(b) a sin t
(c) – a sin t
(d) a cos t

Answer: A

Question. The length of the subnormal at the point (1, 3) of the curve, y = x² + x + 1 is:
(a) 9
(b) 1
(c) 3
(d) 12

Answer: A

Question. The length of the subtangent to the curve, √x+√y=3 at the point (4, 1) is:
(a) 2
(b) 3
(c) 5
(d) 4

Answer: A

Question. If the line ax + by + c = 0 is a normal to the curve xy =1, then (1) a > 0, b > 0 (2) a > 0, b < 0 (3) a < 0, b < 0 (4) a < 0, b > 0
(a) 2 and 4 are correct
(b) 1, 2 and 3 are correct
(c) 1 and 2 are correct
(d) 1 and 3 are correct

Answer: A

Question. Gradient of line passing through the point (2, 8) and touching the curve y = x³, can be
(1) 3   (2) 6
(3) 12 (4) 9
(a) 1 and 3 are correct
(b) 1 and 2 are correct
(c) 1, 2 and 3 are correct
(d) 2 and 4 are correct

Answer: A

Question. Let f (x) = 1/1+ x². Let m be the slope, a be the x-intercept and b be the y-intercept of a tangent to y = f (x), then Value of b for the tangent drawn to the curve y = f (x) whose slope is greatest, is –
(a) 9/8
(b) 1/8
(c) 3/8
(d) 5/8

Answer: A

Question. Let f (x) = 1/1+ x². Let m be the slope, a be the x-intercept and b be the y-intercept of a tangent to y = f (x), then Value of a for the tangent drawn to the curve y = f (x) whose slope is greatest, is –
(a) -√ 3
(b) –1
(c) 1
(d) √3

Answer: A

Question. Statement 1 : The tangent at x = 1 to the curve y = x³ – x² – x + 2 again meets the curve at x = – 2.
Statement 2 : When a equation of a tangent solved with the curve, repeated roots are obtained at point of tangency.
(a) Statement -1 is False, Statement-2 is True.
(b) Statement -1 is True, Statement-2 is False.
(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 True; Statement-2 is a correct explanation for Statement-1.

Answer: A

Question. Statement 1 : Tangent drawn at the point (0, 1) to the curve y = x³ – 3x + 1 meets the curve thrice at one point only.
Statement 2 : Tangent drawn at the point (1, –1) to the curve y = x³ – 3x + 1 meets the curve at one point only.
(a) Statement -1 is True, Statement-2 is False.
(b) Statement -1 is False, Statement-2 is True.
(c) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(d) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

Answer: A

Question. If x + y =16 and x² + y² is minimum, then the values of x and y are
(a) 8, 8
(b) 4, 12
(c) 3 , 13
(d) 6, 10

Answer: A

Question. Maximum value of the function (1/x)^x is
(a) (e)^{1/ e}
(b) (e)^e
(c) (e)^-e
(d) None of these

Answer: A

Question. The largest term in the sequence a_n= n²/n³+200 is given by
(a) 49/543
(b) 529/49
(c) 8/89
(d) None

Answer: A

Question. The minimum value of 4e^2x + 9e^-2x is
(a) 12
(b) 14
(c) 11
(d) 10

Answer: A

Question. If P = (1,1), Q = (3,2) and R is a point on x - axis then the value of PR + RQ will be minimum at
(a) (5/3,0)
(b) (3,0)
(c) (1,0)
(d) None of the options

Answer: A

Question. If f '(x) = (x – a)²ⁿ (x – b)^2p+1 where n and p are positive integers, then :
(a) x is not a point of maximum or minimum
(b) none of these
(c) x = a is a point of maximum
(d) x = a is a point of minimum

Answer: A

Question.  The coordinates of the points on the curve,  f (x) =x/1+x² where the tangent to the curve has greatest slope is:
(a) (0, 0)
(b) (1, 1)
(c) (0, 1)
(d) (0, 2)

Answer: A

Question. The perimeter of a given rectangle is x, its area will maximum when its side are:
(a) x/4, x/4
(b) x/3, x/3
(c) x/2, x/2
(d) None of the options

Answer: A

Question. The minimum value of px + qy when xy = r² is:
(a) 2r √pq
(b) 2pq√ r
(c) -2r√ pq
(d) None of the options

Answer: A

Question. The maximum value of 3cosx + 4 sin x + 5 is :
(a) 7
(b) 9
(c) 5
(d) None of the options

Answer: A