JEE Mathematics Application of Derivatives MCQs Set E

Question:

• a)

  

• b)

  

• c)

  

• d) None of these

Answer:

Question:

• a) None of these
• b) 5/18 cm/sec
• c) 1/4 cm/sec
• d) 5/16 cm/sec

Answer: None of these

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: as the radius

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)

  

• b)

  

• c)

  

• d)

  

Answer:

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: varies as the distance from the axis of y

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: (0,b)

Question: The angle between curves y² = 4x and x² + y² = 5 at (1, 2) is (1, 2) is

• a) tan^-1(3)
• b)

  

• c)

  

• d) tan^-1(2)

Answer: tan^-1(3)

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:

• a) ST = SN
• b) ST = 2SN
• c) ST² = aSN³
• d) ST³ = aSN

Answer: ST = SN

Question:

• a) 3x + 2y = 3√2
• b) 2x + 3y = 3√2
• c) 2x -3y = 3√2
• d) 3x - 2y = 3√2

Answer: 3x + 2y = 3√2

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: 2asin(t / 2) tan(t / 2)

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: (1, 0)

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: p = 2, q = – 7

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: x + 2y = 0

Question: The angle of intersection to the curve y = x², 6y = 7 – x3 at (1, 1) is :

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

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 cos t

Question:

• a) xy = x² + x + 1
• b) y = x² + x + 1
• c) xy = x + 1
• d) none of these

Answer:   xy = x² + x + 1

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: 9

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: 2

Question: The point(s) on the curve y³ + 3x² = 12 y where the tangent is vertical, is (are)

• a)

  

• b) (0,0)
• c)

  

• d)

  

Answer:

Question:

• a)

  

• b)

  

• c) x + 2y = 1
• d) none of these

Answer:

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: 2 and 4 are correct

Question:

then
(1) a > 0, b > 0 (2) a < 0, b < 0
(3) a > 0, b < 0 (4) a < 0, b > 0

• a) 1 and 2 are correct
• b) 1 and 3 are correct
• c) 1, 2 and 3 are correct
• d) 2 and 4 are correct

Answer: 1 and 2 are correct

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: 1 and 3 are correct

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 Abscissa of the point of contact of the tangent for which m is greatest is –

• a)

  

• b) 1
• c) –1
• d) None of these

Answer:

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: 9/8

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: -√ 3

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: Statement -1 is False, Statement-2 is True.

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: Statement -1 is True, Statement-2 is False.

Question:

Statement 1 : The ratio of length of tangent to length of normal is inversely proportional to numerical value of the ordinate of the point of tangency at the curve y² = 4ax.
Statement 2 : Length of normal & tangent to a curve

• 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 False, Statement-2 is True.
• d) Statement -1 is True, Statement-2 is False.

Answer: Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Question:

• a) p = q = r
• b) q≠ r
• c) p ≠ q
• d) p ≠ q

Answer: p = q = r

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: 8, 8

Question: If two sides of a triangle be given, then the area of the triangle will be maximum if the angle between the given sides be

• a)

  

• b)

  

• c)

  

• d)

  

Answer:

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: (e)^{1/ e}

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: 49/543

Question: One point of maximum of sin ^p x cos^q x is

• a)

  

• b)

  

• c) x = tan^-1( p / q)
• d) x = tan^-1(q / p)

Answer:

Question: If a²x⁴ + b² y⁴ = c⁶ , then maximum value of xy is

• a)

  

• b)

  

• c) Both
• d) None

Answer:

Question:

• a) 1
• b) e
• c) 1/ e
• d) 0

Answer: 1

Question: The minimum value of 4e^2x + 9e^-2x is

• a) 12
• b) 14
• c) 11
• d) 10

Answer: 12

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 these

Answer: (5/3,0)

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: x is not a point of maximum or minimum

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: (0, 0)

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 these

Answer: x/4, x/4

Question: The minimum value of px + qy when xy = r² is:

• a) 2r √pq
• b) 2pq√ r
• c) -2r√ pq
• d) none of these

Answer: 2r √pq

Question: The maximum value of 3cosx + 4 sin x + 5 is :

• a) none of these
• b) 9
• c) 5
• d) 7

Answer: none of these