Applications of Derivatives Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. The point P of curve y
2
 = 2x
3
  such that the tangent at P is perpendicular
to the line 4x – 3y + 2 = 0 is given by :
(a) (2, 4)
(b) (1, )
(c)
(d)
2. f (x) = 2x
2 
– log | x | (x ? 0) is monotonic increasing in the interval
(a) (1/2, 8)
(b) (– 8, –1/2) ? (1/2, 8)
(c) (– 8, –1/2) ? (0, 1/2)
(d) (–1/2,  0) ? (1/2, 8)
3. The minimum value of the function
Page 2


PART-I (Single Correct MCQs)
1. The point P of curve y
2
 = 2x
3
  such that the tangent at P is perpendicular
to the line 4x – 3y + 2 = 0 is given by :
(a) (2, 4)
(b) (1, )
(c)
(d)
2. f (x) = 2x
2 
– log | x | (x ? 0) is monotonic increasing in the interval
(a) (1/2, 8)
(b) (– 8, –1/2) ? (1/2, 8)
(c) (– 8, –1/2) ? (0, 1/2)
(d) (–1/2,  0) ? (1/2, 8)
3. The minimum value of the function
f (x) =  for all permissible real x, is
(a) – 10
(b) –6
(c) –7
(d) – 8
4. A particle moves along the curve 6y = x
3
 + 2. The point ‘P’ on the
curve at which the y-coordinate is changing 8 times
as fast as the x-coordinate, are (4, 11) and .
(a) x-coordinates at the point P are 4
(b) y-coordinates at the point P are 11 and 
(c) Both (a) and (b)
(d) None of the above
5. Which of the following statements is false?
(a) The length of sub-tangent to the curve x
2
y
2
 = 16a
4
 at the point (–2a, 2a)
is 2a.
(b) x + y = 3 is a normal to the curve x
2
 = 4y
(c) Curves y =  –4x
2
 and y = e
–x/2
 are orthogonal.
(d) If a ? (–1, 0), then tangent at each point of the curve
 makes an acute angle with the positive
direction of x-axis.
6. Find the coordinates of a point of the parabola
y = x
2
 + 7x + 2
which is closest to the straight line y = 3x – 3.
(a) (–2, –8)
(b) (2, 8)
(c) (–2, 8)
(d) (2, –8)
7. The set of all values of a for which the function
Page 3


PART-I (Single Correct MCQs)
1. The point P of curve y
2
 = 2x
3
  such that the tangent at P is perpendicular
to the line 4x – 3y + 2 = 0 is given by :
(a) (2, 4)
(b) (1, )
(c)
(d)
2. f (x) = 2x
2 
– log | x | (x ? 0) is monotonic increasing in the interval
(a) (1/2, 8)
(b) (– 8, –1/2) ? (1/2, 8)
(c) (– 8, –1/2) ? (0, 1/2)
(d) (–1/2,  0) ? (1/2, 8)
3. The minimum value of the function
f (x) =  for all permissible real x, is
(a) – 10
(b) –6
(c) –7
(d) – 8
4. A particle moves along the curve 6y = x
3
 + 2. The point ‘P’ on the
curve at which the y-coordinate is changing 8 times
as fast as the x-coordinate, are (4, 11) and .
(a) x-coordinates at the point P are 4
(b) y-coordinates at the point P are 11 and 
(c) Both (a) and (b)
(d) None of the above
5. Which of the following statements is false?
(a) The length of sub-tangent to the curve x
2
y
2
 = 16a
4
 at the point (–2a, 2a)
is 2a.
(b) x + y = 3 is a normal to the curve x
2
 = 4y
(c) Curves y =  –4x
2
 and y = e
–x/2
 are orthogonal.
(d) If a ? (–1, 0), then tangent at each point of the curve
 makes an acute angle with the positive
direction of x-axis.
6. Find the coordinates of a point of the parabola
y = x
2
 + 7x + 2
which is closest to the straight line y = 3x – 3.
(a) (–2, –8)
(b) (2, 8)
(c) (–2, 8)
(d) (2, –8)
7. The set of all values of a for which the function
does not possess critical points is
(a) [1, 8)
(b) (0, 1) ? (1, 4)
(c) (–2, 4)
(d) (1, 3) ? (3, 5)
8. Let F(x) = x
3
 + ax
2
 + bx + 5 sin
2
 x be an increasing function in the set of
real number R. Then a and b satisfy the condition.
(a) a
2
 – 3b – 15 > 0
(b) a
2
 – 3b + 15 > 0
(c) a
2
 + 3b – 15 < 0
(d) a > 0 and b > 0
9. The equation of the tangent to the curve  at the point where the
curve cuts the line x = 1 is
(a) e(x + y) = 1
(b) y + ex = 1
(c) y + x = e
(d) None of these
10. The straight line  +  = 2 touches the curve
 +  = 2 at the point (a, b) for
(a) n = 1, 2
(b) n = 3, 4, –5
(c) n = 1, 2, 3
(d) Any value of n
11. The function f (x) = 3 cos
4
x + 10 cos
3
x + 6 cos
2
x – 3,  (0 = x = p) is –
(a) Increasing in 
(b) Increasing in 
Page 4


PART-I (Single Correct MCQs)
1. The point P of curve y
2
 = 2x
3
  such that the tangent at P is perpendicular
to the line 4x – 3y + 2 = 0 is given by :
(a) (2, 4)
(b) (1, )
(c)
(d)
2. f (x) = 2x
2 
– log | x | (x ? 0) is monotonic increasing in the interval
(a) (1/2, 8)
(b) (– 8, –1/2) ? (1/2, 8)
(c) (– 8, –1/2) ? (0, 1/2)
(d) (–1/2,  0) ? (1/2, 8)
3. The minimum value of the function
f (x) =  for all permissible real x, is
(a) – 10
(b) –6
(c) –7
(d) – 8
4. A particle moves along the curve 6y = x
3
 + 2. The point ‘P’ on the
curve at which the y-coordinate is changing 8 times
as fast as the x-coordinate, are (4, 11) and .
(a) x-coordinates at the point P are 4
(b) y-coordinates at the point P are 11 and 
(c) Both (a) and (b)
(d) None of the above
5. Which of the following statements is false?
(a) The length of sub-tangent to the curve x
2
y
2
 = 16a
4
 at the point (–2a, 2a)
is 2a.
(b) x + y = 3 is a normal to the curve x
2
 = 4y
(c) Curves y =  –4x
2
 and y = e
–x/2
 are orthogonal.
(d) If a ? (–1, 0), then tangent at each point of the curve
 makes an acute angle with the positive
direction of x-axis.
6. Find the coordinates of a point of the parabola
y = x
2
 + 7x + 2
which is closest to the straight line y = 3x – 3.
(a) (–2, –8)
(b) (2, 8)
(c) (–2, 8)
(d) (2, –8)
7. The set of all values of a for which the function
does not possess critical points is
(a) [1, 8)
(b) (0, 1) ? (1, 4)
(c) (–2, 4)
(d) (1, 3) ? (3, 5)
8. Let F(x) = x
3
 + ax
2
 + bx + 5 sin
2
 x be an increasing function in the set of
real number R. Then a and b satisfy the condition.
(a) a
2
 – 3b – 15 > 0
(b) a
2
 – 3b + 15 > 0
(c) a
2
 + 3b – 15 < 0
(d) a > 0 and b > 0
9. The equation of the tangent to the curve  at the point where the
curve cuts the line x = 1 is
(a) e(x + y) = 1
(b) y + ex = 1
(c) y + x = e
(d) None of these
10. The straight line  +  = 2 touches the curve
 +  = 2 at the point (a, b) for
(a) n = 1, 2
(b) n = 3, 4, –5
(c) n = 1, 2, 3
(d) Any value of n
11. The function f (x) = 3 cos
4
x + 10 cos
3
x + 6 cos
2
x – 3,  (0 = x = p) is –
(a) Increasing in 
(b) Increasing in 
(c) Decreasing in 
(d) All of above
12. The diagonal of a square is changing at the rate of 0.5 cm/sec. Then the
rate of change of area, when the area is400 cm
2
, is equal to
(a)
(b)
(c)
(d)
13. The difference between the greatest and least values of the function f(x)
= sin 2x – x, on  is
(a)
(b) p
(c)
(d)
14. The largest distance of the point (a, 0) from the curve2x
2
 + y
2
 – 2x = 0,
is given by
(a)
(b)
(c)
(d)
Page 5


PART-I (Single Correct MCQs)
1. The point P of curve y
2
 = 2x
3
  such that the tangent at P is perpendicular
to the line 4x – 3y + 2 = 0 is given by :
(a) (2, 4)
(b) (1, )
(c)
(d)
2. f (x) = 2x
2 
– log | x | (x ? 0) is monotonic increasing in the interval
(a) (1/2, 8)
(b) (– 8, –1/2) ? (1/2, 8)
(c) (– 8, –1/2) ? (0, 1/2)
(d) (–1/2,  0) ? (1/2, 8)
3. The minimum value of the function
f (x) =  for all permissible real x, is
(a) – 10
(b) –6
(c) –7
(d) – 8
4. A particle moves along the curve 6y = x
3
 + 2. The point ‘P’ on the
curve at which the y-coordinate is changing 8 times
as fast as the x-coordinate, are (4, 11) and .
(a) x-coordinates at the point P are 4
(b) y-coordinates at the point P are 11 and 
(c) Both (a) and (b)
(d) None of the above
5. Which of the following statements is false?
(a) The length of sub-tangent to the curve x
2
y
2
 = 16a
4
 at the point (–2a, 2a)
is 2a.
(b) x + y = 3 is a normal to the curve x
2
 = 4y
(c) Curves y =  –4x
2
 and y = e
–x/2
 are orthogonal.
(d) If a ? (–1, 0), then tangent at each point of the curve
 makes an acute angle with the positive
direction of x-axis.
6. Find the coordinates of a point of the parabola
y = x
2
 + 7x + 2
which is closest to the straight line y = 3x – 3.
(a) (–2, –8)
(b) (2, 8)
(c) (–2, 8)
(d) (2, –8)
7. The set of all values of a for which the function
does not possess critical points is
(a) [1, 8)
(b) (0, 1) ? (1, 4)
(c) (–2, 4)
(d) (1, 3) ? (3, 5)
8. Let F(x) = x
3
 + ax
2
 + bx + 5 sin
2
 x be an increasing function in the set of
real number R. Then a and b satisfy the condition.
(a) a
2
 – 3b – 15 > 0
(b) a
2
 – 3b + 15 > 0
(c) a
2
 + 3b – 15 < 0
(d) a > 0 and b > 0
9. The equation of the tangent to the curve  at the point where the
curve cuts the line x = 1 is
(a) e(x + y) = 1
(b) y + ex = 1
(c) y + x = e
(d) None of these
10. The straight line  +  = 2 touches the curve
 +  = 2 at the point (a, b) for
(a) n = 1, 2
(b) n = 3, 4, –5
(c) n = 1, 2, 3
(d) Any value of n
11. The function f (x) = 3 cos
4
x + 10 cos
3
x + 6 cos
2
x – 3,  (0 = x = p) is –
(a) Increasing in 
(b) Increasing in 
(c) Decreasing in 
(d) All of above
12. The diagonal of a square is changing at the rate of 0.5 cm/sec. Then the
rate of change of area, when the area is400 cm
2
, is equal to
(a)
(b)
(c)
(d)
13. The difference between the greatest and least values of the function f(x)
= sin 2x – x, on  is
(a)
(b) p
(c)
(d)
14. The largest distance of the point (a, 0) from the curve2x
2
 + y
2
 – 2x = 0,
is given by
(a)
(b)
(c)
(d)
15. LL’ is the latus rectum of the parabola  and PP’ is double
ordinate drawn between the vertex and the latus rectum. The area of the
trapezium PP’L’L is maximum when the distnace of PP’ from the
vertex is
(a) 1
(b) 4
(c) 9
(d) 36
16. The value(s) of ‘a’ for which the area of the triangle included between the
axes and any tangent to the curve x
a
 y = ?
a
 is a constant is/are :
(a)
(b) 1
(c)
(d) 2
17. The minimum value of the function
 in (0, a) is :
(a) a + b
(b)
(c)
(d)
18. If  is a decreasing function of x in R then the set
of possible values of a (independent of x) is
(a)
(b)