Aod q-1
Aod q-1
Aod q-1
APPLICATION OF DERIVATIVES
3. P is a point on the curve f (x) = x x2 such that abscissae of P lies in the interval (0,
1). The maximum area of the triangle POA, where O and A are the points (0, 0) and
(1, 0) is
1 1
(A) sq. units (B) sq. units
8 4
1
(C) sq. units (D) none of these
2
4. If f(x) > 0, x R, f(3) = 0 and g(x) = f (tan2 x –2 tan x + 4), 0 < x < , then g (x) is
2
increasing in
(A) 0, (B) ,
4 6 3
(C) 0, (D) none of these
3
1
8. A solid cylinder of height H has a conical portion of same height and radius rd of
3
height removed from it. Rain water is falling in the cylinder with rate equal to times
the instantanes radius of water surface inside hole, the time after which hole will fill
up with water is
H2
(A) (B) H2
3
H2 H2
(C) (D)
6 4
3 | x k |, xk
9. If f (x) = 2 sin(x k) has minimum at x = k, then
a 2 x k , x k
10. If the curve y = x2 + bx + c touches the straight line y = x at the point (1, 1), then b
and c are given by
(A) –1, 1 (B) –1, 2
(C) 2, 1 (D) 1, 1
11. The tangent and normal to the curve y = 2 sin x + sin 2x are drawn at P x ,
3
then area of the quadrilateral formed by the tangent, the normal at P and the
coordinate axis is
(A) (B) 3
3
3
(C) (D) none of these
2
12. If a = cos1 x , b = cos–1 x and c = (cos–1 x)2 and a > b > c, then x lies in the interval
(A) (cos 1, 1) (B) (0, cos 1)
(C) (–1, 1) (D) (–1, cos 1)
13. The curve x + y – ln (x + y) = 2x + 5 has a vertical tangent at the point (, ). Then
+ is equal to
(A) –1 (B) 1
(C) 2 (D) –2
ax 2 1 y 2 ax 2
14. If x2 y2 1and x, y 0 then the least value of , where a > 0, is
y2 a x2
2
(A) 2 (B)
a
(C) 3 (D) none of these
3
15. x1 and x2 are two solutions of the equation ex cosx = 1. The minimum number of the
solutions of the equation ex sinx = 1, lying between x1 and x2 can be
(A) 0 (B) 1
(C) 3 (D) none of these
2
d esin x aesin x
16. Let F(x) and a be real. The value of , for which dx F 16 F 1 , is
dx x 1
x
equal to
(A) 4 (B) 2
1
(C) (D) none of these
2
17. The values of the parameter ‘a’ so that the line (3 – a)x + ay + a2 – 1 = 0 is a normal
to the curve xy = 1, is /are ;
(A) (3, ) (B) (-, 3)
(C) (0, 3) (D) none of these
2
19. If f(x) = ( tan-1x)2 + , then f is decreasing in
x2 1
(A) (0, ) (B) [1, 10]
(C) [3, 5] (D) None of these
20. Let f (sin x) < 0 and f (sin x ) > 0, x (0, ) and g(x) = f (sin x) + f (cos x), then
2
g(x) is decreasing in
(A) , (B) 0,
4 2 4
(C) 0, (D) ,
2 6 2
1 1 b1 b2
(A) (B) 1
a1a2 b1b2 a1 a2
b1 b2 b1b2
(C) (D) None of these
a1 a2 a1a2
4
22. If the length of sub–normal at any point ‘t’ to the parabola y2 kx is 2 units, the value
of k is
(A) 1 (B) 2
(C) 3 (D) 4
2
The minimum value of x2 x1
2
25. 1 x12 4 x 22 is equal to
(A) 13 (B) 6
(C) 4 (D) 1
8
26. If the function f : 0, 8 R is differentiable then for 0 , 2, f(t)dt is equal to
0
x
f(x)
28. If f : [0, 3] R, f (x) = 2x3 9x2 + 12x + 6 and g (x) = f (x) + , then {where [.]
20
denotes G.I.F.}
(A) g (x) = 0 has one real root in (0, 1)
(B) g (x) =0 has one real root in (1, 2)
3 3
(C) g (x) = 0 has exactly one real root in 0, and exactly one in , 3
2 2
(D) none of these
5
29. If P is any point on the curve x2 3y2 3xy 1 whose centre is at O, then minimum
value of OP is
2 2
(A) (B)
2 3 2 13
2
(C) (D) None of these
4 13
30. Let f(x) = sin3 x + sin2 x for <x< and > 0. The interval in which should
2 2
lie in order that f(x) has exactly one minimum and one maximum is
3
(A) 0 < < (B) 0 < < 1
2
(C) 1 < < 2 (D) 1 < < 3/2
31. If the parabola y = f (x), having axis parallel to y-axis, touches the line y = x at (1, 1)
then;
(A) 2f (0) + f (0) = 1 (B) 2f (0) + f (0) = 1
(C) 2f (0) - f (0) = 1 (D) 2f (0) - f (0) = 1
xy y
dy
xy y
2
dy 2
32. Solution of sin 2
cos
cos 2
sin
= 0, is
x dx x dx
y
tan 1
n y
(A) e x n
(kx ) , n I+ (B) tan 1 = n ln kx, n I+
x n
(C) y = nx tan–1 n ln kx, n I+ (D) none of these
1 1
33. If f (x) = xn sin + xm cos , then
x 2x
1 2
(A) atleast one root of f (x) = 0 will lie in interval ,
1 1
(B) atleast one root of f (x) = 0 will lie in interval ,
3
1 1
(C) atleast one root of f (x) = 0 will lie in interval ,
2
(D) none of these
34. ABCD is a rectangle with AB = 3a, BC = a. P and Q are some points on AB and CD
respectively. Quadrilateral PBCQ is cut and joined with PB coincident with DQ to
form a new hexagon then maximum perimeter of this hexagon is
1 5 1 5
(A) 2a (B) 2a
3 2 3 2
1 5
(C) (D) none of these
3 2
6
35. If y = x lnx + sin ( lnx) x [e, e2], then which of the following
lines is always a tangent to y for some x [e, e2]
(A) e (y 2x) + (x y) = c (B) e (y + 2x) + (x y) = c
(C) e (y + 2x) (x + y) = c (D) nothing can be said
37. Let x and y be real numbers satisfying the equation x2 –4x + y2 + 3 = 0. If the
maximum and minimum values of x2 + y2 are M and m respectively then M-m is
equal to
(A) 10 (B) 9
(C) 8 (D) 7
39. A function f(x) is defined as f(x) = mx3 – 12x + a, where a is local maximum value
of
(mx3 – 12x) and m > 0, the function f(x) = 0 has a root where
40. If f (x) > 0, x R, f (3) = 0 and g (x) = f (tan2 x –2 tan x + 4), 0 <x < , then g (x)
2
is increasing in
(A) 0, (B) ,
4 6 3
(C) 0, (D) none of these
3
7
41. The set of values of ‘a’ for which all the solutions of the equation 4sin4 x + asin2x + 3
= 0 are real and distinct.
(A) (2, 6) (B) (2, 4)
(C) (0, 1) (D) [-7, -4 3 )
x2
42. Given g(x) = and the line 3x + y – 10 = 0, then the line is
x 1
(A) tangent to g(x) (B) normal to g(x)
(C) chord of g(x) (D) none of these
43. The angles at which the curve y = kekx intersect the y-axis is/are
(A) tan-1(k2) (B) cot-1(k2)
1
(C) sin-1 (D) sec 1 1 k 2
1 k 2
45.
If f x 2 4x 3 0 , x 2, 3; then f(sin x) is increasing on
(A) 2n, 4n 1 2
nI
(B) (4n 1) 2 ,2n
nI
(C) R (D) None of these
46. Let f (x) = sin 2x + x –[x] ([.] denotes the greatest integer function). Then number of points
in [0, 10] at which f (x) take its local maximum value is
(A) 0 (B) 10
(C) 20 (D) infinite