Mathematics - Application of Derivatives
Exam Duration: 45 Mins Total Questions : 30
The line y = mx + 1 is a tangent to the curve y2= 4x, if the value of m is
- (a)
1
- (b)
2
- (c)
3
- (d)
1/2
The interval on which \(f(x)=5x^3-15x^2-120x+3\) is strictly increasing or decreasing are respectively
- (a)
\((-\infty, + 2) \cup (4, \infty) and (-2,3)\)
- (b)
\((-\infty, -1) (4, \infty) and (-2,5)\)
- (c)
\((-\infty, - 2) \cup (4, \infty) and (-2,4)\)
- (d)
\((-\infty, -3) \cup (5, \infty) and (-2,4)\)
Match the columns
Column I | Column II |
A. A circular plate is expanded by heat from radius 6 cm to 6.06 cm. Approximate increase in the area is | p. 5 |
B. If an edge of a cube increases by 2%, then the percentage increase int he volume is | q. 0.72\(\pi\) |
C. If the rate of decreases of \({x^2\over 2}-2x+5\) is thrice the rate of decrease of x, then x is equal to (rate of decrease is non-zero) |
r. 6 |
D. The rate of increase in the area of an equilateral triangles of side 30 cm, when each side increases at the rate of 0.1 cm/s is | s. \(3\sqrt3\over 2\) |
- (a)
A B C D p q r s - (b)
A B C D s r q p - (c)
A B C D r q s q - (d)
A B C D q r p s
How many real solutions does the equation \(x^7+14x^5+16x^3+30x-560=0\) have?
- (a)
5
- (b)
7
- (c)
1
- (d)
3
Tangents are drawn from the origin to the curve y=sinx, then their to the curve y=sinx. then their point of contact lie on the curve
- (a)
\(x^{ 2 }+y^{ 2 }=1\)
- (b)
\(x^{ 2 }-y^{ 2 }=1\)
- (c)
\(\frac { 1 }{ x^{ 2 } } +\frac { 1 }{ y^{ 2 } } =1\)
- (d)
\(\frac { 1 }{ y^{ 2 } } -\frac { 1 }{ x^{ 2 } } =1\)
The equation of the tangent to the curve y = 1- ex/2 at the point of intersection with the Y: axis is
- (a)
x + 2y = 0
- (b)
2x + y = 0
- (c)
x - y = 2
- (d)
none of these
The equation of the tangent to the curve \(y=e^{ -|x| }\) at the point where the curve cuts the line x = 1 is
- (a)
x + y = e
- (b)
e(x + y) = 1
- (c)
y + q; = 1
- (d)
none of these
The acute angles between the curves y =|x2-1| and y =|x3-1| at their points of intersection is
- (a)
\(\pi /4\)
- (b)
\(tan^{ -1 }(4\sqrt { 2 } /7)\)
- (c)
\(tan^{ -1 }(4\sqrt { 7 } )\)
- (d)
none of these
If the tangent at any point on the curve X4 + y4 = a4 cuts off intercepts p and q on the coordinate axes, the value of p -4/3 + q-4/3 is
- (a)
a-4/3
- (b)
a-1/2
- (c)
a 1/2
- (d)
none of these
If y = 4x - 5 is a tangent to the curve \(y^{ 2 }+p^{ 2 }+q\) at (2,3) then
- (a)
= 2, q = - 7
- (b)
p = - 2, q = 7
- (c)
p = - 2, q = - 7
- (d)
p = 2, q"= 7
The tangent to the graph of the function y = f(x) at the point with abscissa x = 1 form an angle of \(\pi /6\) and at the point x = 2 an angle of \(\pi /3\) and at the point x = 3 an angle \(\pi /4\) The value of \(\int _{ 1 }^{ 3 }{ f(x)f^{ '' }(x)dx+ } \int _{ 2 }^{ 3 }{ f(x)dx } \)f (x) is suppose to be continuous) is
- (a)
\(\frac { 4\sqrt { 3 } -1 }{ 3\sqrt { 3 } } \)
- (b)
\(\frac { 3\sqrt { 3 } -1 }{ 2 } \)
- (c)
\(\frac { 4-\sqrt { 3 } }{ 3 } \)
- (d)
none of these
If the parametric equation of a curve given by x=etcost y=et sint then the tangent to the curve at the point t = \(\pi \)/4 makes with axis of x is
- (a)
0
- (b)
\(\pi /4\)
- (c)
\(\pi /3\)
- (d)
\(\pi /2\)
the radius of a cylinder is increasing at the rate of 3m/s and its height is decreasing at the rate of m/s. The rate of change of volume when radius is 4m and height is 6m is, is
- (a)
80πcu m/s
- (b)
144πcu m/s
- (c)
80cu m/s
- (d)
64cu m/s
If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing, is
- (a)
a constant
- (b)
proportional to the radius
- (c)
inversely proportional to the radius
- (d)
inversely proportional to the surface area
The function f(x)=log(1+x)-\({2x\over 2+x}\) is increasing on
- (a)
(-1, ∞)
- (b)
(-∞,0)
- (c)
(-∞,∞)
- (d)
None of these
If \(f(x)={x\over sin}\) and \(g(x)={x\over tanx}, 0<x\le 1,\) then in the interval
- (a)
both f(x) and g(x0 are increasing function
- (b)
both f(x) and g(x)are decreasing function
- (c)
f(x) is an increasing function
- (d)
g(x) is an increasing function
The function \(f(x) = tan^{-1}(sin\ x + cos x)\) is an increasing function in
- (a)
\(({\pi\over 4},{\pi\over 2})\)
- (b)
\((-{\pi\over 2},{\pi\over 2})\)
- (c)
\((0,{\pi\over 2})\)
- (d)
none of these
Tangent to the curve y=x3+3x at x=-1 and x=1 are
- (a)
parallel
- (b)
intersecting obliquely but not at an angle of 450
- (c)
intersecing at right angles
- (d)
intersecting at an angle 450
The point on the curve y2=x, where tangent makes an angle of 450 with the X-axis
- (a)
\(\left({1\over2},{1\over4}\right)\)
- (b)
\(\left({1\over4},{1\over2}\right)\)
- (c)
(4,2)
- (d)
(2,-2)
If there is an error of 2% in measuring the length of a simple pendulum, then percentage error in its period is
- (a)
1%
- (b)
2%
- (c)
3%
- (d)
4%
Sinp θ cosq θ attains a maximum, when θ=
- (a)
\(tan^{-1}\sqrt{p\over q}\)
- (b)
\(tan^{-1}\left(p\over q\right)\)
- (c)
tan-1q
- (d)
\(tan^{-1}\left(q\over p\right)\)
Find both the maximum and minimum values respectively of 3x4-8x3+12x2-48x+1 on the intervals [1,4]
- (a)
-63,-257
- (b)
257,-40
- (c)
257,-63
- (d)
63,-257
Amongst all pairs of positive numbers with product 256, find those whose sum is the least
- (a)
16,14
- (b)
16,16
- (c)
64,4
- (d)
32,8
A beam of length l is supported at one end. If W is the uniform load per unit length the bending moment M at a distance x from the end is given by
\(m={1\over2}lx={1\over 32}Wx^2\)
The distance of point on the bending moment has the maximum value
- (a)
l/2W
- (b)
2W/l
- (c)
l/W
- (d)
W/l
An open box with a square base is to be made out of a given quantity of a cardboard of area c2 square units. The maximum volume of the box is (in cubicunits)
- (a)
\(c^2\over 2\sqrt3\)
- (b)
\(c^2\over 6\sqrt3\)
- (c)
4/5
- (d)
6c2
The height of the closed cylinder of given surface and maximum volume, is equal to the
- (a)
diameter
- (b)
radius
- (c)
half of radius
- (d)
twice of diameter
The curve y = x1/5 at (0, 0) has
- (a)
a vertical tangent (parallel to y-axis)
- (b)
a horizontal tangent (parallel to x-axis)
- (c)
an oblique tangent
- (d)
no tangent
Statement-I: A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m, then length 1.782m and breadth 2.812 m of the rectangle will produce the largest area of the window.
Statement-II: For maximum or minimum f(x) = 0
- (a)
If both Statement-I and Statement-If are true and Staternent -Il is the correct explanation of Statement-1
- (b)
If both Statement-I and Staternent -Il are true but Statement-Il is not the correct explanation of Statement-I
- (c)
If Staternent -I is true but Staternent -Il is false.
- (d)
If Statement-I is false and Staternent -Il is true.