IISER 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 normal to the curve x2 = 4y passing through (1,2) is
- (a)
x + y = 3
- (b)
x - y = 3
- (c)
x + y = 1
- (d)
x - y = 1
The points on the the curve \(y=x^3-x^2-x+3\), where the tangents are parallel to the X-axis, are
- (a)
\((+{1\over 3},{-88\over 27}) and (1,2)\)
- (b)
\((-{1\over 3},{86\over 27}) and (1,2)\)
- (c)
\(({-1\over 3},{86\over 27}) and (-1,-2)\)
- (d)
\(({-1\over 3},{-88\over 27}) and (-1,2)\)
If the line ax + by + c = 0 is normal to curve xy + 5 = 0, then
- (a)
a + b = 0
- (b)
a > 0
- (c)
a < 0, b < 0
- (d)
a = - 2b
The length of subnormal to the curve \(y = {x\over 1-x^2}\) at the point having abscissa \(\sqrt 2\) is
- (a)
\(5\sqrt2\)
- (b)
\(3\sqrt3\)
- (c)
\(\sqrt3\)
- (d)
\(3\sqrt2\)
Let c be the curve \(y = x^2, \forall\quad x\in R\). The tangent at A meets the curve again at B. If the gradient at A is k times the gradient at B, the number of integral values of k is
- (a)
4
- (b)
5
- (c)
3
- (d)
6
The equation of the tangent to the curve \(y = x +{4\over x^2}\), that is parallel to the X-axis, is
- (a)
y = 0
- (b)
y = 1
- (c)
y = 2
- (d)
y = 3
The function \(f(x)= {x\over 2}+{2\over x}\) has a local minimum at
- (a)
\(x=-2\)
- (b)
\(x=0\)
- (c)
\(x=1\)
- (d)
\(x=2\)
Angle between the tangents to the curve \(y=x^2-5x+6\) at the points (2,0) and (3,0) is
- (a)
\(\pi\over 2\)
- (b)
\(\pi\over 6\)
- (c)
\(\pi\over 4\)
- (d)
\(\pi\over 3\)
The normal to the curve \(x=a (cos\theta+\theta sin\theta),\) \(y=a (sin\theta-\theta cos\theta)\) at any point \(\theta\) is such that
- (a)
it is at a constant distance from the origin
- (b)
it passes through \(({a\pi\over 2},-a)\)
- (c)
it makes angle \({\pi\over2}-\theta\) with the X-axis
- (d)
it passes through the origin
The approximate value of (0.007)1/3 is
- (a)
\(\frac { 21 }{ 120 } \)
- (b)
\(\frac { 23 }{ 120 } \)
- (c)
\(\frac { 29 }{ 120 } \)
- (d)
\(\frac { 31 }{ 120 } \)
The tangent to the curve x= \(\sqrt { cos2\theta } cos\theta ,y=a\sqrt { cos2\theta } \) at the point corresponding to \(\theta =\pi /6\)
- (a)
parallel to the X-axis
- (b)
parallel to the y-axis
- (c)
parallel to line y=x
- (d)
none of these
The point of intersection of the tangents drawn to the curve x2y = 1 - Y at the points where it is meet by the curve xy = 1 - y, is given by
- (a)
(0,- 1)
- (b)
(1,1)
- (c)
(0, 1)
- (d)
none of these
The curve y - exy + x = 0 has a vertical tangent at the point
- (a)
(1, 1)
- (b)
at no point
- (c)
(0,1)
- (d)
(1, 0)
The sides of an equilateral triangle are increasing at the rate of 2cm/s. The rate at which the area increases, when the side is 20cm is
- (a)
√3cm2/s
- (b)
20cm2/s
- (c)
\(20\sqrt3cm^2/s\)
- (d)
\({20\over\sqrt3}cm^2/s\)
The distance 's' metres covered by a body in t seconds, is given by s=3t2-8t+5. The body will stop after
- (a)
1s
- (b)
3/4 s
- (c)
4/3s
- (d)
4s
The length of the longest interval, in which the function 3sinx-4sin3x is increasing, is
- (a)
π/3
- (b)
π/2
- (c)
3π/2
- (d)
π
The normal to the curve 2y=3-x2 at (1,1) is
- (a)
x+y=0
- (b)
x+y+1=0
- (c)
x-y+1=0
- (d)
x-y=0
If there is an error of a% in measuring the edge of a cube then percentage error in its surface area is
- (a)
2a%
- (b)
a/2 %
- (c)
3a%
- (d)
None of these
If the radius of a sphere is measured as 9cm with an error od 0.03cm then find the approximating error in calculating its volume
- (a)
2.46πcm3
- (b)
8.62πcm3
- (c)
9.72πcm3
- (d)
7.46πcm3
the two positive numbers whose sum is 15 and the sum of whose squares is minimum are
- (a)
\({15\over2},{13\over2}\)
- (b)
\({13\over2},{11\over2}\)
- (c)
\({15\over2},{15\over2}\)
- (d)
\({17\over2},{13\over2}\)
The distance of that point on y=x4+3x2 +2x which is nearest to the line y=2x-1 is
- (a)
\(3\over \sqrt5\)
- (b)
\(4\over \sqrt5\)
- (c)
\(2\over \sqrt5\)
- (d)
\(1\over \sqrt5\)
If two positive numbers x and y are such that x+y=60 and xy3 is maximum, then the numbers x and y are respectively
- (a)
13,47
- (b)
28,32
- (c)
12,48
- (d)
15,45
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. The cost of the material will be last when depth of the tank is
- (a)
twice of its width
- (b)
half of its width
- (c)
equal to its width
- (d)
None of these
The equation of normal to the curve 3x2 - y2 = 8 which is parallel to the line x + 3y = 8 is
- (a)
3x - y = 8
- (b)
3x + y + 8 = 0
- (c)
x + 3y ± 8 = 0
- (d)
x + 3y = 0
If y = x4 - 10 and if x changes from 2 to 1.99, what is the change in y
- (a)
0.32
- (b)
0.032
- (c)
5.68
- (d)
5.968
The two curves x3 - 3xy2 + 2 = 0 and 3x2y -y3- 2 = 0 intersect at an angle of
- (a)
π/4
- (b)
π/3
- (c)
π/2
- (d)
π/6
Which of the following functions is decreasing on \(\left(0,{\pi\over 2}\right)\) ?
- (a)
sin2x
- (b)
·tanx
- (c)
cosx
- (d)
cos3x
Statement-I: The graph y = x3 + ax2 + bx + e has extremum, if a2 < 3b.
Statement-II: y is either increasing or decreasing ∀ x ∈ R.
- (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.
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.