JEE Main Mathematics - Application of Derivatives
Exam Duration: 60 Mins Total Questions : 30
The line \({x\over a}+{y \over b}=2\) touches then curve \(({X\over a})^n=({y\over b})^n=2\) at the point (a, b) for
- (a)
n = 2 only
- (b)
n = -3 only
- (c)
n is any real number
- (d)
None of the above
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)
\(3\sqrt2\)
The function f(x)=tanx-4x is strictly decreasing on
- (a)
\(({-{\pi\over 3},{\pi\over 3}})\)
- (b)
\([{-{\pi\over 3},{\pi\over 3}})\)
- (c)
\([{-{\pi\over 3},{\pi\over 2}})\)
- (d)
\([{-{\pi\over 3},{\pi\over 3}}]\)
The difference between the greatest and least values of the function \(f(x)=sin2x-x,\ on\ [{-{\pi\over 2}},{\pi\over 2}]\) is
- (a)
\(\pi\)
- (b)
\(2\pi\)
- (c)
\(3\pi\)
- (d)
\(\pi\over2\)
If \(f(x)=\int^x_0{(e^t-1)}(t-1)^2(t+1)^3\ dt\), then f(x) has
- (a)
maximum at x = 5
- (b)
minimum at x = 0
- (c)
minimum at x = -1
- (d)
maximum at x = -2
If m is the slope of a tangent to the curve \(e^y=1+x^2, then\)
- (a)
\(|m| \le 1\)
- (b)
m > -1
- (c)
m > 1
- (d)
|m| > 1
If x is real, then maximum value of \(3x^2+9x+17\over 3x^2+9x+7\) is
- (a)
41
- (b)
1
- (c)
\(17\over7\)
- (d)
\(1\over 4\)
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
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 distance between the origin and the normal to the curve y=\(e^{ 2x }+x^{ 2 }\) at x=0 is
- (a)
2/\(\sqrt { 3 } \)
- (b)
\(2\sqrt { 5 } \)
- (c)
\(2\sqrt { 7 } \)
- (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 slope of the normal at the point with abscissa x = 2 of the graph of the function j(x) = I X2 - x] is
- (a)
-1/6
- (b)
-1/3
- (c)
1/6
- (d)
1/3
The slope of the tangent to the curve y \(\int _{ x }^{ x^{ 2 } }{ cos } ^{ -1 }t^{ 2 }dt\quad x=\frac { 1 }{ \sqrt { 2 } } \)
- (a)
\(\left( \frac { \sqrt [ 4 ]{ 8 } }{ 2 } -\frac { 3 }{ 4 } \right) \pi \)
- (b)
\(\left( \frac { \sqrt [ 4 ]{ 8 } }{ 2 } -\frac { 1 }{ 4 } \right) \pi \)
- (c)
\(\left( \frac { \sqrt [ 5 ]{ 8 } }{ 4 } -\frac { 1 }{ 3 } \right) \pi \)
- (d)
none of these
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 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
Find the approximate value of f(3.02), where f(x)=3x^2+5x+3
- (a)
45.46
- (b)
45.47
- (c)
44.76
- (d)
44.46
Find the local minimum value of the function
\(f(x)=sin^4x+cos^4x, 0<x<{\pi\over2}\)
- (a)
1/√2
- (b)
1/2
- (c)
√3/2
- (d)
0
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume and of radius r
- (a)
r
- (b)
2r
- (c)
r/2
- (d)
3πr/2
State T for true and F for false.
(i) At (3, 2) and (-1,2) on the curve x2 +y2 - 2x - 4y + 1= 0,the tangents are parallel to y-axis.
(ii) f(x) = tan-1 (sin x + cos x) is an increasing function on (0, π/4).
(iii) f(x) = sinx + √3 cos x has maximum value at \(\pi\over 3\)
- (a)
(i) (ii) (iii) T T F - (b)
(i) (ii) (iii) T F F - (c)
(i) (ii) (iii) F T F - (d)
(i) (ii) (iii) F F T
Match the following.
Column I | Column II |
---|---|
(i) The function f(x) = tan-1 x is | (p) k2=8 |
(ii) The function \(f(x) = sin^{-1}\left(2x\over 1+x^2\right)\)is where x ∈ (1, ∞) | (q) decreasing |
(iii) The curves 2x =y2 and 2xy = k cut orthogonally if | (r) increasing |
- (a)
(i) ⟶ (q), (ii) ⟶ (r), (iii) ⟶ (p)
- (b)
(i) ⟶ (r), (ii) ⟶ (p), (iii) ⟶ (q)
- (c)
(i) ⟶ (r), (ii) ⟶ (q), (iii) ⟶ (p)
- (d)
(i) ⟶ (q), (ii) ⟶ (p), (iii) ⟶ (r)
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
The funetionf(x) = 4 sin3x - 6 sin2-x + 12 sinx + 100 is strictly
- (a)
increasing in \(\left(\pi, {3\pi \over2}\right)\)
- (b)
decreasing in \(\left({\pi\over 2},\pi\right)\)
- (c)
decreasing in \(\left[{-\pi\over 2},{\pi\over 2}\right]\)
- (d)
decreasing in \(\left[{0},{\pi\over 2}\right]\)
The smallest value of the polynomial x3 - 18x2 + 96x in [0,9] is
- (a)
126
- (b)
0
- (c)
135
- (d)
160
Statement -I : Letf: R ⟶ R be a function such that f(x) = x3 + x2 + 3x + sin x. Then,fis increasing function.
Statement-II: If f'(x) < 0, then f(x) is decreasing function.
- (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.