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Application of Derivatives

Multiple Choice Questions

291.

The function y = 2x³ - 9x² + 12x - 6 is monotonic decreasing when

1 < x < 2
x > 2
x < 1
None of these

292.

Maximum slope of the curve y = - x³ + 3x² + 9x - 27 is

293.

The points on the curve x² = 2y which are closest to the point (0, 5) are

(2, 2), (- 2, 2)
$(2 \sqrt{2}, 4), (- 2 \sqrt{2}, 4)$
$(\sqrt{6}, 3), (- 6 \sqrt{3}, 3)$
$(2 \sqrt{3}, 6), (- 2 \sqrt{3}, 6)$

294.

The point on the curve y = 2x² - 4x + 5, at which the tangent is parallel to x-axis, will be

(1, 3)
(- 1, 3)
(1, - 3)
(- 1, - 3)

295.

The interval, inwhich the function f(x) = x²e^-x is an increasing function, will be

$(- \infty, \infty)$
(- 2, 0)
$(2, \infty)$
(0, 2)

296.

The minimum distance from the point (4, 2) to the parabola y² = 8x is

$\sqrt{2}$
$2 \sqrt{2}$
2
$3 \sqrt{2}$

297.
A particle moves so that the space described in time t is square root of a quadratic function of t. Then,

$v \propto (\frac{1}{s})$

$acceleration \propto \frac{1}{s^{3}}$

$acceleration \propto s^{3}$

None of these

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$acceleration \propto \frac{1}{s^{3}}$

$Let s = \sqrt{{at}^{2} + bt + c} \Rightarrow v = \frac{ds}{dt} = \frac{(2 a + b)}{2 \sqrt{{at}^{2} + bt + c}} . . . (i) \Rightarrow \frac{d^{2} s}{{dt}^{2}} = \frac{1}{2} \frac{[\sqrt{{at}^{2} + bt + c} . (2 a) - \frac{{(2 a + b)}^{2}}{2 \sqrt{{at}^{2} + bt + c}}]}{({at}^{2} + bt + c)} \Rightarrow \frac{d^{2} s}{{dt}^{2}} = \frac{(2 a) 2 ({at}^{2} + bt + c) - {(2 at + b)}^{2}}{4 {({at}^{2} + bt + c)}^{\frac{3}{2}}} \Rightarrow \frac{d^{2} s}{{dt}^{2}} = \frac{4 a^{2} t^{2} + 4 abt + 4 ac - 4 a^{2} t^{2} - b^{2} - 4 abt}{4 {({at}^{2} + bt + c)}^{\frac{3}{2}}} \Rightarrow \frac{d^{2} s}{{dt}^{2}} = \frac{ac}{{(\sqrt{{at}^{2} + bt + c})}^{3}} \Rightarrow \frac{d^{2} s}{{dt}^{2}} = \frac{ac}{s^{3}} \Rightarrow \frac{d^{2} s}{{dt}^{2}} \propto (\frac{1}{s}) acceleration \propto \frac{1}{s^{3}}$

298.

The points at which the tangent to the curve y = x³ + 5 is perpendicular to the line x + 3y = 2 are

(6, 1), (- 1, 4)
(1, 6), (1, 4)
(6, 1), (4, - 1)
(1, 6), (- 1, 4)

299.

Let f(x) = a - (x - 3)^8/9, then maxima of f(x) is

3
a - 3
a
None of these

300.

If the total cost C(x) in rupees associated with the production of x units of an item is given by C (x) = 3x³ - 2x² + x + 100. Then, the marginal change in cost, when x = 5, is

Previous Year Papers

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Multiple Choice Questions

297. A particle moves so that the space described in time t is square root of a quadratic function of t. Then, v ∝ 1s acceleration ∝ 1s3 acceleration ∝ s3 None of these

297.
A particle moves so that the space described in time t is square root of a quadratic function of t. Then,

$v \propto (\frac{1}{s})$

$acceleration \propto \frac{1}{s^{3}}$

$acceleration \propto s^{3}$

None of these