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The least positive integral value of $k$ for which there exists at least one line that is tangent to the graph of the curve $y = x^{3} - k x$ at one point and normal to the graph at another point is :

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Important Questions on Application of Derivatives

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangent at a point

P

, with parameter

t

, on the curve

x = 4 t^{2} + 3, y = 8 t^{3} - 1, t \in R,

meets the curve again at a point

Q

, then the coordinates of

Q

are :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

A mirror in the first quadrant is in the shape of hyperbola whose equation is

x y = 1 .

A light source in the second quadrant emits a beam of light that hits the mirror at the point

(2, \frac{1}{2})

If the reflected ray is parallel to the

y

-axis, the slope of the incident beam is-

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Consider

f (x) = \tan^{- 1} (\sqrt{\frac{1 + \sin x}{1 - \sin x}}), x \in (0, \frac{π}{2}) .

A normal to

y = f (x)

x = \frac{π}{6}

also passes through the point

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The tangent to the curve,

y = x e^{x^{2}}

passing through the point

(1, e)

also passes through the point:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Suppose the tangent to the parabola

y = x^{2} + p x + q

(0, 3)

has slope

- 1

. Then

p + q

equals

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangent to the curve,

y = x^{3} + a x - b

at the point

(1, - 5)

is perpendicular to the line,

- x + y + 4 = 0,

then which one of the following points lies on the curve?

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Let C be a curve given by

y (x) = 1 + \sqrt{4 x - 3}

x > \frac{3}{4}

. If

P

is a point on C, such that the tangent at

P

has slope

\frac{2}{3}

, then a point through which the normal at

P

passes, is :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of a common tangent to the curves,

y^{2} = 16 x

and

x y = - 4,

is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The normal to the curve

y (x - 2) (x - 3) = x + 6

at the point where the curve intersects the

y

-axis passes through the point:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The point on the curve

y = \sqrt{x - 1}

where the tangent is perpendicular to the line

2 x + y - 5 = 0

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangents on the ellipse

4 x^{2} + y^{2} = 8

at the points

(1, 2)

and

(a, b)

are perpendicular to each other, then

a^{2}

is equal to

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The distance from the origin, of the normal to the curve,

x = 2 \cos t + 2 t \sin t, y = 2 \sin t - 2 t \cos t a t t = \frac{π}{4},

is :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Let

S

be the set of all values of

x

for which the tangent to the curve

y = f (x) = x^{3} - x^{2} - 2 x

(x, y)

is parallel to the line segment joining the points

(1, f (1))

and

(- 1, f (- 1)),

then

S

is equal to

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Let the normal at a point

P

on the curve

y^{2} - 3 x^{2} + y + 10 = 0

intersect the

y

-axis at

(0, \frac{3}{2}) .

m

is the slope of the tangent at

P

to the curve, then

|m|

is equal to ___________.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangent to the curve

y = \frac{x}{x^{2} - 3}, x \in R, (x \neq \pm \sqrt{3}),

at a point

(α, β) \neq (0, 0)

on it is parallel to the line

2 x + 6 y - 11 = 0,

then:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of the tangent to

y = - 2 x^{2} + 3

x = 1

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The length of the perpendicular from the origin, on normal to the curve,

x^{2} + 2 x y - 3 y^{2} = 0

, at the point

(2, 2)

, is.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The tangent at the point

(2, - 2)

to the curve,

x^{2} y^{2} - 2 x = 4 (1 - y)

does not pass through the point:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of the tangent and normal to the ellipse

x^{2} + 2 y^{2} + 2 x - 4 y - 14 = 0

(2, - 1)

HARD

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of a normal to the curve,

\sin y = x \sin (\frac{π}{3} + y) at x = 0,

is:

The least positive integral value of k for which there exists at least one line that is tangent to the graph of the curve y=x3-kx at one point and normal to the graph at another point is :

Important Questions on Application of Derivatives

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The least positive integral value of $k$ for which there exists at least one line that is tangent to the graph of the curve $y = x^{3} - k x$ at one point and normal to the graph at another point is :