For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows: Statement I: If g x is a differentiable function g 1 ≠ 0 , g - 1 ≠ 0 and Rolle's theorem is not applicable f x = x 2 - 1 g x in - 1, 1 then g x has atleast one root in - 1, 1 . Statement II: If f a = f b , then Rolle's theorem is applicable for x ∈ a , b .

Statement I is true. Statement II is false.

Statement I is true. Statement II is also true; Statement II is the correct explanation of Statement I.

Statement I is true. Statement II is also true; Statement II is not the correct explanation of Statement I.

Statement I is false, statement II is true.

The parabola yx2pxq intercepts the straight line y2x3 at a point with abscissa 1If the distance between the vertex of the parabola and the xaxis is least then

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 33

The abscissa of the point on the curve

\sqrt{x y} = a + x

, the tangent at which cuts off equal intercepts from the coordinate axes is

(a > 0)

:

EASY

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 34

If the side of a triangle vary slightly $(d a, d b, d c)$ in such a way that its circumradius remains constant, then $\frac{d a}{\cos A} + \frac{d b}{\cos B} + \frac{d c}{\cos C}$ is equal to

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 35

Let

f (x)

satisfy the requirements of Lagrange’s mean value theorem in

[0, 1]

,

f (0) = 0

and

f^{'} (x) \leq 1 - x, \forall x \in (0, 1)

, then:

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 36

For function

f (x) = \frac{l n x}{x}

, which of the following statements are true?

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 37

Equation of a line which is tangent to both the curves

y = x^{2} + 1

and

y = - x^{2}

is

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 38

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Statement I: If

g (x)

is a differentiable function

g (1) \neq 0, g (- 1) \neq 0

and Rolle's theorem is not applicable

f (x) = \frac{x^{2} - 1}{g (x)}

in

[- 1, 1]

then

g (x)

has atleast one root in

(- 1, 1)

.
Statement II: If

f (a) = f (b)

, then Rolle's theorem is applicable for

x \in (a, b) .

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 39

For the following questions, choose the correct answer from the options

(a), (b), (c)

and

(d)

defined as follows:
Statement I: Shortest distance between

|x| + |y| = 2

and

x^{2} + y^{2} = 16

is

4 - \sqrt{2}

.
Statement II: Shortest distance between the two smooth curves lies along the common normal.

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 40

For the following questions, choose the correct answer from the options (a), (b), (c) and (d) defined as follows:
Statement I: If

α_{1}, α_{2}, α_{3}, \dots, α_{n}

the

n

real roots of a polynomial equation of

n^{th}

degree with real coefficients such that sum of the roots taken

r (1 \leq r \leq n)

at a time is positive,
Statement II: The number of times sign of coefficients change while going left to right of a polynomial equation is the number of maximum positive roots.

The parabola $y = x^{2} + p x + q$ intercepts the straight line $y = 2 x - 3$ at a point with abscissa $1 .$ If the distance between the vertex of the parabola and the $x -$ axis is least, then:

Important Questions on dy/dx as a Rate Measurer & Tangents, Normals

Learn and Prepare for any exam you want !

The parabola y=x2+px+q intercepts the straight line y=2x-3 at a point with abscissa 1. If the distance between the vertex of the parabola and the x-axis is least, then:

Important Questions on dy/dx as a Rate Measurer & Tangents, Normals

Learn and Prepare for any exam you want !

The parabola $y = x^{2} + p x + q$ intercepts the straight line $y = 2 x - 3$ at a point with abscissa $1 .$ If the distance between the vertex of the parabola and the $x -$ axis is least, then: