If $y=f(x)$ be the equation of the line touching the line $y=2 x+3$ at $x=2$, then

$f(2\rangle+f^{\prime}(2)+f^{\prime \prime}(2)=2$

The angle between the curves $y = \sin x\,and\,\;y = \cos x$ is

${\tan ^{ - 1}}\left( {3\sqrt 2 } \right)$

${\tan ^{ - 1}}\left( {3\sqrt 3 } \right)$

${\tan ^{ - 1}}\left( {5\sqrt 2 } \right)$

Back to TANGENTS & NORMALS

Your Progress

0 / 164 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • TANGENTS & NORMALS

Tangents & normals practice · Easy

Choose an answer to check it, or open “View answer” to study the explanation. Sign in to save progress to your account.

Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

Take a chapter quiz →

Showing 151-164 of 164 questions

Preparing practice and saved progress…

151 Easy

151 of 164

Understanding +4 | -1

At any point of a curve (subtangent)

\times

(subnormal) is equal to the square of the

View answer and explanation

Correct option: D

Ordinate of that point

Option A (1) is correct because as x approaches 4 from the right, (x-4) is positive, so the absolute value simplifies to (x-4), and the fraction equals 1.

152 Easy

152 of 164

Analyzing +4 | -1

A tangent to the curve

y=\int\limits_{0}^x|x| d t

, which is parallel to the line

y=x

, cuts off an intercept from the

y

-axis is equal to

View answer and explanation

Correct option: B

\frac{-1}{2}, \frac{1}{2}

Option B (

\frac{{-1}}{{2}}, \frac{{1}}{{2}}

) is correct because it represents the y-intercepts of the tangent lines parallel to y=x.

153 Easy

153 of 164

Application +4 | -1

The point on the curve

y=12 x-x^2

where the tangent is parallel to

x

-axis, is

View answer and explanation

Correct option: D

None of these

The correct answer is None of these because the point where the tangent is parallel to the x-axis is (6, 36), which is not among the options.

154 Easy

154 of 164

Application +4 | -1

The curve

y=a x^3+b x^2+c x+8

touches

x

-axis at

P(-2,0)

and cuts the

y

-axis at point

Q

where its gradient is 3 . The values of

a, b, c

are respectively

View answer and explanation

Correct option: D

-1 / 4,0,3

Option D (-1/4, 0, 3) is correct because these values satisfy the given conditions of the curve touching the x-axis and having a specific gradient at the y-axis intercept.

155 Easy

155 of 164

Application +4 | -1

y=f(x)

be the equation of the line touching the line

y=2 x+3

x=2

, then

View answer and explanation

Correct option: B

2 f(2)=7 f^{\prime}(2)

Option B (

2 f(2)=7 f^{\prime}(2)

) is correct because, since the curve touches the line at x=2, the derivative of f(x) at x=2 is equal to the slope of the line, and f(2) is equal to the y-value of the line at x=2.

156 Easy

156 of 164

Application +4 | -1

The length of the subtangent at any point

\left( {{x_1},\;{y_1}} \right)

on the curve

y = {a^x},\;(a > 0)

View answer and explanation

Correct option: B

\frac{1}{{\log a}}

Option B (

\frac{1}{{\log a}}

) is correct because it represents the length of the subtangent calculated using the derivative of the curve and the formula for subtangent length.

157 Easy

157 of 164

Application +4 | -1

The length of the tangent to the curve

x = a{\sin ^3}t,y = a{\cos ^3}t(a > 0)

at any arbitrary is

View answer and explanation

Correct option: A

a{\cos ^2}t

Option A (

a{\cos ^2}t

) is correct because it's the length of the tangent calculated using derivatives and the formula for tangent length.

158 Easy

158 of 164

Application +4 | -1

The angle between the curves

y = \sin x\,and\,\;y = \cos x

View answer and explanation

Correct option: A

{\tan ^{ - 1}}\left( {2\sqrt 2 } \right)

Option A (

{\tan ^{ - 1}}( {2\sqrt 2 } )

) is correct because it represents the angle between the two curves, calculated using the slopes of the tangent lines at their intersection point.

159 Easy

159 of 164

Application +4 | -1

The normal to the curve

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

, at

(1,1)

View answer and explanation

Correct option: B

Meets the curve again the fourth quadrant.

The correct answer is 'Meets the curve again the fourth quadrant' because the normal at (1,1) intersects the curve again at (3,-1), which lies in the fourth quadrant.

160 Easy

160 of 164

Application +4 | -1

m

is the slope of the tangent to the curve

e^y=1+x^2

, then

View answer and explanation

Correct option: D

|m| \leq 1

Option D (

|m| \leq 1

) is correct because the slope of the tangent,

|m|

, is always less than or equal to 1, as derived using AM-GM inequality.

161 Easy

161 of 164

Analyzing +4 | -1

The two curves

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

and

3 x^2 y-y^3=2

View answer and explanation

Correct option: A

Cut at right angles

Option A (Cut at right angles) is correct because the product of the slopes of the tangents to the two curves at their point of intersection is -1.

162 Easy

162 of 164

Analyzing +4 | -1

At which point the line

\frac{x}{a}+\frac{y}{b}=1

, touches the curve

y=b e^{-x i a}

View answer and explanation

Correct option: C

(0, b)

Option C ((0, b)) is correct because it is the point where the line and the curve have the same slope, indicating tangency.

163 Easy

163 of 164

Application +4 | -1

The point on the curve

y=(x-1)(x-2)

at which the tangent makes an angle of

135^{\circ}

with the positive direction of

x

-axis has coordinates

View answer and explanation

Correct option: A

(1,0)

Option A ((1,0)) is correct because at this point, the derivative of the curve equals -1, which corresponds to a tangent angle of 135 degrees with the x-axis.

164 Easy

164 of 164

Analyzing +4 | -1

If the parabolas

{y^2} = 4x

and

{x^2} = 32y

intersect at

\left( {16,8} \right)

at an angle

\theta ,

then

\theta

is equal to

View answer and explanation

Correct option: A

{\tan ^{ - 1}}\left( {3/5} \right)

Option A (

{\tan ^{ - 1}}\left( {3/5} \right)

) is correct because it is the angle between the tangents to the parabolas at the point of intersection, calculated using the slopes of the tangents.

Revision notes and key takeaways

Review TANGENTS & NORMALS in your textbook or class notes, then use this set to check your understanding.

Showing 151-164 of 164 questions

Your Progress

Tangents & normals practice · Easy

Continue Your Practice

Your practice