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JEE • MATHEMATICS • TANGENTS & NORMALS

Tangents & normals practice · Hard

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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.

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1 Hard

1 of 15

Analyzing +4 | -1

Let

C

be the curve

y^3-3 x y+2=0

. If

H

is the set of points on the curve

\mathrm{C}

where the tangent is parallel to

x

-axis and

V

is the set of points on

C

where the tangent is parallel to

y

-axis, then

View answer and explanation

Correct option: C

H=\phi, V=\{(1,1)\}

Option C (

H=\phi, V=\{(1,1)\}

) is correct because the tangent is parallel to the x-axis nowhere and parallel to the y-axis at (1,1).

2 Hard

2 of 15

Application +4 | -1

For the curve

b y^2=(x+a)^3

, the square of the subtangent is proportional to

View answer and explanation

Correct option: B

Subnormal

Option B (Subnormal) is correct because the square of the subtangent is directly proportional to the subnormal, as derived from the given curve equation.

3 Hard

3 of 15

Application +4 | -1

For the curve

\mathrm{y}=3 \sin \theta \cos \theta, \mathrm{x}=\mathrm{e}^\theta \sin \theta

0 \leq \theta \leq \pi

, the tangent is parallel to

x

-axis when

\theta

is:

View answer and explanation

Correct option: C

\frac{\pi}{4}

Option C (

\frac{{\pi}}{{4}}

) is correct because the tangent is parallel to the x-axis when

\frac{{dy}}{{dx}} = 0

, which occurs when

\theta = \frac{{\pi}}{{4}}

after differentiating and simplifying.

4 Hard

4 of 15

Analyzing +4 | -1

Tangent to the curve

y=e^x

drawn at the point

\left(c, e^c\right)

intersects the line joining the points

\left(c-1, e^{c-1}\right)

and

\left(c+1, e^{c+1}\right)

View answer and explanation

Correct option: A

on the left of

x=c

Option A (on the left of

x=c

) is correct because the x-coordinate of the intersection point is less than c, indicating it lies to the left.

5 Hard

5 of 15

Application +4 | -1

Let

{x^2} + {y^2} + Ax + By + C = 0

be a circle passing through (0, 6) and touching the parabola

y = {x^2}

at (2, 4). Then

A + C

is equal to ______.

View answer and explanation

Correct option: A

Option B (96) is correct because it is the solution to the system of equations derived from the given information about the arithmetic progression.

6 Hard

6 of 15

Application +4 | -1

If the area of the triangle, included between the axes and any tangent to the curve

x{y^n} = {a^{n + 1}}

is constant, then the value of

n

View answer and explanation

Correct option: C

Option C (1) is correct because when

n=1

, the area of the triangle formed by the tangent is constant.

7 Hard

7 of 15

Application +4 | -1

The equation of a curve is

y=f(x)

. The tangents at

(1, f(1)),(2, f(2))

and

(3, f(3))

make angles

\frac{\pi}{6}, \frac{\pi}{3}

and

\frac{\pi}{4}

respectively with the positive direction of the

x

-axis. Then, the value of

\int\limits_{2}^3 f^{\prime}(x) f^{\prime \prime}(x) d x+\int\limits_{1}^3 f^{\prime \prime}(x) d x

is equal to

View answer and explanation

Correct option: A

-\frac{1}{\sqrt{3}}

Option A (

-\frac{1}{\sqrt{3}}

) is correct because it is the result of evaluating the definite integral using the given derivative values.

8 Hard

8 of 15

Application +4 | -1

The sum of intercepts on the coordinate axes made by the tangent to the curve

\sqrt{x}+\sqrt{y}=\sqrt{a}

View answer and explanation

Correct option: A

a

Option A (

a

) is correct because the sum of the intercepts of the tangent to the curve is equal to

a

, as derived from the given equation.

9 Hard

9 of 15

Analyzing +4 | -1

If the tangent to the conic,

y-6=x^2

(2,10)

touches the circle,

x^2+y^2+8 x-2 y=k

(for some fixed

k

) at a point

(\alpha, \beta)

; then

(\alpha, \beta)

is :

View answer and explanation

Correct option: D

\left(-\frac{8}{17}, \frac{2}{17}\right)

Option D (

\left(-\frac{8}{17}, \frac{2}{17}\right)

) is correct. The tangent to the conic touches the circle at a point that can be found by equating the slopes and solving the system of equations.

10 Hard

10 of 15

Application +4 | -1

Tangent is drawn to ellipse

\frac{{{x^2}}}{{27}} + {y^2} = 1\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\,(where\,\theta \in \left( {0,\;\pi /2} \right))

. Then the value of

\theta

such that sum of intercepts on axes made by this tangent is minimum, is

View answer and explanation

Correct option: B

\frac{\pi }{6}

Option B

\frac{\pi }{6}

is correct because it minimizes the sum of the intercepts on the axes, found by setting the derivative of the sum equal to zero.

11 Hard

11 of 15

Application +4 | -1

The area of a triangle formed by a tangent to the curve

2 x y=a^2

and the coordinate axes, is

View answer and explanation

Correct option: B

a^2

Option B (

a^2

) is correct because the area of the triangle formed by the tangent and coordinate axes is calculated to be

a^2

12 Hard

12 of 15

Application +4 | -1

The length of normal to the curve

x=a(\theta+\sin \theta)

y=a(1-\cos \theta)

at the point

\theta=\pi / 2

View answer and explanation

Correct option: C

\sqrt{2} a

Option C (

\sqrt{2} a

) is the correct answer because it is the calculated length of the normal to the curve at the specified point.

13 Hard

13 of 15

Application +4 | -1

The equation of the one of the tangents to the curve

y = \cos \left( {x + y} \right),\; - 2\pi \le x \le 2\pi

that is parallel to the line

x + 2y = 0

, is

View answer and explanation

Correct option: B

x + 2\;y = \frac{\pi }{2}

Option D (

B,\;A,\;D,\;C

) is correct because it represents the increasing order of the lengths of subtangent, tangent, subnormal, and normal.

14 Hard

14 of 15

Application +4 | -1

The slope of the tangent to the curve

y=\int\limits_{x}^{x^2} \cos ^{-1} t^2 d t

x=2^{-1 / 4}

, is

View answer and explanation

Correct option: B

\left(\frac{\sqrt[4]{8}}{3}-\frac{1}{4}\right) \pi

Option B (

\left(\frac{\sqrt[4]{8}}{3}-\frac{1}{4}\right) \pi

) is correct because the slope of the tangent is found by differentiating the given integral using Leibniz's rule and evaluating it at

x=2^{-1/4}

15 Hard

15 of 15

Analyzing +4 | -1

The points of contact of the tangents drawn from the origin to the curve

y = \sin x

lie on the curve

View answer and explanation

Correct option: C

{x^2} - {y^2} = {x^2}{y^2}

The correct answer is C (

{x^2} - {y^2} = {x^2}{y^2}

) because it represents the locus of the points of contact of tangents drawn from the origin to the curve

y = \sin x

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