If $y=\cos ^{-1}(\cos x)$, then $y^{\prime}(x)$ is equal to

1 in $2^{\text {nd }}$ and $3^{\text {rd }}$ quadrant

-1 in $3^{\text {rd }}$ and $4^{\text {th }}$ quadrant

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JEE • MATHEMATICS • TANGENTS & NORMALS

Tangents & normals practice · Mixed difficulty

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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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Showing 1-30 of 281 questions

1 Easy

1 of 281

Analyzing +4 | -1

The complete set of value of '

a

' for which there exists at least one line that is tangent to the graph of the curve

y=x^3-a x

at one point and normal to the graph at another point is given by

View answer and explanation

Correct option: C

[4 / 3, \infty)

Option C (

[4 / 3, \infty)

) is correct because it represents the complete set of values of 'a' for which there exists at least one line that is tangent to the graph of the curve at one point and normal at another.

2 Easy

2 of 281

Analyzing +4 | -1

The point of intersection of the tangents drawn to the curve

x^2 y=1-y

at the points where it is met by the curve

x y=1-y

, is given by

View answer and explanation

Correct option: C

(0,1)

Option C ((0,1)) is correct because it is the point of intersection of the tangents to the curve at the points where it intersects another curve.

3 Hard

3 of 281

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).

4 Easy

4 of 281

Application +4 | -1

The curve

y-e^{x y}+x=0

has a vertical tangent at the point:

View answer and explanation

Correct option: D

(1,0)

Option D ((1,0)) is correct because at this point the derivative dx/dy equals zero, indicating a vertical tangent.

5 Easy

5 of 281

Understanding +4 | -1

If normal to the curve

y = f\left( x \right)

is parallel to

x

-axis, then

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because if the normal to the curve is parallel to the x-axis, then the tangent to the curve is parallel to the y-axis, implying that dx/dy = 0.

6 Easy

6 of 281

Application +4 | -1

The tangents to the curve

x = a\left( {\theta - \sin \theta } \right),\;y = a\left( {1 + \cos \theta } \right)

at the points

\theta = \left( {2k + 1} \right)\pi ,\;k \in Z

are parallel to:

View answer and explanation

Correct option: C

y = 0

Option C (

y = 0

) is correct because the slope

\frac{{dy}}{{dx}}

is 0 at

\theta = (2k+1)\pi

, indicating a horizontal tangent.

7 Easy

7 of 281

Analyzing +4 | -1

The angle of intersection between the curves

{x^2} = 4\left( {y + 1} \right)

and

{x^2} = - 4\left( {y + 1} \right)

View answer and explanation

Correct option: C

Option C (0) is correct because the slopes of both curves at the point of intersection are 0, indicating a tangential intersection.

8 Medium

8 of 281

Analyzing +4 | -1

The angle between the tangents at those points on the curve

x = {t^2} + 1

and

y = {t^2} - t - 6

where it meets

x

-axis is

View answer and explanation

Correct option: C

\pm {\tan ^{ - 1}}\left( {\frac{{10}}{{49}}} \right)

Option C (

\pm {\tan ^{ - 1}}\left( {\frac{{10}}{{49}}} \right)

) is correct because it represents the angle between the tangents at the points where the curve intersects the x-axis.

9 Easy

9 of 281

Application +4 | -1

The tangent to the curve

y = 2{x^2} - x + 1

at a point

P

is parallel to

y = 3x + 4

, then the coordinates of Pare

View answer and explanation

Correct option: B

(1, 2)

Option B (1, 2) is correct because it is the point where the slope of the tangent to the curve equals the slope of the given line.

10 Medium

10 of 281

Application +4 | -1

The point

(0,5)

is closest to the curve

x^{2}=2 y

View answer and explanation

Correct option: D

(2 \sqrt{2}, 4)

Option D (

(2 \sqrt{{2}}, 4)

) is correct because by minimizing the distance between the point (0,5) and a general point on the curve, we find the closest point.

11 Medium

11 of 281

Application +4 | -1

An equation of the tangent to the curve

y=x^4

from the point

(2,0)

not on the curve is

View answer and explanation

Correct option: A

y=0

Option A (

y=0

) is the correct answer because it represents a tangent line to the curve

y=x^4

passing through the point (2,0).

12 Easy

12 of 281

Application +4 | -1

The equation of the tangent to curve

y=b e^{-x / \alpha}

at the point where it crosses

y

-axis is

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it represents the equation of the tangent to the curve

y = be^{-x/a}

at the point where it crosses the y-axis (0, b).

13 Easy

13 of 281

Application +4 | -1

The abscissa of the points, where the tangent to curve

y = {x^3} - 3{x^2} - 9x + 5

is parallel to

x

-axis , are

View answer and explanation

Correct option: D

x = - 1\;and\;3

Option D (

x = - 1\;and\;3

) is correct because these are the x-coordinates where the derivative of the curve is zero, indicating a horizontal tangent.

14 Easy

14 of 281

Analyzing +4 | -1

Let

\theta

be the acute angle between the tangents to the ellipse

\frac{{{x^2}}}{9} + \frac{{{y^2}}}{1} = 1

and the circle

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

at their point of intersection in the first quadrant. Then,

\tan \,\,\theta

is equal to

View answer and explanation

Correct option: B

\frac{2}{{\sqrt 3 }}

Option B (

\frac{2}{{\sqrt 3 }}

) is correct because calculating the slopes of the tangents and using the angle between two lines formula gives

\tan \theta = \frac{2}{{\sqrt 3 }}

15 Easy

15 of 281

Application +4 | -1

The equation of the tangent to the curve

y = x + \frac{4}{{{x^2}}}

that is parallel to the x-axis, is

View answer and explanation

Correct option: D

Y = 3

Option D (

Y = 3

) is correct because it represents the equation of the tangent line that is parallel to the x-axis, found by setting the derivative to zero and solving for y.

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16 Easy

16 of 281

Application +4 | -1

f(x)=\log _x(\ln x)

, then at

x=e, f^{\prime}(x)

equals-

View answer and explanation

Correct option: D

1 / e

Option D (1/e) is correct because after applying the change of base formula and differentiating, the value of f'(e) is 1/e.

17 Medium

17 of 281

Application +4 | -1

The equation of the normal to the curve

y = {e^{ - 2\left| x \right|}}

at the point where the curve cuts the line

x = 1/2

, is

View answer and explanation

Correct option: B

2e\left( {ex - 2y} \right) = {e^2} - 4

Option B (

2e(ex - 2y) = {e^2} - 4

) is correct because it is the equation of the normal to the curve at the specified point, found by calculating the derivative and using the point-slope form.

18 Easy

18 of 281

Application +4 | -1

The angle of intersection of the curves

y = {x^2},\;6y = 7 - {x^3}

at (1, 1) is

View answer and explanation

Correct option: C

\frac{\pi }{2}

Option C (

\frac{\pi }{2}

) is correct because the product of the slopes of the tangents at the point of intersection is -1, indicating that the curves intersect at right angles.

19 Medium

19 of 281

Application +4 | -1

If the two curves

y=a^x

and

y=b^x

intersect at an angle

\alpha

, then tan

\alpha

equals -

View answer and explanation

Correct option: A

\frac{\log b-\log a}{1+\log a \log b}

Option A (

\frac{{\log b - \log a}}{{1 + \log a \log b}}

) is correct because the tangent of the angle between the two curves is found by calculating the derivatives of the curves at their point of intersection and then using the formula for the tangent of the angle between two lines.

20 Easy

20 of 281

Analyzing +4 | -1

y=\cos ^{-1}(\cos x)

, then

y^{\prime}(x)

is equal to

View answer and explanation

Correct option: A

1 for all

x

Option A (1 for all

x

) is correct because the derivative of the inverse cosine of cosine x simplifies to 1.

21 Easy

21 of 281

Application +4 | -1

The equation of the normal line to the curve

y = x{\log _e}x

parallel to

2x - 2y + 3 = 0

View answer and explanation

Correct option: C

x - y = 3{e^{ - 2}}

Option C (

x - y = 3{e^{ - 2}}

) is correct because it represents the normal line to the curve that is parallel to the given line, found by equating slopes and solving for the point of tangency.

22 Easy

22 of 281

Analyzing +4 | -1

A curve with equation of the form

y=a x^4+b x^3+c+c x+d

has zero gradient at the point

(0,1)

and also touches the

x

-axis at the point

(-1,0)

then the values of the

x

for which the curve has a negative gradient are:

View answer and explanation

Correct option: C

x < -1

Option C (

x < -1

) is correct because it identifies the range of x-values where the derivative of the curve is negative, indicating a negative gradient.

23 Easy

23 of 281

Understanding +4 | -1

The normal to a given curve is parallel to

x

-axis, if

View answer and explanation

Correct option: C

\frac{d x}{d y}=0

Option C (

\frac{d x}{d y}=0

) is correct because if the normal is parallel to the x-axis, the tangent is perpendicular to the x-axis, implying

\frac{dx}{dy} = 0

24 Easy

24 of 281

Application +4 | -1

The slope of tangent to the curve

x=t^2+3 t-8

y=2 t^2-2 t-5

at the point

(2,-1)

View answer and explanation

Correct option: B

\frac{6}{7}

Option B (

\frac{6}{7}

) is correct because it is the value of

\frac{dy}{dx}

at t=2, which is found by solving for t and then substituting into the derivative.

25 Easy

25 of 281

Application +4 | -1

At any point of a curve

\sqrt{\frac{\text { subnormal }}{\text { subtangent }}}

is equal to

View answer and explanation

Correct option: C

Slope of the tangent at that point

Option C (Slope of the tangent at that point) is correct because the square root of the ratio of subnormal to subtangent is equal to the slope of the tangent.

26 Easy

26 of 281

Application +4 | -1

Let

C

be a curve given by

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

x > \frac{3}{4}

. If

P

is a point on

\mathrm{C}

such that the tangent at

P

has slope

\frac{2}{3}

, then a point through which the normal at

P

passes, is

View answer and explanation

Correct option: B

(1,7)

Option B (1,7) is correct because the normal at P(3,4) passes through the point (1,7).

27 Medium

27 of 281

Application +4 | -1

If the curve

y = a{x^3} + b{x^2} + cx

is inclined at

{45^0}

x

-axis at (0, 0) but touches

x

-axis at (1, 0), then

View answer and explanation

Correct option: A

a = 1,\;b = - 2,\;c = 1

Option A (

a = 1,\;b = - 2,\;c = 1

) is correct because these values satisfy the given conditions of the curve's inclination and tangency.

28 Medium

28 of 281

Application +4 | -1

If the tangent at a point on the ellipse

\frac{x^2}{27}+\frac{y^2}{3}=1

meets the coordinate axes at

A

and

B

and,

O

is the origin, the minimum area (in sq. units). of the triangle

O A B

, is

View answer and explanation

Correct option: A

Option A (9) is correct because the minimum area of the triangle OAB is found to be 9 square units when sin(2*theta) is maximized.

29 Easy

29 of 281

Application +4 | -1

The equation of one of the tangents to the curve

y=\cos (x+y),-2 \pi \leq x \leq 2 \pi

that is parallel to the line

x+2 y=0

, is

View answer and explanation

Correct option: B

x+2 y=\pi / 2

The correct answer is

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

because it represents the tangent to the curve

y = \cos(x+y)

that is parallel to the line

x+2y=0

30 Hard

30 of 281

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.

Revision notes and key takeaways

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Key Takeaways

✓ This set contains 281 single-correct-answer MCQs on TANGENTS & NORMALS.
✓ Try each question before opening its answer. Read the explanation where one is available.
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