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 · Easy

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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 164 questions

1 Easy

1 of 164

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 164

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 Easy

3 of 164

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.

4 Easy

4 of 164

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.

5 Easy

5 of 164

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.

6 Easy

6 of 164

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.

7 Easy

7 of 164

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.

8 Easy

8 of 164

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).

9 Easy

9 of 164

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.

10 Easy

10 of 164

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 }}

11 Easy

11 of 164

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.

12 Easy

12 of 164

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.

13 Easy

13 of 164

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.

14 Easy

14 of 164

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.

15 Easy

15 of 164

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.

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16 Easy

16 of 164

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.

17 Easy

17 of 164

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

18 Easy

18 of 164

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.

19 Easy

19 of 164

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.

20 Easy

20 of 164

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).

21 Easy

21 of 164

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

22 Easy

22 of 164

Application +4 | -1

The distance between the origin and the tangent to the curve

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

drawn at the point

x=0

View answer and explanation

Correct option: A

\frac{1}{\sqrt{5}}

Option A (

\frac{1}{\sqrt{5}}

) is correct because it represents the perpendicular distance from the origin to the tangent line calculated at x=0.

23 Easy

23 of 164

Application +4 | -1

he length of the subnormal to the parabola

{y^2} = 4\;ax

at any point is equal to

View answer and explanation

Correct option: D

2a

Option D (

2a

) is correct because the length of the subnormal to the parabola

{y^2} = 4ax

is constant and equal to

2a

24 Easy

24 of 164

Application +4 | -1

If the curve

y = a{x^2} + bx + c,

x \in R,

passes through the point

(1,\,\,2)

and the tangent line to this curve at origin is

y = x,

then the possible values of

a,\,\,b,\,\,c

are

View answer and explanation

Correct option: C

a = 1,\,\,\,\,b = 1,\,\,\,c = 0

Option C (

a = 1,\,\,\,b = 1,\,\,\,c = 0

) is the correct set of values because the curve passes through (1,2) and has a tangent line y=x at the origin.

25 Easy

25 of 164

Application +4 | -1

The slope of the tangent to the curve

y=\int\limits_{0}^x \frac{1}{1+t^3} d t

at the point where

x=1

, is

View answer and explanation

Correct option: A

\frac{1}{2}

Option A (

\frac{1}{2}

) is correct because the slope of the tangent is found by differentiating the given integral and evaluating it at x=1.

26 Easy

26 of 164

Application +4 | -1

The point on the curve

{y^2} = x

, the tangent at which makes an angle

{45^0}

with

x

-axis is

View answer and explanation

Correct option: A

\left( {\frac{1}{4},\frac{1}{2}} \right)

Option A (

\left( {\frac{1}{4},\frac{1}{2}} \right)

) is correct because it represents the point on the curve where the tangent has a slope of 1, corresponding to a 45-degree angle with the x-axis.

27 Easy

27 of 164

Analyzing +4 | -1

The point in the interval

\left[ {0,\;2\pi } \right]

where

f\left( x \right) = {e^x}\sin x

has maximum slope is

View answer and explanation

Correct option: B

\frac{\pi }{2}

Option B (

\frac{\pi }{2}

) is correct because the maximum slope occurs where the second derivative is zero and the third derivative is negative.

28 Easy

28 of 164

Analyzing +4 | -1

If the line

ax + by + c = 0

is a tangent to the curve

xy = 4,

then

View answer and explanation

Correct option: C

a < 0,b < 0

Option C (

a < 0, b < 0

) is correct because for the line to be tangent to the curve, the product of a and b must be positive, and since the slope is negative, both a and b must be negative.

29 Easy

29 of 164

Application +4 | -1

The equation of the normal to the curve

y=e^{-2|x|}

at the point where the curve cuts the line

x=-\frac{1}{2}

, is

View answer and explanation

Correct option: A

2 e(e x+2 y)=4-e^2

Option A (

2 e(e x+2 y)=4-e^2

) is correct because it represents the equation of the normal to the curve at the given point, derived using the derivative and the negative reciprocal of the slope.

30 Easy

30 of 164

Application +4 | -1

The equation of the tangent to the curve

{x^2} - 2xy + {y^2} + 2x + y - 6 = 0

at (2,2) is

View answer and explanation

Correct option: A

2x + y - 6 = 0

Option A (

2x + y - 6 = 0

) is correct because it represents the tangent line to the curve at the point (2,2) with the calculated slope.

Revision notes and key takeaways

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

Key Takeaways

✓ This set contains 164 single-correct-answer MCQs on TANGENTS & NORMALS.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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