# RD Sharma Solution CLass 12 Mathematics Chapter 16 Tangents and Normals

Question 1
Solution 1

Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27

Question 28
Solution 28

Question 29
Solution 29
Question 30
Solution 30

## Chapter 16 – Tangents and Normals Exercise Ex. 16.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12

Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15

Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19

Find the equations of
the tangent and the normal to the given curves at the indicated points:

Solution 19

Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22

Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26

Find the equations of the tangent
and the normal to the following curves at the indicated points:

X = 3 cosθ
– cos3θ , y = 3 sinθ
– sin3θ

Solution 26

Question 27
Solution 27
Question 28
Solution 28

Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33

Question 34
Solution 34

Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37

Question 38
Solution 38
Question 39
Solution 39
Question 40

Solution 40

Question 41

Find the equation of the tangents to
the curve
3x2 – y2 = 8, which passes through the point (4/3, 0).

Solution 41

## Chapter 16 – Tangents and Normals Exercise Ex. 16.3

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4

Question 5
Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9

Find the angle of intersection of
the following curves:

Y = 4 – x2 and y = x2

Solution 9

Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16

Solution 16

Question 17

Solution 17

Question 18

Prove that the curves xy =
4 and x2 + y2 = 8 touch each other.

Solution 18

Question 19

Prove that the curves y2 = 4x and x2
+ y2 – 6x + 1 = 0 touch each other at the point (1, 2)

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

## Chapter 16 – Tangents and Normals Exercise MCQ

Question 1

The equation to the normal to the curve y = sin x at (0,0) is

1. x = 0
2. y = 0
3. x + y = 0
4. x – y = 0
Solution 1

Correct option: (c)

Question 2

The equation of the normal to the curve y = x + sin x cos x at x = π/2 is

1. x = 2
2. x = π
3. x + π = 0
4. 2x = π
Solution 2

Correct option: (d)

Question 3

The equation of the normal to the curve y = x (2-x) at the point (2,0) is

1. x – 2y = 2
2. x – 2y + 2 = 0
3. 2x + y = 4
4. 2x + y – 4 = 0
Solution 3

Correct option: (a)

Question 4

The point on the curve y2 = x where tangent makes 45° angles with x-axis is

1. (1/2,1/4)
2. (1/4,1/2)
3. (4,2)
4. (1,1)
Solution 4

Correct option: (b)

Question 5

If the tangent to the curve x = at2, y=2at is perpendicular to x-axis , then its point of contact is

1. (a, a)
2. (0, a)
3. (0, 0)
4. (a, 0)
Solution 5

Correct option: (c)

Question 6

The point on the curve y = x2 – 3x + 2 where tangent is perpendicular to y = x is

1. (0,2)
2. (1,0)
3. (-1,6)
4. (2,-2)
Solution 6

Correct option: (b)

Question 7

The point on the curve y2 = x where tangent makes 45° angle with x-axis is

1. (1/2,1/4)
2. (1/4,1/2)
3. (4,2)
4. (1,1)
Solution 7

Correct option:(b)

Question 8

The point on the curve y = 12x – x2 where the slope of the tangent is zero will be

1. (0,0)
2. (2,16)
3. (3,9)
4. (6,36)
Solution 8

Correct option: (d)

Question 9

The angle between the curves y2 = x and x2 = y at (1,1) is

Solution 9

Correct option: (b)

Question 10

The equation of the normal to the curve 3x2 – y2 = 8 which is parallel to x + 3y = 8 is

1. x – 3y = 8
2. x – 3y + 8 = 0
3. x + 3y ± 8 = 0
4. x + 3y = 0
Solution 10

Correct option: (c)

Question 11

The equation of tangent at those points where the curve y = x2 – 3x + 2 meets x-axis are

1. x – y + 2 = 0 = x – y – 1
2. x + y – 1 = 0 = x – y – 2
3. x – y – 1 = 0 = x – y
4. x – y = 0 = x + y
Solution 11

Correct option: (b)

Question 12

The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at point (2,-1) is

1. 22 /7
2. 6/7
3. -6
4. 7/6
Solution 12

Correct option: (b)

Question 13

The what points the slope of the tangent to the curve x2 + y2 – 2x – 3 = 0 is zero

1. (3,0), (-1,0)
2. (3,0), (1,2)
3. (-1,0) , (1,2)
4. (1,2), (1,-2)
Solution 13

Correct option: (d)

Question 14

The angle of intersection of the curves xy = a2 and x2 – y2 = 2a2 is

1. 45°
2. 90°
3. 30°
Solution 14

Correct option: (c)

Question 15

If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1,1), then a is equal to

1. 1
2. -6
3. 6
4. 0
Solution 15

Correct option: (c)

Question 16

If the line y = x touches the curve y = x2 + bx + c at a point (1,1) then

1. b = 1, c = 2
2. b =-1, c = 1
3. b = 2, c = 1
4. b = -2, c = 1
Solution 16

Correct option: (b)

Question 17

The slope of the tangent to the curve x = 3t2 + 1, y = t3-1 at x = 1 is

1. 1/2
2. 0
3. -2
Solution 17

Correct option: (b)

Question 18

The curves y = aex and y = be-x cut orthogonally, if

1. a = b
2. a = -b
3. ab = 1
4. ab =2
Solution 18

Correct option: (c)

Question 19

The equation of the normal to the curve x = a cos3θ, y = a sin3θ at the point θ = π/4 is

1. x = 0
2. y = 0
3. x = y
4. x + y = a
Solution 19

Correct option: (c)

Question 20

If the curves y = 2 ex and y = ae-x intersect orthogonally, then a =

1. 1/2
2. -1/2
3. 2
4. 2e2
Solution 20

Correct option: (a)

Question 21

The point on the curve y = 6x – x2 at which the tangent to the curve is inclined at π/4 to the line x + y = 0

1. (-3,-27)
2. (3,9)
3. 7/2, 35/4
4. (0,0)
Solution 21

Correct option: (b)

Question 22

The angle of intersection of the parabolas y2 = 4 ax and x2 = 4ay at the origin is

1. π/6
2. π/3
3. π/2
4. π/6
Solution 22

Correct option: (c)

Question 23

The angle of intersection of the curve y = 2 sin2 x and y = cos 2 x at x

1. π/4
2. π /2
3. π /3
4. π /6
Solution 23

Correct option: (c)

Question 24

Any tangent to the curve y = 2x7 + 3x + 5

1. is parallel to x-axis
2. is parallel to y -axis
3. makes an acute angle with x- axis
4. makes an obtuse angle with x -axis
Solution 24

Correct option: (c)

Question 25

The point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts with the axis is

1. (4, ±8/3)
2. (-4, 8/3)
3. (-4,-8/3)
4. (8/3,4)
Solution 25

Correct option: (a)

Question 26

The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,-1) is

1. 22/7
2. 6/7
3. 7/6
4. -6/7
Solution 26

Correct option: (b)

Question 27

The line y = mx + 1 is a tangent to the curve y2 = 4x, if the value of m is

1. 1
2. 2
3. 3
4. 1/2
Solution 27

Correct option: (a)

Question 28

The normal at the point (1,1) on the curve 2y + x2 = 3 is

1. x + y = 0
2. x – y = 0
3. x + y +1 = 0
4. x – y = 1
Solution 28

Correct option: (b)

Question 29

The normal to the curve x2 = 4y passing through (1,2) is

1. x + y = 3
2. x – y = 3
3. x + y = 1
4. x – y = 1
Solution 29

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