RD SHARMA Solutions for Class 12-science Maths Chapter 16 - Tangents and Normals

Chapter 16 - Tangents and Normals Exercise MCQ

Question 30

The abscissa of the point on the curve   the normal at which passes through the origin is

a. 1

b.   

c. 2

d.   

Solution 30

Let (p, q) be the point on the given curve   at which the normal passes through the origin.

Therefore,

  

 

Now, the equation of normal at (p, q) is passing through origin.

Therefore,

  

As p = 1 satisfies the equation, therefore the abscissa is 1.

Question 31

Two curves   

a. touch each other

b. cut at right angle

c. cut at an angle

d. cut at an angle

Solution 31

The curves are

  … (i)

  … (ii)

Differentiating (i) w.r.t x, we get

  

Differentiating (ii) w.r.t x, we get

  

Now,

So, both the curves are cut each other at right angle.

Question 32

The tangent to the curve   makes with x-axis an angle

a. 0

b.   

c.   

d.   

Solution 32

Given:

Differentiating 'x' and 'y' w.r.t t, we get

  

Dividing (ii) by (i), we get

  

Hence, the tangent to the curve makes an angle of   with x-axis.

Question 33

The tangent to the curve   at the point (0, 1) meets x-axis at:

a. (0, 1)

b.   

c. (2, 0)

d. (0, 2)

Solution 33

Given curve is

  

Therefore, slope of tangent is 2.

The equation of tangent is y - 1 = 2(x - 0)

i.e. y = 2x + 1

This equation of tangent meets x-axis when y = 0

  

Thus, the required point is

Question 34

The equation of tangent to the curve   where it crosses x-axis is

a. x + 5y = 2

b. x - 5y = 2

c. 5x - y = 2

d. 5x + y = 2

Solution 34

Given curve is

  

The curve crosses x-axis when y = 0

Therefore, x = 2

So, the tangent touches the curve at point (2, 0).

  

The equation of tangent at (2, 0) is

  

Question 35

The points at which the tangents to the curve   are parallel to x-axis are

a. (2, -2) (-2, -34)

b. (2, 34) (-2, 0)

c. (0, 34) (-2, 0)

d. (2, 2) (-2, 34)

Solution 35

Given curve is

  

As the tangents are parallel to x-axis, their slope will be 0.

  

When x = 2, y = 23 - 12 × 2 + 18 = 2

When x = -2, y = (-2)3 - 12(-2) + 18 = 34

So, the points are (2, 2) and (-2, 34).

Question 36

The curve   has at (0, 0)

a. a vertical tangent

b. a horizontal tangent

c. an oblique tangent

d. no tangent

Solution 36

Given curve is

  

At (0, 0), we have

  

Thus, the given curve   has vertical tangent, which is parallel to y-axis, at (0, 0).

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

 

NOTE: Options are incorrect.

Chapter 16 - Tangents and Normals Exercise Ex. 16.1

Question 1(i)

Solution 1(i)


Question 1(ii)
Solution 1(ii)
Question 1(iii)
Solution 1(iii)
Question 1(iv)
Solution 1(iv)
Question 1(v)

Find the slopes of the tangent and the normal to the curve x = a(θ - sinθ), y =a(1 + cos θ) at θ = begin mathsize 12px style negative straight pi over 2 end style.

Solution 1(v)

Question 1(vi)
Solution 1(vi)
Question 1(vii)
Solution 1(vii)
Question 1(viii)
Solution 1(viii)
Question 1(ix)
Solution 1(ix)
Question 1(x)
Solution 1(x)
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

Chapter 16 - Tangents and Normals Exercise Ex. 16.2

Question 1
Solution 1
Question 2
Solution 2
Question 3(i)
Solution 3(i)
Question 3(ii)

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

y=x4 - 6x3 + 13x2 - 10x + 5 at (x = 1)

Solution 3(ii)

Question 3(iii)
Solution 3(iii)
Question 3(iv)
Solution 3(iv)
Question 3(v)
Solution 3(v)
Question 3(vi)
Solution 3(vi)
Question 3(vii)
Solution 3(vii)
Question 3(viii)
Solution 3(viii)
Question 3(ix)
Solution 3(ix)
Question 3(x)
Solution 3(x)

Question 3(xi)
Solution 3(xi)

Question 3(xii)
Solution 3(xii)
Question 3(xiii)
Solution 3(xiii)
Question 3(xiv)
Solution 3(xiv)
Question 3(xv)
Solution 3(xv)
Question 3(xvi)

Find the equation of the normal to curve y2 = 4x at the point (1, 2) and also find the tangent.

Solution 3(xvi)

The equation of the given curve is y2 = 4x . Differentiating with respect to x, we have: 

begin mathsize 12px style 2 straight y dy over dx equals 4
rightwards double arrow dy over dx equals fraction numerator 4 over denominator 2 straight y end fraction equals 2 over straight y
therefore right enclose dy over dx end enclose subscript open parentheses 1 comma space 2 close parentheses end subscript equals 2 over 2 equals 1
Now comma space the space slope space at space point space left parenthesis 1 comma space 2 right parenthesis space is space 1 over right enclose begin display style dy over dx end style end enclose subscript open parentheses 1 comma space 2 close parentheses end subscript equals fraction numerator negative 1 over denominator 1 end fraction equals negative 1.
therefore Equation space of space the space tangent space at space left parenthesis 1 comma space 2 right parenthesis space is space straight y minus 2 space equals space minus 1 left parenthesis straight x minus 1 right parenthesis.
rightwards double arrow straight y minus 2 equals negative straight x plus 1
rightwards double arrow straight x plus straight y minus 3 equals 0
Equation space of space the space normal space is comma
straight y minus 2 equals negative left parenthesis negative 1 right parenthesis left parenthesis straight x minus 1 right parenthesis
straight y minus 2 equals straight x minus 1
straight x minus straight y plus 1 equals 0
end style

Question 3(xix)

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

Solution 3(xix)

Question 4
Solution 4

Question 5(i)
Solution 5(i)
Question 5(ii)

Solution 5(ii)

From (A)

Equation of tangent is

begin mathsize 12px style open parentheses straight y minus straight a over 5 close parentheses equals 13 over 16 open parentheses straight x minus fraction numerator 2 straight a over denominator 5 end fraction close parentheses
16 straight y minus fraction numerator 16 straight a over denominator 5 end fraction equals 13 straight x minus fraction numerator 26 straight a over denominator 5 end fraction
13 straight x minus 16 straight y minus 2 straight a equals 0
Equation space of space normal space is comma
open parentheses straight y minus straight a over 5 close parentheses equals 16 over 13 open parentheses straight x minus fraction numerator 2 straight a over denominator 5 end fraction close parentheses
13 straight y minus fraction numerator 13 straight a over denominator 5 end fraction equals negative 16 straight x plus fraction numerator 32 straight a over denominator 5 end fraction
16 straight x plus 13 straight y minus 9 straight a equals 0 end style

Question 5(iii)
Solution 5(iii)
Question 5(iv)
Solution 5(iv)
Question 5(v)
Solution 5(v)
Question 5(vi)

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 5(vi)

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 21

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

Solution 21

   

Question 3(xvii)

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

  

Solution 3(xvii)

Given equation curve is

Differentiating w.r.t x, we get

  

Slope of tangent at   is

  

Slope of normal will be

  

Equation of tangent at   will be

  

Equation of normal at   is

  

Question 3(xviii)

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

  

Solution 3(xviii)

Given equation curve is

Differentiating w.r.t x, we get

  

Slope of tangent at   is

  

Slope of normal will be

  

Equation of tangent at   will be

  

Equation of normal at   is

  

Question 20

At what points will tangents to the curve   be parallel to x-axis? Also, find the equations of the tangents to the curve at these points.

Solution 20

Given equation curve is

Differentiating w.r.t x, we get

  

As tangent is parallel to x-axis, its slope will be m = 0

  

As this point lies on the curve, we can find y

  

Or

  

So, the points are (3, 6) and (2, 7).

Equation of tangent at (3, 6) is

y - 6 = 0 (x - 3)

y - 6 = 0

Equation of tangent at (2, 7) is

y - 7 = 0 (x - 2)

y - 7 = 0

Chapter 16 - Tangents and Normals Exercise Ex. 16.3

Question 1(i)
Solution 1(i)
Question 1(ii)
Solution 1(ii)
Question 1(iii)
Solution 1(iii)
Question 1(iv)
Solution 1(iv)

Question 1(v)

Find the angle of intersection of the folloing curves

begin mathsize 12px style straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space straight x squared space plus space straight y squared space equals space ab end style

Solution 1(v)


Question 1(vi)
Solution 1(vi)
Question 1(vii)
Solution 1(vii)
Question 1(viii)
Solution 1(viii)
Question 1 (ix)

Find the angle of intersection of the following curves:

Y = 4 - x2 and y = x2

Solution 1 (ix)

Question 2(i)
Solution 2(i)
Question 2(ii)
Solution 2(ii)
Question 2(iii)
Solution 2(iii)
Question 3(i)
Solution 3(i)
Question 3(ii)
Solution 3(ii)
Question 3(iii)
Solution 3(iii)
Question 4

Solution 4

Question 5

Solution 5

Question 6

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

Solution 6

 

Question 7

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

Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 9

Solution 9

Question 10

Solution 10

Chapter 16 - Tangents and Normals Exercise Ex. 16VSAQ

Question 1

Solution 1

Question 2

Find the slope of the tangent to the curve x = t2 + 3t - 8, y = 2t2 - 2t - 5 at t = 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

Write the equation of the tangent drawn to the curve y = sin x at the point (0, 0).

Solution 18

Given curve is y = sin x

  

Slope of tangent at (0, 0) is

  

So, the equation of tangent at (0, 0) is

y - 0 = 1 (x - 0)

y = x

Question 19

Find the slope of the tangent to the curve   at

Solution 19

Given curve is

  

Slope of tangent at (0, 0) is

  

Hence, slope of tangent at   is 0.