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RD Sharma solutions for Class 12 Mathematics chapter 16 - Tangents and Normals

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) Chapter 16: Tangents and Normals

Ex. 16.1Ex. 16.2Ex. 16.3Others

Chapter 16: Tangents and Normals Exercise 16.1 solutions [Pages 10 - 11]

Ex. 16.1 | Q 1.01 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point $y = \sqrt{x^3} \text { at } x = 4$ ?

Ex. 16.1 | Q 1.02 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point $y = \sqrt{x} \text { at }x = 9$ ?

Ex. 16.1 | Q 1.03 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  y = x3 − x at x = 2 ?

Ex. 16.1 | Q 1.04 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x2 + 3 sin x at x = 0 ?

Ex. 16.1 | Q 1.05 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?

Ex. 16.1 | Q 1.06 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  x = a cos3 θ, y = a sin3 θ at θ = π/4 ?

Ex. 16.1 | Q 1.07 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  x = a (θ − sin θ), y = a(1 − cos θ) at θ = π/2 ?

Ex. 16.1 | Q 1.08 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  y = (sin 2x + cot x + 2)2 at x = π/2 ?

Ex. 16.1 | Q 1.09 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  x2 + 3y + y2 = 5 at (1, 1)  ?

Ex. 16.1 | Q 1.1 | Page 10

Find the slope of the tangent and the normal to the following curve at the indicted point  xy = 6 at (1, 6) ?

Ex. 16.1 | Q 2 | Page 10

Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?

Ex. 16.1 | Q 3 | Page 10

If the tangent to the curve y = x3 + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?

Ex. 16.1 | Q 4 | Page 10

Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?

Ex. 16.1 | Q 5 | Page 10

Find the points on the curve y = x3 − 2x2 − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?

Ex. 16.1 | Q 6 | Page 10

Find the points on the curve y2 = 2x3 at which the slope of the tangent is 3 ?

Ex. 16.1 | Q 7 | Page 10

Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?

Ex. 16.1 | Q 8 | Page 10

Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?

Ex. 16.1 | Q 9 | Page 10

At what points on the circle x2 + y2 − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?

Ex. 16.1 | Q 10 | Page 10

At what point of the curve y = x2 does the tangent make an angle of 45° with the x-axis?

Ex. 16.1 | Q 11 | Page 10

Find the points on the curve y = 3x2 − 9x + 8 at which the tangents are equally inclined with the axes ?

Ex. 16.1 | Q 12 | Page 10

At what points on the curve y = 2x2 − x + 1 is the tangent parallel to the line y = 3x + 4?

Ex. 16.1 | Q 13 | Page 10

Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is $- \frac{1}{6}$  ?

Ex. 16.1 | Q 14 | Page 11

Find the points on the curve x2 + y2 = 13, the tangent at each one of which is parallel to the line 2x + 3y = 7 ?

Ex. 16.1 | Q 15 | Page 11

Find the points on the curve 2a2y = x3 − 3ax2 where the tangent is parallel to x-axis ?

Ex. 16.1 | Q 16 | Page 11

At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?

Ex. 16.1 | Q 17.1 | Page 11

Find the points on the curve $\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangent is parallel to the x-axis ?

Ex. 16.1 | Q 17.2 | Page 11

Find the points on the curve$\frac{x^2}{4} + \frac{y^2}{25} = 1$ at which the tangent is  parallel to the y-axis ?

Ex. 16.1 | Q 18 | Page 11

Find the points on the curve x2 + y2 − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Ex. 16.1 | Q 19.1 | Page 11

Find the points on the curve $\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangent is  parallel to x-axis ?

Ex. 16.1 | Q 19.2 | Page 11

Find the points on the curve $\frac{x^2}{9} + \frac{y^2}{16} = 1$ at which the tangent is  parallel to y-axis ?

Ex. 16.1 | Q 20 | Page 11

Who that the tangents to the curve y = 7x3 + 11 at the points x = 2 and x = −2 are parallel ?

Ex. 16.1 | Q 21 | Page 11

Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?

Chapter 16: Tangents and Normals Exercise 16.2 solutions [Pages 27 - 29]

Ex. 16.2 | Q 1 | Page 27

Find the equation of the tangent to the curve $\sqrt{x} + \sqrt{y} = a$ at the point $\left( \frac{a^2}{4}, \frac{a^2}{4} \right)$ ?

Ex. 16.2 | Q 2 | Page 27

Find the equation of the normal to y = 2x3 − x2 + 3 at (1, 4) ?

Ex. 16.2 | Q 3.01 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point x4 − bx3 + 13x2 − 10x + 5 at (0, 5)  ?

Ex. 16.2 | Q 3.02 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1 ?

Ex. 16.2 | Q 3.03 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  y = x2 at (0, 0) ?

Ex. 16.2 | Q 3.04 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y = 2x2 − 3x − 1 at (1, −2) ?

Ex. 16.2 | Q 3.05 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)$ ?

Ex. 16.2 | Q 3.06 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y = x2 + 4x + 1 at x = 3  ?

Ex. 16.2 | Q 3.07 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)$ ?

Ex. 16.2 | Q 3.08 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)$ ?

Ex. 16.2 | Q 3.09 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point y2 = 4ax at $\left( \frac{a}{m^2}, \frac{2a}{m} \right)$ ?

Ex. 16.2 | Q 3.1 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)$ ?

Ex. 16.2 | Q 3.11 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point xy = c2 at $\left( ct, \frac{c}{t} \right)$ ?

Ex. 16.2 | Q 3.12 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)$ ?

Ex. 16.2 | Q 3.13 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)$ ?

Ex. 16.2 | Q 3.14 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  $x^\frac{2}{3} + y^\frac{2}{3}$ = 2 at (1, 1) ?

Ex. 16.2 | Q 3.15 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  x2 = 4y at (2, 1) ?

Ex. 16.2 | Q 3.16 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  y2 = 4x at (1, 2)  ?

Ex. 16.2 | Q 3.17 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?

Ex. 16.2 | Q 3.18 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point  y2 = 4ax at (x1, y1)?

Ex. 16.2 | Q 3.19 | Page 27

Find the equation of the tangent and the normal to the following curve at the indicated point $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)$ ?

Ex. 16.2 | Q 4 | Page 27

Find the equation of the tangent to the curve x = θ + sin θ, y = 1 + cos θ at θ = π/4 ?

Ex. 16.2 | Q 5.1 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = $\frac{\pi}{2}$ ?

Ex. 16.2 | Q 5.2 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points $x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}$ ?

Ex. 16.2 | Q 5.3 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = at2, y = 2at at t = 1 ?

Ex. 16.2 | Q 5.4 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points  x = asect, y = btant at t ?

Ex. 16.2 | Q 5.5 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?

Ex. 16.2 | Q 5.6 | Page 28

Find the equation of the tangent and the normal to the following curve at the indicated points x = 3cosθ − cos3θ, y = 3sinθ − sin3θ

Ex. 16.2 | Q 6 | Page 28

Find the equation of the normal to the curve x2 + 2y2 − 4x − 6y + 8 = 0 at the point whose abscissa is 2 ?

Ex. 16.2 | Q 7 | Page 28

Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?

Ex. 16.2 | Q 8 | Page 28

The equation of the tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x − 5. Find the values of a and b ?

Ex. 16.2 | Q 9 | Page 28

Find the equation of the tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?

Ex. 16.2 | Q 10 | Page 28

Find an equation of normal line to the curve y = x3 + 2x + 6 which is parallel to the line x+ 14y + 4 = 0 ?

Ex. 16.2 | Q 11 | Page 28

Determine the equation(s) of tangent (s) line to the curve y = 4x3 − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?

Ex. 16.2 | Q 12 | Page 28

Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0 ?

Ex. 16.2 | Q 13.1 | Page 28

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?

Ex. 16.2 | Q 13.2 | Page 28

Find the equation of the tangent line to the curve y = x2 − 2x + 7 which perpendicular to the line 5y − 15x = 13. ?

Ex. 16.2 | Q 14 | Page 28

Find the equations of all lines having slope 2 and that are tangent to the curve $y = \frac{1}{x - 3}, x \neq 3$ ?

Ex. 16.2 | Q 15 | Page 28

Find the equations of all lines of slope zero and that are tangent to the curve $y = \frac{1}{x^2 - 2x + 3}$ ?

Ex. 16.2 | Q 16 | Page 28

Find the equation of the tangent to the curve  $y = \sqrt{3x - 2}$ which is parallel to the 4x − 2y + 5 = 0 ?

Ex. 16.2 | Q 17 | Page 28

Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?

Ex. 16.2 | Q 18 | Page 29

Prove that $\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2$ touches the straight line $\frac{x}{a} + \frac{y}{b} = 2$ for all n ∈ N, at the point (a, b) ?

Ex. 16.2 | Q 19 | Page 29

Find the equation of the tangent to the curve x = sin 3ty = cos 2t at

$t = \frac{\pi}{4}$ ?

Ex. 16.2 | Q 20 | Page 29

At what points will be tangents to the curve y = 2x3 − 15x2 + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?

Ex. 16.2 | Q 21 | Page 29

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

Chapter 16: Tangents and Normals Exercise 16.3 solutions [Pages 40 - 41]

Ex. 16.3 | Q 1.1 | Page 40

Find the angle of intersection of the following curve y2 = x and x2 = y  ?

Ex. 16.3 | Q 1.2 | Page 40

Find the angle of intersection of the following curve  y = x2 and x2 + y2 = 20  ?

Ex. 16.3 | Q 1.3 | Page 40

Find the angle of intersection of the following curve  2y2 = x3 and y2 = 32x ?

Ex. 16.3 | Q 1.4 | Page 40

Find the angle of intersection of the following curve x2 + y2 − 4x − 1 = 0 and x2 + y2 − 2y − 9 = 0 ?

Ex. 16.3 | Q 1.5 | Page 40

Find the angle of intersection of the following curve $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and x2 + y2 = ab ?

Ex. 16.3 | Q 1.6 | Page 40

Find the angle of intersection of the following curve  x2 + 4y2 = 8 and x2 − 2y2 = 2 ?

Ex. 16.3 | Q 1.7 | Page 40

Find the angle of intersection of the following curve  x2 = 27y and y2 = 8x ?

Ex. 16.3 | Q 1.8 | Page 40

Find the angle of intersection of the following curve x2 + y2 = 2x and y2 = x ?

Ex. 16.3 | Q 1.9 | Page 40

Find the angle of intersection of the following curve y = 4 − x2 and y = x2 ?

Ex. 16.3 | Q 2.1 | Page 40

Show that the following set of curve intersect orthogonally y = x3 and 6y = 7 − x?

Ex. 16.3 | Q 2.2 | Page 40

Show that the following set of curve intersect orthogonally x3 − 3xy2 = −2 and 3x2y − y3 = 2 ?

Ex. 16.3 | Q 2.3 | Page 40

Show that the following set of curve intersect orthogonally x2 + 4y2 = 8 and x2 − 2y2 = 4 ?

Ex. 16.3 | Q 3.1 | Page 40

Show that the following curve intersect orthogonally at the indicated point x2 = 4y and 4y + x2 = 8 at (2, 1) ?

Ex. 16.3 | Q 3.2 | Page 40

Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?

Ex. 16.3 | Q 3.3 | Page 40

Show that the following curve intersect orthogonally at the indicated point y2 = 8x and 2x2 +  y2 = 10 at  $\left( 1, 2\sqrt{2} \right)$ ?

Ex. 16.3 | Q 4 | Page 40

Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?

Ex. 16.3 | Q 5 | Page 40

Show that the curves 2x = y2 and 2xy = k cut at right angles, if k2 = 8 ?

Ex. 16.3 | Q 6 | Page 40

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

Ex. 16.3 | Q 7 | Page 40

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

Ex. 16.3 | Q 8.1 | Page 41

Find the condition for the following set of curve to intersect orthogonally $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2$ ?

Ex. 16.3 | Q 8.2 | Page 41

Find the condition for the following set of curve to intersect orthogonally $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ ?

Ex. 16.3 | Q 9 | Page 41

Show that the curves $\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1$ intersect at right angles ?

Ex. 16.3 | Q 10 | Page 41

If the straight line xcos $\alpha$ +y sin $\alpha$ = p touches the curve  $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ then prove that a2cos2 $\alpha$ $-$ b2sin$\alpha$ = p?

Chapter 16: Tangents and Normals solutions [Pages 41 - 42]

Q 1 | Page 41

Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?

Q 2 | Page 41

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

Q 3 | Page 41

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of $\frac{dy}{dx}$ ?

Q 4 | Page 41

Write the value of $\frac{dy}{dx}$ , if the normal to the curve y = f(x) at (x, y) is parallel to y-axis ?

Q 5 | Page 41

If the tangent to a curve at a point (xy) is equally inclined to the coordinates axes then write the value of $\frac{dy}{dx}$ ?

Q 6 | Page 41

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of $\frac{dx}{dy}$ ?

Q 7 | Page 41

Find the slope of the normal at the point 't' on the curve $x = \frac{1}{t}, y = t$ ?

Q 8 | Page 41

Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle $\frac{\pi}{4}$ with x-axis  ?

Q 9 | Page 41

Write the angle made by the tangent to the curve x = et cos t, y = et sin t at $t = \frac{\pi}{4}$ with the x-axis ?

Q 10 | Page 42

Write the equation of the normal to the curve y = x + sin x cos x at $x = \frac{\pi}{2}$ ?

Q 11 | Page 42

Find the coordinates of the point on the curve y2 = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?

Q 12 | Page 42

Write the equation on the tangent to the curve y = x2 − x + 2 at the point where it crosses the y-axis ?

Q 13 | Page 42

Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?

Q 14 | Page 42

Write the angle between the curves y = e−x and y = ex at their point of intersections ?

Q 15 | Page 42

Write the slope of the normal to the curve $y = \frac{1}{x}$  at the point $\left( 3, \frac{1}{3} \right)$ ?

Q 16 | Page 42

Write the coordinates of the point at which the tangent to the curve y = 2x2 − x + 1 is parallel to the line y = 3x + 9 ?

Q 17 | Page 42

Write the equation of the normal to the curve y = cos x at (0, 1) ?

Q 18 | Page 42

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

Chapter 16: Tangents and Normals solutions [Pages 42 - 44]

Q 1 | Page 42

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

• x = 0

• y = 0

• x + y = 0

• x − y = 0

Q 2 | Page 42

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

• = 2

• x = π

• x + π = 0

• 2x = π

Q 3 | Page 42

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

• x − 2y = 2

• x − 2y + 2 = 0

• 2x +  y = 4

• 2x + y − 4 = 0

Q 4 | Page 42

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

• (1/2, 1/4)

• (1/4, 1/2)

• (4, 2)

• (1, 1)

Q 5 | Page 42

If the tangent to the curve x = a t2, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .

• (a, a)

• (0, a)

• (0, 0)

• (a, 0)

Q 6 | Page 42

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

• (0, 2)

• (1, 0)

• (−1, 6)

• (2, −2)

Q 7 | Page 42

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

• (1/2, 1/4)

• (1/4, 1/2)

• (4, 2)

• (1, 1)

Q 8 | Page 42

The point at the curve y = 12x − x2 where the slope of the tangent is zero will be _____________ .

• (0, 0)

• (2, 16)

• (3, 9)

• none of these

Q 9 | Page 42

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

• $\tan^{- 1} \frac{4}{3}$

• $\tan^{- 1} \frac{3}{4}$

• 90°

• 45°

Q 10 | Page 43

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

• x + 3y = 8

• x + 3y + 8 = 0

• x + 3y ± 8 = 0

• x + 3y = 0

Q 11 | Page 43

The equations of tangent at those points where the curve y = x2 − 3x + 2 meets x-axis are _______________ .

• x − y + 2 = 0 = x − y − 1

• x + y − 1 = 0 = x − y − 2

• x − y − 1 = 0 = x − y

• x − y = 0 = x + y

Q 12 | Page 43

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

• 22/7

• 6/7

• -6

• none of these

Q 13 | Page 43

At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero

• (3, 0), (−1, 0)

• (3, 0), (1, 2)

• (−1, 0), (1, 2)

• (1, 2), (1, −2)

Q 14 | Page 43

The angle of intersection of the curves xy = a2 and x2 − y2 = 2a2 is ______________ .

• 45°

• 90°

• none of these

Q 15 | Page 43

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

• 1

• -6

• 6

• 0

Q 16 | Page 43

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

• b = 1, c = 2

• b = −1, c = 1

• b = 2, c = 1

• b = −2, c = 1

Q 17 | Page 43

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

• 1/2

• 0

• -2

Q 18 | Page 43

The curves y = aex and y = be−x cut orthogonally, if ___________ .

• a = b

• a = −b

• ab = 1

• ab = 2

Q 19 | Page 43

The equation of the normal to the curve x = a cos3 θ, y = a sin3 θ at the point θ = π/4 is __________ .

• x = 0

• y = 0

• x = y

• x + y = a

Q 20 | Page 43

If the curves y = 2 ex and y = ae−x intersect orthogonally, then a = _____________ .

• 1/2

• −1/2

• 2

• 2e2

Q 21 | Page 43

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 is __________ .

• (−3, −27)

• (3, 9)

• (7/2, 35/4)

• (0, 0)

Q 22 | Page 43

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

• π/6

• π/3

• π/2

• π/4

Q 23 | Page 43

The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at $x = \frac{\pi}{6}$ is ____________ .

• π/4

• π/2

• π/3

• none of these

Q 24 | Page 43

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

• is parallel to x-axis

• is parallel to y-axis

• makes an acute angle with x-axis

• makes an obtuse angle with x-axis

Q 25 | Page 43

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

(a) $\left( 4, \frac{8}{3} \right)$

(b) $\left( - 4, \frac{8}{3} \right)$

(c) $\left( 4, - \frac{8}{3} \right)$

(d) none of these

Q 26 | Page 43

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

• $\frac{22}{7}$

• $\frac{6}{7}$

• $\frac{7}{6}$

• $- \frac{6}{7}$

Q 27 | Page 43

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

• 1

• 2

• 3

• 1/2

Q 28 | Page 44

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

• x + y = 0

• x − y = 0

• x + y + 1 = 0

• x − y = 1

Q 29 | Page 44

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

• x + y = 3

• x − y = 3

• x + y = 1

• x − y = 1

• none of these

Chapter 16: Tangents and Normals

Ex. 16.1Ex. 16.2Ex. 16.3Others

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) RD Sharma solutions for Class 12 Mathematics chapter 16 - Tangents and Normals

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Concepts covered in Class 12 Mathematics chapter 16 Tangents and Normals are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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