# 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Ex 6.3

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## Karnataka 2nd PUC Maths Question Bank Chapter 6 Application of Derivatives Ex 6.3

### 2nd PUC Maths Application of Derivatives NCERT Text Book Questions and Answers Ex 6.3

Question 1.
Find the slope of the tangent to the curve y = 3x4 – 4x at x = 4. Question 2.
Find the slope of the tangent to the curve
$$y=\frac{x-1}{x-2}, x \neq 2 \text { at } x=10$$ Question 3.
Find the slope of the tangent to curve y = x3 – x + 1 at the point whose x- coordinate is 2. Question 4.
Find the slope of the tangent to the curve y = x3 – 3x + 2 at the point whose x-coordinate is 3.
$$\frac{d y}{d x}$$ = 3x2 – 3 dx
slope at x = 3 is 3 (9) – 3 = 24. Question 5.
Find the slope of the normal to the curve
$$x=a \cos ^{3} \theta, y=a \sin ^{3} \theta \text { at } \theta=\frac{\pi}{4}$$  Question 6.
Find the slope of the normal to the curve
$$x=1-a \sin \theta, y=b \cos ^{2} \theta \text { at } \theta=\frac{\pi}{2}$$ Question 7.
Find points at which the tangent to the curve y = x3 – 3x2 – 9x + 7 is parallel to
the x-axis.
$$\frac{d y}{d x}$$ = 3x2 – 6x – 9, slope of the tangent
since the tangent is parallel to x-axis $$\frac{d y}{d x}$$ = 0
3x2 – 6x – 9 = 0
3 (x + 1) (x – 3) = 0, x = 3, x = -1
when x = 3, y = 27 – 27 – 27 4- 7 = -20
when x = -1, y = -1 -3 + 9 + 7 = 12
The points at which the tangent parallel to x – axis are (3, -20) and (-1, 12).

Question 8.
Find a point on the curve y = (x – 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).  Question 9.
Find the point on the curve y = x3 – 11x + 5 at which the tangent is y = x – 11.
Slope of the line y = x – 11 is 1
slope of the curve $$\frac{d y}{d x}$$ = 3x2 – 11
∴ 3x2 – 11 = 1                   ‘
3x2 = 12 ⇒ x2 = 4, x = +2
when x = 2, y = (2)3 – 11 (2) + 5 = -9
when x = -2, y = -8 + 22 + 5 = 19
points are (2, -9) and (-2, 19)
equation of tangent at (2, -9) and slope is 1
y + 9 = 1 (x – 2)
y = x- 11 equation of tangent at (-2, 19)
y – 19 = 1 (x + 2) ⇒ y = x + 211
∴ (2, -9) is the only point at which the tangent is y = x – 11.10. Question 10.
Find the equation of all lines having slope – 1 that are tangents to the curve
$$y=\frac{1}{x-1}, x \neq 3$$ x = 2, x = 0
when x = 2, y = 1
when x = 0, y = -1
The points are (2, 1) and (0,-1)
equation of line though (2,1) having slope = -1
is y – 1 = -1 x (x – 2) ⇒ y + x = 3
or x + y – 3 = 0 and equation of line through (0,- 1)y + 1 = -1 (x – 0)
∴  y + x + 1 = 0.

Question 11.
Find the equation of all lines having slope 2 which are tangents to the curve
$$y=\frac{1}{x-3}, x \neq 3$$
slope of the line = 2
slope of the curve = $$\frac{-1}{(x-3)^{2}}$$
$$\frac{-1}{(x-3)^{2}}$$ =2
⇒ 2 (x -3)2 = -1
⇒ 2 (x2 – 6x + 9) = -1
⇒ 2x2 – 12x + 19 = 0
which has no real roots b2 – 4ac < 0
hence there is no point on the curve.

Question 12.
Find the equations of all lines having slope 0 which are tangent to the curve
$$y=\frac{1}{x^{2}-2 x+3}$$ Question 13.
Find the points of curve $$\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$$ which the tangents are
(i) parallel to x-axis
(ii) parallel to y-axis.
16x2 + 9y2 = 144
16 x 2 x + 9 x 2y x $$\frac{d y}{d x}$$ = 0 (0,4) and (0, -4) are the points on the curve at which tangent is parallel to x – axis. (ii) For tangents parallel to y – axis $$\frac{d y}{d x}$$ = $$\frac{1}{0}$$
$$\frac{-16 x}{9 y}=\frac{1}{0} \Rightarrow y=0$$
when y = 0, x2 = 9x = ± 3 point is (3,0) and (-3,0)
(3, 0) and (-3, 0) are the points at which the curve is parallel to y – axis.

Question 14.
Find the equations of the tangent and normal to the given curves at the indicated points:
(i) y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5)
y = x4 – 6x3 + 13x2 – 10x + 5 at (0, 5) dy
$$\frac{d y}{d x}$$ = 4x3 – 18x2 + 26x – 10 dx
slope at (0,5) = -10
∴ Equation of tangent at (0, 5) is
y – 5 = -10 (x – 0)
y – 5 = -10 x 10 x + y – 5 = 0
slope of the normal at (0,5)
$$(0,5)=\frac{-1}{-10}=\frac{1}{10}$$
∴ equation of normal is
y – 5 =$$\frac{1}{10}$$ (x – 0) = 10y – 50 = x
x – 10y + 50 = 0

(ii) y = x4 – 6x3 + 13x2 – 10x + 5 at (1, 3)
$$\frac{d y}{d x}$$ = 4x3 – 18x2 + 26x – 10 dx
slope of the tangent at x = 1

(iii) y = x3 at (1,1)
$$\frac{d y}{d x}=3 x^{2}$$
∴ slope of the tangent at x = 1 = 3
∴ Equation of tangent is y – 1 = 3(x – 1)
3x – y – 2 = 0
∴ slope of normal = $$\frac{-1}{3}$$
∴ equation of normal is y – 1 = $$\frac{-1}{3}$$(x – 1)
3y – 3 = – (x – 1)
x + 3y – 4 = 0. (iv) y = x2 at (0, 0)
$$\frac{d y}{d x}=2 x$$
∴ slope at x = 0 =0
∴ Equation of tangent is y – 0 = 0 (x – 0) ⇒ y = 0
slope of the normal = -1/(0)
∴ equation of normal $$y – 0=\frac{-1}{0}(x-0) \Rightarrow x=0$$

(v) x = cos t, y = sin t at $$\frac{\pi}{4}$$  Question 15.
Find the equation of the tangent line to the curve y = x2 – 2x +7 which is
(a) parallel to the line 2x – y + 9 = 0
$$\frac{d y}{d x}$$ = 2x – 2 also slope = 2
2 x – 2 = 2 ⇒2x = 4 ⇒ x = 2
when x = 2, y = (2)2 – 2 (2) + 7 = 7
∴ equation of tangent is y – 7 = 2 (x – 2)
2x – y + 3 = 0 ⇒ 2x – y + 3 = 0.

(b) perpendicular to the line 5y – 15x = 13. 12x + 36y – 227 = 0

Question 16.
Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = – 2 are parallel.  Question 17.
Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.
y = x3
$$\frac{d y}{d x}$$ = 3x2,
Given that $$\frac{d y}{d x}$$ = y = x3
∴ 3x2 = x3 ⇒ x2 (3 – x) = 0 ⇒ x = 0 or x = 3
when x = 3, y = 33 = 27, when x = 0, y = 0
∴ The required points are (0, 0), (3, 27).

Question 18.
For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.
y = 4x3 – 2x5 dy
$$\frac{d y}{d x}$$ = 12x2 – 10x4 dx
Let (a, b) be the point on the curve at which the tangent passes through the origin.
∴ Equation of tangent is
y – b = (12a2 – 10a2) (x – a)
but this passes through the origin
∴ 0 – b = (12a2 – 10a4) (-a) b = 12a3 – 10a5 ….(1)
Also from the equation b = 4a3 – 2a5 …. (2)
from (1) and (2) 12a3 – 10a5 = 4a3 – 2a5
8a3 = 8a5
a3 (1 – a2) = 0 ⇒ a = 0, a = + 1
when a = 0, b = 0, (0, 0)
a = 1,b= 12(1)- 10(1) = 2, (1,2)
a = -1, b = 12 (-1) -10 (-1) = -2, (-1, -2)
Hence the required points are (0,0), (1,2), (-1,-2).

Question 19.
Find the points on the curve x2 + y2 – 2x – 3 = 0 at which the tangents are parallel to the x-axis.  Question 20.
Find the equation of the normal at the point (am2,am3) for the curve ay2 = x3. Question 21.
Find the equation of the normal to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
y = x3 + 2x + 6
$$\frac{d y}{d x}$$ = 3x2+ 2 Question 22.
Find the equations of the tangent and normal to the parabola y2 = 4ax at the
point (at2, 2at).  Question 23.
Prove that the curves x = y2 and xy = k cut at right angles* if 8k2 = 1. Question 24.
Find the equations of the tangent and normal to the hyperbola
$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ at the point (x0, y0)  Question 25.
Find the equation of the tangent to the curve
$$y=\sqrt{3 x-2}$$ which is parallel to the line 4x – 2y +5 = 0   Choose the correct answer in Exercises 26 and 27.

Question 26.
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
(A) 3
(B) 1/3
(C) -3
(D) -1/3   