Class 12

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Additional Exercise

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
Write the maximum value of the function y = x⁵ in the interval [1, 5].
(a) 125
(b) 3125
(c) 625
(d) 225
Solution:
(b) 3125

Question 2.
Differentiate sin^-1 (cos x) with respect to x.
(a) -1
(b) 0
(c) 1
(d) None of the above
Solution:
(a) -1

Question 3.
Write the equation of the tangent to the curve y = |x| at the point (-2, 2).
(a) 2x + 2y = 0
(b) 2x + y = 0
(c) x + y = 1
(d) x + y = 0
Solution:
(d) x + y = 0

Question 4.
Write the value of \(\frac{d}{d x}\)(sin x)
(a) tan x
(b) sin x
(c) cos x
(d) cot x
Solution:
(c) cos x

Question 5.
Write the value of \(\frac{d}{d x}\)(cos x)
(a) cos x
(b) -sin x
(c) sin x
(d) cot x
Solution:
(b) -sin x

Question 6.
Write the value of \(\frac{d}{d x}\)(tan x)
(a) tan x
(b) sec² x
(c) sin x
(d) cot x
Solution:
(c) sin x

Question 7.
Write the value of \(\frac{d}{d x}\)(sec x)
(a) sin x cos x
(b) sin x tan x
(c) cot x tan x
(d) sec x tan x
Solution:
(d) sec x tan x

Question 8.
Write the value of \(\frac{d}{d x}\)(cot x) = -cosec²x
(a) -sin² x
(b) -cos² x
(c) -cosec² x
(d) -tan² x
Solution:
(c) -cosec² x

Question 9.
Write the value of \(\frac{d}{d x}\)(cosec x)
(a) -cosec x cot x
(b) -sin x cot x
(c) -tan x cot x
(d) -cos x cot x
Solution:
(a) -cosec x cot x

Question 10.
Write the value of \(\frac{d}{d x}\)(e^x) = ex
(a) e
(b) x
(c) e^x
(d) e^2x
Solution:
(c) e^x

Question 11.
Write the value of \(\frac{d}{d x}\)(a^x)
(a) a^2x log a
(b) a^x log a
(c) log a
(d) log a^x
Solution:
(b) a^x log a

Question 12.
Write the value of \(\frac{d}{d x}\)(log x)
(a) x
(b) \(\frac{1}{2 x}\)
(c) \(\frac{1}{x^2}\)
(d) \(\frac{1}{x}\)
Solution:
(d) \(\frac{1}{x}\)

Question 13.
Write the value of \(\frac{d}{d x}\)(log_a x)
(a) \(\frac{x}{\log a}\)
(b) \(\frac{1}{x \log a}\)
(c) \(\frac{1}{x^2 \log a}\)
(d) \(\frac{1}{2 x^2 \log a}\)
Solution:
(b) \(\frac{1}{x \log a}\)

Question 14.
Write the value of \(\frac{d}{d x}\)(cot^-1 x) = \(\frac{-1}{1+x^2}\)
(a) \(\frac{1}{1-x^2}\)
(b) \(\frac{1}{1+x^2}\)
(c) \(\frac{-1}{1-x^2}\)
(d) \(\frac{-1}{1+x^2}\)
Solution:
(d) \(\frac{-1}{1+x^2}\)

Question 15.
Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) for √x + √y = √c.
(a) \(\frac{\sqrt{y}}{\sqrt{x}}\)
(b) \(\frac{\sqrt{x}}{\sqrt{y}}\)
(c) –\(\frac{\sqrt{y}}{\sqrt{x}}\)
(d) –\(\frac{\sqrt{x}}{\sqrt{y}}\)
Solution:
(c) –\(\frac{\sqrt{y}}{\sqrt{x}}\)

Question 16.
If f(x) = \(\log _{x^2}\) (log x), then find f'(e).
(a) \(\frac{1}{e}\)
(b) \(\frac{1}{\mathrm{e}^2}\)
(c) \(\frac{1}{2 \mathrm{e}}\)
(d) \(\frac{2}{\mathrm{e}}\)
Solution:
(c) \(\frac{1}{2 \mathrm{e}}\)

Question 17.
If y = \(\sqrt{\sin x+y}\), then what is \(\frac{\mathrm{dy}}{\mathrm{dx}}\)?
(a) \(\frac{\cos x}{2 y+1}\)
(b) \(\frac{\cos x}{2 y-1}\)
(c) \(\frac{\sin x}{2 y-1}\)
(d) \(\frac{\sin x}{2 y+1}\)
Solution:
(b) \(\frac{\cos x}{2 y-1}\)

Question 18.
If y = 10^x and z = 100^x/2 then find \(\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{y}^2}{\mathrm{z}}\right)\)
(a) 10^x log 10
(b) 10^2x log 10
(c) x log 10
(d) x² log 10
Solution:
(a) 10^x log 10

Question 19.
Find the differential of y = sin² x.
(a) 2sinx cot x dx
(b) 2sinx tan x dx
(c) 2sinx cosx dx
(d) 2cosx sinx dx
Solution:
(c) 2sinx cosx dx

Question 20.
Differentiate \(\sqrt{\sin \sqrt{x}}\)
(a) \(\frac{\cos \sqrt{x}}{4 \sqrt{x \sin \sqrt{x}}}\)
(b) \(\frac{\sin x}{4 \sqrt{x \cos \sqrt{x}}}\)
(c) \(\frac{\sin \sqrt{x}}{4 \sqrt{x \cos \sqrt{x}}}\)
(d) \(\frac{\sin x}{4 \sqrt{x \cos x}}\)
Solution:
(a) \(\frac{\cos \sqrt{x}}{4 \sqrt{x \sin \sqrt{x}}}\)

Question 21.
If f(x) = [tan² x], what is f'(0)?
(a) 0
(b) 1
(c) -1
(d) None of the above
Solution:
(a) 0

Question 22.
If xy = e^x-y then find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
(a) \(\frac{\log x}{(1-\log x)^2}\)
(b) \(\frac{\log x}{(1+\log x)^2}\)
(c) \(\frac{\log x}{(1+\log x)}\)
(d) \(\frac{\log x}{(1-\log x)}\)
Solution:
(b) \(\frac{\log x}{(1+\log x)^2}\)

Question 23.
Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) if y = cot^-1\(\left(\frac{1+\cos x}{1-\cos x}\right)^{\frac{1}{2}}\)
(a) 0
(b) \(\frac{1}{2}\)
(c) 1
(d) \(\frac{1}{4}\)
Solution:
(b) \(\frac{1}{2}\)

Question 24.
Find the derivative of sin x° w.r.t. x.
(a) \(\frac{\pi}{90}\) cos x°
(b) π cos x°
(c) \(\frac{\pi}{180}\) cos x°
(d) π sin x°
Solution:
(c) \(\frac{\pi}{180}\) cos x°

Question 25.
The least value of a such that f(x) = x² + ax + 1 is strictly increasing on (1, 2) is
(a) -2
(b) -4
(c) 2
(d) 4
Solution:
(a) -2

Question 26.
If y = sec^-1\(\left(\frac{x+1}{x-1}\right)\) + sin^-1\(\left(\frac{x-1}{x+1}\right)\) then find \(\frac{\mathrm{dy}}{\mathrm{dx}}\).
(a) 0
(b) -1
(c) 1
(d) None of the above
Solution:
(a) 0

Question 27.
Let f(x) = e^x g(x), g(0) = 2 and g'(0) = 1, then find f'(0).
(a) 1
(b) 2
(c) 3
(d) 0
Solution:
(c) 3

Question 28.
If x ∈ (0, π/2) then find \(\frac{\mathrm{d}}{\mathrm{dx}}\)[sin x]
(a) 0
(b) 1
(c) 7C
(d) \(\frac{\pi}{2}\)
Solution:
(a) 0

Question 29.
If f(x) = [x²] then f’\(\left(\frac{3}{2}\right)\) = ________.
(a) 1
(b) \(\frac{3}{2}\)
(c) 0
(d) –\(\frac{3}{2}\)
Solution:
(c) 0

Question 30.
A differentiable function f defined for all x > 0 satisfies f(x²) = x³ for all x > 0. What is f'(b)?
(a) 3
(b) 4
(c) 6
(d) 12
Solution:
(c) 6

Question 31.
What is the derivative of f'(In x) w.r.t x where f(x) = ln x.
(a) \(\frac{1}{x^2 \ln x}\)
(b) \(\frac{1}{2 x \ln x}\)
(c) \(\frac{1}{x \ln x^2}\)
(d) \(\frac{1}{x \ln x}\)
Solution:
(d) \(\frac{1}{x \ln x}\)

Question 32.
For what value of a, f(x) = log_a (x) is increasing on R?
(a) a > 1
(c) a < 1
(b) a = 1
(d) a > 1
Solution:
(a) a ≥ 1

Question 33.
Write the points at which tangent to the curve y = x² – 3x is parallel to x – axis.
(a) \(\left(\frac{3}{2}, \frac{9}{4}\right)\)
(b) \(\left(\frac{3}{2}, \frac{-9}{4}\right)\)
(c) \(\left(\frac{-3}{2}, \frac{9}{4}\right)\)
(d) \(\left(\frac{-3}{2}, \frac{-9}{4}\right)\)
Solution:
(b) \(\left(\frac{3}{2}, \frac{-9}{4}\right)\)

Question 34.
If y = e^x + e^-x + 2 has a tangent parallel to x – axis at (α, β) then find the value of α.
(a) 0
(b) 2
(c) 1
(d) -1
Solution:
(a) 0

Question 35.
Write slope of the tangent to the curve y = √3 sin x + cos x at (\(\frac{\pi}{3}\), 2)
(a) 1
(b) -1
(c) 0
(d) None of the above
Solution:
(c) 0

Question 36.
Find the value of x for which f(x) is either a local maximum or a local minimum when
f(x) = x³ – 3x² – 9x + 6.
(a) (3, -1)
(b) (3, 1)
(c) (-1, 3)
(d) (1, -3)
Solution:
(a) (3, -1)

Question 37.
Find the x – coordinates of the extreme points of the function.
y = cos x + sin x , x ∈ [0, \(\frac{\pi}{2}\)]
(a) \(\frac{\pi}{4}\)
(b) \(\frac{2 \pi}{3}\)
(c) \(\frac{\pi}{2}\)
(d) π
Solution:
(a) \(\frac{\pi}{4}\)

Question 38.
Find the equation of tangent to the curve x = y² – 2 at the points where slope of the normal equal to (-2).
(a) x + y – 1 = 0
(b) 2x + y – 1 = 0
(c) 2x – y + 1 = 0
(d) x – y + 1 = 0
Solution:
(b) 2x + y – 1 = 0

Question 39.
f(x) = x⁴ – 62x² + ax + 9 attains its maximum value at x = 1 in the interval [0, 2]. Find the value of ‘a’.
(a) 12
(b) 120
(c) 1
(d) 1200
Solution:
(b) 120

Question 40.
If sin (x + y) = log (x + y) then \(\frac{d y}{d x}\) is:
(a) 2
(b) -2
(c) 1
(d) -1
Solution:
(d) -1

Question 41.
If f(x) = 2x and g(x) = \(\frac{x^2+1}{2}\) then which of the following can be a discontinuous function?
(a) F(x) + g(x)
(b) f(x) – g(x)
(c) f(x) . g(x)
(d) \(\frac{f(x)}{g(x)}\)
Solution:
(d) \(\frac{f(x)}{g(x)}\)

Question 42.
If f(x) = x² sin (‘) where x ≠ 0 then the value of the function f at x = 0, so that the function is continuous at x = 0 is
(a) 0
(b) -1
(c) 1
(d) none of these
Solution:
(a) 0

Question 43.
The derivative of cos^-1 (2x² – 1) with respect to cos^-1 x
(a) 2
(b) \(\frac{-1}{2 \sqrt{1-x^2}}\)
(c) \(\frac{2}{x}\)
(d) 1 – x²
Solution:
(a) 2

Question 44.
If y = \(\sqrt{\sin x+y}\) then \(\frac{d y}{d x}\) is equal to:
(a) \(\frac{\cos x}{2 y-1}\)
(b) \(\frac{\cos x}{1-2 y}\)
(c) \(\frac{\sin x}{1-2 y}\)
(d) \(\frac{\sin x}{2 y-1}\)
Solution:
(a) \(\frac{\cos x}{2 y-1}\)

Question 45.
Find the value of \(\frac{d y}{d x}\) at θ = \(\frac{\pi}{3}\) if x = asec³ θ and x = atan³ θ is:
(a) \(\frac{\sqrt{3}}{2}\)
(b) –\(\frac{\sqrt{3}}{2}\)
(c) \(\frac{1}{2}\)
(d) 1
Solution:
(a) \(\frac{\sqrt{3}}{2}\)

Question 46.
If x²y = e^x-y then \(\frac{d y}{d x}\) is:
(a) \(\frac{1+x}{1+\log x}\)
(b) \(\frac{1-x}{1+\log x}\)
(c) \(\frac{x}{1+\log x}\)
(d) \(\frac{x}{(1+\log x)^2}\)
Solution:
(d) \(\frac{x}{(1+\log x)^2}\)

Question 47.
Differential coefficient of sec (tan^-1 x) is:
(a) \(\frac{x}{1+x^2}\)
(b) \(x \sqrt{1+x^2}\)
(c) \(\frac{x}{\sqrt{1+x^2}}\)
(d) \(\frac{1}{\sqrt{1+x^2}}\)
Solution:
(c) \(\frac{x}{\sqrt{1+x^2}}\)

Question 48.
The function f(x) = x^x is decreasing in the interval:
(a) (0, e)
(b) (0, \(\frac{1}{e}\))
(c) (0, 1)
(d) none of these
Solution:
(b) (0, \(\frac{1}{e}\))

(B) Very Short Type Questions With Answers

Question 1.
If f'(2⁺) = 0 and f'(2^–) = 0, then is f(x) continuous at x = 2?
Solution:
f'(2⁺) = 0 and f'(2^–) = 0
⇒ f is differentiable at x = 2
⇒ f is continuous at x = 2.

Question 2.
If Φ(x) = f(x) + f(1 – x), f'(x) = 0 for 0 < x < 1, then is x = a point of maxima or minima of Φ(x)?
Solution:
Let Φ(x) = f(x) + f(1 – x)
⇒ Φ’ = f'(x) – f'(1 – x)
Φ”(x) = f”(x) + f”(1 – x)
Now Φ'(\(\frac{1}{2}\)) = f'(\(\frac{1}{2}\)) – f'(\(\frac{1}{2}\)) = 0
⇒ \(\frac{1}{2}\) is a critical point.
Φ”(\(\frac{1}{2}\)) = f”(\(\frac{1}{2}\)) – f”(\(\frac{1}{2}\)) = 0
⇒ x = \(\frac{1}{2}\) is neither a point of local maxima nor a point of local minima.

Question 3.
Write the interval in which the function f(x) = sin^-1 (2 – x) is differentiable.
Solution:
f(x) = sin^-1 (2 – x) is differentiable for
1 – (2 – x)² > 0.
⇒ (2 – x)² < 1
⇒ -1 < 2 – x < 1
⇒ -3 < -x < -1
⇒ 3 > x < 1
⇒ 1 < x < 3
⇒ x ∈ (1, 3)
∴ f is differentiable on (1, 3)

Question 4.
Write the set of values of x for which the function f(x) = sin x – x is increasing.
Solution:
f(x) = sin x – x
⇒ f'(x) = cos x – 1 ≤ 0 for all x ∈ R
∴ f(x) is increasing for all x ∈ R

Question 5.
Write the differential coefficient of e^{[x] ln (x + 1)} where 3 ≤ x < 4 with respect to x.
Solution:
Let y = e
= e^3ln(x+1) (∵ 3 ≤ x < 4)
= \(e^{\ln (x+1)^3}\) = (x + 1)³
\(\frac{d y}{d x}\) = 3(x + 1)²

Question 6.
Write a logarithmic function which is differentiable only in the open interval (-1, 1).
Solution:
f(x) = \(\left\{\begin{array}{l}
\ln \left|\sin ^{-1} x\right|, x \neq 0 \\
1, x=0
\end{array}\right.\) is differentiable only on (-1, 1).

Question 7.
Write a function which has both relative and absolute maximum at the point (1, 2).
Solution:
f(x) = -x² + 2x + 1 on [0, 1] has relative and absolute maximum at (1, 2).

Question 8.
Write the derivation of e^3logx with respect to x².
Solution:
Let u = e^3lnx = e^lnx = \(e^{\ln x^3}\)= x³ and v = x²
∴ \(\frac{d u}{d x}\) = 3x² and \(\frac{d v}{d x}\) = 2x
∴ Derivative of u w.r.t. v is
\(\frac{d u}{d v}\) = \(\frac{3 x^2}{2 x}\) = \(\frac{3 x}{2}\)

Question 9.
Write the maximum value of the function y = x⁵ in the interval [1, 5]:
Solution:
y = x⁵
⇒ \(\frac{d y}{d x}\) = 5x⁴ > 0 ∀ x ∈ [1, 5]
i.e y is strictly increasing in the given interval.
Thus the maximum value of
y = x⁵ = 5⁵ = 3125

Question 10.
Differentiate a^lnx with respect to x
Solution:
\(\frac{d}{d x}\)(a^{ln x}) = a^{ln x} (ln a)\(\frac{d}{d x}\)ln x
= \(\frac{a^{\ln x} \ln a}{x}\)

Question 11.
Mention the values of x for which x the function f(x) = x³ – 12x is increasing.
Solution:
Given f(x) = x³ – 12x
f is increasing if f'(x) > 0
⇒ 3x² – 12 > 0
⇒ x² > 4
⇒ -2 > x > 2
⇒ x ∈ (-∞, -2) ∪ (2, ∞)

Question 12.
Differentiate sin^-1 (cos x) with respect to x.
Solution:
\(\frac{d}{d x}\)sin^-1 (cos x) = \(\frac{d}{d x}\)sin^-1 sin (\(\frac{\pi}{2}\) – x)
= \(\frac{d}{d x}\) (\(\frac{\pi}{2}\) – x) = 1

Question 13.
Write the equation of the tangent to the curve y = |x| at the point (-2, 2).
Solution:
y = |x| = \(\left\{\begin{aligned}
x, & x \geq 0 \\
-x, & x<0
\end{aligned}\right.\)
When x < 0, \(\frac{d y}{d x}\) = -1
which is the slope of the tangent at (-2, 2).
Equation of the required tangent is
y – 2 = (-1) (x + 2)
⇒ y = -x
⇒ x + y = 0.

Question 14.
What is the derivative of sec^-1 x with respect to x if x < -1?
Solution:
Let y = sec^-1 x
⇒ \(\frac{d y}{d x}\) = \(\frac{1}{|x| \sqrt{x^2-1}}\)
when x < (-1) we have |x| = -x
∴ \(\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right]_{\mathrm{x}<(-1)}\) = \(\frac{(-1)}{x \sqrt{x^2-1}}\)

Question 15.
Write the set of points where the function f(x) = x³ has relative (local) extrema.
Solution:
f(x) = x³
⇒ f'(x) = 3x²
f'(x) = 0
⇒ x = 0
But f”(x) = 6x = 0 when x = 0
∴ x = 0 is not a point of local extrema.
∴ Hence the set of points where f has relative extrema = Φ.

Question 16.
In which interval of x the function \(\frac{\ln \mathbf{x}}{x}\) is decreasing?
Solution:
Let f(x) = \(\frac{\ln x}{x}\)
⇒ f'(x) = \(\frac{x \cdot \frac{1}{x}-\ln x}{x^2}\) = \(\frac{1-\ln x}{x^2}\)
f is decreasing for
f'(x) < 0 ⇒ \(\frac{1-\ln x}{x^2}\) < 0
⇒ 1 – ln x < 0 (∵ x² > 0)
⇒ 1 < ln x
⇒ ln x > 1
⇒ x e (e, ∞).

(C) Short Type Questions With Answers

Question 1.
If y = \(e^{x^{e^{e^{e^{x^x…..}}}}}\), then find \(\frac{d \mathbf{y}}{\mathbf{d x}}\).
Solution:

Question 2.
If \(\frac{\mathbf{d}^2 \mathbf{y}}{\mathbf{d x}^2}\), if x = a cosΦ and y = b sinΦ.
Solution:
x = a cosΦ, y = b sinΦ

Question 3.
Find the point on the curve x² + y² – 4xy + 2 = 0, where the normal to the curve is parallel to the x – axis.
Solution:
Given equation of the curve is
x² + y² – 4xy + 2 = 0 … (1)
Differentiang we get

Question 4.
Find the intervals in which the function y = \(\frac{\ln x}{x}\) is increasing and decreasing.
Solution:

Question 5.
Differentiate y = tan^-1\(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\)
Solution:

Question 6.
Differentiate y = (sin y)^{sin 2x}
Solution:
y = (sin y)^{sin 2x}
⇒ In y = sin 2x ln (sin y)

Question 7.
Test the differentiability and continuity of the following function at x = 0:
f(x) = \(\left\{\begin{array}{cc}
\frac{1-e^{-x}}{x} & x \neq 0 \\
1 & x=0
\end{array}\right.\)
Solution:
Given function is


As LHD = RHD, we have the given function is differentiable at x = 0 and f'(0) = –\(\frac{1}{2}\).

Question 8.
Show that the sum of intercepts on the co-ordinate axes of any tangent to the curve
√x + √y = √a is constant.
Solution:
Given curve √x + √y = √a … (1)

Question 9.
Find the equation of the normal to the curve y = (log x)² at x = \(\frac{1}{e}\).
Solution:
Given equation of the curve is
y = (log x)²

Question 10.
If y = x + \(\frac{1}{x+\frac{1}{x+\cdots \cdots \infty}}\), find \(\frac{\mathbf{d y}}{\mathbf{d x}}\), the rhs being a valid expression.
Solution:

Question 11.
Differentiate sec^-1 (\(\frac{1}{2 x^2-1}\)) with respect to \(\sqrt{1-x^2}\).
Solution:

Question 12.
Find \(\frac{d \mathbf{y}}{\mathbf{d t}}\) , when y = sin^-1 \(\left(\frac{2 \sqrt{t}-1}{t^2}\right)\)
Solution:

Question 13.
Find \(\frac{\mathbf{d y}}{\mathbf{d x}}\) , if x^myⁿ = \(\left(\frac{\mathbf{x}}{\mathbf{y}}\right)^{\mathrm{m}+\mathrm{n}}\)
Solution:

Question 14.
If x = a sec θ, y = b tan θ, prove that:
\(\frac{d^2 y}{d x^2}\) = –\(\frac{b^4}{a^2 y^3}\)
Solution:

Question 15.
Find the interval where the following function is increasing:
y = sin x + cos x, x ∈ [0, 2π].
Solution:
y = sin x + cos x, x ∈ [0, 2π].

Question 16.
Find \(\frac{d y}{d x}\), when y^x = x^{sin y}.
Solution:
y^x = x^{sin y}
Taking log of both sides we get,
x ln y = (sin y) ln x
Differentiating with respect to x we get,

Question 17.
Show that \(\frac{d y}{d x}\) is independent of t,
if cos x = \(\sqrt{\frac{1}{1+t^2}}\), sin y = \(\frac{2 t}{1+t^2}\)
Solution:
cos x = \(\sqrt{\frac{1}{1+t^2}}\), sin y = \(\frac{2 t}{1+t^2}\)
Let t = tan θ
∴ cos x = cos θ, sin y = sin 2θ
⇒ y = 2θ, x = θ
⇒ y = 2x
⇒ \(\frac{d y}{d x}\) = 2 which is independent of t.

Question 18.
Show that 2sin x + tan x > 3x for all x ∈ (0, \(\frac{\pi}{2}\))
Solution:
Let f(x) = 2sin x + tan x – 3x
⇒ f'(x) = 2cos x + sec² x – 3
Let g(x) = f'(x) = 2cos x + sec² x – 3
⇒ g'(x) = -2sin x + 2sec² x tan x
= 2sin x (sec³ x – 1) ≥ 0
for all x ∈ (0, \(\frac{\pi}{2}\))
∴ g is an increasing function on (0, \(\frac{\pi}{2}\))
But g(0) = 0
∴ g(x) ≥ 0 for all x ∈ (0, \(\frac{\pi}{2}\))
⇒ f(x) ≥ 0 for all x ∈ (0, \(\frac{\pi}{2}\))
⇒ f is an increasing function on (0, \(\frac{\pi}{2}\))
But f(0) = 0
∴ f(x) ≥ 0 for all x ∈ (0, \(\frac{\pi}{2}\))
⇒ 2sin x + tan x ≥ 3x for x ∈ (0, \(\frac{\pi}{2}\))

Question 19.
Show that no two normals to a parabola are parallel.
Solution:
Let us consider the parabola y² = 4ax – (1) and A(at₁², 2at₁) and B (at₂², 2at₂) are any two points on its from (1)

As t₁ ≠ t₂ we have no two normals of a parabola are parallel.

Question 20.
Examine the differentiability of In x² for all real values of x.
Solution:
Let f(x) = In (x²) = 2 ln |x|
Clearly Dom f = R – {0}
Let any a ∈ Dom f.

Question 21.
Find the derivative of x^{sin x} with respect to x.
Solution:
Let y = x^{sin x} = e^{(sin x)/nx}

Question 22.
Differentiate sin^-1 \(\left(\frac{2 x}{1+x^2}\right)\) with respect to cos^-1 \(\left(\frac{1-x^2}{1+x^2}\right)\).
Solution:

Question 23.
Find the slope of the tangent to the curve x = 2(t – sin t), y = 2(1 – cos t) at t = \(\frac{\pi}{4}\).
Solution:

Question 24.
If cos y = x cos(a + y), then show that \(\frac{d y}{d x}\) = \(\frac{\cos ^2(a+y)}{\sin a}\).
Solution:

Question 25.
Find the extreme values of the function y = x + \(\frac{y}{x}\).
Solution:

Question 26.
Find the intervals where the following function is (i) increasing, (ii) decreasing:
f(x) = \(\begin{cases}\mathbf{x}^2+1, & x \leq-3 \\ x^3-8 x+13, & x>-3\end{cases}\)
Solution:
Given f(x) = \(\begin{cases}\mathbf{x}^2+1, & x \leq-3 \\ x^3-8 x+13, & x>-3\end{cases}\)
Clearly f is not differentiable at x = -3.
and f ‘(x) = \(\begin{cases}2 \mathrm{x}, & \mathrm{x}>-3 \\ 3 \mathrm{x}^2-8 & \mathrm{x}<-3 \\ \text { does not exist } & \text { for } \mathrm{x}=-3\end{cases}\)
Case – 1: x > -3
Clearly f'(x) > 0 for x > 0 and f'(x) < 0 for -3 < x < 0
∴ f is increasing in (0, ∞) and decreasing in (-3, 0).
Case – 2: x < -3
Clearly for x < -3, f(x) = 3x² – 8 > 0
i.e., f is increasing in (-∞, -3)
Thus f is increasing in (-∞, -3) ∪ (0, ∞) and decreasing in (-3, 0).

Question 27.
Prove that:
y = In tan \(\left(\frac{\pi}{4}+\frac{x}{2}\right)\) ⇒ \(\frac{d y}{d x}\) = sec x
Solution:

Question 28.
Differentiate with respect to x:
y = \(2^{x^2}\) + tan^-1 \(\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)\)
Solution:

Question 29.
Find the equation of the tangent to the curve x = y² – 1 at the point where the slope of the normal to the curve is (-2).
Solution:
Given equation of the curve is x = y² – 1.

Question 30.
Find \(\frac{d y}{d x}\) if y = \(\log _{\left(x^2\right)}\)3.
Solution:

Question 31.
Write why the function sin^-1 \(\frac{1}{\sqrt{1-x^2}}\) cannot be differentiated anywhere.
Solution:

⇒ 1 – x² ≥ 1
⇒ x = 0
∴ DOM of {0 3}
But f is not defined in a deleted interval of x = 0.
Hence f is not differentiable at x = 0 and hence not differentiable anywhere.

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(c)

Find derivatives of the following functions.
Question 1.
(x² +5)⁸
Solution:
y = (x² +5)⁸
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(x² +5)⁸
= 8(x² +5)⁷ × \(\frac{d}{d x}\)(x² +5) by chain rule
= 8(x² +5)⁷ . 2x
= 16x (x² +5)⁷

Question 2.
\(\frac{1}{\left(x^3+\sin x\right)^2}\)
Solution:

Question 3.
In (√x+1)
Solution:

Question 4.
sin 5x + cos 7x
Solution:
sin 5x + cos 7x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(sin 5x) + \(\frac{d}{d x}\)(cos 7x)
= cos 5x . \(\frac{d}{d x}\)(5x) – sin 7x . \(\frac{d}{d x}\)(7x)
= 5 cos 5x – 7 sin 7x

Question 5.
e^{sin t}
Solution:
y = e^{sin t}
\(\frac{d y}{d x}\) = \(\frac{d}{d t}\)(e^{sin t}) = e^{sin t} . \(\frac{d}{d t}\)(sin t)
= e^{sin t} . cos t

Question 6.
\(\sqrt{a x^2+b x+c}\)
Solution:
y = \(\sqrt{a x^2+b x+c}\)

Question 7.
\(\left(\frac{x+1}{x^2+3}\right)^{-3}\)
Solution:

Question 8.
sec (tan θ)
Solution:
y = sec (tan θ)
\(\frac{d y}{d θ}\) = \(\frac{d}{d θ}\) {sec (tan θ)}
= sec (tan θ) . tan (tan θ) . \(\frac{d}{d θ}\)(tan θ)
= sec (tan θ) . tan (tan θ) . sec² θ

Question 9.
sin \(\left(\frac{1-x^2}{1+x^2}\right)\)
Solution:

Question 10.
\(\sqrt{\tan (3 z)}\)
Solution:

Question 11.
tan³ x
Solution:
y = tan³x = (tan x)³
\(\frac{d y}{d x}\) = 3(tan x)². \(\frac{d}{d x}\)(tan x)
= 3tan² x . sec² x

Question 12.
sin⁴ x
Solution:
y = sin⁴x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(sin⁴ x)
= 4 (sin x)³ . \(\frac{d}{d x}\)(sin x)
= 4 sin³ x . cos x

Question 13.
sin² x cos² x
Solution:
y = sin² x cos² x = \(\frac{1}{4}\)sin² 2x
\(\frac{d y}{d x}\) = \(\frac{1}{4}\) \(\frac{d}{d x}\)(sin 2x)²
= \(\frac{1}{4}\). 2sin 2x . \(\frac{d}{d x}\)(sin 2x)
= \(\frac{1}{2}\) sin 2x . cos 2x . \(\frac{d}{d x}\)(2x)
= \(\frac{1}{2}\) sin 2x cos 2x . 2 = sin 2x . cos 2x

Question 14.
sin 5x cos 7x
Solution:
y = sin 5x cos 7x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(sin 5x) . cos 7x + \(\frac{d}{d x}\)sin 5x . (cos 7x)
= cos 5x . \(\frac{d}{d x}\)(5x) . cos 7x + sin 5x . (-sin 7x) . \(\frac{d}{d x}\)(7x)
= 5 cos 5x . cos 7x – 7 sin 5x . sin 7x

Question 15.
tan x cot 2x
Solution:
y = tan x. cot 2x
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(tan x) . cot 2x + tan x . \(\frac{d}{d x}\)(cot 2x)
= sec² x / cot 2x + tan x . (-cosec² x) \(\frac{d}{d x}\)(2x)
= sec² x . cot 2x – 2 tan x . cosec² x

Question 16.
\(\sqrt{\sin \sqrt{x}}\)
Solution:

Question 17.
\(\sqrt{\sec (2 x+1)}\)
Solution:

Question 18.
cosec (ax + b)²
Solution:
y = cosec (ax + b)²
\(\frac{d y}{d x}\) = – cosec (ax + b)² cot (ax + b)² . \(\frac{d}{d x}\)(ax + b)²
[ ∵ \(\frac{d}{d x}\)(cosec u) = -cosec u . cot u . \(\frac{d u}{d x}\)
= – cosec (ax + b)² . cot (ax + b)² . 2(ax + b) . \(\frac{d}{d x}\)(ax + b)
[ ∵ \(\frac{d}{d x}\)(u²) = 2u \(\frac{d u}{d x}\)
= – cosec (ax + b)² . cot (ax + b)² . 2(ax + b) . a
= – 2a (ax + b) cosec (ax + b)² cot (ax + b)²

Question 19.
a^{In x}
Solution:
y = a^{In x} . In a . \(\frac{d}{d x}\)(In x)
[ ∵ \(\frac{d}{d x}\)(a^u) = a^u . In a . \(\frac{d u}{d x}\)
= a^{In x} . In a . \(\frac{1}{x}\) = \(\frac{a^{\ln x} \ln a}{x}\)

Question 20.
\(a^{x^2} b^{x^3}\)
Solution:

Question 21.
In tan x
Solution:

Question 22.
\(5^{\sin x^2}\)
Solution:

Question 23.
In tan\(\left(\frac{\pi}{4}+\frac{x}{2}\right)\)
Solution:

Question 24.
\(\sqrt{\left(a^{\sqrt{x}}\right)}\)
Solution:

Question 25.
In (e^nx + e^-nx)
Solution:
y = In (e^nx + e^-nx)
\(\frac{d y}{d x}\) = \(\frac{1}{e^{n x}+e^{-n x}}\) . \(\frac{d}{d x}\) (e^nx + e^-nx)
= \(\frac{n\left(e^{n x}-e^{-n x}\right)}{e^{n x}+e^{-n x}}\)

Question 26.
\(e^{\sqrt{a x}}\)
Solution:

Question 27.
\(\sqrt{\log x}\)
Solution:
y = \(\sqrt{\log x}\)
\(\frac{d y}{d x}\) = \(\frac{1}{2 \sqrt{\log x}}\) . \(\frac{d}{d x}\)(log x)
= \(\frac{1}{2 \sqrt{\log x}}\) . \(\frac{1}{x}\)

Question 28.
e^{sin x} – a^{cos x}
Solution:
y = e^{sin x} – a^{cos x}
\(\frac{d y}{d x}\) = \(\frac{d}{d x}\)(e^{sin x}) – \(\frac{d}{d x}\)(a^{cos x})
= e^{sin x} . \(\frac{d}{d x}\)(sin x) – a^{cos x} . In a . \(\frac{d}{d x}\)(cos x)
= e^{sin x} . cos x + a^{cos x} . In a . sin x

Question 29.
\(\frac{e^{3 x^2}}{\ln \sin x}\)
Solution:

Question 30.
Prove that
\(\frac{d}{d x}\left[\frac{1-\tan x}{1+\tan x}\right]^{\frac{1}{2}}\) = 1 / \(\sqrt{\cos 2 x}\) (cos x + sin x)
Solution:

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(c)

Question 1.
Find the intervals where the following functions are (a) increasing and (b) decreasing.
(i) y = sin x, x ∈ [0, 2π]
Solution:

(ii) y = In x, x ∈ R₊
Solution:
⇒ \(\frac{d y}{d x}\) = \(\frac{1}{x}\) > 0
for all x ∈ R₊.
Thus y = In x is increasing in (0, ∞) and decreasing nowhere.

(iii) y = a^x, a > 0, x ∈ R
Solution:
⇒ \(\frac{d y}{d x}\) = a^x In a > 0 for all x ∈ R provided a > 1.
The function is increasing in (-∞, ∞) is a > 1.
Again \(\frac{d y}{d x}\) = a^x In a < 0 if 0 < a < 1.
Thus the function is decreasing in (-∞, ∞) if (0 < a < 1)

(iv) y = sin x + cos x, x ∈ [0, 2π]
Solution:

(v) y = 2x³ + 3x² – 36x – 7
Solution:
\(\frac{d y}{d x}\) = 6x² – 6x – 36
y is increasing if \(\frac{d y}{d x}\) > 0
⇒ x² + x – 6>0
⇒ x² + 3x – 2x – 6>0
⇒ x (x + 3) -2 (x + 3) > 0
⇒ (x + 3) (x – 2) > 0
for x > 2 or x < -3 and (x + 3 ) (x – 2) < 0 for -3 < x < 2
∴ The function is increasing in (-∞, -3) ∪ (2, ∞) and is decreasing in (-3, 2).

(vi) y = \(\frac{1}{x-1^{\prime}}\) x ≠ 1
Solution:
\(\frac{d y}{d x}\) = \(\frac{-1}{(x-1)^2}\)
The function is increasing if \(\frac{d y}{d x}\) > 0
⇒ \(\frac{-1}{(x-1)^2}\) > 0 which is impossible because \(\frac{-1}{(x-1)^2}\) is always -ve for all x ≠ 1. So the function is increasing nowhere. It is decreasing in R – {1}

(vii) y = \(\left\{\begin{array}{cc}
x^2+1, & x \leq-3 \\
x^3-8 x+13, & x>-3
\end{array}\right.\)
Solution:

(viii) y = 4x² + \(\frac{1}{x}\)
Solution:

(ix) y = (x – 1)² (x + 2)
Solution:
\(\frac{d y}{d x}\) = 2 (x – 1) (x + 2) + (x – 1)²
= (x – 1) (2x + 4 + x – 1)
= (x – 1 ) (3x + 3)
= 3 (x – 1) (x + 1 )
The function is increasing if \(\frac{d y}{d x}\) > 0.
⇒ (x + 1) (x – 1) > 0
⇒ x < -1 or x > 1
∴ The function is increasing in (-∞, -1) ∪ (1, ∞)
It is decreasing in (-1, 1).

(x) y = \(\frac{\ln x}{x}\), x > 0
Solution:

(xi) y = tan x – 4 (x – 2), x ∈ (-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\))
Solution:

(xii) y = sin 2x – cos 2x, x ∈ [0, 2π]
Solution:

Question 2.
Give a rough sketch of the functions given in question 1.
Solution:
Do yourself.

Question 3.
Show that the function \(\frac{e^x}{x^p}\) is strictly increasing for x > p > 0.
Solution:

Question 4.
Show that 2 sin x + tan x ≥ 3x for all x ∈ (0, \(\frac{\pi}{2}\)).
Solution:
Let f(x) = 2 sin x + tan x – 3x
Then f(x) = 2 cos x + sec² x – 3

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Ex 6(d) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 6 Probability Exercise 6(d)

Question 1.
State which of the following is the probability distribution of a random variable X with reasons to your answer:
(a)

X = x	0	1	2	3	4
p(x)	0.1	0.2	0.3	0.4	0.1

(b)

X = x	0	1	2	3
p(x)	0.15	0.35	0.25	0.2

(c)

X = x	0	1	2	3	4	5
p(x)	0.4	R	0.6	R2	0.7	0.3

Solution:
(a) As ∑p_i = 1.1 > 1, the given distribution is not a probability distribution.
(b) As ∑p_i = 0.95 < 1, the given distribution is not a probability distribution.
(c) As the value of R is not known, the given distribution is not a probability distribution.

Question 2.
Find the probability distribution of number of doublets in four throws of a pair of dice. Find also the mean and the variance of the number of doublets.
Solution:
Let the random variable X represents the number of doublets in 4 throws of a pair of dice.
X can take values 0, 1, 2, 3, 4
In a single throw of two dice
X can take values 0, 1, 2, 3, 4
In a single throw of two dice.
P(doublet) = \(\frac{6}{36}\)
P(non-doublet) = 1 – P (doublet) = \(\frac{5}{36}\)
Clearly the given experiment is a binomial experiment with n = 4,
p = \(\frac{1}{6}\), q = \(\frac{5}{6}\)

N.B. We can use the definition to find mean and variance.

Question 3.
Four cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Find the probability distribution of the number of aces. Calculate the mean and variance of the number of aces.
Solution:
Let the random variable X denotes the number of aces.
Thus X can take values 0, 1, 2, 3, 4.
Clearly the given experiment is a binomial experiment with n = 4

N.B. We can use the definition to find mean and variance.

Question 4.
Find the probability distribution of
(a) number of heads in three tosses of a coin.
Solution:
Let the random variable X denotes the number of heads in three tosses of a coin.
X can take values 0, 1, 2, 3
In one toss p(H) = \(\frac{1}{2}\), p(T) = \(\frac{1}{2}\)
This experiment is a binomial experiment with n = 3, p = \(\frac{1}{2}\) and q = \(\frac{1}{2}\)

(b) number of heads in simultaneous tosses of four coins.
Solution:
When 4 coins tossed simultaneously, let the random variable X denotes the number of heads.
X can take values 0, 1. 2, 3, 4
This experiment is a binomial experiment with n = 4, p = \(\frac{1}{2}\) and q = \(\frac{1}{2}\)

Question 5.
A biased coin where the head is twice as likely to occur as the tail is, tossed thrice. Find the probability distribution of number of heads.
Solution:
Let in a toss P(T) = x
According to the question P(H) = 2x
Now 2x + x = 1
⇒ x = \(\frac{1}{3}\)
Thus P(T) = \(\frac{1}{3}\), P(H) = \(\frac{3}{3}\)
Clearly the given experiment is a binomial experiment with n = 3, p = P(H) = \(\frac{2}{3}\), q = P(T) = \(\frac{1}{3}\)
Let the random variable X denotes the number of heads in three throws of that coin.
X can take values 0, 1, 2, 3

Question 6.
Find the probability distribution of the number of aces in question no. 3 if the cards are drawn successively without replacement.
Solution:
Total number of cards = 52
Number of aces = 4
Number of cards drawn = 4 (one by one without replacement)
Let X = the random variable of number of aces drawn.
X can take values 0, 1, 2, 3 or 4

Question 7.
From a box containing 32 bulbs out of which 8 are defective 4 bulbs are drawn at random successively one after another with replacement. Find the probability distribution of the number of defective bulbs.
Solution:
Total number of bulbs = 32
Number of defective bulbs = 8
∴ Number of nondefective bulbs = 24
Number of bulbs drawn = 4 (one after another with replacement)
Clearly the experiment is a binomial experiment with n = 4
p = p ((drawing a defective bulb) = \(\frac{8}{32}\) = \(\frac{1}{4}\)
q = p (drawing a non defective bulb) = \(\frac{3}{4}\)
Let the random variable X denotes the number of defective bulbs.
∴ X can take values 0, 1, 2, 3, 4.

Question 8.
A random variable X has the following probability distribution.

X = x	0	1	2	3	4	5
p(x)	0	R	2R	3R	3R	R

Determine
(a) R
(b) P(X < 4)
(d) P(2 ≤ X ≤ 5)
(c) P(X ≥ 2)
Solution:
(a) Clearly ∑P_i = 1
⇒ R + 2R + 3R + 3R + R = 1

Question 9.
Find the mean and the variance of the number obtained on a throw of an unbiased coin.
Solution:
When an unbiased coin is tossed we donot get any number.
p(getting a number) = 0
Thus mean = variance = 0
If instead of coin it would be an unbiased die
Then let X = The number obtained in the throw.
X can take values 1, 2, 3, 4, 5 or 6.
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4)

Question 10.
A pair of coins is tossed 7 times. Find the probability of getting
(i) exactly five tails
(ii) at least five tails
(iii) at most five tails
Solution:
Clearly the given experiment is a binomial experiment with

Question 11.
If a pair of dice is thrown 5 times then find the probability of getting three doublets.
Solution:
In a single through of a pair of dice
p (a doublet) = \(\frac{6}{36}\) = \(\frac{1}{6}\)
p (a non doublet) = 1 – p (a doublet) = 1 – \(\frac{1}{6}\) = \(\frac{5}{6}\)
Clearly the given experiment is a binomial experiment with n = 5
p = \(\frac{1}{6}\) and q = \(\frac{5}{6}\)
p(3 doublets in 5 throw) = ⁵C₃ p³q²
= 10. \(\left(\frac{1}{6}\right)^3\) \(\left(\frac{5}{6}\right)^2\) = \(\frac{250}{6^5}\) = \(\frac{125}{3888}\)

Question 12.
Four cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that:
(i) all the four cards are diamonds
(ii) only two cards are diamonds
(iii) none of the cards is a diamond.
Solution:
Out of 52 cards there are 13 diamonds. 4 cards are drawn one by one with replacement.
When we draw a card

Question 13.
In an examination, there are twenty multiple-choice questions each of which is supplied with four possible answers. What is the probability that a candidate would score 80% or more in the answers to these questions?
Solution:
Total number of questions = 20 for each question
p (correct answer) = \(\frac{1}{4}\)
p (wrong answer) = \(\frac{3}{4}\)
p (the score is > 80%)
= p (no. of correct answer > 16)
= p (16 correct answers)
+ p(17 correct answers)
+ p (18 correct answers)
+ p (19 correct answers)
+ p (20 correct answers)

Question 14.
A bag contains 7 balls of different colours. If five balls are drawn successively with replacement then what is the probability that none of the balls drawn is white?
Solution:
Total number of balls = 7 (The colours are different)
Number of balls drawn = 5 (one by one with replacement)
Case – 1
(If a white coloured ball is not present)
p(non is white) = 1
Case – 2
(If one ball is white).
In one draw p(white ball) = \(\frac{1}{7}\)
p (non white ball) = \(\frac{6}{7}\)
When 5 balls are drawn
p (none of the balls drawn is white) = ⁵C₀ \(\left(\frac{1}{7}\right)^0\) \(\left(\frac{6}{5}\right)^5\) = \(\left(\frac{6}{7}\right)^5\)

Question 15.
Find the probability ofthrowing at least 3 sixes in 5 throws of a die.
Solution:
In one throw of a die

Question 16.
The probability that a student securing first division in an examination is \(\). What is the probability that out of 100 students twenty pass in first division?
Solution:
Clearly the given experiment is a binomial experiment with

Question 17.
Sita and Gita throw a die alternatively till one of them gets a 6 to win the game. Find their respective probability of winning if Sita starts first.
Solution:
In one throw of a die
p (getting a 6) = \(\frac{1}{6}\)
p (not getting a 6) = \(\frac{5}{6}\)
If Sita begins the game she may win in first round or second round or third round, etc.
Thus p(Sita wins)

Question 18.
If a random variable X has a binomial distribution B(8, \(\frac{1}{2}\)) then find X for which the outcome is the most likely. [Hint: Find X=x for which P(X = x) is the maximum, x = 0, 1, 2, 3,…….. 8.]
Solution:
Given binomial distribution is B(8, \(\frac{1}{2}\))
Thus n = 8, p = \(\frac{1}{2}\), q = \(\frac{1}{2}\)
We shall find X which is most likely i.e. for which p(x = r) is maximum where r = 0, 1, 2, ….., 8
As p = q, the probability is maximum when ⁸Cr is maximum
But ⁸Cr, r = 0, 1, 2, ….. ,8 is maximum when r = 4.
Thus the most likely outcome is x = 4.

Odisha State Board Elements of Mathematics Class 12 CHSE Odisha Solutions Chapter 7 Continuity and Differentiability Ex 7(j) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 7 Continuity and Differentiability Exercise 7(j)

Test differentiability and continuity of the following functions.
Question 1.
\(\left|1-\frac{1}{x}\right|\) at x = 1
Solution:

Question 2.
x² |x| at x = 0.
Solution:
Let f(x) = x² |x|
Then f(0) = 0

As L.H.D.=R.H.D., f(x) is differentiable at x = 0. We know that every differentiable function is continuous. So f(x) is also continuous at x = 0.

Question 3.
f(x) = tan x at x = \(\frac{\pi}{2}\)
Solution:
f(x) = tan x
f(\(\frac{\pi}{2}\)) = tan \(\frac{\pi}{2}\) which does not exist.
So f(x) is neither continuous not differentiable.

Question 4.
f(x) = cot x at x = \(\frac{\pi}{2}\).
Solution:

Question 5.
f(x) = |sin x| at x = π.
Solution:
Differentiability:
f(x) = |sin x|, x = π
f(π) |sin π| = 0

Question 6.
f(x) = latex]\frac{x}{1+|x|}[/latex] at x = 0
Solution:

As L.H.D. = R.H.D., f(x) is differentiable at x = 0.
Every differentiable function is continuous.
So f(x) is also continuous at x = 0.

Question 7.
f(x) = \(\begin{cases}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{cases}\) at x = 0
Solution:

Question 8.
f(x) = \(\begin{cases}\frac{1-e^{-x}}{x}, & x \neq 0 \\ 1 & x=0\end{cases}\) at x = 0
Solution:
f(0) = 1

As L.H.D. = R.H.D., f(x) is differentiable at the origin. Again every differentiable function is continuous. So f(x) is continuous at the origin.

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Ex 8(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 8 Application of Derivatives Exercise 8(b)

Question 1.
Find the equations to the tangents and normals to the following curves at the indicated points.
(i) y = 2x² + 3 at x = -1
Solution:
y = 2x² + 3 at x = -1
⇒ \(\frac{d y}{d x}\) = 4        ∴\(\left.\frac{d y}{d x}\right]_{-1}\) = -4
Thus slope of the tangent at x = -1 is -4.
Angin for x = -1 , y = 2 + 3 = 5
The point of contact is (-1, 5).
Equation of the tangent is y – 5 = -4 (x + 1)
⇒ 4x + y – 1 = 0
Slope of the normal = \(\frac{1}{4}\)
Eqn. of the normal is y – 5 = \(\frac{1}{4}\) (x + 1)
⇒ 4y – 20 = x + 1
⇒ x – 4y + 21 = 0

(ii) y = x³ – x at x = 2
Solution:
y = x³ – x at x = 2
When x = 2, y = 2³ – 2 = 6
The point is (2, 6).
Again \(\frac{d y}{d x}\) = 3x² – 1
\(\left.\frac{d y}{d x}\right]_{x=2}\) = 12 – 1 = 11
Slope of the tangent = 11.
Slope of the normal = –\(\frac{1}{11}\)
Equation of the tangent is
y – 6 = 11 (x – 2)
⇒ 11x – y – 16 = 0
Equation of the normal is
y – 6 = –\(\frac{1}{11}\)
⇒ 11y – 66 = – x + 2
⇒ x + 11y – 68 = 0

(iii) y = √x + 2x + 6 at x = 4
Solution:
y = √x + 2x + 6 at x = 4
For x = 4, y = 2 + 8 + 6 = 16
The point is (4, 16)
Again \(\frac{d y}{d x}\) = \(\frac{1}{2 \sqrt{x}}\) + 2
\(\left.\frac{d y}{d x}\right]_{x=4}\) = \(\frac{1}{4}\) + 2 = \(\frac{9}{4}\)
Slope of the tangent = \(\frac{9}{4}\)
Slope of the normal = –\(\frac{4}{9}\)
Equation of the tangent is
y – 16 = \(\frac{9}{4}\)(x – 4)
⇒ 9x – 4y + 28 = 0
Equation of the normal is
y – 16 = – \(\frac{4}{9}\)(x – 4)
⇒ 9y – 144 = -4x + 16
⇒ 4x + 9y – 160 = 0

(iv) y = √3 sin x + cos x at x = \(\frac{\pi}{3}\)
Solution:

(v) y = (log x)² at x = \(\frac{1}{e}\)
Solution:

(vi) y = \(\frac{1}{\log x}\) at x = 2
Solution:

(vii) y = xe^-x at x = 0
Solution:
y = xe^-x at x = 0
For x = 0, y = 0
The point is (0, 0).
\(\frac{d y}{d x}\) = e^-x + x . (-e^-x)
\(\left.\frac{d y}{d x}\right]_{x=0}\) = 1
Slope of the tangent = 1
Slope of the normal = -1
Equation of the tangent is y = x and equation of the normal is y = -x

(viii) y = a (θ – sin θ), y = a (1 – cos θ) at θ = \(\frac{\pi}{2}\)
Solution:
y = a (θ – sin θ), y = a (1 – cos θ) at θ = \(\frac{\pi}{2}\)

(ix) \(\left(\frac{x}{a}\right)^{2 / 3}\) + \(\left(\frac{y}{b}\right)^{2 / 3}\) = 1 at (a cos³ θ, b sin³ θ)
Solution:
\(\left(\frac{x}{a}\right)^{2 / 3}\) + \(\left(\frac{y}{b}\right)^{2 / 3}\) = 1 at (a cos³ θ, b sin³ θ)

Question 2.
Find the point on the curve y² – x² + 2x – 1 = 0 where the tangent is parallel to the x – axis.
Solution:
Given curve is y² – x² + 2x – 1 = 0 … (1)

Putting x = 1 in (1) we get
y² – 1 + 2 – 1 = 0 ⇒ y = 0
∴ The point is (1, 0).

Question 3.
Find the point (s) on the curve x = \(\frac{3 a t}{1+t^2}\), y = \(\frac{3 a t^2}{1+t^2}\) where the tangent is perpendicular to the line 4x + 3y + 5 = 0.
Solution:

Question 4.
Find the point on the curve x² + y² – 4xy + 2 = 0 where the normal is parallel to the x axis.
Solution:
x² + y² – 4xy + 2 = 0

Question 5.
Show that the line y = mx + c touches the parabola y² = 4ax if c = \(\frac{a}{m}\).
Solution:
Given line is y = mx + c … (1)
Given parabola is y² = 4ax

Question 6.
Show that the line y = mx + c touches the ellipse
\(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1 if c² = a²m² + b²
[Hints: Find equation to tangent at a point (x’, y’) of the curve and compare it with
y = mx + c].
Solution:

Question 7.
Show that the sum of the intercepts on the coordinate axes of any tangent to the curve
√x + √y = √a is constant.
Solution:

Question 8.
Show that the curves y = 2^x and y = 5^x intersect at an angle
tan^-1 \(\left|\frac{\ln \left(\frac{5}{2}\right)}{1+\ln 2 \ln 5}\right|\)
(Note: Angle between two curves is the angle between their tangents at the point of intersection)
Solution:
Given curves are
y = 2^x … (1)
and y = 5^x … (2)
Differentiating (1) we get \(\frac{d y}{d x}\) = 2^x . In 2
Differentiating (2) we get \(\frac{d y}{d x}\) = 5^x . In 2
Slope of the tangent to the first curve at (x, y) = 2^x . In 2
Slope of the tangent to the second curve at (x, y) = 5^x . In 2
Solving (1) and (2) we get 2^x = 5^x ⇒ x = 0
For x = 0, y = 1
∴ The point of intersection is (0, 1).
At (0, 1) slope of the 1st tangent = In 2
Slope of the second tangent = In 5
If θ is the angle between two tangents
then \(\frac{\ln 5-\ln 2}{1+\ln 5 \cdot \ln 2}\) = \(\frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\)
= tan^-1 \(\left(\frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\right)\)
We know that angle between two curves is the angle between their tangents at the point of intersection.
Hence the two curves intersect at an angle.
\(\left(\tan ^{-1} \cdot \frac{\ln \frac{5}{2}}{1+\ln 5 \cdot \ln 2}\right)\)

Question 9.
Show that the curves ax² + by² =1 and a’x² + b’y² = 1. Intersect at right angles if
\(\frac{1}{a}\) – \(\frac{1}{a}\) = \(\frac{1}{a^{\prime}}\) – \(\frac{1}{b^{\prime}}\).
Solution:
Given curves are
ax²+ by²= 1 … (1)
and a’x² + b’y² = 1 … (2)
Differentiating (1) we get

Question 10.
Find the equation of the tangents drawn from the point (1, 2) to the curve.
y² – 2x³ – 4y + 8 = 0
Solution:
Given curve is
y² – 2x³ – 4y + 8 = 0 … (1)

⇒ -(k – 2)² = 3h² – 3h³
⇒ (k – 2)² = 3h³ – 3h²
Putting it in (2) we get
k² – 2h³ – 4k + 8 = 0
⇒ k² – 4k + 4 – 2h³ + 4 = 0
⇒ (k – 2)² – 2h³ + 4 = 0
⇒ 3h³ – 3h² – 2h³ + 4 = 0
⇒ h³ – 3h² + 4 = 0
⇒ h³ + h² – 4h² + 4 = 0
⇒ h² (h + 1) – 4 (h² – 1) = 0
⇒ h² (h + 1) – 4 (h + 1) (h – 1) = 0
⇒ (h + 1) (h² – 4h + 4) = 0
⇒ (h + 1) (h – 2)² = 0
⇒ h = -1, 2.
For h = -1, k is imaginary.
For h = 2, k = 2 + 2√3
The point at which the tangent is drawn is (2, 2 + 2√3).
Slope of the tangents
= \(\frac{3 h^2}{k-2}\) = \(\frac{12}{\pm 2 \sqrt{3}}\) = ± 2√3
Equations of the tangental
y – (2 ± 2√3) = ± 2√3(x – 2)

Question 11.
Show that the equation to the normal to x^2/3 + y^2/3 = a^2/3 is y cos θ – x sin θ = a cos 2θ where θ is the inclination of the normal to x – axis.
Solution:

The point is (a² sin³ θ, a² cos³ θ).
Equation of the normal is
y – a² cos³ θ = tan θ (x – a² sin³ θ)
⇒ y cos θ – a² cos⁴ θ = x sin θ – a² sin⁴ θ
⇒ y cos θ – x sin θ = a² (cos⁴ θ – sin⁴ θ)
= a² (cos² θ + sin² θ) (cos² θ – sin² θ)
= a² cos 2θ.
∴ Equation of the normal is
y cos θ – x sin θ = a² cos 2θ.(Proved)

Question 12.
Show that the length of the portion of the tangent to x^2/3 + y^2/3 = a^2/3 intercepted between the axes is constant.
Solution:

Question 13.
Find the tangent to the curve y = cos (x + y), 0 < x ≤ 2π which is parallel to the line
x + 2y = 0.
Solution:
y = cos (x + y) … (1)

Question 14.
If tangents are drawn from the origin to the curve y = sin x then show that the locus of the points of contact is x²y² = x² – y²
Solution:
Given curve is
y = sin x … (1)
\(\frac{d y}{d x}\) = cos x
Slope of the tangent to the cure (1) at any point (x, y) is cos x.
Let (h, k) be the point of contact.
Then slope of the tangent at (h, k) = cos h.
Equation of the tangent to the curve (1) at (h, k) is
y – k = cos h (x – h) … (2)
If the tangent is drawn from the origin then -k = -h cos h
⇒ k = h cos h … (3)
As (h, k) is the poitn of contact then we have
k = sin h … (4)
From (3) we get,
k = h\(\sqrt{1-\sin ^2 h}\) = h\(\sqrt{1-k^2}\) by (4).
Squaring both sides we get
k² = h² (1 – k²)
⇒ k² = h² – h²k²
⇒ h²k² = h² – k²
The locus of (h, k) is x².y² = x² – y² (Proved)

Question 15.
Find the equation of the normal to the curve given by
x = 3 cos θ – cos³ θ
y = 3 sin θ – sin³ θ at θ = \(\frac{\pi}{4}\)
Solution:
x = 3 cos θ – cos³ θ
y = 3 sin θ – sin³ θ
Differentiating we get
\(\frac{d x}{d \theta}\) = -3 sin θ + 3 cos² θ . sin θ
\(\frac{d x}{d \theta}\) = 3 cos θ – 3 sin² θ . cos θ

Question 16.
If x cos α + y sin α = p is a tangent to the curve \(\left(\frac{x}{a}\right)^{\frac{n}{n-1}}\) + \(\left(\frac{y}{b}\right)^{\frac{n}{n-1}}\) = 1
then show that (a cos α)ⁿ + (b sin α)ⁿ = pⁿ
Solution:
Given straight line is x cos α + y sin α = p … (1)
Given curve is

Question 17.
Show that the tangent to the curve
x = a (t – sin t), y = at (1 + cos t) at t = \(\frac{\pi}{2}\) has slope. (1 – \(\frac{\pi}{2}\))
Solution:
Given curve is x = a (t – sin t)
y = at (1 + cos t)

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 6 The Past Simple and the Past Perfect Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 6 The Past Simple and the Past Perfect

SECTION – 1

Look at the sentences below.
(a) I reached the hostel in the morning and found that somebody had broken into my room during the night.
(b) She said that her friend had published a book.
(c) He had lived in this town for ten years; then he migrated to Japan.
Can you find the Past Perfect Tense in each sentence? Note that the sentence in which it occurs refers to two actions — the action expressed by the Past Perfect and another action expressed by the Past Simple. Of the two actions, which takes place earlier and which takes place later? List them below.

(a) (1) I reached the hotel in the morning and found. (later).
(2) that somebody had broken (earlier action) into my room during the night.
(b) (1) She said _________(later action).
(2) that her friend had published a book _________(earlier action).
(c) (1) _________then he migrated to Japan _________(later action).
(2) He had lived in this town for ten years, (earlier action).
Can you answer now? Which action does the Past Perfect refer to — the earlier one or the later one? Which action does the Past Simple refer to?
Answer:
The earlier action refers to Past Perfect and the later one refers to Past Simple Tense.

Activity – 23
Combine each pair of sentences below into a single sentence, using the Past Perfect to show which action took place earlier. (You may have to use words like after; when etc.
(a) (i) I finished my homework.
(ii) Then I went to buy a pen.
_____________________________________.
(b) (i) Then the doctor gave some medicine to the patient,
(ii) Then the patient regained his senses.
_____________________________________.
(c) (i) I read a few pages from the book.
(ii) After that I returned it to the librarian.
_____________________________________.
(d) (i) I worked in the garden for some time.
(ii) After that I had my breakfast.
____________________________
(e) (i) He left the place in a hurry.
(ii) After that his friend arrived.
____________________________
(f) (i) The young girl finished shopping.
(ii) Then she met with an accident.
____________________________
(g) (i) The thief ran away with the gold.
(ii) After that the police arrived.
____________________________
Answer:
(a) After I had finished my homework, I went to buy a pen.
(b) After the doctor had given some medicine to the patient, the patient regained his senses.
(c) After I had read a few pages from the book, I returned to the librarian.
(d) I had worked in the garden for some time before I had my breakfast.
(e) When he had left the place in a hurry. his friend arrived.
(f) After the young girl had finished shopping. she met with an accident.
(g) The thief had run away with the gold before the police arrived.

Activity — 24
Jatin arrived late at different places yesterday. What did he find when he arrived
at each place?
Example — When he arrived at the cricket stadium, the game had ended.
(a) the hank it / already I close.
____________________________
(b) his uncle’s house his uncle I go the sleep.
____________________________
(c) the bus stops the bus I already / leave.
____________________________
(d) book shop the book he wanted/sold out already.
_________________________________
(e) the club his friends/leave.
_________________________________
(f) the hostel everyone / go to bed.
_________________________________
Answer:
(a) When he arrived at the bank, it had already closed.
(b) When he got to his uncle’s house, his uncle had gone to sleep.
(c) When he reached the bus stop, the bus had already left.
(d) When he came to the bookshop, the book he wanted had been sold out already.
(e) When he arrived at the club, his friends had left.
(f) When he came to the hostel, everyone had gone to bed.

Activity – 25
Use the verb supplied in brackets in the appropriate form.
(a) We went to Anil’s house and _____________(knock) on the door but there _____________ (be) no answer. Either he _____________ (go) out or he _____________ (not want) to see anyone.
(b) Sadhan _____________(go) for a walk yesterday because the doctor _____________(tell) him last week that he _____________(need) exercise.
(c) A : _____________(Seema / arrive) at the party in time last night ?
B: No, she was late. By the time, we got there, everyone _____________(leave).
Answer:
(a) We went to Anil’s house and knocked on the door but there was no answer. Either he had gone out or he did not want to see anyone.
(b) Sadhan went for a walk yesterday because the doctor told him last week that he needed exercise.
(c) A: Did Seema arrive at the party in time last night?
B: No, she was late. By the time, we got there, everyone had left.

Odisha State Board CHSE Odisha Class 12 Invitation to English 3 Solutions Writing Reporting Events and Business Matters Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Writing Reporting Events and Business Matters

8.1 Reporting Events

Generally, we come across events in daily newspapers. Reporting events requires a special skill in presenting the event accurately, concisely, and authentically. Various types of events take different forms of expression.

Now read the following news report from a staff reporter of a newspaper.

Answer these questions on the news item :

Question 1.
Who has been sentenced?
Answer:
A doctor belonging to Ranchi has been sentenced.

Question 2.
How long is the sentence?
Answer:
The sentence is for five years’ rigorous imprisonment.

Question 3.
Why has he been sentenced?
Answer:
He has been sentenced because he was caught in the act of removing a kidney of one Nasir Ali when he was unconscious.

Question 4.
When was the case detected?
Answer:
The case was detected by the police on May 8, 2008.

Question 5.
What was the doctor doing?
Answer:
The doctor was trying to remove the kidney of the unconscious Nasir Ali.

Activity – 1

A report about smuggling of fake currency notes appeared in The Times of India on 2 March 2009. The bare facts are given below. Write the report, using these facts.
Three Pakistani nationals have been arrested.
They were arrested on Monday.
Place of arrest, Amritsar.
They were attempting to smuggle fake currency notes.
The value of the currency notes was Rs. 20 lakh.
They were travelling by the Samjhauta Express.
The customs officials seized the fake currency notes.
This was the fifth major seizure.

Answer:
Fake Currency Notes Siezed The Times of India News Service:
Amritsar March 2 : Custom officials seized currency notes worth Rs. 20 lakhs from three Pakistani nationals who were travelling by Samjhauta Express. They were arrested on Monday at Amritsar. This was the custom officials fifth major seizure.

Activity – 2

A newspaper reporter sent a report of an incident on March 12, 2009 over telephone. A sub-editor in the newspaper office, who received the telephone message, took the following notes. Use them to write up a report for the newspaper.
Five pick-pockets caught red-handed
In north Calcutta
Wednesday
Robbing passengers on a private bus
45 wallets and jewellery recovered
police arrested them.

Answer:
Pick-pockets caught Red-handed
Calcutta March 12 : Police caught five pick-pockets in north Calcutta this Wednesday while they were trying to flee after having robbed passengers on a private bus. 45 wallets and jewellery were recovered from them. Police arrested them.

Activity – 3

There was a train accident in the area where you work as a news reporter. You went to the spot and talked to different people including some of the passengers. You also met the railway officials. The following are the points you noted down.
Train accident at Retang at 7.30 p.m. on 11 April.
Three bogies of the East Coast Express derailed.
7 bodies recovered so far
25 injured sent to hospital in Cuttack; 5 serious cases.
Rescue operations still on.
More bodies suspected trapped inside wreakage.

Answer:
Train Derailed : Many Feared Dead, 25 Injured.
Retang, April 1 1 : Three bogies of the East Coast Express derailed here at 7 30 p.m. this morning. 7 bodies have been recovered so far but many are feared to be still trapped under the wreckage. The injured have been sent to the SC B Hospital Cuttack Oi 25 injured, five are in an extremely serious condition. Rescue operations are still on.

Activity – 4

Below you can see a picture of an incident that happened in front of Badanibadi Bus Stand, Cuttack. Report the incident, using the hints given below the picture.

Hints :
Badanibadi, Cuttack  12 December, 2008.  Morning 9
bulls ready to fight traffic held up
Answer:
Bulls Hold up Traffic
Cuttack 12 : Traffic was held up for almost an hour from 9.00 am this morning at Badanibadi as two bulls were poised to fight in the middle of the road. The bull had locked horns while bystanders cheered them.

Activity – 5

Very often, as a students ’ representative, you may have to read out a report in a function, ceremony, meeting etc. At that time, you don’t have to make the mention of the place, date, etc. as is done in writing a news report. It is normally written like an essay.

A. The Minister of Education was the Chief Guest in the Annual Day Celebration of your college. Write a news report to be sent to a newspaper.
Answer:

To,
The Chief Editor,
The Times of Odisha, Bhubaneswar

Report on Annual Day Celebration

Cuttack: 12 March
The Annual Day celebration of A.K.B. College, Govindpur came off smoothly. The Hon’ble Education Minister Sri N. K. Patra was the Chief Guest. The celebration started with the national anthem and devotional song. The local Collector and Dist. Magistrate presided over the meeting. After a short speech made by the Hon’ble Chief Guest on the all-round development of the college, gave away the prizes for curricular and extracurricular activities of the students. The celebration was a grand success.

B. During the Annual Day celebration, you were asked to present a report on the students’ activities during the year. Draft your report.
Answer:

A Report on Students’ Activities

The A.K.B. College, Govindpur has already been recognized as a unique educational institution imparting teaching on all branches/streams in the State. Elections to the students’ union, the dramatic society, the day-scholars’ Association, the +2 Cultural Association, etc. were held without any untoward incident. The performance of the students in curricular activities was excellent. Most of the students were awarded gold medals for their outstanding results in different streams. The mention of the co-operation and mutual understanding among the students in the college and outside would not be out of place here,

8.2 Business Reports

Business reports are mainly based on market condition, market surveys or market- analyses. We usually take into account the background information, method of investigation, findings, and recommendations while preparing business reports. Besides, we follow direct and factual language based on the market-surveys and market analyses. There is no scope of rendering personal feelings or attributing subjective interpretations.

Activity – 6

Here is a report about the introduction of a new mosquito-repellant. Read the report, paying attention to various parts.

Quality Marketing Agency
27 Janpath, Bhubaneswar

4 March, 2013

To
Mr. M. Pradhan
Managing Director
Home Products India Ltd.
Industrial Estate
Mancheswar, Bhubaneswar.
Dear Mr. Pradhan,
As requested by you, vide your letter No. MD/NS/2233 dated 2.2.2013, we have carried out a market survey to test the public acceptance of the mosquito repellant which your company plans to manufacture. We conducted an opinion poll covering 1000 families in the coastal districts of Odisha. Forty percent of these families use mosquito repellants, but most of them are unhappy with the existing products in the market. They find the electronic repellants too expensive while the coil-based ones emit too much smoke.

The preference is for a less expensive product, preferably one that produces no smoke. Our study suggests there may be a good market for a new repellant, provided these requirements are kept in mind. We recommend that your company should concentrate on manufacturing an improved kind of smoke-free mosquito coil, preferably one that produces a pleasant fragrance.
Yours sincerely,
S.K. Patnaik
Director of Research
Quality Marketing Agency

You must have observed that the report has been written in the format of a business letter. However, it could be written in a different format as given below.

Date : 4 March 20
To : Mr M Pradhan, Managing Director, Home Products India Ltd. Industrial Estate, Mancheswar, Bhubaneswar
From : S.K. Patnaik, Director of Research, Quality Marketing Agency 27 Janpath, Bhubaneswar
Subject : Survey report on the Introduction of a new mosquito repellant.

A Report On The Introduction Of A New Mosquito Repellant

A market survey to test the public’s acceptance of a new mosquito repellant was conducted in the coastal districts of Odisha on 20 February, 20 by Quality Marketing
Agency, Bhubaneswar.
An Opinion poll covering 100 families ……………………. etc.
______________________________________
______________________________________
______________________________________
An improved kind of smoke-free mosquito coil, preferably one that produces a pleasant fragrance is likely to be widely accepted by the public.
Sd/-
(Director of Research)

Activity – 7

Imagine that you are the President of the Literary Society of your College. Your Society plans to publish a journal. You have asked the Secretary of the Society to contact all the printing firms in town and to select one of them to print your journal.
Answer:

Literary Society, Kulailey College, Kailey

5 February 2012

To,
Prof. B. Pujari
President, Literary Society
Sir,
As desired by you a team consisting of the Secretary and Assistant Secretary of the society contacted all five printing firms in town and obtained quotations from them for printing of the proposed journal. All the firms quoted the same price, that is Rs. 5000/- for 1000 copies. Rasmita Printers, however, offered a discount of ten per cent, provided we allowed them an extra period of fifteen days for printing. Since we do not need the copies of the journal till a month later, we could consider the offer of Rasmita Printers as it will cost us Rs. 500/- less than the offers quoted by other printing firms.
Yours faithfully,
S. Pujari
Secretary

Activity – 8

A customer approached a bank for a house building loan. Before sanctioning the loan, the Branch Manager asked the Field Officer to examine the application and suggest whether the loan should be sanctioned.

The following is the report that the Field Officer wrote. Some parts of the report are missing. Re-write the missing parts, using the hints supplied. (See the letter at textbook page 75)
Answer:

SBI PD BRANCH
CRP Square
Bhubaneswar

3 March, 2012

To,
Prof. M. Mishra
Branch Manager
SBI PD Branch
CRP Square
Bhubaneswar
Sir,

1. As desired by you in your letter No. 254 dt. 24 February 2010, I examined the application of Mr. J.K. Panda for a house-building loan. I also personally inspected the site, interviewed Mr Panda and examined the documents relating to the plot.
2. My examination of the application and the relevant documents reveal that the site is an undisputed one. Till date all land cess has been paid and the plot is litigation free. Mr J.K. Panda is the owner of the plot and he has clear papers certifying its ownership. The plot is 112 decimals in size and its market value is around Rs. 8.00 lakhs. Mr Panda also has a regular income of Rs. 15,000 and has no outstanding loan to his account.
3. As Mr Panda is a deserving party, the sanction of the loan is recommended
Yours faithfully,
K.C. Panigrahy
Field Officer

Activity – 9

Imagine that you are the foreman in a factory. There has been a fire in the Factory and one of the workers has been badly burnt and is in hospital. Your General Manager has asked you to send him a report on the fire. Write a report.
Answer:

JAYASHREE CHEMICALS
Corporate Office, Metro Towers
Bhubaneswar – 751007

28.3. 2012

To,
Mr J.B. Sahu
General Manager
Jayashree Chemicals
Corporate Office, Bhubaneswar
Sir,
1. As desired by you vide your letter No. JC 289 dt. 20.3.2012, I conducted an enquiry into the fire that ravaged the factory at the Mancheswar Industrial Estate.
2. On inspection of the fire ravaged site and after interviewing the workers present during the incident, I discovered that the fire was caused by the Careless throwing of a lighted cigarette into the boiler room where it ignited a dry rope. The rope was lying in contact with a jar of petrol kept by an employee for his personal use. The jar of petrol caught fire immediately and broke into high flames that ravaged the electrical circuit. Fortunately, there was no one in the boiler room as it was lunch time except Mr Lingaraj Patra who was cleaning some machine part.
3. He has received 40 per cent bums in his body and has been admitted at the Kalinga hospital. He is presently out of danger and the personnel department has already sanctioned Rs. 50,000 for his treatment.
4. Damage to the boiler section of the factory was limited to the minimum by the quick action of our fire personnel. A few chairs and benches must be replaced and the Electrical circuit done up again.
This is for your kind information.
Yours faithfully
D.P. Dehury
Chief Security Officer
Jayashree Chemicals

Activity – 10

Your club wants to buy a CTV (Colour TV) set. You have been asked to contact various firms marketing such sets and make your recommendation on the brand to be bought. Make a comparative study of the price, quality and durability of different brands of CTV and make a report.
Answer:

YOUNG FOLKS CLUB
213 Kharvela Nagar
Bhubaneswar

21 April 2012

Mr P. Khuntia
The Secretary
Young Folks Club
213 Kharavela Nagar
Bhubaneswar
Dear Mr Khuntia,

As requested by you I contacted 10 dealers of colour television in the town and personally inspected several makes of colour televisions in their showrooms. LG, Samsung, Philips and Onida companies all offer 25” CTVs at competitive prices. Of course L.G. is the costliest at Rs. 58,000 but it employs the golden eye technology which is truly soothing to the eyes. Samsung and Onida brands are also good. The unique thing about the Samsung T V. is that it offers two games in its set but we cannot afford members fiddling with the game program of the T.V. while others want to watch programs.

Except for its woofer impacted quality the Videocon CTV’s picture quality is no match for those of other brands. The Onida brand is good. It commands a price tag of Rs. 50,000. However its after sales-service is doubtful as the dealer here does not guarantee qualified personnel from the company to give service. However with a price tag of Rs. 48,000/- and a 5 year warranty contract, quality picture and sound, the Philips flatron CTV is perhaps the best that our town has to offer.

The concerned dealer is offering a free Antennae along with booster as well as 10 VCD’s with the T.V. He has also agreed to transport the CTV and to install it in our conference Hall free of cost. It is therefore recommended that the Shanti Electronics, 105, Bapuji Nagar, Bhubaneswar be contacted and the Philips Flatron CTV be bought from them at the price as well as the terms and conditions offered by them.
Yours faithfully
Girish Mishra
Joint Secretary
Young Folks Club

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 5 Past Simple and Past Progressive Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 5 Past Simple and Past Progressive

SECTION – 1

Study the sentences below :
(a) It started to rain while we were walking home.
(b) My sister was tidying my room when I saw your letter.
(c) Anita was walking along the road when suddenly she heard footsteps behind her. Someone was following her. She was frightened and started to run.
What do you think the use of the Past Simple and the Past Progressive indicates in these sentences?
(Hint: Think of a point of time and a duration of time in the past and relate them to the action.)

Note:
The Past Simple tells us that the work/action started and finished in the past. The speaker has a definite time in mind. But in Past Progressive the time of beginning or completion of the activity is not mentioned. The activity was in progress for that hour.

Activity – 21
Put the verbs into the correct form, Past Progressive or Past Simple.
(a) My friend ______________(meet) Anima and Amiya at the bus stop four days ago. They ______________(go) to Paradeep and my friend ______________ (go) to Bolangir. They ______________(have) a chat while they ______________(wait) for their buses.
(b) My brother ______________ (cycle) to school last Monday when suddenly an old woman ______________(step) out into the road in front of him. He ______________(go) quite fast but luckily he ______________ (manage) to stop in time and ______________(not / hit) her.
Answer:
(a) My friend met Anima and Amiya at the bus stop four days ago. They were going to Paradeep and my friend was going to Bolangir. They had a chat while they were waiting for their buses.
(b) My brother was cycling to school last Monday when suddenly an old woman stepped out into the road in front of him. He was going quite fast but luckily he managed to stop in time and did not hit her.

Activity – 22
Here is a true story.
An old couple …………………………living in a flat in Bhubaneswar. ………………………… locked up in one room ………………………….. Some unknown people took away everything ………………………… police arrived ………………………… climbed ………………………… rescued ………………………… broke open a door ………………………… one dacoit was killed ………………………… detective was called …………………………interviewed a witness.
Imagine that you are being questioned by the police as if you were a witness to the crime. A police Inspector is recording your statements in a notebook. Think about the situation and write the appropriate answers.
Inspector: Where were you standing at that time?
Answer: _____________________________.
Inspector: Why did you come here?
Answer: _____________________________.
Inspector: What was the old man doing at the time?
Answer: _____________________________.
Inspector: How did you see that?
Answer: _____________________________.
Inspector: How long were you standing there?
Answer: _____________________________.

Answer:
An old couple _____ living in a flat in Bhubaneswar, _____ Orissa locked up in one room _____. Some unknown people took away everything. _____ police arrived _____, limbed _____ rescued _____ , broke open a door _____one dacoit killed, _____detective was called _____ interviewed a witness.
Inspector: Where were you standing at that time?
Answer: I was standing near the flat.
Inspector: Why did you come here?
Answer: I came here to play.
Inspector: What was the old man doing at that time?
Answer: The old man was shouting and trembling out of fear.
Inspector: How did you see that?
Answer: I heard him shouting and saw him trembling.
Inspector: How long were you standing there?
Answer: I stood there till your arrival.

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Tense Patterns Unit 4 Present Perfect and Past Simple Textbook Activity Questions and Answers.

CHSE Odisha 12th Class English Grammar Tense Patterns Unit 4 Present Perfect and Past Simple

Study the dialogue given below.
Susant: Have you ever ridden a horse?
Subir: Yes, I have.
Susant: When was that?
Subir: I rode one last summer.
Susant : What was it like?
Subir: Oh, it was awful.
Susant : Why? What happened?
Subir: I fell off and hurt my back.
Identify the Present Perfect and Past Simple sentences and examine their use carefully. How are they different in meaning?
{Hint: One of them answers the question ‘When’? and the other does not)
Except for the first two i.e. “Have you ever ridden a horse ?” and “Yes, I have”, the rest of the sentences in the above dialogue belong to Past Simple constructions.
When an action/event took place in the past but its result is still operative at the present moment of speaking/time, we generally use a Present Perfect tense and the Past Simple means that the action/happening occurred before the present moment.

Activity – 18
1. Complete the dialogue using the hints given.
(i) A: ever / see /a lion _______________?
B: Yes, _______________.
A: Where _______________?
B: In the zoo _______________.
A: What/look _______________?
B: terrible _______________.
A: You / afraid _______________?
B: No, _______________?

(ii) A : ever / be to / Dhauligiri ………………………?
B: Yes, _______________.
A: What! see /there _______________?
B: A temple I top/hill _______________.
A See / the inscriptions _______________?
B: Yes, _______________?
A: Able to read the inscriptions _______________?
B: No, _______________.

Answer:
(i) A: Have you ever seen a lion?
B: Yes, I have.
A: Where did you see it?
B: I saw it in the zoo.
A: What did it look like?
B: Yes, it was very terrible to look at.
A: Were you afraid?
B: No, I wasn’t.

(ii) A: Have you ever been to Dhauligiri?
B: Yes, I have been two times.
A: What did you see there?
B: I saw a temple at the top of the hill.
A: Did you see the inscriptions there?
B: Yes, I saw the inscriptions there.
A: Were you able to read the inscriptions there?
B: No, I wasn’t.

Activity – 19
Choose the right verb for each blank space and put it into the correct tense.
(do, wear, carry, ask, say, think)
A : _______________your grandfather _______________ something really crazy ?
B: He _______________ something really silly last summer. One one of the hottest days he _______________ a raincoat and _______________ an umbrella. Everyone _______________ him why. He _______________he _______________ it was going to rain.
Answer:
A: Did your grandfather wear something really crazy?
B: He wore something really silly last summer. On one of the hottest days, he wore a raincoat and carried an umbrella. Everyone asked him why. He said he thought it was going to rain.

Activity – 20
Complete the sentences, using the verbs in brackets either in Past Simple or Present Perfect form.
(a) She _______________ up her mind (made). She’s going to look for another college.
(b) Amulya : _______________me his pen but I’m afraid I _______________ it. (give, lose)
(c) A: It’s a little bit noisy in here, isn’t it?
B: Pardon? I can’t hear. What _______________ you _______________? (say)
(d) Where is my bike? It _______________ outside the classroom. It _______________! (be, disappear)
(e) Did you know that Umesh _______________ a new scooter? (buy)
(f) I did Sanskrit at school but I _______________ most of it. (forget)
(g) A : Sima, this is Rajesh.
B: Hello, Rajesh. Actually, we know each other. We _______________ already ___________ (meet).

Answer:
(a) She has made up her mind. She’s going to look for another college.
(b) Amulya gave me his pen but I’m afraid I have lost it.
(c) A: It’s a little bit noisy in here, isn’t it?
B: Pardon? I can’t hear. What did you say?
(d) Where is my bike? It was outside the classroom. It has disappeared!
(e) Did you know that Umesh has bought a new scooter?
(f) I did Sanskrit at school but I have forgotten most of it.
(g) A : Sima, this is Rajesh.
B: Hello, Rajesh. Actually, we know each other. We have already met.

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(d) Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(d)

Question 1.
(i) ∫\(\frac{d x}{\sqrt{11-4 x^2}}\)
Solution:

(ii) ∫\(\frac{e^{3 x}}{\sqrt{4-e^{6 x}}}\)dx
Solution:

(iii) ∫\(\frac{d x}{\sqrt{25-(\ln x)^2}}\)
Solution:

(iv) ∫\(\frac{\cos \theta}{\sqrt{4-\sin ^2 \theta}}\)dθ
Solution:

(v) ∫\(\frac{x^2}{\sqrt{36-x^6}}\)dx (x³ = z)
Solution:

(vi) ∫\(\frac{x+3}{\sqrt{9-x^2}}\)dx
Solution:

(vii) ∫\(\frac{d x}{\sqrt{5-x^2-4 x}}\) (x + 2 = z)
Solution:

(viii) ∫\(\frac{x+3}{\sqrt{5-x^2-4 x}}\)dx
Solution:

Question 2.
(i) ∫\(\frac{d x}{3 x^2+7}\)
Solution:

(ii) ∫\(\frac{e^{4 x}}{e^{8 x}+4}\)dx (e^4x = z)
Solution:

(iii) ∫\(\frac{d x}{x\left\{(\ln x)^2+25\right\}}\)
Solution:

(iv) ∫\(\frac{\sec \theta \tan \theta}{\sec ^2 \theta+4}\)dθ
Solution:

(v) ∫\(\frac{x^9}{x^{20}+4}\)dx (x¹⁰ = z)
Solution:

(vi) ∫\(\frac{3 x+4}{x^2+4}\)dx
Solution:

(vii) ∫\(\frac{d x}{x^2+6 x+13}\) (x + 3 = z)
Solution:

(viii) ∫\(\frac{x+5}{x^2+6 x+13}\)dx
Solution:

Question 3.
(i) ∫\(\frac{d x}{x \sqrt{4 x^2-9}}\)
Solution:

(ii) ∫\(\frac{d x}{\sqrt{e^{4 x}-5}}\) (e^2x = z)
Solution:

(iii) ∫\(\frac{d x}{x \ln x \sqrt{(\ln x)^2-4}}\)
Solution:

(iv) ∫\(\frac{\sec \theta d \theta}{\sin \theta \sqrt{3 \tan ^2 \theta-1}}\)
Solution:

(v) ∫\(\frac{d x}{x \sqrt{x^{14}-b^2}}\) (x⁷ = z)
Solution:

(vi) ∫\(\frac{x^2+3}{x \sqrt{x^2-4}}\)dx
Solution:

(vii) ∫\(\frac{d x}{(x+1) \sqrt{x^2+2 x-3}}\) (x + 1 = z)
Solution:

(viii) ∫\(\frac{x^2+2 x+4}{(x+1) \sqrt{x^2+2 x-3}}\) dx
Solution:

Question 4.
(i) ∫\(\frac{d x}{\sqrt{3 x^2+4}}\)
Solution:

(ii) ∫\(\frac{4 e^x}{\sqrt{3 e^x+4}}\)dx
Solution:

(iii) ∫\(\frac{d x}{x \sqrt{(\ln x)^2+8}}\)
Solution:

(iv) ∫\(\frac{d \theta}{\sin ^2 \theta \sqrt{\cot ^2 \theta+2}}\)
Solution:

(v) ∫\(\frac{x^2}{\sqrt{x^6+a^6}}\)dx
Solution:

(vi) ∫\(\frac{3 x+4}{\sqrt{5 x^2+8}}\)dx
Solution:

(vii) ∫\(\frac{e^x \cos e^x}{\sqrt{\sin ^2 e^x+9}}\)dx
Solution:

(viii) ∫\(\frac{2 x+11}{\sqrt{x^2+10 x+29}}\)dx (x + 5 = z)
Solution:

Question 5.
(i) ∫\(\frac{d x}{\sqrt{4 x^2-6}}\)
Solution:

(ii) ∫\(\frac{e^{5 x}}{\sqrt{e^{10 x}-4}}\)dx
Solution:

(iii) ∫\(\frac{d x}{x \sqrt{(\ln x)^2-4}}\); x > e²
Solution:

(iv) ∫\(\frac{\cos \theta d \theta}{\sin ^2 \theta \sqrt{{cosec}^2 \theta-4}}\)
Solution:

(v) ∫\(\frac{d x}{\sqrt{x} \sqrt{x-a^2}}\)
Solution:

(vi) ∫\(\frac{x-2}{\sqrt{3 x^2-8}}\)dx
Solution:

(vii) ∫\(\frac{3 x+4}{x \sqrt{2 x^2-5}}\)dx
Solution:

(viii) ∫\(\frac{x^2+2 x+2}{x \sqrt{x^2-4}}\)dx
Solution:

(ix) ∫\(\frac{d x}{\sqrt{x^2+8 x}}\)
Solution:

(x) ∫\(\frac{x+7}{\sqrt{x^2+8 x}}\)dx
Solution: