2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus Ex 18.6

Students can Download Basic Maths Exercise 18.6 Questions and Answers, Notes Pdf, 2nd PUC Basic Maths Question Bank with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka 2nd PUC Basic Maths Question Bank Chapter 18 Differential Calculus Ex 18.6

Part – A

2nd PUC Basic Maths Differential Calculus Ex 18.6 Two or Three Marks Questions and Answers

Question 1.
Find \(\frac{d y}{d x}\) if
(a) x = \(a\left(t-\frac{1}{t}\right), y=a\left(t+\frac{1}{t}\right) 1\)
(b) x = e^2t , y = log(2t + 1)
(c) x = log(1 + t), y = \(\frac{1}{1+t}\)
(d) x = log t, y = \(\frac{1}{t}\)
(e) x = 4t , y = \(\frac{4}{t}\)
(f) x = a sec θ, y = b tan θ.
(g) x = a(θ – sin θ), y = a(1 – cos θ)
(h) x = a cos(log t), y = a log(cos t)
Answer:
(a) Given x = \(a\left(t-\frac{1}{t}\right), y=a\left(t+\frac{1}{t}\right) 1\)
Differentiate both w.r.t

(b) Given x = e^2t , y = log(2t + 1)
Differentiate both w.r.t we get

(c) Given x = log (1+t), y = \(\frac{1}{1+t}\)
Differentiate both w.r.t x t, we get

(d) Given x = log t, y = \(\frac{1}{t}\)
Differentiate both w.r.t t, we get

(e) Given x = 4t y = \(\frac{4}{t}\)
Differentiate both w.r.t t, we get

(f) Given a = secθ, y = b tan θ
Differentiate both w.r.t. θ we get

(g) Given x = a (θ – sin θ), y = a(1 – cosθ)
Differentiate both w.r.t θ

(h) Given x = a cos(logt) y = a log(cos t)
Differentiate both w.r.t t

Question 2.
Differentiate tan² w.r.t cos²x.
Answer:
Let u = tan²x, v = cos²x
Differentiate both w.r.t x
\(\frac{d u}{d x}\) = 2tan x. sec²x,
\(\frac{d v}{d x}\) = 2 cosx(-sinx)
∴ \(\frac{d u}{d v}\) = \(\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 \tan x \cdot \sec ^{2} x}{2 \sin x \cdot \cos x}\)
= \(\frac{1}{\sec ^{4} x}\)

Question 3.
Differentiate sin²x w.r.t. x²
Answer:
Let u = sin²x, v = x²
Differentiate both w.r.t. x
\(\frac{d u}{d x}\) = 2 sin x. cos x,
\(\frac{d v}{d x}\) = 2x
∴ \(\frac{d u}{d v}\) = \(\frac{\frac{d u}{d x}}{\frac{d v}{d x}}=\frac{2 \sin x \cos x}{2 x}\)
= \(\frac{\sin x \cos x}{x}\)

Question 4.
Differentiate tan \(\sqrt{x}\) w.r.t \(\sqrt{x}\)
Answer:
Let u = tan \(\sqrt{x}\) v = \(\sqrt{x}\)
Differentiate both w.r.t x

Question 5.
Differentiate log x w.r.t \(\frac { 1 }{ x }\)
Answer:
Let u = log x, v = \(\frac { 1 }{ x }\)

Question 6.
Differentiate log sin x w.r.t \(\sqrt{\cos x}\)
Answer:
Let u = log (sin x) v = \(\sqrt{\cos x}\)

Question 7.
If x = e^{log cos 4θ} , y = e^{log sin 4θ} show that \(\frac{d y}{d x}=\frac{-x}{y}\)
Answer:
Given x = cos 4θ, y = sin 4θ
\(\frac{d x}{d \theta}\) = -4sin 4θ
\(\frac{d y}{d \theta}\) = + 4cos 4θ

Question 8.
If x = a cos⁴θ, y = a^{sin 4θ} show that \(\frac{d y}{d x}\) = -tan²θ
Answer:
Given x = a cos⁴θ, y = a sin 4θ
\(\frac{d x}{d \theta}\) = a(4cos³θ( – sinθ),) \(\frac{d y}{d \theta}\) = 4a sin³θ cos θ
∴ \(\frac{d y}{d x}=\frac{4 a \sin ^{3} \theta \cos \theta}{-4 a \cos ^{3} \theta \sin \theta}=-\tan ^{2} \theta\)

Question 9.
If x = e^t(cos t + sin t), y = e^t(cos t – sin t). show that \(\frac{d y}{d x}\) = -tan t.
Answer:
Given x = e^t(cos t + sin t), y = e^t(cos t – sin t)

Question 10.
If x = a log sec θ, y = a(tanθ – 1) show that \(\frac{d y}{d x}\) = 2 cosec2θ
Answer:
Given x = a log sec θ, y = a(tanθ – 1)