Students can Download Basic Maths Exercise 18.3 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 18.3

Part – A

**2nd PUC Basic Maths Differential Calculus Ex 18.3 One or Two Marks Questions and Answers**

Question 1.

3x^{2} + 4y^{2} = 10

Answer:

Given 3x^{2} + 4y^{2} = 10

Diff w.r.t x

6x + 8y \(\frac{d y}{d x}\) = 0

\(\Rightarrow \quad \frac{d y}{d x}=\frac{-8 y}{6 x}=\frac{-4 y}{3 x}\)

Question 2.

\(\sqrt{x}+\sqrt{y}=3\)

Answer:

Question 3.

y^{2} = 4ax.

Answer:

Given y^{2} = 4ax.

Differentiate with respect to x

2y \(\frac{d y}{d x}\) = 4a.1 ⇒ \(\frac{d y}{d x}\) = \(\frac{4 a}{2 y}=\frac{2 a}{y}\)

Question 4.

\(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\)

Answer:

Question 5.

x^{2} = 4ay

Answer:

Given x^{2} = 4ay

Differentiate with respect to x, 2x = 4a \(\frac{d y}{d x}\) ⇒ \(\frac{d y}{d x}\) = \(\frac{2 x}{4 a}=\frac{x}{2 a}\)

Question 6.

\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

Answer:

Question 7.

x^{3} + y^{3} = 3axy

Answer:

Given x^{3} + y^{3} = 3axy

3x^{2} + 3y^{2} \(\frac{d y}{d x}\) = 3a \(\left(x \cdot \frac{d y}{d x}+y \cdot 1\right)\)

\(\frac{d y}{d x}\) (3y^{2} – 3ax) = 3ay – 3x^{2} = \(\frac{d y}{d x}=\frac{a y-x^{2}}{y^{2}-a x}\)

Question 8.

x – y = 0

Answer:

Given x – y = 0

Differentiate with respect to x,

1 – \(\frac{d y}{d x}\) = 0 ⇒ \(\frac{d y}{d x}\) = 1

Question 9.

x^{2} – y^{2} = a^{2}

Answer:

Given x^{2} – y^{2} = a^{2}

Differentiate with respect to x we get,

2x – 2y. \(\frac{d y}{d x}\) = 0 ⇒ \(\frac{d y}{d x}\) = \(\frac{2 x}{2 y}=\frac{x}{y}\)

Question 10.

x + \(\sqrt{x y}\) = x^{2}.

Answer:

Part-B

**2nd PUC Basic Maths Differential Calculus Ex 18.3 Three marks Questions and Answers**

Question 1.

log(xy) = x^{2} + y^{2}

Answer:

Given log(xy) = x^{2} + y^{2}

log x + log y = x^{2} + y^{2} differentiate w.r.t x

Question 2.

2^{x} + 2^{y} = 2^{x+y}

Answer:

Given 2^{x} + 2^{y} = 2^{x+y}

Differentiate w.r.t. x we get

2^{x} log 2 + 2^{y} log 2 \(\frac{d y}{d x}\)

Question 3.

x^{y} = y^{x}.

Answer:

Given x^{y} = y^{x}., taking log^{m} both sides

y log x = x log y differentiate

Both sides w.r.t x

Question 4.

sin xy = cos(x + y).

Answer:

Given sin xy = cos(x + y), diff w.r.t x.

cos(xy) \(\left[\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right]\)

\(\frac{d y}{d x}\) [sin(x + y) + xcos (xy)]

= -sin(x + y) – y cos (xy)

\(\frac{d y}{d x}=\frac{-[\sin (x+y)+\cos x y]}{(\sin (x+y)+x \cos x y)}\)

Question 5.

y = 4^{x+y}

Answer:

Given y = 4^{x+y}, diff. w r.t. x

\(\frac{d y}{d x}\) = 4^{x+y} log 4(1 + \(\frac{d y}{d x}\)) = 4^{x+y}

\(\frac{d y}{d x}\)(1 – 4^{x+y} log 4) = 4^{x+y} log 4

∴ \(\frac{d y}{d x}=\frac{4^{x+y} \cdot \log 4}{1-4^{x+y} \cdot \log 4}\)

Part-C

**2nd PUC Basic Maths Differential Calculus Ex 18.3 Five Marks Questions and Answers.**

Question 1.

If \(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}\) = a, Prove that x . \(\frac{d y}{d x}\) = y.

Answer:

Question 2.

If x^{y} = e^{y – x}, show that \(\frac{d y}{d x}\) = \(\frac{2-\log x}{(1-\log x)^{2}}\)

Answer:

Given x^{y} = e^{y – x }. Taking log both sides

y log x = (y – x)log e^{e}

x = y (1 – log x) ∵ log e^{e} = 1

Question 3.

If cos y = x cos(a + y). show that \(\frac{d y}{d x}\) = \(\frac{\cos ^{2}(a+y)}{\sin a}\)

Answer:

Question 4.

If e^{x} = y^{x} show that \(\frac{d y}{d x}\) = \(\frac{(\log y)^{2}}{\log y-1}\)

Answer:

Given e^{x} = y^{x} Taking log^{m} both sides

y log e^{e} = x log y

y = x log y differentiate w.r.t x

Question 5.

If e^{x+y} = xy show that \(\frac{d y}{d x}\) = \(\frac{y(1-x)}{x(y-1)}\)

Answer:

Given ye^{x+y} = xy

Taking log m both sides

(x + y) loge = log(xy)

x+y = log x + log y diff w.r.t x

Question 6.

If y^{x} = x^{y} show that \(\frac{d y}{d x}\) = \(\frac{y(y=x \log y)}{x(x-y \log x)}\)

Answer:

Given y^{x} = x^{y}, Taking log^{m} both sides

x log y = y log x, diff w.r.t x

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