Students can Download Basic Maths Chapter 3 Theory of Indices Questions and Answers, Notes Pdf, 1st 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 1st PUC Basic Maths Question Bank Chapter 3 Theory of Indices

Question 1.

3(5^{-2}) + \(\left(\frac{5}{3}\right)^{-4}\) + (a + 5)^{0}

Answer:

Question 2.

Show that \(\left(\frac{x^{b}}{x^{c}}\right)^{a}-\left(\frac{x^{c}}{x^{a}}\right)^{b} \cdot\left(\frac{x^{a}}{x^{b}}\right)^{c}=1\)

Answer:

LHS = (x^{b-c})^{a} (x^{c-a})^{3} (x^{a-b})^{c}

= x^{ab-bc} .x^{bc-ab} .x^{ac-bc}

= x^{ab-bc+bc+-ab+ac-bc}

= x^{0}

= 1

= RHS

Question 3.

If pqr = 1 Show that \(\frac{1}{1+p+q^{-1}}+\frac{1}{1+q+r^{-1}}+\frac{1}{1+r+p^{-1}}=1\)

Answer:

Question 4.

If a^{x} = b^{y} = and b^{2} = ac. Show that \(\frac{1}{x}+\frac{1}{z}=\frac{2}{y}\)

Answer:

Let a^{x} = b^{y} = c^{z} = k (say)

⇒ a^{x} = k, b^{y} = k, C^{z} = k

⇒ a = k^{1/x}; b = k^{1/y}; c = k^{1/z}

By data b^{2} = ac

(k^{1/y})^{2} = k^{1/x} .k^{1/z}

k^{2/y} = k^{1/x+1/z}

⇒ \(\frac{2}{y}=\frac{1}{x}+\frac{1}{z}\) (∵bases are same)

Question 5.

If 65^{1/x} = 13^{1/y} = 5^{1/z} show that x=y+z.

Answer:

Let 65^{1/x} ⇒ k = 65 k; 13^{1/y} = k ⇒ 13 = k; 5^{1/z} = k ⇒ 5 = k

We know that 65 = 13 × 5 k^{x} = k^{y}. k^{z} ⇒ x = y + z