model 3 based on positive and negative exponent Section-Wise Topic Notes With Detailed Explanation And Example Questions

Answer:(c)

0.42 × $100^k$ = 42

$42/100 × 100^k$ = 42

$100^k={42×100}/42=100^1$

k=1

Answer:(c)

$3^{x + y}$ = 81 and $81^{x - y}$ = 3

$3^{x + y}$= 81

$3^{x + y} = 3^4$

x + y = 4 ...(i)

and $81^{x - y}$ = 3

$(3^4)^{x - y}$ = 3

$3^{4x - 4y} = 3^1$

4x - 4y = 1 ...(ii)

By equation (i) × 4 + (ii) we have,

4x + 4y = 16…(iii)

4x-4y=1…(iv)

Solving these two equations(iii),(iv)

8x=17 ⇒ $x = 17/8$

Answer:(b)

Expression = $√{ √{2} × √{3}}$

= $(√2 ×√3)^{1/2}=(2^{1/2} × 3^{1/2})^{1/2$

= $(6)^{1/2× 1/2} = (6)^{1/4}$

Answer:(a)

Sum of the cubes of first n natural numbers = $({n(n + 1)}/2)^2$

Their average = ${n(n+1)^2}/4$

Required average when n = 7,

= ${7(7+ 1)^2}/4 = {7 × 8×8}/4$ = 112

Answer:(d)

$x = (3)^{1/3} - (3)^{-1/3}$

On cubing both sides,

$x^3 = ((3)^{1/3} - (3)^{-1/3})^3$

= $((3)^{1/3})^3-((3)^{-1/3})^3-3×3^{1/3}×3^{-1/3}(3^{1/3}-3^{-1/3})$

=$3-3^{-1}-3×x=3-1/3-3x$

$x^3 + 3x = 3- 1/3={9- 1}/3$

$x^3 + 3x = 8/3$

$3x^3 + 9x = 8/3 ×3 =8$

Answer:(a)

$3^10 × 27^2 = 9^2 × 3^n$

$3^10 × (3^3)^2 = (3^2)^2 × 3^n$

$3^10 × 3^6 = 3^4 × 3^n$

$3^{10+6} = 3^4 × 3^n$

$3^16 = 3^{4 + n}$

4 + n = 16

n = 16 - 4 = 12