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Calculate the percentage increase in speed from 18 km/h to 30 km/h. Solve the equation 5(3z − 5) = 70 for z. Simplify (x⁻⁴ / x2)⁻³.

Mathematics

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Q.1.10
To find the percentage increase in speed, we use the formula:

Percentage Increase=New Speed−Original SpeedOriginal Speed×100%\text{Percentage Increase} = \frac{\text{New Speed} - \text{Original Speed}}{\text{Original Speed}} \times 100\%

  1. Identify the original and new speeds.

    Original Speed =18 km/h= 18 \text{ km/h}
    New Speed =30 km/h= 30 \text{ km/h}

  2. Calculate the change in speed.

    Change in Speed=30 km/h−18 km/h=12 km/h\text{Change in Speed} = 30 \text{ km/h} - 18 \text{ km/h} = 12 \text{ km/h}

  3. Calculate the percentage increase.

    Percentage Increase=12 km/h18 km/h×100%\text{Percentage Increase} = \frac{12 \text{ km/h}}{18 \text{ km/h}} \times 100\% Percentage Increase=23×100%\text{Percentage Increase} = \frac{2}{3} \times 100\% Percentage Increase≈0.6666...×100%\text{Percentage Increase} \approx 0.6666... \times 100\% Percentage Increase≈66.67%\text{Percentage Increase} \approx 66.67\%

    The correct option is (3).

    The final answer is \boxed{\text{66.67%}}.

    Q.1.11
    To determine the value of zz in the expression 5(3z−5)=705(3z - 5) = 70:

  4. Divide both sides of the equation by 5.

    5(3z−5)5=705\frac{5(3z - 5)}{5} = \frac{70}{5} 3z−5=143z - 5 = 14

  5. Add 5 to both sides of the equation.

    3z−5+5=14+53z - 5 + 5 = 14 + 5 3z=193z = 19

  6. Divide both sides by 3.

    3z3=193\frac{3z}{3} = \frac{19}{3} z=193z = \frac{19}{3}

    The correct option is (4).

    The final answer is z=193\boxed{z = \frac{19}{3}}.

    Q.1.12
    To determine the answer to the exponent problem (x−4x2)−3\left(\frac{x^{-4}}{x^2}\right)^{-3}:

  7. Simplify the expression inside the parentheses using the quotient rule for exponents, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

    x−4x2=x−4−2=x−6\frac{x^{-4}}{x^2} = x^{-4-2} = x^{-6}

  8. Apply the outer exponent using the power rule for exponents, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.

    (x−6)−3=x(−6)×(−3)(x^{-6})^{-3} = x^{(-6) \times (-3)} =x18= x^{18}

    The correct option is (3).

    The final answer is x18\boxed{x^{18}}.

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