ACT · Practice Sheet 30

1. 1.

   Mathematics

   Find the binomial coefficient C(6, 3) — the coefficient of x^3 in (1 + x)^6.

   Formula:Binomial Coefficient

   $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$

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2. 2.

   Mathematics

   Find the sum of the first 10 terms of an AP with first term 8 and common difference 1.

   Formula:AP Sum of n Terms

   $S_n = \dfrac{n}{2}\,[2a+(n-1)d]$

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3. 3.

   Mathematics

   Find the sum of the first 7 terms of a GP with first term 2 and common ratio 2.

   Formula:GP Sum of n Terms

   $S_n = a\,\dfrac{r^{n}-1}{r-1}$

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4. 4.

   Physics

   A projectile is launched at 37 m/s at 30° to the horizontal. Find its range (g = 9.8 m/s²).

   Formula:Projectile Range

   $R = \dfrac{u^{2}\sin 2\theta}{g}$

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5. 5.

   Physics

   Find the force required to accelerate a 7 kg mass at 2 m/s².

   Formula:Newton's Second Law (F = ma)

   $F = ma$

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6. 6.

   Mathematics

   Evaluate log base 5 of 3,125.

   Formula:Logarithm Evaluation

   $\log_{b}(b^{k}) = k$

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7. 7.

   Quantitative Aptitude

   A value changes from 700 to 875. Find the percentage increase.

   Formula:Percentage Change

   $\%\,\text{change} = \dfrac{|new-old|}{old}\times 100$

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8. 8.

   Mathematics

   If f(x) = 4x³ + 1x² + 2x, find f′(3).

   Formula:Derivative of a Polynomial

   $\dfrac{d}{dx}x^{n} = n\,x^{\,n-1}$

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9. 9.

   Physics

   A force of 521 N acts on an area of 2 m². Find the pressure.

   Formula:Pressure

   $P = \dfrac{F}{A}$

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10. 10.

    Physics

    A body moving at 7 m/s accelerates at 4 m/s² for 10 s. Find the distance travelled.

    Formula:Equation of Motion (s = ut + ½at²)

    $s = ut + \tfrac{1}{2}at^{2}$

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11. 11.

    Physics

    Find the voltage across a 22 Ω resistor carrying a current of 19 A.

    Formula:Ohm's Law

    $V = IR$

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12. 12.

    Physics

    Find the density of an object with mass 247 g and volume 13 cm³.

    Formula:Density

    $\rho = \dfrac{m}{V}$

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13. 13.

    Physics

    Find the voltage across a 39 Ω resistor carrying a current of 15 A.

    Formula:Ohm's Law

    $V = IR$

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14. 14.

    Quantitative Aptitude

    A value changes from 400 to 440. Find the percentage increase.

    Formula:Percentage Change

    $\%\,\text{change} = \dfrac{|new-old|}{old}\times 100$

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15. 15.

    Mathematics

    Find the sum of the first 4 terms of a GP with first term 6 and common ratio 2.

    Formula:GP Sum of n Terms

    $S_n = a\,\dfrac{r^{n}-1}{r-1}$

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16. 16.

    Mathematics

    For the quadratic 1x² + 8x − 8 = 0, find the sum and product of its roots.

    Formula:Vieta's Formulas (roots)

    $\alpha+\beta=-\dfrac{b}{a},\;\alpha\beta=\dfrac{c}{a}$

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17. 17.

    Mathematics

    Find the sum of the first 18 terms of an AP with first term 19 and common difference 10.

    Formula:AP Sum of n Terms

    $S_n = \dfrac{n}{2}\,[2a+(n-1)d]$

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18. 18.

    Quantitative Aptitude

    Divide $1,746 between two people in the ratio 4:2. How much does each receive?

    Formula:Sharing in a Ratio

    $\text{Share} = \text{Total}\times\dfrac{\text{part}}{\text{sum of parts}}$

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19. 19.

    Quantitative Aptitude

    A value changes from 800 to 1,000. Find the percentage increase.

    Formula:Percentage Change

    $\%\,\text{change} = \dfrac{|new-old|}{old}\times 100$

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20. 20.

    Quantitative Aptitude

    In how many ways can 3 objects be arranged out of 6 distinct objects? (Find ⁿᴘᵣ for n = 6, r = 3.)

    Formula:Permutations

    $^{n}P_{r} = \dfrac{n!}{(n-r)!}$

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