ACT · Practice Sheet 49

1. 1.

   Quantitative Aptitude

   Two fair dice are rolled. What is the probability that the sum of the numbers is 6?

   Formula:Probability (two dice)

   $P = \dfrac{\text{favourable}}{\text{total}}$

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2. 2.

   Quantitative Aptitude

   Pipe A fills a tank in 3 hours and pipe B in 4 hours. If both are opened together, how long to fill the tank?

   Formula:Pipes & Cisterns

   $T = \dfrac{xy}{x+y}$

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3. 3.

   Quantitative Aptitude

   The sum of the ages of two people is 86 years, and one is 16 years older than the other. Find their ages.

   Formula:Ages (linear equations)

   $x + (x+d) = S$

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4. 4.

   Mathematics

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

   Formula:Derivative of a Polynomial

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

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5. 5.

   Quantitative Aptitude

   In how many ways can 2 objects be chosen from 9 distinct objects? (Find ⁿᴄᵣ for n = 9, r = 2.)

   Formula:Combinations

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

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6. 6.

   Quantitative Aptitude

   Two fair dice are rolled. What is the probability that the sum of the numbers is 9?

   Formula:Probability (two dice)

   $P = \dfrac{\text{favourable}}{\text{total}}$

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7. 7.

   Mathematics

   Find the sum of the first 15 terms of an AP with first term 7 and common difference 2.

   Formula:AP Sum of n Terms

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

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8. 8.

   Physics

   Find the momentum of a 4 kg body moving at 23 m/s.

   Formula:Linear Momentum

   $p = mv$

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9. 9.

   Mathematics

   Find the binomial coefficient C(9, 4) — the coefficient of x^4 in (1 + x)^9.

   Formula:Binomial Coefficient

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

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10. 10.

    Quantitative Aptitude

    Find the average of these 6 numbers: 71, 33, 57, 36, 87, 20.

    Formula:Arithmetic Mean

    $\bar{x} = \dfrac{\sum x}{n}$

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11. 11.

    Quantitative Aptitude

    In how many ways can 3 objects be chosen from 7 distinct objects? (Find ⁿᴄᵣ for n = 7, r = 3.)

    Formula:Combinations

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

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12. 12.

    Mathematics

    For the quadratic 1x² + 5x + 3 = 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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13. 13.

    Mathematics

    Find the sum of the first 13 terms of an AP with first term 19 and common difference 4.

    Formula:AP Sum of n Terms

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

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14. 14.

    Quantitative Aptitude

    Two fair dice are rolled. What is the probability that the sum of the numbers is 5?

    Formula:Probability (two dice)

    $P = \dfrac{\text{favourable}}{\text{total}}$

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15. 15.

    Physics

    If 1,070 J of work is done in 7 s, find the power.

    Formula:Power

    $P = \dfrac{W}{t}$

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16. 16.

    Mathematics

    Evaluate log base 3 of 27.

    Formula:Logarithm Evaluation

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

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17. 17.

    Physics

    An object starts at 18 m/s and accelerates at 9 m/s² for 11 s. Find its final velocity.

    Formula:Equation of Motion (v = u + at)

    $v = u + at$

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18. 18.

    Quantitative Aptitude

    In how many ways can 4 objects be arranged out of 9 distinct objects? (Find ⁿᴘᵣ for n = 9, r = 4.)

    Formula:Permutations

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

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19. 19.

    Mathematics

    Find the slope of the line through (-4, 8) and (1, -6).

    Formula:Slope of a Line

    $m = \dfrac{y_2-y_1}{x_2-x_1}$

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20. 20.

    Quantitative Aptitude

    In how many ways can 3 objects be chosen from 8 distinct objects? (Find ⁿᴄᵣ for n = 8, r = 3.)

    Formula:Combinations

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

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