ACT · Practice Sheet 19

1. 1.

   Mathematics

   Evaluate ∫ from 2 to 3 of 2x^1 dx.

   Formula:Definite Integral of a Power

   $\int_{p}^{q} a x^{n}\,dx = \dfrac{a}{n+1}\big[x^{n+1}\big]_{p}^{q}$

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2. 2.

   Quantitative Aptitude

   Find the average of these 6 numbers: 54, 83, 50, 59, 26, 41.

   Formula:Arithmetic Mean

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

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3. 3.

   Physics

   Find the voltage across a 12 Ω resistor carrying a current of 11 A.

   Formula:Ohm's Law

   $V = IR$

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4. 4.

   Quantitative Aptitude

   Find the average of these 10 numbers: 77, 50, 47, 21, 24, 87, 30, 84, 49, 68.

   Formula:Arithmetic Mean

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

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5. 5.

   Quantitative Aptitude

   Two fair dice are rolled. What is the probability that the sum of the numbers is 7?

   Formula:Probability (two dice)

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

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6. 6.

   Mathematics

   Find the sum of the first 3 terms of a GP with first term 6 and common ratio 3.

   Formula:GP Sum of n Terms

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

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7. 7.

   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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8. 8.

   Quantitative Aptitude

   In how many ways can 2 objects be arranged out of 8 distinct objects? (Find ⁿᴘᵣ for n = 8, r = 2.)

   Formula:Permutations

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

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9. 9.

   Quantitative Aptitude

   Two varieties costing $37 and $80 per kg are mixed to give a mixture worth $60 per kg. Find the mixing ratio.

   Formula:Alligation (Mixtures)

   $\dfrac{q_1}{q_2} = \dfrac{p_2-m}{m-p_1}$

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10. 10.

    Quantitative Aptitude

    The marked price of an item is $1,300. After a 30% discount, find the selling price.

    Formula:Discount & Selling Price

    $SP = MP\left(1-\dfrac{d}{100}\right)$

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11. 11.

    Quantitative Aptitude

    Pipe A fills a tank in 12 hours and pipe B in 13 hours. If both are opened together, how long to fill the tank?

    Formula:Pipes & Cisterns

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

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12. 12.

    Physics

    An object starts at 11 m/s and accelerates at 9 m/s² for 2 s. Find its final velocity.

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

    $v = u + at$

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13. 13.

    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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14. 14.

    Quantitative Aptitude

    In how many ways can 2 objects be chosen from 7 distinct objects? (Find ⁿᴄᵣ for n = 7, r = 2.)

    Formula:Combinations

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

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15. 15.

    Physics

    Calculate the potential energy of a 50 kg object at a height of 44 m (g = 9.8 m/s²).

    Formula:Gravitational Potential Energy

    $PE = mgh$

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16. 16.

    Physics

    Find the electrical power when 31 V drives a current of 5 A.

    Formula:Electrical Power

    $P = VI$

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17. 17.

    Mathematics

    In an arithmetic progression with first term 16 and common difference 10, find term number 5.

    Formula:AP nth Term

    $a_n = a + (n-1)d$

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18. 18.

    Physics

    Find the density of an object with mass 147 g and volume 4 cm³.

    Formula:Density

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

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19. 19.

    Quantitative Aptitude

    The sum of the ages of two people is 70 years, and one is 2 years older than the other. Find their ages.

    Formula:Ages (linear equations)

    $x + (x+d) = S$

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20. 20.

    Quantitative Aptitude

    Find the simple interest on $3,000 at 12% per annum for 4 year(s).

    Formula:Simple Interest

    $SI = \dfrac{P\times R\times T}{100}$

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