ACT · Practice Sheet 4

1. 1.

   Quantitative Aptitude

   A can finish a job in 8 days and B in 16 days. Working together, how long will they take?

   Formula:Time & Work (combined)

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

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2. 2.

   Physics

   Find the density of an object with mass 10 g and volume 29 cm³.

   Formula:Density

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

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3. 3.

   Quantitative Aptitude

   An item is bought for $900 and sold for $1,260. Find the profit percentage.

   Formula:Profit Percentage

   $\text{Profit\%} = \dfrac{SP-CP}{CP}\times 100$

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4. 4.

   Mathematics

   Find the slope of the line through (7, 7) and (-3, 7).

   Formula:Slope of a Line

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

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5. 5.

   Physics

   A constant force of 79 N moves an object 38 m in the direction of the force. Calculate the work done.

   Formula:Work Done

   $W = F\,d$

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6. 6.

   Mathematics

   For the quadratic 4x² + 3x + 9 = 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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7. 7.

   Quantitative Aptitude

   Divide $1,500 between two people in the ratio 7:3. 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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8. 8.

   Mathematics

   In an arithmetic progression with first term 14 and common difference 2, find term number 16.

   Formula:AP nth Term

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

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9. 9.

   Physics

   Find the momentum of a 31 kg body moving at 35 m/s.

   Formula:Linear Momentum

   $p = mv$

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10. 10.

    Quantitative Aptitude

    Two varieties costing $36 and $91 per kg are mixed to give a mixture worth $68 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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11. 11.

    Mathematics

    Evaluate ∫ from 1 to 3 of 5x^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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12. 12.

    Mathematics

    Find the sum of the first 6 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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13. 13.

    Quantitative Aptitude

    In how many ways can 3 objects be arranged out of 5 distinct objects? (Find ⁿᴘᵣ for n = 5, r = 3.)

    Formula:Permutations

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

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14. 14.

    Quantitative Aptitude

    Two varieties costing $15 and $67 per kg are mixed to give a mixture worth $56 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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15. 15.

    Physics

    Find the force required to accelerate a 25 kg mass at 18 m/s².

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

    $F = ma$

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16. 16.

    Mathematics

    Evaluate log base 3 of 243.

    Formula:Logarithm Evaluation

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

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17. 17.

    Physics

    Find the momentum of a 15 kg body moving at 9 m/s.

    Formula:Linear Momentum

    $p = mv$

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18. 18.

    Mathematics

    Find the sum of the first 11 terms of an AP with first term 18 and common difference 7.

    Formula:AP Sum of n Terms

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

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19. 19.

    Physics

    A constant force of 93 N moves an object 39 m in the direction of the force. Calculate the work done.

    Formula:Work Done

    $W = F\,d$

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20. 20.

    Quantitative Aptitude

    Find the HCF and LCM of 27 and 9.

    Formula:LCM & HCF

    $\text{LCM}\times\text{HCF} = a\times b$

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