ACT · Practice Sheet 33

1. 1.

   Mathematics

   Find the distance between the points (-4, -1) and (4, -6).

   Formula:Distance Between Two Points

   $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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2. 2.

   Mathematics

   Evaluate log base 10 of 1,000.

   Formula:Logarithm Evaluation

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

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3. 3.

   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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4. 4.

   Physics

   Find the force required to accelerate a 16 kg mass at 13 m/s².

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

   $F = ma$

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5. 5.

   Mathematics

   In an arithmetic progression with first term 18 and common difference 9, find term number 5.

   Formula:AP nth Term

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

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6. 6.

   Physics

   Find the electrical power when 30 V drives a current of 19 A.

   Formula:Electrical Power

   $P = VI$

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7. 7.

   Quantitative Aptitude

   Two varieties costing $40 and $72 per kg are mixed to give a mixture worth $58 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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8. 8.

   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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9. 9.

   Mathematics

   If f(x) = 3x³ + 6x² + 4x, find f′(1).

   Formula:Derivative of a Polynomial

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

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10. 10.

    Quantitative Aptitude

    The marked price of an item is $1,300. After a 20% discount, find the selling price.

    Formula:Discount & Selling Price

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

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11. 11.

    Physics

    Find the density of an object with mass 94 g and volume 26 cm³.

    Formula:Density

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

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12. 12.

    Mathematics

    Find the distance between the points (-4, -6) and (3, 1).

    Formula:Distance Between Two Points

    $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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13. 13.

    Mathematics

    For the quadratic 2x² − 3x + 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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14. 14.

    Quantitative Aptitude

    Find the average of these 8 numbers: 32, 46, 44, 50, 84, 44, 37, 35.

    Formula:Arithmetic Mean

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

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15. 15.

    Mathematics

    Find the sum of the first 11 terms of an AP with first term 12 and common difference 5.

    Formula:AP Sum of n Terms

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

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16. 16.

    Quantitative Aptitude

    Two varieties costing $29 and $71 per kg are mixed to give a mixture worth $64 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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17. 17.

    Mathematics

    Find the sum of the first 6 terms of a GP with first term 4 and common ratio 3.

    Formula:GP Sum of n Terms

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

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18. 18.

    Quantitative Aptitude

    An item is bought for $900 and sold for $1,125. Find the profit percentage.

    Formula:Profit Percentage

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

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19. 19.

    Quantitative Aptitude

    In how many ways can 2 objects be arranged out of 6 distinct objects? (Find ⁿᴘᵣ for n = 6, r = 2.)

    Formula:Permutations

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

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20. 20.

    Quantitative Aptitude

    Find the HCF and LCM of 34 and 8.

    Formula:LCM & HCF

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

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