A-Level Maths · Practice Sheet 18

1. 1.

   Quantitative Aptitude

   A can finish a job in 17 days and B in 4 days. Working together, how long will they take?

   Formula:Time & Work (combined)

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

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2. 2.

   Mathematics

   If f(x) = 1x³ + 6x² + 5x, find f′(3).

   Formula:Derivative of a Polynomial

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

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3. 3.

   Quantitative Aptitude

   What is 75% of 320?

   Formula:Percentage of a Number

   $\text{Value} = \dfrac{P}{100}\times N$

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4. 4.

   Mathematics

   If f(x) = 2x³ + 5x² + 8x, find f′(4).

   Formula:Derivative of a Polynomial

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

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5. 5.

   Mathematics

   Find the distance between the points (5, 1) and (-2, -7).

   Formula:Distance Between Two Points

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

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6. 6.

   Quantitative Aptitude

   The sum of the ages of two people is 44 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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7. 7.

   Quantitative Aptitude

   The marked price of an item is £1,900. After a 50% discount, find the selling price.

   Formula:Discount & Selling Price

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

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8. 8.

   Quantitative Aptitude

   Divide £590 between two people in the ratio 6:4. 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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9. 9.

   Mathematics

   Find term number 3 of a geometric progression with first term 3 and common ratio 2.

   Formula:GP nth Term

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

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10. 10.

    Quantitative Aptitude

    Find the simple interest on £39,000 at 4% per annum for 5 year(s).

    Formula:Simple Interest

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

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11. 11.

    Quantitative Aptitude

    A vehicle covers 168 km in 4 hours. Find its average speed.

    Formula:Speed, Distance & Time

    $\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}$

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12. 12.

    Quantitative Aptitude

    In how many ways can 2 objects be chosen from 8 distinct objects? (Find ⁿᴄᵣ for n = 8, r = 2.)

    Formula:Combinations

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

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13. 13.

    Mathematics

    Find the binomial coefficient C(9, 2) — the coefficient of x^2 in (1 + x)^9.

    Formula:Binomial Coefficient

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

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14. 14.

    Mathematics

    Evaluate log base 3 of 27.

    Formula:Logarithm Evaluation

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

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15. 15.

    Quantitative Aptitude

    A invests £5,000 and B invests £6,000 for the same period. If the total profit is £38,000, find A's share.

    Formula:Partnership (profit share)

    $\text{Share}_A = \text{Profit}\times\dfrac{A}{A+B}$

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16. 16.

    Quantitative Aptitude

    Divide £780 between two people in the ratio 3:3. 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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17. 17.

    Mathematics

    In an arithmetic progression with first term 20 and common difference 6, find term number 6.

    Formula:AP nth Term

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

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18. 18.

    Mathematics

    For the quadratic 1x² − 3x − 4 = 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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19. 19.

    Quantitative Aptitude

    In how many ways can 2 objects be arranged out of 9 distinct objects? (Find ⁿᴘᵣ for n = 9, r = 2.)

    Formula:Permutations

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

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20. 20.

    Quantitative Aptitude

    A invests £9,000 and B invests £4,000 for the same period. If the total profit is £11,000, find A's share.

    Formula:Partnership (profit share)

    $\text{Share}_A = \text{Profit}\times\dfrac{A}{A+B}$

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