11+ · Practice Sheet 43

1. 1.

   Mathematics

   Find the binomial coefficient C(8, 3) — the coefficient of x^3 in (1 + x)^8.

   Formula:Binomial Coefficient

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

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2. 2.

   Mathematics

   Find the sum of the first 16 terms of an AP with first term 7 and common difference 4.

   Formula:AP Sum of n Terms

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

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3. 3.

   Mathematics

   Evaluate ∫ from 0 to 4 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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4. 4.

   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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5. 5.

   Quantitative Aptitude

   An item is bought for £800 and sold for £1,000. Find the profit percentage.

   Formula:Profit Percentage

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

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6. 6.

   Mathematics

   Find the distance between the points (7, -1) and (8, 0).

   Formula:Distance Between Two Points

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

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7. 7.

   Quantitative Aptitude

   A vehicle covers 324 km in 4 hours. Find its average speed.

   Formula:Speed, Distance & Time

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

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8. 8.

   Mathematics

   Find term number 7 of a geometric progression with first term 4 and common ratio 2.

   Formula:GP nth Term

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

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9. 9.

   Mathematics

   Find the sum of the first 7 terms of a GP with first term 5 and common ratio 3.

   Formula:GP Sum of n Terms

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

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10. 10.

    Mathematics

    If f(x) = 1x³ + 5x² + 4x, find f′(3).

    Formula:Derivative of a Polynomial

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

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11. 11.

    Quantitative Aptitude

    What is 10% of 900?

    Formula:Percentage of a Number

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

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12. 12.

    Mathematics

    If f(x) = 3x³ + 4x² + 9x, find f′(3).

    Formula:Derivative of a Polynomial

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

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13. 13.

    Mathematics

    For the quadratic 2x² − 2x + 1 = 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

    What is 10% of 440?

    Formula:Percentage of a Number

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

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15. 15.

    Quantitative Aptitude

    Divide £2,552 between two people in the ratio 7:4. 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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16. 16.

    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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17. 17.

    Quantitative Aptitude

    A value changes from 700 to 980. Find the percentage increase.

    Formula:Percentage Change

    $\%\,\text{change} = \dfrac{|new-old|}{old}\times 100$

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18. 18.

    Quantitative Aptitude

    What is 60% of 300?

    Formula:Percentage of a Number

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

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19. 19.

    Quantitative Aptitude

    A vehicle covers 414 km in 9 hours. Find its average speed.

    Formula:Speed, Distance & Time

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

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20. 20.

    Quantitative Aptitude

    Find the compound interest on £2,000 at 10% per annum compounded annually for 3 years.

    Formula:Compound Interest

    $A = P\left(1+\dfrac{R}{100}\right)^{T}$

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