Partial fractions and reduction formulae

Partial fractions and reduction formulae Quiz

AQA A-Level Further Maths: Quiz

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  1. 1. Which is the form of the partial fractions of 1(x+1)(x2+4)\dfrac{1}{(x+1)(x^2+4)}?

    • A. Ax+1+Bx2+4\dfrac{A}{x+1}+\dfrac{B}{x^2+4}
    • B. Ax+1+Bx+2+Cx−2\dfrac{A}{x+1}+\dfrac{B}{x+2}+\dfrac{C}{x-2}
    • C. Ax+1+Bx+Cx2+4\dfrac{A}{x+1}+\dfrac{Bx+C}{x^2+4}
    • D. Ax+Bx+1+Cx2+4\dfrac{Ax+B}{x+1}+\dfrac{C}{x^2+4}

  2. 2. Find ∫341(x−1)(x−2) dx\displaystyle\int_3^4\frac{1}{(x-1)(x-2)}\,dx.

    • A. ln⁡43\ln\dfrac43
    • B. ln⁡34\ln\dfrac34
    • C. ln⁡2\ln2
    • D. ln⁡32\ln\dfrac32

  3. 3. Find ∫xx2+5 dx\displaystyle\int\frac{x}{x^2+5}\,dx.

    • A. ln⁡(x2+5)+c\ln(x^2+5)+c
    • B. 2ln⁡(x2+5)+c2\ln(x^2+5)+c
    • C. 15arctan⁡x5+c\dfrac{1}{\sqrt5}\arctan\dfrac{x}{\sqrt5}+c
    • D. 12ln⁡(x2+5)+c\dfrac12\ln(x^2+5)+c

  4. 4. Given In=e−nIn−1I_n=\mathrm{e}-nI_{n-1} and I0=e−1I_0=\mathrm{e}-1, find I1I_1.

    • A. e−1\mathrm{e}-1
    • B. 11
    • C. e−2\mathrm{e}-2
    • D. 2e−12\mathrm{e}-1

  5. 5. Given In=n−1nIn−2I_n=\dfrac{n-1}{n}I_{n-2} for In=∫0π/2sin⁡nx dxI_n=\displaystyle\int_0^{\pi/2}\sin^nx\,dx, find I4I_4.

    • A. 3π16\dfrac{3\pi}{16}
    • B. π4\dfrac{\pi}{4}
    • C. 3π8\dfrac{3\pi}{8}
    • D. 38\dfrac38

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