Q56 / 90 · KVPY 2015 (SB/SX) · Mathematics

If $p(x)$ is a cubic polynomial with $p(1)=3, p(0)=2$ and $p(-1)=4$, then $\int_{-1}^{1} p(x) d x$ is

  • A
    2
  • B
    3
  • C
    4
  • Correct answer
    D
    5

Solution

$\mathrm{p}(\mathrm{x})=\mathrm{ax}^{3}+\mathrm{b} \mathrm{x}^{2}+\mathrm{cx}+\mathrm{d}$ $\mathrm{p}(0)=2$ $\mathrm{~d}=2$ $\mathrm{p}(1)=3$ $\mathrm{a}+\mathrm{b}+\mathrm{c}+2=3$ $\mathrm{a}+\mathrm{b}+\mathrm{c}=1$ $\mathrm{p}(-1)=4$ $-a+b-c+2=4$ $-a+b-c=2$ $2 b=3 \Rightarrow b=\frac{3}{2}$ $P(x)=a x^{3}+\frac{3}{2} x^{2}+c x+2$ $\int_{-1}^{1} p(x) d x=\int_{-1}^{1}\left(a x^{2}+\frac{3}{2} x^{2}+c x+2\right) d x=5$

Hence, the correct answer is option: 'D'.

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