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Value of k (or constants) making a continuous p.d.f. valid

This type of question, asked 3 times: 22 Apr 2025 S2, 11 Apr 2026 S1, 18 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    22 Apr 2025 S2

    124
    If a random variable XX has p.d.f. f(x)={ax22+bx, if 1≤x≤30, otherwisef(x)=\begin{cases}\dfrac{ax^{2}}{2}+bx & ,\text{ if }1\le x\le 3\\ 0 & ,\text{ otherwise}\end{cases} and f(2)=2f(2)=2, then the values of aa and bb are, respectively

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  2. Question 2

    11 Apr 2026 S1

    99
    If the following function is probability density function of r.v. X f(x)=kx2(1−x)f(x) = kx^2(1 - x), for 0<x<1=00 < x < 1 = 0, otherwise, then the value of kk is

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  3. Question 3

    18 Apr 2026 S2

    99
    If the p. d. f. of a continuous random variable X is given by f(x)={k(9+8x−x2),for −1≤x≤40,otherwisef(x) = \begin{cases} k(9 + 8x - x^2), & \text{for } -1 \le x \le 4 \\ 0, & \text{otherwise} \end{cases} then the value of kk is …\ldots

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