Match the rational functions with the correct form of the partial fractions.

[ $p,q,r,a,b,c,A,B,C$ are arbitrary constants]

Match List-I with List-II

List-I (Rational Function) List-II (Partial Fraction)
(A) $\dfrac{px^2+qx+r}{(x-a)(x^2+bx+c)}$ (I) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$
(B) $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}, a\neq b$ (II) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$
(C) $\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}, a\neq b\neq c$ (III) $\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+bx+c}$
(D) $\dfrac{px+q}{(x-a)^2}$ (IV) $\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{x-c}$

Choose the correct answer from the options given below:

Question

Match the rational functions with the correct form of the partial fractions. [ $$p,q,r,a,b,c,A,B,C$$ are arbitrary constants] Match **List-I** with **List-II** | List-I (Rational Function) | List-II (Partial Fraction) | |---|---| | (A) $$\dfrac{px^2+qx+r}{(x-a)(x^2+bx+c)}$$ | (I) $$\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$$ | | (B) $$\dfrac{px^2+qx+r}{(x-a)^2(x-b)}, a
eq b$$ | (II) $$\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$$ | | (C) $$\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}, a
eq b
eq c$$ | (III) $$\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+bx+c}$$ | | (D) $$\dfrac{px+q}{(x-a)^2}$$ | (IV) $$\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{x-c}$$ | Choose the **correct** answer from the options given below:

Accepted Answer

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

Answer

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

Answer

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

Answer

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

List I	List II
A. $\int \frac{dx}{x^2 - a^2}$ is equal to	I. $\frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$
B. $\int \frac{dx}{\sqrt{x^2 + a^2}}$ is equal to	II. $\frac{1}{2a}\log\left\lvert \frac{x-a}{x+a}\right\rvert + C$
C. $\int \sqrt{a^2 - x^2} \, dx$ is equal to	III. $\frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + C$
D. $\int \sqrt{a^2 + x^2} \, dx$ is equal to	IV. $\log\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C$

List-I	List-II
Integral	Value of the integral
(A) $\displaystyle\int_{1}^{2} e^{2\log x} \, dx$	(I) $\log 2$
(B) $\displaystyle\int_{-1}^{1} \left\vert x \right\vert \, dx$	(II) $\dfrac{1}{2}$
(C) $\displaystyle\int_{0}^{1} \log e^x \, dx$	(III) $\dfrac{7}{3}$
(D) $\displaystyle\int_{1}^{2} \dfrac{1}{x} \, dx$	(IV) $1$

List-I (Rational Function)	List-II (Partial Fraction)
(A) $\dfrac{px^2+qx+r}{(x-a)(x^2+bx+c)}$	(I) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}+\dfrac{C}{x-b}$
(B) $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}, a\neq b$	(II) $\dfrac{A}{x-a}+\dfrac{B}{(x-a)^2}$
(C) $\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}, a\neq b\neq c$	(III) $\dfrac{A}{x-a}+\dfrac{Bx+C}{x^2+bx+c}$
(D) $\dfrac{px+q}{(x-a)^2}$	(IV) $\dfrac{A}{x-a}+\dfrac{B}{x-b}+\dfrac{C}{x-c}$

Solution

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