Let a, b,and c be such that frac{1}{(1-x)(1-2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given the partial fraction decomposition: $\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}$.Multiplying both sides b

Vedclass · Accepted Answer

$\frac{1}{15}$

Vedclass · Answer

(A) Given the partial fraction decomposition: $\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}$.Multiplying both sides by $(1-x)(1-2x)(1-3x)$,we get:$1 = a(1-2x)(1-3x) + b(1-x)(1-3x) + c(1-x)(1-2x)$.To find $a$,put $x = 1$:$1 = a(1-2)(1-3) + 0 + 0 \Rightarrow 1 = a(-1)(-2) \Rightarrow a = \frac{1}{2}$.To find $b$,put $x = \frac{1}{2}$:$1 = 0 + b(1-\frac{1}{2})(1-\frac{3}{2}) + 0 \Rightarrow 1 = b(\frac{1}{2})(-\frac{1}{2}) \Rightarrow b = -4$.To find $c$,put $x = \frac{1}{3}$:$1 = 0 + 0 + c(1-\frac{1}{3})(1-\frac{2}{3}) \Rightarrow 1 = c(\frac{2}{3})(\frac{1}{3}) \Rightarrow c = \frac{9}{2}$.Now,calculate $\frac{a}{1} + \frac{b}{3} + \frac{c}{5}$:$\frac{1/2}{1} + \frac{-4}{3} + \frac{9/2}{5} = \frac{1}{2} - \frac{4}{3} + \frac{9}{10}$.Finding a common denominator of $30$:$\frac{15}{30} - \frac{40}{30} + \frac{27}{30} = \frac{15 - 40 + 27}{30} = \frac{2}{30} = \frac{1}{15}$.

List$-I$	List$-II$
$(P)$ Passing $H _2 S$ in the presence of $NH _4 OH$	$(1) \ Cu ^{2+}$
$(Q) \ \left( NH _4\right)_2 CO _3$ in the presence of $NH _4 OH$	$(2) \ Al ^{3+}$
$(R) \ NH _4 OH$ in the presence of $NH _4 Cl$	$(3) \ Mn ^{2+}$
$(S)$ Passing $H _2 S$ in the presence of dilute $\text{HCl}$	$(4) \ Ba ^{2+}$
	$(5) \ Mg ^{2+}$

Let $a, b$,and $c$ be such that $\frac{1}{(1-x)(1-2x)(1-3x)} = \frac{a}{1-x} + \frac{b}{1-2x} + \frac{c}{1-3x}$. Then $\frac{a}{1} + \frac{b}{3} + \frac{c}{5}$ is equal to

Similar Questions

The $pK_a$ values of four carboxylic acids are given below. Identify the weakest carboxylic acid.

Which one of the following statements is not correct?

As cells in the cortex are loosely packed,what is the effect on water movement?

Dispersive power depends on the following:

Vedclass Test Series

Exam Paper Generator

Online Exam Module