NIMCET 2007 — Mathematics PYQ

To solve this problem, we will first find the individual values of the constants $a$, $b$, and $c$ using the Heaviside Cover-up Method for partial fractions, and then substitute them into the target expression.

Step 1: Find the value of $a$

To isolate $a$, multiply both sides of the identity by $(1-x)$ and evaluate at the root where $1-x=0$ (which means $x=1$):

$$a={\left[\frac{1}{(1-2x)(1-3x)}\right]}_{x=1}$$

Substitute $x=1$:

$$a=\frac{1}{(1-2(1))(1-3(1))}=\frac{1}{(1-2)(1-3)}$$

$$a=\frac{1}{(-1)(-2)}=\frac{1}{2}$$

Step 2: Find the value of $b$

To isolate $b$, multiply both sides by $(1-2x)$ and evaluate at the root where $1-2x=0$ (which means $x=\frac{1}{2}$):

$$b={\left[\frac{1}{(1-x)(1-3x)}\right]}_{x=\frac{1}{2}}$$

Substitute $x=\frac{1}{2}$:

$$b=\frac{1}{\left(1-\frac{1}{2}\right)\left(1-3\left(\frac{1}{2}\right)\right)}=\frac{1}{\left(\frac{1}{2}\right)\left(1-\frac{3}{2}\right)}$$

$$b=\frac{1}{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)}=\frac{1}{-\frac{1}{4}}=-4$$

Step 3: Find the value of $c$

To isolate $c$, multiply both sides by $(1-3x)$ and evaluate at the root where $1-3x=0$ (which means $x=\frac{1}{3}$):

$$c={\left[\frac{1}{(1-x)(1-2x)}\right]}_{x=\frac{1}{3}}$$

Substitute $x=\frac{1}{3}$:

$$c=\frac{1}{\left(1-\frac{1}{3}\right)\left(1-2\left(\frac{1}{3}\right)\right)}=\frac{1}{\left(\frac{2}{3}\right)\left(1-\frac{2}{3}\right)}$$

$$c=\frac{1}{\left(\frac{2}{3}\right)\left(\frac{1}{3}\right)}=\frac{1}{\frac{2}{9}}=\frac{9}{2}$$

Step 4: Evaluate the Target Expression

We need to calculate the value of:

$$E=\frac{a}{1}+\frac{b}{3}+\frac{c}{5}$$

Substitute the values $a=\frac{1}{2}$, $b=-4$, and $c=\frac{9}{2}$ into the expression:

$$E=\frac{\left(\frac{1}{2}\right)}{1}+\frac{-4}{3}+\frac{\left(\frac{9}{2}\right)}{5}$$

$$E=\frac{1}{2}-\frac{4}{3}+\frac{9}{10}$$

Find a common denominator for $2$, $3$, and $10$, which is $30$:

$$E=\frac{1\times 15}{30}-\frac{4\times 10}{30}+\frac{9\times 3}{30}$$

$$E=\frac{15-40+27}{30}$$

$$E=\frac{42-40}{30}$$

$$E=\frac{2}{30}=\frac{1}{15}$$

Conclusion

The value of the expression is $\frac{1}{15}$.

Correct Option: (a) $1/15$