NIMCET 2007 — Mathematics PYQ

Step 1: Set up the equations using A.P. properties

• Since the $m^{\text{th}}$ term of the H.P. is $n$, the $m^{\text{th}}$ term of the corresponding A.P. is $\frac{1}{n}$:

  $$a+(m-1)d=\frac{1}{n}\text{— (Equation 1)}$$

• Since the $n^{\text{th}}$ term of the H.P. is $m$, the $n^{\text{th}}$ term of the corresponding A.P. is $\frac{1}{m}$:

  $$a+(n-1)d=\frac{1}{m}\text{— (Equation 2)}$$

Step 2: Solve for the common difference ($d$)

Subtract Equation 2 from Equation 1 to eliminate $a$:

$$[a+(m-1)d]-[a+(n-1)d]=\frac{1}{n}-\frac{1}{m}$$

$$(m-1-n+1)d=\frac{m-n}{mn}$$

$$(m-n)d=\frac{m-n}{mn}$$

Dividing both sides by $(m-n)$:

$$d=\frac{1}{mn}$$

Step 3: Solve for the first term ($a$)

Substitute the value of $d$ back into Equation 1:

$$a+(m-1)\left(\frac{1}{mn}\right)=\frac{1}{n}$$

$$a+\frac{m}{mn}-\frac{1}{mn}=\frac{1}{n}$$

$$a+\frac{1}{n}-\frac{1}{mn}=\frac{1}{n}$$

Subtract $\frac{1}{n}$ from both sides:

$$a=\frac{1}{mn}$$

Step 4: Find the $(m+n{)}^{\text{th}}$ term of the H.P.

First, let us find the $(m+n{)}^{\text{th}}$ term of the corresponding A.P. ($T_{m+n}$):

$$T_{m+n}=a+(m+n-1)d$$

Substitute the values of $a$ and $d$:

$$T_{m+n}=\frac{1}{mn}+(m+n-1)\left(\frac{1}{mn}\right)$$

$$T_{m+n}=\frac{1+m+n-1}{mn}$$

$$T_{m+n}=\frac{m+n}{mn}$$

Since the terms of an H.P. are the reciprocals of the terms of its corresponding A.P., the $(m+n{)}^{\text{th}}$ term of the H.P. is:

$${\text{Term}}_{\text{H.P.}}=\frac{1}{T_{m+n}}=\frac{mn}{m+n}$$

Correct Answer

The correct option is (a) $\frac{mn}{m+n}$.