NIMCET 2010 — Mathematics PYQ

1. Analyze the given progressions:

• Since $a,b,c$ are in Arithmetic Progression (A.P.):

  $$2b=a+c$$

• Since $p,q,r$ are in Harmonic Progression (H.P.):

  $$q=\frac{2pr}{p+r}$$

• Since $ap,bq,cr$ are in Geometric Progression (G.P.):

  $$(bq{)}^2=(ap)(cr)$$
  $$b^2{q}^2=apcr⟹q^2=\frac{apcr}{b^2}$$

2. Substitute the values of $b$ and $q$ into the G.P. equation:

From the H.P. condition, we have $q=\frac{2pr}{p+r}$. Substituting this into the G.P. relation:

Cancel $pr$ from both sides (assuming $p,r\ne 0$):

3. Incorporate the A.P. condition ($2b=a+c⟹b=\frac{a+c}{2}$):

Substitute $b^2=\frac{(a+c{)}^2}{4}$ into the equation:

4. Simplify and solve for the required expression:

Dividing both sides by 4:

Invert both sides:

Expand the squares:

Separate the terms:

Subtract 2 from both sides:

Correct Option:

(b) $\frac{a}{c}+\frac{c}{a}$