MAH-CET 2026 — Mathematics PYQ

Step 1: Analyze the Arithmetic Progression (AP)

Let $d$ be the common difference of the AP.

We are given:

• First term, $a_1=2$

• Tenth term, $a_{10}=3$

Using the general formula for the $n$-th term of an AP, $a_n=a_1+(n-1)d$:

$$a_{10}=a_1+9d$$

$$3=2+9d⟹9d=1⟹d=\frac{1}{9}$$

Now, let's find the required terms $a_7$ and $a_{19}$:

$$a_7=a_1+6d=2+6\left(\frac{1}{9}\right)=2+\frac{2}{3}=\frac{8}{3}$$

$$a_{19}=a_1+18d=2+18\left(\frac{1}{9}\right)=2+2=4$$

Step 2: Analyze the Geometric Progression (GP)

Let $r$ be the common ratio of the GP.

We are given:

• First term, $q_1=2$

• Tenth term, $q_{10}=3$

Using the general formula for the $n$-th term of a GP, $q_n=q_1\cdot r^{n-1}$:

$$q_{10}=q_1\cdot r^9$$

$$3=2\cdot r^9⟹r^9=\frac{3}{2}⟹r={\left(\frac{3}{2}\right)}^{\frac{1}{9}}$$

Now, let's find the required term $q_7$:

$$q_7=q_1\cdot r^6=2\cdot {\left[{\left(\frac{3}{2}\right)}^{\frac{1}{9}}\right]}^6=2\cdot {\left(\frac{3}{2}\right)}^{\frac{2}{3}}$$

Step 3: Verify the Given Options

• Checking Option (a): Evaluate $a_7{a}_{19}$

  $$a_7{a}_{19}=\left(\frac{8}{3}\right)\cdot 4=\frac{32}{3}$$

  Since $\frac{32}{3}$ is a fraction, $a_7{a}_{19}$ is not an integer. This statement is true.

• Checking Option (b): Evaluate $a_{19}{q}_7$

  $$a_{19}{q}_7=4\cdot 2\cdot {\left(\frac{3}{2}\right)}^{\frac{2}{3}}=8\cdot {\left(\frac{3}{2}\right)}^{\frac{2}{3}}$$

  This contains an irrational radical component, so it is clearly not an integer.

• Checking Option (c): Compare $a_7{a}_{19}$ and $a_{19}{q}_{10}$

  $$a_7{a}_{19}=\frac{32}{3}$$

  $$a_{19}{q}_{10}=4\cdot 3=12$$

  Since $\frac{32}{3}\ne 12$, this option is incorrect.

Correct Answer

(a) $a_7{a}_{19}$ is not an integer