If in the expansion of (1+px)q, q is an element of N, the coefficients of x and x^2 are 21 and 189 respectively, then p and q are respectively

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q17 – AP-EAMCET 2026

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Home Maths Binomial TheoremQuestion #32

If in the expansion of

(1 + p x)^{q}

q \in N

, the coefficients of

x

and

x^{2}

are 21 and 189 respectively, then

p

and

q

are respectively

Clue 1 of 3: Binomial Expansion

Recall the general terms in the binomial expansion of $(1 + X)^{n}$ .

🧠 Full Solution Path

Step 1: Recall the Binomial Expansion✦ Active

The binomial expansion of $(1 + X)^{n}$ is given by $1 + n X + \frac{n (n - 1)}{2!} X^{2} + \dots$ . For the given expansion $(1 + p x)^{q}$ , we replace $X$ with $p x$ and $n$ with $q$ .

(1 + p x)^{q} = 1 + q (p x) + \frac{q (q - 1)}{2!} (p x)^{2} + \dots

Simplifying the terms, we get:

(1 + p x)^{q} = 1 + (q p) x + \frac{q (q - 1) p^{2}}{2} x^{2} + \dots

💡 Teacher's Secret Hint

Remember that the term $2!$ in the denominator is simply 2. Don't confuse it with other factorials.

Step 2: Formulate Equations from Given Coefficients○ Expand

From the problem statement, the coefficient of $x$ is 21. Comparing this with our expansion:

q p = 21 (1)

Also, the coefficient of $x^{2}$ is 189. Comparing this with our expansion:

\frac{q (q - 1) p^{2}}{2} = 189 (2)

💡 Teacher's Secret Hint

Always ensure you correctly identify the full coefficient, including any constants like $\frac{1}{2}$ or $p^{2}$ with the binomial term.

Step 3: Solve the System of Equations for q○ Expand

From equation (1), we can express $p$ in terms of $q$ : $p = \frac{21}{q}$ . Substitute this expression for $p$ into equation (2):

\frac{q (q - 1)}{2} {(\frac{21}{q})}^{2} = 189

Simplify the equation:

\frac{q (q - 1)}{2} \frac{441}{q^{2}} = 189

Cancel one $q$ term from the numerator and denominator:

\frac{(q - 1) 441}{2 q} = 189

Now, solve for $q$ :

441 (q - 1) = 189 \times 2 q

441 q - 441 = 378 q

441 q - 378 q = 441

63 q = 441

q = \frac{441}{63} = 7

💡 Teacher's Secret Hint

Be careful with algebraic manipulation, especially when squaring terms and simplifying fractions. Double-check your arithmetic.

Step 4: Find the Value of p○ Expand

Substitute the value of $q = 7$ back into equation (1) to find $p$ :

q p = 21

7 p = 21

p = \frac{21}{7} = 3

Therefore, the values are $p = 3$ and $q = 7$ .

💡 Teacher's Secret Hint

Always perform a quick check by substituting both $p$ and $q$ values back into the original coefficient expressions to ensure they satisfy both conditions.

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Maths Question 17 – AP-EAMCET 2026

Clue 1 of 3: Binomial Expansion

🧠 Full Solution Path