EquationsMCQRTP Sep 24Question 1104 of 221
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If α\displaystyle α\displaystyle \alpha and β\displaystyle β\displaystyle \beta are the roots of the equation x2+7x+12=0\displaystyle x^2 + 7x + 12 = 0, then the equation whose roots (α+β)2\displaystyle (α+β\displaystyle \alpha + \beta)^2 and (α−β)2\displaystyle (α−β\displaystyle \alpha - \beta)^2 will be:

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option c — x2−50x+49=0\displaystyle x^2 - 50x + 49 = 0

All Options:

  • Ax2−14x+49=0\displaystyle x^2 - 14x + 49 = 0
  • Bx2−24x+144=0\displaystyle x^2 - 24x + 144 = 0
  • Cx2−50x+49=0\displaystyle x^2 - 50x + 49 = 0
  • Dx2−19x+144=0\displaystyle x^2 - 19x + 144 = 0

Detailed Solution & Explanation

For the quadratic equation x2+7x+12=0\displaystyle x^2 + 7x + 12 = 0, we can find its roots α\displaystyle α\displaystyle \alpha and β\displaystyle β\displaystyle \beta by factoring:
x2+7x+12=0x^2 + 7x + 12 = 0
(x+3)(x+4)=0(x + 3)(x + 4) = 0
So the roots are α=−3\displaystyle α\displaystyle \alpha = -3 and β=−4\displaystyle β\displaystyle \beta = -4.

We want to find the equation whose roots are R1=(α+β)2\displaystyle R_1 = (α+β\displaystyle \alpha + \beta)^2 and R2=(α−β)2\displaystyle R_2 = (α−β\displaystyle \alpha - \beta)^2.

1. **Compute the new roots:**
R1=(−3+(−4))2=(−7)2=49R_1 = (-3 + (-4))^2 = (-7)^2 = 49
R2=(−3−(−4))2=(1)2=1R_2 = (-3 - (-4))^2 = (1)^2 = 1

2. **Find the sum (S\displaystyle S) and product (P\displaystyle P) of the new roots:**
S=R1+R2=49+1=50S = R_1 + R_2 = 49 + 1 = 50
P=R1⋅R2=49⋅1=49P = R_1 ⋅\displaystyle \cdot R_2 = 49 ⋅\displaystyle \cdot 1 = 49

3. **Form the quadratic equation:**
The required quadratic equation is:
x2−Sx+P=0x^2 - Sx + P = 0
x2−50x+49=0x^2 - 50x + 49 = 0

Hence, the correct option is **Option (c)**.

About This Chapter: Equations

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Linear, Quadratic and Cubic Equations

This chapter covers Linear, Quadratic and Cubic Equations and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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