EquationsRTP Sep 24Question 1104 of 155
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If α\displaystyle \alpha and 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 (\alpha + \beta)^2 and (α−β)2\displaystyle (\alpha - \beta)^2 will be:

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
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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 \alpha and 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 \alpha = -3 and β=−4\displaystyle \beta = -4.

We want to find the equation whose roots are R1=(α+β)2\displaystyle R_1 = (\alpha + \beta)^2 and R2=(α−β)2\displaystyle R_2 = (\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 \cdot R_2 = 49 \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.

More Questions from Equations

If 3x+y+2x−y=−1\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = -1 and 1x+y−1x−y=43\displaystyle \frac{1}{x+y} - \frac{1}{x-y} = \frac{4}{3} then (x,y)\displaystyle (x, y) is:

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The cost of 2\displaystyle 2 oranges and 3\displaystyle 3 apples is 28\displaystyle 28. If the cost of an apple is doubled then the cost of 3\displaystyle 3 oranges and 5\displaystyle 5 apples is 75\displaystyle 75. The original cost of 7\displaystyle 7 oranges and 4\displaystyle 4 apples (in Rs) is:

In a multiple choice question paper consisting of 100\displaystyle 100 questions of 1\displaystyle 1 mark each, a candidate gets 60%\displaystyle 60\% marks. If the candidate attempted all questions and there was a penalty of 0.25\displaystyle 0.25 marks for wrong answers is:

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