EquationsMCQMTP June 24 Series IIQuestion 1037 of 221
All Questions

Find the positive value of k\displaystyle k for which the equations: x2+kx+64=0\displaystyle x^2 + kx + 64 = 0 & x2−8x+k=0\displaystyle x^2 - 8x + k = 0 will have real roots:

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

✅ Option a — 12

All Options:

  • A12
  • B16
  • C18
  • D22

Detailed Solution & Explanation

For a quadratic equation Ax2+Bx+C=0\displaystyle Ax^2 + Bx + C = 0 to have real roots, its discriminant D=B2−4AC\displaystyle D = B^2 - 4AC must be greater than or equal to 0 (D≥0\displaystyle D ≥\displaystyle \ge 0). 1. For the first equation, x2+kx+64=0\displaystyle x^2 + kx + 64 = 0: D1=k2−4(1)(64)≥0  ⟹  k2−256≥0  ⟹  k2≥256D_1 = k^2 - 4(1)(64) ≥\displaystyle \ge 0 \implies k^2 - 256 ≥\displaystyle \ge 0 \implies k^2 ≥\displaystyle \ge 256 Since k\displaystyle k is a positive quantity, we have: k≥16— (i)k ≥\displaystyle \ge 16 \quad — (i)\displaystyle \text{--- (i)} 2. For the second equation, x2−8x+k=0\displaystyle x^2 - 8x + k = 0: D2=(−8)2−4(1)(k)≥0  ⟹  64−4k≥0  ⟹  4k≤64  ⟹  k≤16— (ii)D_2 = (-8)^2 - 4(1)(k) ≥\displaystyle \ge 0 \implies 64 - 4k ≥\displaystyle \ge 0 \implies 4k ≤\displaystyle \le 64 \implies k ≤\displaystyle \le 16 \quad — (ii)\displaystyle \text{--- (ii)} Combining equations (i) and (ii) for a positive value of k\displaystyle k: 16≤k≤16  ⟹  k=1616 ≤\displaystyle \le k ≤\displaystyle \le 16 \implies k = 16 Mathematically, the positive value of k\displaystyle k is $16$ (Option b). However, according to the provided key, the answer is marked as Option a ($12$). Based on the provided key: **Option a**

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 3x+y+2x−y\displaystyle \frac{3}{x+y} + \frac{2}{x-y} = -1 and 1x+y−1x−y=43\displaystyle 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 $28$. If the cost of an apple is doubled then the cost of $3orangesand\displaystyle oranges and5$ apples is $75$. The original cost of 7\displaystyle 7 oranges and 4\displaystyle 4 apples (in Rs) is:

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