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Questions › IIT JAM › IIT JAM 2021 › Q38
2 marksMathematics › Quantitative aptitude and data reasoning › Equations and inequalitiesHardabout 3 min

IIT JAM 2021 Question 38: Quantitative aptitude and data reasoning

Let k∈Rk \in \mathbb{R}k∈R. Which of the following is/are correct for the roots of x2+2(k+1)x+9k−5=0x^2 + 2(k+1)x + 9k - 5 = 0x2+2(k+1)x+9k−5=0?

Select all that apply.

Answer and worked solution

Answer: (B) and (D)

The discriminant is 4[(k+1)2−(9k−5)]=4(k−1)(k−6)4[(k+1)^2 - (9k-5)] = 4(k-1)(k-6)4[(k+1)2−(9k−5)]=4(k−1)(k−6), so the roots are real when k≤1k \le 1k≤1 or k≥6k \ge 6k≥6 and complex when 1<k<61 < k < 61<k<6. Thus for 2≤k≤42 \le k \le 42≤k≤4 the roots are complex, giving (B).

The sum of roots is −2(k+1)<0-2(k+1) < 0−2(k+1)<0 and the product is 9k−59k - 59k−5. For k≥6k \ge 6k≥6 the roots are real with negative sum and positive product, hence both negative, giving (D). For k≤1k \le 1k≤1 the roots are real but their sum is negative and product positive, so they are negative, not positive, so (A) is wrong; and 4<k<64 < k < 64<k<6 gives complex roots, so (C) is wrong. The correct statements are (B) and (D).

The trap: (A) If k≤1k \le 1k≤1, then the roots are real and positive

For k≤1k \le 1k≤1 the roots are real but negative (negative sum, positive product), so the claim of positive roots in (A) fails.

Plain-text version

Let k ∈ ℝ. Which of the following is/are correct for the roots of x² + 2(k+1)x + 9k - 5 = 0? (A) If k ≤ 1, then the roots are real and positive (B) If 2 ≤ k ≤ 4, then the roots are complex (C) If 4 < k < 6, then the roots are real and opposite in sign (D) If k ≥ 6, then the roots are real and negative

Also written as: Let k in R. Which of the following is/are correct for the roots of x^2 + 2(k+1)x + 9k - 5 = 0?

Quantitative aptitude and data reasoning has come up 92 times in 6 exams since 2008.

What this question tests

This question is on Equations and inequalities, part of Quantitative aptitude and data reasoning in Mathematics.

Solving quadratic and other equations and inequalities.

TestStop has 26 past-paper questions on this subtopic.

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Question from the IIT JAM 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.