NDA 2026 Mathematics PYQ — Consider the following statements: I. has two irrational roots. I… | Mathem Solvex | Mathem Solvex
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NDA 2026 — Mathematics PYQ
NDA | Mathematics | 2026
Consider the following statements:
I. x+x+1=0 has two irrational roots.
II. 5x−x−4=0 has two rational roots.
Which of the statements given above is/are correct?
निम्नलिखित कथनों पर विचार कीजिए:
I. x+x+1=0के दो अपरिमेय मूल हैं
II. 5x−x−4=0के दो परिमेय मूल हैं
उपर्युक्त कथनों में से कौन-सा/कौन-से सही है/हैं?
Choose the correct answer:
A.
I only
B.
II only
(Correct Answer)
C.
Both I and II
D.
Neither I nor II
Correct Answer:
II only
Explanation
Let t=x, where t≥0 and x=t2.
Statement I: x+x+1=0
Substituting t:
t+t2+1=0⟹t2+t+1=0
To find the roots, we use the quadratic formula t=2a−b±b2−4ac:
t=2(1)−1±12−4(1)(1)=2−1±−3
The roots for t are complex numbers. Since we defined t=x, t must be a real, non-negative number. Therefore, this equation has no real roots for x. Statement I is incorrect.
Statement II: 5x−x−4=0
Substituting t:
5t−t2−4=0⟹t2−5t+4=0
Factoring the quadratic equation:
(t−4)(t−1)=0
This gives us two solutions for t:
t=4ort=1
Since t=x, we solve for x:
If x=4, then x=16.
If x=1, then x=1.
Both 16 and 1 are rational numbers. Therefore, the equation has two rational roots. Statement II is correct.
Conclusion: Only statement II is correct.
Correct Option: (b) II only
Explanation
Let t=x, where t≥0 and x=t2.
Statement I: x+x+1=0
Substituting t:
t+t2+1=0⟹t2+t+1=0
To find the roots, we use the quadratic formula t=2a−b±b2−4ac:
t=2(1)−1±12−4(1)(1)=2−1±−3
The roots for t are complex numbers. Since we defined t=x, t must be a real, non-negative number. Therefore, this equation has no real roots for x. Statement I is incorrect.
Statement II: 5x−x−4=0
Substituting t:
5t−t2−4=0⟹t2−5t+4=0
Factoring the quadratic equation:
(t−4)(t−1)=0
This gives us two solutions for t:
t=4ort=1
Since t=x, we solve for x:
If x=4, then x=16.
If x=1, then x=1.
Both 16 and 1 are rational numbers. Therefore, the equation has two rational roots. Statement II is correct.
