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If the expression \( \Large \left(mx-1+\frac{1}{x}\right) \) is always nonnegative, then the minimum value of m must be:

A) \( \Large -\frac{1}{2} \)
B) 0
C) \( \Large \frac{1}{4} \)
D) \( \Large \frac{1}{2} \)

Correct Answer:



C) \( \Large \frac{1}{4} \)

Description for Correct answer:

We know that \( \Large ax^{2}+bx+c\ge 0 \) if a > 0 and \( \Large b^{2}-yac \le 0 \).

Now \( \Large mx-1+\frac{1}{x}=0 => \frac{mx^{2}-x+1}{x}\ge 0 \)

=> \( \Large mx^{2}-x+1\ge 0\ and\ x > 0 \)

Now, \( \Large mx^{2}-x+1\ge 0,\) if m > 0 and \( \Large1-4 m \le 0\) or if m>o and \( \Large m\ge \frac{1}{4} \)

Thus the minimum value of m is \( \Large \frac{1}{4} \).

Part of solved Quadratic Equations questions and answers : >> Elementary Mathematics >> Quadratic Equations

Comments

Similar Questions

1). The value of x in the given equation \( \Large 4^{x}-3^{x-\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1} \) is:

A). \( \Large \frac{4}{3} \)

B). \( \Large \frac{3}{2} \)

C). \( \Large \frac{2}{1} \)

D). \( \Large \frac{5}{3} \)

2). The harmonic mean of the roots of equation \( \Large \left(5+\sqrt{2}x^{2}-14+\sqrt{5}\right)x+8+2\sqrt{5}=0 \) is:

A). 2

B). 4

C). 6

D). 8

3). For what value of \( \Large \lambda \) the sum of the squares of the roots of \( \Large x^{2}+ \left(2+\lambda\right)n-\frac{1}{2} \left(1+\lambda\right)=0 \) is minimum?

A). \( \Large \frac{3}{2} \)

B). 1

C). \( \Large \frac{1}{2} \)

D). \( \Large \frac{11}{4} \)

4). Given that \( \Large \tan \alpha \) and \( \Large \tan \beta \) are the roots of \( \Large x^{2}-px+q=0 \), then the value of \( \Large \sin^{2} \left(\alpha + \beta \right) \) is equal to:

A). \( \Large \frac{p^{2}}{p^{2}+ \left(1-q\right)^{2} } \)

B). \( \Large \frac{p^{2}}{p^{2}+q^{2}} \)

C). \( \Large \frac{q^{2}}{p^{2} \left(1-q\right)^{2} } \)

D). \( \Large \frac{p^{2}}{ \left(p+q\right)^{2} } \)

5). The number of values of K for which the system of equations \( \Large \left(K+1\right)x+84=4K \) and \( \Large Kx+ \left(K+3\right)y=3 K-1 \) has infinitely many solution, is:

A). 0

B). 1

C). 2

D). infinite

6). The set of all real numbers x for which \( \Large x^{2}-|x+2|+x>0 \) is:

A). \( \Large \left(-\infty, -2\right) \ \left(2, \infty\right) \)

B). \( \Large \left(-\infty, -\sqrt{2}\right) \ \left(\sqrt{2}, \infty\right) \)

C). \( \Large \left(-\infty, -1\right) \ \left(1, \infty\right) \)

D). \( \Large \left(\sqrt{2}, \infty\right) \)

7). Let a, b, c be positive numbers, the following systems of equations in x, y and z \( \Large \frac{x^{2}}{a^{2}} +\ \frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1; \ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}} +\ \frac{z^{2}}{c^{2}}=1 and\ \frac{-x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \) has;

A). no solutions

B). unique solution

C). infinitely many solution

D). finitely many solution.

8). If \( \Large 2 \sin^{2}\frac{ \pi }{8} \) is a root of equation\( \Large x^{2}+ax+b=0 \), where a and b are rational numbers, then a - b is equal to

A). \( \Large -\frac{5}{2} \)

B). \( \Large -\frac{3}{2} \)

C). \( \Large -\frac{1}{2} \)

D). \( \Large \frac{1}{2} \)

9). If \( \Large \sin \theta +cosec \theta =2,\ then\ \sin^{2} \theta + cosec^{2} \theta \) is equal to

A). 1

B). 4

C). 2

D). none of these

10). The value of \( \Large \frac{\cot^{2} \theta +1}{\cot^{2} \theta -1} \) is equal to:

A). \( \Large \sin 2 \theta \)

B). \( \Large \cos 2 \theta \)

C). \( \Large \cosh 2 \theta \)

D). \( \Large \sec 2 \theta \)