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The number of solution of \( \Large \frac{log 5 + log \left(x^{2}+1\right) }{log \left(x-2\right) }=2 \)

A) 2
B) 3
C) 1
D) none of these

Correct Answer:




D) none of these

Description for Correct answer:

We have, \( \Large \frac{log5 + log \left(x^{2}+1\right) }{log \left(x-2\right) } = 2 \)

=> \( \Large log\{ 5 \left(x^{2}+1\right) \} = log \left(x-2\right)^{2} \)

=> \( \Large 5 \left(x^{2}+1\right)= \left(x-2\right)^{2}=4x^{2}+4x+1=0 \)

=> \( \Large x = -\frac{1}{2} \)

But for \( \Large x = -\frac{1}{2},\ log \left(x-z\right) \) is not meaningful, it has no root.

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

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Similar Questions

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

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

3). 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

4). 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

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

6). 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

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

8). 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.

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

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A). 1

B). 4

C). 2

D). none of these