6
Q:

# The number of solutions of the equation $\inline \log_{\frac{x}{2}}x^{2}+40\log_{4x}\sqrt{x}-14\log_{16x}x^{3}=0$ is:

 A) 0 B) 1 C) 2 D) 3

Explanation:

By changing the base to 2 the given equation becomes

$\inline&space;\frac{\log_{2}x^{2}}{\log_{2}x/2}+\frac{40\log_{2}\sqrt{x}}{\log_{2}4x}-14\frac{\log_{2}x^{3}}{\log_{2}16x}=0$

$\inline&space;\Rightarrow$     $\inline&space;\frac{2\log_{2}x}{\log_{2}x-1}+20\frac{\log_{2}x}{2+\log_{2}x}-42\frac{\log_{2}x}{4+\log_{2}x}=0$

Let $\inline&space;t=\log_{2}x$, then we have

$\inline&space;2t(4+t)(2+t)-42t(t-1)(t+2)+20t(t-1)(t+4)=0$

$\inline&space;\Rightarrow$    $\inline&space;2t[t^{2}+6t+8-21t^{2}-21t+42+10t^{2}+30t-40]=0$

$\inline&space;\Rightarrow$    $\inline&space;t[2t^{2}-3t-2]=0$

$\inline&space;\Rightarrow$     t = 0, t = 2, t = -1/2

$\inline&space;\Rightarrow$     x = 1, x = 4, x = $\inline&space;\frac{1}{\sqrt{2}}$

Q:

Solve the equation $\inline \fn_jvn \small \left ( \frac{1}{2} \right )^{2x+1} =1$  ?

 A) -1/2 B) 1/2 C) 1 D) -1

Explanation:

Rewrite equation as $\inline \fn_jvn \small \left ( \frac{1}{2} \right )^{2x+1} = (\frac{1}{2})^{0}$

Leads to 2x + 1 = 0

Solve for x : x = -1/2

6 722
Q:

If $\inline \fn_jvn \log _{7}2$ = m, then $\inline \fn_jvn \log _{49}28$ is equal to ?

 A) 1/(1+2m) B) (1+2m)/2 C) 2m/(2m+1) D) (2m+1)/2m

Explanation:

$\inline \fn_jvn \log _{49}28 = \frac{1}{2\log _{7}(7x4)}$

= $\inline \fn_jvn \frac{1}{2}(1+\log _{7}4)$
= $\inline \fn_jvn \frac{1}{2}+\frac{1}{2}(2\log _{7}2)$
$\inline \fn_jvn \frac{1}{2}+\log _{7}2$
$\inline \fn_jvn \frac{1}{2}+m$

$\inline \fn_jvn \frac{(1+2m)}{2}$.

11 972
Q:

If $\fn_jvn \small a^{2}+b^{2}=c^{2}$ , then $\inline \fn_jvn \frac{1}{\log_{c+a}b} + \frac{1}{\log_{c-a}b}= ?$

 A) 1 B) 2 C) 4 D) 8

Explanation:

Given $\fn_jvn \small a^{2}+b^{2}=c^{2}$

Now $\inline \fn_jvn \frac{1}{\log _{c+a}b}+\frac{1}{\log _{c-a}b}$ = $\inline \fn_jvn \log _{b}(c+a)+\log _{b}(c+a)$

$\inline \fn_jvn \log _{b}(c^{2}-a^{2})$

$\inline \fn_jvn \log _{b}b^{2}=2\log _{b}b = 1$

9 939
Q:

If log 64 = 1.8061, then the value of log 16 will be (approx)?

 A) 1.9048 B) 1.2040 C) 0.9840 D) 1.4521

Explanation:

Given that, $\inline \fn_jvn \small \log 64=1.8061$

$\inline \fn_jvn \small i.e. \log 4^{3}=1.8061$

$\inline \fn_jvn \small \Rightarrow 3\log 4=1.8061$

$\inline \fn_jvn \small \Rightarrow \log 4=0.6020$

$\inline \fn_jvn \small \Rightarrow 2\log 4=1.2040$

$\inline \fn_jvn \small \Rightarrow \log 4^{2}=1.2040$

$\inline \fn_jvn \small \Rightarrow \log 16=1.2040(approx)$

16 2524
Q:

A fast moving superfast express crosses another pasenger train in 20 seconds. The speed of faster train is 72 km/hr and speeds of slower train is 27 km/h. Also the length of faster ntrain is 100m, then find the length of the slower train if they are moving in the same direction.

 A) 100 m B) 125 m C) 150 m D) 175 m

$\inline \fn_jvn Time = \frac{sum\: of\: length\: of\: the\: two\: train}{Difference\: in\: speeds}$
$\inline \fn_jvn 20 = \frac{(100+x)}{25/2}$
$\inline \fn_jvn \Rightarrow x= 150\: m$