algebra

If $P e^x=Q e^{-x}$ for all real values of $x$, which one of the following statements is true?

If two distinct non-zero real variables x and y are such that (x + y) is proportional to (x - y) then the value of $\frac{x}{y}$

For positive non-zero real variables x and y, if $\ln\left(\frac{x+y}{2}\right) = \frac{1}{2}[\ln (x) + \ln (y)]$ then, the value of $\frac{x}{y} + \frac{y}{x}$ is

For positive non-zero real variables p and q, if log (p² + q²) = log p + log q + 2 log 3, then, the value of $\frac{p^4+q^4}{p^2q^2}$ is

If two distinct non-zero real variables $x$ and $y$ are such that $(x + y)$ is proportional to $(x - y)$ then the value of $\frac{x}{y}$

For positive non-zero real variables $p$ and $q$, if $\log \left(p^2 + q^2\right) = \log p + \log q + 2 \log 3$, then, the value of $\frac{p^4 + q^4}{p^2 q^2}$ is

Consider the operators $\diamond$ and $\square$ defined by $a \diamond b=a+2 b, a \square b=a b$, for positive integers. Which of the following statements is/are TRUE?

For positive non-zero real variables $x$ and $y$, if $\ln \left( \frac{x + y}{2} \right) = \frac{1}{2} [ \ln (x) + \ln (y) ]$ then, the value of $\frac{x}{y} + \frac{y}{x}$ is

Let r be a root of the equation x 2 + 2x + 6 = 0. Then the value of the expression (r + 2) (r + 3) (r + 4) (r + 5) is

If $${\left( {x - {1 \over 2}} \right)^2} - {\left( {x - {3 \over 2}} \right)^2} = x + 2$$, then the value of x is :

Ten friends planned to share equally the cost of buying a gift for their teacher. When two of them decided not to contribute, each of the other friends had to pay Rs 150 more. The...

If $$pqr \ne 0$$ and $${p^{ - x}} = {1 \over q},{q^{ - y}} = {1 \over r},\,{r^{ - z}} = {1 \over p},$$ what is the value of the product $$𝑥𝑦𝑧$$?

Consider a quadratic equation x² - 13x + 36 = 0 with coefficients in a base b. The solutions of this equation in the same base b are x = 5 and x = 6. Then b =

The expression $\frac{(x+y)-|x-y|}{2}$ is equal to :

If $$f\left( x \right) = 2{x^7} + 3x - 5,$$ which of the following is a factor of $$f(x)$$?

In a quadratic function, the value of the product of the roots $$\left( {\alpha ,\beta } \right)$$ is $$4.$$ Find the value of $$${{{\alpha ^n} + {\beta ^n}} \over {{\alpha ^{ - n}...

If $$g(x)=1-x$$ & $$h\left( x \right) = {x \over {x - 1}}\,\,$$ then $$\,\,{{g\left( {h\left( x \right)} \right)} \over {h\left( {g\left( x \right)} \right)}}\,\,\,$$ is

A function $$f(x)$$ is linear and has a value of $$29$$ at $$x=-2$$ and $$39$$ at $$x=3.$$ Find its value at $$x=5.$$

Given $$\sqrt {\left( {224} \right),} = {\left( {13} \right)_r},$$ The value of the radix' $$r$$ is:

Let R denote the set of real numbers. Let f: $$R\,x\,R \to \,R\,x\,R\,$$ be a bijective function defined by f (x, y ) = (x + y, x - y). The inverse function of f is given by

If a, b and c are constants, which of the following is a linear inequality?