If in the expansion of $(1 + x)^{n},$ the coefficient of pth and qth terms are equal and the relation between p and q is p-q =n . Find the value of $p^{2} + q^{2}$ . where $p \neq q .$

Find the term independent of x in the expansion of ${(2 x - \frac{1}{x})}^{10} .$

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The polynomial (x+y)9 is expanded in decreasing powers of x. The second and third terms have equal values when evaluated at x=p and y=q, where p and q are positive numbers whose sum is one. The value of p is

The correct option is B 45General term, Tr+1=9Cr x9−ryr Given, 9C1 x8y=9C2 x7y2 ⇒9x=36y ⇒x=4y ⇒p=4q, where p+q=1 Hence, p+p4=1 ⇒p=45

The polynomial $(x + y)^{9}$ is expanded in decreasing powers of $x .$ The second and third terms have equal values when evaluated at $x = p$ and $y = q,$ where $p$ and $q$ are positive numbers whose sum is one. The value of $p$ is

The correct option is B $\frac{4}{5}$
General term, $T_{r + 1} =^{9} C_{r} x^{9 - r} y^{r}$
Given, $^{9} C_{1} x^{8} y =^{9} C_{2} x^{7} y^{2}$
$\Rightarrow 9 x = 36 y$
$\Rightarrow x = 4 y$
$\Rightarrow p = 4 q,$ where $p + q = 1$
Hence, $p + \frac{p}{4} = 1$
$\Rightarrow p = \frac{4}{5}$