In ABC with the usual notations prove that (a-b)^2 ^2 ( C 2 )+(a+b)^2 ^2 ( C 2 )=c^2

LHS =(a-b)^2 ^2 (C 2 )+(a+b)^2 ^2 (C 2 ) =a^2 [ ^2 (C 2 )+ ^2 (C 2 ) ]+b^2 [ ^2 (C 2 )+ ^2 (C 2 ) ]-2 a b [ ^2 (C 2 )- ^2 (C 2 ) ] =a^2+b^2-a^2-b^2+c^2 =c^2 =RHS Hence proved

In ABC with the usual notations prove that (a-b)^2 ^2 ( C 2 )+(a+…

Evaluate : $\int_0^{\pi / 2} \frac{1}{1+\sqrt[3]{\tan x}} \cdot d x$

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If $x=f(t)$ and $y=g(t)$ are differentiable function of $t$ so that $y$ is a differentiable function of $x$ and $\frac{d x}{d t} \neq 0$, then prove that :
$\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
Hence find $\frac{d y}{d x}$ if $x=\sin t$ and $y=\operatorname{cost} t$.

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Evaluate: $\int_0^\pi x \cdot \sin x \cdot \cos ^2 x \cdot d x$

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Find the expected value, variance and standard derivation of random variable $X$ whose probability mass function $(p.m.f.)$ is given below

$X =x$	$1$	$2$	$3$
$P ( X =x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

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Evaluate: $\int_0^{\frac{\pi}{2}} x \sin x d x$

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Solve the following differential equations:

$(x+y) \frac{d y}{d x}=1$

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Using truth table prove that p ˅ (q ˄ r) ≡ (p ˅ q) ˄ (p ˅ r)

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