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Let \((2x^{2} + 3x +4)^{10} = \displaystyle \sum_{r = 0}^{20} a_{r}x^{r}\), where \(a_0, a_1, \ldots , a_{20} \) are constants.

Find the value of ...

An ideal AC voltage source with frequency \(f_0\) supplies a load consisting of a resistor and an inductor in series. At frequency \(f_0\), the load current lags the source voltage ...

\[ ln(p)=\lim_{n\to\infty}\dfrac{1}{n} \left(\sum_{p=1}^{n}\sum_{k=1}^{p-1} \int_{k}^{k+1} \dfrac{k+1}{x(x+1)}dx - n\ln(n+1)\right) \]

Consider a triangular region bounded by the \(x\)-axis, the \(y\)-axis, and the line \(y = 1 - x\). The parabola \(y = a \, x^2\) divides this region into two parts ...

For a quadratic function \(f(x)=\dfrac{3x-x^2}{2},\) a function \(g(x)\) defined in the interval \([0,~\infty)\) satisfies the below conditions.

(1) For \(0\le x<1,\) ...

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