80010 If the function \(f\) defined by
\(f(x)=\left\{\begin{array}{l}\cos x, \text { if } x \leq 0 \\ 3 x+\alpha, \text { if } 0\lt x\lt 2 \\ \beta x+3, \text { if } 2 \leq x \leq 4 \\ 11, \quad \text { if } x>4\end{array}\right.\)
Where, \(\alpha\) and \(\beta\) are real constants, is continuous on \(R\), then \(\alpha^{2}+\beta^{2}=\)

80011 If \(f:[-2,2] \rightarrow R\) is defined by \(f(x)=\left\lvert\, \begin{array}{cc}\frac{\sqrt{1+c x}-\sqrt{1-c x}}{\frac{x}{x+3}} & \text { for }-2 \leq x\lt 0 \\ \frac{x+1}{x+} & \text { for } 0 \leq x \leq 2\end{array}\right.\)
continuous on \([-2,2]\) then \(\mathrm{c}\) is equal to

80012 \(f(x)= \begin{cases}\frac{72^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}} x \neq 0 \\ k \log 2 \log 3 x=0\end{cases}\)
Find the value of ' \(k\) ' for which the function \(f\) is continuous.

80013 If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then
\(f(x)=\left\{\begin{array}{l}x+a \sqrt{2}(\sin x), 0 \leq x\lt \frac{\pi}{4} \\ 2 x(\cot x)+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a(\cos 2 x)-b(\sin x), \frac{\pi}{4}\lt x \leq \pi\end{array}\right.\)

80010 If the function \(f\) defined by
\(f(x)=\left\{\begin{array}{l}\cos x, \text { if } x \leq 0 \\ 3 x+\alpha, \text { if } 0\lt x\lt 2 \\ \beta x+3, \text { if } 2 \leq x \leq 4 \\ 11, \quad \text { if } x>4\end{array}\right.\)
Where, \(\alpha\) and \(\beta\) are real constants, is continuous on \(R\), then \(\alpha^{2}+\beta^{2}=\)

80011 If \(f:[-2,2] \rightarrow R\) is defined by \(f(x)=\left\lvert\, \begin{array}{cc}\frac{\sqrt{1+c x}-\sqrt{1-c x}}{\frac{x}{x+3}} & \text { for }-2 \leq x\lt 0 \\ \frac{x+1}{x+} & \text { for } 0 \leq x \leq 2\end{array}\right.\)
continuous on \([-2,2]\) then \(\mathrm{c}\) is equal to

80012 \(f(x)= \begin{cases}\frac{72^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}} x \neq 0 \\ k \log 2 \log 3 x=0\end{cases}\)
Find the value of ' \(k\) ' for which the function \(f\) is continuous.

80013 If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then
\(f(x)=\left\{\begin{array}{l}x+a \sqrt{2}(\sin x), 0 \leq x\lt \frac{\pi}{4} \\ 2 x(\cot x)+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a(\cos 2 x)-b(\sin x), \frac{\pi}{4}\lt x \leq \pi\end{array}\right.\)

80010 If the function \(f\) defined by
\(f(x)=\left\{\begin{array}{l}\cos x, \text { if } x \leq 0 \\ 3 x+\alpha, \text { if } 0\lt x\lt 2 \\ \beta x+3, \text { if } 2 \leq x \leq 4 \\ 11, \quad \text { if } x>4\end{array}\right.\)
Where, \(\alpha\) and \(\beta\) are real constants, is continuous on \(R\), then \(\alpha^{2}+\beta^{2}=\)

80011 If \(f:[-2,2] \rightarrow R\) is defined by \(f(x)=\left\lvert\, \begin{array}{cc}\frac{\sqrt{1+c x}-\sqrt{1-c x}}{\frac{x}{x+3}} & \text { for }-2 \leq x\lt 0 \\ \frac{x+1}{x+} & \text { for } 0 \leq x \leq 2\end{array}\right.\)
continuous on \([-2,2]\) then \(\mathrm{c}\) is equal to

80012 \(f(x)= \begin{cases}\frac{72^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}} x \neq 0 \\ k \log 2 \log 3 x=0\end{cases}\)
Find the value of ' \(k\) ' for which the function \(f\) is continuous.

80013 If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then
\(f(x)=\left\{\begin{array}{l}x+a \sqrt{2}(\sin x), 0 \leq x\lt \frac{\pi}{4} \\ 2 x(\cot x)+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a(\cos 2 x)-b(\sin x), \frac{\pi}{4}\lt x \leq \pi\end{array}\right.\)

80010 If the function \(f\) defined by
\(f(x)=\left\{\begin{array}{l}\cos x, \text { if } x \leq 0 \\ 3 x+\alpha, \text { if } 0\lt x\lt 2 \\ \beta x+3, \text { if } 2 \leq x \leq 4 \\ 11, \quad \text { if } x>4\end{array}\right.\)
Where, \(\alpha\) and \(\beta\) are real constants, is continuous on \(R\), then \(\alpha^{2}+\beta^{2}=\)

80011 If \(f:[-2,2] \rightarrow R\) is defined by \(f(x)=\left\lvert\, \begin{array}{cc}\frac{\sqrt{1+c x}-\sqrt{1-c x}}{\frac{x}{x+3}} & \text { for }-2 \leq x\lt 0 \\ \frac{x+1}{x+} & \text { for } 0 \leq x \leq 2\end{array}\right.\)
continuous on \([-2,2]\) then \(\mathrm{c}\) is equal to

80012 \(f(x)= \begin{cases}\frac{72^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}} x \neq 0 \\ k \log 2 \log 3 x=0\end{cases}\)
Find the value of ' \(k\) ' for which the function \(f\) is continuous.

80013 If the function \(f(x)\), defined below is continuous in the interval \([0, \pi]\), then
\(f(x)=\left\{\begin{array}{l}x+a \sqrt{2}(\sin x), 0 \leq x\lt \frac{\pi}{4} \\ 2 x(\cot x)+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a(\cos 2 x)-b(\sin x), \frac{\pi}{4}\lt x \leq \pi\end{array}\right.\)