Continuity and Differentiability
17 Board Maths previous year questions on Continuity and Differentiability — options free on every question; 2 include the answer & explanation free, the rest unlock with PYQ Pass.
If \({\mathrm{xe}}^{\mathrm{y}}=1\), then the value of \(\frac{\mathrm{dy}}{\mathrm{d}x}\) at \(\mathrm{x}=1\) is :
-1
\(\frac{d}{dx}\left(x{e}^{y}\right)=\frac{d}{dx}(1)\)
\({e}^{y}+x{e}^{y}\frac{dy}{dx}=0\)
\(\frac{dy}{dx}=\frac{-{e}^{y}}{x{e}^{y}}\)
\(\frac{dy}{dx}=-\frac{1}{x}\)
\({\left.\frac{dy}{dx}\right|}_{x=1}=-\frac{1}{1}=-1\)
Question. Number of points of discontinuity of the function
\(f(x)=\frac{1}{x−5},x\neq 5\)
is :
1
A function is discontinuous at points where:
- It is not defined.
- The left-hand limit (LHL) and right-hand limit (RHL) do not match or are not finite.
The given function is undefined at \(x=5\) as the denominator becomes zero, leading to an infinite discontinuity.
The function is well-defined and continuous everywhere else because it is a simple rational function(except \(x=5\))
Step 3: ConclusionSince there is only one point where the function is not defined (\(x=5\)x=5), the number of points of discontinuity is 1.
The value of \(k\), for which \(\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{matrix}\frac{\sqrt{3}\mathrm{cosx}+\mathrm{sinx}}{3\mathrm{x}+\frac{\pi }{2}}, & \mathrm{x}\neq -\frac{\pi }{3} \\ \mathrm{k}, & \mathrm{x}=-\frac{\pi }{3}\end{matrix}\right.\) is continuous at \(\mathrm{x}=-\frac{\pi }{3}\), is :
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Which of the following statements is true for the function \(f(x)=\left\{\begin{array}{cc}x^2+3, & x \neq 0 \\ 1, & x=0\end{array} ?\right.\)
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The derivative of \(\tan ^{-1}\left(x^2\right)\) w.r.t. \(x\) is :
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The number of points of discontinuity of \(f(x)=\left\{\begin{aligned}|x|+3, & \text { if } x \leq-3 \\ -2 x, & \text { if }-3
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Derivative of \({\mathrm{e}}^{{\sin }^{2}x}\) with respect to \(\cos x\) is :
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The derivative of \(\sin \left(x^2\right)\) w.r.t. \(x\), at \(x=\sqrt{\pi}\) is :
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The value of k, for which \(\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{matrix}\frac{\sqrt{3}\mathrm{cosx}+\mathrm{sinx}}{3\mathrm{x}+\frac{\pi }{2}}, & \mathrm{x}\neq -\frac{\pi }{3} \\ \mathrm{k}, & \mathrm{x}=-\frac{\pi }{3}\end{matrix}\right.\) is continuous at \(\mathrm{x}=-\frac{\pi }{3}\), is :
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The derivative of \(\sin \left(x^2\right)\) w.r.t. \(x\), at \(x=\sqrt{\pi}\) is :
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The number of points of discontinuity of \(f(x)=\left\{\begin{aligned}|x|+3, & \text { if } x \leq-3 \\ -2 x, & \text { if }-3
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If \({\mathrm{xe}}^{\mathrm{y}}=1\), then the value of \(\frac{\mathrm{dy}}{\mathrm{d}x}\) at \(\mathrm{x}=1\) is :
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Which of the following statements is true for the function \(f(x)=\left\{\begin{array}{cc}x^2+3, & x \neq 0 \\ 1, & x=0\end{array} ?\right.\)
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Derivative of \({\mathrm{e}}^{{\sin }^{2}x}\) with respect to \(\cos x\) is :
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Question. Number of points of discontinuity of the function
\(f(x)=\frac{1}{x−5},x\neq 5\)
is :
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The derivative of \(\tan ^{-1}\left(x^2\right)\) w.r.t. \(x\) is :
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If \(y=x\cdot {\log }_{e}x\), then the value of \(\frac{{d}^{2}y}{d{x}^{2}}\) will be -
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