While calculating the mean and variance of 10 readings,
Question: While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the c$\overline{\mathrm{x}}=\frac{\sum \mathrm{x}_{\mathrm{i}}}{\mathrm{n}}$orrect mean and the variance. Solution: Given while calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25 . He obtained the mean and variance as 45 and 16 respective...
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Question: Evaluate $\lim _{x \rightarrow 2}\left(\frac{3^{x}-3^{3-x}-12}{3^{3-x}-3^{x / 2}}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 2}\left(\frac{3^{x}-3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}\right)$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\lim _{x \rightarrow a} \frac{\mathrm{f}^{\prime}(\mathrm{x})}...
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Question: Evaluate $\lim _{x \rightarrow 1}\left(\frac{x^{4}-3 x^{2}+2}{x^{3}-5 x^{2}+3 x+1}\right)$ Solution: To Evaluate: $\lim _{x \rightarrow 1}\left(\frac{x^{4}-3 x^{2}+2}{x^{3}-5 x^{2}+3 x+1}\right)$ L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\m...
Read More →$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{x-\frac{\pi}{4}}$
[question] Question. $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{x-\frac{\pi}{4}}$ [/question] [solution] solution: Given $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{x-\frac{\pi}{4}}$ We have $\sin x-\cos x=\sqrt{2}\left(\frac{\sin x}{\sqrt{2}}-\frac{\cos x}{\sqrt{2}}\right)=\sqrt{2}\left(\sin x \cos \left(\frac{\pi}{4}\right)-\cos x \sin \left(\frac{\pi}{4}\right)\right)$ By using this formula in given equation we get $\Rightarrow \sqrt{2}\left(\sin x \cos \left(\fra...
Read More →Mean and standard deviation of 100 observations
Question: Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation. Solution: Given mean and standard deviation of 100 observations were found to be 40 and 10 , respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively Now we have to find the corre...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{\sqrt{1+x^{2}}-\sqrt{1+x}}{\sqrt{1+x^{3}}-\sqrt{1+x}}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{1+x^{2}}-\sqrt{1+x}}{\sqrt{1+x^{3}}-\sqrt{1+x}}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x ...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{\sqrt{a+x}-\sqrt{a}}{x \sqrt{a(a+x)}}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{a+x}-\sqrt{a}}{x \sqrt{a(a+x)}}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(...
Read More →Following are the marks obtained,
Question: Following are the marks obtained, out of 100, by two students Ravi and Hashina in 10 tests. Who is more intelligent and who is more consistent? Solution: Given the marks obtained, out of 100, by two students Ravi and Hashina in 10 tests Now we have to find who is more intelligent and who is more consistent Case 1: For Ravi The marks of Ravi being taken separately and finding other values can be tabulated as shown below, Here we have assumed 45 as mean. Total there are marks of 10 subje...
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Question: Evaluate $\lim _{x \rightarrow 4}\left(\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 4}\left(\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}\right)$ Formula used: Multiplying numerator and denominator with conjugates of numerator and denominator i.e $(1+\sqrt{5-x})(3+\sqrt{5+x})$ $\lim _{x \rightarrow 4}\left(\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}\right)=\lim _{x \rightarrow 4}\left(\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}\right)\left(\frac{1+\sqrt{5-x}}{1+\sqrt{5-x}}\ri...
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Question: Evaluate $\lim _{x \rightarrow 2}\left(\frac{x^{2}-4}{\sqrt{x+2}-\sqrt{3 x-2}}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 2} \frac{x^{2}-4}{\sqrt{x+2}-\sqrt{3 x-2}}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{...
Read More →Determine mean and standard deviation
Question: Determine mean and standard deviation of first n terms of an A.P. whose first term is a and common difference is d. Solution: Given first $n$ terms of an A.P. whose first term is a and common difference is $d$ Now we have to find mean and standard deviation The given AP in tabular form is as shown below, Here we have assumed a as mean. Given the AP have $n$ terms. And we know the sum of all the terms of AP can be written as, $\sum x_{i}=\frac{n}{2}[2 a+(n-1) d]$ Now we will calculate t...
Read More →The weights of coffee in 70 jars are shown in the following table:
Question: The weights of coffee in 70 jars are shown in the following table: Determine variance and standard deviation of the above distribution. Solution: Given the weights of coffee in 70 jars Now we have to find the variance and standard deviation of the distribution Let us make a table of the given data and append other columns after calculations Here mean, ${ }^{\bar{x}}=\frac{\sum f_{1} x_{1}}{N}=\frac{14137}{70}=201.9$ So the above table with more columns is as shown below, And we know st...
Read More →$\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2} \frac{\pi}{3}-x}$
[question] Question. $\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2} \frac{\pi}{3}-x}$ [/question] [solution] solution: Given $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}$ Now by using the formula $\cos 6 x=1-2 \sin ^{2} 3 x$ Then the above equation becomes, $\Rightarrow$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-\cos 6 x}}{\sqrt{2}\left(\frac{\pi}{3}-x\right)}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1-...
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Question: Evaluate $\lim _{x \rightarrow 1}\left(\frac{\sqrt{3+x}-\sqrt{5-x}}{x^{2}-1}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 1} \frac{\sqrt{3+x}-\sqrt{5-x}}{x^{2}-1}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \r...
Read More →Determine the mean and standard deviation for the following distribution:
Question: Determine the mean and standard deviation for the following distribution: Solution: Given the frequency distribution Now we have to find the mean and standard deviation Let us make a table of the given data and append other columns after calculations Here mean, $\bar{x}=\frac{\sum f_{\mathrm{f}} \mathrm{x}_{\mathrm{i}}}{\mathrm{N}}=\frac{229}{40}=6.02=6$ So the above table with more columns is as shown below, And we know standard deviation is $\sigma=\sqrt{\frac{\sum f_{i} d_{i}^{2}}{n...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{2 x}{\sqrt{a+x}-\sqrt{a-x}}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{2 x}{\sqrt{a+x}-\sqrt{a-x}}$ Formula used: Multiplying numerator and denominator by $\sqrt{a+x}+\sqrt{a-x}$ $\lim _{x \rightarrow 0} \frac{2 x}{\sqrt{a+x}-\sqrt{a-x}}=\lim _{x \rightarrow 0} \frac{2 x}{\sqrt{a+x}-\sqrt{a-x}}\left(\frac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}\right)$ $\lim _{x \rightarrow 0} \frac{2 x}{\sqrt{a+x}-\sqrt{a-x}}=\lim _...
Read More →Calculate the mean deviation from the median of the following data:
Question: Calculate the mean deviation from the median of the following data: Solution: Given the frequency distribution Now we have to find the mean deviation from the median Let us make a table of the given data and append other columns after calculations Now, here N=20, which is even. Now, here $\mathrm{N}=20$, which is even. Here median class $=\frac{\mathrm{N}}{2}=10^{\text {th }}$ term This observation lie in the class interval 12-18, so median can be written as, $\mathrm{M}=\mathrm{l}+\fr...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{\sqrt{3-x}-1}{2-x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{3-x}-1}{2-x}$ Formula used: We have, $\lim _{x \rightarrow a} f(x)=f(a)$ As $\mathrm{X} \rightarrow 0$, we have $\lim _{x \rightarrow 0} \frac{\sqrt{3-x}-1}{2-x}=\frac{\sqrt{3}-1}{2}$ Thus, the value of $\lim _{x \rightarrow 0} \frac{\sqrt{3-x}-1}{2-x}$ is $\frac{\sqrt{3}-1}{2}$...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{\sqrt{1+x+x^{2}}-1}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{1+x+x^{2}}-1}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathr...
Read More →$\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$
[question] Question. $\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$ [/question] [solution] solution: Given $\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$ Here cos mx can be written as $\Rightarrow \cos m x=1-2 \sin ^{2} \frac{m x}{2}$ And similarly $\Rightarrow \operatorname{cosn} x=1-2 \sin ^{2} \frac{n x}{2}$ Using these two identities in given equation we get $\Rightarrow$$\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}=\lim _{x \rightarrow 0} \frac{\left[1-\left(1-2 \...
Read More →Calculate the mean deviation about the mean
Question: Calculate the mean deviation about the mean for the following frequency distribution: Solution: Given the frequency distribution Now we have to find the mean deviation about the mean Let us make a table of the given data and append other columns after calculations Here mean, $\overline{\mathrm{X}}=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}=\frac{230}{25}=9.2$ Now we have to find mean deviation Hence Mean Deviation becomes, $M . D=\frac{\su...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{\sqrt{2-x}-\sqrt{2+x}}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{2-x}-\sqrt{2+x}}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(...
Read More →Find the mean and variance of the frequency
Question: Find the mean and variance of the frequency distribution given below: Solution: Given the frequency distribution Now we have to find the mean and variance Converting the ranges of x to groups, the given table can be rewritten as shown below, And we know variance can be written as $\sigma^{2}=\frac{\sum f_{i} x_{i}^{2}}{n}-\left(\frac{\sum f_{i} x_{i}}{n}\right)^{2}$ Substituting values from above table, we get $\sigma^{2}=\frac{340.25}{16}-\left(\frac{66.5}{16}\right)^{2}$ On simplifyi...
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Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{\sqrt{1+x}-1}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\lim ...
Read More →If for a distribution ∑ (x −5)=3,
Question: If for a distribution (x 5)=3, (x 5)2 = 43 and the total number of item is 18, find the mean and standard deviation. Solution: Given for a distribution (x 5) = 3, (x 5)2= 43 and the total number of item is 18 Now we have to find the mean and standard deviation. As per given criteria, Number of items, n=18 And given (x 5) = 3, And also given, (x 5)2= 43 But we know mean can be written as, $\overline{\mathrm{X}}=\mathrm{A}+\frac{\sum(\mathrm{x}-5)}{\mathrm{n}}$ Here assumed mean is 5, so...
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