Mathematics

# Let $f(x)= \left| \begin{matrix} \sec { x } & \cos { x } & \sec { ^{ 2 }x+\cot { x\csc { x } } } \\ \cos { ^{ 2 }x } & \cos { ^{ 2 }x } & \csc { ^{ 2 }x } \\ 1 & \cos { ^{ 2 }x } & \cos { ^{ 2 }x } \end{matrix} \right|$. Then ${\int}_{0}^{\pi/2}f(x)dx$ is equal to

$\dfrac{15\pi}{60}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
If $n\rightarrow \infty$ then the limit of series in $n$ can be evaluated by following the rule : $\displaystyle \lim_{n\rightarrow \infty}\sum_{r=an+b}^{cn+d}\frac{1}{n}f\left ( \frac{r}{n} \right )=\int_{a}^{c}f(x)dx,$ where in $LHS$, $\dfrac{r}{n}$ is replaced by $x$, $\dfrac{1}{n}$ by $dx$ and the lower and upper limits are $\lim_{n\rightarrow \infty }\dfrac{an+b}{n}\, and \, \lim_{n\rightarrow \infty }\dfrac{cn+d}{n}$ respectively. Then answer the following question.
$\lim_{n\rightarrow \infty }\left \{ \dfrac{1}{\sqrt{4n-1^{2}}}+\dfrac{1}{\sqrt{8n-2^{2}}} +...+\dfrac{1}{\sqrt{3}n}\right \}$ equals?
• A. $\dfrac{\pi }{6}$
• B. $\dfrac{4\pi }{6}$
• C. $\dfrac{\pi }{4}$
• D. $\dfrac{\pi }{3}$

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Medium
Let $I = \int_1^3 | (x - 1) (x - 2) (x - 3) |dx$. The value of $I^{-1}$ is

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
Let $S_{n}= \sum_{k= 1}^{n}\displaystyle \frac{n}{n^{2}+kn+k^{2}}, T_{n}= \sum_{k= 0}^{n-1}\displaystyle \frac{n}{n^{2}+kn+k^{2}}$ for $n= 1, 2, 3...$ Then
• A. $S_{n}> \displaystyle \frac{\pi }{3\sqrt{3}}$
• B. $T_{n}< \displaystyle \frac{\pi }{3\sqrt{3}}$
• C. $S_{n}< \displaystyle \frac{\pi }{3\sqrt{3}}$
• D. $T_{n}> \displaystyle \frac{\pi }{3\sqrt{3}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Given, $f(x) = \begin{vmatrix} 0 & {x^2 - \sin x} & {\cos x - 2} \\ {\sin x -x^2} & 0 & {1 - 2x} \\ {2 - \cos x} & {2x - 1} & 0 \end{vmatrix}$, then $\displaystyle \int f(x) dx$ is equal to
• A. $\frac{x^3}{3} - x^2 \sin x + \sin 2x + C$
• B. $\frac{x^3}{3} - x^2 \sin x + \cos 2x + C$
• C. $\frac{x^3}{3} - x^2 \cos x + \cos 2x + C$
• D. None of the above

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
If $A=\displaystyle \int_{0}^{1}{\dfrac{{e}^{t}}{1+t}}dt$ then $\displaystyle \int_{0}^{1}{{e}^{t}ln(1+t)}dt=$
• A. $e\ ln{2}+A$
• B. $Ae\ ln{2}$
• C. $A\ ln{2}$
• D. $e\ ln{2}-A$