Tag: business maths

Questions Related to business maths

The value of the definite integral $\int _{ 0 }^{ \pi /2 }{ \sin { x } \sin { 2x } \sin { 3x } dx } $ is equal to:

business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. $\cfrac{1}{3}$

  2. $-\cfrac{2}{3}$

  3. $-\cfrac{1}{3}$

  4. $\cfrac{1}{6}$

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Correct Option: D
Explanation:

$\int _0^{\pi/2}\sin x\sin 2x\sin 3x dx$


$\Rightarrow \dfrac{1}{2}\int _0^{\pi/2}2\sin x\sin 2x\sin 3x dx$

$\Rightarrow \dfrac{1}{2}\int _0^{\pi/2}2\sin x\sin 3x\sin 2x dx$

We know that       $2\sin A \sin B=\cos(A-B)-\cos (A+B)$

$\Rightarrow \dfrac{1}{2}\int _0^{\pi/2}(\cos (x-3x)-\cos(x+3x))\sin 2x dx$

$\Rightarrow \dfrac{1}{2}\int _0^{\pi/2}(\cos 2x-\cos 4x)\sin 2x dx$

$\Rightarrow \dfrac{1}{2}\int _0^{\pi/2} \sin 2x \cos 2x dx-\dfrac{1}{2}\int _0^{\pi /2}\sin 2x \cos 4x dx$

$\Rightarrow \dfrac{1}{4}\int _0^{\pi/2} 2\sin 2x \cos 2x dx-\dfrac{1}{4}\int _0^{\pi /2}2\sin 2x \cos 4x dx$

We know that   $2\sin A \cos B=\sin(A+B)+\sin (A-B)$

$\Rightarrow \dfrac{1}{4}\int _0^{\pi/2} (\sin (2x+2x)+\sin (2x-2x))dx-\dfrac{1}{4}\int _0^{\pi /2}(\sin (2x+4x)+\sin (2x-4x)) dx$

$\Rightarrow \dfrac{1}{4}\int _0^{\pi/2} \sin 4xdx-\dfrac{1}{4}\int _0^{\pi /2}(\sin 6x-\sin 2x) dx$

$\Rightarrow \dfrac{1}{4}\int _0^{\pi/2} \sin 4xdx-\dfrac{1}{4}\int _0^{\pi /2}\sin 6x dx+\dfrac{1}{4}\int _{0}^{\pi/4}\sin 2x dx$

$\Rightarrow \dfrac{1}{4}[\dfrac{\sin 4x}{4}] _0^{\pi/4}-\dfrac{1}{4}[\dfrac{-\sin 6x}{6}] _0^{\pi/4}+\dfrac{1}{4}[\dfrac{-\cos 2x}{2}] _0^{\pi/4}$

$\Rightarrow \dfrac{-1}{12}+\dfrac{1}{4}=\dfrac{1}{6}$

The value of the integral $\displaystyle\int{\sin{x}{\cos}^{4}{x}dx}$ where $x\in\left[-1,\,1\right]$ is 

business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. 1

  2. 1\2

  3. 0

  4. 4

Show answer
Correct Option: C
Explanation:
$f\left(x\right)=\sin{x}{\cos}^{4}{x},$

$f\left(-x\right)=\sin{\left(-x\right)}{\cos}^{4}{\left(-x\right)}=-f\left(x\right)$

Since $f\left(x\right)$ is an odd function, $\displaystyle\int _{-1}^{1}\sin{x}{\cos}^{4}{x}dx=0$

If $\Delta (x)=\left| \begin{matrix} 1+x+2{ x }^{ 2 } & x+3 & 1 \ x+2{ x }^{ 2 } & x & 3 \ 3x+6{ x }^{ 2 } & 3x+11 & 9 \end{matrix} \right| $ then $\displaystyle \int^{1} _{0}\Delta (x)dx$ is

business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. $\dfrac {176}{5}$

  2. $-\dfrac {176}{3}$

  3. $\dfrac {186}{3}$

  4. $-\dfrac {192}{3}$

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Correct Option: A
If $\phi{\left(x\right)}={\phi}^{\prime}{\left(x\right)}$ and $\phi{\left(1\right)}=2$ then $\phi{\left(3\right)}$  is equal to
business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. ${ \phi  }^{ 2 }$

  2. $2{ \phi  }^{ 2 }$

  3. $3{ \phi  }^{ 2 }$

  4. $2{ \phi  }^{ 3 }$

Show answer
Correct Option: A
Explanation:
$ \phi(x) = \phi '(x) \Rightarrow  \phi(x) = \frac{2\phi(x)}{dx} $

$ \Rightarrow  $  $\int  dx = \int \frac{d(\phi (x))}{\phi(x)}$

$ \Rightarrow x+c = ln \phi (x) \Rightarrow \phi (x) = k.e^{x}$

$ \phi(1) = 2 \Rightarrow  2 = k.e^{1}  $ $ \Rightarrow k=2e^{-1}$

$  \therefore \phi (x) = 2.e^{x-1}$

$ \phi(3) = 2.e^{2} = \phi^{2}$

$ \phi(3) = \phi^{2}$


$\displaystyle \int _{1}^{4}\frac{\mathrm{x}\mathrm{d}\mathrm{x}}{\sqrt{2+4\mathrm{x}}}=$

business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. $\displaystyle \frac{1}{2}$

  2. $\displaystyle \frac{1}{\sqrt{2}}$

  3. $\displaystyle \frac{3}{2}$

  4. $\displaystyle \frac{3}{\sqrt{2}}$

Show answer
Correct Option: D
Explanation:

$\int _{1}^{4}\dfrac{x   dx}{\sqrt{2 + 4x}}=\dfrac{1}{2}\int _{1}^{4}\dfrac{x   dx}{\sqrt{x+\dfrac{1}{2}}}$
$=\dfrac{1}{2}\left [ \int _{1}^{4}\dfrac{(x+\dfrac{1}{2})dx}{\sqrt{x+\dfrac{1}{2}}}-\int _{1}^{4}\dfrac{\dfrac{1}{2}dx}{\sqrt{x+\dfrac{1}{2}}} \right ]$
$=\dfrac{1}{2}\left [ \int _{1}^{4} \sqrt{x+\dfrac{1}{2}} dx-\int _{1}^{4}(x+\dfrac{1}{2})^{\dfrac{1}{2}} \int _{1}^{4} \right ]$
$=\dfrac{1}{2} \left [ \dfrac{2}{3} (x+\dfrac{1}{2})^{\dfrac{3}{2}} \int _{1}^{4}-(x+\dfrac{1}{2})^{\dfrac{4}{2}} \int _{1}^{4} \right ]$
$=\dfrac{1}{3} \left [ \left ( \dfrac{9}{2} \right )^{\dfrac{3}{2}}-\left ( \dfrac{3}{2} \right )^{\dfrac{3}{2}} \right ] -\dfrac{1}{2} \left [ \left ( \dfrac{9}{2} \right )^{\dfrac{1}{2}}-\left ( \dfrac{3}{2} \right )^{\dfrac{1}{2}} \right ]$
$=\dfrac{1}{3} \left [ \left ( \dfrac{9}{2} \right )\left ( \dfrac{9}{2} \right )^{\dfrac{1}{2}}-\left ( \dfrac{3}{2} \right )\left ( \dfrac{3}{2} \right )^{\dfrac{1}{2}} \right ] - \dfrac{1}{2} \left [ \left ( \dfrac{3}{\sqrt{2}} \right )-\dfrac{\sqrt{3}}{\sqrt{2}} \right ]$
$=\dfrac{3}{\sqrt{2}}$

The value of $\displaystyle \int _{0}^{2}(x-\log _{2}a)dx=2\log _{2}(\frac{2}{a})$ for which of the following conditions?

business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. $\mathrm{a}>0$

  2. $\mathrm{a}>2$

  3. $\mathrm{a}=4$

  4. $\mathrm{a}=8$

Show answer
Correct Option: A
Explanation:

$\int _{ 0 }^{ 2 }{ (x-\log _{ 2 }a } )dx=2\log _{ 2 }(\cfrac { 2 }{ a } )$

The solution exists only if function is defined.
$ x-\log _{ 2 }a\longrightarrow$ defined
$ x\longrightarrow$ is defined for all values 
But $\log _{ 2 }a\longrightarrow$ defined for all values
But $ \log _{ 2 }a\longrightarrow$ defined for only a>0$
$\therefore \log (0)$ and $\log \text {(negative values)} )\longrightarrow$ not defined
Hence, required condition is $a>0$.

Evaluate $I = \displaystyle \int _{\pi /6}^{\pi /3}\sin x:dx$

business maths definite integral and it applications to areas fundamental theorem of calculus interpretation of define integral as an area fundamental theorem of integral calculus

  1. $\displaystyle \frac{1-\sqrt{3}}{2}$

  2. $\displaystyle \frac{\sqrt{3}+1}{2}$

  3. $\displaystyle \frac{\sqrt{3}-1}{2\sqrt{3}}$

  4. None of these

Show answer
Correct Option: A
Explanation:
Given : $I = \displaystyle \int _{\pi /6}^{\pi /3}\sin x\:dx$

Integeration of $\sin x dx$ is $-cos x dx + c$

$I = -cos x dx$

Substuting the upper and lower limit values we get,

$I = -cos\dfrac{\pi}{3}+cos\dfrac{\pi}{6}$

$I = \dfrac{-\sqrt{3}}{2} + \dfrac{1}{2}$

$I = \dfrac{1-\sqrt{3}}{2}$

A shopkeeper may sell an article for less than the marked price for some reason. The difference between the marked price and the lowered selling price is called the ________.

business maths commission,brokerage and discount advantages of bill of exchange definition, characteristics and parties of bills of exchange simple transactions related to bills of exchange

  1. Commission

  2. Discount

  3. Selling price

  4. Cost

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Correct Option: A
Explanation:

discount

Tarun got $30%$ concession on the labelled price of an article and sold it for $Rs. 8750$ with $25%$ profit on the price he bought. What was the labelled price ?

business maths commission,brokerage and discount advantages of bill of exchange definition, characteristics and parties of bills of exchange simple transactions related to bills of exchange

  1. $Rs. 10,000$

  2. $Rs. 12,000$

  3. $Rs. 16,000$

  4. Data inadequate

  5. None of these

Show answer
Correct Option: A
Explanation:
Let the labelled price be $x$ Rs
$\therefore $ Price at which $=x-\dfrac{30}{10}\times x$
he bought $=\left(\dfrac{7x}{10}\right)$ Rs
Also profit is $25\%$
$\therefore  125\% $ of $\dfrac{7x}{10}=8750$ Rs
$\therefore \dfrac{5}{4}\times \dfrac{7x}{10}=8750$ Rs
$\therefore x=\dfrac{8750\times 4\times 10}{5\times 7}$
$\therefore x=10000$ Rs
$\therefore $ Labelled price is $10000$ Rs

An article marked at Rs. 450 is sold at a discount of 20%. Find the discount given,

business maths commission,brokerage and discount advantages of bill of exchange definition, characteristics and parties of bills of exchange simple transactions related to bills of exchange

  1. Rs. $90$

  2. Rs. $100$

  3. Rs. $110$

  4. Rs. $120$

Show answer
Correct Option: A
Explanation:

Discount $ = \dfrac {20}{100} \times 450 = Rs 90 $