Mathematics

# One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\dfrac{m+\sqrt{n}}{r}$. When 'm' is non zero integer and n and r relatively prime natural numbers. Then $\dfrac{m+n+r}{100}=?$

$200$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int { \cfrac { { \sin }^{ 6 }x+{ \cos }^{ 6 }x }{ { \sin }^{ 2 }x{ \cos }^{ 2 }x } } dx$
• A. $\sin x-\cot x+C$
• B. $\cos x-\cot x+C$
• C. All of the above
• D. $\tan x-\cot x-3x+C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve $\displaystyle \int {\frac{x}{{1 - {x^3}}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate the integral $\displaystyle \int_0^{\tfrac {\pi}{2}}\frac {\sin x}{1+\cos^2x}dx$   using substitution.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle \int { co\sec { x } \log { \left( co\sec { x } -\cot { x } \right) } } dx\quad$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
In calculating a number of integrals we had to use the method of integration by parts several times in succession.
The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts
$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$
where  $\displaystyle v_{1}\left ( x \right )=\int v\left ( x \right )dx,v_{2}\left ( x \right )=\int v_{1}\left ( x \right )dx ..., v_{n}\left ( x \right )= \int v_{n-1}\left ( x \right )dx$
Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when  calculating $\displaystyle \int P_{n}\left ( x \right )Q\left ( x \right )dx,$ where $\displaystyle P_{n}\left ( x \right )$ is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n+1 times.