If \(\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}\), then the larger value of \(K\) for which \(f(K)+A+B+C=1\), is

If $\int \frac{1-{\left(\cot x\right)}^{2019}}{\tan x+{\left(\cot x\right)}^{2020}}dx=\frac{1}{n}ln\left|{\left(f\left(x\right)\right)}^n+{\left(g\left(x\right)\right)}^n\right|+c,$ then the value of $n{\left[{\left(f\left(x\right)\right)}^4+{\left(g\left(x\right)\right)}^4\right]}_{{}_{x=\frac{\pi }{3}}}=$

If the length and breadth of a rectangle of maximum area that can be inscribed in an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $8\sqrt{2}$ and $4\sqrt{2}$ respectively, then the eccentricity of that ellipse is

In \(\triangle A B C,(a+b+c)\left(\tan \frac{A}{2}+\tan \frac{B}{2}\right)\) is equal to

If \(P, Q, R\) are the mid-points of the sides \(A B, B C\) and \(C A\) of \(\triangle A B C\) respectively, then \(\mathbf{P C}-\mathbf{B Q}=\)

The radius of the circle having three chords along \(y\)-axis, the line \(y=x\) and the line \(2 x+3 y=10\) is

If \([x]\) denotes the greatest integer not exceeding \(x\) and if the function \(f\) defined by \(f(x)= \begin{cases}\frac{a+2 \cos x}{x^2} & , x < 0 \ b \tan \frac{\pi}{[x+4]} & , x \geq 0\end{cases}\) is continuous at \(x=0\), then the ordered pair \((a, b)\) is equal to

The monomers of buna-S rubber are

Among the oxides \(\mathrm{SiO}_2, \mathrm{SO}_2, \mathrm{Al}_2 \mathrm{O}_3\) and \(\mathrm{P}_2 \mathrm{O}_3\), the correct order of acidic strength is

An elevator of mass \(500 \mathrm{~kg}\) is ascending upwards with a constant acceleration \(a=2 \mathrm{~m} / \mathrm{s}^2\). What is the work done by the tension in the elevator cable during its climb by \(12 \mathrm{~m}\) ? (Take, \(g=10 \mathrm{~m} / \mathrm{s}^2\) )

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

If 2 cards drawn at random from a well shuffled pack of 52 playing cards are from the same suit, then the probability of getting a face card and a card having a prime number is

Which of the following results is not true about photoelectric effect?