The maximum area (in sq. units) of a rectangle having its base on the \(x-\) axis and its other two vertices on the parabola, \(y = 12 -x^2\) such that the rectangle lies inside the parabola, is

The  real number \(k\) for which  the equation, \(2{x^2} + 3x + k = 0\) has two distinct real roots in \([0, 1]\) 

If \(2 x^y+3 y^x=20\), then \(\frac{d y}{d x}\) at \((2,2)\) is equal to

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

If \(n \geq 2\) is a positive integer, then the sum of the series \({ }^{ n +1} C _{2}+2\left({ }^{2} C _{2}+{ }^{3} C _{2}+{ }^{4} C _{2}+\ldots+{ }^{ n } C _{2}\right)\) is ...... .

Let \(p(x)\) be a quadratic polynomial such that \(p(0)= 1\) . If \(p(x)\) leaves remainder \(4\) when divided by \(x-1\) and it leaves remainder \(6\) when divided by \(x+ 1\) ; then

If \( f(x)=\left\{\begin{array}{cc} \frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x} & , x \neq 0 \\ b & , x=0 \end{array}\right. \) is continuous at \( x=0 \) then \( a+b \) is equal to:

If \(\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\dfrac{\pi}{3}\right)\cot\left(x+\dfrac{\pi}{3}\right)+1\right)dx = \alpha\log_e(\sqrt{3}-1)\), then \(9\alpha^2\) is equal to ________.

If the system of linear equations \(x + y + z = 5\) ; \(x = 2y + 2z = 6\) ; \(x + 3y + \lambda z = u (\lambda \, \mu \in R)\), has infinitely many solutions then the value of \(\lambda  + \mu \) is

The domain of the function \(\operatorname{cosec}^{-1}\left(\frac{1+\mathrm{x}}{\mathrm{x}}\right)\) is :

Let f: \(\mathrm{R} \rightarrow \mathrm{R}\) be defined as \(f(x) \rightarrow \frac{\lambda\left|x^{2}-5 x+6\right|}{\mu\left(5 x-x^{2}-6\right)}, x<2\) \(\quad\quad\quad\quad e^{\frac{\tan (x-2)}{x-[x]}}, \quad x>2\) \(\quad\quad\quad\quad \mu \quad\quad\quad\quad x=2\) Where \([x]\) is the greatest integer less than or equal to \(x\). If \(f\) is continuous at \(x=2\), then \(\lambda+\mu\) is equal to:

Let \(P\) be a point on the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let the line passing through \(P\) and parallel to \(y\)-axis meet the circle \(x^2+y^2=9\) at point \(Q\) such that \(P\) and \(Q\) are on the same side of the \(x\)-axis. Then, the eccentricity of the locus of the point \(R\) on \(P Q\) such that \(P R: R Q=4: 3\) as \(P\) moves on the ellipse, is :

Let \(H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N\). Let \(k\) be the smallest even value of \(n\) such that the eccentricity of \(H _{ k }\) is a rational number. If \(l\) is length of the latus return of \(H _{ k }\), then \(21 l\) is equal to \(.......\).