Consider the function $f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by
$$f(x) = (|x| + |x-1|) \sin x + \left[ x \sin x \right],$$
where $\left[ x \sin x \right]$ is the greatest integer less than or equal to $x \sin x$.
Let $\alpha$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT differentiable.
Then the value of $\alpha + \beta$ is ____________.
The number of ways to distribute 10 identical red pens and 14 identical blue pens among four persons such that each person gets 6 pens, is ______________.
Let
$$ \alpha = \left( 1 - 2\cos\left(\frac{\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{3\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{9\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{27\pi}{11}\right) \right) \left( 1 - 2\cos\left(\frac{81\pi}{11}\right) \right). $$
Then the value of $5 - \alpha^2$ is ______________.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List-I | List-II |
|---|---|
| (P) If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2026}}$ and $\frac{1}{(\beta+1)^{2026}}$ is | (1) $x^2 + x + 1 = 0$ |
| (Q) If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2027}}$ and $\frac{1}{(\beta+1)^{2027}}$ is | (2) $x^2 - x + 1 = 0$ |
| (R) If $\gamma$ and $\delta$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $\frac{1}{(\gamma-1)^{2026}} + \frac{1}{(\delta-1)^{2026}}$ is | (3) $x^2 + x - 1 = 0$ |
| (S) If $p$ and $r$ are the distinct roots of the equation $x^2 + x - 1 = 0$, then the value of $\frac{1}{(p+1)^3} + \frac{1}{(r+1)^3}$ is | (4) $-1$ |
| (5) $-4$ |
JEE Advanced Papers
All year-wise previous year question papers