JEE (Advanced) 2019 Paper-1

Let $$\alpha and \beta$$ be the roots of $$x^2 — x — 1 = 0,$$ with \alpha > \beta$$. For all positive integers n, define
$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},$$ $$n \geq 1,$$
$$b_1 = 1$$ and $$b_n = a_{n - 1} + a_{n + 1},$$ $$n \geq 2.$$
Then which of the following options is/are correct?

Let
$$M = \begin{bmatrix}0 & 1 & a \\1 & 2 & 3 \\3 & b & 1 \end{bmatrix}$$ and $$adj M = \begin{bmatrix}-1 & 1 & -1 \\8 & -6 & 2 \\-5 & 3 & -1 \end{bmatrix}$$
where a and b are real numbers. Which of the following options is/are correct?

There are three bags $$B_1, B_2$$ and $$B_3$$. The bag $$B_1$$ contains 5 red and 5 green balls, $$B_2$$ contains 3 red and 5 green balls, and $$B_3$$ contains 5 red and 3 green balls. Bags $$B_1, B_2$$ and $$B_3$$ have probabilities $$\frac{3}{10}, \frac{3}{10}$$ and $$\frac{4}{10}$$ respectively of being chosen. A bag is selected at random anda ball is chosen at random from the bag. Then which of the following options is/are correct?

In a non-right-angled triangle $$\triangle PQR$$,let p, g,r denote the lengths of the sides opposite to the angles at P,Q,R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at FE, and RS and PE intersect at O. If $$p = \sqrt 3, q = 1,$$ and the radius of the circumcircle of the $$\triangle PQR$$ equals 1, then which of the following options is/are correct?

Define the collections {E_1, E_2, E_3, ... } of ellipses and {R_1, R_2, R_3, ... } of rectangles as follows:
$$E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1$$;
$$R_1$$: rectangle of largest area, with sides parallel to the axes, inscribed in $$E_1$$;
$$E_n$$: ellipse $$\frac{x^2}{a_n^2} + \frac{y^2}{b_n^2} = 1$$ of largest area inscribed in $$R_{n - 1}, n > 1$$
$$R_n$$: rectangle of largest area, with sides parallel to the axes, inscribed in $$E_n, n > 1$$
Then which of the following options is/are correct?

Let $$f:R \rightarrow R$$ be given by
$$f(x) = \begin{cases}x^5 + 5x^4 + 10x^3 + 10x^2 + 3x +1 & x < 0;\\x^2 - x + 1, & 0 \leq x \leq 1; \\\frac{2}{3}x^3 - 4x^2 + 7x - \frac{8}{3}, & 1 \leq x < 3;\\ (x - 2) \log_e(x - 2) - x + \frac{10}{3}, & x \geq 3. \end{cases}$$
Then which of the following options is/are correct?

Let $$\lceil$$ denote a curve y = y(x) which is in the first quadrant and let the point (1,0) lie on it. Let the tangent to $$\lceil$$ at a point P intersect the y-axis at $$Y_P$$. If $$PY_P$$ has length 1 for each point P on $$\lceil$$ then which of the following options is/are correct?

Let $$L_1$$ and $$L_2$$ denote the lines
$$\overrightarrow{r} = \widehat{i} + \lambda (-\widehat{i} + 2\widehat{j} + 2\widehat{k}), \lambda \in R$$ and
$$\overrightarrow{r} = \mu (2\widehat{i} - \widehat{j} + 2\widehat{k}), \mu \in R$$
respectively. If $$L_3$$ is a line which is perpendicular to both $$L_1$$ and $$L_2$$ and cuts both of them, then which of the following options describe(s) $$L_3$$?

Let $$\omega \neq 1$$ be a cube root of unity. Then the minimum of the set
{$$|a + b \omega + c\omega^2|^2$$ : a, b, c distinct non-zero integers}
equals __________

Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If
$$AP(1; 3) \cap AP(2; 5) \cap AP(3; 7) = AP(a; d)$$
then a + d equals __________