JEE (Advanced) 2020 Paper-1

Let $$C_{1}$$ and $$C_{2}$$ be two biased coins such that the probabilities of getting head in a single toss are $$\frac{2}{3}$$ and $$\frac{1}{3}$$ , respectively. Suppose $$\alpha$$ is the number of heads that appear when $$C_{1}$$ is tossed twice, independently, and suppose $$\beta$$ is the number of heads that appear when $$C_{2}$$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $$x^{2} − \alpha x + \beta$$ are real and equal, is

Consider all rectangles lying in the region

$$\left\{(x, y) \epsilon R \times R : 0 \leq x \leq \frac{\pi}{2} and 0 \leq y \leq 2\sin(2x)\right\}$$

and having one side on the $$x$$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

Let the function $$f: R \rightarrow R$$ be defined by $$f(x) = x^{3} − x^{2} + (x − 1) \sin x$$ and let $$g: R \rightarrow R$$ be an arbitrary function. Let $$fg: R \rightarrow R$$ be the product function defined by $$(fg)(x) = f(x)g(x)$$. Then which of the following statements is/are TRUE?

Let 𝑀 be a $$3 \times 3$$ invertible matrix with real entries and let I denote the $$3 \times 3$$ identity matrix. If $$𝑀^{−1} = adj (adj 𝑀)$$, then which of the following statements is/are ALWAYS TRUE?

Let 𝑆 be the set of all complex numbers z satisfying $$|z^{2} + z + 1| = 1$$. Then which of the following statements is/are TRUE?

Let 𝑥, 𝑦 and 𝑧 be positive real numbers. Suppose 𝑥, 𝑦 and 𝑧 are the lengths of the sides of a triangle
opposite to its angles 𝑋, 𝑌 and 𝑍, respectively. If

$$\tan \frac{X}{2} + \tan \frac{Z}{2} = \frac{2y}{x + y + z}$$,

then which of the following statements is/are TRUE?

Let $$𝐿_{1}$$ and $$𝐿_{2}$$ be the following straight lines.

$$L_{1}: \frac{x-1}{1} = \frac{y}{-1} = \frac{z-1}{3}$$ and $$L_{2}: \frac{x-1}{-31} = \frac{y}{-1} = \frac{z-1}{1}$$.

Suppose the straight line

$$L: \frac{x - \alpha}{l} = \frac{y - 1}{m} = \frac{z - \gamma}{-2}$$

lies in the plane containing $$L_{1}$$ and $$L_{2}$$, and passes through the point of intersection of $$L_{1}$$ and $$L_{2}$$. If the line 𝐿 bisects the acute angle between the lines $$L_{1}$$ and $$L_{2}$$, then which of the following statements is/are TRUE?

Which of the following inequalities is/are TRUE?

Let 𝑚 be the minimum possible value of $$\log_{3}(3^{y_{1}} + 3^{y_{2}} + 3^{y_{3}})$$, where $$y_{1}, y_{2}, y_{3}$$ are real numbers for which $$y_{1} + y_{2} + y_{3} = 9$$. Let 𝑀 be the maximum possible value of $$(log_{3}x_{1} + log_{3}x_{2} + log_{3}x_{3})$$, where $$x_{1}, x_{2}, x_{3}$$ are positive real numbers for which $$x_{1} + x_{2} + x_{3} = 9$$. Then the value of $$log_{2}(m^{3}) + log_{3}(m^{2})$$ is _____

Let $$a_{1}, a_{2}, a_{3}$$, … be a sequence of positive integers in arithmetic progression with ommon difference 2. Also, let $$b_{1}, b_{2}, b_{3}$$, … be a sequence of positive integers in geometric progression with common ratio 2. If $$a_{1} = b_{1} = c$$, then the number of all possible values of c, for which the equality

$$2(a_{1} + a_{2} + ... + a_{n}) = (b_{1} + b_{2} + ... + b_{n}$$

hold for some positve integer n, is _______