NTA JEE Main 25th January 2023 Shift 2

For the two positive numbers $$a, b$$, if $$a, b$$ and $$\frac{1}{18}$$ are in a geometric progression, while $$\frac{1}{a}$$, 10 and $$\frac{1}{b}$$ are in an arithmetic progression, then, $$16a + 12b$$ is equal to _____.

The remainder when $$(2023)^{2023}$$ is divided by 35 is

If $$m$$ and $$n$$ respectively are the numbers of positive and negative value of $$\theta$$ in the interval $$[-\pi, \pi]$$ that satisfy the equation $$\cos 2\theta \cos \frac{\theta}{2} = \cos 3\theta \cos \frac{9\theta}{2}$$, then $$mn$$ is equal to _____.

A triangle is formed by X-axis, Y-axis and the line $$3x + 4y = 60$$. Then the number of points $$P(a, b)$$ which lie strictly inside the triangle, where $$a$$ is an integer and $$b$$ is a multiple of $$a$$, is _____.

Points $$P(-3, 2)$$, $$Q(9, 10)$$ and $$R(a, 4)$$ lie on a circle $$C$$ with $$PR$$ as its diameter. The tangents to $$C$$ at the points $$Q$$ and $$R$$ intersect at the point $$S$$. If $$S$$ lies on the line $$2x - ky = 1$$, then $$k$$ is equal to _____.

If $$\int_{1/3}^{3} |\log_e x| dx = \frac{m}{n} \log_e\left(\frac{n^2}{e}\right)$$, where $$m$$ and $$n$$ are coprime natural numbers, then $$m^2 + n^2 - 5$$ is equal to _____.

The shortest distance between the lines $$x + 1 = 2y = -12z$$ and $$x = y + 2 = 6z - 6$$ is

If the shortest distance between the line joining the points $$(1, 2, 3)$$ and $$(2, 3, 4)$$, and the line $$\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-2}{0}$$ is $$\alpha$$, then $$28\alpha^2$$ is equal to _____.

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $$N - 2, \sqrt{3N}, N + 2$$ are in geometric progression be $$\frac{k}{48}$$. Then the value of $$k$$ is

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer then a non-smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $$\frac{k}{10}$$. Then the value of $$k$$ is _____.