NTA JEE Main 10th January 2019 Shift 2

Consider the following three statements:
P: 5 is a prime number
Q: 7 is a factor of 192
R: LCM of 5 and 7 is 35
Then the truth value of which one of the following statements is true?

If the mean and standard deviation of 5 observations $$x_1, x_2, x_3, x_4, x_5$$ are 10 and 3, respectively, then the variance of 6 observations $$x_1, x_2, \ldots, x_5$$ and $$-50$$ is equal to:

With the usual notation in $$\triangle ABC$$, if $$\angle A + \angle B = 120^{\circ}$$, $$a = \sqrt{3} + 1$$ units and $$b = \sqrt{3} - 1$$ units, then the ratio $$\angle A : \angle B$$ is:

Let $$A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$$, where $$b \gt 0$$. Then the minimum value of $$\frac{\det(A)}{b}$$ is:

The number of values of $$\theta \in (0, \pi)$$ for which the system of linear equations
$$x + 3y + 7z = 0$$
$$-x + 4y + 7z = 0$$
$$(\sin 3\theta)x + (\cos 2\theta)y + 2z = 0$$
has a non-trivial solution, is:

Let $$a_1, a_2, a_3, \ldots, a_{10}$$ be in G.P. with $$a_i > 0$$ for $$i = 1, 2, \ldots, 10$$ and $$S$$ be the set of pairs $$(r, k)$$, $$r, k \in N$$ (the set of natural numbers) for which $$\begin{vmatrix} \log_e a_1^r a_2^k & \log_e a_2^r a_3^k & \log_e a_3^r a_4^k \\ \log_e a_4^r a_5^k & \log_e a_5^r a_6^k & \log_e a_6^r a_7^k \\ \log_e a_7^r a_8^k & \log_e a_8^r a_9^k & \log_e a_9^r a_{10}^k \end{vmatrix} = 0$$. Then the number of elements in $$S$$, is:

The value of $$\cot\left(\sum_{n=1}^{19} \cot^{-1}\left(1 + \sum_{p=1}^{n} 2p\right)\right)$$ is:

Let $$N$$ be the set of natural numbers and two functions $$f$$ and $$g$$ be defined as $$f, g: N \to N$$ such that $$f(n) = \begin{cases} \frac{n+1}{2}, & \text{if n is odd} \\ \frac{n}{2}, & \text{if n is even} \end{cases}$$ and $$g(n) = n - (-1)^n$$. Then $$fog$$ is:

Let $$f: (-1, 1) \to R$$ be a function defined by $$f(x) = \max\left\{-|x|, -\sqrt{1-x^2}\right\}$$. If $$K$$ be the set of all points at which $$f$$ is not differentiable, then $$K$$ has exactly:

A helicopter is flying along the curve given by $$y - x^{3/2} = 7$$, $$(x \geq 0)$$. A soldier positioned at the point $$\left(\frac{1}{2}, 7\right)$$, who wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is: