JEE (Advanced) 2021 Paper-1

For $$3 \times 3$$ matrix M, let $$\mid M \mid$$ denote the determinant of M. Let
$$E = \begin{bmatrix}1 & 2 & 3 \\2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}, P = E = \begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$ and $$F = \begin{bmatrix}1 & 3 & 2 \\8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}$$
If Q is a nonsingular matrix of order $$3 \times 3$$, then which of the following statements is (are) TRUE ?

Let $$f : R \rightarrow R$$ be defined by
$$f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$$
Then which of the following statements is (are) TRUE?

Let E, F and G be three events having probabilities
$$P(E) = \frac{1}{8}, P(F) = \frac{1}{6}$$ and $$P(G) = \frac{1}{4}$$, and let $$P(E \cap F \cap G) = \frac{1}{10}$$.
For any event $$H$$, if $$H^c$$ denotes its complement, then which of the following statements is (are) TRUE ?

For any $$3 \times 3$$ matrix M, let $$\mid M \mid$$ denote the determinant of M. Let I be the $$3 \times 3$$ identity matrix. Let E and F be two $$3 \times 3$$ matrices such that $$(I − EF)$$ is invertible. If $$G = (I − EF) − 1$$, then which of the following statements is (are) TRUE ?

For any positive integer n, let $$S_n : (0, \infty) \rightarrow R$$ be defined by
$$S_n(x) = \sum_{k=1}^n \cot^{-1}\left(\frac{1 + k(k + 1)x^2}{x}\right)$$,
where for any $$x \in R, \cot^{-1}(x) \in (0, \pi)$$ and $$\tan^{-1}(x) \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$. Then which of the following statements is (are) TRUE ?

For any complex number $$w = c + id$$, let $$arg(w) \in (−\pi, \pi]$$, where $$i = \sqrt{-1}$$ . Let $$\alpha$$ and $$\beta$$ be real numbers such that for all complex numbers $$z = x + iy$$ satisfying $$arg \left(\frac{z + \alpha}{z + \beta}\right) = \frac{\pi}{4}$$, the ordered pair (𝑥,𝑦) lies on the circle
$$x^2 + y^2 + 5x − 3y + 4 = 0$$
Then which of the following statements is (are) TRUE ?

For $$x \in R$$, the number of real roots of the equation
$$3x^2 - 4 \mid x^2 - 1 \mid + x - 1 = 0$$ is .........

In a triangle ABC, let $$AB = \sqrt{23}, BC = 3$$ and $$CA = 4$$. Then the value of
$$\frac{\cot A + \cot C}{\cot B}$$ is ........

Let $$\overrightarrow{u},\overrightarrow{v}$$ and $$\overrightarrow{w}$$ be vectors in three-dimensional space, where $$\overrightarrow{u}$$ and $$\overrightarrow{v}$$ re unit vectors which are not perpendicular to each other and
$$\overrightarrow{u}.\overrightarrow{w} = 1, \overrightarrow{v}.\overrightarrow{w} = 1, \overrightarrow{w}\overrightarrow{w} = 4$$
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors $$\overrightarrow{u},\overrightarrow{v}$$ and $$\overrightarrow{w}$$, is $$\sqrt{2}$$, then the value of $$\mid 3\overrightarrow{u} + 5 \overrightarrow{v} \mid$$ is ___ .

The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is