NTA JEE Main 18th March 2021 Shift 2

Define a relation $$R$$ over a class of $$n \times n$$ real matrices $$A$$ and $$B$$ as "$$ARB$$" iff there exists a non-singular matrix $$P$$ such that $$PAP^{-1} = B$$. Then which of the following is true?

Let the system of linear equations
$$4x + \lambda y + 2z = 0$$
$$2x - y + z = 0$$
$$\mu x + 2y + 3z = 0$$, $$\lambda, \mu \in R$$
has a non-trivial solution. Then which of the following is true?

Let $$f : R - \{3\} \to R - \{1\}$$ be defined by $$f(x) = \frac{x-2}{x-3}$$. Let $$g : R \to R$$ be given as $$g(x) = 2x - 3$$. Then, the sum of all the values of $$x$$ for which $$f^{-1}(x) + g^{-1}(x) = \frac{13}{2}$$ is equal to

Let $$f : R \to R$$ be a function defined as
$$$f(x) = \begin{cases} \frac{\sin(a+1)x + \sin 2x}{2x}, & \text{if } x < 0 \\ b, & \text{if } x = 0 \\ \frac{\sqrt{x+bx^3} - \sqrt{x}}{bx^{5/2}}, & \text{if } x > 0 \end{cases}$$$
If $$f$$ is continuous at $$x = 0$$, then the value of $$a + b$$ is equal to :

Let $$g(x) = \int_0^x f(t)dt$$, where $$f$$ is continuous function in $$[0, 3]$$ such that $$\frac{1}{3} \le f(t) \le 1$$ for all $$t \in [0, 1]$$ and $$0 \le f(t) \le \frac{1}{2}$$ for all $$t \in (1, 3]$$.
The largest possible interval in which $$g(3)$$ lies is :

The area (in sq. unit) bounded by the curve $$4y^2 = x^2(4-x)(x-2)$$ is equal to

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = (y+1)\left((y+1)e^{x^2/2} - x\right)$$, $$0 < x < 2.1$$, with $$y(2) = 0$$. Then the value of $$\frac{dy}{dx}$$ at $$x = 1$$ is equal to

In a triangle $$ABC$$, if $$|\overrightarrow{BC}| = 8$$, $$|\overrightarrow{CA}| = 7$$, $$|\overrightarrow{AB}| = 10$$, then the projection of the vector $$\overrightarrow{AB}$$ on $$\overrightarrow{AC}$$ is equal to :

Let $$\vec{a}$$ and $$\vec{b}$$ be two non-zero vectors perpendicular to each other and $$|\vec{a}| = |\vec{b}|$$, If $$|\vec{a} \times \vec{b}| = |\vec{a}|$$, then the angle between the vectors $$\left(\vec{a} + \vec{b} + (\vec{a} \times \vec{b})\right)$$ and $$\vec{a}$$ is equal to :

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :