NTA JEE Main 26th June 2022 Shift 1

Let $$f(x) = \frac{x-1}{x+1}, x \in R - \{0, -1, 1\}$$. If $$f^{n+1}(x) = f(f^n(x))$$ for all $$n \in N$$, then $$f^6(6) + f^7(7)$$ is equal to

$$f, g : R \to R$$ be two real valued functions defined as $$f(x) = \begin{cases} -|x+3| & x < 0 \\ e^x & x \geq 0 \end{cases}$$ and
$$g(x) = \begin{cases} x^2 + k_1 x & x < 0 \\ 4x + k_2 & x \geq 0 \end{cases}$$, where $$k_1$$ and $$k_2$$ are real constants. If $$gof$$ is differentiable at $$x = 0$$, then $$gof(-4) + gof(4)$$ is equal to

The sum of the absolute minimum and the absolute maximum values of the function
$$f(x) = |3x - x^2 + 2| - x$$ in the interval $$[-1, 2]$$ is

Let $$S$$ be the set of all the natural numbers, for which the line $$\frac{x}{a} + \frac{y}{b} = 2$$ is a tangent to the curve $$\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$$ at the point $$(a, b), ab \neq 0$$. Then

Let $$f(x) = 2\cos^{-1}x + 4\cot^{-1}x - 3x^2 - 2x + 10, x \in [-1, 1]$$. If $$[a, b]$$ is the range of the function, then $$4a - b$$ is equal to

The area bounded by the curve $$y = |x^2 - 9|$$ and the line $$y = 3$$ is

If $$\vec{a} \cdot \vec{b} = 1, \vec{b} \cdot \vec{c} = 2$$ and $$\vec{c} \cdot \vec{a} = 3$$, then the value of $$\left[\vec{a} \times (\vec{b} \times \vec{c}), \vec{b} \times (\vec{c} \times \vec{a}), \vec{c} \times (\vec{b} \times \vec{a})\right]$$ is

If the two lines $$l_1 : \frac{x-2}{3} = \frac{y+1}{-2}, z = 2$$ and $$l_2 : \frac{x-1}{1} = \frac{2y+3}{\alpha} = \frac{z+5}{2}$$ are perpendicular, then an angle between the lines $$l_2$$ and $$l_3 : \frac{1-x}{3} = \frac{2y-1}{-4} = \frac{z}{4}$$ is

Let the plane $$2x + 3y + z + 20 = 0$$ be rotated through a right angle about its line of intersection with the plane $$x - 3y + 5z = 8$$. If the mirror image of the point $$(2, -\frac{1}{2}, 2)$$ in the rotated plane is $$B(a, b, c)$$, then

Let a biased coin be tossed 5 times. If the probability of getting 4 heads is equal to the probability of getting 5 heads, then the probability of getting atmost two heads is