For the following questions answer them individually
Let $$f(x) = x^5 + 2x^3 + 3x + 1, x \in \mathbb{R}$$, and $$g(x)$$ be a function such that $$g(f(x)) = x$$ for all $$x \in \mathbb{R}$$. Then $$\frac{g(7)}{g'(7)}$$ is equal to :
If the function $$f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}, x \in \mathbb{R}$$, is continuous at $$x = 0$$, then $$f(0)$$ is equal to :
Let a rectangle $$ABCD$$ of sides 2 and 4 be inscribed in another rectangle $$PQRS$$ such that the vertices of the rectangle $$ABCD$$ lie on the sides of the rectangle $$PQRS$$. Let $$a$$ and $$b$$ be the sides of the rectangle $$PQRS$$ when its area is maximum. Then $$(a + b)^2$$ is equal to :
For the function $$f(x) = \sin x + 3x - \frac{2}{\pi}(x^2 + x)$$, where $$x \in [0, \frac{\pi}{2}]$$, consider the following two statements : (I) $$f$$ is increasing in $$(0, \frac{\pi}{2})$$. (II) $$f'$$ is decreasing in $$(0, \frac{\pi}{2})$$. Between the above two statements,
The value of $$\int_{-\pi}^{\pi} \frac{2y(1+\sin y)}{1+\cos^2 y} dy$$ is :
The integral $$\int_0^{\pi/4} \frac{136\sin x}{3\sin x + 5\cos x} dx$$ is equal to :
If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + 2y = \sin(2x), y(0) = \frac{3}{4}$$, then $$y\left(\frac{\pi}{8}\right)$$ is equal to:
If the line $$\frac{2-x}{3} = \frac{3y-2}{4\lambda+1} = 4-z$$ makes a right angle with the line $$\frac{x+3}{3\mu} = \frac{1-2y}{6} = \frac{5-z}{7}$$, then $$4\lambda + 9\mu$$ is equal to :
Let $$d$$ be the distance of the point of intersection of the lines $$\frac{x+6}{3} = \frac{y}{2} = \frac{z+1}{1}$$ and $$\frac{x-7}{4} = \frac{y-9}{3} = \frac{z-4}{2}$$ from the point $$(7, 8, 9)$$. Then $$d^2 + 6$$ is equal to :
The coefficients $$a, b, c$$ in the quadratic equation $$ax^2 + bx + c = 0$$ are chosen from the set $$\{1, 2, 3, 4, 5, 6, 7, 8\}$$. The probability of this equation having repeated roots is :
