For the following questions answer them individually
Let $$f: R \rightarrow R$$ be a function defined $$f(x) = \frac{x}{(1+x^4)^{1/4}}$$ and $$g(x) = f(f(f(f(x))))$$ then $$18\int_0^{\sqrt{2\sqrt{5}}} x^2 g(x) \, dx$$
Let $$a$$ and $$b$$ be real constants such that the function $$f$$ defined by $$f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x > 1 \end{cases}$$ be differentiable on $$R$$. Then, the value of $$\int_{-2}^{2} f(x) \, dx$$ equals
Let $$f: R - \{0\} \rightarrow R$$ be a function satisfying $$f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$$ for all $$x, y$$, $$f(y) \neq 0$$. If $$f'(1) = 2024$$, then
Let $$f(x) = x+3^{2}x-2^3$$, $$x \in [-4, 4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4, 4]$$, then the value of $$M - m$$ is:
Let $$y = f(x)$$ be a thrice differentiable function in $$(-5, 5)$$. Let the tangents to the curve $$y = f(x)$$ at $$(1, f(1))$$ and $$(3, f(3))$$ make angles $$\frac{\pi}{6}$$ and $$\frac{\pi}{4}$$, respectively with positive x-axis. If $$27\int_1^3 \{f'(t)\}^2 + 1\} f''(t) \, dt = \alpha + \beta\sqrt{3}$$ where $$\alpha, \beta$$ are integers, then the value of $$\alpha + \beta$$ equals
Let $$f: R \rightarrow R$$ be defined $$f(x) = ae^{2x} + be^x + cx$$. If $$f(0) = -1$$, $$f'(\log_e 2) = 21$$ and $$\int_0^{\log 4}(f(x) - cx) \, dx = \frac{39}{2}$$, then the value of $$|a + b + c|$$ equals:
Let $$\vec{a} = \hat{i} + \alpha\hat{j} + \beta\hat{k}$$, $$\alpha, \beta \in R$$. Let a vector $$\vec{b}$$ be such that the angle between $$\vec{a}$$ and $$\vec{b}$$ is $$\frac{\pi}{4}$$ and $$|\vec{b}|^2 = 6$$. If $$\vec{a} \cdot \vec{b} = 3\sqrt{2}$$, then the value of $$(\alpha^2 + \beta^2)|\vec{a} \times \vec{b}|^2$$ is equal to
Let $$\vec{a}$$ and $$\vec{b}$$ be two vectors such that $$|\vec{b}| = 1$$ and $$|\vec{b} \times \vec{a}| = 2$$. Then $$|(\vec{b} \times \vec{a}) - \vec{b}|^2$$ is equal to
Let $$L_1: \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})$$, $$\lambda \in R$$, $$L_2: \vec{r} = (\hat{j} - \hat{k}) + \mu(3\hat{i} + \hat{j} + p\hat{k})$$, $$\mu \in R$$ and $$L_3: \vec{r} = \delta(l\hat{i} + m\hat{j} + n\hat{k})$$, $$\delta \in R$$ be three lines such that $$L_1$$ is perpendicular to $$L_2$$ and $$L_3$$ is perpendicular to both $$L_1$$ and $$L_2$$. Then the point which lies on $$L_3$$ is
Bag $$A$$ contains 3 white, 7 red balls and bag $$B$$ contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag $$A$$, if the ball drawn is white, is:
