For the following questions answer them individually
Let $$f(x) = \frac{1}{7 - \sin 5x}$$ be a function defined on $$\mathbb{R}$$. Then the range of the function $$f(x)$$ is equal to :
Suppose for a differentiable function $$h$$, $$h(0) = 0$$, $$h(1) = 1$$ and $$h'(0) = h'(1) = 2$$. If $$g(x) = h(e^x)e^{h(x)}$$, then $$g'(0)$$ is equal to:
If the function $$f(x) = \left(\frac{1}{x}\right)^{2x}; x > 0$$ attains the maximum value at $$x = \frac{1}{e}$$ then :
If $$\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx = \frac{1}{12} \tan^{-1}(3 \tan x) +$$ constant, then the maximum value of $$a \sin x + b \cos x$$, is :
If the area of the region $$\{(x, y) : \frac{a}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2, 0 < a < 1\}$$ is $$(\log_e 2) - \frac{1}{7}$$ then the value of $$7a - 3$$ is equal to:
Suppose the solution of the differential equation $$\frac{dy}{dx} = \frac{(2+\alpha)x - \beta y + 2}{\beta x - 2\alpha y - (\beta\gamma - 4\alpha)}$$ represents a circle passing through origin. Then the radius of this circle is :
Let $$\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$$, $$\vec{b} = ((\vec{a} \times (\hat{i} + \hat{j})) \times \hat{i}) \times \hat{i}$$. Then the square of the projection of $$\vec{a}$$ on $$\vec{b}$$ is :
Let $$\vec{a} = 6\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. If $$\vec{c}$$ is a vector such that $$|\vec{c}| \geq 6$$, $$\vec{a} \cdot \vec{c} = 6|\vec{c}|$$, $$|\vec{c} - \vec{a}| = 2\sqrt{2}$$ and the angle between $$\vec{a} \times \vec{b}$$ and $$\vec{c}$$ is $$60°$$, then $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ is equal to:
Let $$P(\alpha, \beta, \gamma)$$ be the image of the point $$Q(3, -3, 1)$$ in the line $$\frac{x-0}{1} = \frac{y-3}{1} = \frac{z-1}{-1}$$ and $$R$$ be the point $$(2, 5, -1)$$. If the area of the triangle $$PQR$$ is $$\lambda$$ and $$\lambda^2 = 14K$$, then $$K$$ is equal to :
If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:
