For the following questions answer them individually
Let the system of equations $$x + 2y + 3z = 5$$, $$2x + 3y + z = 9$$, $$4x + 3y + \lambda z = \mu$$ have infinite number of solutions. Then $$\lambda + 2\mu$$ is equal to:
If the domain of the function $$f(x) = \frac{\sqrt{x^2 - 25}}{4 - x^2} + \log_{10}(x^2 + 2x - 15)$$ is $$(-\infty, \alpha) \cup [\beta, \infty)$$, then $$\alpha^2 + \beta^3$$ is equal to:
Let $$f(x) = |2x^2 + 5|x| - 3|$$, $$x \in R$$. If $$m$$ and $$n$$ denote the number of points where $$f$$ is not continuous and not differentiable respectively, then $$m + n$$ is equal to:
The value of $$\int_0^1 (2x^3 - 3x^2 - x + 1)^{1/3} dx$$ is equal to:
If $$\int_0^{\pi/3} \cos^4 x \, dx = a\pi + b\sqrt{3}$$, where $$a$$ and $$b$$ are rational numbers, then $$9a + 8b$$ is equal to:
Let $$\alpha$$ be a non-zero real number. Suppose $$f: R \to R$$ is a differentiable function such that $$f(0) = 1$$ and $$\lim_{x \to -\infty} f(x) = 1$$. If $$f'(x) = \alpha f(x) + 3$$, for all $$x \in R$$, then $$f(-\log_e 2)$$ is equal to:
Consider a $$\triangle ABC$$ where $$A(1, 3, 2)$$, $$B(-2, 8, 0)$$ and $$C(3, 6, 7)$$. If the angle bisector of $$\angle BAC$$ meets the line BC at D, then the length of the projection of the vector $$\vec{AD}$$ on the vector $$\vec{AC}$$ is:
If the mirror image of the point $$P(3, 4, 9)$$ in the line $$\frac{x-1}{3} = \frac{y+1}{2} = \frac{z-2}{1}$$ is $$(\alpha, \beta, \gamma)$$, then $$14(\alpha + \beta + \gamma)$$ is:
Let P and Q be the points on the line $$\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$$ which are at a distance of 6 units from the point $$R(1, 2, 3)$$. If the centroid of the triangle PQR is $$(\alpha, \beta, \gamma)$$, then $$\alpha^2 + \beta^2 + \gamma^2$$ is:
Let Ajay will not appear in JEE exam with probability $$p = \frac{2}{7}$$, while both Ajay and Vijay will appear in the exam with probability $$q = \frac{1}{5}$$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is:
