For the following questions answer them individually
The lines $$L_1, L_2, \ldots, L_{20}$$ are distinct. For $$n = 1, 2, 3, \ldots, 10$$ all the lines $$L_{2n-1}$$ are parallel to each other and all the lines $$L_{2n}$$ pass through a given point P. The maximum number of points of intersection of pairs of lines from the set $$\{L_1, L_2, \ldots, L_{20}\}$$ is equal to:
If three successive terms of a G.P. with common ratio $$r$$ $$(r > 1)$$ are the length of the sides of a triangle and $$\lfloor r \rfloor$$ denotes the greatest integer less than or equal to r, then $$3\lfloor r \rfloor + \lfloor -r \rfloor$$ is equal to:
Let ABC be an isosceles triangle in which A is at $$(-1, 0)$$, $$\angle A = \frac{2\pi}{3}$$, $$AB = AC$$ and B is on the positive x-axis. If $$BC = 4\sqrt{3}$$ and the line BC intersects the line $$y = x + 3$$ at $$(\alpha, \beta)$$, then $$\frac{\beta^4}{\alpha^2}$$ is:
Let $$A = I_2 - 2MM^T$$, where M is real matrix of order $$2 \times 1$$ such that the relation $$M^TM = I_1$$ holds. If $$\lambda$$ is a real number such that the relation $$AX = \lambda X$$ holds for some non-zero real matrix X of order $$2 \times 1$$, then the sum of squares of all possible values of $$\lambda$$ is equal to:
If $$y = \frac{\sqrt{x+1}x^2 - \sqrt{x}}{x\sqrt{x} + x + \sqrt{x}} + \frac{1}{15}(3\cos 2x - 5\cos 3x)$$, then $$96y'\left(\frac{\pi}{6}\right)$$ is equal to:
Let $$f: [0, \infty) \to R$$ and $$F(x) = \int_0^x tf(t) \, dt$$. If $$F(x^2) = x^4 + x^5$$, then $$\sum_{r=1}^{12} f(r^2)$$ is equal to:
Three points $$O(0,0)$$, $$P(a, a^2)$$, $$Q(-b, b^2)$$, $$a > 0$$, $$b > 0$$, are on the parabola $$y = x^2$$. Let $$S_1$$ be the area of the region bounded by the line PQ and the parabola, and $$S_2$$ be the area of the triangle OPQ. If the minimum value of $$\frac{S_1}{S_2}$$ is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to:
The sum of squares of all possible values of $$k$$, for which area of the region bounded by the parabolas $$2y^2 = kx$$ and $$ky^2 = 2(y - x)$$ is maximum, is equal to:
If $$\frac{dx}{dy} = \frac{1 + x - y^2}{y}$$, $$x(1) = 1$$, then $$5x(2)$$ is equal to:
Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = -\hat{i} - 8\hat{j} + 2\hat{k}$$ and $$\vec{c} = 4\hat{i} + c_2\hat{j} + c_3\hat{k}$$ be three vectors such that $$\vec{b} \times \vec{a} = \vec{c} \times \vec{a}$$. If the angle between the vector $$\vec{c}$$ and the vector $$3\hat{i} + 4\hat{j} + \hat{k}$$ is $$\theta$$, then the greatest integer less than or equal to $$\tan^2 \theta$$ is:
