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Instructions

For the following questions answer them individually

Question 1

Question 2

Question 3

Question 4

Let $$\theta_1, \theta_2, ......., \theta_{10}$$ be positive valued angles (in radian) such that $$\theta_1 + \theta_2 + ..... + \theta_{10} = 2 \pi$$. Define the complex numbers $$Z_1 = e^{i \theta_1}, Z_k = Z_{k - 1}e^{i \theta_k}$$ for $$k = 2, 3, ...., 10,$$ where $$i = \sqrt{-1}$$. Consider the statements P and Q given below:$$P : \mid Z_2 - Z_1 \mid + \mid Z_3 - Z_2 \mid + .... + \mid Z_{10} - Z_9 \mid + \mid Z_1 - Z_{10} \mid \leq 2 \pi$$$$Q : \mid Z_2^2 - Z_1^2 \mid + \mid Z_3^2 - Z_2^2 \mid + .... + \mid Z_{10}^2 - Z_9^2 \mid + \mid Z_1^2 - Z_{10}^2 \mid \leq 4 \pi$$Then

Instructions

Question Stem
Three numbers are chosen at random, one after another with replacement, from the set $$S = \left\{1, 2, 3, …, 100 \right\}$$. Let $$p_1$$ be the probability that the maximum of chosen numbers is at least 81 and $$p_2$$ be the probability that the minimum of chosen numbers is at most 40.

Question 5

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Question 6

The value of $$\frac{125}{4} p_2$$ is .........

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Instructions

Question Stem
Let $$\alpha, \beta$$ and $$\gamma$$ be real numbers such that the system of linear equations
$$x + 2y + 3z = \alpha$$
$$4x + 5y + 6z = \beta$$
$$7x + 8y + 9z = \gamma - 1$$
is consistent. Let $$\mid M \mid$$ represent the determinant of the matrix
$$M = \begin{bmatrix}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$$
Let P be the plane containing all those $$\left(\alpha, \beta, \gamma \right)$$ for which the above system of linear equations is consistent, and 𝐷 be the square of the distance of the point (0, 1, 0) from the plane P.

Question 7

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Question 8

The value of D is .........

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Instructions

Question Stem
Consider the lines $$L_1$$ and $$L_2$$ defined by
$$L_1 : x \sqrt{2} + y - 1 = 0$$ and $$L_2 : x \sqrt{2} - y + 1 = 0$$
For a fixed constant $$\lambda$$, let C be the locus of a point P such that the product of the distance of P from $$L_1$$ and the distance of P from $$L_2$$ is $$\lambda^2$$. The line $$y = 2x + 1$$ meets C at two points R and S, where the distance between R and S is $$\sqrt{270}$$.

Let the perpendicular bisector of RS meet C at two distinct points $$R′$$ and $$S′$$. Let D be the square of the distance between $$R′$$ and $$S′$$.

Question 9

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Question 10

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