Group A (Very Short Question) - Attempt ALL the questions.
[10×1 = 10]a. Write the negation of the statement "if 3 < 4, then 2 is a rational number".
b. If p and q be two simple statements, write the conjunction of the following statements:
p: 3 is a natural number, q: \(\frac{3}{5}\) is a rational number.
c. Solve: \(|2x - 5| \leq 11\).
d. State the inequality relation between the Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM) of any two unequal positive numbers.
e. Find the rank of the matrix \(A = \begin{bmatrix} 2 & 4 & 6 \\ -1 & -2 & -3 \end{bmatrix}\).
f. Find the directrix of the parabola \(y^2 = 10x\).
g. If \(\vec{a} = (1,\ 2,\ 2)\), then find the unit vector along the direction of \(\vec{a}\).
h. Show that the two vectors \(\vec{a} = 2\hat{i} + 3\hat{j} - 8\hat{k}\) and \(\vec{b} = 2\hat{i} + 4\hat{j} + 2\hat{k}\) are perpendicular to each other.
i. For any vectors u, v, w \(\in\) V over a field F, write the associative property of vector addition in V.
j. How many different words can be formed from the letters of the word 'ARROMA'?
Group B - Attempt any SIX questions.
[6×5 = 30]If p and q be any two statements, prove that \((p \wedge q) \Rightarrow (p \vee q)\) is a tautology.
Define bijective function. If \(f: R \to R\) be defined by the function \(f(x) = 2x + 1\), show that f is bijective.
If the sum of an infinite geometric series is 15 and the sum of their squares is 45, then find the series.
Prove that \(\begin{vmatrix} 1 & x & x^3 \\ 1 & y & y^3 \\ 1 & z & z^3 \end{vmatrix} = (x - y)(y - z)(z - x)(x + y + z)\).
Find the eccentricity, foci and the directrix of the ellipse \(6x^2 + 9y^2 = 54\).
For any triangle ABC, if \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\), prove that \(\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}\).
In how many ways can 4 girls and 4 boys be arranged in a round table if:
a. they may sit anywhere?
b. two girls always sit together?
Group C - Attempt any TWO questions.
[2×10 = 20]a. Find the \(n^{th}\) term and sum to n terms of the series: \(1.3 + 2.4 + 3.5 + \ldots\)
b. Show that \(u = \begin{bmatrix} 6 \\ 5 \end{bmatrix}\) is an eigenvector, but \(v = \begin{bmatrix} 3 \\ -2 \end{bmatrix}\) is not an eigenvector of the matrix \(A = \begin{bmatrix} 1 & 6 \\ 5 & 2 \end{bmatrix}\).
a. If \(A = \begin{pmatrix} 2 & -4 \\ 3 & -2 \\ 0 & 1 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & -1 & 2 \\ -2 & 1 & 0 \end{pmatrix}\), show that \((AB)^T = B^T A^T\). [5+5]
b. Find the equation of the circle which is concentric with the circle \(x^2 + y^2 - 8x + 12y + 15 = 0\) and passes through (5, 4).
a. Find the volume of a parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) where \(\vec{a} = (6, 3, -1)\), \(\vec{b} = (0,\ 1,\ 2)\) and \(\vec{c} = (4,\ -2,\ 5)\).
b. A committee of 5 is to be formed from 6 boys and 4 girls. In how many ways can this be done so as to include at least two girls?
