OPT 3

Group 'A' \([10 \times 1 = 10]\)

What is the minimum value of \(y = \cos x\) function?
What is the geometric mean between two numbers \(m\) and \(n\)?
Write \(-1 \le x < 2\) in interval notation.
If matrix \(A = [a \text{ } -b]\), then what is the value of \(|A|\)?
What is the formula of angle between the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\)?
If the intersection plane is parallel to the axis of cone, then what conic does it form?
Express \(\cos 2A\) in terms of \(\tan A\).
Express \(\sin 2M + \sin 2N\) into product form.
If \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}|\), then write the angle between \(\vec{a}\) and \(\vec{b}\).
Define inversion transformation.

Group 'B' \([8 \times 2 = 16]\)

If the functions \(f(x) = 2x + 3\) and \((fog)(x) = 5x - 1\), then find the function \(g(x)\).
Find the vertex of the parabola \(y = x^2 - 3x + 2\).
Find the value of \(D_x\) and \(D_y\) using Cramer's rule from the equations \(4x - 5y = 4\) and \(3x + 4y = 10\).
Find the acute angle \(\theta\) between the pair of lines represented by the equation \(2x^2 + 7xy + 3y^2 = 0\).
(Prove that): \(\frac{1 + \cos\theta + \cos\frac{\theta}{2}}{\sin\theta + \sin\frac{\theta}{2}} = \cot \frac{\theta}{2}\).
(Solve): \(2\sin^2\theta - \sqrt{3}\sin\theta = 0\), \((0^\circ \le \theta \le 90^\circ)\).
If \(\vec{a} + 2\vec{b}\) and \(5\vec{a} - 4\vec{b}\) are perpendicular to each other and \(\vec{a}\) and \(\vec{b}\) are unit vectors, find the angle between \(\vec{a}\) and \(\vec{b}\).
In a data, value of first quartile is \(2p\) and quartile deviation is \(p\). Find third quartile and coefficient of quartile deviation.

Group 'C' \([11 \times 3 = 33]\)

(Solve): \(3x^3 - 13x^2 + 16 = 0\).
The sum of three consecutive terms in arithmetic series is \(15\) and their product is \(120\), find the three numbers.
Examine the continuity or discontinuity of the function. \[ f(x) = \begin{cases} \frac{x^2 - 5x}{x - 5} & x \neq 5 \\ 10 & \text{at } x = 5 \end{cases} \]
Solve by matrix method: \(2x + 7y = 25\) and \(x - 4y = -10\).
Find the equation of the perpendicular bisector of the line segment joining the points \((1, 2)\) and \((5, 4)\).
(Prove that): \(\frac{\sec 4\theta - 1}{\sec \theta - 1} = \tan 4\theta \cot \theta\).
If \(A + B + C = 180^\circ\), Prove that: \(\cos(B + C - A) + \cos(C + A - B) + \cos(A + B - C) = 1 + 4\cos A \cos B \cos C\).
From the top of the \(60 \text{ m}\) high building, the angle of elevation of the top of a tower is \(30^\circ\) and the angle of depression of the bottom is \(45^\circ\). Find the height of the tower.
An unit square \(\begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) is transformed by the matrix \(\begin{pmatrix} a & 1 \\ 1 & b \end{pmatrix}\) to get \(\begin{pmatrix} 0 & 3 & q & 1 \\ 0 & p & 3 & 2 \end{pmatrix}\). Find the values of \(a\), \(b\), \(p\) and \(q\).
Compute the mean deviation from median of the data given data:

Class 0-2 2-4 4-6 6-8 8-10

Frequency 4 2 1 3 2
Find the standard deviation of the given data.

Mid-Value 5 10 15 20 25

Frequency 3 2 1 3 4

Group 'D' \([4 \times 4 = 16]\)

Maximize \(P = 10x + 2y\) under the following constraints: \(x + y \le 6\), \(x - y \ge -2\), \(x \ge 0\), \(y \ge 0\).
Find the equation of a circle concentric with the circle \(2x^2 + 2y^2 + 4x - 2y + 1 = 0\) passing through the point \((4, -2)\).
Prove by vector method that the circumference angle \(APB\) of a semi circle with a diameter \(AB\) is a right angle.
The triangle \(\triangle ABC\) whose vertices are \(A(2, 3)\), \(B(4, 1)\) and \(C(3, 5)\) is first reflected in the line \(y = 0\) to form \(\triangle A'B'C'\) and then \(\triangle A'B'C'\) is rotated through \(90^\circ\) about origin to form \(\triangle A''B''C''\). Write down the co-ordinates of \(\triangle A'B'C'\) and \(\triangle A''B''C''\). Draw \(\triangle ABC\) and its images on the same graph paper.

From

Class	0-2	2-4	4-6	6-8	8-10
Frequency	4	2	1	3	2

Mid-Value	5	10	15	20	25
Frequency	3	2	1	3	4