OPT 4

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

Define quadratic function.
What is geometric mean between two numbers \(p\) and \(q\)?
What is the meaning of \([x, y]\)?
If a matrix is given and its determinant is zero, then what type of matrix is this?
If two lines are \(ax + by = c\) and \(mx + ny = p\) and the relation of their slope is \(an = mb\), then, what is the relationship between these two lines?
In which condition circle is formed?
Express \(\cos 8D\) in terms of \(\tan 4D\).
Express \(2\cos 30^\circ \cos 20^\circ\) in term of sine or cosine in the form of sum.
In which condition two vectors are parallel to each other?
In an inversion circle \(OP = 8 \text{ cm}\) and \(OP' \times OP = 100 \text{ cm}^2\), then find the value of \(OP'\).

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

If \((x - 4)\) is a factor of polynomial \(f(x) = x^3 - 4x^2 + (z + 6)x + 18\), then find the value of \(z\).
If the parabola \(y = x^2 - 2x - 3\) and its vertex is \((1, a)\), then find the value of \(a\).
If \(A = \begin{pmatrix} 2 & 4 \\ 5 & 6 \end{pmatrix}\), \(B = \begin{pmatrix} 1 & 2 \\ m & 4 \end{pmatrix}\), and \(\det(AB) = 0\), then find the value of \(m\).
Two equations \(px + 7y = 10\) and \(7x - 5y = 8\) are normal (perpendicular) to each other, then find the value of \(p\).
(Prove that): \(\frac{1 + \cos 8\theta}{1 - \cos 8\theta} = \cot^2 4\theta\).
(Solve): \(\sin A \cos A = 1\). (Up to \(180^\circ\)).
If \(|5\vec{a} - \vec{b}| = |5\vec{a} + \vec{b}|\), then prove that two vectors \(\vec{a}\) and \(\vec{b}\) are orthogonal to each other.
In a continuous data \(\sum fm = 200\), \(N = 20\) and \(\sum f|m - \bar{x}| = 150\), then find mean deviation from mean and its coefficient.

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

Function \(f(x) = 2x\), \((gof)(x) = 4x + 5\), then find \(g(x)\). Also find the value of \(p\) if \((fog)(5) = 3p\).
Split \(69\) into three parts such that they are in arithmetic series and the product of two smaller parts is \(483\).
Prove that function \(f(x) = 4x + 5\) is continuous at \(x = 2\) by finding left hand side limit, right hand side limit and functional value.
Solve by using Cramer's rule: \(2x + y = 3\), \(3x + 2y = 2\).
Find the separate equations of two lines represented by the equation \(x^2 + 5xy + 4y^2 = 0\) and also find the angle between those lines.
(Prove that): \((2\cos A - 1)(2\cos A + 1)(2\cos 2A - 1) = 2\cos 4A + 1\).
If \(A + B + C = 180^\circ\), then prove that: \(\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C\).
The angles of elevation of the top of a tower as observed from the distance of \(49 \text{ meter}\) and \(64 \text{ meter}\) from the foot of the tower are found to be complementary. Find the height of the tower.
Find the \(2 \times 2\) transformation matrix in which unit square is transformed into the parallelogram \(A(-1, 0)\), \(B(3, 1)\), \(C(2, 2)\), \(D(-2, 1)\).
Find the quartile deviation of the data:

\(x\) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90

\(f\) 3 7 10 12 14 17 20 24
Find the standard deviation of the data:

\(x\) 2-4 4-6 6-8 8-10 10-12 12-14 14-16

\(f\) 1 2 3 4 5 6 7

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

Find the maximum value of objective function \(F = 3x + 2y\) under the following constraints: \(x + y \le 4\), \(x - y \le 2\), \(x \ge 0\) and \(y \ge 0\).
If the end points of the diameter of circle are \(A(6, 0)\) and \(B(0, 4)\), find the equation of circle. Does the point \((1, -1)\) satisfy the equation of circle? Give reason with calculation. Does the point lie on the circumference of the same circle?
Prove that vectorically diagonals of the parallelogram are bisect to each other.
The vertices of triangle \(\triangle ABC\) are \(A(1, 1)\), \(B(3, 1)\) and \(C(2, 5)\). Rotate the triangle about the origin through \(-90^\circ\). Again reflect the triangle in \(x = 1\) line and show object and image in the same graph paper.

From

\(x\)	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90
\(f\)	3	7	10	12	14	17	20	24

\(x\)	2-4	4-6	6-8	8-10	10-12	12-14	14-16
\(f\)	1	2	3	4	5	6	7