OPT 17

Group A

a) What type of algebraic function is \( f(x) = 27 \)? Write it.
b) The first term of an AP is, number of terms be 'n', common difference be 'd' and last term is \( t_n \), then write the relation between a, n, d and \( t_n \).
a) Write the interval notation of the numbers lying between \( -\infty \) to \( \infty \).
b) In which condition the inverse matrix cannot be defined? Write it.
a) Write down the slope of the line which is parallel to the line \( y = mx + c \)
b) Which geometrical figure will be formed when a plane intersects a cone parallel to the generator? Write it.
a) Express \( \cos A \) in terms of \( \cos \frac{A}{2} \)
b) Express \( 2\sin 15^\circ \cos 5^\circ \) in sum or difference.
a) If \( \vec{a} = k\vec{b} \) what is the angle between \( \vec{a} \) and \( \vec{b} \) where k is any scalar quantity.
b) Where is the inverse of P if point P lies outside the circle of inversion?

Group B

a) If \( f = \{(3, m), (1, 1), (2, 1)\} \) and \( g = \{(a, 1), (b, 2), (c, 3)\} \), then show the function \( f \circ g \) in arrow diagram then find it in ordered pair form.
b) Find the vertex of the parabola \( y = x^2 - 2x + 3 \).
a) If the inverse of the matrix \( \begin{pmatrix} 2 & 4 \\ 4 & k \end{pmatrix} \) is \( \begin{pmatrix} \frac{-7}{2} & 2 \\ 2 & -1 \end{pmatrix} \), then find the value of \( k \).
b) If the pair of line represented by the equation \( (m + 2)^2 x^2 - 5xy - 16y^2 = 0 \) are perpendicular to each other then find the value of \( m \).
a) Prove that: \( \frac{1 - \tan^2 \left( \frac{\pi}{4} - \frac{\theta}{4} \right)}{1 + \tan^2 \left( \frac{\pi}{4} - \frac{\theta}{4} \right)} = \sin \frac{\theta}{2} \)
b) Solve: \( 3\sec\theta - 4\cos\theta = 0 \) [\( 0^\circ \leq \theta \leq 90^\circ \)]
a) If \( \vec{a} \cdot \vec{b} = 12 \), \( |\vec{b}| = 6 \) and the angle between \( \vec{a} \) and \( \vec{b} \) is \( 60^\circ \) then find the value of \( |\vec{a}| \).
b) In a continuous data the first quartile and quartile deviation are 20 and 10, find the coefficient of quartile deviation.

Group C

Solve: \( 2x^3 - 3x^2 - 8x - 3 = 0 \)
The arithmetic and geometric mean of two positive numbers are 5 and 4 respectively, find the numbers.
A function \( f(x) \) is defined as follows.
\( f(x) = \begin{cases} kx + 3 & \text{for } x \geq 2 \\ -3x - 1 & \text{for } x < 2 \end{cases} \)
If the function \( f(x) \) is continuous at \( x = 2 \) then find the value of \( k \)
Solve by Cramer's rule: \( 2x + 3y = 13 \) and \( 4x - y = 5 \)
Find the equation of the right bisector joining the points \( (-1, 4) \) and \( (5, 2) \).
Prove that: \( 4[\cos^3 20^\circ + \sin^3 50^\circ] = 3[\cos 20^\circ + \sin 50^\circ] \)
If \( A + B + C = \pi^c \), prove that \( \cos A + \cos B + \cos C = 1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \)
From the top of a cliff 200m high, the angles of depression of the top and bottom of a house are observed to be \( 30^\circ \) and \( 60^\circ \) respectively. Find the height of the building.
Find \( 2 \times 2 \) matrix that transforms the unit square \( \begin{pmatrix} 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \) into the parallelogram \( \begin{pmatrix} 0 & 3 & 4 & 1 \\ 0 & 1 & 3 & 2 \end{pmatrix} \).
If the median of the given data is 30, find the mean deviation from median of given data.

Class 0–10 10–20 20–30 30–40 40–50

Frequency 2 3 5 6 4
Find the Standard deviation of the given data.

Class 0–4 4–8 8–12 12–16 16–20

Frequency 2 3 4 1 2

Class	0–10	10–20	20–30	30–40	40–50
Frequency	2	3	5	6	4

Class	0–4	4–8	8–12	12–16	16–20
Frequency	2	3	4	1	2

Group D

Maximize the objective function \( z = 20x + 5y \) subject to the constraints:
\( x + 2y \leq 4 \), \( x - y \leq 4 \), \( x \geq 0 \) and \( y \geq 0 \).
Find the equation of the circle having centre \( (-4, 3) \) and passing through the point of intersection of the lines: \( x + y + 2 = 0 \) and \( x - 3y - 2 = 0 \).
Prove by vector method that the diagonals of a parallelogram bisect each other.
On a graph paper draw a triangle PQR having the vertices \( P(2, 4) \), \( Q(5, 1) \) and \( R(6, 5) \). Find the image of \( \triangle PQR \) by stating coordinates and graphing them after successive reflection in the y-axis followed by a rotation through \( -90^\circ \) about origin.

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