OPT 13

Group A

1. In which condition the functions \( f \) and \( g \) are inverse to each other?
2. State remainder theorem.
1. Write in a sentence: \(\lim_{x \to 5} f(x)\)
2. Define a singular matrix.
1. What is the slope of a line perpendicular to the line making \( 60^\circ \) with the x-axis?
2. Write the formula for the equation of a circle with endpoints of a diameter \((x_1, y_1)\) and \((x_2, y_2)\).
1. If \( A + B + C = \pi^c \) which transformation ratio is equal to \(\cot(2A + 2B)\)?
2. What is the value of \( 2\sin 15^\circ \cos 15^\circ \)?
1. What is the feasible region in linear programming?
2. If \( \hat{i} \) is the standard unit vector along the x-axis, what is the value of \( \hat{i} \cdot \hat{i} \)?

Group B

1. If \( f(x) = ax + 3 \) and \( f(f(x)) = x^2 \), then find the value of \( a \).
2. If one factor of \( p(x) = (x + 2)^7 + (2x + k)^3 \) is \( (x + 2) \), then what is the value of \( k \).
3. If \( A = \begin{bmatrix} 3 & 4 \\ -2 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix} \), find the determinant of \( 3A - 4B \).
1. If the line \( l_1 x + m_1 y + n_1 = 0 \) and \( l_2 x + m_2 y + n_2 = 0 \) are perpendicular to each other, prove that: \( l_1 l_2 + m_1 m_2 = 0 \).
2. Define standard deviation and its coefficients.
1. Solve: \( \sin \alpha - \sin \alpha + \frac{1}{4} = 0 \) \([0^\circ \leq \alpha \leq 90^\circ]\)
2. If \( \cos \frac{\alpha}{2} = \frac{1}{2} \), find the value of \( \sin \alpha \).
3. If \( \vec{p} \) and \( \vec{q} \) are unit vectors such that \( (\vec{p} + 2\vec{q}) \) and \( (5\vec{p} - 4\vec{q}) \) are perpendicular to each other, find the angle between \( \vec{p} \) and \( \vec{q} \).

Group C

Solve by using the factor theorem: \((z - 2)(z^2 - 9z + 13) + 5 = 0\).
If \( a, 30, b, 50, \) and \( c \) are in AP, find the values of \( a, b, \) and \( c \).
Prove that the function \( f(x) = \frac{x^2 + x - 6}{x + 3} \); \( x \in \mathbb{R} \) is continuous at a point \( x = -3 \).
Solve the following matrix equations by matrix method: \( \begin{pmatrix} 4 & -3 \\ 5 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 9 \\ 18 \end{pmatrix} \).
Find the equations of two lines represented by the equation \( x^2 - 2xy\cot 2\alpha - y^2 = 0 \) and the angle between them.
Prove that: \( \cos^3 X \cdot \sin^2 X = \frac{1}{16} (2\cos X - \cos 3X - \cos 5X) \).
If \( A + B = \pi^c \), prove that: \( 2(1 - \sin A \cdot \sin B) = \cos^2 A + \cos^2 B \).
The angle of depression of the top of a 5 m high pole observed from the top of the house is 60° and the angle of elevation of the top of the pole from the foot of the house is 30° then find the height of the house.
Find the equation of the straight lines passing through the point (4, 5) and making an angle of 45° with the line \( x + y - 7 = 0 \).
Construct a frequency distribution table taking 0–4 as a class interval and find the mean deviation.
1, 3, 2, 3, 4, 5, 6, 7, 7, 10, 9, 15

Find the variance of the data given below.

Class	0≤x<10	10≤x<20	20, 30	30≤x<40	40, 50
Frequency	1	5	4	5	6

Group D

The amount of a sum of Rs. x is \( y = x + \frac{xTR}{100} \) where T=10 years and R=5%
1. Find \( f(5,000) \).
2. Find \( f^{-1}(x) \) and \( f^{-1}(7,500) \).
3. What does \( f^{-1}(x) \) represent?
Obtain the vertices of the feasible region (convex polygonal region) of the given constraints \( x + y \leq 3 \), \( x \leq 2 \), \( y \leq 1 \), \( x \geq 0 \), \( y \geq 0 \).
Find the equation of the circle which passes through the points (5, 7), (6, 6) and (2, –2).
ΔABC with vertices A(1, 2), B(4, –1), and C(2, 5) is reflected successively in the line \( y = x \) and x-axis. Find by stating coordinates and graphically represent images under these transformations. State also the single transformation given by the combination of these transformations.

From