Group 'A'[10*1=10]

What type of the function is represented by \( f(x) = x^{2} + 5x + 6 \)? Write it.
Write the statement of Remainder Theorem.
In which condition a function \( f(x) \) is continuous at the point \( x = a \)? Write it.
What is the determinant value of a singular matrix? Write it.
Write the condition of parallelism of two lines having slopes \( m_{1} \) and \( m_{2} \).
In which condition a circle is formed when a plane surface intersects a cone? Write it.
Express \( \cos 2\theta \) in terms of \( \tan \theta \).
If \( \cos A = 1 \), write the acute angle value of \( A \).
If \( \beta \) is the angle between two vectors \( \vec{p} \) and \( \vec{q} \), then write the formula to find the value of \( \cos \beta \).
The inversion point of a point \( P \) is \( P' \) in a circle having centre \( O \) and radius \( r \). If the distance of the point \( P \) and \( P' \) from the centre \( O \) are \( OP \) and \( OP' \), write the relationship among \( OP \), \( OP' \), and \( r \).

Group 'B'[8*2=16]

If \( f(x) = \dfrac{x - 3}{5} \) and \( g(x) = 5x + 3 \), find the value of \( (f \circ g)(2) \).
Find the vertex of the parabola formed from the equation \( y = x^{2} - 5x + 6 \).
Find the inverse matrix of the matrix \( A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix} \)
The line passing through the points \( (-2, -5) \) and \( (1, a) \) is perpendicular to the line having equation \( 2x - y + 5 = 0 \). Find the value of \( a \).
Prove that: \( \sqrt{1 - \sin A} = \sin\left(\frac{A}{2}\right) - \cos\left(\frac{A}{2}\right), \quad \text{where } 0 \le A \le \pi. \)
Solve: \( 2\cos^{2}\theta - 2\sin^{2}\theta = 1 \)
Prove that \( \vec{a} = 3\vec{i} + 2\vec{j} \) and \( \vec{b} = 4\vec{i} - 6\vec{j} \) are perpendicular to each other.
The interquartile range of a continuous data is \( 20 \) and the first quartile \( (Q_{1}) = 10 \). Find the coefficient of quartile deviation.

If \( f(x) = 3x - 5 \), \( g(x) = \dfrac{2x + 7}{3} \) and \( (f \circ g^{-1})(x) = f(x) \), find the value of \( x \).
If the third term and sixth term of a geometric series are \( 12 \) and \( 96 \) respectively, what is the arithmetic mean between the first term and the sum of the first four terms of the series? Find it.
If \( f(x) = 3x - 5 \) is a real valued function.
Find \( f(3.9) \), \( f(3.99) \), \( f(4.01) \) and \( f(4.001) \).
Find \( \lim_{x \to 4^{+}} f(x) \) and \( \lim_{x \to 4^{-}} f(x) \).
Is the function \( f(x) \) continuous at the point \( x = 4 \)? Give reason.
Solve using Cramer's rule: \( \begin{aligned} 4x + 3y &= -18 \\ 2x - 5y &= 4 \end{aligned} \)
Find the angle between a pair of lines represented by the equation \( 2x^{2} + 7xy + 3y^{2} = 0 \).
Prove that: \( \cos 20^\circ \cdot \cos 40^\circ \cdot \cos 60^\circ \cdot \cos 80^\circ = \frac{1}{16}. \)
If \( A + B + C = \pi \), then prove that: \( \sin 2A + \sin 2B + \sin 2C = 4 \sin A \cdot \sin B \cdot \sin C. \)
From the top of a tower, the angles of depression of the roof and basement of a building \( 20 \) meter high are \( 30^\circ \) and \( 45^\circ \) respectively. Find the height of the tower.
A triangle \( PQR \) having the vertices \( P(1,2) \), \( Q(4,1) \) and \( R(2,5) \) is transformed by a \( 2 \times 2 \) matrix so that the coordinates of the image are \( P'(5,2) \), \( Q'(6,1) \) and \( R'(12,5) \). Find the \( 2 \times 2 \) matrix.
Find the mean deviation from mean of the given data.

Marks obtained	5–15	15–25	25–35	35–45	45–55
Frequency	3	5	4	5	3

Class Interval	2–4	4–6	6–8	8–10
Frequency	3	4	2	1

Find the maximum value of the objective function \( P = 4x + 6y \) under the constraints: \( x + 2y \le 8,\quad 3x + 2y \le 12,\quad x \ge 0,\; y \ge 0. \)
One end of a diameter of the circle having equation \( x^{2} + y^{2} - 4x - 6y - 12 = 0 \) is \( (5,4) \). Find the coordinates of the other end of the diameter.
In triangle \( XYZ \), the midpoints of the sides \( XY \) and \( YZ \) are \( A \) and \( B \) respectively. Prove by vector method that \( AB \parallel XZ \).
A triangle \( ABC \) with vertices \( A(1,2) \), \( B(4,-1) \) and \( C(2,5) \) is reflected successively on the lines \( x = 5 \) and \( y = -2 \). Find vertices of the images so obtained. Plot the triangle and images on the same graph paper.