OPT 11

Group A

Under what condition the inverse function of a function is possible? Write it.
If the first term of a geometric series is 'a' and the common ratio is 'r', what is the n^th term? Write it.
In which condition the function \( f(x) \) is continuous at the point \( x=a \)? Write in symbol.
What is the determinant of a unit matrix of order \( 2 \times 2 \)? Write it.
Define circle on the basis of conic section.
Write the formula to find the angle between the straight lines \( y = m_1 x + c_1 \) and \( y = m_2 x + c_2 \).
If \( A + B + C = \pi^c \) express \( \cos(A + B) \) in terms of angle \( C \).
Write \( \cos P - \cos Q \) and \( \sin P - \sin Q \) in terms of product of sine or cosine.
If the angle between \( \vec{m} \) and \( \vec{n} \) is \( \theta \), then write the scalar product of \( \vec{m} \) and \( \vec{n} \).
If the inversion circle is \( x^2 + y^2 = a^2 \) and the point \( P(x, y) \) is outside the circle, then write the co-ordinates of the inversion point \( P' \) of the point \( P(x, y) \).

Group B

Find the value of \( Q(x) \) and \( R \) in \( 2x^2 - 4x + 3 = (x - 2) \cdot Q(x) + R \) using synthetic division method.
Find the co-ordinates of the vertex of a quadratic function of the equation \( y = x^2 + 2x - 3 \).
If the matrix \(\begin{bmatrix} 4 & x \\ 2 & 2 \end{bmatrix}\) does not define its inverse, then find the value of \( x \).
Find the slope of the line segment which is perpendicular to the line segment joining the points (2, 3) and (5, 7).
Prove that: \(\frac{1 + \cos \theta + \sin \theta}{1 - \cos \theta + \sin \theta} = \cot \frac{\theta}{2}\).
Express \(\sqrt{2 + \sqrt{2 + 2\cos 4\theta}}\) in terms of \(\cos \theta\).
M is the mid-point of a line segment AB. If \(\overrightarrow{OA} = 3\mathbf{i} + 5\mathbf{j}\) and \(\overrightarrow{OM} = 2\mathbf{i} - \mathbf{j}\), find the position vector of the point B.
In a continuous data, if mean value \((\bar{x}) = 28\), sum of frequencies \((N) = 20\) and \(\Sigma f|D| = 208\), find the coefficient of mean deviation.

Group C

If \( f(x) = 2x - 3 \), \( g(x) = \frac{2x - 7}{3} \) and \( f(x) = g^{-1}(x) \), find the value of \( x \).
The sum of fourth, fifth and sixth terms of an arithmetic series is zero. If the first term is 6, find the sum of the first 12 terms of the series.
If the function \( f(x) = 2x + 3 \) is defined:
1. Find the values of \( f(1.999) \) and \( f(2.001) \).
2. Write the values of \( \lim_{x \to 2^-} f(x) \) and \( \lim_{x \to 2^+} f(x) \).
3. Is the function \( f(x) \) continuous at \( x = 2 \)? Give reason.
Solve by matrix method: \( 3x + \frac{4}{y} = 7 \), \( 4x - \frac{10}{y} = 17 \).
The equations of two diameters of a circle passing through a point \( A(5, 1) \) are given by \( x - y = 3 \) and \( 2x + y = 21 \). Find the equation of the circle.
Prove that: \( 16 \cos 10^\circ \cos 20^\circ \cos 50^\circ \cos 70^\circ = 3 \).
Solve: \( \sin 4x + \sin 2x = \cos 0 \) \( 0 \leq x \leq 360^\circ \).
From the roof of a house 40 m high, the angle of elevation and depression of the top and foot of a tower are found to be \( 60^\circ \) and \( 30^\circ \) respectively. Find the height of the tower and distance between the house and the tower.
A unit square is transformed by a matrix \( \begin{bmatrix} 3 & b \\ c & 1 \end{bmatrix} \) to get \( \begin{bmatrix} 0 & a & 5 & 2 \\ 0 & 1 & 2 & d \end{bmatrix} \). Find the values of \( a, b, c \) and \( d \).

Find the quartile deviation from the data given below:

Class interval	0–20	20–40	40–60	60–80	80–100
Frequency	2	5	4	5	4

Find the standard deviation and its coefficient from the data given below:

Marks obtained	0–20	0–20	0–30	0–40	0–50
No. of students	1	5	12	15	20

Group D

Minimize the objective function \( z = 2x - 3y \) under the following constraints: \( x + y \geq 0 \), \( x - y \leq 0 \), \( z \geq -1 \), \( y \leq 2 \).
A straight line is represented by an equation \( 2x^2 + 7xy + ky^2 = 0 \) is \( 45^\circ \) and the positive value of \( k \). Then also find the separate equation of the pair of lines.
Prove by vector method that the angle in semi circle is a right angle.
A triangle with vertices A(1, 2), B(4, -1) and C(2, 5) is reflected successively in the line \( x = 5 \) and \( y = -2 \). Find by stating co-ordinates and graphically represent the images under these transformations. State also the single transformation given by the combination of these transformations.

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