OPT 12

Group A

1. Which function is the function \( f(x) = 5 \)? Write its name.
2. Write the formula to calculate the general term of a G.P.
1. Write the notation of left hand limit of the continuity of function \( f(x) \) at the point \( x = a \).
2. Write the determinant of matrix \( \begin{pmatrix} p & s \\ -r & q \end{pmatrix} \).
1. Write the condition of perpendicularity for the pair of lines \( ax^2 + 2hxy + by^2 = 0 \).
2. Write the condition of generating an ellipse when a cone is cut by a plane surface.
1. Write a formula of \( \sin A \) in terms of \( \sin \frac{A}{3} \).
2. Write \( \cos A + \cos B \) in the product or sum of Sine and Cosine.
1. Define scalar product of two vectors.
2. If the inversion point of \( A(x, y) \) with respect to a circle with center \( O(0, 0) \) and radius \( r \) units is \( A'(x', y') \), write the formula to find \( A' \).

Group B

1. If \( f(x) = 2x - 3 \) and \( f^{-1}(x) = 5 \), find the value of \( x \).
2. Show that \( (x - 3) \) is factor of \( x^2 - 5x - 12 \).
1. If the matrices \(\begin{bmatrix} 2a & 7 \\ 5 & 9 \end{bmatrix}\) and \(\begin{bmatrix} 9 & b \\ -5 & 4 \end{bmatrix}\) are inverse to each other, find the values of \(a\) and \(b\).
2. Find the obtuse angle between a pair of straight lines having equation \(2x^2 - 7xy + 3y^2 = 0\).
1. Prove that: \(\cos 20^\circ - \cos 70^\circ = \sqrt{2} \sin 25^\circ\).
2. Solve: \(4\cos^2 A - 3 = 0\) \((0^\circ \leq A \leq 180^\circ)\).
1. For what value of \( k \), the vectors \( \vec{p} = k\hat{i} - 6\hat{j} \) and \( \vec{q} = 3\hat{i} - 4\hat{j} \) are orthogonal to each other? Find it.
2. The values of the first quartile and the third quartile of a continuous data are 15.83 and 41.67 respectively. Find the quartile deviation and the coefficient of quartile deviation.

Group C

Solve: \( 6x^3 + 13x^2 + x - 2 = 0 \).
Solve by graphically: \( x^2 - 3x + 2 = 0 \).
Test the continuity or not of the function \( f(x) = 4x + 31 \) calculating left hand limit, the right hand limit and the functional value at \( x = 2 \).
Solve by using Cramer’s rule: \( 4x + 7y = 26 \), \( 2x - 4 = y \).
The equation of two diameters of a circle are \( x + 4y = 10 \) and \( x - 3y = 7 \) respectively. If the circle passes through a point (5, –3), find the equation of the circle.
Prove: \( (2\cos 2A - 1)(2\cos A + 1)(2\cos A - 1) = 2\cos 4A + 1 \).
If \( A + B + C = \pi^c \), Prove that: \( \cos^2 A + \cos^2 B + 2\cos A \cos B \cos C = \sin^2 C \).
From the roof of a house 15 m high, a man observes a tower which is on the same plane and he finds the angle of elevation 45° to the top of the tower and the angle of depression 30° to the bottom of the tower. Find the height of the tower.
Find the 2×2 transformation matrix which transforms a parallelogram ABCD with vertices A(0, 0), B(3, 0), C(4, 1) and D(1, 1) into a unit square.

Find the mean deviation of the data given below from median.

Obtained marks	0–10	10–20	20–30	30–40	40–50
No. of students	8	12	16	10	4

Compute the coefficient of variation from the given data.

Class	0–8	8–16	16–24	24–32	32–40
Frequency	7	3	5	3	2

Group D

The sum of three numbers of an A.P. is 24. If 1, 4 and 13 are added to them respectively, the resulting numbers are in G.P. Find the numbers.
Two straight rods which are equally inclined to the straight beam having equation \( 4x + 5y = 30 \) are perpendicular to each other at a point (3, 4). Find the equations which represent the rods.
Prove by vector method that the line joining the mid point of the sides of a quadrilateral is formed a parallelogram.
\( R_1 \) represents the reflection on the line \( x = -1 \) and \( R_2 \) represents the reflection on the line \( x = 3 \). State the single transformation given by the combinations of \( R_1 \) and \( R_2 \). A triangle with vertices \( P(2, 2) \), \( Q(4, -1) \) and \( R(6, 5) \) is transformed by that single transformation. Find by stating co-ordinates of the image and represent the given triangle and the image graphically under these transformations.

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