OPT 6

1. a) Under what condition are the functions \(f\) and \(g\) inverses of each other?
   b) How many terms are there in the arithmetic progression if 7 arithmetic means are inserted between 1 and 101?
2. a) Write \(\lim_{x \to 4} f(x)\) in words.
   b) If \(\begin{vmatrix} x & 0 \\ 3 & 1 \end{vmatrix} = 0\), what is the value of \(x\)?
3. a) What type of conic section does the given figure represent?
   b) What is the slope of a line perpendicular to the line that makes a \(60^\circ\) angle with the x-axis?
4. a) What is the value of \(2\sin15^\circ \cdot \cos15^\circ\)?
   b) If \(A + B + C = n\pi\), which trigonometric ratio is equal to \(\tan(2A + 2B)\)?
5. a) What is the value of \(\vec{i} \cdot \vec{j}\)?
   b) Write the angle and the coordinates of the center of the rotation that represents the combined reflection in the x-axis followed by reflection in the line \(y = -x\).
6. a) For what value of \(k\) is the polynomial \(2x^4 + 3x^3 + 2kx^2 + 3x + 6\) exactly divisible by \(x + 2\)?
   b) In a geometric sequence, if the product of the first five terms is 32, find the third term.
7. a) What is the value of \(\begin{vmatrix} \sin x & \cos x \\ -\cos x & \sin x \end{vmatrix}\)? Find.
   b) If the line passing through (3, -4) and (-2, a) is parallel to the line having by the equation \(y + 2x + 3 = 0\), find the value of 'a'.
8. a) If \(\cos \frac{\alpha}{3} = \frac{1}{2}\) find the value of \(\sin \alpha\).
   b) Solve: \(\sqrt{3} \tan \theta - 3 = 0\) \([0^\circ \leq \theta \leq \pi^c]\)
9. a) Find the angle between unit vector \(\vec{i}\) and \(\vec{d} = \sqrt{3} \vec{i} + \vec{j}\).
   b) Define standard deviation and its coefficient?
10. Solve: \(6x^3 - 5x^2 - 3x + 2 = 0\)
11. The sum of the first ten terms of an arithmetic series is 50 and its fifth term is 3 times the second term. Calculate the sum of the first twenty terms of the series.
12. Draw the graph of the given function \(y = |x|, x \in \mathbb{Z}\) and then identify whether it is continuity or (discontinuity).
13. Solve (by Cramer's rule): \(2x = y - 3, 3x + 2y = 20\)
14. If AP ⊥ BC and the coordinates of A, B and C are (-2, 6), (-6, 0) and (2, 0) respectively, find the gradient of BC and equation of AP.
15. Prove that: \(\cos10^\circ \cdot \cos50^\circ \cdot \cos70^\circ = \frac{\sqrt{3}}{8}\)
16. If A, B and C are the angles of a triangle, prove that: \(\cos(B + C - A) + \cos(C + A - B) + \cos(A + B - C) = 1 + 4\cos A \cdot \cos B \cdot \cos C\)
17. From the top of a tower 50 m high the measures of the angles of depression of two objects due east of the tower are found to be 45° and 60°. Find the distance between the objects.
18. A square ABCD with vertices A(2,0), B(5,1), C(4,4) and D(1,3) is mapped to a parallelogram A', B', C' and D' by a 2×2 matrix. If the vertices of the parallelogram are A'(2,2), B'(7,3), C'(12,-4) and D'(7,-5), find the 2×2 matrix.
19. Find semi inter-quartile range and its coefficient for the data given below.
    

    (Marks)	0-10	10-20	20-30	30-40	40-50
    (Frequency)	2	1	2	3	5

20. Calculate the mean deviation from the mean and its coefficient for the data given below.
    

    (Marks)	0-4	4-8	8-12	12-16	16-20
    (Frequency)	12	8	10	6	4

21. Find the maximum value of \(P = 2x + 3y + 5\) under the following constraints: \(x + 2y \geq 1\), \(x + y \leq 5\), \(x \geq 0\) and \(y \geq 0\).
    Also, find the equation of the circle whose centre is (4, 5) and passes through the centre of the circle \(x^2 + y^2 + 4x + 6y - 12 = 0\).
22. Prove vectorially that the midpoint of the hypotenuse of a right angled triangle is equidistant from its vertices.
23. A rectangle with the vertices A(3,1), B(6,1), C(6,5) and D(3,5) is reflected in the line \(x = 1\) and then rotated about the origin through \(90^\circ\) in the positive direction. Find the coordinates of the images of the rectangle and show them on the same graph paper.