OPT 9

1. What is the period of the given trigonometric function? Write it.
2. Write the arithmetic mean between two numbers \(\theta\) and \(\beta\).
3. Check the continuity or discontinuity of a function \(f(x) = \frac{x + 2}{x - 3}\) at \(x = 3\).
4. Define singular matrix.
5. Which geometrical figure will be formed if a plane intersects a right circular cone parallel to the generator of the cone?
6. Write the condition when the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\) are perpendicular to each other.
7. Express \(\sin 3A\) in terms of \(\sin A\).
8. Express \(2\sin A \cos B\) in the form of sum or differences of sine or cosine.
9. If \(\vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\), write the scalar product of \(\vec{a}\) and \(\vec{b}\).
10. In the given figure, O is the centre and r is the radius of inversion circle. If P' is the inversion point of the point P, write the relation between OP, OP' and r.
11. If \(f(x) = x^3 + kx^2 - 4x + 12\) and \(x + 2\) is a factor of polynomial \(f(x)\), find the value of \(k\).
12. Find the inequality represented by the shaded region in the given graph.
13. If the equations \(9x - 8y = 12\) and \(mx + 3y = 17\) have \(D = 43\), find the value of 'm' using Cramer's rule.
14. Find the obtuse angle between the two lines \(\sqrt{3}x - y = 5\) and \(x - \sqrt{3}y = 15\).
15. Prove that: \(\frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} = 1 + \frac{1}{2} \sin \theta\)
16. Solve: \(2\cos^2 \theta = 3\sin \theta\) \([0^\circ \leq \theta \leq 180^\circ]\)
17. In the given figure, if P is the mid point of the side YZ of \(\triangle XYZ\), and the position vectors of the points X and P are \(3\vec{i} + 5\vec{j}\) and \(3\vec{i} - 4\vec{j}\) respectively, find the position vector of the centroid (G) of \(\triangle XYZ\).
18. In the continuous data, the coefficient of quartile deviation is \(\frac{1}{3}\) and quartile deviation is 10, find the value of \(Q_3\).
19. If \(f(x) = \frac{2x + 5}{8}\), \(g(x) = 3x - 4\) and \((f \circ g)^{-1}(x)\) is an identity function, find the value of \(x\).
20. The sum of three numbers in AP is 15. If 1 and 5 are respectively added to the second and the third numbers then the first number together with these two numbers are in GP. Find the original numbers.
21. Check the continuity or discontinuity of the given function at \(x = 3\): \(f(x) = \begin{cases} \frac{2}{5 - x} & \text{for } x \leq 3 \\ 5 - x & \text{for } x > 3 \end{cases}\)
22. Solve by inverse matrix method: \(\frac{3x + 5y}{8} = \frac{5x - 2y}{3} = 3\)
23. If the angle between the pair of lines represented by the equation \(2x^2 + kxy + 3y^2 = 0\) is 45°, find the value of \(k\).
24. If \(2\tan A = 3\tan B\), prove that: \(\tan(A + B) = \frac{5\sin 2B}{5\cos 2B - 1}\)
25. If \(A + B + C = 180^\circ\), prove that: \(\sin^2 A - \sin^2 B + \sin^2 C = 2\sin A \cdot \cos B \cdot \sin C\)
26. A ladder 9 m long reaches at a point 9 m below the top of a vertical flagstaff. From the foot of the ladder, the angle of elevation of the flagstaff is 60°. Find the height of the flagstaff.
27. If the matrix \(\begin{pmatrix} a & 2 \\ b & 2 \end{pmatrix}\) transform a unit square to the parallelogram \(\begin{pmatrix} 0 & 4 & c & 2 \\ 0 & 1 & 3 & d \end{pmatrix}\), find the values of \(a\), \(b\), \(c\) and \(d\).
28. Find the mean deviation from mean and its coefficient for the given data.
    

    Marks Obtained	0-10	10-20	20-30	30-40	40-50
    Frequency	5	8	15	16	6

29. Find the standard deviation and coefficient of variation from the given data.
    

    Marks Obtained	5-15	15-25	25-35	35-45	45-55
    Frequency	2	6	8	12	2

30. Find the equation of parabola shown in the given figure.
31. Find the equation of circle passing through the points (3,7) and (5,5) and centre lies on the line \(x - 4y = 1\).
32. Prove by vector method that the diagonals of a rectangle are equal.
33. \(\triangle RAM\) with vertices R(2,3), A(4,5) and M(1,4) is reflected about the x-axis and then image so formed is rotated through 180° in anticlockwise direction about origin. Write the co-ordinates of vertices of the images thus obtained and present \(\triangle RAM\) and its images in the same graph paper.