OPT 8

Group A
1. For what value of \(m\) in \(y = mx + c\) makes it a constant function?
2. What is the common ratio of G.P. having \(n\) mean between 'a' and 'b'?
3. What is the meaning of the interval [a, b]?
4. If \(A = \begin{bmatrix} 1 & 0 \\ v & e \end{bmatrix}\), then what does \(\begin{bmatrix} e & -0 \\ -v & 1 \end{bmatrix}\) denote?
5. Write the formula to find angle between the pair of lines \(ax^2 + 2hxy + by^2 = 0\).
6. Define circle with respect to the conic section.
7. If \(A = B\), then which formula will be formed from \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)?
8. If \(A + B + C = \pi^c\), express \(\sin\left(\frac{A}{2} + \frac{B}{2}\right)\) in terms of angle \(\frac{C}{2}\).
9. If O be the origin and \(\vec{OA} + \vec{OB} = 6\vec{i} + 10\vec{j}\), then find the position vector of M which is the mid-point of AB.
10. Where is the inversion of P if point P is outside the circle of inversion?
Group B
11. If \(f(x) = x - 1\) and \(g(x) = x^2\), find the value of \(gf^{-1}(x)\).
12. Find the vertex of parabola, which is formed from \(f(x) = 4x^2 - 3\).
13. If \(A = \begin{bmatrix} 3 & -x \\ -4 & 12 \end{bmatrix}\) is a singular matrix, find the value of \(x\).
14. Find the equation of pair of straight lines represented by the equation \(x^2 - 2xy \sec \theta + y^2 = 0\).
15. If \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), find the value of \(\sin 15^\circ\).
16. Solve: \(3\sec^2 \theta - 3 = \tan 45^\circ\) \([0^\circ \leq \theta \leq 180^\circ]\)
17. If \(\vec{a} = \begin{pmatrix} -2 \\ p + 1 \end{pmatrix}\), \(\vec{b} = \begin{pmatrix} 4 \\ -5 \end{pmatrix}\) and \(\vec{a}.\vec{b} = -23\), then find the value of \(p\).
18. In a data, quartile deviation and first quartile are 20 and 35 respectively. Find the third quartile and the coefficient of quartile deviation.
19. Solve: \(x^3 - 4x^2 - 7x + 10 = 0\)
20. Find the minimum value of the objective function \(z = x + y + 2\) under the given constraints: \(x + y \leq 4\), \(x \geq 0\) and \(y \geq 0\).
21. Examine the continuity or discontinuity of the function \(f(x) = \begin{cases} x^2 + 1 & x \leq 3 \\ 2x + 4 & x > 3 \end{cases}\) at \(x = 3\).
22. Solve by Cramer's rule: \(3x + 4y = 17\), \(2x + 3y = 12\)
23. Find the equation of the line passing through the centroid of \(\triangle PQR\) with vertices P(3,3), Q(-2,-6) and R(5,-3) and parallel to the line QR.
24. Prove that: \(cosec50^\circ + \sqrt{3}\sec50^\circ - 4 = 0\)
25. If \(X + Y + Z = n^c\), then prove that: \(\cos2X - \cos2Y + \cos2Z = 1 - 4\sin X \cos Y \sin Z\)
26. The angle of elevation of the top of a building from 10m away in the same level is 45°. By what height should the building be raised to get the angle of elevation of the new top to be 60°?
27. Write down the 2×2 transformation matrix which represents the rotation about the origin through +90° and use it to find the image of \(\triangle ABC\) with vertices A(2,-3), B(4,5) and C(6,2).
28. Find the mean deviation of the following data.
    

    x	0-10	10-20	20-30	30-40	40-50
    f	2	3	6	5	4

29. Find the standard deviation.
    

    Height (in cm)	10-20	10-30	10-40	10-50	10-60
    No. of plant	8	20	35	44	50

30. a) If she continues her savings in this sequence, in how many months will she save Rs.2,000?
    b) If instead of saving Rs.4 more each month starting with Rs.32, the girl decided to double her savings each month, how many months will it take for her to save Rs.2,000 or more? Compare this with the number of months in both cases.
31. If \(x - y = 2\) is the equation of a chord of the circle \(x^2 + y^2 + 2y = 0\), find the equation of the circle if this chord is a diameter. Which circle occupies more area and why?
32. In the adjoining figure, what type of triangle is ABC? If \(\angle B = 90^\circ\), \(AM = CM\), then using vector method, establish the distance relation of M from A, B and C.
33. E is the enlargement \([0, 0], 2]\) and \(R\) are reflections on the line \(y = -x\). If \(\triangle CAT\) with vertices C(2,5), A(-1,3) and T(4,1) is mapped to \(\triangle C''A''T''\) under the transformation EOR, then find the co-ordinates of \(\triangle C''A''T''\) and plot all the triangles on the same graph. Does the order of transformations EOR or ROE matter in this case? Justify your answer with an example by applying the transformations in a different order to point A.