OPT 2

Group 'A' \([10 \times 1 = 10]\)

1. What is the nature of a constant function in the graph?
2. What is the common ratio of GS having 'n' means between 'a' and 'b'?
3. Write \(\lim_{x \to a} f(x)\) in sentence.
4. Define unit square matrix.
5. What geometric figure will be formed if a plane intersects a cone parallel to its base?
6. Write the equation of the straight line orthogonal to the line \(px + qy - r = 0\).
7. Write \(\tan2A\) in terms of \(\tan A\).
8. Express \(2\sin A \cos B\) in term of sum and difference.
9. If \(\vec{a} = k\vec{b}\) and where \(k\) is a scalar quantity, what is the angle between \(\vec{a}\) and \(\vec{b}\)?
10. If \(P(x, y)\) is image of \(P'(x', y')\) and \(r\) is radius of circle with centre \((h, k)\) in an inversion transformation, write the value of \(y'\).

Group 'B' \([8 \times 2 = 16]\)

11. A polynomial \(f(x)\) is divided by \((4x + 3)\) to get the quotient \(2x^2 - 3x + 1\) and remainder \(4\), find the polynomial \(f(x)\).
12. Find the vertex of the parabola having equation \(y = x^2 - 6x + 5\).
13. For what value \(x\), the matrix \(\begin{pmatrix} x & -2 \\ x & x + 2 \end{pmatrix}\) has no inverse? Find it.
14. If \(\frac{x}{b} + \frac{y}{4} = 1\) and \(3x + 4y = 1\) are parallel to each other then find the value of \(b\).
15. (Prove that): \(\frac{1 + \tan^2(45^\circ - \frac{A}{2})}{1 - \tan^2(45^\circ - \frac{A}{2})} = cosec A\).
16. (Solve): \(cosec^2\theta - 2 = 0\). \([0^\circ \le \theta \le 90^\circ]\)
17. If \(\vec{p} + \vec{q} + \vec{r} = \vec{0}\), \(|\vec{p}| = 6\), \(|\vec{q}| = 10\) and \(\vec{p} \cdot \vec{q} = 30\), then find \(|\vec{r}|\).
18. The third quartile of a data is \(15\). If the coefficient of quartile deviation is \(\frac{1}{14}\), find the first quartile and inter-quartile range of the data.

Group 'C' \([11 \times 3 = 33]\)

19. If \(g(x) = \frac{3x-1}{2}\) and \(f(x) = 2x + 5\) and \((fog^-1)(x) = f(x)\), find the value of \(x\).
20. The sum of three terms in an arithmetic series is \(36\). If \(1\), \(4\), and \(43\) are added to them respectively, the results are in geometric series. Find the terms.
21. For a real valued function \(f(x) = 2x - 3\), find the values of \(f(1.999)\), \(f(2.001)\) and \(f(2)\). Is this function continuous at \(x = 2\)?
22. Solve by using Cramer's rule: \(2x + \frac{4}{y} = 3\) and \(3x - 12 = \frac{6}{y}\).
23. If the lines represented by the equation \(3x^2 + 8xy + my^2 = 0\) are perpendicular to each other, find the separate equations of two lines.
24. (Prove that): \(\cos^2(B + 120^\circ) + \cos^2(B - 120^\circ) + \cos^2 B = \frac{3}{2}\).
25. If \(A + B + C = 180^\circ\), Prove that: \(\cos(B + C - A) + \cos(C + A - B) + \cos(A + B - C) = 1 + 4\cos A \cos B \cos C\).
26. The angle of elevation of the top of a tower observed from \(27 \text{ m}\) and \(75 \text{ m}\) away from its foot on the same side are found to be complementary. Find the height of the tower.
27. Find the inverse point of \(A(-1, -3)\) with respect to the circle \((x - 2)^2 + (y + 1)^2 = 16\).
28. Compute the mean deviation from mean and its coefficient from the data given below:

    Age (in years)	0-4	0-8	0-12	0-16	0-20
    No. of boys	12	20	30	36	40

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

    Class Interval	0-10	10-20	20-30	30-40	40-50
    Frequency	5	8	15	16	6

Group 'D' \([4 \times 4 = 16]\)

30. Maximize \(P = 6x + 5y\) under the following constraints: \(x + y \le 6\), \(x - y \ge -2\), \(x \ge 0\), \(y \ge 0\).
31. The equations of two diameters of a circle passing through the point \((3, 4)\) are \(x + y = 14\) and \(2x - y = 4\). Find the equation of the circle.
32. If \(S\) is mid-point of \(QR\) and \(PQ = PR\), then prove by vector method that \(PS \perp QR\). (\(PQR\) is an isosceles triangle.)
33. State the single transformation equivalent to the combination of reflections on \(X\)-axis and \(Y\)-axis respectively. Using this single transformation, find the coordinates of the vertices of the image of \(\triangle ABC\) having vertices \(A(4, 3)\), \(B(1, 1)\) and \(C(5, -1)\). Also, draw the object and image on the same graph paper.