OPT 10

1. Write the range scale of the function \(y = \sin x\).
2. Find the mean between the terms p and q having in the geometric series.
3. Express \(\lim_{x \to a^+} f(x)\) in the language form.
4. Write the definition of singular matrix.
5. Which geometrical figure will be formed if a plane intersects parallel to the base of a cone?
6. Express \(\cos 3A\) in terms of \(\cos A\).
7. Define altitude of the sun.
8. In which condition \(\vec{a}\) and \(\vec{b}\) are parallel to each other?
9. If B' is the image of B and r is the radius of circle with Centre O in an inversion transformation, write the relation of OB, OB' and r.
10. If the polynomial \(x^3 + 6x^2 + kx + 10\) is divided by \((x + 2)\) with a remainder of 4, find the value of 'k' using the remainder theorem.
11. Present the given inequality in graph paper: \(4x + 5y \geq 20\)
12. If the equations \(mx - 5y = 3\) and \(4x + 3y = 4\) have \(D = 29\), then find the value of 'm' by using Cramer's rule.
13. If the straight lines \(3x - 4y = 10\) and \(4x + ky = 12\) are orthogonal, find the value of \(k\).
14. Solve: \(\sin \theta - \cos \theta = 0\) \([0^\circ \leq \theta \leq 90^\circ]\)
15. Prove: \(cosec \theta - \cot \theta = \tan \frac{\theta}{2}\)
16. The position vectors of the points A and B are \(8\vec{i} + 6\vec{j}\) and \(3\vec{i} + \vec{j}\) respectively. Find the position vector of the point C which divides AB in the ratio 2:3 internally.
17. In a series, the first quartile (\(Q_1\)) is 17.5 and the quartile deviation is 20, find the third quartile and coefficient of quartile deviation.
18. If \(g \circ f(4) = -13\), then find the value of \(a\) and \(g^{-1}(x)\).
19. Solve: \(x^3 - 4x^2 + x + 6 = 0\)
20. Examine the continuity or discontinuity of \(f(x) = \begin{cases} 4x - 1, & x < 1 \\ 7x, & x \geq 1 \end{cases}\) at \(x = 1\) by calculating left hand limit, right hand limit and functional value.
21. Solve by matrix method: \(3x + \frac{4}{y} = 7x + \frac{1}{y} = 3\)
22. The angle between the pair of lines represented by the equation \(2x^2 + kxy + 3y^2 = 0\) is 45°. Find the positive value of \(k\) and find the separate equations of the pair of lines.
23. Prove that: \(\frac{1 + \sin 2A + \cos 2A}{1 + \sin 2A - \cos 2A} = \cot A\)
24. For \(A + B + C = 180^\circ\), prove that: \(\sin 2A - \sin 2B + \sin 2C = 2\cos A \cdot \sin B \cdot \cos C\)
25. Two posts are 120 feet apart and the height of one is double that of the other. From the middle of the line joining their feet, an observer finds the angular elevations of tops to be complementary. Find the height of the posts.
26. Find 2×2 square matrices that transform a square ABCD having the vertices A(2,3), B(4,3), C(4,5) and D(2,5) into a square A'B'C'D' containing the vertices A'(3,2), B'(3,4), C'(5,4) and D'(5,2).