OPT 7

Group A
1. a) If \(f: A \to B\) and \(g: B \to C\) are two functions, then what will denote the composition function from \(A \to C\)?
   b) If \(p(x)\) is a factor of polynomial \(p(x - m)\), what is the value of \(p(m)\)?
2. a) What is the minimum value of \(y = \sin x\)?
   b) Find the value of determinant of an identity matrix.
3. a) If the intersection plane is parallel to the axis of cone then what conic does it form?
   b) Write the formula to calculate angle between the lines \(y = m_1x + c_1\) & \(y = m_2x + c_2\).
4. a) Write the condition of coincident of a pair of lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\).
   b) What is the relation between \(\sin3A\) and \(\sin A\)?
5. a) Express \(2\cos A \sin B\) in terms of sum or difference.
   b) If \(\vec{a} = 4\vec{i}\) then find \(\vec{a}^2\).
Group B
6. a) If \(f(x) = 3x - 2\) and \(f \circ g(x) = 5x - 2\), find \(g(x)\).
   b) If the polynomial \(x^3 + 6x^2 + kx + 10\) is exactly divisible by \((x + 2)\) with a remainder of 4, find the value of \(k\).
7. a) According to Cramer's rule, find the value of \(x\) in the equations \(ax + by = 1\) and \(bx + ay = 1\).
   b) If a line passing through (3, 4) and (-2, k) is perpendicular to the line having the equation \(5x + y + 3 = 0\), find the value of \(k\).
8. a) If \(3\tan(A + B) = 4\tan(A - B) = 12\), then find the value of \(\tan2A\).
   b) If \(\tan \frac{A}{2} = \frac{1}{2}\), then find the value of \(\sin A\).
9. a) If \(\vec{a} + 2\vec{b}\) and \(5\vec{a} - 4\vec{b}\) are orthogonal to each other and \(\vec{a}\) & \(\vec{b}\) are unit vectors, find the angle between \(\vec{a}\) & \(\vec{b}\).
   b) In a continuous series \(N = 47\), \(\sum fm = 770\), \(\sum fm^2 = 16450\) and \(\bar{x} = 16.38\), then find the standard deviation and its coefficient.
Group C
10. If \(f(x) = 2x + 5\), \(g(x) = \frac{3x - 1}{2}\) and \(fg^{-1}(x) = f(x)\), find the value of \(x\).
11. Solve: \(3x^3 = 7x^2 - 4\)
12. Solve by matrix method: \(\frac{3}{x} = \frac{2}{y} + 2\), \(\frac{6}{x} - \frac{2}{y} = 1\)
13. Find the equation of the circle having centre (1, 2) and passing through the point of intersection of the lines \(x + 2y = 3\) and \(3x + y = 4\).
14. An angle between a pair of lines represented by an equation \(2x^2 + 7xy + ky^2 = 0\) is 45°. Find the positive value of \(k\), then also find the separate equation of the pair of lines.
15. Prove that: \(2\tan2\theta + 4\cot4\theta = \cot\theta - \tan\theta\)
16. If \(A + B + C = 200^\circ\) and \((\cos A + \cos B)(\cos A - \cos B) - \cos^2 C = 2\cos A \sin B \sin C - 1\), then prove that: \(\cos A + \cos B + \cos C = 1\).
17. A rope dancer was walking on a loose rope tied to the tops of two equal posts of height 9m. When he was 2.7m above the ground, it was found that the stretched length of the rope made angles of 30° and 60° with horizontal plane parallel to the ground. Find the length of the rope.
18. Solve: \(\sqrt{3} \cos x + \sin x = \sqrt{3}\) \([0 \leq x \leq 2\pi]\)
19. Find the quartile deviation and its coefficient.
    

    Height	60-62	62-64	64-66	66-68	68-70	70-72
    No. of Students	4	6	8	12	7	2

20. Find the mean deviation from median.
    

    Class interval	5-15	15-25	25-35	35-45	45-55
    Frequency	7	3	6	4	5

Group D
21. If 5 times the square of a number is added to the cube of the number, then it is 24 more than two times the number. Find the possible number.
22. Find the equation of the straight line which is perpendicular to \(2x + 3y = 11\) and such that the sum of the intercepts on axes is 15.
23. Prove vectorially that the diagonals of a rectangle are equal to each other.
24. The circular lid of a water tank is fixed to have an equation \(x^2 + y^2 + 2x - 6y = 0\). If the lid is moved right side by 4 units parallel to x-axis, find the equation of the circular lid in the new position.