OPT 5

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

1. Define trigonometric function.
2. Define remainder theorem.
3. What is the meaning of \(-3 < x < 3\)?
4. According to the Cramer's rule \(D = 5\) and \(D_x = 10\), then what is the value of \(x\)?
5. In which condition parabola is formed?
6. What is the relation between the lines \(2x + 3y = 7\) and \(4x + 6y = 10\)? Write with reason.
7. Express \(\cos 4B\) in terms of \(\tan 2B\).
8. Express \(2\sin 2A \cos 2B\) in terms of sum or difference of sine or cosine.
9. In which condition two vectors are perpendicular to each other?
10. Write the formula to find \(x'\) if \(P'(x', y')\) is the inversion point of \(P(x, y)\) of a circle whose centre is \((h, k)\) and radius '\(a\)'?

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

11. If \((x - 3)\) is a factor of polynomial \(f(x) = x^3 - 5x^2 + 7x + 10 + k\), then find the value of \(k\).
12. Find the vertex of the parabola \(y = x^2 - 4x + 3\).
13. If \(A = \begin{pmatrix} 5 & 6 \\ 7 & 2 \end{pmatrix}\), \(B = \begin{pmatrix} 3 & 4 \\ 2 & 1 \end{pmatrix}\), and \(\det(AB) = 4x + 5\), then find the value of \(x\).
14. If two equations \(px + 3y = 8\) and \(4x + 6y + 8 = 0\) are parallel to each other then find the value of \(P\).
15. Prove that: \(\tan\frac{\theta}{2} + \cot\frac{\theta}{2} = 2cosec\theta\).
16. Solve: \(2\sin^2 A - 1 = 0\) (Range up to \(180^\circ\)).
17. If \(\vec{a} = 2\vec{i} + 3\vec{j}\) and \(\vec{b} = 6\vec{i} + 4\vec{j}\) are two vectors then prove that these two vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.
18. In a continuous data has \(\sum fm = 120\), \(N = 10\) and \(\sum f|m - \bar{x}| = 34\) Find mean deviation and its coefficient from mean.
19. If \(f(x) = 3x + 5\), \(g^{-1}(x) = 3x - 4\), \(g\circ f(2) + f\circ g(1) = 2m\) then find the value of \(m\).
20. The sum of three terms of AP is \(15\). If the product of first and last term is \(16\) then find the terms.
21. For real function \(f(x) = 3x + 5\), find the values of \(f(2.9)\), \(f(3.01)\) and \(f(3)\). Is this function continuous at \(x=3\)?
22. Solve by matrix method: \(3x + 5y = 24\), \(5x - 9 = 2y\).
23. Find the separate equations of line represented by the equation \(x^2 + 5xy + 4y^2 = 0\). Also find the angle between them.
24. Prove that: \(\mathrm{Cosec}\,2A + \mathrm{Cot}\,4A = \mathrm{Cot}\,A - \mathrm{Cosec}\,4A\).
25. If \(P + Q + R = 200^\circ\) then prove that: \(\sin2P + \sin2Q + \sin2R = 4\sin P\sin Q\sin R\).
26. The shadow of a tower on the ground is found to be \(10.5\mathrm{m}\) longer when sun's altitude is \(45^\circ\) then when it is \(60^\circ\). Find the height of tower.
27. \(\mathrm{ABCD}\) is a parallelogram with vertices \(A(0, 0)\), \(B(3, 0)\), \(C(4, 1)\) and \(D(1, 1)\). Find \(2\times 2\) transformation matrix.
28. Find the quartile deviation and coefficient of quartile deviation from the data:

    \(x\)	\(0-10\)	\(10-20\)	\(20-30\)	\(30-40\)	\(40-50\)	\(50-60\)
    \(f\)	\(2\)	\(5\)	\(8\)	\(11\)	\(14\)	\(15\)

29. Find the standard deviation and coefficient of variation of the data:

    Marks	\(4-8\)	\(8-12\)	\(12-16\)	\(16-20\)	\(20-24\)	\(24-28\)	\(28-32\)
    No. of students	\(3\)	\(4\)	\(7\)	\(8\)	\(10\)	\(11\)	\(13\)

30. Find the maximum value of the objective function: \(F(x) = x + y - 3\); subject to constraints: \(x - y \le 4\), \(x + y \ge 6\) and \(x \ge 0\), \(y \ge 0\).
31. In a circular meadow, Ram, Hari and Shyam are standing at the points \(A(0, 0)\), \(B(2, 0)\) and \(C(0, 4)\) respectively, then
    1. If Bindalal is setting at the centre of meadow then find the centre of meadow.
    2. Are they standing equidistant from the centre of meadow? If so how far they are standing from centre of meadow?
    3. Find the equation of circular meadow.
32. Prove vectorially that the diagonals of rectangle PQRS are always equal to each other.
33. \(H(2, 2)\), \(A(6, 2)\), \(R(7, 4)\) and \(I(3, 4)\) are the vertices of a parallelogram \(HARI\). Find the coordinates of vertices of images of \(HARI\) under the rotation of positive \(90^\circ\) about origin followed by enlargement \(E((0, 0); 3)\). Represent the object and image in the same graph paper.