OPT 19

Group A

1. 1. If \( f: A \to B \) and \( g: B \to C \) be the two functions then what will denote the composite function from \( A \to C \)?
   2. What will be the remainder if the polynomial \( f(x) \) is divided by \( -dx + c \)?
2. 1. Write the geometrical form of a numbers lying between \( -\infty \) to \( \infty \)?
   2. Show that \( A = \begin{pmatrix} 5 & 6 \\ 4 & 2 \end{pmatrix} \) is non singular matrix.
3. 1. Define parabola
   2. Find the slope of the line which is perpendicular to the line \( 3x + 5y + 1 = 0 \).
4. 1. Express \( \sin A \) in terms of \( \sin \frac{A}{3} \).
   2. Express \( 2\sin A \sin B \) in the form of sum or difference
5. 1. If the angle between \( \vec{p} \) and \( \vec{q} \) is \( \theta \), then write the scalar product of \( \vec{p} \) and \( \vec{q} \).
   2. If the inversion of circle \( (x - h)^2 + (y - k)^2 = r^2 \), then write the coordinates of the line inversion point \( P' \) of the point \( P(x, y) \)

Group B

6. 1. If \( f(x) = 3x - 2 \) and \( fog^{-1}(x) = 6x - 2 \) find \( g^{-1}(x) \)
   2. Find the co-ordinate of the vertex of a quadratic function \( y = x^2 + 2x - 3 \)
7. 1. Solve by crammer's rule \( 3x + 2y = 13 \) and \( 2y + 5 = 3x \)
   2. If the lines \( a_1 x + b_1 y + c_1 = 0 \) and \( a_2 x + b_2 y + c_2 = 0 \) are perpendicular to each other, prove that \( \frac{a_1}{b_1} + \frac{a_2}{b_2} = 0 \).
8. 1. Prove that: \( \frac{1 - \tan^2 (45^\circ - A)}{1 + \tan^2 (45^\circ - A)} = \sin 2A \)
   2. Solve \( 7\sin^2 x + 3\cos^2 x = 4 \) \( (0^\circ \leq x \leq 180^\circ) \)
9. 1. In the adjoining figure, the position vector of A and D are \( 3\hat{i} + 2\hat{j} \) and \( 3\hat{i} + 5\hat{j} \) respectively. Find the position vector of centroid G of \( \triangle ABC \).
   2. In a data, the quartile deviation and its coefficient are 14 and \( \frac{7}{22} \) respectively. Find the third quartile.

Group C

10. Solve: \( x^3 - 21x + 20 = 0 \).
11. Find the maximum value of \( P = 3x + 2y \) in the following condition \( x + y \geq 0 \), \( x - y \leq 0 \), \( y \leq 2 \) and \( x \geq -1 \).
12. A function \( f(x) \) is defined as \( f(x) = \begin{cases} \dfrac{3x^2 - 12}{x - 2} & x \neq 2 \\ k + 10 & x = 2 \end{cases} \). Find the value of \( k \) so that \( f(x) \) is continuous at \( x = 2 \).
13. Solve by matrix method \( x - \dfrac{2}{y} = 4 \) and \( \dfrac{3}{y} + 2x - 1 = 0 \).
14. Find the equation of circle having equation of two diameters \( 3x + 2y = 1 \) and \( 3x - 2y = 5 \) which touches the straight line of equation \( 4x + 2y + 7 = 0 \).
15. Prove that: \( \tan A + 2\tan 2A + 4\tan 4A + 8\cot 8A = \cot A \)
16. If \( A + B + C = \pi = 180^\circ \), then prove that: \( \frac{\sin(B+C-A) + \sin(C+A-B) + \sin(A+B-C)}{4 \cos\frac{A}{2} \cos\frac{B}{2} \cos\frac{C}{2}} = 8 \sin\frac{A}{2} \sin\frac{B}{2} \sin\frac{C}{2} \)
17. Two lamp posts are in equal height. A boy standing mid way between them observes the angle of elevation of either post to be 30°. After walking 15 ft towards one of them he observes its elevation to be 60°. Find the height of post and the distance between them.
18. Find 2×2 matrix which transforms unit square to a parallelogram \begin{pmatrix} 0 & 3 & 4 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix}
19. Find the mean deviation from median and its coefficient of the given data.

    Class	0-10	10-20	20-30	30-40	40-50
    Frequency	2	3	6	5	4

20. Find the coefficient of S.D. of the given data.

    Age	0-8	8-16	16-24	24-32	32-40	40-48
    No. of Persons	2	3	8	2	4	2

Group D

21. The Sum of the first four terms in a geometric series is 30 and the sum of last four terms is 960. If the first and last terms are 2 and 512, find the common ratio.
22. Define homogenous equation. Find the single equation of a pair of straight lines passing through the origin and perpendicular to the lines represented by \( ax^2 + 2hxy + by^2 = 0 \).
23. By using vector method, prove that quadrilateral formed by joining the mid-point of adjacent sides of a quadrilateral is a parallelogram.
24. If \( R_1 \) represents the reflection on the line \( x = 0 \) and \( R_2 \) represents the reflection on the line \( x = 3 \) then which single transformation does the combine transformation \( R_1 \circ R_2 \) represent? Write it using single transformation, find the image of \( \triangle ABC \), where A(3, 1) B(1, 3) and C(2, -1). Also present the object and the image in same graph.