OPT 16

Group A

1. a) Define composite function.
   b) What will be the remainder if the polynomial \( f(x) \) is divided by \( x-c \)?
2. a) Under what condition, the function \( f(x) \) is said to be continuous at the point \( x=a \)? Write.
   b) Find the determinant of the matrix \( \begin{pmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{pmatrix} \).
3. a) Define conic section.
   b) Prove the lines represented by \( 4x^2 + 6xy - 4y^2 = 0 \) are perpendicular to each other.
4. a) Express \( \cos 3A \) in terms of \( \cos A \).
   b) Express \( 2\sin A \cos B \) in the form of difference or sum.
5. a) If \( \vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} \), find the value of \( \vec{a} \cdot \vec{b} \)
   b) If the inversion circle is \( x^2 + y^2 = r^2 \), then write the co-ordinates of the inversion point \( P' \) of the point \( p(x, y) \)

Group B

6. a) If \( x - 3 \) is a factor of polynomial \( x^3 + 4x^2 + kx - 30 \), find the value of \( k \).
   b) Which term of the series \( 5 + 9 + 13 + \dots \) is 85? Find it.
7. a) If the matrix \( \begin{pmatrix} x & 5 \\ 2 & -1 \end{pmatrix} \) and \( \begin{pmatrix} 2 & -5 \\ y & 1 \end{pmatrix} \) are inverse to each other, find \( x \) and \( y \).
   b) If the angle between the pairs of lines represented by the equation \( px^2 - 7xy - 15y^2 = 0 \) is 45°, find the value \( p \).
8. a) Prove: \( \frac{1 - \cos 2A}{\sin 2A} = \tan A \)
   b) Prove: \( 2 \cos(45^\circ + A) \cdot \sin(45^\circ - A) = \cos 2A \)
9. a) The vertex \( A(2, -1) \) and mid-point of opposite side is \( M(2, -4) \) of triangle ABC. Find the position vector of centroid of that triangle.
   b) In continuous data, first quartile is 20 and quartile deviation is 2.5 find the third quartile and the coefficient of quartile deviation.

Group C

10. Solve: \( 6x^3 - 5x = 6 + 17x^2 \)
11. If the sum of first two terms in GP is 9 and sum of first 4 terms is 45. Find the first term and common ratio. Also, find the 6th term.
12. Find the value of \( k \) if the function \( f(x) \) is continuous at \( x = 2 \) which is defined as \( f(x) = \begin{cases} 3x^2 - 2 & \text{for } x < 2 \\ kx & \text{for } x > 2 \end{cases} \)
13. Solve by matrix method: \( \frac{x}{2} + \frac{5y}{3} = -3 \) and \( \frac{x}{4} - 3y = 10 \)
14. If the equation of three sides AB, BC, CA of \( \triangle ABC \) are \( 5x - 3y + 2 = 0 \), \( x - 3y - 2 = 0 \) and \( x + y - 6 = 0 \) respectively. Find the equation of the line that passes through A and perpendicular to BC.
15. Prove: \( 16 \sin 10^\circ \cdot \sin 30^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = 1 \)
16. If \( A + B + C = \pi^c \), prove that \( \sin(B + C - A) + \sin(C + A - B) + \sin(A + B - C) = 4 \sin A \cdot \sin B \cdot \sin C \)
17. The angles of elevation of the top of a tower observed from two points on the same plane are found to be complementary. If the points are at a distance of 20m and 45m from the foot of the tower, find the height of tower.
18. Find a \( 2 \times 2 \) transformation matrix which transform a square ABCD with vertices \( A(2, 3) \), \( B(4, 3) \), \( C(4, 5) \) and \( D(2, 5) \) into a square \( A'B'C'D' \) with \( A'(3, 2) \), \( B'(3, 4) \), \( C'(5, 4) \), and \( D'(5, 2) \).
19. Find the value of quartile deviation of the given data:

    Class interval	0–8	8–16	16–24	24–32	32–40
    Frequency	12	13	15	14	11

20. Find the coefficient of standard deviation of the given data:

    Class interval	30–40	40–50	50–60	60–70	70–80
    Frequency	2	3	6	5	4

Group D

21. Solve the equation \( y = x^2 - 2x - 3 \) and \( y = -3 \) drawing the graph. Also, write the equation of symmetric line.
22. Find the equation of circle having equation of two diameters \( x + 2y = 5 \) and \( 3x - y = 1 \) which touches the straight line of equation \( 3x + 4y + 4 = 0 \).
23. Prove by vector method that mid-point of hypotenuse of a right angled triangle is equidistant from the all vertices.
24. A triangle with vertices \( A(3, 0) \), \( B(4, -1) \) and \( C(5, 2) \) are reflected on \( x=0 \) and \( y=3 \) line. Find the image of this combined transformation and represent on the same graph. Also, find the single transformation for this combined transformation.