OPT 15

Group A

1. a) What is y-intercept of \( y = \cos x \)?
   b) If the first term \( = a \), last term \( = b \), common difference \( = d \) and number of means \( = n \), write the formula to find \( d \).
2. a) Write the interval notation of \( \{x: -3 < x < 4\}\)
   b) If the matrix A is an identity matrix of \( 2 \times 2 \) then what is the determinant of A?
3. a) If two straight lines \( y = a_1 x + c_1 \) and \( y = a_2 x + c_2 \) are perpendicular to each other then what is the value of \( a_1 \times a_2 \)?
   b) If the intersection plane is parallel to its generator of a right circular cone, which conic section does it form? Write it.
4. a) Write the formula of \( \tan \alpha \) in term of \( \tan \frac{\alpha}{3} \).
   b) \( A + B + C = \pi \), express \( \cos(A+B) \) in term of the angle \( C \).
5. a) \( \vec{a} \cdot \vec{b} = 0 \) then what is the relation between \( \vec{a} \) and \( \vec{b} \)?
   b) Find the distance of inversion of a point Q which is at distance 2 units from the center O of a circle with radius 4 units.

Group B

6. a) If \( \vec{f} = ((1, 3), (0, 0), (-1, -3)) \) and \( \vec{g} = ((0, 2), (-3, -1), (3, 5), (2, 2)) \), write \( \vec{g} \circ \vec{f} \) in ordered pair form by representing in a mapping diagram.
   b) Find the vertex of the parabola \( y = x^2 + 4x + 3 \)
7. a) If the inverse of the matrix \( \begin{pmatrix} 8 & 5 \\ 3 & x \end{pmatrix} \) is the matrix \( \begin{pmatrix} 2 & -5 \\ -y & 8 \end{pmatrix} \), find the value of \( x \) and \( y \).
   b) Find the obtuse angle between the lines \( 12x^2 - 23xy + 5y^2 = 0 \)
8. a) If \( \cos \theta = \frac{1}{2} \left( m + \frac{1}{m} \right) \), show that \( \cos 2\theta = \frac{1}{2} \left( \frac{m^4 + 1}{m^2} \right) \)
   b) Solve: \( \sin^2 \theta - \sin \theta + \frac{1}{4} = 0 \)    [\( 0^\circ \leq \theta \leq 90^\circ \)]
9. a) If \( \vec{a} + 2\vec{b} \) and \( 5\vec{a} - 4\vec{b} \) are two vectors such that their scalar product is -1 and \( \vec{a} \) and \( \vec{b} \) unit vectors, find \( \vec{a} \cdot \vec{b} \).
   b) If the first quartile of any data is 43 and the quartile deviation is 6.5, find the coefficient of quartile deviation.

Group C

10. Solve: \( x^3 - 11x^2 + 31x - 21 = 0 \).
11. Find the maximum value of \( P = 4x + 2y \) under the constraints:
    \( x + y \leq 4 \), \( x - 2y \leq 1 \), \( x \geq 0 \), \( y \geq 0 \)
12. Examine the continuity or discontinuity of the function:
    \( f(x) = \begin{cases} 2x - 1 & \text{for } x < 2 \\ 3 & \text{for } x = 3 \\ x + 1 & \text{for } x > 2 \end{cases} \) at \( x = 2 \) by calculating left hand limit, right hand limit and functional value.
13. Solve the equations by matrix method: \( x + 2y = 4 \) and \( 5y + 2x = 9 = 0 \).
14. The equation of diagonal XZ in rhombus WXYZ is \( 4x - 3y + 10 = 0 \) and coordinates of W are \( (3, -2) \) then find the equation of other diagonal WY.
15. Without using table or calculator, find the value of: \( \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ \)
16. If \( A + B + C = \pi \), Prove that:
    \( \sin^2 \frac{A}{2} + \sin^2 \frac{B}{2} + \sin^2 \frac{C}{2} = 1 - 2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \)
17. The angles of depression of top and foot of a lamp post observed from the roof of a 60m high house are found to be \( 30^\circ \) and \( 60^\circ \) respectively. Find the height of the lamp post and its distance from the house.
18. If the matrix \( \begin{pmatrix} a & -3 \\ 4 & b \end{pmatrix} \) transforms a unit square to the parallelogram \( \begin{pmatrix} 0 & c & -1 & -3 \\ 0 & 4 & d & 5 \end{pmatrix} \) then find the value of \( a, b, c, d \).
19. Find the mean deviation from the coefficient from median of the given data:

    Class interval	10–20	20–30	30–40	40–50	50–60
    Frequency	5	8	10	4	5

20. Find the standard deviation and coefficient of variation from the following data:

    Class interval	0–10	10–20	20–30	30–40	40–50
    Frequency	4	6	10	20	6

21. Mr. Suyog surveys the number of pens which are sold in two stationary shops. The pens of both shops are sold in a week. In the first shop, 60 pens are sold in the first day and 6 pens are sold more in everyday as comparison of previous day. Similarly, in the second shop, 5 pens are sold in the first day and double numbers of pens are sold in everyday as comparison of previous day. On the basis of the information answer the following questions.
    A. How many pens are sold in first shop on the last day of the week?
    B. How many pens are sold in second shop on the fifth day of the week?
    C. In which shop and how many pens are sold more? Find it.
22. Manya, Kusum and Serena are sitting in a playground to make a circular path. The coordinates of their position on the circumference on the circle are \( (5, 7) \), \( (-1, 7) \) and \( (5, -1) \). Find the coordinates of the point equidistance from their position. Also find the equation of the locus.
23. Prove by vector method that the middle point of hypotenuse of a right angled triangle is equidistance from its vertices.
24. A triangle with vertices \( A(1, 2) \), \( B(4, -1) \) and \( C(2, 5) \) is rotated about origin through \( 90^\circ \) in anti-clockwise direction. The image so obtained is reflected on the line \( x=0 \). Find the vertices of image triangles and present all the triangles on the same graph paper.