OPT 1

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

    1. Define trigonometric function.
    2. What is arithmetic mean between two numbers 'a' and 'b'?
    3. Write the name of the set of numbers which is continuous.
    4. If matrix A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), what is the value of |A|?
    5. If the angle between two straight lines is θ and their slopes are m₁ and m₂ respectively, write the formula to find the value of tanθ.
    6. Which geometric figure will be formed if a plane intersects a cone parallel to its base? Write.
    7. Express sin2A in terms of tanA.
    8. Define angle of elevation.
    9. What is the scalar product of two vectors \(\vec{a}\) and \(\vec{b}\) if the angle between them is θ?
    10. If P' is the image of P and r is radius of circle with centre O in an inversion transformation, write the relation of OP, OP' and r.

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

    11. If \(2x^3 - 7x^2 + x + 10 = (x - 1).Q(x) + R\), find the remainder R and quotient Q(x).
    12. Write the inequality represented by the shaded region in the adjoining figure.
    13. Find the determinants \(D_1\) and \(D_2\) of coefficient of \(x\) and \(y\) by using Cramer's rule from the equations \(4x - 5y = 2\) and \(3x + 4y = 48\).
    14. Find the slopes of two straight lines having equations \(3x + 4y + 5 = 0\) and \(6x + 8y + 7 = 0\) and write the relationship between them.
    15. Convert \(\sin 6A \cos 4A\) into sum or difference of sine or cosine.
    16. If \(2\sin 2\theta = \sqrt{3}\), find the value of \(\theta\). \([0^\circ \le \theta \le 180^\circ]\)
    17. If \(O\) is the origin in the given figure, if \(\vec{a}\) and \(\vec{b}\) are the position vectors of points \(A\) and \(B\), show that the position vector of point \(P\) is \(\vec{p} = \frac{1}{2}(\vec{a} + \vec{b})\).
    18. In a series, the first quartile \((Q_1) = 35\) and third quartile \((Q_3) = 75\), find the quartile deviation and its coefficient.

      19. Group 'C' \([11 \times 3 = 33]\)

    19. If two functions are \(f(x) = \frac{2x + 5}{8}\) and \(g(x) = 3x - 4\), find \((fog)^{-1}(3)\).
    20. Solve by graphical method: \(2x^2 + x - 6 = 0\).
    21. For a real valued function \(f(x) = 2x + 3\), find the values of \(f(2.99)\), \(f(3.01)\) and \(f(3)\). Is this function continuous at \(x = 3\)?
    22. Use matrix method to solve the following systems of equations: \(3x + 5y = 11\), \(2x - 3y = 1\).
    23. Find the equations of the pair of lines represented by the equation \(6x^2 - xy - y^2 = 0\) and also find the angle between them.
    24. (Prove that): \(\tan A + 2\tan 2A + 4\cot 4A = \cot A\).
    25. If \(A + B + C = \pi^c\), prove that: \(\sin 2A - \sin 2B + \sin 2C = 2\sin A \cos B \sin C\).
    26. From a place at the ground level in front of a tower, the angle of elevations of the top and bottom of flag staff \(6\text{m}\) high situated at the top of a tower are \(60^\circ\) and \(45^\circ\) respectively. Find the height of the tower and the distance between the base of the tower and point of observation.
    27. Find the \(2 \times 2\) matrix which transforms unit square \(\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) to a parallelogram \(\begin{pmatrix} 0 & 3 & 4 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix}\).
    28. Find the mean deviation and its coefficient of the given data.
      Marks obtained 0-10 10-20 20-30 30-40 40-50
      No. of students 2 3 6 5 4
    29. Find the standard deviation from given data.
      Age 0-10 10-20 20-30 30-40 40-50 50-60
      No. of Persons 4 6 10 20 6 4

      30. Group 'D' \([4 \times 4 = 16]\)

    30. The sum of three terms in an arithmetic series is \(24\). If \(1\), \(6\) and \(18\) are added to them respectively, the results are in geometrical series, find the terms.
    31. In the given figure, \(X\) and \(Y\) are the center of circles \(A\) and \(B\) respectively. Circle \(A\) passes through the center \(Y\) of the circle \(B\). If the equation of the circle \(B\) is \(x^2 + y^2 - 4x + 6y - 12 = 0\) and the coordinates of \(X\) is \((-4, 5)\), find the equation of the circle \(A\).
    32. By using vector method, prove that the quadrilateral formed by joining the midpoints of adjacent sides of a quadrilateral is a parallelogram.
    33. The image of the triangle \(A\) is \(A'\) and image of \(A'\) is \(A''\) in the given graph.
      1. By what transformation is triangle \(A\) mapped to \(A'\)? Write with reason.
      2. By what transformation is the image of triangle \(A'\) mapped to \(A''\)? Write with reason.
      3. Write the name of transformation which denotes the combined transformation of above two transformations? Write with reason.

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