OPT 18

    Group A

  1. a) When a polynomial \( p(x) \) is divided by linear polynomial \( g(x) = x - m \), in what condition \( (x - m) \) is a factor of \( p(x) \)?
    b) Write down the inequality represented by the given shaded portion in the diagram alongside when a straight line meets the axes at \( A(3, 0) \) and \( B(0, -2) \) respectively.
  2. a) Write down \( \lim_{x \to a^-} f(x) \) in a sentence.
    b) Under what condition, a square matrix has no inverse?
  3. a) If \( m_1 \) and \( m_2 \) be the slopes of the straight equation \( ax^2 + 2hxy + by^2 = 0 \) and \( m_1 \cdot m_2 = -1 \), what is the relation between 'a' & 'b'?
    b) Which conic section is created when a cone is cut parallel to its generator not passing through the vertex?
  4. a) If \( Q \leq \cos A \leq R \), 'Q' and 'R' represent the whole numbers and 'A' represents the standard angle, what are the extreme possible values of 'Q' and 'R'?
    b) Express \( \cos 8M - \cos 8N \) into product form.
  5. a) If \( \hat{i} \) and \( \hat{j} \) the unit vectors along X-axis and Y-axis respectively; find the value of \( \hat{i} \cdot \hat{j} \) and also \( \hat{i} \cdot \hat{i} \)
    b) If the center of inversion circle is \( O(0, 0) \) radius of the circle is 'r'; any point is \( P(x, y) \) and inversion point is \( P'(x', y') \); Write the co-ordinates of \( P'(x', y') \).

    6. Group B

  6. a) If \( f = \{(1, 3), (0, 0), (-1, -3)\} \) and \( g = \{(0, 2), (-3, -1), (3, 5)\} \), show the function \( g \circ f \) in an arrow diagram and find it in the form of ordered pair.
    b) If \( 2x^3 - 4x + 3 = (x - 3) \cdot Q(x) + R \) find the values of \( Q(x) \) and \( R \).
  7. a) If the sum of two numbers \( x \) and \( y \) is 20 and their difference is 10, express this relation in matrix form for finding the values of \( x \) and \( y \). Also find the determinant of the associated matrix.
    b) If the straight lines \( (a^2 - b^2)x = (p + q)y \) and \( (p^2 - q^2)x + (a + b)y = 0 \) are orthogonal; prove that: \( (a - b)(p - q) = 1 \).
  8. a) Prove that: \( \frac{1 + \tan^2 (45^\circ - \theta)}{1 - \tan^2 (45^\circ - \theta)} = cosec 2\theta \).
    b) For \( 0^\circ \leq x \leq 360^\circ \), find the degree measure of \( x \) when \( \cos 2x - \sin x = 0 \).
  9. a) Find the angle between two vectors \( \vec{a} \) and \( \vec{b} \) when \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \).
    b) In a continuous data if the mean value \( (\bar{x}) = 58 \), sum of the frequencies \( (\Sigma f) = 20 \) and \( \Sigma f|D| = 580 \); find the coefficient of mean deviation.

    10. Group C

  10. If \( f(x) = x^2 - 2x, g(x) = 2x + 3 \) and \( fog^{-1}(x) = 3 \), find the value of \( x \).
  11. \( x + \frac{2}{3}x + \frac{4}{9}x + \dots + \frac{128}{2187}x \) has 8 terms. Find \( x \) and also find the numerical sum of the series.
  12. Given that: \( f(x) = \begin{cases} 2x^2 + 1, & x < 3 \\ 5, & x = 3 \\ 6x + 1, & x > 3 \end{cases} \) Investigate the continuity; if not continuous, suggest what can continue the function \( f(x) \)?
  13. Solve by using Cramer's rule \( \frac{2}{3}x + y = 1 \) and \( \frac{1}{2}x - y = -\frac{7}{2} \).
  14. Find a pair of equation of straight lines represented by \( x^2 - 2xycosec\theta + y^2 = 0 \) and also prove that the angle between them is \( \frac{\pi}{2} \pm \theta \).
  15. Prove that: \( \cos^8 \frac{A}{2} + \sin^8 \frac{A}{2} = 1 - \sin^2 A + \frac{1}{8} \sin^4 A \).
  16. If \( A + B + C = \pi \), prove that: \(\cos^2 A + \cos^2 B + 2\cos A \cos B \cos C = \sin^2 C\).
  17. The angle of elevation to the top of an incomplete tower from the distance of 100 m from its base is found to be 45°. What height should the tower be raised so that the elevation may changes to 60°?
  18. Find a 2×2 matrix which transforms a quadrilateral SPDF of vertices S(1, 0), P(0, 1), D(2, 4), F(3, 2) into a new position S'P'D'F' with vertices S'(1, 0) P'(2, -1), D'(10, -4) & F'(-7, -2).
  19. Find the quartile deviation from the given data:
    Class 20-30 30-40 40-50 50-60 60-70 70-80
    Frequency 10 20 11 30 14 15
  20. Find out the standard deviation and its co-efficient from the data given below:
    Weight in kg. 30-35 35-40 40-45 45-50 50-55
    No. of students 14 20 25 16 5

    21. Group D

  21. A quadratic equation is given by: \( x^2 - 2x - 3 = 0 \)
    1. Find the co-ordinates of the vertex of the given parabola.
    2. Find the equation of symmetry line.
    3. Solve the quadratic equation graphically to find the values of \( x \).
  22. A circle has its radius 4 units. A straight line \( 3x + 4y + 28 = 0 \) is tangent to the circle. If the center of the circle lies on the line \( 13x + 4y - 32 = 0 \), find the equation of the circle.
  23. ABCD is a quadrilateral with equal sides. AC and BD are its two diagonals. Prove by vector method that AC and BD bisect at right angles.
  24. \( E_1 \) denotes an enlargement about the center (3, 4) with a scale factor of 2 and \( R_2 \) denotes a reflection of the line \( Y \)-axis. \( \triangle PQR \) with the vertices P(1, 2), Q(5, 4) and R(4, -2) is mapped on the \( \triangle P_2Q_2R_2 \) under the transformation \( E_1 \circ R_2 \). Find the coordinates of the vertices of the images in both cases. Show the necessary sketches on the same graph paper.

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OPT 21

Group A If a polynomial \( f(x) \) divided by \( (px + q) \). What is its remainder? What is the Geometric mean between the number 'a' and 'b'? Write a set of number which is continuous in number line. Define singular matrix, with example. Which geometrical figure will be formed if a plane intersects a cone parallel to its base? What is the angle between a pair of line represented by the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) Express \(\cos A\) in terms of \(\tan \frac{A}{2}\). If \(\cos \theta = \frac{4}{5}\), find the value of \(\cos 3\theta\). If \( \vec{a} = (x_1, y_1) \) and \( \vec{b} = (x_2, y_2) \) what is the scalar product of \( \vec{a} \) and \( \vec{b} \). In a circle with centre 'O' and radius \( r \), P is the inversion point...
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