SEE 2081_RE2021A_OPT Math

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Group 'A' [10 * 1 = 10]

  1. Write the definition of cubic function.
  2. If the first term of an arithmetic series is 'a', common difference 'd' and the number of terms 'n', then write the formula to find the sum of the first n terms of the series.
  3. Write the name of the set of numbers which is continuous.
  4. In which condition does the inverse matrix of the given matrix does not exist? Write it.
  5. If the slopes of two lines are \(m_1\) and \(m_2\) respectively, then in which condition these two lines are perpendicular to each other? Write it.
  6. Which geometrical figure is formed if a plane surface intersects a cone parallel to its base? Write it.
  7. Express \(\sin 3A\) in terms of \(\sin A\).
  8. Express \(\sin A - \sin B\) in the product form.
  9. If j is a unit vector along Y-axis, what is the value of \((j)^2\)? Write it.
  10. The radius of the inversion circle is 'r' and the centre is 'O'. If the image of a point A is A', what is equal to \(OA \times OA'\)? Write it.

Group 'B' [8 * 2 = 16]

  1. If \(f(x) = x^3 + mx^2 - x + 7\) is divided by \(x - 3\), the remainder is 4. Find the value of m using remainder theorem.
  2. Present the inequality \(x - y \geq 2\) in the graph.
  3. If \(D = 10\), \(D_x = \begin{vmatrix} 4 & 2 \\ 2 & 6 \end{vmatrix}\) and \(D_y = \begin{vmatrix} 4 & 1 \\ -2 & 7 \end{vmatrix}\) then find the values of x and y.
  4. If two lines having equations \(3x - y + 7 = 0\) and \(kx + 2y + 8 = 0\) are parallel, find the value of k.
  5. Prove that: \(\frac{\sin 50^\circ - \sin 30^\circ}{\cos 30^\circ - \cos 50^\circ} = \cot 40^\circ\)
  6. If \(3 \tan^2 \theta - 9 = 0\) then find the value of \(\theta\). (\(0^\circ \leq \theta \leq 90^\circ\))
  7. If \(\vec{a} \cdot \vec{b} = 48\), \(|\vec{a}| = 6\sqrt{2}\) and \(|\vec{b}| = 8\) then find the angle between \(\vec{a}\) and \(\vec{b}\).
  8. If the value of first quartile (\(Q_1\)) of any data is 43 and the quartile deviation (Q.D.) is 6.5, find the coefficient of quartile deviation.

Group 'C' [11 * 3 = 33]

  1. If two functions are \(f(x) = \frac{2x - 5}{3}\) and \(g(x) = x + 4\), find \((f \circ g)^{-1}(3)\).
  2. Solve quadratic equation \(x^2 + 2x - 3 = 0\) by graphical method.
  3. If the function \(f(x) = \begin{cases} 3x - 1 & \text{for } x \geq 2 \\ x + 3 & \text{for } x < 2 \end{cases}\) is defined, is the function \(f(x)\) continuous at \(x = 2\)? Give reason.
  4. Solve by matrix method: \(4x - 3y = 11\) and \(3x + 7y + 1 = 0\).
  5. Find the equation of a circle having centre (3, 5) and passing through the centre of the circle \(x^2 + y^2 + 4x - 6y - 36 = 0\).
  6. Prove that: \(\frac{\cos^2 A - \sin^2 B}{\sin A \cos A + \sin B \cos B} = \cot(A + B)\)
  7. If \(A + B + C = 180^\circ\) then prove that: \(\cos A + \cos B + \cos C = 1 + 4 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}\)
  8. An observer observes the roof of the house 20 m high to the top and bottom of the television tower and found the angle of elevation and angle of depression to be \(45^\circ\) and \(30^\circ\) respectively. Find the height of the tower.
  9. The centre and scale factor of the enlargement E are (-3, -4) and 2 respectively. R represents the reflection in the line \(y = 0\).
  10. In which point the combined transformation \(E \circ R\) transforms a point P(x, y)? Find it.
  11. Transform the \(\triangle ABC\) with vertices A(2, 0), B(3, 1) and C(1, 1) using the combined transformation \(E \circ R\).
  12. Present the \(\triangle ABC\) and the image in the same graph paper.

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