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 of P. If \( OP = 5 \) cm, \( OP' = 20 \) cm. find the value of \( r \).

Group B

If \( f(x) = 4x, g(x) = x + 1 \) and \( fog(x) = 20 \), then find 'x'.
If \( (4x + 5) = 12x + 18 \), find \( f^{-1}(x) \).
If \( A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}, B = \begin{pmatrix} 2 & m \\ -1 & 2 \end{pmatrix} \) and \( AB = I \), where \( I \) is 2×2 Identity matrix. Find value of m.
Find the angle between the pair of straight line represented by \( x^2 - 2xy cosec x + y^2 = 0 \)
Prove that: \( \cos^2 \left( \frac{\pi}{4} - \frac{\theta}{4} \right) - \sin^2 \left( \frac{\pi}{4} - \frac{\theta}{4} \right) = \sin \frac{\theta}{2} \)
Solve: \( 2 \cos^2 \theta + \sin \theta = 2 \) \([0^\circ \leq \theta \leq 180^\circ]\)
If \( \vec{a} + 2\vec{b} \) and \( 5\vec{a} - 4\vec{b} \) perpendicular to each other and \( \vec{a} \) and \( \vec{b} \) are unit vectors. Find the angle between \( \vec{a} \) and \( \vec{b} \).
If the first quartile of a grouped data is 15 and the quartile deviation is 30. Then find the coefficient of the Q.D.

Group C

Solve: \( x(x^2 + 3) = 2(3x^2 - 5) \)
In a geometric series if the sixth term is 16 times the second term and sum of first seven terms is \( \frac{127}{4} \) Find the positive common ratio and first term of the series
Prove that the function \( f(x) = \begin{cases} 3x - 1 & x < 2 \\ 5 & x = 2 \\ 2x + 1 & x > 2 \end{cases} \) continuous at \( x = 2 \).
Solve equations using Cramer’s rule \( \dfrac{6}{y} + \dfrac{10}{x} = 6 \) and \( \dfrac{3}{y} - \dfrac{21}{x} = -10 \)
The vertex of right angled isosceles triangle ABC is A (4, -1) and the equation of the base BC is \( 2x - 3y = 5 \) Find the equations of equal sides.
Prove that: \( \sin(60^\circ - \theta) \sin(60^\circ + \theta) = \frac{1}{4} \sin 3\theta \)
If \( A + B + C = \pi^c \) then prove that: \( \sin 3A + \sin 3B + \sin 3C = -4 \cos \frac{3A}{2} \cdot \cos \frac{3B}{2} \cdot \cos \frac{3C}{2} \)
The shadow of a tower on the ground is found to be 10.5 m longer when sun's altitude is 45° than when it was 60°. Find the height of the tower.
Find a 2×2 transformation matrix which transform a unit square to a parallelogram \begin{pmatrix} 0 & 2 & 5 & 3 \\ 0 & 4 & 6 & 2 \end{pmatrix}.

Find S.D from following data.

Mark Obtained	0-10	10-20	20-30	30-40	40-50
No. of Students	3	5	12	6	4

Find mean deviation and its coefficient.

x	0-20	20-40	40-60	60-80	80-100
f	2	3	4	5	6

Group D

Find the maximum value of the objective function \( z = 5x + 4y \) under the following constraints: \( x + y \geq 0, x - y \leq 0, x \geq 0, y \leq 2 \)
In \( \triangle ABC \), \( \angle ABC = 90^\circ \), and O is the mid point of side AC. Then, prove by vector method that \( OA = OB = OC \)
The center of a circle which passes through the origin and the point (4, 2) lies on the line \( x + y = 1 \), find the equation of the circle.
E denote the enlargement about the centre (3, 1) with a scale factor of 2 and R denotes a reflection on the line \( y = x \). Find the image of \( \triangle ABC \) having the vertices A(2, 3), B(4, 5) and C(1, -2) under the combined transformation EoR. Draw both figures on the same graph paper.

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