OPT 20

Group A

Under what condition the function \( f(x) = mx + c \) become an identity function.
What is the sum of n-natural numbers?
Which set of number is continuity?
What types of equations are solved by Cramer’s rule?
Find the slope of straight line parallel to y-axis.
Define hyperbola conic section.
Express \(\cos 2A\) in terms of \(\tan A\).
Find the acute angle of equation \(\sec \theta = \sqrt{2}\)
From the given figure write the relation between \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\).
Write inversion point A \((x, y)\) with respect to the circle having radius \(r\) and centre at \((h, k)\)

Group B

Find the quadratic equation having roots 6 and -7.
If 4, m, n, k, 1024 are in G.P., then find the value of m, n, and k.
For what value of 'a' the matrix \( P = \begin{pmatrix} 3a + 2 & 7 \\ 4 & a - 1 \end{pmatrix} \) is a singular matrix?
Find the separate equations of straight lines represented by the equation \( x^2 - 2xy \cos 2\alpha + y^2 = 0 \).
If \( \cos \frac{\alpha}{3} = \frac{1}{2} \left( m + \frac{1}{m} \right) \) then find the value of \( \cos \alpha \).
Prove that: \( \frac{\sin 80^\circ + \cos 80^\circ}{\sin 80^\circ - \cos 80^\circ} = \cot 35^\circ \)
If \( \hat{a} + \hat{b} + \hat{c} = 0, |\hat{a}| = 3, |\hat{b}| = 3 \) and \( |\hat{c}| = 7 \) then find the angle between the \( \hat{a} \) and \( \hat{b} \).
The third quartile of a data is 58. If the coefficient of quartile deviation is \( \frac{7}{22} \), find the first quartile and inter-quartile range of the data.

Group C

Solve for x: \( (x + 2)(x^2 - 6x + 13) - 20 = 0 \)
The sum of three numbers in geometric sequence is 56. If 1, 7 and 22 are subtracted from them respectively, the resulting numbers will form A.P. Find the numbers.
A function \( f(x) = \begin{cases} x^2 + 2 & \text{for } 0 \leq x \leq 3 \\ 4x - 1 & \text{for } x > 3 \end{cases} \) is \( f(x) \) is continuous at point \( x = 3 \)? Examine it.
Solve by matrix method: \( \frac{3x+5}{4} = \frac{7x+3y}{5} = 4 \)
Find the equation of perpendicular of the line segment joining the point (3, 5) and (-1, 3).
Prove that: \( \sec 4\theta + \operatorname{cosec} 8\theta + \operatorname{cosec} 16\theta = \cot 2\theta - \cot 16\theta \)
If \( A + B + C = \pi^c \) then Prove that: \( \cos \frac{A}{2} - \cos \frac{B}{2} + \cos \frac{C}{2} = 4 \cos \left( \frac{\pi + A}{4} \right) \cos \left( \frac{\pi - B}{4} \right) \cos \left( \frac{\pi + C}{4} \right) \)
A tower of radio station is divided by a point in the ratio of 9:1. From the top if both the parts of radio station is subtends equal angle at a point on ground level 600 m away from its bottom, then find the height of tower.
Find a 2×2 transformation matrix which transform a square PQRS with vertex P(2, 3), Q(4, 3), R(4, 5) and S(2, 5) into a square P'Q'R'S' with vertices P'(3, 2), Q'(3, 4), R'(5, 4) and S'(5, 2).

Find the mean deviation from median of given data:

C.I.	20-30	30-40	40-50	50-60	60-70
Frequency	4	6	10	8	2

Find the coefficient of variation of given data:

Marks obtained	20-30	30-40	40-50	50-60	60-70
No. of Students	5	11	12	10	2

Group D

Find the maximum of objective function: \( Z = 3x + 5y - 2 \) subject to the constraints: \( x + 2y \leq 10, 2x + y \leq 14, x \geq 0 \) and \( y \geq 0 \)
There are two points P(3, 7) and Q(5, 5) on the edge of circular pond. A bridge is made through the centre of the pond and the bridge represented by the equation \( x - 3y = 1 \). Find the locus of circular pond.
Using vector methods prove that the diagonals of rectangle are equal in length.
A \( \triangle LMN \) with vertices L(1, 2), M(3, 2) and N(3, 4) is reflected in y-axis. The triangle L'M'N' is reflected in the line \( x = 3 \) to give a second image \( \triangle L''M''N'' \). Find the coordinates of the image of \( \triangle LMN \). Show the \( \triangle LMN \) and \( \triangle L''M''N'' \) in same graph paper. Also state that the single transformation to represent this combination of reflection.

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