Opt Math

Define trigonometric function.

What is arithmetic mean between two numbers a and b.

Write the name of the set of numbers which is continuous.

If matrix A = (a b / c d), what is the value of |A|?

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.

Which geometric figure will be formed if a plane intersects a cone parallel to its base? Write.

Express sin2A in terms of tanA.

Define angle of elevation.

What is the scalar product of two vectors a and b if the angle between them is θ?

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.

If 2x³ − 7x² + x + 10 = (x−1)Q(x) + R, find the remainder R and quotient Q(x).

Write down the inequality represented by the shaded region in the adjoining figure.

Find the determinants D₁ and D₂ of coefficient of x and y by using Cramer’s rule from the equations 4x − 5y = 2 and 3x + 4y = 48.

Find the slopes of two straight lines having equations 3x + 4y + 5 = 0 and 6x + 8y + 7 = 0 and write the relationship between them.

Convert sin6A · cos4A into sum or difference of sine or cosine.

If 2sin2θ = √3, find the value of θ. (0° ≤ θ ≤ 180°)

O is the origin in the given figure. If ā and b̄ are the position vector of the points A and B, show that the position vector of point P is p= ½(a+b).

In a series, the first quartile (Q₁) = 35 and third quartile (Q₃) = 75, find the quartile deviation and its coefficient.

If two functions are f(x) = (2x+5)/8 and g(x) = 3x − 4, find (f ∘ g)⁻¹(3).

Solve by graphical method:
2x² + x − 6 = 0.

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?

Use matrix method to solve the following systems of equations:
3x + 5y = 11, 2x − 3y = 1

Find the equations of the pair of lines represented by the equation 6x² − xy − y² = 0 and also find the angle between them.

Prove that: tanA + 2tan2A + 4cot4A = cotA

If A + B + C = π°, prove that: sin²A − sin²B + sin²C = 2sinA cosB sinC

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 m high situated at the top of a tower are observed 60° and 45° respectively. Find the height of the tower and the distance between the base of the tower and point of observation.

27. 

Find the mean deviation and its coefficient of the given data.

Find the standard deviation from given data.

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.

In the given figure, X and Y are the center of circles A and B respectively. Circle A passes through the centre Y of the circle B. If the equation of the circle B is x² + y² − 4x + 6y − 12 = 0 and the coordinates of X is (−4, 5), find the equation of the circle A.

By using vector method, prove that the quadrilateral formed by joining the midpoints of adjacent sides of a quadrilateral is a parallelogram.

The image of the triangle A is A' and image of A' is A'' in the given graph.

(a) By what transformation the image of the triangle A is A'? Write with reason.
(b) By what transformation the image of the triangle A' is A''? Write with reason.
(c) Write the name of transformation which denotes the combined transformation of above two transformations? Write with reason.