Using integration find the area of the triangle formed by positive *x*-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

Concept: Area Under Simple Curves

Find the value of constant ‘k’ so that the function f (x) defined as

f(x) = `{((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}`

is continous at x = -1

Concept: Concept of Continuity

Show that the function f(x) = `{(x^2, x<=1),(1/2, x>1):}` is continuous at x = 1 but not differentiable.

Concept: Concept of Continuity

Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`

Concept: Area Under Simple Curves

If 1, `omega` and `omega^2` are the cube roots of unity, prove `(a + b omega + c omega^2)/(c + s omega + b omega^2) = omega^2`

Concept: Basic Concepts of Differential Equation

Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = pi/2, x != 0`

Concept: Basic Concepts of Differential Equation

Given that the observations are: (9, -4), (10, -3), (11, -1), (12, 0), (13, 1), (14, 3), (15, 5), (16, 8). Find the two lines of regression and estimate the value of y when x = 13·5.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression

Solve the following initial value problem:-

\[y' + y = e^x , y\left( 0 \right) = \frac{1}{2}\]

Concept: Basic Concepts of Differential Equation

Evaluate : `int sec^2x/(cosec^2x)dx`

Concept: Introduction of Integrals

Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.

Concept: Concept of Continuity

Evaluate : `int(x1+x^2)/(1+x^4)dx`

Concept: Introduction of Relations and Functions

Solve the differential equation `dy/dx = (x + y+2)/(2(x+y)-1)`

Concept: Introduction of Relations and Functions

Evaluate: `int_-6^3 |x+3|dx`

Concept: Introduction of Relations and Functions

Solve the following system of linear equation using matrix method:

`1/x + 1/y +1/z = 9`

`2/x + 5/y+7/z = 52`

`2/x+1/y-1/z=0`

Concept: Introduction of Matrices

Evaluate: `int_0^x (xtan x)/(sec x + tan x) dx`

Concept: Introduction of Integrals

Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person choose a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.

Concept: Introduction of Integrals

Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

Concept: Direction Cosines and Direction Ratios of a Line

Draw a rough sketch and find the area bounded by the curve x^{2} = y and x + y = 2.

Concept: Area Under Simple Curves

Find the equation of the regression line of y on x, if the observations (x, y) are as follows :

(1,4),(2,8),(3,2),(4,12),(5,10),(6,14),(7,16),(8,6),(9,18)

Also, find the estimated value of y when x = 14.

Concept: Lines of Regression of X on Y and Y on X Or Equation of Line of Regression

Write the negation of the following statements :

(a) Radha likes tea or coffee.

(b) `∃x cc` R such that x + 3 ≥ 10.

Concept: Introduction of Matrices