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

Concept: Introduction of Integrals

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

Concept: Introduction of Integrals

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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

Evaluate: `int 1/"x"^2 "sin"^2 (1/"x") "dx"`

Concept: Introduction of Integrals

Evaluate: `int_0^(pi/4) "log" (1 + "tan" theta) "d" theta`

Concept: Introduction of Integrals

Evaluate: ` int tan^3x "dx"`

Concept: Introduction of Integrals

Using De Moivre’s theorem, find the least positive integer n such that `((2i)/(1+i))^n` is a positive integer.

Concept: Introduction of Integrals

A company manufactures two types of toys A and B. A toy of type A requires 5 minutes for cutting and 10 minutes for assembling. A toy of type B requires 8 minutes for cutting and 8 minutes for assembling. There are 3 hours available for cutting and 4 hours available for assembling the toys in a day. The profit is ₹ 50 each on a toy of type A and ₹ 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit? Use linear programming to find the solution.

Concept: Introduction of Linear Programming

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Concept: Equation of a Line in Space

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 p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l_{1}.

Concept: Equation of a Line in Space

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: Differential Equations

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

Concept: Differential Equations

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: Differential Equations

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

Concept: Concept of Continuity