If for any 2 x 2 square matrix A, `A(adj A) = [(8,0), (0,8)]`, then write the value of |A|

Concept: Types of Matrices

Determine the value of 'k' for which the follwoing function is continuous at x = 3

`f(x) = {(((x+3)^2-36)/(x-3), x != 3), (k, x =3):}`

Concept: Concept of Continuity

Find `int (sin^2 x - cos^2 x)/(sin xcosx) dx`

Concept: Indefinite Integral by Inspection

Find the distance between the planes 2x - y + 2z = 5 and 5x - 2.5y + 5z = 20

Concept: Shortest Distance Between Two Lines

If A is a skew symmetric matric of order 3, then prove that det A = 0

Concept: Symmetric and Skew Symmetric Matrices

Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`

Concept: Increasing and Decreasing Functions

The Volume of cube is increasing at the rate of 9 cm 3/s. How fast is its surfacee area increasing when the length of an edge is 10 cm?

Concept: Rate of Change of Bodies Or Quantities

Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R

Concept: Increasing and Decreasing Functions

The x-coordinate of a point of the line joining the points P(2,2,1) and Q(5,1,-2) is 4. Find its z-coordinate

Concept: Vector and Cartesian Equation of a Plane

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

Concept: Independent Events

Two tailors, A and B, earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP

Concept: Linear Programming Problem and Its Mathematical Formulation

Find `int dx/(5 - 8x - x^2)`

Concept: Properties of Indefinite Integral

If `tan^(-1) (x- 3)/(x - 4) + tan^(-1) (x +3)/(x + 4) = pi/4`, then find the value of x.

Concept: Indefinite Integral by Inspection

Using properties of determinants, prove that

`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`

Concept: Properties of Determinants

Find matrix A such that `((2,-1),(1,0),(-3,4))A = ((-1, -8),(1, -2),(9,22))`

Concept: Order of a Matrix

if `x^y + y^x = a^b`then Find `dy/dx`

Concept: Derivatives of Implicit Functions

If e^{y} (x + 1) = 1, show that `(d^2y)/(dx^2) =((dy)/(dx))^2`

Concept: Second Order Derivative

Find `int (cos theta)/((4 + sin^2 theta)(5 - 4 cos^2 theta)) d theta`

Concept: Properties of Indefinite Integral

Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x) dx`

Concept: Definite Integral as the Limit of a Sum

Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`

Concept: Properties of Definite Integrals

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

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations

Show that the points A, B, C with position vectors `2hati- hatj + hatk`, `hati - 3hatj - 5hatk` and `3hati - 4hatj - 4hatk` respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle

Concept: Introduction of Product of Two Vectors

Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.

Concept: Scalar Triple Product of Vectors

There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean 'and variance of X.

Concept: Random Variables and Its Probability Distributions

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chos~n at random from the school and he was found ·to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer

Concept: Baye'S Theorem

Maximise Z = x + 2y subject to the constraints

`x + 2y >= 100`

`2x - y <= 0`

`2x + y <= 200`

Solve the above LPP graphically

Concept: Graphical Method of Solving Linear Programming Problems

Determine the product `[(-4,4,4),(-7,1,3),(5,-3,-1)][(1,-1,1),(1,-2,-2),(2,1,3)]` and use it to solve the system of equations x - y + z = 4, x- 2y- 2z = 9, 2x + y + 3z = 1.

Concept: Types of Matrices

Consider `f:R - {-4/3} -> R - {4/3}` given by f(x) = `(4x + 3)/(3x + 4)`. Show that f is bijective. Find the inverse of f and hence find `f^(-1) (0)` and X such that `f^(-1) (x) = 2`

Concept: Inverse of a Function

Let A = Q x Q and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∈ A. Determine, whether * is commutative and associative. Then, with respect to * on A

1) Find the identity element in A

2) Find the invertible elements of A.

Concept: Concept of Binary Operations

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

Concept: Maxima and Minima

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4 , 1), B (6, 6) and C (8, 4).

Concept: Area Under Simple Curves

Find the area enclosed between the parabola 4y = 3x^{2} and the straight line 3x - 2y + 12 = 0.

Concept: Area Under Simple Curves

Find the particular solution of the differential equation `(x - y) dy/dx = (x + 2y)` given that y = 0 when x = 1.

Concept: Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

Find the coordinates of the point where the line through the points (3, - 4, - 5) and (2, - 3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2,- 3) and (0, 4, 3)

Concept: Plane - Equation of a Plane in Normal Form

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is `1/x^2 + 1/y^2 + 1/z^2 = 1/p^2`

Concept: Plane - Intercept Form of the Equation of a Plane