Date & Time: 19th March 2017, 12:30 pm

Duration: 3h

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):}`

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.03] Matrices

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

Chapter: [0.11] Three - Dimensional Geometry

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

Chapter: [0.07] Integrals

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

Chapter: [0.07] Integrals

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

Chapter: [0.12] Linear Programming

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.

Chapter: [0.13] Probability

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

Chapter: [0.11] Three - Dimensional Geometry

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

Chapter: [0.06] Applications of Derivatives

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

Chapter: [0.06] Applications of Derivatives

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

Chapter: [0.03] Matrices

The volume of a sphere is increasing at the rate of 8 cm^{3}/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.

Chapter: [0.06] Applications of Derivatives

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.

Chapter: [0.13] Probability

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

Chapter: [0.1] Vectors

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

Chapter: [0.13] Probability

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

Chapter: [0.07] Integrals

Using properties of determinants, prove that

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

Chapter: [0.04] Determinants

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

Chapter: [0.03] Matrices

if x^{x}+x^{y}+y^{x}=a^{b}, then find `dy/dx`.

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.07] Integrals

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

Chapter: [0.07] Integrals

Solve the following linear programming problem graphically :

Maximise Z = 7x + 10y subject to the constraints

4x + 6y ≤ 240

6x + 3y ≤ 240

x ≥ 10

x ≥ 0, y ≥ 0

Chapter: [0.12] Linear Programming

Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`

Chapter: [0.07] Integrals

if `veca = 2hati - hatj - 2hatk " and " vecb = 7hati + 2hatj - 3hatk`, , then express `vecb` in the form of `vecb = vec(b_1) + vec(b_2)`, where `vec(b_1)` is parallel to `veca` and `vec(b_2)` is perpendicular to `veca`

Chapter: [0.1] Vectors

Find the general solution of the differential equation `dy/dx - y = sin x`

Chapter: [0.09] Differential Equations

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).

Chapter: [0.08] Applications of the Integrals

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

Chapter: [0.08] Applications of the Integrals

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

Chapter: [0.09] 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)

Chapter: [0.11] Three - Dimensional Geometry

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`

Chapter: [0.11] Three - Dimensional Geometry

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`

Chapter: [0.01] Relations and Functions

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.

Chapter: [0.01] Relations and Functions

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find *A*^{−1}. Using A^{−1} solve the system of equations

2x – 3y + 5z = 11

3x + 2y – 4z = – 5

x + y – 2z = – 3

Chapter: [0.04] Determinants

A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Chapter: [0.06] Applications of Derivatives

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