Board Paper of Class 12-Humanities 2010 Maths (SET 2) - Solutions

Evaluate:

If, then for what value of α is A an identity matrix?

What is the principal value of?

What is the cosine of the angle which the vector makes with y-axis?

Write a vector of magnitude 15 units in the direction of vector

What is the range of the function?

Find the minor of the element of second row and third column (a₂₃) in the following determinant:

Write the vector equation of the following line:

What is the degree of the following differential equation?

If, then write the value of k.

Find all points of discontinuity of f, where f is defined as follows:

OR

Find, if

Prove the following:

OR

Prove the following:

On a multiple choice examination with three possible answers (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

Let * be a binary operation on Q defined by

Show that * is commutative as well as associative. Also find its identify element, if it exists.

Using elementary row operations, find the inverse of the following matrix:

Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, − 1, 2) and parallel to the line.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, externally in the ratio 1:2. Also, show that P is the midpoint of the line segment RQ.

Evaluate:

Evaluate:

OR

Evaluate:

Find the equations of the normals to the curve y = x³ + 2x + 6 which are parallel to the line x + 14y + 4 = 0

Find the particular solution of the differential equation satisfying the given conditions:

x²dy + (xy + y²) dx = 0; y = 1 when x = 1.

Find the general solution of the differential equation

OR

Find the particular solution of the differential equation satisfying the given conditions:

, given that y = 1 when x = 0

Evaluateas limit of sums.

OR

Using integration, find the area of the following region:

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring the 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 90, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically.

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn at random and are found to both clubs. Find the probability of the lost card being of clubs.

OR

From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.

Using properties of determinants, show the following:

 

Find the values of xfor which f(x) = [x(x− 2)]²is an increasing function. Also, find the points on the curve, where the tangent is parallel to x-axis.

Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is equal to the radius of the base.

Write the vector equations of the following lines and hence determine the distance between them: