# 12th Maths Paper Solutions Set 1 : CBSE Delhi Previous Year 2011

General Instructions:
i. All questions are compulsory.
ii. The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each, and Section C comprises of 7 questions of six marks each.
iii. All questions in section A are to be answered in one word, one sentence or as per the exact requirements of the question.
iv. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
v. Use of calculators is not permitted.
Q1 :

State the reason for the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} not to be transitive.

It is known that a relation R in a set A is transitive if (a1, a2) âˆˆR and (a2, a3) âˆˆ R â‡’ (a1, a3) âˆˆ R &mnForE; a1, a2, a3âˆˆ A.

It can be observed that (1, 2), (2, 1) âˆˆ R, but (1, 1) âˆ‰ R.

Thus, the relation R in the set {1, 2, 3} is not transitive.

Q2 :

Write the value of

Thus, the value of the given expression is 1.

Q3 :

For a 2 x 2 matrix, A = [aij] whose elements are given by , write the value of a12.

Any element in the ith row and jth column is

For a12, the value of i = 1 and j = 2.

Thus, the value of a12 is .

Q4 :

For what value of x, the matrix is singular?

Q5 :

Write A-1 for

Q6 :

Write the value of

Q7 :

Write the value of

Q8 :

For what value of ‘a’ the vectors and are collinear?

Q9 :

Write the direction cosines of the vector

Q10 :

Write the intercept cut off by the plane 2x + yz = 5 on x-axis.

Q11 :

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min. {a, b}. Write the operation table of the operation *.

Q12 :

Prove the following:

OR

Find the value of

Q13 :

Using properties of determinants, prove that

Q14 :

Find the value of ‘a’ for which the function f defined as

is continuous at x = 0

Q15 :

Differentiate.

OR

If x = a (Î¸ - sinÎ¸), y = a (1 + cosÎ¸), find .

Q16 :

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

OR

Find the points on the curve x2 + y2 − 2x− 3 = 0 at which the tangents are parallel to x-axis.

Q17 :

Evaluate:

OR

Evaluate:

Q18 :

Solve the following differential equation:

extan y dx + (1 − ex) sec2y dy = 0

Q19 :

Solve the following differential equation:

Q20 :

Find a unit vector perpendicular to each of the vector and where

Q21 :

Find the angle between the following pair of lines:

and check whether the lines are parallel or perpendicular.

Q22 :

Probabilities of solving a specific problem independently by A and B are and respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

Q23 :

Using matrix method, solve the following system of equations:

OR

Using elementary transformations, find the inverse of the matrix

Q24 :

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Q25 :

Using integration find the area of the triangular region whose sides have equations y = 2x + 1, y = 3x + 1 and x = 4.

Q26 :

OR

Q27 :

Find the equation of the plane which contains the line of intersection of the planes and which is perpendicular to the plane

Q28 :

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman”™s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman”™s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman”™s time.

If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve graphically.