# 12th Maths Paper Solutions Set 3 : CBSE Delhi Previous Year 2015

General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
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Q1 :

Find the differential equation representing the family of curves + B, where A and B are arbitrary constants.

The equation of the family of curves is , where A and B are arbitrary constants.
We have

Differentiating both sides with respect to r, we get

Again, differentiating both sides with respect to r, we get

This is the differential equation representing the family of the given curves.

Q2 :

Find the integrating factor of the differential equation
.

We have

It is in the form , where P and Q are the constants or functions of x.
Thus, the integrating factor of the given differential equation is

Q3 :

If then find the projection of .

Given:

The projection of on is given by

Q4 :

Find Î», if the vectors are coplanar.

Q5 :

If a line makes angles 90Â°, 60Â° and Î¸ with x, y and z-axis respectively, where Î¸ is acute, then find Î¸.

Q6 :

Write the element a23 of a 3 âœ• 3 matrix A = (aij) whose elements aij are given by

Q7 :

If x = a cos Î¸ + b sin Î¸, y = a sin Î¸ âˆ’ b cos Î¸, show that

Q8 :

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Q9 :

Find :

Q10 :

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:

 School Article A B C Hand-fans 40 25 35 Mats 50 40 50 Plates 20 30 40

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation.

Q11 :

If find and hence find a matrix X such that
OR
If

Q12 :

If , using properties of determinants find the value of f(2x) âˆ’ f(x).

Q13 :

Find :

OR

Integrate the following w.r.t. x

Q14 :

Evaluate :

Q15 :

A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

OR
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.

Q16 :

Q17 :

Find the distance between the point (âˆ’1, âˆ’5, âˆ’10) and the point of intersection of the line and the plane x âˆ’ y + z = 5.

Q18 :

If sin [cotâˆ’1 (x+1)] = cos(tanâˆ’1x), then find x.

OR

If (tanâˆ’1x)2 + (cotâˆ’1x)2 = , then find x.

Q19 :

If then find .

Q20 :

If A and B are two independent events such that then find P(A) and P(B).

Q21 :

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2Ï€.

Q22 :

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :

Q23 :

Let N denote the set of all natural numbers and R be the relation on N Ã— N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

Q24 :

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

OR

Evaluate as a limit of a sum.

Q25 :

Solve the differential equation :

OR

Find the particular solution of the differential equation given that y = 1, when x = 0.