# 12th Maths Paper Solutions Set 1 : CBSE All India Previous Year 2005

General Instructions:
(i) The question paper consists of three sections A, B and C Section A is compulsory for all students. In addition to Section A, every student has to attempt either Section B OR Section C.
(ii) For Section A -
Question numbers 1 to 8 are of 3 marks each.
Question numbers 9 to 15 are of 4 marks each.
Question numbers 16 to 18 are of 6 marks each.<o:p></o:p></span></p>
(iii) For Section B/Section C
Question numbers 19 to 22 are of 3 marks each.
Question numbers 23 to 25 are of 4 marks each.
Question number 26 is of 6 marks.
(iv) All questions are compulsory.
(v) Internal choices have been provided in some questions. You have to attempt only one of the choices in such questions.
(vi) Use of calculator is not permitted. However, you may ask for logarithmic and statistical tables, if required.
Q1 :

If and f(x) = x2- 2x- 3, show that f(A) = O.

We are given that and f(x) = x2- 2x- 3.

We have to prove that f(A) = O

Now, f(A) = A2- 2A- 3I, where I is the identity matrix Thus, we have f(A) = O.

Q2 :

Using the properties of determinants, solve for x: The given determinant is On applying C1 â†’C1 + C2 + C3, we have On applying R2 â†’R2 - R1 and R3 â†’R3 - R1, we have Now, Î”= 0 gives

4x2(3a- x) = 0

x= 0 or x = 3a

Q3 :

An integer is chosen at random from the first 200 positive integers. Find the probability that it is divisible by 6 or 8.

Let A and Bbe the following events.

A: Out of the first 200 positive integers, an integer is divisible by 6.

B: Out of the first 200 positive integers, an integer is divisible by 8.

Clearly, A âˆª B andA âˆ© B are the following events.

A âˆª B: Out of the first 200 positive integers, an integer is divisible by 6 or 8.

A âˆ© B: Out of the first 200 positive integers, an integer is divisible by 6 and 8.

We know that the probability of an event Eis given by From the first 200 positive integers, the numbers which are divisible by 6 are

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198

Thus, P(A) = From the first 200 positive integers, the numbers which are divisible by 8 are

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200

Thus, P(B) = From the first 200 positive integers, the numbers which are divisible by both 6 and 8 are

24, 48, 72, 96, 120, 144, 168, 192

Thus, P(A âˆ© B) = We know that Thus, the probability of getting a number which is divisible by 6 or 8 is Q4 :

X is taking up subjects-Mathematics, Physics and Chemistry in the examination, His probability of getting Grade A in these subjects are 0.2, 0.3, and 0.5 respectively. Find the probability that he gets.

(i)   Grade A in all subjects

(ii)  Grade A in no subject

(iii) Grade A in two subjects

Q5 :

Evaluate: Q6 :

Evaluate: Q7 :

Solve the following differential equation: Q8 :

Solve the differential equation:

(x2+ xy)dy= (x2+ y2)dx

Or

Solve the following differential equation: , given that , when x = 0

Q9 :

Test the validity of the following argument:

S1: p âˆ§ q; S2: âˆ¼p; S: âˆ¼q

Or

If B is a Boolean algebra and x, y âˆˆB, then show the following:

(x+ y) + (x”™ y”™) = 1

Q10 :

Evaluate: .

Q11 :

Differentiate with respect to xfrom the first principle.

Q12 :

If , then prove that Q13 :

Find the intervals in which the function f(x) = 2x3− 15x2+ 36x + 1 is strictly increasing or decreasing. Also find the points on which the tangents are parallel to the x-axis.

Q14 :

Evaluate: .

Q15 :

Evaluate: , where Q16 :

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.

Or

Prove that the curves x= y2and xy = kcut at right angle if 8k2= 1.

Q17 :

Using matrix method, solve the following system of linear equations:

x+ yz = 1

xyz= −1

3x+ y − 2z = 3

Q18 :

Find the area of the region bound by the curve y= x2and the line y= x.

Or

Find the area enclosed by the parabola y2= x and the line y+ x = 2 and the first quadrant.

Q19 :

If , and are three mutually perpendicular vectors of equal magnitude, find the angle between and Q20 :

Show that the four points A, B, C and D, whose position vectors are and respectively are co-planer

Q21 :

A body moves for 3 seconds with a uniform acceleration and describes a distance of 108 m. At that point the acceleration ceases and the body covers distance of 126 m in the next 3 seconds. Find the initial velocity and acceleration of the body.

Q22 :

A body is projected with a velocity of 24 m/sec at an angle of 60Â°with horizontal. Find
(a) The equation of its path;
(b) The time of flight; and
(c) The maximum height attained by it.

Or

A particle is projected so as to graze the tops of two walls, each of height 10 m, at 15 m and 45 m, respectively from the point of projection. Find the angle of projection.

Q23 :

Find the equation of the line passing through the point P (-1, 3, -2) and perpendicular to the lines and .

Q24 :

The resultant of forces and acting on a particle is . If is doubled, is doubled. If is reversed, is again doubled. Prove that Or

A and B are two fixed points in a horizontal line at a distance 50 cm apart. Two strings AC and BC of lengths 30 cm and 40 cm respectively support a weight W at C. Show that the tensions in the strings CA and CB are in the ratio of 4 : 3.

Q25 :

The resultant of two unlike parallel forces of 18N and 10 N acts along a line at distance of 12 cm from the line of action of the smaller force. Find the distance between the lines of action of the two given forces.

Q26 :

Find the equation of the sphere passing through the points (1, -3, 4), (1, -5, 2), (1, -3, 0) and having its centre on the plane .

Q27 :

A speaks the truth 8 times out of 10 times. A die is tossed. He reports that it was 5. What is the probability that it was actually 5?

Q28 :

A coin is tossed 4 times. Find the mean and variance of the probability distribution of the number of heads.

Or

For a Poisson distribution, it is given that P(X= 1) = P(X= 2). Find the value of the mean distribution. Hence, find P(X= 0) and P(X= 4).

[Use e−2 = 0.13534]

Out of current Syllabus

Q29 :

If the banker's gain on a bill be of the banker's discount, the rate of interest being 10% per annum, find the expired period of the bill.

Out of the current syllabus

Q30 :

A bill of Rs. 5300, drawn on 16thJanuary, 2003 for 8 months was discounted on 12thFebruary, 2003 at 10% per annum. Find the banker's gain and discounted value of the bill.

Out of the current syllabus

Q31 :

In a business partnership, A invests half of the capital for half of the period, B invests one-third of the capital for one-third of the period, and C invests the rest of the capital for the whole period. Find the share of each in the total profit of Rs. 190000.

Out of the current syllabus

Q32 :

A plans to buy a new flat after 5 years, which will cost him Rs. 552000. How much money should he deposit annually to accumulate this amount, if he gets interest 5% per annum compounded annually? [Use (1.05)5=1.276]

Out of the current syllabus

Q33 :

The cost function of a firm is given by C(x) = , where xstands for the output.

Calculate:
(a) The output at which the marginal cost is the minimum.
(b) The output at which the average cost is equal to the minimum cost.

Out of the current syllabus

Or

The total cost and the total revenue of a firm that produces and sells xunits of its products daily are expressed as

C(x) = 5x + 350 and R(x) = 50x - x2

Calculate:
(a) The break-even points, and
(b) The number of units the firm will produce which will result in loss.

Out of the current syllabus