Board Paper of Class 12-Science 2005 Maths All India(SET 1) - Solutions

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

Using the properties of determinants, solve for x:

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

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

Evaluate:

Evaluate:

Solve the following differential equation:

Solve the differential equation:

(x²+ xy)dy= (x²+ y²)dx

Or

Solve the following differential equation:

, given that, when x = 0

Test the validity of the following argument:

S₁: p ∧ q; S₂: ∼p; S: ∼q

Or

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

(x + y) + (x’ y’) = 1

Evaluate: .

Differentiatewith respect to xfrom the first principle.

If, then prove that

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.

Evaluate:.

Evaluate:, where

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 = y² and xy = k cut at right angle if 8k² = 1.

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

x+ y − z = 1

x− y− z= −1

3x+ y − 2z = 3

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.

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

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

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.

A body is projected with a velocity of 24 m/sec at an angle of 60°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. 

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

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. 

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. 

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 .

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?

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

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

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

Out of the current syllabus

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

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

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

A manufacturer produces two types of steel trunks. He has two machines A and B. The first type of trunk requires 3 hours on machine A and 3 hours on machine B. The second type of trunk requires 3 hours on machine A and 2 hours on machine B. Machines A and B are run daily for 18 hours and 15 hours respectively. There is a profit of Rs. 30 on the first type of trunk and Rs.25 on the second type of trunk. How many trunks of each type should be produced and sold to make maximum profit?