# 12th Maths Paper Solutions Set 1 : CBSE Delhi Previous Year 2004

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 , show that It is given that  On adding equations (1), (2), and (3), we obtain Hence, Q2 :

Using properties of determinants, solve for x We have: Q3 :

An urn contains 7 white, 5 black and 3 red balls. Two balls are drawn at random. Find the probability that
(i) both the balls are red
(ii) one ball is red, the other is black
(iii) one ball is white

Total number of balls in the urn = 7 + 5 + 3 = 15

Outof these 15 balls, two balls can be drawn in ways.

(i) Let Abe the event that both the balls drawn are red.

Now, there are 3 red balls out of which 2 red balls can be drawn in ways.

Thus, P(A) = (ii) Let Bbe the event that of the two balls drawn, one is red and the other is black.

There are 3 red balls out of which one red ball can be drawn in 3C1ways and there are 5 black balls out of which one black ball can be drawn in 5C1ways.

Thus, P(B) = (iii) Let C be the event that of the two balls drawn, one is white.

There are 7 white balls and 8 non-white balls. Two balls can be drawn out of them, one ball being white and the other being non-white, in ways.

Thus, P(C) = Q4 :

A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of X.

Q5 :

Evaluate: Q6 :

Evaluate: Q7 :

Form the differential equation corresponding to where a andb are arbitrary constants.

Q8 :

Solve the differential equation: given that y = 1, when x= 0.

OR

Solve the differential equation: Q9 :

Show that in a Boolean algebra, B:
(i) (ii) Q10 :

Evaluate: Q11 :

Differentiate w.r.t. x from first principles.

Q12 :

Differentiate w.r.t. x

Q13 :

Find the equations of the tangent and the normal to the curve at .

Q14 :

Evaluate: Q15 :

Evaluate: Q16 :

Using matrix method, solve the following system of linear equations: Q17 :

Show that a right circular cylinder, which is open at the top and has a given surface area, will have the greatest volume if its height is equal to the radius of its base.

Q18 :

Using integration, find the area of the circle x2+ y2= 16, which is exterior to the parabola y2=6x.

OR

Find the area of the smaller region bounded by the ellipse and the line .

Q19 :

If , then show that the vectors and are orthogonal.

Or

Find x such that the four points A(3, 2, 1), B(4, x, 5), C(4, 2, -2), and D(6, 5, - 1) are coplanar.

Q20 :

Prove that: Q21 :

Find the Cartesian and vector equations of a line, which passes through the point (1, 2, 3) and is parallel to the line Q22 :

Find the Cartesian equation of the sphere which has the points A (2, −3, 4) and B (−5, 6, −7) as the end points of one of its diameters. Also find its centre and radius.

Q23 :

Show that the lines and intersect. Find the point of intersection also.

Q24 :

Three forces acting on a point are in equilibrium. If the angle between be double the angle between , prove that Out of current syllabus

Q25 :

Two unlike parallel forces act at two points units apart. Show that if the direction of be reversed, the resultant will be displaced by a distance units.

Out of current syllabus

Q26 :

A particle starting with some initial velocity and moving with uniform acceleration acquires a velocity of 20 cm/sec after moving through 10 cm from a point P to Q and a velocity of 30 cm/sec after further moving 20 cm from Q to R in the same direction. Find
(i) its velocity at the point P
(ii) its acceleration
(iii) the time it will take and the distance

Out of current syllabus

OR

From a point on the ground at a distance from the foot of a vertical wall, a ball is thrown at an angle of 450 which just clears the top of the wall and afterwards strikes the ground at a distance of yon the other side of the wall. Find the height of the wall.

Out of current syllabus

Q27 :

Find the banker's discount and true discount on a bill of Rs 22,800 due 4 months at 4% per annum.

Q28 :

A bill of exchange drawn on January 4, 2003 at 5 months date was discounted on march 26, 2003 at 3% per annum. If the banker's discount is Rs 1207.20, find the face value of the bill.

Q29 :

Three urns A, B, and C contain 6 red and 4 white, 2 red and 6 white, and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.

Q30 :

The mean and variance of a binomial distribution are 4 and respectively. Find P(X â‰¥1).

OR

If the sum of the mean and variance of a binomial distribution for 5 trials be 1.8, find the distribution.

Q31 :

A, B, and C are partners in a business. A, being a working partner, receives 10% of the total profit as salary. The remaining profit is distributed among them in the ratio of 2 : 3 : 4. If A gets Rs 3,00,000 in all, find the shares of B and C.

Q32 :

Find the present value of an annuity of Rs 1,200 payable at the end of each of the 6 months for 3 years, when the interest is 8% per annum, compounded semi-annually.

[Use (1.04)6= 1.2653]

Q33 :

The total cost and demand function of an item are given by and p = 100 - xrespectively.
Write the total revenue function and the profit function. Find the number of items when the profit will be maximum. Find the maximum profit also.

Q34 :

An oil company requires 13,000, 20,000 and 15,000 barrels of high grade, medium grade and low grade oil respectively. Refinery A produces 100, 300 and 200 barrels per day of high, medium and low grade oil respectively whereas refinery B produces 200, 400 and 100 barrels per day respectively. If A costs Rs 400 per day and B costs Rs 300 per day to operate, how many days should each be run to minimize the cost of requirement?

OR

A firm makes items A and B and the total number of items it can make in a day is 24. It takes one hour to make an item of A and only half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on an item of A is Rs 300 and on one item of B is Rs 160. How many items of each type should be produced to maximize the profit? Solve the problem graphically.