# 12th Maths Paper Solutions Set 1 : CBSE All India 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, prove that, n N.

It is given that

We will prove the required result by the principal of mathematical induction.

For n = 1, we have

Thus, the result is true for n= 1.

Now let the result be true for n= k, where k N. Thus,

â€¦ (1)

Now, we will prove the result for n= k + 1.

That is, we have to prove that

Thus,

Therefore, the result is true for n= k + 1 also.

Hence, by principal of mathematical induction, the result is true for all n N. That is,

, n N

Q2 :

Using the properties of determinants, prove that

We have:

By applying C1 â†’C1 - C2 and C2 â†’C2 - C3, we obtain

Expanding the determinant along R1, we obtain

Thus,

Q3 :

From a bag containing 20 tickets, numbered from 1 to 20, two tickets are drawn at random. Find the probability that
(i) Both the tickets have prime numbers on them
(ii) On one there is prime number and on the other there is a multiple of 4

It is given that from a bag containing 20 tickets numbered from 1 to 20, two tickets are drawn at random.

Out of 20 tickets, two tickets can be drawn in 20C2ways.

Therefore, total number of elementary events = 20C2= 190

(i) The prime numbers amongst the numbers from 1 to 20 are

2, 3, 5, 7, 11, 13, 17, 19

Out of these 8 prime numbers, two numbers can be selected in 8C2ways.

Thus, number of outcomes favourable to the event â€œboth the tickets have prime numbers on themâ€ = 8C2= 28

Therefore, required probability =

(ii) The numbers which are multiples of 4 from amongst the numbers from 1 to 20 are

4, 8, 12, 16, 20

Out of these 5 numbers, one can be selected in 5C1ways.

Out of 8 prime numbers from 1 to 20, one prime number can be selected in 8C1ways.

Thus, the number of favourable outcomes for the event â€œon one there is prime number and on the other there is a multiple of 4â€ = 5C1 x 8C1= 5 x8 = 40

Therefore, required probability =

Q4 :

Two dice are tossed once. Find the probability of getting an even number on the first die or a total of 8.

Or

From a lot of 30 bulbs, which includes 6 defective bulbs, a sample of 3 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Q5 :

Evaluate:

Q6 :

Evaluate:

Q7 :

Solve the differential equation:

Q8 :

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

Q9 :

For each x in a Boolean Algebra B, prove that

Q10 :

Evaluate:

Or

Evaluate:

Q11 :

Differentiate with respect to x:

Q12 :

Differentiate with respect to xfrom first principles.

Q13 :

The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. Find the rate of change of its surface area at the instant when its radius is 5 cm.

Q14 :

Evaluate:

Q15 :

Prove that:

Or

Prove that:

Q16 :

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

x+ y - z = 1

3x+ y - 2z = 3

x- y- z= -1

Or

If, prove that A2- 4A- 5I= O. Hence, find A-1

Q17 :

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius cm is 500πcm3.

Q18 :

Find the area bounded by the circle x2+ y2= 16 and the line y= x in the first quadrant.

Q19 :

If the vectors are coplanar, then find the value of Î».

Q20 :

Prove that:

Or

The volume of the parallelopiped whose edges are is 546 cubic units. Find the value of Î».

Q21 :

The Cartesian equations of a line are 3x+ 1 = 6y − 2 = 1 − z. Find the fixed point through which it passes, its direction ratios, and also its vector equation.

Q22 :

Find the equation of the plane passing through the points (0, −1, 0), (1, 1, 1) and (3, 3, 0).

Q23 :

Prove that the plane x+ 2y z = 4 intersects the sphere x2+ y2+ z2 x+ z − 2 = 0 in a circle of radius unity. Also, find the centre of the circle.

Q24 :

Three forces and act along OA, OB and OC, where O is the orthocenter of Î”ABC. If the forces be in equilibrium, prove that P: Q: R= a: b: c.

Q25 :

The resultant of two like parallel forces and acting at A and B, 8 cm apart, is 40 N. If the resultant passes through C, where AC = 3 cm, find the magnitude of the forces.

Q26 :

A particle moving in a straight line with uniform acceleration describes successive equal distance in times t1, t2and t3. Prove that.

Or

A stone is projected at an angle Î±with the horizontal. Given its velocity when it attains half of the maximum height (it can attain) is times of the velocity at the maximum height. Prove that

Q27 :

The true discount and the banker”™s gain on a certain bill of exchange due after a certain period of time are respectively Rs 700 and Rs 17.50. Find the face of the bill.

Out of current syllabus

Q28 :

A bill of Rs 1,000 drawn on May 7, 2003 for six months was discounted on August 29, 2003 for a cash payment of Rs 988. Find the rate of interest charged by the bank.

Out of current syllabus

Q29 :

A company has two plants to manufacture T.V.s. The first plant manufactures 70% of the T.V.s. and the rest are manufactured by the other plant. 80% of the T.V.s. manufactured by the first plant is rated of standard quality while that of the second plant, only 70% are of standard quality. If a T.V. chosen at random is found to be of standard quality, find the probability that it was produced by the first plant.

Q30 :

A pair of dice is thrown 7 times. If getting the total 7 is considered a success, find the probability of (i) no success (ii) at least 6 successes.

Or

If the probability that an individual suffers from reaction by an injection is, find the probability that out of 5000 individuals given that injection (i) exactly 5 will suffer from reaction (ii) no one will suffer from reaction.

Given that

Out of current syllabus

Q31 :

X and Y entered into a joint business with their capitals in the ratio 3 : 2. At the end of 3 months, X took out one-third of his capital, but after another 3 months Y put in a sum equal to what X had taken out. If at the end of the year Y gets Rs 11,000 more than what X got as profit, find
(i) the total profit in the business
(ii) the profit share of X in the business

Out of current syllabus

Q32 :

A television set is available for Rs 20, 000 cash or Rs 5,000 as cash down payment followed by 6 equal annual instalments, the first to be paid one year after the date of purchase. If the rate of interest under the instalment plan is 10% per annum, determine the amount of instalment. [Given that (1.1)−6= 0.5644]

Out of current syllabus

Q33 :

If the cost function of an article manufactured by a company is given by

C(x) =, where xstands for the output.

Find the output at which:

(i) The marginal cost is minimum
(ii) The average cost is minimum

Out of current syllabus