Board Paper of Class 12-Humanities 2004 Maths All India(SET 1) - Solutions

If the vectors are coplanar, then find the value of λ.

Prove that:

Or

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

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.

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

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.

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

Out of current syllabus
 

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.

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 αwith 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

Out of current syllabus
 

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

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

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. 

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

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

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

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

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

Find the output at which:

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

Out of current syllabus

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 calories. Two foods X and Y are available at a cost of Rs 4 and Rs 3 per unit respectively. One unit of the food X contains 200 units of vitamins, 1 unit of minerals and 40 calories, whereas one unit of food Y contains 100 units of vitamins, 2 units of minerals and 40 calories. Find what combination of X and Y should be used to have least cost, satisfying the requirements.