Class 12 - Mathematics

Sample Papers

Q&A
Question:

Find the vector and cartesian equations of the line through the point (1,2, – 4) and perpendicular to the two lines.

Sample Papers - Questions
2.

A dealer in rual area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit ? Make it as a LPP and solve it graphically.

3.

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below:
2x + 4y â‰¤ 8
3x + y â‰¤ 6
x + y â‰¤ 4
xâ‰¥ 0, y â‰¤ 0.6

5.

Find the local maxima and local minima, of the function f(x) = sin x – cos x, 0 < x < 2Ï€. Also find the local maximum and local minimum values.

10.

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.

12.

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a . spade. [6]

13.

Using integration find the area of the triangle formed by positive x-axis and tangent and normal to the circle x2 + y2= 4 at (1, âˆš3).

15.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3. Also find maximum volume in terms of volume of the sphere.

16.

Find the coordinate of the point P where the line through A(3, -4,-5) and B (2, – 3,1) crosses the plane passing through three points L (2,2,1), M (3,0,1) and N (4, -1,0). Also, find the ratio in which P divides the line segment AB.

17.

Solve the differential equation: (tan-1 y-x)dy = (1 + y2) dx.
Solution. Same as solution Q. 23 (OR) Set 1 (Outside Delhi) up to eq.
x = tan-1 y -1 + ce -tan-1 y
OR

18.

A manufacturer produces two products A and 6. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

19.

Three Schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:

Find the fund collected by each school separately by selling the above articles. Also find the total funds collected for the purpose. Write one value generated by the above situation.

23.

Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctually. The school P wants to award Rs x each, y each and Rs z each for the three respective values to its 3,2 and 1 students with a total award money of Rs 1,000. School Q wants to spend Rs 1,500 to award its 4,1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs 600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

24.

Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R(c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

26.

A bag ‘A’ contains 4 black and 6 red balls and bag ‘B’ contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

28.

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

29.

If sin [cot-1 (x +1)] = cos(tan-1 x), then find x.

30.

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

31.

An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.

32.

Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < Ï€, is strictly increasing or strictly decreasing.

33.

Using integration find the area of the region {(x, y): x2 +y2 â‰¤2ax, y2 â‰¥ ax, x,yâ‰¥ 0}

34.

Let/: N -> N be a function defined as/(x) = 9x2 + 6x-5. Show that: Nâ€”>S, where S is the range off, is invertible. Find the inverse of f and hence find f-1 (43) and f-1(163).

35.

Find the general solution of the following differential equation:

40.

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game’ then find the probability that B wins.

41.

Find the differential equation representing the family A of curves Ï… = A/r + B, where A and B are arbitrary r constants.

48.

Find the particular solution of the differential equation (1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.

General Knowledge Quiz
3001 Bizarre Facts
Dictionary
Elementary English Grammar Test
Standard English Grammar Test
Daily English Quiz
English Test
Grammar Examination
ESL / EFL Test
Grammar Test
Antonyms
Synonyms
Homonyms
Vocabulary Examination
Word Analogy
Vocabulary Flashcards
Idioms
Idioms Quiz
Quantitative Aptitude Test
iQ Quiz Game
Verbal Reasoning Quiz - Mind Game