Time Allowed: 3 Hours Maximum Marks: 100

http://www.cbseguess.com/

Guess Paper – 2007 Class – XII Mathematics

Time Allowed: 3 hours Maximum Marks: 100

General Instructions:

(i) The questions paper consist of three sections A, B and C. Sections 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 number 1 to 8 are of 3 marks each. Question number 9 to 15 are of 4 marks each. Questions number 16 to 18 are of 6 marks each. (iii) For Section B/ Section C. Question number 19 to 22 are of 3 marks each. Question number 23 to 25 are of 4 marks each. Question number 26 is of 6 marks. (iv) All questions are compulsory. (v) Internal choice have been provided in some questions. You have to attempt only one of the choice in such questions. (vi) Use of calculator is not permitted. However you may ask for logarithmic and Statistical tables if required.

SECTION - A

轾1 3 2 轾 1 犏犏 Q. 1. Find the value of x such that [1x 1]犏 2 5 1 犏 2= 0 臌犏15 3 2 臌犏x Ans. x = -14

1 0 1 0 轾轾 2 If A = 犏 and I = 犏 , then find k so that A =8A+kI. 臌-1 7 臌0 1 Ans. k = - 7

Q. 2. Using properties of determinant show that:

------1 www.cbseguess.com Other Educational Portals www.icseguess.com | www.ignouguess.com | www.dulife.com | www.magicsense.com http://www.cbseguess.com/

x+ y x x 5x+ 4 y 4 x 2 x = x3 10x+ 8 y 8 x 3 x

Q. 3. Two cards are drawn without replacement from a well shuffled pack of 52 cards. Find the probability that one is a spade and other is a queen of red colour. 1 Ans. 51

Q. 4. A speak truth in 60% of the cases and B in 90% of the cases. In what percentage of cases are they likely to contradict each other in stating the same fact? 21 Ans. 50

Q. 5. Evaluate: x2tan- 1 xdx x3 1 1 Ans. tan-1x- x 2 + log x 2 + 1 + C 3 6 6

Evaluate:

x 骣2+ sin 2x e琪 dx 桫1+ cos 2x Ans. ex tan x+ C

Q. 6. Evaluate: tanq+ tan3 q dq 1+ tan3 q

1 12 1- 1 骣 2 tanq - 1 Ans. -log 1 + tanq + log tan q - tan q + 1 + tan 琪 + c 3 6 3桫 3

Q. 7. Solve the differential equation: dy +ysec x = tan x dx Ans. y(sec x+ tan x) = sec x + tan x - x + C

Q. 8. Solve the differential equation x2 dy+ y( x + y) dx = 0 , given that y =1 when x =1. 2x 1 Ans. y= , x 贡 3x2 - 1 3

Show that the differential equation representing one parameter family of curve 2 ( x2- y 2) = c( x 2 + y 2 ) is ( x3-3 xy 2) dx =( y 3 - 3 x 2 y) dy .

Q. 9. Find the value of the constant k so that function given below is continuous at x=0 1- cos 2x ,x 0 f( x ) = 2x2 k, x = 0 Ans. k=1

Or Evaluate: ( x+ y)sec( x + y) - x sec x lim y 0 y Ans. xtan x sec x+ sec x

2 d2 y dy y轾log x x2 1 2 Q. 10. If = + + , show that (1+x) 2 + x = 2 . 臌犏 ( ) dx dx

Q. 11. Simplify the Boolean Expression: x( x+ y) +臌轾( y' + x) y ' Ans. 1

Q. 12. Differentiate cos x w.r.t. x from first principle method. sin x Ans. - 2 x

Q. 13. Evaluate: p 2 sin2 x dx 0 sinx+ cos x

1 Ans. -log 2 - 1 2 ( )

Evaluate: p 2 cot x dx 0 cotx+ tan x p Ans. 4

Q. 14. Evaluate: tanqd q

Q. 15. Find the interval in which the function f(x) is (a) increasing, (b) decreasing: f( x )= 2 x3 - 9 x 2 + 12 x + 15 Ans. For increasing (-ト,1) ( 2, ) For decreasing (1,2)

1 Using the differentials, find the approximate value of (66)3 . Ans. 4.0416

轾-4 4 4 轾 1 - 1 1 犏犏 Q. 16. Determine the product 犏-7 1 3 犏 1 - 2 - 2 and the given product is 臌犏5- 3 - 1 臌犏 2 1 3 system of equations: x- y + z = 4 x-2 y - 2 z = 9 2x+ y + 3 z = 1 Ans. x=3, y = - 2, z = - 1

Q. 17. Show that semi-vertical angle of right circular cone of given surface area and

-1 骣1 maximum volume is sin 琪 . 桫3

Find the point on the curve y2 = 4 x which is nearest to point (2,-8) Ans. (4,-4)

2 2 2 Q. 18. Find the area of the region 臌轾( x, y) : y� 4 x ,4 x 4 y 9 .

2 9p 9-1 骣 1 Ans. + - sin 琪 sq. units 6 8 4桫 3

Evaluate the following. Definite Integral as limit of sums: 3 (2x2 + x + 9) dx 1 118 Ans. 3

SECTION-C

Q. 19. A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus, scooter or by other means of 3 1 1 2 transport are, respectively, , , and . The probabilities that he will be late 10 5 10 5 1 1 1 are , and , if he comes by train, bus and scooter, respectively, but if he 4 3 12 comes by other means of transport, then he will not be late. When he arrives, he is late. What is the probabilities that he comes by train? 1 Ans. 2

Q. 20. At a busy traffic intersection, the probabilities p of an individual car having an accident is very small, say, p=0.0001. However, during a certain part of the day, a large number of cars, say, 1000 pass through the intersection. Under these conditions, what is the probability of two or more accidents occurring during this period? [e-0.1=0.905] Ans. 0.0045

Q. 21. Solve the following problem graphically. Maximize Z = 2.5x + y subject to the constraints

3x + 5y 15 5x + 2y 10 x, y 0

Q. 22. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding / cutting machine and a sprayer. It takes 2 hours on grinding/ cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp, while it takes 1 hour on the grinding / cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding / cutting machine for at, the most 12 hours. The profit from the sale of a lamp is Rs 5 and that from a shade is Rs 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit? Ans. Maximum profit=Rs. 32, Lamps =4, Shades = 4

Q. 23. Three persons A, B and C are partners in a business. They invested their capital in the ratio of 20 : 15 : 12. A withdraws half of his capital at the end of 6 months and two-third of the remaining after 3 months. B withdraws one fourth of his capital after 9 months. After a year they divided a profit of Rs 18,910 in the ratio of their adjusted capitals. Find each person’s share. Ans. A’s share of profit: Rs. 6,400; B’s share of profit: Rs. 6,750; C’s share of profit Rs. 5,760

Q. 24. A house is sold for Rs 50000 down and 10 semi-annual payments of Rs 5000 each, the first due 3 years hence. Find the cash price of the house if money is worth 6% per annum compounded semi-annually. Ans. Rs. 86,791

Q. 25. A monopolist’s demand function for one of its products is p(x) = ax + b He knows that he can sell 1400 units when the price is Rs 4 per unit and he can sell 1800 units at a price of Rs 2 per unit. Find the total, average and marginal revenue functions. Also find the price per unit when the marginal revenue is zero. x2 x x Ans. TR=11 x - ; AR = 11 - ; MR = 11 - ; Price at MR=0 is 5.50. 200 200 100

Q. 26. The Banker’s discount and banker’s gain on a certain bill of exchange due after certain time are, respectively, Rs 1250 and Rs 50. Find the face value of the bill. Ans. Rs. 30,000

Sunil tutorials PH: 9810105034

------7 www.cbseguess.com Other Educational Portals www.icseguess.com | www.ignouguess.com | www.dulife.com | www.magicsense.com