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Guess Paper – 2008
CLASS – XII
Subject – Mathematics
Time: 3 Hours
GENERAL INSTRUCTIONS
For Part A
Question numbers 1 to 8 are of 3 marks each
Questions number 9 to 15 are of 4 marks each
Questions number 16 to 18 are of 6marks each
For Part B/C
Question numbers 19 to22 are of 3 marks each
Questions number 23 to 25 are of 4 marks each
Questions number 26 is of 6 marks
Max Marks: 100
PART A
Q. 1. Using properties of determinants, show that
1 x  y x2  y 2
1 y  z y 2  z 2   x  y  y  z  z  x 
1 z  x z 2  x2
 a b   2  5
Q. 2. Find the values of a and b for which the following holds: 
    
 a 2b   1  4
Q. 3. If P( A)  0.3, P( B) 0.7 and P( B/ A) 0.5 , find P  A / B  and P  A  B 
Q. 4. A box contains100 bolts and 50 nuts. It is given that 50% bolts and 50% nuts are rusted. Two
objects are selected from the box random. Find the probability that both are both or are rusted.
sin 2 x
dx
Q. 5. Evaluate: 
sin 5 x sin 3 x
sin x
Q. 6. Evaluate: 
dx
sin  x  a 
2
d 2 y  dy 
Q. 7. Show that the differential equation of all parabolas is given by y 2     0
dx  dx 
dy
Q. 8. Solve the following differential equation: 2 x  y  6 x 3
dx
2x
Q. 9. Evaluate:  2
dx
( x + 1) ( x 2  2)
 /2
Q. 10. Evaluate:

0
x
dx
sin x  cos x
Q. 11. Find the intervals in which the function f given by f ( x)  sin x  cos x, 0  x  2
(i) increasing(ii)decreasing.

d2y
Q. 12. If x  a   sin   , y  a 1  cos  find
at   .
2
2
dx
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2
Q. 13. Differentiate e x w.r.t. x using first principle
x  y  1
Q. 14. For each x in a Boolean algebra B,
  y  x'
x. y  0 
Q. 15. Evaluate: lim
1
x
2


1  tan  sin x 
x  cos sin 1 x
1
Q. 16. Find the area of the following region: {( x, y) : y 2  4 x, 4 x 2  4 y 2  9} .
Q. 17. A window is in form of a rectangle surmounted by a semi-circle. If the perimeter of the window is
100m, find the dimensions of the window so that maximum light enters through the window.
 4 4 4 
1 1 1 


Q. 18. Given that A  7 1 3 and B  1 2 2




 5 3 1
 2 1 3 
Find AB. Use this to solve the following system of linear equations:
x y z  4
x  2 y  2z  9
2 x  y  3z  1
SECTION - B
Q. 19. For what value of 'a' are the four points A (3, 2, 1) B (4, a, 5 ), C (4, 2,-2) and D (6, 5,-1)
coplanar?
Q. 20. Using vectors prove that an angle in a semicircle is a right angle.
Q. 21. Show that the sphere x 2  y 2  z 2  6 x  12 y  8z  45  0 touches the plane 2 x  2 y  z  10
Q. 22. Find the equation of the line through the point (-1, 2, 3) which is perpendicular to the lines
x y 1 z  2
x  3 y  2 z 1


and


.
2
3
2
1
2
3
Q. 23. Prove that the equation of the plane making intercepts a, b and c on the co-ordinate axes, is of the
x y z
form    1
a b c
Q. 24. A body of weight 5 kg is suspended by two strings 7 cm and 24 cm long. Their other ends being
fastened to the end points of a rod of length 25 cm. If the rod be so held that the body hangs immediately
below its mid-point, find the tension in the strings.
Q. 25. Two forces, when acting at right angles, have their resultant of magnitude 10N and when they
act at 60 , the magnitude of the resultant is 13N Find the magnitude of the two forces.
Q. 26. A particle is projected upwards with a velocity of 4 m/sec and after t seconds, another particle is
projected upwards from the same point with the same velocity. Prove that the particles will meet at a
t u
4u 2  g 2t 2
height of
m after a time period of    seconds start.
8g
2 g
Or
A particle is projected so as to graze the tops of two walls, each of height 10 m at distances of 10 m and
60 m respectively from the point of projection. Find the angles of projection.
SECTION - C
Q. 19.A bill of exchange drawn on May 1, 2003 for Rs. 73,200 for 6 months was discounted by a bank
at 6% per annum for Rs. 72,443,20. On what day was the bill discounted?
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Q. 20. The difference between Bankers discount and True Discount on a bill legally due 3 months hence
at 5 %per annum is Rs. 50. Find the face value, Bankers Discount and True Discount.
Q. 21. From a well shuffled pack of 52 cards. 3 cards are draw one-by-one with replacement. Find the
probability distribution of number of queens.
Q. 22. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers and
their probability of accidents are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with
an accident. Find the probability that he is a scooter driver.
Q. 23. How much should a company set aside at the end of each year if it has to buy a machine expected
to cost Rs. 2,00,000 at the end of 6 years. When the rate of interest is 10% per annum compounded
annually? [Use (1.1) 6 = 1.772]
Q. 24. A, B and C enter into a partnership business. A invests one-fourth of the total capital for half of
the total time, B invests one-third of the total capital for one-fourth of the total time and C invests the
remaining capital for full time. Find the share of each in a profit of Rs. 30,000.
20
2
Q. 25. The cost function of a firm is given by C  200 x  x 2  x3
3
9
Where x is the output. Find the output at which the
(i) Average cost is minimum
(ii) Marginal cost is minimum
(iii) Marginal cost is the equal to average cost.
Q. 26. A company manufactures, two types of toys-A and B. Toy A require 4 minutes for cutting and 8
minutes for assembling and Toy B requires 8 minutes for assembling. There are 3 hours and 20 minutes
available in a day for cutting and 4 hours for assemble. The profit on a piece of toy A is Rs. 50 and that
on toy B is Rs. 60. How many toys of each type should be made daily to have maximum profit? Solve
the problem graphically.
BHASKARACHARYA MATHEMATICS
SHYAM SUNDER KAUSHIK
9868875979
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