1. CARMELAGIRI CMI PUBLIC SCHOOL
SECOND MODEL EXAMINATION - 2014
MATHEMATICS
CLASS: XII MAX.MARK:100
TIME: 3Hrs.
SECTION A
Question numbers 1 to 10 carry 1 mark each
1. Find the projection of the vector 푖⃗ +푗⃗ +7푘⃗⃗ on the vector 2푖⃗ - 3푗⃗ + 6푘⃗⃗
2. Write a unit vector in the direction of the sum of vectors 푎⃗ = 2 푖 ⃗⃗⃗⃗ - 푗⃗ + 2푘⃗⃗ , 푏⃗⃗ = - 푖⃗ + 푗 ⃗⃗⃗ +3푘⃗⃗
3. Write the value of( 푘 ⃗⃗⃗⃗x 푗) ⃗⃗⃗⃗ . 푖⃗ + 푗⃗ .푘⃗⃗.
4. Write the value of 푝⃗ for which 푎⃗ = 3 푖⃗ + 2 푗⃗ +9푘⃗⃗ and 푏⃗⃗ = 푖⃗ +p 푗⃗ +3푘⃗⃗ are parallel vectors.
5. Write the distance of the following plane from the origin: 2 x –y +2z +1 =0
6. Write the degree of the differential equation: ∫ (푑2 푦
푑 푥2)
3
+ 푦 (푑푦
푑푥
)
4
+ 푥 3 =0
7. Evaluate ∫ cos−1( sin 푥)푑푥
8. Evaluate ∫
2 cos푥
3푠푖푛2 푥
푑푥
9. Evaluate ∫
1
√1−푥2
1
√2
0
푑푥
휋
4
−휋
4
10. Evaluate : ∫ 푠푖푛3 푥 푑푥
SECTION - B
Question numbers 11 to 22 carry 4 marks each.
11. Evaluate ∫
cos2푥 −cos 2훼
cos 푥−cos 훼
푑푥

2. 12. Evaluate ∫
푥
푥3−1
푑푥
OR
13. Evaluate ∫
2
(1−푥) (1+푥2)
푑푥
14. Evaluate ∫
5푥+3
√푥2+4푥 +10
푑푥
15. Evaluate ∫(sin−1 푥) 2 푑푥
휋
2
0
16. Evaluate ∫ log sin 푥 푑푥
17. Find the differential equation of all circles in the first quadrant which touch the
Co ordinate axes.
18. Find the particular solution of the following differential equation
푑푦
푑푥
= 1 + 푥 2 + 푦2 + 푥 2푦2 , given that y=1, when y=1, when x=o
19. The dot product of a vectors with the vectors 푖 ⃗⃗ -3 푗⃗ , 푖⃗ - 2 푘⃗⃗ , 푖⃗ +푗⃗ +4푘⃗⃗ are 0, 5, and 8
Respectively. Find the vector.
20. Find the equation of the perpendicular drawn from the point P (2, 4,-1) to the line
푥+5
1
= 푦 +3
4
=
푧− 6
−9
OR
Find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the
Line joining the points (1, 4, 2) and (2, 3, 5) also find the perpendicular distance of the
Plane from the origin.
21. Find the shortest distance between the lines
푥+1
7
= 푦±1
−6
= 푧 +1
1
And
푥−3
1
=
푦−5
−2
=
푧−7
1
22. A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases
Cases are they likely to contradict each other in stating the same fact?

3. Do you think that statement of B is true?
SECTION –C
Question numbers 23 to 29 carry 6 marks each.
23. Evaluate:∫
푥2+1
(푥−1)2(푥+3)
푑푥
24. Draw a rough sketch of the region enclosed between the circles 푥 2 + 푦2 =4 and
(푥 − 2)2 +푦2 =1. Using integration, find the area of the enclosed region.
25. Find the particular solution of the differential equation
( 푥푑푦 − 푦푑푥 ) 푦 sin (푦
푥
) = (푦푑푥 + 푥푑푦)푥 cos 푦
푥
26. Find the equation of the plane passing through the line of intersection of the planes
2푥 + 푦 − 푧 = 3 And 5푥 − 3푦 + 4푧 + 9 = 0 and parallel to the lines
푥_1
2
=
푦−3
4
=
푧_ 5
5
OR
Find the distance of the point (2, 12, 5) from the point of intersection of the line
푟 ⃗⃗⃗ = 2푖⃗ -4푗⃗ +2푘⃗⃗ +휇 (3 푖⃗+ 4푗⃗+ 2푘⃗⃗ ) and to the plane푟 ⃗⃗⃗. (푖 ⃗⃗-2푗⃗ +푘⃗⃗ ) = 0.
27. A merchant plans to sell two types of personal computers a desktop model and a portable
Model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the monthly
Demand of computers will not exceed 250 units. Determine the number of units of each
Type of computers which the merchant should stock to get maximum profit if he does not
Want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and
On portable model is Rs 5000.
28. Suppose a girl throws a die. If she gets a 5, or 6 , she tosses a coin 3 times that notes the
Number of heads. If she gets 1, 2, 3, or4 she tosses a coin once and whether a head or tail
Is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3?

4. Or 4 with the die?
29. In a test, an examinee either knows the answer or guesses or copies the answer to a
Multiple choice questions with your choices. The probability that he makes a guess is
1
6
And the probability that he copies the answer is
1
9
. The probability that his answer is
Correct, given that he copied, is
1
8
. Find the probability that he knew the answer to the
Question, given that he correctly answered it.
Do the result of his question indicates that most of the students believe in the value of
Passing an examination with honesty and self knowledge?
OR
A person wants to construct a hospital in a village for welfare. The probabilities are 0.04
That some bad element opposes this work, .08 that the hospital will be completed if here
Is no oppose of any bad elements and 0.03 that the hospital will be completed if bad
Element oppose. Determine the probability that the construction of hospital will be
Completed. How the hospital is necessary for village area? Justify your answer.