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Guess Paper – 2012
Class – XII
Subject – Mathematics
REVISION-VEC-3D-PROB
SECTION- A
1.Find the angle between the lines
x  35 y  55
x  3 y  20

, z  2; and

 2z
2
1
2
3
2.If a  5 ; b  13 and a  b  25 , find a . b
3. Find the direction cosine of the line which is perpendicular to lines whose direction cosines are proportional to
(1,-2,-2) and (0,2,1).
4. Find a vector parallel to the sum of vectors i  2 j  3k and  i  3 j  k
5. Find the value of  if i  3 j   k and i  j  8k are parallel
6. Find the equation of the line which passes through the po int s (2,3,4) and ( 3,1,2) .
7. If a  2, b  7 a  b = 3iˆ  2 ˆj  6kˆ , find the angle between a and b .
8. Find the distance of the point (-2, 3, 5) form xy and yz planes.




9. Find a unit vector perpendicular to a and b where a  i  j  k , b  i 2 j  3k
10. Find mean of the probability distribution of the random variable ‘no. of tails’ when three coins are tossed.

11. Find the shortest distance between the lines whose vector equations are given by r  (4iˆ  4 ˆj  kˆ)   (iˆ  ˆj  kˆ) and

r  (3iˆ  8 ˆj  3kˆ)   (2iˆ  3 ˆj  3kˆ)
12.A box contains 4 gold and 3 silver coins .Another box contains 3 gold and 5 silver coins . A box is choosen at
random and a coin is drawn from it .If the selected coin is a gold coin find the probability that it was drawn from the
second box.
13. Find the equation of the plane through the points (3,4,2) and (7,0,6) and is perpendicular to plane 2x-5y=15
14.The probability of A hitting a target is 4/5 and that of B hitting it is 2/3. They both fire at the target .find the probability
that (i) At least one of them will hit the target (ii) Only one of them will hit the target.
Find the equation of theline pas sin g through (1,2,3 ) and perpendicu lar to the lines
15
and
x  8 y  19 z  10


3
 16
7
x  15 y  29 z  5


3
8
5
16. Find the equation of the plane which contains the line of intersection of the planesx+2y+3z -4=0 and 2x+y-z+5=0 and
perpendicular to the plane 5x+3y+6z+8=0.
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

17.If a and b are two vectors such that a.b = a.c , a  b  a  c and a  0 , then prove that b  c .
18.A die is thrown ten times. If getting a prime number is considered as, success finds the probability of getting (i) at
most eight (ii) atleast 8 (iii) exactly 8 successes.
19. Find the equation of the plane containing the line
x 1 y 1 z  3
=
=
and perpendicular to plane x+2y+z-12=0
1
4
2
20. Find image of the point (0, 2, 3) in the line r= j+2k + ‫( ג‬i+2j+3k).



 










21. If a = i +4 j +2 k b =3 i -2 j +7 k , c =2 i - j +4 k Find a vector d which is perpendicular to a and b and


c  d =15
22. Three vectors a , b ,and c satisfy the condition a  b  c  0 .Evaluate   a.b  b.c  c.a ,if a  1 , b  4, c  2
23. Find the foot of perpendicular from (1, 2, 3 ) to the line
x6 y7 z7


and also the equation Of the plane
3
2
2
containing the line and the point (1, 2, 3).
24. Prove that the image of the point (3, -2, 1) in the plane 3x – y + 4z = 2 lies on the plane x+y+z+4=0
25. Find the distance of the point (3, 4, 5) from the plane x + y + z = 2 measured parallel to the line 2x = y = z
26. Show that the lines
x 1 y  3  z
x  4 1 y z 1




and
are coplanar. Also find the equation of the
2
4
1
3
2
1
plane containing the lines.
27. By examining the chest X- rays, the probability that T.B is detected when a person actually suffering is o.99.The
probability that the doctor diagnoses incorrectly that a person has T.B on the basis of X-ray is 0.001. In a certain city 1 in
1000 person suffers from T.B. A person is selected at random and is diagnosed to have T.B.what is the chance that he
actually has T.B
28. A manufacture has three machine operators A, B, and C.The first operatorA produces 1% defectiveitems ,
where as the other two operators B and Cproduce 5% and 7% defectiveitem respectively . A is on the job for
50% of the time , B is on the job for 30% of the time and C is on the job for 20% of the time . A defectiveitem is
produced , what is the probability that it was produced by A ?
29. A line makes angle α, β, ,  with the four diagonals of a cube Prove thatcos2 α + cos2 β + cos2 + cos2  =4/3.
30. A box contains 12 bulbs of which 3 are defective. If 3 bulbs are drawn from the box at random,find the probability
distribution of X, the number of defective bulbs drawn. Hence compute the mean of X.
2x  1 4  y z  1


. Find the direction cosines of line AB.
2
7
2
2x  1 y  2 z  3
2. The Cartesian equation of a line AB is
. Find equation of a line parallel to AB and trough the point


2
3
3
1. Cartesian equations of a line AB are
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(-1,0,-3)
  
 

  
3. If a  i  j  k; b  2 i - j  3k and c  i - 2 j  k , find a unit vector parallel to the vector 2a  b  3c
4 Ten eggs are drawn successively with replacement from a lot containing 10% defective eggs. Find the probability that there is at
least one defective egg.
5. Find direction cosines of a line which is equally inclined with the axes.
6 Find value of λ so that the line x  2  y  1  z  3 is perpendicular to the plane 3x – y – 2z = 7.
9

6
7. If a line makes angles ∏ /4 with each of X –axis and Y-axis, then what angle does it makes with the Z- axis.
8. If | a + b | = | a - b | then find the angle between a and b
9. If a is a unit vector and ( x - a ).( x + a ) = 8 , then find | x |
10. .Reduce the equation 2x-6y+3z=24 to the intercept form.
11. If
p is a unit vector and ( x - p) . ( x + p) = 80, then find the x








2
, then find the angle between a and b such that a x b is a unit
3
12. Let the vectors a and b be such that a  3 and b 
vector.
13. If a  2,
b 7
a  b = 3iˆ  2 ˆj  6kˆ , find the angle between a and b
14. The Cartesian equations of a line AB are
2x  1 4  y z  1


. Find the dr’s and dc’s of line AB.
2
7
2
15. Find the equation of the plane through the point (2,3,4) and parallel to the plane r.(2i-3j+5k)+7=0.








16. Find the shortest distance between the lines r  (1  t )i  (t  2) j  (3  2t )k and r  ( s  1)i  (2s  1) j  (2 s  1)k
x 1 y  2 z  3
x2 y 4 z 5




and
2
3
4
4
4
5

18. Find the length and foot of perpendicular from the point (1,1,2) to the plane r .(2iˆ  2 ˆj  4kˆ)  5  0
17. Find the shortest distance between the lines
19. Find the Cartesian as well as the vector equations of the planes passing through the intersection of the planes


r .(2iˆ  6 ˆj )  12  0 and r .(3iˆ  ˆj  4kˆ)  5  0 which are at unit distance from origin.
20.Find the foot of perpendicular from (1,2,−3) to the line
21Show that the lines
x2 y 3
z


Also find the length of perpendicular.
2
2
1
x  4 y  6 z 1


and 3x-2y+z+5=0=2x+3y+4z-4 intersect. Find the equation of plane in which they
3
5
2
lie and also their point of intersection.
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22.Find the equation of the perpendicular from the point (3,-1,11 ) to the line
x y 2 z 3


. Also find the foot of the
2
3
4
perpendicular and length of the perpendicular.
23. Find the shortest dis tan ce between the lines
r  ( 1  t ) i  ( t  2 ) j  ( 3  2t )k and
r  ( s  1) i  ( 2 s  1) j  ( 2 s  1) k
24.Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured parallel to the line
x 1 y  3 z 1


2
3
6
25.In a bolt factory, three machines A, B and C manufacture 25%, 35% and 40% of the total bolts manufactured. Of their
output, 5%, 4% and 2% are defective respectively. A bolt is drawn at random and is found to be defective. Find the
probability that it was manufactured by (i) Machine A or C. (ii) machine B.
26.Let X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of
kx

getting admission in x number of colleges. It is given that p ( x  x)  2kx
k (5  x)

if x  0 or i
if x  2
, K is  ve cons tan t.
if x  3 or 4
(a) Find the value of k.(b) What is the probability that you will get admission in exactly two colleges?(c) Find the mean
and variance of the probability distribution.
27.A bag contains 3 red and 7 black balls .Two balls are drawn one after the other without replacement at random from
the bag. If the second selection is given to be red , what is the probability that the first is also red .
28.Find the coordinate of the foot of perpendicular drawn from the point(1,2,1) to the line joining the points (1,4,6)
and (5,4,4).
29.A speaks truth in 60 percent of the cases and B in 90 percent of the cases .In what percentage of cases , are they
likely to contradict each other in stating the same fact.
30.Find the equation of the plane passing through the points (3,2,1) and (0,1,7) and parallel to the line r=2i-j+k+ ‫( ג‬i-j-k).
31. Find the distance of the point (2, 3, 4) from the plane 3x+2y+2z+5=0.


 

32.If a , b and c be three vectors a  3 , b  4 and c  5 and each one of them being perpendicular to sum of the
other two, find
 

a b c
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33.Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point (1,3,4) from the plane
2x – y + z +3 = 0. Find also the image of the point in the plane.
34.A card from a pack of 52 cards is dropped. From the remaining cards two cards are drawn and are found to be club.
Find the prob that the dropped card is club.
35.Find the angle between the lines x − 2y + z = 0 = x + 2y − 2z and x + 2y + z = 0 = 3x + 9y + 5z .
36. Find the distance of point (2,3,4) from the plane 3x + 2y + 2z + 5 = 0 , measured parallel to the line
x3 y2 z


3
6
2
37.Find the shortest distance and vector equation of the line of shortest distance between the lines given by


r  (4iˆ  4 ˆj  kˆ)   (iˆ  ˆj  kˆ) and r  (3iˆ  8 ˆj  3kˆ)   (2iˆ  3 ˆj  3kˆ)








38. Find the shortest distance between the lines r  (1  t )i  (t  2) j  (3  2t )k and r  ( s  1)i  (2s  1) j  (2 s  1)k
x 1 y  2 z  3
x2 y 4 z 5




and
2
3
4
4
4
5

40. Find the length and foot of perpendicular from the point (1,1,2) to the plane r .(2iˆ  2 ˆj  4kˆ)  5  0
39.Find the shortest distance between the lines
41.Find the Cartesian as well as the vector equations of the planes passing through the intersection of the planes


r .(2iˆ  6 ˆj )  12  0 and r .(3iˆ  ˆj  4kˆ)  5  0 which are at unit distance from origin.
x2 y 3
z


Also find the length of perpendicular.
2
2
1
x6 y7 z7


43.Find the foot of perpendicular from (1,2,3) to the line
Also obtain the equation of plane containing
3
2
2
42.Find the foot of perpendicular from (1,2,−3) to the line
the line and the point (1,2,3)
44.Prove that the image of the point ( 3, -2, 1 ) in the plane 3x – y + 4z = 2 lies on the plane, x + y + z + 4 =0
45.Find the distance of the point ( 3,4,5) from the plane x + y + z = 2 measured parallel to the line 2x = y = z
46.Show that the lines
x 1 y  3  z
x  4 1 y z 1




and
are coplanar . Also find the eq of the plane containing
2
4
1
3
2
1
the lines.
47.Find the eq of the plane through the points (3,4,2) and (7,0,6) and is perpendicular to the plane
2x – 5y = 15

  


48.Find the eq of the plane passing through the points 2i  j  k and  i  3 j  4k and perpendicular to the plane


 
r .(i  2 j  4k )  10
A box is selected at random and then a ball is randomly drawn
from the selected box. The colour of the ball is back, what is
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the probability that ball drawn is from the box III.
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box
Colour
Black
White
Red
Blue
I
3
4
5
6
II
2
2
2
2
III
1
2
3
1
IV
4
3
1
5
50.Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets
1, 2, 3, or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is
the probability that she threw 1,2,3 or 4 with the die?
51.Find the angle between the lines x − 2y + z = 0 = x + 2y − 2z and x + 2y + z = 0 = 3x + 9y + 5z .
52.A pair of dice is thrown 3 times. If getting a total of 10 is
considered a success, find the probability distribution of the number of successes. Also find the mean and variance of X.
Paper Submitted By:
Name ESTHER.P
Email [email protected]
Phone No. 0505342999
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