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SAMPLE TEST PAPER 07 FOR CLASS 12
3D GEOMETRY, LINEAR PROGRAMMING &
PROBABILITY
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This question paper consists of 29 questions divided into four sections – A, B, C and D
Section A contains 4 questions of 1 mark each
Section B contains 8 questions of 2 marks each
Section C contains 11 questions of 4 marks each
Section D contains 6 questions of 6 marks each
Ques
Question
No.
Write the coordinates of the foot of the perpendicular from the point (1,2,3) on yaxis.
1
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[1 mark]
Write the distance of a point P(a, b, c) from xaxis.
2
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[1 mark]
¯¯¯
A
3 5 1
3 If P(A) = , P(B) = &P(A∩B)= , then P( )=
8 8 4 B
3612735
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[1 mark]
If the direction ratios of a line are ( − 18, 12, − 4), then direction cosines of a line are
4
3868338
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[1 mark]
Find the vector and the Cartesian equations of the line through the point (5, 2, 4) and which is parallel to
5 2612
ˆ
ˆ ˆ
the vector 3i + 2j − 8k .
[2
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marks]
Find the vector equation for the line passing through the points (1, 0, 2) and (3, 4, 6) .
6 2611
[2
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marks]
7 A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability
1340552 that the number 4 has appeared at least once?
[2
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marks]
In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in
class XII. What is the probability that a student chosen randomly studies in Class XII given that the chosen student is
8 2793
a girl?
[2
marks] Watch Free Video Solution on Doubtnut
2 1 1
Three events A , B and C have probabilities , and respecively. Given tat
9 5 3
2
1 1
26209
P(A∩C)= andP(B∩C)= , find the values of P(C/B)andP(A∩C)
.
5 4
[2
marks]
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1 7 1
10 If A and B are two events such that P(A) = , P(B) = and P(not A or not B) = . State whether A and
2 12 4
2770
B are independent?
[2
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marks]
11
Show that the function f:R → R given by f(x) = cosx for all x ∈ R , is neither oneone nor onto.
1457020
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[2
marks]
−−→ −−→
12
If f : {5, 6}2, 3andg:{2, 3}5, 6 are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, find fog
.
18915
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[2
marks]
y + 4
x−5 z − 6
13
The cartesian equation of a line is = = . Write its vector form.
2558
3 7 2
[4
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marks]
If O is the origin and the coordinates of A are (a, b, c) . Find the direction cosines of OA and the equation of the
14
27134
plane through A at right angles to OA.
[4
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marks]
Find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points
15
27229
(1, 4, 2)and(2, 3, 5) find also the perpendicular distance of the origin from this plane.
.
[4
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marks]
.
16
→
ˆ ˆ ˆ ˆ ˆ ˆ
Find the image of the point having position vector i + 3j + 4k in the plane r 2i − j + k + 3 = 0.
27168
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[4
marks]
17
Find the length and the foot of perpendicular from the point (1, 3/2, 2) to the plane 2x − 2y + 4z + 5 = 0.
27192
[4
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marks]
The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly
18
¯¯¯ ¯¯¯
one of A, B occurs is q then prove that P(A) + P(B) = 2 − 2p + q
26261
[4
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marks]
19 A committee of 4 students is selected at random from a group consisting of 8 boys and 4 girls. Given that there is
atleast one girl in committee, calculate the probability that there are exactly 2 girls in the committee.
11918
[4 Watch Free Video Solution on Doubtnut
marks]
A and B are two candidates seeking admission in a college. The probability that A is selected is 0.7 and the
20
probability that exactly one of them is selected is 0.6. Find the probability that B is selected.
26187
[4 Watch Free Video Solution on Doubtnut
marks]
21 A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What
is the probability that at least one of the three marbles drawn be black, if the first marble is red?
26387
[4 Watch Free Video Solution on Doubtnut
marks]
A and B throw alternately a pair of dice. A wins if he throws 6 before B throws 7 and B wins if he throws 7 before A
22
throws 6. Find their respective chance of winning, if A begins.
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[4 Watch Free Video Solution on Doubtnut
marks]
23 Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is 2headed. If
you take one out, toss it and get a head, what is the probability that it was fair coin?
26255
[4 Watch Free Video Solution on Doubtnut
marks]
Find the vector equation of the plane passing through the points (2, 1, 1) and (1, 3, 4) and perpendicular to the
24 plane x − 2y + 4z = 10 . Also show that the plane thus obtained contains the line
10755 →
ˆ ˆ
ˆ ˆ ˆ ˆ
r = − i +3j +4k+λ (3i −2j − 5k)
.
[6
marks]
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One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from
25
first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball dawn is
26331
white.
[6
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marks]
A bag contains (2n + 1) coins. It is known that n of these coins have a head on both sides whereas the rest of the
coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a
26
31
26328
head is , determine the value of n .
42
[6
marks]
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27 Three machinesE1, E2 and E3 in a certain factory producing electric bulbs, produce 50%, 25% and 25%
13430 respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines
E1 and E2 are defective and that 5% of those produced by machine E3are defective. If one bulb is picked up at
[6 random from a day’s production, calculate the probability that it is defective.
marks]
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A manufacturer produces two products A and B. 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
28
requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on
13463
second machine. Each unit of product A is sold at Rs. 7 profit and that of B at a profit of Rs. 4. Find the production
level per day for maximum profit graphically.
[6
marks]
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Two godowns A and B have gram capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration
shops, D. E and F whose requirements are 60, 50 and 40 quintals respectively. The cost of transportation per quintal
29
from the godowns to the shops are given in the following table: How should the supplies be transported in order that
2655
the transportation cost is minimum? What is the minimum cost?
[6
marks] Watch Free Video Solution on Doubtnut
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