# Free Objective Test 02 Practice Test - 11th and 12th

### Question 1

A set of integers is given as (3,6,8,14,17). What is the probability that a triangle can be constructed.?

#### SOLUTION

Solution :B

Any 3 points can be selected in 5C3 i.e 10 ways.

For forming a triangle sum of two sides should be greater than the 3rd side.

Hence following set of integers can be selected

(3,6,8)

(6,8,17)

(8,14,17)

Hence P(Triangle is formed)= 310

### Question 2

Letters of the word "EDUCATION" is arranged. What is the probability that vowels and consonants are in alphabetical order?

#### SOLUTION

Solution :A

There are 5 vowels and 4 consonants.

Total ways to arrange: 9!

Vowels can be arranged in 5! ways out of which in 15! they are in alphabetical order.

Similarly for consonants.

Total Favorable ways: 9!(5!×4!)

Total ways: 9!

P(E)= 1(5!×4!)

### Question 3

There are 4 torn books in a lot consisting of 10 books. If 5 books are seleceted at random what is the probability of presence of 2 torn books among the selected?

#### SOLUTION

Solution :C

Total Sample space: 10C5=252

Favorable cases: 4C2×6C3=120

Hence P(E)= 120252=1021

### Question 4

Of the 10 prizes 5 prizes are of category Platinum 3 of gold and 2 of silver and they are placed in an enclosure for an olympiad contest. The prizes are awarded by allowing winners to select randomly from the prizes remaining. When the 8th participant goes to collect the prize what the probability that last 3 prizes are 1 of platinum 1 of gold and 1 of silver?

#### SOLUTION

Solution :B

Sample space: 10C7. because 7 prizes have already been selected= 120.

Favorable: 5C4×3C2×2C1=30.

P(E)= 30120=14

### Question 5

A fortune teller has numbers from 1-7 in his box. He draws 3 numbers from it. What is the probability that the number picked is of alternate order as odd-even-odd or even-odd-even?

#### SOLUTION

Solution :B

Sample Space: 7×6×5=210

odd-even-odd= 4×3×3=36

even-odd-even= 3×4×2=24

Total ways: 60

P(E)= 60210=27

### Question 6

Two cubes have their face painted as either red or blue color. 1st cube has 5 faces red and 1 face blue. The Probability that both the faces show same color on rolling is 12. How many red faces are present on the 2nd cube.

#### SOLUTION

Solution :C

Suppose x faces of cube 2 is red then 6-x will be blue.

P(RR or BB)= 56×x6+16×(6−x)6=12

5x36+(6−x)36=12

4x+6=18

x=3

Hence number of red faces is 3.

### Question 7

The probability that a 'P and C' question will be asked in IIT JEE is 25 and probability question is uploaded is 47. If the probability of getting at least 1 is 23 what is the probability that questions from both the topics are asked.

#### SOLUTION

Solution :A

P(A U B)= P(A)+ P(B)- P(A intersection B)

23=(25)+(47)−x

x=17105

### Question 8

A maze with following pattern is given. An insect has to reach point R.It can only land only on R if it lands on the adjacent cubes. What is the probability of it reaching R.

#### SOLUTION

Solution :A

P(It lands on R)= P( S to R)+ P(Q to R)

From Q it can move in 2 directions.

P(Landing on R)= 16×1+16×12=14

### Question 9

CSK lost the toss 13 out of 14 times. What is the probability of this event?

#### SOLUTION

Solution :C

13 losses from 14 can be selected in 14C13 ways i.e.14

At any of the tosses only 2 events can happen win or loss hence total sample space: 214

P(E)= 14214

### Question 10

A bag contains 'x' red balls '2x' white balls and '3x' black balls. 3 balls are drawn at random. The probability that all the balls drawn are of different colors is 0.3 .How many white balls are present in the bag?

#### SOLUTION

Solution :A

P(All different)= (xC1)×(2xC1)×(3xC1)/(6xC3)

P(E)= (6x3)×66x×[(6x−1)×(6x−2)]=310

or, 3x2[(6x−1)×(6x−2)]=310

10x2=18x2−9x+1

or, (x-1)(8x-1)=0

x can not be a fraction hence x=1

Total number of white balls= 2x=2

### Question 11

Urn *A* contains 6 red and 4 black balls and urn *B* contains 4 red and 6 black balls. One ball is drawn at random from urn *A* and placed in urn *B*. Then one ball is drawn at random from urn *B* and placed in urn *A*. If one ball is now drawn at random from urn *A*, the probability that it is found to be red, is

#### SOLUTION

Solution :A

Let the events are

R1 = A red ball is drawn from urn A and placed in B

B1 = A black ball is drawn from urn A and placed in B

R2 = A red ball is drawn from urn B and placed in A

B2 = A black ball is drawn from urn B and placed in A

R = A red ball is drawn in the second attempt from A

Then the required probability

= P(R1R2R)+(R1B2R)+P(B1R2R)+P(B1B2R)

= P(R1)P(R2)P(R)+P(R1)P(B2)P(R)+P(B1)P(R2)P(R)+P(B1)P(B2)P(R)

= 610×511×610+610×611×510+410×411×710+410×711×610

= 3255

### Question 12

The probability of a bomb hitting a bridge is 12 and two direct hits are needed to destroy it. The least number of bombs required so that the probability of the bridge beeing destroyed is greater then 0.9, is

#### SOLUTION

Solution :A

Let n be the least number of bombs required and x the number of bombs that hit the bridge. Then x follows a binomial distribution with parameter n and p=12

Now, P(X≥2)>0.9⇒1−P(X<2)>0.9

⇒P(X=0)+P(X=1)<0.1

⇒nC0(12)n+nC1(12)n−1(12)<0.1⇒10(n+1)<2n

This gives n≥8

### Question 13

One of the two events must occur. If the chance of one is 23 of the other, then odds in favour of the other are

#### SOLUTION

Solution :D

Let p be the probability of the other event, then the probability of the first event is 23 p. Since two events are totally exclusive, we have p+(23)p=1⇒p=35

Hence odds in favour of the other are 3 : 5 – 3, i.e, 3 : 2.

### Question 14

The probability of happening an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then the probability of happening neither A nor B is

#### SOLUTION

Solution :B

P(¯¯¯¯A∩¯¯¯¯B)=P(¯¯¯¯A∪¯¯¯¯B)−1−P(A∪B)

Since A and B are mutually exclusive,

P(A∪B)=P(A)+P(B)

Hence required probability =1–(0.5+0.3)=0.2

### Question 15

If the integers m and n are chosen at random between 1 and 100, then the probability that a number of the form 7m+7n is divisible by 5 equals

#### SOLUTION

Solution :A

Since m and n are selected between 1 and 100, hence sample space = 100×100.

Also 71=7,72=49,73=343,74=2401,75=16807 etc. Hence 1, 3, 7 and 9 will be the last digits in the powers of 7. Hence for favorable cases

n m→ ↓1,1 1,2 1,3.... 1,1002,1 2,2 2,3... 2,100...........................................100,1 100,2 100,3 100,100

For m = 1; n = 3, 7, 11….. 97

∴ Favorable cases = 25

Similarly for every m, favorable n are 25.

∴ Total favourable cases = 100×25

Hence required probability = 100×25100×100=14