Find out which of the following sentences are statements and which are not. Justify your answer.

Listen to me, Ravi!

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Every set is a finite set.

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Two non-empty sets have always a non-empty intersection.

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The cat pussy is black.

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Are all circles round?

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All triangles have three sides.

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Every rhombus is a square.

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x² + 5|x| + 6 = 0 has no real roots.

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This sentences is a statement

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Is the earth round?

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Go!

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The real number x is less than 2.

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There are 35 days in a month.

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Mathematics is difficult.

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All real numbers are complex numbers.

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The product of (-1) and 8 is 8.

Give three examples of sentences which are not statements. Give reasons for the answers.

A coin is tossed once. Write its sample space.

If a coin is tossed two times, describe the sample space associated to this experiment.

If a coin is tossed three times (or three coins are tossed together), then describe the sample space for this experiment.

Write the sample space for the experiment of tossing a coin four times.

Two dice are thrown. Describe the sample space of this experiment.

What is the total number of elementary events associated to the random experiment of throwing three dice together?

A coin is tossed and then a die is thrown. Describe the sample space for this experiment.

A coin is tossed and then a die is rolled only in case a head is shown on the coin. Describe the sample space for this experiment.

A coin is tossed twice. If the second throw results I a tail, a die is thrown. Describe the sample space for this experiment.

An experiment consists of tossing a coin and then tossing it second time if head occurs. If a tail occurs on the first toss, then a die is tossed once. Find the sample space.

A coin is tossed. If it shows tail, we draw a ball from a box which contains 2 red 3 black balls; it shows head, we throw a die. Find the sample space of this experiment.

A coin is tossed repeatedly until a tail comes up for the first time. Write the sample space for this experiment.

A box contains 1 red and 3 black balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

A pair of dice is rolled. If the outcome is a doublet, a coin is tossed. Determine the total number of elementary events associated to this experiment.

A coin is tossed twice. If the second draw results in a head, a die is rolled. Write the sample space for this experiment.

A bag contains 4 identical red balls and 3 identical black balls. The experiment consists of drawing one ball, then putting it into the bag and again drawing a ball. What are the possible outcomes of the experiment?

In a random sampling three items are selected from a lot. Each item is tested and classified as defective (D) or non-defective (N). Write the sample space of this experiment.

An experiment consists of boy-girl composition of families with 2 children.

(i) What is the sample space if we are interested in knowing whether it is boy or girl in the order of their births?

(ii) What is the sample space if we are interested in the number of boys in a family?

There are three coloured dice of red, white and black colour. These dice are placed in a bag. One die is drawn at random from the bag and rolled, its colour and the number on its uppermost face is noted. Describe the sample space for this experiment.

2 boys and 2 girls are in room P and 1 boy 3 girls are in room Q. Write the sample space for the experiment in which a room is selected and then a person.

A bag contains one white and one red ball. A ball is drawn from the bag. If the ball drawn is white it is replaced in the bag and again a ball is drawn. Otherwise, a die is tossed. Write the sample space for this experiment.

A box contains 1 white and 3 identical black balls. Two balls are drawn at random in succession without replacement. Write the sample space for this experiment.

An experiment consists of rolling a die and then tossing a coin once it number on the die is even. If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment.

A die is thrown repeatedly until a six comes up. What is the sample space for this experiment.

Write the negation of the following statement:

Banglore is the capital of Karnataka.

Write the negation of the following statement:

It rained on July 4, 2005.

Write the negation of the following statement:

Ravish is honest.

Write the negation of the following statement:

The earth is round.

Write the negation of the following statement:

The sun is cold.

All birds sing.

Some even integers are prime.

There is a complex number which is not a real number.

I will not go to school

Both the diagonals of a rectangle have the same length.

All policemen are thieves.

Are the following pairs of statements are a negation of each other:

(i) The number x is not a rational number.

The number x is not an irrational number.

(ii) The number x is not a rational number.

The number x is an irrational number.

Write the negation of the following statements:

p : For every positive real number x, the number (x – 1) is also positive.

Write the negation of the following statements:

q : For every real number x, either x > 1 or x < 1.

Write the negation of the following statements:

r : There exists a number x such that 0 < x < 1.

Check whether the following pair of statements is a negation of each other. Give reasons for your answer.

(i) a + b = b + a is true for every real number a and b.

(ii) There exist real numbers a and b for which a + b = b + a.

A coin is tossed. Find the total number of elementary events and also the total number of events associated with the random experiment.

List all events associated with the random experiment of tossing of two coins. How many of them are elementary events?

Three coins are tossed once. Describe the following events associated with this random experiment:

A = Getting three heads, B = Getting two heads and one tail, C = Getting three tails, D = Getting a head on the first coin.

(i) Which pairs of events are mutually exclusive?

(ii) Which events are elementary events?

(iii) Which events are compound events?

In a single throw of a die describe the following events:

(i) A = Getting a number less than 7

(ii) B = Getting a number greater than 7

(iii) C = Getting a multiple of 3

(iv) D = Getting a number less than 4

(v) E = Getting an even number greater than 4.

(vi) F = Getting a number not less than 3.

Also, find A ∪ B, A ∩ B, B ∩ C, E ∩ F, D ∩ F and .

Three coins are tossed. Describe

(i) two events A and B which are mutually exclusive.

(ii) three events A, B and C which are mutually exclusive and exhaustive.

(iii) two events A and B which are not mutually exclusive.

(iv) two events A and B which are mutually exclusive but not exhaustive.

A die is thrown twice. Each time the number appearing on it is recorded. Describe the following events:

(i) A = Both numbers are odd.

(ii) B = Both numbers are even

(iii) C = sum of the numbers is less than 6.

Also, find A ∪ B, A ∩ B, A ∪ C, A ∩ C. Which pairs of events are mutually exclusive.

Two dice are thrown. The events A, B, C, D, E and F are describes as follows.

A = Getting an even number on the first die.

B. Getting an odd number on the first die.

C = Getting at most 5 as sum of the numbers on the two dice.

D = Getting the sum of the numbers on the dice greater than 5 but less than 10.

E = Getting at least 10 as the sum of the numbers on the dice.

F = Getting an odd number on one of the dice.

(i) Describe the following events: A and B, B or C, B and C, A and E, A or F, A and F

(ii) State true or false:

(a) A and B are mutually exclusive.

(b) A and B are mutually exclusive and exhaustive events

(c) A and C are mutually exclusive events

(d) C and D are mutually exclusive and exhaustive events

(e) C, d and E are mutually exclusive and exhaustive events

(f) A’ and B’ are mutually exclusive events

(g) A, B, F are mutually exclusive and exhaustive events

The numbers 1, 2, 3 and 4 are written separately on four slips of paper. The slips are then put in a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the following events:

A = The number on the first slip is larger than the one on the second slip.

B = The number on the second slip is greater than 2

C = The sum of the numbers on the two slips is 6 or 7

D = The number on the second slips is twice that on the first slip.

Which pair (s) of events is (are) mutually exclusive?

A card is picked up from a deck of 52 playing cards.

(i) What is the sample space of the experiment?

(ii) What is the event that the chosen card is a black faced card?

Find the component statements of the following compound statements:

(i) The sky is blue, and the grass is green.

(ii) The earth is round, or the sun is cold.

(iii) All rational numbers are real, and all real numbers are complex.

(iv) 25 is a multiple of 5 and 8.

For each of the following statements, determine whether an inclusive “OR” o exclusive “OR” is used. Give reasons for your answer.

(i) Students can take Hindi or Sanskrit as their third language.

(ii) To entry a country, you need a passport or a voter registration card.

(iii) A lady gives birth to a baby boy or a baby girl.

(iv) To apply for a driving license, you should have a ration card or a passport.

Write the component statements of the following compound statements and check whether the compound statement is true or false:

(i) To enter into a public library children need an identification card from the school or a letter from the school authorities.

(ii) All rational numbers are real and all real numbers are not complex.

(iii) Square of an integer is positive or negative.

(iv) x = 2 and x = 3 are the roots of the equation 3x² – x – 10 = 0.

(v) The sand heats up quickly in the sun and does not cool down fast at night.

Determine whether the following compound statements are true or false:

(i) Delhi is in India and 2 + 2 = 4

(ii) Delhi is in England and 2 + 2 = 4

(iii) Delhi is in India and 2 + 2 = 5

(iv) Delhi is in England and 2 + 2 = 5

Which of the following cannot be valid assignment of probability for elementary events or outcomes of sample space S = {w₁, w₂, w₃, w₄, w₅, w₆, w₇}:

A die is thrown. Find the probability of getting:

(i) a prime number

(ii) 2 or 4

(iii) a multiple of 2 or 3

In a simultaneous throw of a pair of dice, find the probability of getting:

(i) 8 as the sum

(ii) a doublet

(iii) a doublet of prime numbers

(iv) an even number on first

(v) a sum greater than 9

(vi) an even number on first

(vii) an even number on one and a multiple of 3 on the other

(viii) neither 9 nor 11 as the sum of the numbers on the faces

(ix) a sum less than 6

(x) a sum less than 7

(xi) a sum more than 7

(xii) neither a doublet nor a total of 10

(xiii) odd number on the first and 6 on the second

(xiv) a number greater than 4 on each die

(xv) a total of 9 or 11

(xvi) a total greater than 8

In a single throw of three dice, find the probability of getting a total of 17 or 18

Three coins are tossed together. Find the probability of getting:

(i) exactly two heads

(ii) at least two heads

(iii) at least one head and one tail

What is the probability that an ordinary year has 53 Sundays?

What is the probability that a leap year has 53 Sundays and 53 Mondays?

A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that:

(i) All the three balls are white

(ii) All the three balls are red

(iii) One ball is red and two balls are white

In a single throw of three dice, find the probability of getting the same number on all the three dice

Two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10

A card is drawn at random from a pack of 52 cards. Find the probability that the card drawn is ;

(i) a black king

(ii) either a black card or a king

(iii) black and a king

(iv) a jack, queen or a king

(v) neither an ace nor a king

(vi) spade or an ace

(vii) neither an ace nor a king

(viii) a diamond card

(ix) not a diamond card

(x) a black card

(xi) not an ace

(xii) not a black card

In shutting a pack of 52 playing cards, four are accidently dropped; find the chance that the missing cards should be one from each suit

From a deck of 52 cards, four cards are drawn simultaneously, find the chance that they will be the four honors of the same suit

Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is a multiple of 3 or 7?

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that one is red, one is white and one is blue

A bag contains 7 white, 5 black and 4 red balls. If two balls are drawn at random, find the probability that:

(i) both the balls are white

(ii) one ball is black and the other red

(iii) both the balls are of the same colour

A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that:

(i) one is red and two are white

(ii) two are blue and one is red

(iii) one is red

Five cards are drawn from a pack of 52 cards. What is the chance that these 5 will contain:

(i) just one ace

(ii) at least one ace?

The face cards are removed from a full pack. Out of the remaining 40 cards, 4 are drawn at random. What is the probability that they belong to different suits?

There are four men and six women on the city councils. If one council member is selected for a committee at random, how likely is that it is a women?

A box contains 100bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that:

(i) all 10 are defective

(ii) all 10 are good

(iii) at least one is defective

(iv) none is defective

Find the probability that in a random arrangement of the letters of the word ‘SOCIAL’ vowels come together

The letters of the word ‘CLIFTON’ are placed at random in a row. What is the chance that two vowels come together?

The letters of the word “FORTUNATES’ are arranged at random in a row. What is the chance that the two ‘T’ come together?

A committee of two persons is selected from two men and two women. What is the probability that the committee will have (i) no man? (ii) one man? (iii) two men?

If odds in favour of an event be 2: 3, find the probability of occurrence of this event

If odds against an events be 7: 9, find the probability of non-occurrence of this event.

Two balls are drawn at random from a bag containing 2 white, 3 red, 5 green and 4 black balls, one by one without, replacement. Find the probability that both the balls are of different colours.

Two unbiased dice are thrown. Find the probability that:

(i) neither a doublet nor a total of 8 will appear

(ii) the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3

A bag contains 8 red, 3 white and 9 blue balls. If three balls are drawn at random, determine the probability that (i) all the three balls are blue balls 9ii) all the balls are of different colours

A bag contains 5 red, 6 white and 7 black balls. Two balls are drawn at random. What is the probability that both balls are red or both are black?

If a letter is chosen at random from the English alphabet, find the probability that the letter is (i) a vowel (ii) a constant

In a lottery, a person chooses six different numbers at random from 1 to 20, and if these six numbers match with six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game?

20 cards are numbered from 1 to 20. One card is drawn at random. What is the probability that the number on the cards is:

(i) a multiple of 4?

(ii) not a multiple of 4?

(iii) odd?

(iv) greater than 12?

(v) divisible by 5?

(vi) not a multiple of 6?

Two dice are thrown. Find the odds in favour of getting the sum

(i) 4 (b) 5

(iii) What are the odds against getting the sum 6?

What are the odds in favour of getting a spade if a card is drawn from a well-shuffled deck of cards? What are the odd in favour of getting a king?

A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn at random. From the box, what is the probability that:

(i) al are blue?

(ii) at least one is green?

A box contains 6 red marbles numbered 1 through 6 and 4 white marbles numbered from 12 through 15. Find the probability that a marble drawn is (i) white (ii) white and odd numbered (iii) even numbered (iv) red or even numbered.

A class consists of 10 boys and 8 girls. Thee students are selected at random. What is the probability that the selected group has (i) all boys? (ii) all girls? (iii) 1 boy and 2 girls? (iv) at least one girl? (v) at most one girl?

Five cards are drawn from a well-shuffled pack of 52 cards. Find the probability that all the five cards are hearts.

A bag contains tickets numbered from 1 to 20. Two tickets are drawn. Find the probability that (i) both the tickets have prime numbers on them (ii) on one there is a prime number and on the other, there is a multiple of 4.

An urn contains 7 white, 5 black and 3 red balls. Two balls are drawn at random. Find the probability that 9i) both the balls are red (ii) one ball is red and the other is black (iii) one ball is white.

A and B throw a pair of dice. If A throws 9, find B’s chance of throwing a higher number

In a hand at Whist, what is the probability that four kings are held by a specified player?

Find the probability that in a random arrangement of the letters of the word ‘UNIVERSITY’, the two I’s do not come together.

Write the negation of each of the following statements:

(i) For every x ∈ N, x + 3 > 10

(ii) There exists x ∈ N, x + 3 = 10

Negate each of the following statements:

(i) All the students complete their homework.

(ii) There exists a number which is equal to its square.

Write each of the following statements in the form “if p, then q.”

(i) You can access the website only if you pay a subscription fee.

(ii) There is traffic jam whenever it rains.

(iii) It is necessary to have a passport to log on to the server.

(iv) It is necessary to be rich in order to be happy.

(v) The game is canceled only if it is raining.

(vi) It rains only if it is cold.

(vii) Whenever it rains, it is cold.

(viii) It never rains when it is cold.

State the converse and contrapositive of each of the following statements:

If it is hot outside, then you feel thirsty.

State the converse and contrapositive of each of the following statements:

I go to a beach whenever it is a sunny day.

State the converse and contrapositive of each of the following statements:

A positive integer is prime only if it has no divisions other than 1 and itself.

State the converse and contrapositive of each of the following statements:

If you live in Delhi, then you have winter clothes.

State the converse and contrapositive of each of the following statements:

If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Rewrite each of the following statements in the form “p if and only is q.”

(i) p : If you watch television, then your mind is free, and if your mind is free, then you watch television.

(ii) q : If a quadrilateral is equiangular, then it is a rectangle, and if a quadrilateral is a rectangle, then it is equiangular.

(iii) r : For you to get an A grade, it is necessary and sufficient that you do all the homework you regularly.

(iv) s : If a tumbler is half empty, then it is half full, and if a tumbler is half full, then it is half empty.

Determine the contrapositive of each of the following statements:

(i) If Mohan is a poet, then he is poor.

(ii) Only if Max studies will he pass the test.

(iii) If she works, she will earn money.

(iv) If it snows, then they do not drive the car.

(v) It never rains when it is cold.

(vi) If Ravish skis, then it snowed.

(vii) If x is less than zero, then x is not positive.

(viii) If he has courage he will win.

(ix) It is necessary to be strong in order to be a sailor.

(x) Only if he does not tire will he win.

(xi) If x is an integer and x² is odd, then x is odd.

Check the validity of the following statements:

(i) p : 100 is a multiple of 4 and 5.

(ii) q : 125 is a multiple of 5 and 7.

(iii) r : 60 is a multiple of 3 or 5.

Check whether the following statement is true or not:

p : If x and y are odd integers, then x + y is an even integer.

Check whether the following statement is true or not:

q : if x, y are integer such that xy is even, then at least one of x and y is an even integer.

Show that the statement

p : “If x is a real number such that x³ + x = 0, then x is 0” is true by

(i) Direct method

(ii) method of contrapositive

(iii) method of contradiction

Show that the following statement is true by the method of the contrapositive

p : “If x is an integer and x² is odd, then x is also odd.”

Show that the following statement is true

“The integer n is even if and only if n² is even”

By giving a counter example, show that the following statement is not true.

p : “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”

Which of the following statements are true and which are false? In each case give a valid reason for saying so

(i) p : Each radius of a circle is a chord of the circle.

(ii) q : The centre of a circle bisect each chord of the circle.

(iii) r : Circle is a particular case of an ellipse.

(iv) s : If x and y are integers such that x > y, then – x < - y.

Determine whether the argument used to check the validity of the following statement is correct:

p: “If x² is irrational, then x is rational.”

The statement is true because the number x² = π² is irrational, therefore x = π is irrational.