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The CORRECT choice is (d) There are 32 DAYS in this month

The best I can explain: When we talk about this, that, here, there, we are not sure about what we are talking about so “There are 27 days in this month” is not a statement.

“FEBRUARY has 28 days” and “February has 29 days” both are not statements because February may have 28 days or 29 das based on the year. “There are 32 days in this month” is a statement as it is FALSE. We cannot have 32 days in a month.

Right OPTION is (C) FALSE, false

To explain I would SAY: In contrapositive method, we assume p to be false and use it to prove that q is also false.

There is no case for p to be true in direct method.

Correct answer is (B) EXCLUSIVE Or, TRUE

The EXPLANATION: Square is a QUADRILATERAL is true so the statement is true. Square is not a pentagon so it is Exclusive Or.

Correct choice is (d) 100 is DIVISIBLE by 2 and 5

Easiest explanation: PRIME number is not divisible by 2 always. Two and two makes our not five. All integers are not positive. 100 is divisible by 2 and 5 both. So, COMPONENT statements of “100 is divisible by 2 and 5” are true.

Correct answer is (c) It is false that Delhi is not in India

To explain: “Delhi is not in India”, “It is false that Delhi is in India”, “It is not the CASE that Delhi is in India” are negations of the GIVEN statement. Negation of a statement can be FORMED by adding ‘it is false that’ or ‘it is not the case that’. In statement “It is false that Delhi is not in India” two negation phrases are USED which is wrong so it is not the negation statement.

Correct choice is (a) Inclusive Or

For EXPLANATION I would SAY: If all the COMPONENT STATEMENTS can be true simultaneously then ‘Or’ is SAID to be ‘Inclusive Or’.

Correct choice is (c) Sum of a and b is 5

For EXPLANATION I would say: “TWO and two makes four” and “Elephant is heavier than ant” are TRUE so they are statements. “A prime number is ALWAYS odd” is false as prime number may be even so it is a statement. “Sum of and b is 5” is not a statement as it can be true or false based on the values of a and b taken.

The CORRECT answer is (a) True

The best EXPLANATION: If we write “Q if and only if p” then it MEANS q <=> p i.e. p=> q and q=> p.

And if we write “p if and only if q” then it means p <=> q i.e. q=> p and p=> q.

So, both are exactly same.

Right ANSWER is (C) any, false

The best I can explain: The compound statement with ‘And’ can be false if any of its component statements are false. All component statements can be false. In such CASE ALSO, the compound statement will be false.

Right choice is (c) STATEMENTS generally not USE word like sum and product

For explanation I WOULD SAY: Statements generally not use ambiguous words like here, there, this, that, today, tonight, tomorrow.

The correct choice is (c) 81 is a square and cube

For explanation I would SAY: SINCE Delhi and Mumbai both are cities in India so “Delhi and Mumbai are cities of India” is true.

Since Chandigarh is the CAPITAL of both states Punjab and Haryana so “Chandigarh is capital of Punjab and Haryana” is true.

Since 81 is a square and not a cube so “81 is a square and cube” is false.

Since 16 is EVEN number and is also divisible by 8 so “16 is even number and divisible by 8” is true.

Right option is (B) all, FALSE

The best I can EXPLAIN: The compound statement with ‘Or’ can be false if all its COMPONENT statements are false. If this condition is not SATISFIED, the compound statement cannot be false.

The CORRECT choice is (b) true, true

For explanation I would SAY: In direct method, we assume p to be true and use it to PROVE that q is ALSO true.

There is no case for p to be false in direct method.

The CORRECT answer is (d) Japan or China is in INDIA

For explanation I would say: 24 is EVEN NUMBER so “24 is even number or odd number” is correct.15 is divisible by 3 so “15 is prime or divisible by 3” is true.71 is prime as well as odd so “71 is odd number or prime” is true.Neither Japan nor China is in India so “Japan or China is in India” is false.

Correct OPTION is (B) False

Easy explanation: If we write “Q only if p” then it MEANS q => p.

And if we write “p only if q” then it means p => q.

So, both are DIFFERENT.

Right answer is (b) if q then p

Best EXPLANATION: Converse statement for GIVEN statement “if p then q” is “if q then p” i.e. q => p or q IMPLIES p.

The correct OPTION is (b) ~q => ~P

For EXPLANATION: If p then q means p => q or NEGATION of q implies negation of pi.e. ~q => ~p.

The CORRECT OPTION is (c) Inclusive Or

To explain: Since people can have both Aadhar card and VOTER ID. In such case also, they are eligible to apply for RATION card so phrase used is inclusive or.

Right option is (B) 43 is prime as WELL as odd

Explanation: The compound STATEMENT with ‘And’ can be TRUE if all its component statements are true. If this condition is not satisfied, the compound statement cannot be true.

1 is neither less than 2 nor less than 3 so it is false. 43 is prime number as well as odd so it is true.

0 is neither positive number nor negative number. 16 is not divisible by 3 so it is false.

Correct answer is (d) Q, P

To ELABORATE: If p then q means q is NECESSARY condition for p or p => q. It is not same as p is necessary condition for q.

The correct option is (B) p => Q

To elaborate: If p then q means p IMPLIES q i.e. p => q. This does not mean q => p.

Q only if p means q=>p.

The CORRECT ANSWER is (C) p, Q

Explanation: If p then q MEANS p is sufficient condition for q or p=>q. It is not same as q is sufficient condition for p.

Right CHOICE is (a) true, true

To explain I WOULD say: If p then q means if p is true then q must be true. It SAYS nothing when p is FALSE. If p is false then q MIGHT be true or false.

Right choice is (b) Two plus two is three

The explanation: A sentence is called mathematically acceptable statement if it is EITHER TRUE or FALSE but not both.

“Two plus two is three” is false so it is a statement. Rest we cannot decide WHETHER they are true or false.

Correct ANSWER is (a) SUM of 2 and 3 is smaller than 4

For explanation I would say: Negation of greater is ‘not greater’. ‘Not greater’ means either smaller than or equal to. So, “Sum of 2 and 3 is smaller than or equal to 4”, “Sum of 2 and 3 is not greater than 4”, “Sum of 2 and 3 is not greater than 4” are negation of the GIVEN STATEMENT. And “Sum of 2 and 3 is smaller than 4” is not the negation of given statement as it should include equal to also.

Right ANSWER is (B) False

The BEST EXPLANATION: We know, any integer cannot be POSITIVE as well as negative simultaneously so ‘or’ is exclusive. Hence it is false.

The correct answer is (b) Exclusive Or

To explain I WOULD say: Students can take any ONE of English and Hindi as fifth subject so phrase USED in the give statement is Exclusive Or.

The correct option is (B) Exclusive Or

To elaborate: If all the COMPONENT STATEMENTS can be true SIMULTANEOUSLY then ‘Or’ is said to be ‘Exclusive Or’.

Right choice is (c) There EXISTS REAL number a and b for which a*b ≠ b*a

For explanation I WOULD say: Negation of for every is not for every. a*b = b*a not for every real number a and b.

This MEANS There exists real number a and b for which a*b ≠ b*a.

Right choice is (a) A rectangle is a quadrilateral is false

The best EXPLANATION: A rectangle has FOUR SIDES so it is a quadrilateral. All the four sides are not EQUAL as two are equal and other two are equal.

Correct option is (b) No person in India speaks Hindi

The best I can explain: Negation of EVERYONE is not everyone. So, negation of the given STATEMENT should be “Not everyone in India speaks Hindi”, “Someone in India does not speaks Hindi”, “At LEAST ONE person in India who does not speaks Hindi”. “No person in India speaks Hindi” cannot be the negation of the given statement as this means no one speaks Hindi which does not means not everyone.

The correct ANSWER is (b) False

To EXPLAIN I would say: “How far is Delhi from here” is not a statement as we cannot DECIDE what is “here” and from where we are GOING to measure the DISTANCE to Delhi.

Right CHOICE is (a) True

Explanation: We know, square of a natural is always positive. For a STATEMENT connected with ‘Or’, at least ONE of the component statements to be true so it is true.

Right CHOICE is (d) All angles are UNEQUAL in equilateral TRIANGLE

To explain I would SAY: Negation of all angles is ‘not all angles’. So, negation of the given statement is “There is at least one angle different in equilateral triangle”, “It is false that all angles are equal in equilateral triangle”, “It is not the case that all angles are equal in equilateral triangle”. But “All angles are unequal in equilateral triangle” cannot be the negation of given statement as this does not include isosceles triangles.

The CORRECT answer is (a) P, q

To elaborate: If p then q MEANS p only if q or p=>q which is not same as q only if p or q=>p.