Math Ncert Exemplar 2019 Solutions for Class 11 Science Maths Chapter 14 - Mathematical Reasoning

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Page No 261:

Question 1:

Which of the following sentences are statements? Justify
(i) A triangle has three sides.
(ii) 0 is a complex number.
(iii) Sky is red.
(iv) Every set is an infinite set.
(v) 15 + 8 > 23.
(vi) y + 9 = 7.
(vii) Where is your bag?
(viii) Every square is a rectangle.
(ix) Sum of opposite angles of a cyclic quadrilateral is 180°.
(x) sin²x + cos²x = 0

Answer:

A statement is a sentence which is either true or false but not both simultaneously.
(j) A triangle has three sides- It is a statement which is true.
(ii) 0 is a complex number- It is a statement which is true.
(iii) Sky is red- It is a statement which is false.
(iv) Every set is an infinite set- It is a statement which is false.
(v) 15 + 8 > 23- It is a statement which is false.
(vi) y + 9 = 7- It is not a statement as the value of y is not given.
(vii) Where is your bag?- It is a question and not a statement.
(viii) Every square is a rectangle- It is a statement which is true.
(ix) Sum of opposite angles of a cyclic quadrilateral is 180°- It is a statement which is true.
(x) sin²x + cos²x = 0- It is a statement which is false.

Page No 262:

Question 2:

Find the component statements of the following compound statements.
(i) Number 7 is prime and odd.
(ii) Chennai is in India and is the capital of Tamil Nadu.
(iii) The number 100 is divisible by 3, 11 and 5.
(iv) Chandigarh is the capital of Haryana and U.P.
(v) $\sqrt{7}$ is a rational number or an irrational number.
(vi) 0 is less than every positive integer and every negative integer.
(vii) Plants use sunlight, water and carbon dioxide for photosynthesis.
(viii) Two lines in a plane either intersect at one point or they are parallel.
(ix) A rectangle is a quadrilateral or a 5 - sided polygon.

Answer:

(i) p: Number 7 is prime.
q: Number 7 is odd.

(ii) p: Chennai is in India.
q: Chennai is capital of Tamil Nadu.

(iii) p: The number 100 is divisible by 3.
q: The number 100 is divisible by 11.
r: The number100 is divisible by 5.

(iv) p: Chandigarh is capital of Haryana.
q: Chandigarh is capital of U.P.

(v) p: $\sqrt{7}$ is a rational number.
q: $\sqrt{7}$ is an irrational number.

(vi) p: 0 is less than every positive integer.
q: 0 is less than every negative integer.

(vii) p: Plants use sunlight for photosynthesis.
q: Plants use water for photosynthesis.
r: Plants use carbon dioxide for photosynthesis.

(viii) p: Two lines in a plane intersect at one point.
q: Two lines in a plane are parallel.

(ix) p: A rectangle is a quadrilateral.
q: A rectangle is a 5-sided polygon.

Page No 262:

Question 3:

Write the component statements of the following compound statements and check whether the compound statement is true or false.
(i) 57 is divisible by 2 or 3.
(ii) 24 is a multiple of 4 and 6.
(iii) All living things have two eyes and two legs.
(iv) 2 is an even number and a prime number.

Answer:

For a compound statement of the form p $\lor$ q, the statement is true if either p or q or both are true.
Similarly, for a compound statement of the form p $\land$ q, the statement is true if both p and q are true.
(i) The given compound statement is of the form p $\lor$ q.
p: 57 is divisible by 2. [false]
q: 57 is divisible by 3. [true]
Since, one of them is true. Therefore, it is true statement.

(ii) The given compound statement is of the form p $\land$ q.
p: 24 is multiple of 4. [true]
q: 24 is multiple of 6. [true]
Since, both the statements are true. Therefore, it is a true statement.

(iii) The given compound statement is of the form p $\land$ q.
p: All living things have two eyes. [false]
q: All living things have two legs. [false]
Since, both the statements are false. Therefore, it is a false component statement.

(iv) The given compound statement is of the form p $\land$ q.
p: 2 is an even number. [true]
q: 2 is a prime number. [true]
Since, both the statements are true. Therefore, it is a true statement.

Page No 262:

Question 4:

Write the negation of the following simple statements
(i) The number 17 is prime.
(ii) 2 + 7 = 6.
(iii) Violets are blue.
(iv) $\sqrt{5}$ is a rational number.
(v) 2 is not a prime number.
(vi) Every real number is an irrational number.
(vii) Cow has four legs.
(viii) A leap year has 366 days.
(ix) All similar triangles are congruent.
(x) Area of a circle is same as the perimeter of the circle.

Answer:

(i) The number 17 is not prime.
(ii) 2 + 7 ≠ 6.
(iii) Violets are not blue.
(iv) $\sqrt{5}$ is not a rational number.
(v) 2 is a prime number.
(vi) Every real number is not an irrational number.
(vii) Cow does not have four legs.
(viii) A leap year does not have 366 days.
(ix) There exist similar triangles which are not congruent.
(x) Area of a circle is not same as the perimeter of the circle

Page No 262:

Question 5:

Translate the following statements into symbolic form
(i) Rahul passed in Hindi and English.
(ii) x and y are even integers.
(iii) 2, 3 and 6 are factors of 12.
(iv) Either x or x + 1 is an odd integer.
(v) A number is either divisible by 2 or 3.
(vi) Either x = 2 or x = 3 is a root of 3x² – x – 10 = 0
(vii) Students can take Hindi or English as an optional paper.

Answer:

(i) p: Rahul passed in Hindi.
q: Rahul passed in English.
Thus, its symbolic form is p $\land$ q: Rahul passed in Hindi and English.

(ii) p: x is an even integer.
q: y is an even integer.
Thus, its symbolic form is p $\land$ q: x and y are even integers..

(iii) p: 2 is factor of 12.
q: 3 is factor of 12.
r: 6 is factor of 12.
Thus, its symbolic form is p $\land$ q $\land$ r: 2, 3 and 6 are factors of 12.

(iv) p: x is an odd integer.
q: (x + 1) is an odd integer.
Thus, its symbolic form is p $\lor$ q: Either x or (x + 1) is an odd integer..

(v) p: A number is divisible by 2.
q: A number is divisible by 3.
Thus, its symbolic form is p $\lor$ q: A number is either divisible by 2 or 3..

(vi) p: x = 2 is a root of 3x² – x – 10 = 0.
q: x = 3 is a root of 3x² – x – 10 = 0.
Thus, its symbolic form is p $\lor$ q: Either x = 2 or x = 3 is a root of 3x² – x – 10 = 0.

(vii) p: Students can take Hindi as an optional paper.
q: Students can take English as an optional paper.
Thus, its symbolic form is p $\lor$ q: Students can take Hindi or English as an optional paper..

Page No 263:

Question 6:

Write down the negation of following compound statements
(i) All rational numbers are real and complex.
(ii) All real numbers are rationals or irrationals.
(iii) x = 2 and x = 3 are roots of the quadratic equation x² – 5x + 6 = 0.
(iv) A triangle has either 3-sides or 4-sides.
(v) 35 is a prime number or a composite number.
(vi) All prime integers are either even or odd.
(vii) |x| is equal to either x or – x.
(viii) 6 is divisible by 2 and 3.

Answer:

The negation of compound statements is given as:
~ (p $\land$ q) = ~p $\lor$ ~q and
~ (p $\lor$ q) = ~p $\land$ ~q
(i) Let p: All rational numbers are real.
q: All rational numbers are complex.
~ p: All rational numbers are not real.
~ q: All rational numbers are not complex.
Then,
~ (p $\land$ q): Either all rational numbers are not real or not complex.

(ii) Let p: All real numbers are rationals.
q: All real numbers are irrationals.
Then,
~ (p $\lor$ q): All real numbers are not rational and all real numbers are not irrational.

(iii) Let pâ€‹: x = 2 is root of quadratic equation x² – 5x + 6 = 0.
q: x = 3 is root of quadratic equation x² – 5x + 6 = 0.
Then,
~ (p $\land$ q): Either x = 2 is not a root of quadratic equation x² – 5x + 6 = 0 or x = 3 is not a root of the quadratic equation x² – 5x + 6 = 0.

(iv) Let p: A triangle has 3 sides.
q: A triangle has 4 sides.
Then,
~ (p $\lor$ q): A triangle neither has 3 nor 4 sides.

(v) Let p: 35 is a prime number.
q: 35 is a composite number.
Then,
~ (p $\lor$ q): 35 is neither a prime nor a composite number.

(vi) Let p: All prime integers are even.
q: All prime integers are odd.
Then,
~ (p $\lor$ q): All prime integers are not even and all prime integers are not odd.

(vii) Let p: |x| is equal to x.
q: |x| is equal to –x.
Then,
~ (p $\lor$ q): |x| is not equal to x and it is not equal to –x.

(viii) Let p: 6 is divisible by 2.
q: 6 is divisible by 3.
Then,
~ (p $\land$ q): 6 is not divisible by 2 or it is not divisible by 3.

Page No 263:

Question 7:

Rewrite each of the following statements in the form of conditional statements
(i) The square of an odd number is odd.
(ii) You will get a sweet dish after the dinner.
(iii) You will fail, if you will not study.
(iv) The unit digit of an integer is 0 or 5 if it is divisible by 5.
(v) The square of a prime number is not prime.
(vi) 2b = a + c, if a, b and c are in A.P.

Answer:

(i) If the number is odd number, then its square is odd number.

(ii) If you take dinner, then you will get sweet dish.

(iii) If you will not study, then you will fail.

(iv) If an integer is divisible by 5, then its unit digit is 0 or 5.

(v) If the number is prime, then its square is not prime.

(vi) If a, b and c are in A.P, then 2b = a + c.

Page No 263:

Question 8:

Form the biconditional statement p ↔ q, where

(i) p : The unit digit of an integer is zero.

q : It is divisible by 5.

(ii) p : A natural number n is odd.

q : Natural number n is not divisible by 2.

(iii) p : A triangle is an equilateral triangle.

q : All three sides of a triangle are equal.

Answer:

(i) p ↔ q: The unit digit of an integer is zero if and only if it is divisible by 5.

(ii) p ↔ q: A natural number n is odd if and only if it is not divisible by 2.

(iii) p ↔ q: A triangle is an equilateral triangle if and only if all three sides of triangle are equal.

Page No 263:

Question 9:

Write down the contrapositive of the following statements:
(i) If x = y and y = 3, then x = 3.
(ii) If n is a natural number, then n is an integer.
(iii) If all three sides of a triangle are equal, then the triangle is equilateral.
(iv) If x and y are negative integers, then xy is positive.
(v) If natural number n is divisible by 6, then n is divisible by 2 and 3.
(vi) If it snows, then the weather will be cold.
(vii) If x is a real number such that 0 < x < 1, then x² < 1.

Answer:

(i) If x ≠ 3, then x ≠ y or y ≠ 3.

(ii) If n is not an integer, then it is not a natural number.

(iii) If the triangle is not equilateral, then all three sides of the triangle are not equal.

(iv) If xy is not a positive integer, then either x or y is not a negative integer.

(v) If the natural number n is not divisible by 2 or 3, then n is not divisible by 6.

(vi) The weather will not be cold, if it does not snow.

(vii) If x² is not less than 1, then x is not a real number such that 0 < x < 1.

Page No 264:

Question 10:

Write down the converse of following statements :
(i) If a rectangle ‘R’ is a square, then R is a rhombus.
(ii) If today is Monday, then tomorrow is Tuesday.
(iii) If you go to Agra, then you must visit Taj Mahal.
(iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.
(v) If all three angles of a triangle are equal, then the triangle is equilateral.
(vi) If x : y = 3 : 2, then 2x = 3y.
(vii) If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.
(viii) If x is zero, then x is neither positive nor negative.
(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

Answer:

(i) If the rectangle ‘R’ is a rhombus, then it is a square.

(ii) If tomorrow is Tuesday, then today is Monday.

(iii) If you must visit Taj Mahal, you go to Agra.

(iv) If the triangle is right angled, then the sum of squares of two sides of a triangle is equal to the square of the third side of the triangle.

(v) If the triangle is equilateral, then all three angles of the triangle are equal.

(vi) If 2x = 3y, then x : y = 3 : 2.

(vii) If the opposite angles of a quadrilateral S are supplementary, then S is a cyclic quadrilateral.

(viii) If x is neither positive nor negative, then x is zero.

(ix) If the ratio of corresponding sides of two triangles are equal, then the triangles are similar.

Page No 264:

Question 11:

Identify the Quantifiers in the following statements.
(i) There exists a triangle which is not equilateral.
(ii) For all real numbers x and y, xy = yx.
(iii) There exists a real number which is not a rational number.
(iv) For every natural number x, x + 1 is also a natural number.
(v) For all real numbers x with x > 3, x² is greater than 9.
(vi) There exists a triangle which is not an isosceles triangle.
(vii) For all negative integers x, x³ is also a negative integers.
(viii) There exists a statement in above statements which is not true.
(ix) There exists a even prime number other than 2.
(x) There exists a real number x such that x² + 1 = 0.

Answer:

Following are the quantifiers in the above sentences-
(i) There exists
(ii) For all
(iii) There exists
(iv) For every
(v) For all
(vi) There exists
(vii) For all
(viii)There exists
(ix) There exists
(x) There exists

Page No 265:

Question 12:

Prove by direct method that for any integer ‘n’, n³ – n is always even.
[Hint: Two cases (i) n is even, (ii) n is odd.]

Answer:

Case 1: When n is even.
Let n = 2k and k $\in$ N.
$\begin{array}{rcl} \Rightarrow & n^{3} - n = {(2 k)}^{3} - (2 k) \\ = & 2 k (4 k^{2} - 1) \\ = & 2 μ, where μ = k (4 k^{2} - 1) \end{array}$
Thus, n³ – n is even when n is even.

Case 2: When n is odd.
Let n = 2k + 1 and k $\in$ N.
$\begin{array}{rcl} n^{3} - n & = & {(2 k + 1)}^{3} - (2 k + 1) \\ = & (2 k + 1) [{(2 k + 1)}^{2} - 1] \\ = & (2 k + 1) [4 k^{2} + 1 + 4 k - 1] \\ = & (2 k + 1) [4 k^{2} + 4 k] \\ = & 4 k (2 k + 1) [k + 1] \\ = & 2 λ, where λ = 2 k (2 k + 1) [k + 1] \end{array}$
Thus, n³ – n is even when n is odd.

Hence, n³ – n is always even.

Page No 265:

Question 13:

Check the validity of the following statement.
(i) p : 125 is divisible by 5 and 7.
(ii) q : 131 is a multiple of 3 or 11.

Answer:

It is known that
$p \land q$ has truth value F (false) whenever either p or q or both have the truth value F
$p \lor q$ has the truth value T (true) whenever either p or q or both have the truth value T

(i) Given that, p : 125 is divisible by 5 and 7.
Let q: 125 is divisible by 5.
r: 125 is divisible by 7.
Thus, q is true but r is false.
⇒ $q \land r$ is false.
Hence, p is not valid.

(ii) Given that, q : 131 is a multiple of 3 or 11.
Let p: 131 is multiple of 3.
r: 131 is a multiple of 11.
Thus, p is false but r is true.
⇒ $p \lor r$ is true.
Hence, q is valid.

Page No 265:

Question 14:

Prove the following statement by contradication method.
p : The sum of an irrational number and a rational number is irrational.

Answer:

Let the given statement be false. Then, the sum of an irrational and a rational number is rational.

Consider an irrational number $\sqrt{a}$ and a rational number b.
Now, their sum is given as:
$\sqrt{a} + b = r, where r is rational \Rightarrow \sqrt{a} = r - b$

Now, $\sqrt{a}$ is irrational but $(r - b)$ is rational as the difference of two rational numbers is a rational number.
This is a contradiction.
Then, our supposition was wrong.

Hence, p is true.

Page No 265:

Question 15:

Prove by direct method that for any real numbers x, y if x = y, then x² = y².

Answer:

Given that, x, y are any two real numbers such that x = y.
Let p: x = y; x, y $\in$ R.

Squaring both sides,
x² = y²: q
Thus, p ⇒ q.
Hence, proved.

Page No 265:

Question 16:

Using contrapositive method prove that if n² is an even integer, then n is also an even integers.

Answer:

Let
p : n² is an even integer
q : n is also an even integer.
According to the contra positive method, if p → q, then ~q → ~p.
Therefore,
~q : n is not an even integer
~p : n² is not an even integer
Consider ~q : n is not an even integer.
∴ n is odd
= n² is also odd as square of odd is always odd.
⇒ ~p is true.
Thus, ~q → ~p.
Hence, proved

Page No 265:

Question 17:

Choose the correct answer out of the four options:
Which of the following is a statement.
(A) x is a real number.
(B) Switch off the fan.
(C) 6 is a natural number.
(D) Let me go.

Answer:

A statement is a sentence which is either true or false.
Here, "6 is a natural number" is a true statement.

Hence, the correct answer is option (c).

Page No 265:

Question 18:

Choose the correct answer out of the four options:
Which of the following is not a statement
(A) Smoking is injurious to health.
(B) 2 + 2 = 4
(C) 2 is the only even prime number.
(D) Come here.

Answer:

A statement is a sentence which is either true or false.
Here, "come here" is not a statement while all the others are true statements.
Hence, the correct answer is optioned (d).

Page No 265:

Question 19:

Choose the correct answer out of the four options:
The connective in the statement “2 + 7 > 9 or 2 + 7 < 9” is
(A) and
(B) or
(C) >
(D) <

Answer:

The connective in the given statement is "or".
Hence, the correct answer is option (b).

Page No 266:

Question 20:

Choose the correct answer out of the four options:
The connective in the statement
“Earth revolves round the Sun and Moon is a satellite of earth” is
(A) or
(B) Earth
(C) Sun
(D) and

Answer:

The connective in the given statement is "and".

Hence, the correct answer is option (d).

Page No 266:

Question 21:

Choose the correct answer out of the four options:
The negation of the statement “A circle is an ellipse” is
(A) An ellipse is a circle.
(B) An ellipse is not a circle.
(C) A circle is not an ellipse.
(D) A circle is an ellipse.

Answer:

Let p : A circle is an ellipse.
Thus,
~p : A circle is not an ellipse.

Hence, the correct answer is option (c).

Page No 266:

Question 22:

Choose the correct answer out of the four options:
The negation of the statement “7 is greater than 8” is
(A) 7 is equal to 8.
(B) 7 is not greater than 8.
(C) 8 is less than 7.
(D) none of these

Answer:

Let p : 7 is greater than 8.
Thus,
~p : 7 is not greater than 8.

Hence, the correct answer is option (b).

Page No 266:

Question 23:

Choose the correct answer out of the four options:
The negation of the statement “72 is divisible by 2 and 3” is
(A) 72 is not divisible by 2 or 72 is not divisible by 3.
(B) 72 is not divisible by 2 and 72 is not divisible by 3.
(C) 72 is divisible by 2 and 72 is not divisible by 3.
(D) 72 is not divisible by 2 and 72 is divisible by 3.

Answer:

Let p : 72 is divisible by 2 and 3. Thus,
q : 72 is divisible by 2.
r : 72 is divisible by 3.
Now,
~q : 72 is not divisible by 2.
~r : 72 is not divisible by 3.
Now, ~(p ∧ q) = ~p ∨ ~q.
Thus, the negation of the statement is given by ~q ∨ ~r.
Therefore, the negation of the statement is:
72 is not divisible by 2 or 72 is not divisible by 3.

Hence, the correct answer is option (a).

Page No 266:

Question 24:

Choose the correct answer out of the four options:
The negation of the statement “Plants take in CO₂ and give out O₂” is
(A) Plants do not take in CO₂ and do not give out O₂.
(B) Plants do not take in CO₂ or do not give out O₂.
(C) Plants take in CO₂ and do not give out O₂.
(D) Plants take in CO₂ or do not give out O₂.

Answer:

Let p : Plants take in CO₂ and give out O₂.
Now,
q : Plants take in O₂.
r : Plants give out O₂.
The negation of the statement is given by ~q ∨ ~r.
Now,
~q : Plants do not take in CO₂
~r : Plants do not give out O₂
Thus, the negation of the given statement is:
Plants do not take in CO₂ or plants do not give out O₂.

Hence, the correct answer is option (b).

Page No 267:

Question 25:

Choose the correct answer out of the four options:
The negation of the statement “Rajesh or Rajni lived in Bangalore” is
(A) Rajesh did not live in Bangalore or Rajni lives in Bangalore.
(B) Rajesh lives in Bangalore and Rajni did not live in Bangalore.
(C) Rajesh did not live in Bangalore and Rajni did not live in Bangalore.
(D) Rajesh did not live in Bangalore or Rajni did not live in Bangalore.

Answer:

The given statement is of the form p ∨ q.
Now,
~(p ∨ q) = ~p ∧ ~q.
Here,
~p : Rajesh did not live in Bangalore.
~q : Rajni did not live in Bangalore.
Thus, the negation of the given statement is:
Rajesh did not live in Bangalore and Rajni did not live in Bangalore.

Hence, the correct answer is option (c).

Page No 267:

Question 26:

Choose the correct answer out of the four options:
The negation of the statement “101 is not a multiple of 3” is
(A) 101 is a multiple of 3.
(B) 101 is a multiple of 2.
(C) 101 is an odd number.
(D) 101 is an even number.

Answer:

Let
p : 101 is not a multiple of 3.
Thus,
~p : 101 is a multiple of 3.

Hence, the correct answer is option (a).

Page No 267:

Question 27:

Choose the correct answer out of the four options:
The contrapositive of the statement “If 7 is greater than 5, then 8 is greater than 6” is
(A) If 8 is greater than 6, then 7 is greater than 5.
(B) If 8 is not greater than 6, then 7 is greater than 5.
(C) If 8 is not greater than 6, then 7 is not greater than 5.
(D) If 8 is greater than 6, then 7 is not greater than 5.

Answer:

Let
p : 7 is greater than 5
q : 8 is greater than 6
According to the contra positive method, if p → q, then ~q → p.
Now,
~p : 7 is not greater than 5
~q : 8 is not greater than 6
If p is true, then 7 is greater than 5.
i.e. 7 > 5
⇒ 7 + 1 > 5 + 1
⇒ 8 > 6
Thus, p → q.
Now, if ~q is true, then 8 is not greater then than 6.
⇒ 8 âŠ 6
⇒ 8 – 1 âŠ 6 – 1
⇒ 7 âŠ 5
Thus, ~q → ~p.
Therefore, the contrapositive of the given statement is:
If 8 is not greater than 6, then 7 is not greater than 5.

Hence, the correct answer is option (c).

Page No 267:

Question 28:

Choose the correct answer out of the four options:
The converse of the statement “If x > y, then x + a > y + a” is
(A) If x < y, then x + a < y + a.
(B) If x + a > y + a, then x > y.
(C) If x < y, then x + a > y + a.
(D) If x > y, then x + a < y + a.

Answer:

The given statement is:

If x > y, then x + a > y + a.

The converse of this statement is:

If x + a > y + a, then x > y.

Hence, the correct answer is option (b).

Page No 267:

Question 29:

Choose the correct answer out of the four options:
The converse of the statement “If sun is not shining, then sky is filled with clouds” is
(A) If sky is filled with clouds, then the sun is not shining.
(B) If sun is shining, then sky is filled with clouds.
(C) If sky is clear, then sun is shining.
(D) If sun is not shining, then sky is not filled with clouds.

Answer:

The converse of the given statement is:
If sky is filled with clouds, then the sun is not shining.

Hence, the correct answer is option (a).

Page No 268:

Question 30:

Choose the correct answer out of the four options:
The contrapositive of the statement “If p, then q”, is
(A) If q, then p.
(B) If p, then ~ q.
(C) If ~ q, then ~ p.
(D) If ~ p, then ~ q.

Answer:

The given statement is:
If p, then q.
Thus, the contrapositive of statement is:
If ~q, then ~p.

Hence, the correct answer is option (c).

Page No 268:

Question 31:

Choose the correct answer out of the four options:
The statement “If x² is not even, then x is not even” is converse of the statement
(A) If x² is odd, then x is even.
(B) If x is not even, then x² is not even.
(C) If x is even, then x² is even.
(D) If x is odd, then x² is even.

Answer:

The given statement is the converse of "If x is not even, then x² is not even."

Hence, the correct answer is option (a).

Page No 268:

Question 32:

Choose the correct answer out of the four options:
The contrapositive of statement ‘If Chandigarh is capital of Punjab, then Chandigarh is in India’ is
(A) If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.
(B) If Chandigarh is in India, then Chandigarh is Capital of Punjab.
(C) If Chandigarh is not capital of Punjab, then Chandigarh is not capital of India.
(D) If Chandigarh is capital of Punjab, then Chandigarh is not in India.

Answer:

Let
p : Chandigarh is the capital of Punjab.
q : Chandigarh is in India.
~p : Chandigarh is not the capital of Punjab.
~q : Chandigarh is not in India.
Now, p → q is true.
But if Chandigarh is not in India, Chandigarh cannot be the capital of Punjab.
Therefore, ~q → ~p.
Thus, the contrapositive of the statement is:
If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

Hence, the correct answer is option (a).

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Question 33:

Choose the correct answer out of the four options:
Which of the following is the conditional p ↔ q?
(A) q is sufficient for p.
(B) p is necessary for q.
(C) p only if q.
(D) if q, then p.

Answer:

The conditional for p → q is:
p only if q.

Hence, the correct answer is option (c).

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Question 34:

Choose the correct answer out of the four options:
The negation of the statement “The product of 3 and 4 is 9” is
(A) It is false that the product of 3 and 4 is 9.
(B) The product of 3 and 4 is 12.
(C) The product of 3 and 4 is not 12.
(D) It is false that the product of 3 and 4 is not 9.

Answer:

Let
p : The product of 3 and 4 is 9.
~p : The product of 3 and 4 is not 9.
or It is false that the product of 3 and 4 is 9.

Hence, the correct answer is option (a).

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Question 35:

Choose the correct answer out of the four options:
Which of the following is not a negation of “A natural number is greater than zero”
(A) A natural number is not greater than zero.
(B) It is false that a natural number is greater than zero.
(C) It is false that a natural number is not greater than zero.
(D) None of the above

Answer:

Let
p : A natural number is greater than zero.
~p : A natural number is not greater than zero.
or
~p : It is false that a natural number is greater than zero.

Hence, the correct answer is option (c).

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Question 36:

Choose the correct answer out of the four options:
Which of the following statement is a conjunction ?
(A) Ram and Shyam are friends.
(B) Both Ram and Shyam are tall.
(C) Both Ram and Shyam are enemies.
(D) None of the above.

Answer:

Here, two simple statements are joined by "and" in a conjunction.
In the first three options, two simple statements do not make any sense individually.

Hence, the correct answer is option (d).

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Question 37:

State whether the following sentences are statements are not :
(i) The angles opposite to equal sides of a triangle are equal.
(ii) The moon is a satellite of earth.
(iii) May God bless you!
(iv) Asia is a continent.
(v) How are you?

Answer:

(i) It is a true statement.
(ii) It is a true statement.
(iii) It is not a statement.
(iv) It is a true statement.
(v) It is not a statement.

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