NCERT Exemplar Class 11 Maths Solutions Chapter 14 Mathematical Reasoning

NCERT Exemplar Class 11 Maths Solutions Chapter 14

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) ${\sin }^{2}x+{\cos }^{2}x=0$

Answer:

A statement is considered an assertive sentence if it is either true or false, but it shouldn’t be both.

i) The given sentence is a true statement, since we know that it is true that “a triangle has three sides.”
ii) The given sentence is a true statement, since we know that it is true that “0 is a complex number”, because we can also write it as a + ib, where the imaginary part is 0 as a + 0i.
iii) The given sentence is a false statement, since we know that it is not true that “the sky is red.”
iv) The given sentence is a false statement, since we know that it is not true that “every set is an infinite set.”
v) The given sentence is a false statement, since we know that it is not true that $15+8>23$ because $LHS\ne RHS$.
vi) The given sentence is not a statement, since we know that “y + 9 = 7” will be true for some value & false for some. For example, at y = -2 is true & y = 1 or any other value is false.
vii) The given sentence is not a statement since we know that “where is your bag” is a question.
viii) The given sentence is a true statement, since we know that it is true that “ every square is a rectangle.”
ix) The given sentence is a true statement, since we know that “Sum of opposite angles of a cyclic quadrilateral is 180?” is true by the properties f a quadrilateral.
x) The given sentence is a false statement since we know that ${\sin }^{2}x+{\cos }^{2}x=1$ is the correct expression according to the laws of trigonometry.

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:

A compound statement is defined as a combination of two statements or components.

i) Here, the components of the given statement “number 7 is prime and odd, will be –
p: Number 7 is prime &
q: Number 7 is odd
ii) Here, the components of the given statement “Chennai is in India and is the capital of Tamil Nadu” will be –
p: Chennai is in India
q: Chennai is the capital of Tamil Nadu

iii) Here, the components of the given statement “the number 100 is divisible by 3, 11 & 5” will be –
p: 100 is divisible by 3
q: 100 is divisible by 11 &
r: 100 is divisible by 5

iv) Here, the components of the given statement “Chandigarh is the capital of Haryana and U.P.” will be –
p: Chandigarh is the capital of Haryana &
q: Chandigarh is the capital of U.P.

v) Here, the components of the given statement $\sqrt{7}$ is a rational number or an irrational number” will be –
p: $\sqrt{7}$ is a rational number &
q: $\sqrt{7}$ is an irrational number
vi) Here, the components of the given statement “0 is less than every positive integer and every negative integer” will be –
p: 0 is less than every positive integer &
q: 0 is less than every negative integer
vii) Here, the components of the given statement “Plants use sunlight, water and carbon dioxide for photosynthesis” will be –
p: Plants use sunlight for photosynthesis
q: Plants use water for photosynthesis &
r: Plants use carbon dioxide for photosynthesis
viii) Here, the components of the given statement “two lines in a plane either intersect at one point or they are parallel” will be –
p: Two lines in a plane intersect at one point &
q: Two lines in a plane are parallel
ix) Here, the components of the given statement “A rectangle is a quadrilateral or a 5-sided polygon” will be –
p: A rectangle is a quadrilateral
q: A rectangle is a 5-sided polygon.

Question:3

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

Answer:

A compound statement is defined as a combination of two statements or components.

i) The components of the given statement “57 is divisible by 2 or 3”, will be –
p: 57 is divisible by 2 &
q: 57 is divisible by 3
Here, the given statement is true. A compound statement is in the form P V Q, which has truth value T if either P or Q or both will be true.

ii) the components of the given statement “24 is a multiple of 4 & 6”, will be –
p: 24 is a multiple of 4 &
q: 24 is a multiple of 6
Here, the given statement is true as both p & q are true since 24 is a multiple of both 4 & 6.
iii) A statement is considered an assertive sentence if it is either true or false, but it shouldn’t be both. The given sentence is a true statement, since we know that “All living things have two eyes and two legs”, will be –
p: All living things have two eyes &
q: All living things have two legs.
Compound statement is in the form $P\wedge Q$, which has truth value T which will be true only if both the components will be true.
Here, p is false & q is true. Hence, the given statement is false.
iv) the components of the given statement “2 is an even number and a prime number” will be –
p: 2 is an even number &
q: 2 is a prime number.
A compound statement is in the form P Q, which has truth value T which will be true only if both the components will be true. Here p & q both are true.
Hence, the given statement is true.

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:

We know that negation of p is “not p”, viz. symbolized as –p, also its truth value of –p is the opposite of the truth value of p.

i) The number 17 is not prime.
ii) $2+7\ne 6$.
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) A cow does not have four legs.
viii) A leap year does not have 366 days.
ix) All similar triangles are not congruent.
x) Area of a circle is not the same as the perimeter of the circle.

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 $3x^2-x-10=0$
(vii) Students can take Hindi or English as an optional paper.

Answer:

(i) It is a compound statement whose components are –
p: Rahul passed in Hindi &
q: Rahul passed in English
Symbolically, the function is represented as –
$p\wedge q$ – Rahul passed in Hindi & English
(ii) It is a compound statement whose components are –
p: x is an even integer &
q: y is an even integer
Symbolically, the function is represented as –
$p\wedge q-xandy$ are even integers.
(iii) It is a compound statement whose components are –
p: 2 is a factor of 12
q: 3 is a factor of 12 &
r: 6 is a factor of 12
Symbolically, the function is represented as –
$p\wedge q\wedge r:2,3and6$ are factors of 12.
(iv) It is a compound statement whose components are –
p: x is an odd integer &
$q:x+1$ is an odd integer
Symbolically, the function is represented as –
p V q – Either x or x + 1 is an odd integer.
(v) It is a compound statement whose components are –
p: A number is divisible by 2 &
q: A number is divisible by 3
Symbolically, the function is represented as –
p V q – A number is either divisible by 2 or 3.
(vi) It is a compound statement whose components are –
$p:x=2$ is a root of $3x^2-x-10=0$ &
$q:x=3$ is a root of $3x^2-x-10=0$
Symbolically, the function is represented as –
p v q – Either x = 2 or x = 3 is a root of $3x^2-x-10=0$
(vii) It is a compound statement whose components are –
p: Hindi is the optional paper &
q: English is the optional paper
Symbolically, the function is represented as –
p v q – Either Hindi or English is an optional paper.

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 $x^2-5x+6=0$
(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) $\left|x\right|$ is equal to either x or – x.
(viii) 6 is divisible by 2 and 3.

Answer:

(i) It is a compound statement whose components are –
p: all rational numbers are real
-p: all rational numbers are not real
q: All rational numbers are complex
-q: All rational numbers are not complex
Thus,$(p\wedge q)=$All rational numbers are real and complex.
& $-(p\wedge q)=-pV-q=$All rational numbers are neither complex nor real
(ii) It is a compound statement whose components are –
p: all real numbers are rational
-p: all real numbers are not rational
q: All real numbers are not irrational &
-q: All real numbers are not irrational
Thus, (p v q) = All real nos. are either rational or irrational.
& $-(p\wedge q)=-pV-q$= All real numbers are neither rational or irrational.
(iii) It is a compound statement whose components are –
p: x = 2 is a root of Quadratic equation $x^2-5x+6=0$.
-p: x = 2 is not a root of Quadratic equation $x^2-5x+6=0$.
q: x = 3 is a root of Quadratic equation $x^2-5x+6=0$.
-q: x = 3 is not a root of Quadratic equation $x^2-5x+6=0$.
Thus, $(p\wedge q)=x=2andx=3$ are roots of Quadratic equation $x^2-5x+6=0$.
& $-(p\wedge q)=-pV-q$
Neither x = 2 nor x = 3 are roots of Quadratic equation $x^2-5x+6=0$.
(iv) It is a compound statement whose components are –
p: A triangle has 3 sides
-p: A triangle does not have 3 sides
q: A triangle has 4 sides
-q: A triangle does not have 4 sides
Thus, (p V q) = A triangle has either 3 or 4 sides &$-(pVq)=-p\wedge -q$ = A triangle has neither 3 nor 4 sides
(v) It is a compound statement whose components are –
p: 35 is a prime number.
-p: 35 is not a prime no.
q: 35 is a composite no.
-q: 35 is not a composite no.
Thus, (p V q) = 35 is either a prime number or a composite number. & -(p V q) = $-p\wedge q$ = 35 is neither a prime no. nor a composite no.
(vi) It is a compound statement whose components are –
p: All prime integers are even
-p: All prime integers are not even
q: All integers are odd
-q: All integers are not odd
Thus, (p V q) = All prime integers are either even or odd & -(p V q) = -(p –q) = All prime integers are neither even nor odd
(vii) It is a compound statement whose components are –
p: $\left|x\right|$ is equal to x.
-p: $\left|x\right|$ is not equal to x.
q: $\left|x\right|$ is equal to-x.
-q: $\left|x\right|$ is not equal to-x.
Thus, (p V q) = x is either equal to x or-x & -(p V q) = -p ? – = $\left|x\right|$ is neither equal to x nor-x.
(viii) ) It is a compound statement whose components are –
p: 6 is divisible by 2
-p: 6 is not divisible by 2
q: 6 is divisible by 3
-q: 6 is not divisible by 3
Thus, (p ^ q) = 6 is divisible by 2 & 3
& -(p v q) = -p V –q = 6 is neither divisible by 2 nor 3.

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:

The expression in the given conditional statement is- if p, then q.

(I) p: The number is odd
q: The square of the number is odd
Thus, “if the number is odd, then its square is even.
(ii) p: Take the dinner
q: you will get the sweet dish
Thus, “If you take the dinner, then you will get the sweet dish.”
(iii) p: You do not study
q: You will fail
Thus, “If you do not study, you will fail.”
(iv) p: An integer is divisible by 5
q: Unit digits of an integer are 0 or 5.
Thus, “If an integer is divisible by 5, then its unit digits are 0.”
(v) p: Any number is prime
q: The square of no. is not prime
Thus, “If any number is prime, then its square is not prime.”
(vi) p: a, b & c are in AP
q: 2b = a + c
Thus, “If a, b & c are in AP, then 2b = a + c.”

Question:8

Form the biconditional statement $P\leftrightarrow q$, 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:

We use only & only if in biconditional statements.

i) p: The unit digit of an integer is zero.
q: It is divisible by 5.
Thus, p ↔ q = Unit digit of an integer is zero if and only if it is divisible by 5.
ii) p: A natural number. n is odd
q: Natural number n is not divisible by 2.
Thus, p ↔ q = A natural number is odd if and only if it is not divisible by 2.
iii) p: A triangle is an equilateral triangle.
q: All three sides of a triangle are equal.
p ↔ q = A triangle is an equilateral triangle if and only if all three sides of a triangle are equal.

Question:9

Write down the contrapositive of the following statements:
(i) If $x=yandy=3,thenx=3$.
(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 $0<x<1,then{x}^2<1$.

Answer:

Contrapositive definition: A conditional statement is said to be logically equivalent to its contrapositive.

(I) If x ≠ 3, then y ≠x 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 a 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 doesn’t snow.
(vii) If $x^2>1$ then, x is not a real number such that 0 < x < 1.

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:

Converse definition: A conditional statement is said to be not logically equivalent to its converse.

(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 the Taj Mahal, then you should go to Agra.
(iv) If the triangle is a right triangle, then the sum of the squares of the other two sides is equal to the square of the third side.
(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 are supplementary, then S is cyclic.
(viii) If x is neither positive nor negative, then x = 0.
(ix) If the ratio of the corresponding sides of two triangles is equal, then triangles are similar.

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 $x^2+1=0.$

Answer:

Quantifiers: It is a phrase used to make the prepositional statement, for example – ‘there exists’, ‘for all’, ‘for every’, etc.

(I) In the given statement, the quantifier is – “There exists” Thus, there is a quantifier.
(ii) In the given statement, quantifier is – “For all” Thus, there is a quantifier.
(iii) In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(iv) In the given statement, quantifier is – “For every” Thus, there is a quantifier.
(v) In the given statement, quantifier is – “For all” Thus, there is a quantifier.
(vi) In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(vii) In the given statement, quantifier is – “For all” Thus, there is a quantifier.
(viii) In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(ix) In the given statement, quantifier is – “There exist” Thus, there is a quantifier.
(x) In the given statement, quantifier is – “There exist” Thus, there is a quantifier.

Question:12

Prove by direct method that for any integer $n,n^3-n$ is always even.

Answer:

Given : $n^3-n$
Let us consider that, n is even
Thus, $n=2k.......(k-naturalno)$
$$\begin{gathered} {n}^3-n=(2k{)}^3-(2k) \\ {n}^3-n=2k(4k^2-1) \end{gathered}$$
[Let us consider that $k(4k^2-1)=m$]
Thus, $n^3-n=2m$
Thus,
$n^3-n$ is even
Now, we will assume that n is an odd no.
Thus, $n=(2k+1).......(k-naturalno.)$
$$\begin{gathered} {n}^3-n=(2k+1{)}^3-(2k+1) \\ {n}^3-n=(2k+1)[(4k2+4k+1-1)] \end{gathered}$$
$$\begin{gathered} {n}^3-n=(2k+1)[(4k2+4k)] \\ {n}^3-n=4k(2k+1)(k+1) \end{gathered}$$
$$\begin{gathered} {n}^2-n=2.2k(2k+1)(k+1) \\ {n}^3-n=2\lambda \end{gathered}$$
Thus, from this we can say that $n^3-n$ is always even.

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:

i) Here, p: 125 is divisible by 5 & 7
Now let, q: 125 is divisible by 5 &
r: 125 is divisible by 7.
We know that q is true & r is false.
Thus, q ^ r is false..
Therefore, p is no valid.
ii) Here, q: 131 is a multiple of 3 or 11.
Now let, P: 131 is a multiple of 3
& Q: 131 is a multiple of 11.
We know that both P & Q are false.
Thus, P V Q is False & q is not valid.

Question:14

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

Answer:

p: The sum of an irrational number and a rational number is irrational.
We know that the sum of a rational no. & a irrational no. is irrational, p is false.
Let, $n\to rationalno.$
& $\sqrt{\lambda }\to irrationalno.$
Now,
$\sqrt{\lambda }+n=r$
$\sqrt{\lambda }=r-n$
But since, $\sqrt{\lambda }$ is irrational and n is rational viz.contradictiona, our assumption will be false.
Thus, P is true.

Question:15

Prove by direct method that for any real numbers $x,y$ if $x=y$, then $x^2=y^2.$

Answer:

Given: For any real no. $x,y,x=y$.
Let us consider that,
$p:x=y$, where x & y are real no.
If we square both sides,
$q:x^2=y^2$
……. (assumption)
Thus, $p=q$.

Question:16

Using the contra positive method prove that if n² is an even integer, then n is also an even integer.

Answer:

Let us consider that,
p: n² is an even integer
-p: n is not an even integer
Q: n is also an even integer
-q: n is not an even integer.
A conditional statement is said to be logically equivalent to its contrapositive.
Thus, $-q\to -p=$If n is not an even integer, then $n^2$ is not an even integer.
Thus, $-qand\to -p$ are true.

Question:17

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 considered as an assertive sentence if it is either true or false, but it should not be both.
Therefore, (C) 6 is a natural no. is true.

Question:18

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:

Option D
‘Go there’ is not a statement to a given order like ‘Come here’.

Question:19

The connective in the statement $2+7>9or2+7<9$ is
A. and
B. or
C. >
D. <

Answer:

Since the two statements $2+7>92+7<9$ are connected by ‘or’ –
Option (B) is the correct answer.

Question:20

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’
Thus, option (D) is the correct answer.

Question:21

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:

The negation of the given statement will be (C) A circle is not an ellipse

Question:22

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:

The negation of the given statement will be –
(B) 7 is not greater than 8

Question:23

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:

In the given statement-
p: 72 is divisible by 2 & 3
q: 72 is divisible by 2 & -q: 72 is not divisible by 2
r: 72 is divisible by 3 & -r: 72 is not divisible by 3
$-(q\wedge r)=-qV-r$
Thus, 72 is not divisible by 2 or 72 is not divisible by 3.
Thus, opt A is the correct answer.

Question:24

The negation of the statement
“Plants take in $C{O}_2$ and give out $O_2$” is
A. Plants do not take in $C{O}_2$ and do not give out $O_2$.
B. Plants do not take in $C{O}_2$ or do not give out $O_2$.
C. Plants take in $C{O}_2$ and do not give out $O_2$.
D. Plants take in $C{O}_2$ or do not give out $O_2$.

Answer:

In the given statement-
p: Plants take in $C{O}_2$ and give out $O_2$
q: Plants take in $C{O}_2$ & -q: Plants do not take in $C{O}_2$
r: Plants give out $O_2$ & -r: Plants do not give out $O_2$
$-(q\wedge r)=-qV-r$
Thus, Plants do not take in $C{O}_2$ or do not give out $O_2$.
Thus, option B is the correct answer.

Question:25

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:

In the given statement-
p: Rajesh or Rajini lived in Bangalore
q: Rajesh lived in Bangalore & -q: Rajesh dis not lived in Bangalore
r: Rajini lived in Bandalore & -r: Rajini did not lived in Bangalore
$-(q\vee r)=-q\wedge -r$
Thus, Rajesh did not live in Bangalore and Rajini did not live in Bangalore.
Thus, Option C is the correct answer.

Question:26

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:

The negation of the given statement is –
101 is a multiple of 3.
Thus, option A is the correct answer.

Question:27

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:

Here, p: 7 is greater than 5 & -p: 7 is not greater than 5
& q: 8 is greater than 6 & -q: 8 is not greater than 6.
Now,
A conditional statement is said to be logically equivalent to its contrapositive.
Thus, -p → -q = If 8 is not greater than 6, then 7 is not greater than 5.
Thus, option (C) is the correct answer.

Question:28

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:

Here, $p:x>y$
& q: $x+a>y+a$
Thus, $p\to q$, whose converse will be –
$q\to p$ viz., if $x+a>y+a$, then $x+a>y+a$.
Thus, option (B) is the correct answer.

Question:29

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:

Here, p: Sun is not shining
& q: Sky is filled with clouds.
Thus, $p\to q,$ whose converse will be –
$q\to p$ viz., If the sky is filled with clouds, then the sun is not shining.
Thus, option (A) is the correct answer.

Question:30

The contrapositive of the statement
“If p, then q”, is
A. If q, then p.
B. If p, then $\sim q$.
C. If $\sim q$ , then $\sim p$.
D. If $\sim p$, then $\sim q$.

Answer:

Here the statement is in the form – “If p, then q” viz. $p\to q$
whose converse will be $-q\to p$
Thus, if –q, then –p.
Therefore, option (C) is the correct answer.

Question:31

The statement

“If $x^2$ is not even, then x is not even” is converse of the statement
A. If $x^2$ is odd, then x is even.
B. If x is not even, then $x^2$ is not even.
C. If x is even, then $x^2$ is even.
D. If x is odd, then $x^2$ is even.

Answer:

Here, let p: x² is not even & q: x is not even
Thus, $p\to q$, whose converse will be –
$q\to p$ viz., If x is not even, then $x^2$ is not even.
Thus, option B is the correct answer.

Question:32

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:

Here, let us take,
p: Chandigarh is the capital of Punjab, thus –p: Chandigarh is not the capital of Punjab
& q: Chandigarh is in India, thus –q: Chandigarh is not in India.
Now, If (-q), then (-p),
Thus, If Chandigarh is not in India, then Chandigarh is not the capital of Punjab.
Thus, option A is the correct answer.

Question:33

Which of the following is the conditional $p\to q$?
A. q is sufficient for p.
B. p is necessary for q.
C. p only if q.
D. if q, then p.

Answer:

p only if q is the same as $p\to q$.
Thus, option C is the correct answer.

Question:34

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:

The negation of the given statement is –
“It is false that the product of 3 & 4 is 9.”
Thus, option A is the correct answer.

Question:35

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:

We know that the negation of the given statement is false, viz.
“It is false that a natural no. is not greater than zero.”
Thus, option C is the correct answer.

Question:36

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:

None of the given statements is separated by ‘and’, thus, option D is the correct answer.

Question:37

State whether the following sentences are statements or 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!
(vi) Asia is a continent.
(v) How are you?

Answer:

(i) “The angles opposite to equal sides of a triangle are equal” is true, thus, it is clear that it is a statement.
(ii) “The moon is the satellite of the earth” is true, thus, it is clear that it is a statement.
(iii) “May God bless you!” is an exclamation sentence, thus, it is clear that it is not a statement.
(iv) “Asia is a continent” is true, thus, it is clear that it is a statement.
(v) “How are you?” is a question, thus, it is clear that it is not a statement.