Sample QuestionsMathematical Statements questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The contrapositive of the statement
"If Chandigarh is capital of Haryana, then Chandigarh is in India" is :
- A
If Chandigarh is in India, then Chandigarh is capital of Haryana.
- ✓
If Chandigarh is not in India, then Chandigarh is not the capital of Haryana.
- C
If Chandigarh is capital of Haryana, then Chandigarh is not in India.
- D
If Chandigarh is not in India, then Chandigarh is capital of Haryana.
Answer: B.
View full solution →The negation of the statement " 42 is divisible by 2 and $3^{\prime \prime}$ is :
- A
42 is not divisible by 2 and 42 is not divisible by 3 .
- ✓
42 is not divisible by 2 or 42 is not divisible by 3
- C
42 is not divisible by 2 and 42 is divisible by 3
- D
42 is divisible by 2 and 42 is not divisible by 3 .
Answer: B.
View full solution →The contrapositive of the statement "If $p$, then $q$ " is :
- A
If $q$, then $\sim p$
- B
If $\sim p$, then $\sim q$
- C
If $p$, then $\sim q$
- ✓
If $\sim q$, then $\sim p$
Answer: D.
View full solution →The converse of the statement "If $x>y$, then $x+a>y+a^{\prime \prime}$ is :
- A
If $x > y$, then $x+a < y+a$
- B
If $x < y$, then $x+a > y+a$
- ✓
If $x+a > y+a$, then $x > y$
- D
If $x < y$, then $x+a < y+a$
Answer: C.
View full solution →The connective in the statement " $3+5>9$ or $3+5<9^{\prime \prime}$ is
Answer: C.
View full solution →Write the negation of the following statement.
"Set $A$ and $B$ are equal if and only if $A \leq B$ and $B \leq A$ "
View full solution →Write converse and contrapositive of the following statement.
"You cannot comprehend geometry if you do not know how to reason deductively".
Two pair of statements are given below. Combine these two statements using 'if and only if'.
(a) $p$ : If a rectangle is a square, then all its four sides are equal
$q$ : If all the four sides of a rectangle are equal, then rectangle is a square.
(b) $p$ : If the sum of the digits of a number is divisible by 3 , then the number is divisible by 3 .
$q$ : If a number is divisible by 3 , then sum of its digits is divisible by 3 .
View full solution →Prove by the direct method for any integer ' $n$ ', $n^3-n$ is always even.
View full solution →Show that following statement is true $p:$ For any real numbers $x, y$, if $x=y$, then $2 x+a=$ $2 y+a$, when $a \in Z$
View full solution →Write the negation of the following statements and check whether the resulting statements are true:
(i) The sum of 2 and 5 is 9 .
(ii) Every natural number is greater than zero.
Identify the quantifiers and write negation of the following statements.
(i) For all even integers, $x, x^2$ is also even.
(ii) There exists a number which is multiple of 6 and 9 .
View full solution →Write the component statements of the following compound statements and check whether the compound statement is true or false.
(i) $2+4=6$ or $2+4=7$
(ii) A rectangle is a quadrilateral or five-sided polygon
View full solution →Check whether the following sentences are statements? Give reason your answer.
(i) $3+x=5$
(ii) Every set is a finite set.
(iii) The sun is a star.
(iv) $x^2-3 x+2=0$.
View full solution →(a) Check the validity of the statements:
(i) $r: 100$ is multiple of 4 and 5
(ii) $p: 125$ is multiple of 5 and 7.
(b) Prove the following statement by contradiction method:
'The sum of an irrational and a rational number is irrational'.
(c) By giving the counter examples, show that the following statements are not true
(i) If all the angles of a triangle are equal, then triangle is an obtuse angled triangle.
(ii) If $n$ is an odd integer, then $n$ is prime.
View full solution →(a) Show that the following statement is true: Given a positive real number $p$, there exists a rational number $r$ such that $a<r<p$.
(b) Verify by the method of contradiction $p: \sqrt{7}$ is irrational
View full solution →Using the words "necessary and sufficient" rewrite the statement :
The integer $n$ is odd if $n^2$ is odd, and vice-versa Also, check whether the statement is true.
View full solution →Show that the statement $p$ : If $x$ is real number such 'that $x^3+4 x=0$, then $x$ is zero' is true by
(i) Direct method
(ii) Method of contradiction
(iii) Method of contropositive
View full solution →Define basic logical connectives and quantifiers with the help of examples.
Write the converse of the statement : 'If $x$ is zero, then it is neither positive nor negative'.
View full solution →The biconditional statement $p \leftrightarrow q$ is _________________ where
$p$ : The unit digit of an integer is zero
$q$ : It is divisible by 5 .
View full solution →The statement 'You will get a sweet dish after the dinner' in the form of conditional statement is _________________
The negation of the statement 'Ram or Shyam lived in Rajasthan' is _________________
Prime factors of 6 are 2 and 3 is a _________________ statement .
Show that the following statement is true by the method of contrapositive $p$ : If $x$ is an integer and $x^2$ is even, then $x$ is also even.
View full solution →Check whether the following statement is true or not; "If $x$ and $y$ are odd integers, then $x y$ is an odd integer" by (i) Direct method (ii) Contrapositive method.
View full solution →Write the following statement in four different ways, conveying the same meaning.
$p$ : If a triangle is equiangular, then it is an obtuse angled triangle.
View full solution →Verify by method of contradiction $p: \sqrt{11}$ is irrational.
View full solution →| Column-I | Column-II |
| (a) Inverse of implication $p \Rightarrow q$ | (i) $\sim p \wedge \sim q$ |
| (b) Negation of implication $p \Rightarrow q$ | (ii) $\sim p \Rightarrow \sim q$ |
| (c) $\sim(p \vee q)$ is equal to | (iii) $\sim p \vee \sim q$ |
| (d) $\sim(p \wedge q)$ is equal to | (iv) $p \wedge \sim q$ |
| Column-I | Column-II |
| (a) Mars supports life | (i) Interrogative sentence |
| (b) Please bring me a cup of tea | (ii) Declarative sentence |
| (c) How big is the whole fish! | (iii) Imperative sentence |
| (d) What is your age? | (iv) Exclamatory sentence |