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Applied Maths 2 Marks Questions

Mathematical Statements · CBSE STD 11 Science (CBSE - English Medium). 20 questions.

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2 Marks Questions

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Question 12 Marks
Write the negation of the following statement.
"Set $A$ and $B$ are equal if and only if $A \leq B$ and $B \leq A$ "
Answer
Let $p:$ Set $A$ and $B$ are equal
$
q: A \leq B \text { and } B \leq A
$
We know that,$
\sim(p \Leftrightarrow q)=(p \wedge \sim q) \vee(q \wedge \sim p)
$
The negation of the given statement is :
Either $A=B$ and $(A \leq B$ or $B \leq A)$ OR $(A \leq B$ and $B \leq A)$ and $A \neq B$
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Question 22 Marks
Write converse and contrapositive of the following statement.
"You cannot comprehend geometry if you do not know how to reason deductively".
Answer
Converse Statement : If you do not know how to reason deductively, then you cannot comprehend geometry.
Contrapositive Statement : If you know how to reason deductively, then you can comprehend geometry.
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Question 32 Marks
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 .
Answer
For the given pairs, combined statements of these using "if an only if" are respectively as follows:
(a) A rectangle is a square if and only if all its four sides are equal.
(b) A number is divisible by 3 if and only if the sum of its digits is divisible by 3 .
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Question 42 Marks
Prove by the direct method for any integer ' $n$ ', $n^3-n$ is always even.
Answer
Let $n$ be even, then $n=2 m$
$
\therefore \quad n^3-n=n\left(n^2-1\right)=2 m\left(4 m^2-1\right)
$
which is even.
If $n$ is odd, $n=2 m+1$Then,
$
\begin{aligned}
n^3-n & =(2 m+1)^3-(2 m+1) \\
& =(2 m+1)\left(4 m^2+4 m+1-1\right) \\
& =4 m(m+1)(2 m+1),
\end{aligned}
$
Which is also even
Therefore, for any integer $n, n^3-n$ is always even.
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Question 52 Marks
Show that following statement is true $p:$ For any real numbers $x, y$, if $x=y$, then $2 x+a=$ $2 y+a$, when $a \in Z$
Answer
$
\begin{array}{l}
\text { Let } 2 x+a \neq 2 y+a, a \in Z \\
\Rightarrow \quad 2 x \neq 2 y \\
\Rightarrow \quad x \neq y
\end{array}
$
Then given statement is true by contrapositive method.
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Question 62 Marks
Check whether the following pair of statements are negation of each other.
(i) $a+b=b+a$ is true for every real numbers $a$ and $b$.
(ii) There exist real numbers $a$ and $b$ for which $a+b=$ $b+a$.
Answer
No,
Let $p: a+b=b+a$ is true for every real numbers $a$ and $b$.
$\sim p \quad$ : There exist real numbers $a$ and $b$ for which $a+b=b+a$.
Therefore, negation of statement (i) is different from statement (ii).
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Question 72 Marks
Write the negation of the following statements :
(i) for every real number $n, n+1>n$.
(ii) There exists a number which is equal to its square.
Answer
(i) Negation of quantifier 'for every' is 'There exists'. Therefore, let $p$ : For every real number $n, n+1>n$. then $\sim p$ : There exist a real numbers $n, n+1>n .$
(ii) Similarly, the negation of quantifier 'There exists' is 'for every / for all'.
Therefore, let $p$ : There exists a number which is equal to its square.
$\sim p$ : for every real number $x$, we have $x \neq x^2$. or $\sim p$ : There does not exist a number which is equal to its square.
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Question 82 Marks
For each of the following statement, determine whether an inclusive 'OR' or exclusive 'OR' is used Give reasons :
(i) The school is closed if it is a holiday or a Sunday.
(ii) Two lines in a plane intersect at a point or are parallel.
Answer
(i) Here, 'OR' is inclusive since school is closed on holiday and Sunday. If a holiday on Sunday, then also the school remains closed.
(ii) Two lines either intersect at a point or they are parallel, but it is not possible for two lines to intersect and be parallel together. Thus, here 'OR' is exclusive.
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Question 92 Marks
Identify the component statements and the connective word in the following compound statement.
(i) All primes are either even or odd.
(ii) 2 is an even number and prime number.
Answer
(i) The component statements are-
$p$ : All primes are even
$q$ : All primes are odd
The connecting word is 'or'.
(ii) The component statement are-
$p: 2$ is an even number.
$q: 2$ is a prime number.
The connecting word is 'and'.
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Question 102 Marks
Which of the following sentences are statements ?
Give reasons.
(i) Mathematics is fun.
(ii) The number $n$ is a positive integer
Answer
(i) The given sentence is not a statement because Mathematics is fun for some people not for all.
(ii) Unless $n$ is known, this sentence is not a statement. Yes, if $n=5$, then this statement will be a true statement.
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Question 112 Marks
Write the following statement in the form 'if $p$, then $q^{\prime}$.
$p:$ It is necessary to have a password to $\log$ on to the server.
Answer
Statement $p$ can be written as follows :
'If you $\log$ on to the server, then you have a password'.
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Question 122 Marks
Write the negation of the following statement :
'If I will become a doctor, then I will open hospital'.
Answer
Let $p:$ I will become a doctor
$q$ : I will open a hospital
$
\sim(p \Rightarrow q) \equiv p \wedge \sim q
$
Therefore, 'I will become a doctor and I will not open a hospital'.
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Question 132 Marks
By giving the counter example, show that the following statement is false.
"If $n$ is an odd integer, then $n$ is prime"
Answer
Let $n=9$, which is odd but not prime Therefore, given statement is false.
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Question 142 Marks
Write the converse of the statement:
If $3 \times 7=21$, then $3+7=10$
Answer
The converse statement is :
If $3+7=10$, then $3 \times 7=21$
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Question 152 Marks
Give the contrapositive of the statement : "If a number is divisible by $q$, then it is divisible by 3 ."
Answer
The contrapositive statement is :
"If a number is not divisible by 3 , then it is not divisible by $q^{\prime \prime}$.
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Question 162 Marks
State whether the following are statement or not?
(i) 10 is less than 8 .
(ii) All complex number are real.
Answer
(i) Yes, it is a statement and its values is false.
(ii) Yes, it is a statement and its values is false.
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Question 172 Marks
Identify the quantifier in the following statement : "There exists a number which is not real".
Answer
The quantifier is "There exists".
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Question 182 Marks
Identify the quantifier in the following statement :
"For every integer $p, \sqrt{p}$ is a real number."
Answer
The quantifier is "For every".
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Question 192 Marks
Identity the type 'or' (Inclusive or exclusive) used in the following statement.
" $\sqrt{2}$ is a rational number or an irrational number."
Answer
Here, the 'or' used is exclusive, because $\sqrt{2}$ is either rational or irrational but not both.
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Question 202 Marks
Check whether the following compound statement is true or false. Write the component statements.
"A square is a quadrilateral and its four sides are equal."
Answer
The given compound statement is true, Its component statements are-
$p:$ A square is a quadrilateral.
$p$ : The four sides of a square are equal.
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2 Marks Questions - Applied Maths STD 11 Science Questions - Vidyadip