2 Marks Questions

Question 12 Marks

Find the component statements of the following compound statements and check whether they are true or false.
Number 3 is prime or it is odd.

Answer

The component statements are as follows.
p: Number 3 is prime.
q: Number 3 is odd.
Both the statements are true.

View full question & answer→

Question 22 Marks

State the converse and contrapositive of each of the following statements:
p: A positive integer is prime only if it has no divisors other than 1 and itself.

Answer

Statement p can be written as follows.
If a positive integer is prime, then it has no divisors other than 1 and itself.
The converse of the statement is as follows.
If a positive integer has no divisors other than 1 and itself, then it is prime.
The contra positive of the statement is as follows.
If positive integer has divisors other than 1 and itself, then it is not prime.

View full question & answer→

Question 32 Marks

Find the component statements of the following compound statements and check whether they are true or false.
All integers are positive or negative.

Answer

The component statements are as follows.
p: All integers are positive.
q: All integers are negative.
Both the statements are false.

View full question & answer→

Question 42 Marks

Given below are two statements
p : 25 is a multiple of 5.
q : 25 is a multiple of 8.
Write the compound statements connecting these two statements with “And” and “Or”. In both cases check the validity of the compound statement.

Answer

The compound statement with ‘And’ is “25 is a multiple of 5 and 8”.
This is a false statement, since 25 is not a multiple of 8.
The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”.
This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.

View full question & answer→

Question 52 Marks

Which of the following statements are true and which are false? In each case give a valid reason for saying so.
$r:$ Circle is a particular case of an ellipse.

Answer

The equation of an ellipse is,
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$
If we put $a = b = 1,$ then we obtain
$x^2 + y^2 = 1,$ which is an equation of a circle
Therefore, circle is a particular case of an ellipse.
Thus, statement $r$ is true.

View full question & answer→

Question 62 Marks

Write the contrapositive and converse of the following statements.
You cannot comprehend geometry if you do not know how to reason deductively.

Answer

The contra positive is as follows.
If you know how to reason deductively, then you can comprehend geometry.
The converse is as follows.
If you do not know how to reason deductively, then you cannot comprehend geometry.

View full question & answer→

Question 72 Marks

Given statements in $(b).$ Identify the statements given below as contrapositive or converse.
If a quadrilateral is a parallelogram, then its diagonals bisect.

If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Answer

This is the contra positive of the given statement.
This is the converse of the given statement.

View full question & answer→

Question 82 Marks

Write the contrapositive and converse of the following statements.
Something is cold implies that it has low temperature.

Answer

The contra positive is as follows.
If something does not have low temperature, then it is not cold.
The converse is as follows.
If something is at low temperature, then it is cold.

View full question & answer→

Question 92 Marks

For each of the following compound statements first identify the connecting words and then break it into component statements.
Square of an integer is positive or negative.

Answer

Here, the connecting word is ‘or’.
The component statements are as follows.
p: Square of an integer is positive.
q: Square of an integer is negative.

View full question & answer→

Question 102 Marks

Write the following statement in five different ways, conveying the same meaning.
$p:$ If a triangle is equiangular, then it is an obtuse angled triangle.

Answer

The given statement can be written in five different ways as follows.

A triangle is equiangular implies that it is an obtuse$-$angled triangle.
A triangle is equiangular only if it is an obtuse$-$angled triangle.
For a triangle to be equiangular, it is necessary that the triangle is an obtuse$-$angled triangle.
For a triangle to be an obtuse$-$angled triangle, it is sufficient that the triangle is equiangular.
If a triangle is not an obtuse$-$angled triangle, then the triangle is not equiangular.

View full question & answer→

Question 112 Marks

For each of the following compound statements first identify the connecting words and then break it into component statements.
All rational numbers are real and all real numbers are not complex.

Answer

Here, the connecting word is ‘and’.
The component statements are as follows.
p: All rational numbers are real.
q: All real numbers are not complex.

View full question & answer→

Question 122 Marks

By giving a counter example, show that the following statements are not true.

$p:$ If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.
$q:$ The equation $x^2– 1 = 0$ does not have a root lying between $0$ and $2.$

Answer

The given statement is of the form “if $q$ then $r”.$

$q:$ All the angles of a triangle are equal.
$r:$ The triangle is an obtuse-angled triangle.
The given statement $p$ has to be proved false. For this purpose, it has to be proved that if $q,$ then $\sim\text{r}.$
To show this, angles of a triangle are required such that none of them is an obtuse angle.
It is known that the sum of all angles of a triangle is $180^\circ .$ Therefore, if all the three angles are equal, then each of them is of measure $60^\circ ,$ which is not an obtuse angle.
In an equilateral triangle, the measure of all angles is equal. However, the triangle is not an obtuse-angled triangle.
Thus, it can be concluded that the given statement $p$ is false.

The given statement is as follows.

$q:$ The equation $x^2 – 1 = 0$ does not have a root lying between $0$ and $2.$
This statement has to be proved false. To show this, a counter example is required.
Consider $x^2 – 1 = 0$
$x^2 = 1$
$\text{x = }\pm1$
One root of the equation $x^2 – 1 = 0,$ i.e. the root $x = 1,$ lies between $0$ and $2.$
Thus, the given statement is false.

View full question & answer→

Question 132 Marks

Find the component statements of the following compound statements and check whether they are true or false.
100 is divisible by 3, 11 and 5.

Answer

The component statements are as follows.
p: 100 is divisible by 3.
q: 100 is divisible by 11.
r: 100 is divisible by 5.
Here, the statements, p and q, are false and statement r is true.

View full question & answer→

Question 142 Marks

For each of the following compound statements first identify the connecting words and then break it into component statements.
$x = 2$ and $x = 3$ are the roots of the equation $3x^2 – x – 10 = 0.$

Answer

Here, the connecting word is ‘and’.
The component statements are as follows.
$p: x = 2$ is a root of the equation $3x^2 – x – 10 = 0$
$q: x = 3$ is a root of the equation $3x^2 – x – 10 = 0$

View full question & answer→

Question 152 Marks

State the converse and contrapositive of each of the following statements:
q: I go to a beach whenever it is a sunny day.

Answer

The given statement can be written as follows.
If it is a sunny day, then I go to a beach.
The converse of the statement is as follows.
If I go to a beach, then it is a sunny day.
The contra positive of the statement is as follows.
If I do not go to a beach, then it is not a sunny day.

View full question & answer→

Question 162 Marks

For each of the following compound statements first identify the connecting words and then break it into component statements.
The sand heats up quickly in the Sun and does not cool down fast at night.

Answer

Here, the connecting word is ‘and’.
The component statements are as follows.
p: The sand heats up quickly in the sun.
q: The sand does not cool down fast at night.

View full question & answer→

Question 172 Marks

Write the contrapositive and converse of the following statements.
If x is a prime number, then x is odd.

Answer

The contra positive is as follows.
If a number x is not odd, then x is not a prime number.
The converse is as follows.
If a number x is odd, then it is a prime number.

View full question & answer→

Question 182 Marks

Given statements in $(a)$. Identify the statements given below as contrapositive or converse.
If you live in Delhi, then you have winter clothes.

If you do not have winter clothes, then you do not live in Delhi.
If you have winter clothes, then you live in Delhi.

Answer

This is the contra positive of the given statement.
This is the converse of the given statement.

View full question & answer→

Question 192 Marks

Write the contrapositive and converse of the following statements.
x is an even number implies that x is divisible by 4.

Answer

The given statement can be written as follows.
If x is an even number, then x is divisible by 4.
The contra positive is as follows.
If x is not divisible by 4, then x is not an even number.
The converse is as follows.
If x is divisible by 4, then x is an even number.

View full question & answer→

Question 202 Marks

State the converse and contrapositive of each of the following statements:
r: If it is hot outside, then you feel thirsty.

Answer

The converse of statement r is as follows.
If you feel thirsty, then it is hot outside.
The contra positive of statement r is as follows.
If you do not feel thirsty, then it is not hot outside.

View full question & answer→

Question 212 Marks

Write the contrapositive and converse of the following statements.
If the two lines are parallel, then they do not intersect in the same plane.

Answer

The contra positive is as follows.
If two lines intersect in the same plane, then they are not parallel.
The converse is as follows.
If two lines do not intersect in the same plane, then they are parallel.

View full question & answer→

Question 222 Marks

Prove by direct method that for any integer $‘n’, n^3 - n$ is always even.
[Hint: Two cases $(i)$ n is even, $(ii)$ n is odd.$]$

Answer

Case I: When $n$ $s$ even. Let $\text{n} = 2\text{k}, \text{k}\in\text{N}$ $\Rightarrow\text{n}^3-\text{n}=(2\text{k})^3-(2\text{k})=2\text{k}(4\text{k}^2-1)=2\lambda,$ Where $\lambda = \text{k}(4\text{k}^2-1)$Thus, $(n^3 - n)$ is even when n is even.
Case II: When $n$ is odd.
Let $\text{n} = 2\text{k}+1, \text{k}\in\text{N}$
$\Rightarrow\text{n}^3-\text{n}=(2\text{k}+1)^3 -(2\text{k}+1)$ $=(2\text{k}+1)[(2\text{k}+1)^2-1]$ $=(2\text{k}+1)[4\text{k}^2+1+4\text{k}-1]$ $=(2\text{k}+1)[4\text{k}^2+4\text{k}]$ $=4\text{k}(2\text{k}+1)(\text{k}+1)=2\mu,$ Where $\mu=2\text{k}(\text{k}+1)(2\text{k}+1)$ Thus, $n^3 - n$ is even when n is odd. So, $n^3 - n$ is always even.

View full question & answer→

Question 232 Marks

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

Answer

Let p: $\text{x} = \text{y},\text{x},\text{y}\in\text{R}$ On squaring both sides we have
$x^2 = y^2: q ($say$)$
$\Rightarrow p = q$
Hence, proved.

View full question & answer→

Question 242 Marks

Using contrapositive method prove that if $n\ 2$ is an even integer, then $n$ is also an even integers.

Answer

Let $p: n^2$ is an even integer.
$q: n$ is also an even integer.
Let $\sim p$ is true i.e., $n$ is not an even integer.
$\Rightarrow n^2$ is not an even integer. $[$Since square of an odd integer is odd$]$
$⇒ \sim p$ is true.
Therefore, $\sim q $ is true which provides that $\sim p$ is true.
Hence proved.

View full question & answer→

Question 252 Marks

State whether the following sentences are statements are not:

The angles opposite to equal sides of a triangle are equal.
The moon is a satellite of earth.
May God bless you!
Asia is a continent.
How are you?

Answer

It is a statement.
It is a statement,
It is not a statement, since it is an exclamations.
It is a statement.
It is not a statement, since it is a question.

View full question & answer→

Maths 2 Marks Questions

Take a timed test