Mathematical Statement

Q.

Let $r$ be relation from $R$(set of real numbers) to $R$ defined by $r=\left\{\overline{)\left(a,b\right)}a,b\in R\text{and}a-b+\sqrt{3}\text{is an irrational number}\right\}$. then relation $r$ is

1. An equivalence relation

2. Reflective only

3. Symmetric only

4. Transitive only

Q. Prove that sq root 2n+1 +spuare root 2n+3 is irrational for any natural number n

Q. Prove that (p)power 1/n is irrational when p prime and n is greater than one

Q. If the domain of the function $y=f\left(x\right)$ is $[-3,2],$ then the domain of $f\left(|\right[x\left]|\right)$ is
(where [.] denotes the greatest integer function)

1. $[-3,2]$
2. $[-2,3)$
3. $[-2,3]$
4. $[-3,3)$

Q.

If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R-Q) ?

Q. Out of the four given alternatives, choose the one which expresses the given sentence in passive voice.
"I am going to bring him here."

1. He is going to bring here by me.
2. He is being brought here by me.
3. He is going to be brought here by me.
4. He is being gone to be brought here by me.

Q.

What is the abstract noun of honest

Q.

Without expanding the determinant, prove that

Q. In the question below, a sentence has been given in direct/indirect speech. From the given alternatives, choose the one which best expresses the given sentence in indirect/direct speech.

The designer said to her, ‘Will you have the dress ready by tomorrow evening?’

1. The designer asked her if she would have the dress ready by next evening.
2. The designer asked her that she would have the dress ready by next evening.
3. The designer asked her that if she will like to have the dress by next evening.
4. The designer asked her that she will have the dress ready by next evening.

Q. If the polynomial $R\left(x\right)$ is the remainder upon dividing $x^{2020}$ by $x^2-7x+12$. If $R\left(1\right)$ can be expressed as $(p^m-q^n)$, where $m,n\in N,\text{and}p\&q$ are prime numbers, then the value of $n+m-pq$ is

Q. In the question below, a sentence has been given in direct/indirect speech. From the given alternatives, choose the one which best expresses the given sentence in indirect/direct speech.
He said to her, "Are you coming to the party today?"

1. He told her if she was coming to the party today.
2. He asked her if she was coming to the party that day.
3. He asked her if she will be coming to the party.
4. He asked her whether she was coming to the party today.

Q. Which among the following is false
$\left(i\right)$ If $P$ is a statement, then $\sim P$ is also a statement
$\left(ii\right)$ Negation of $P$: ''$x$ is a positive number" is, ''$x$ is a negative number"
$\left(iii\right)$ If $P$ is a statement, then $\sim (\sim P)$ is not a statement

1. $\left(i\right)$ and $\left(ii\right)$
2. Only $\left(iii\right)$
3. Only $\left(ii\right)$
4. $\left(ii\right)$ and $\left(iii\right)$

Q. How many of the following sentences are true statements
$\left(1\right)$ Paris is in France.
$\left(2\right)$ Each prime number has exactly two factors.
$\left(3\right)$ The equation $x^2+5|x|+6=0$ has no real roots.
$\left(4\right)(2+\sqrt{3})$ is a complex number.
$\left(5\right)$ Is $6$ a positive integer $?$
$\left(6\right)$ The product of $-3$ and $-2$ is $-6$.
$\left(7\right)$ The angles opposite to equal sides of an isosceles triangle are equal
$\left(8\right)$ Oh! it is too hot.
$\left(9\right)$ Monika is a beautiful girl.
$\left(10\right)$ Every quadratic equation has at least one real root.

Q. Prove that (p) raise to power 1/n is irrational when p is prime and n >1.

Q. Which among the following is not a statement

1. There are finitely many real numbers.
2. The product of non-zero rational number and irrational number is always irrational.
3. Every square is a rectangle.
4. Kohli is the best cricketer.

Q.

If A = {x: x is a multiple of 3} and B = {x: x is a multiple of 5}, then A - B is

Q. In the following question, one part of the sentence may have an error. Choose which part of the sentence has an error and choose the option corresponding to it. If the sentence is free from error, choose ''No error'' option.
They are looking forward (a)/ to meet (b)/ their parents. (c)/ No error (d)

1. a
2. b
3. c
4. d

Q. If two numbers $a,b$ are such that $a:b=8:3$ and $a-5:b+6=5:3$, then $a-7:b+7$ is equal to

1. $\frac{6}{7}$
2. $\frac{5}{4}$
3. $\frac{2}{3}$
4. $\frac{3}{2}$

Q. In the question below, a sentence has been given in Direct/Indirect speech. From the given alternatives, choose the one which best expresses the given sentence in Indirect/Direct speech.
''You should lock your car as there have been some instances of theft'', said Anil.

1. There have been some instances of theft so I should lock my car was told to me by Anil.
2. Anil advised me that I should lock my car as there had been some instances of theft.
3. There have been some instances of theft so I should have locked my car said Anil.
4. Anil said that I should lock my car as there had been some instances of theft.

Q.

Check the validity of the following statements :

(i) p : 100 is a multiple of 4 and 5.

(ii) q : 125 is a multiple of 5 and 7.

(iii) r : 60 is a multiple of 3 or 5.

Q. Which among the following is a statement

1. $3$ is a composite number.
2. Wash your face.
3. You should be obedient.
4. Hurray! India won the match.

Q.

If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer.

[Hint: write aⁿ = (a – b + b)ⁿ and expand]

Q. Which of the following is not a mathematical statement?

1. Negation of statement $p$ is defined as $\sim p$
2. $x^2-4x+3=0$
3. $y=e^x$ is not continuous in $[0,2]$
4. $5$ is an odd number.

Q. Out of the four given alternatives, choose the one which expresses the sentence below in passive voice.
Why do you tell a lie?

1. Why a lie told by you?
2. Why is a lie be told by you?
3. Why is a lie told by you?
4. Why is a lie being told by you?

Q. If the domain of the function $y=f\left(x\right)$ is $[-3,2],$ then the domain of $f\left(|\right[x\left]|\right)$ is
(where [.] denotes the greatest integer function)

1. $[-2,3]$
2. $[-3,3)$
3. $[-3,2]$
4. $[-2,3)$

Q. L.C.M. of the numbers $2,\sqrt{2},\sqrt{3}$ is

1. $2\sqrt{6}$
2. $12\sqrt{2}$
3. does not exist
4. $6$

Q. Which among the following option's is correct for the sentence ''$\sqrt{17}$ is an irrational number"

1. It is a false statement
2. It is not a statement
3. It is a true statement
4. It is an open statement

Q.

A simple statement can be a combination of 2 or more simple statements

1. True

2. False

Q.

Give three examples of sentence which are not statements. Give reasons for the answers.

Q. The value of $\left|\begin{array}{ccc}\sqrt{6}& 2i& 3+\sqrt{6}\\ \sqrt{12}& \sqrt{3}+\sqrt{8}i& 3\sqrt{2}+\sqrt{6}i\\ \sqrt{18}& \sqrt{2}+\sqrt{12}i& \sqrt{27}+2i\end{array}\right|$

1. a rational number
2. an irrational number
3. an imaginary number
4. an integer