Statement

Q.

What is an abelian group ? Could you please provide an example as well ?

Q. $x^n-y^n$, $n\in N$ is always divisible by $x-y$.

1. True
2. False

Q.

Using mathematical induction prove that $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ for all positive integers n.

Q.

By principle of mathematical Induction show that aⁿ​​​​​​-bⁿ is divisible by a+b when n is a positive integer.

Q.

Write the following cube in expanded form: ${\left(x-\frac{2}{3}y\right)}^3$

Q. $\mathrm{If}A=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\mathrm{and}I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\mathrm{then}\mathrm{which}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{holds}\mathrm{for}\mathrm{all}n\in N?$

1. $A^n=\mathrm{nA}-\left(n-1\right)I$
2. $A^n=2^{n-1}A-\left(n-1\right)I$
3. $A^n=\mathrm{nA}+\left(n-1\right)I$
4. $A^n=2^{n-1}A+\left(n-1\right)I$

Q.

State the second principle of mathematical induction.

Q. At which points the function $f\left(x\right)=\frac{x}{\left[x\right]}$, where [.] is greatest integer function, is discontinuous

1. Only positive integers
2. All positive and negative integers and (0, 1)
3. All rational numbers
4. None of these

Q. For every natural number n, n(n+1) is always

1. odd
2. divisible by 4
3. even
4. divisible by 3

Q.

Let $\mathrm{S}=\left\{\mathrm{x}|\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{positive}\mathrm{multiple}\mathrm{of}3\mathrm{less}\mathrm{than}100\right\}$, $\mathrm{P}=\left\{\mathrm{x}|\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{prime}\mathrm{number}\mathrm{less}\mathrm{than}20\right\}$. Then $\mathrm{n}\left(\mathrm{S}\right)+\mathrm{n}\left(\mathrm{P}\right)$ is

1. $34$

2. $41$

3. $33$

4. $30$

Q.

State the first principle of mathematical induction.

Q.

Prove the following by using the principle of mathematical induction for all n ∈ N:

Q. Prove the following by using the principle of mathematical induction for all $n\in N.$
${10}^{2n-1}+1$ is divisible by $11$.

Q. Where principle of mathematical induction is use in day to day life ?

Q.

Do I need to do Complex numbers, PNC and binomial before starting off with limits

Q. How draw root 5 on number line

Q.

What Is meant by mathematical induction?

Q. Prove the following by using the principle of mathematical induction for all $n\in N:$
$1\cdot 2+2\cdot 3+3\cdot 4+\cdots +n(n+1)=\frac{n(n+1)(n+2)}{3}$

Q. If $x^4+y^4+z^4=0,$ then $\left|\begin{array}{ccc}1& xy& yz\\ zx& 1& xy\\ yz& zx& 1\end{array}\right|=$ (where $x,y,z\in R)$

1. $1$
2. $xyx+2$
3. $0$
4. $x+y+z+3$

Q.

Prove the following by using the principle of mathematical induction for all n ∈ N:

Q. What is set?

Q. Evaluate
$\left(i\right)5!\left(ii\right)7!\left(iii\right)7!-5!$

Q. Fill in the blank in the following:
If $P\left(n\right):2n<n!,nϵN$, then $P\left(n\right)$ is true for all $n\ge$

Q.

If $I(m,n)={\int }_0^1{x}^{m-1}(1-x{)}^{n-1}dx$, then $(m,n\in I,m,n\ge 0)$

1. $I(m,n)={\int }_0^{\infty }\frac{x^{m-1}}{(1+x{)}^{m-n}}dx$
2. $I(m,n)={\int }_0^{\infty }\frac{x^m}{(1+x{)}^{m+n}}dx$
3. $I(m,n)={\int }_0^{\infty }\frac{x^{n-1}}{(1+x{)}^{m+n}}dx$
4. $I(m,n)={\int }_0^{\infty }\frac{x^n}{(1+x{)}^{m+n}}dx$

Q.

Prove the following by using the principle of mathematical induction for all n ∈ N:

Q. Prove the following by using the principle of mathematical induction for all $n\in N.$
$n(n+1)(n+5)$ is a multiple of $3$.

Q. For every natural number $n\ge 2,$
Statement 1: $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}>\sqrt{n}$
Statement 2: $\sqrt{n\left(n+1\right)}<n+1$

1. Statement 1 is false, Statement 2 is true.
2. Statement 1 is true, Statement 2 is true. Statement 2 is a correct explanation for Statement 1.
3. Statement 1 is true, Statement 2 is true. Statement 2 is not a correct explanation for Statement 1.
4. Statement 1 is true, Statement 2 is false.

Q.

Consider the following two statements :

Statement-$1$ : $~\left(p\leftrightarrow ~q\right)$ is equivalent to $p\leftrightarrow q$.

Statement-$2$ : $~\left(p\leftrightarrow ~q\right)$ is a tautology.

Then which one of the following choices is correct $?$

1. Both statement-$1$ and statement-$2$ are true and statement-$2$ is a correct explanation for statement-$1$

2. Both statement-$1$ and statement-$2$ are true and statement-$2$is not a correct explanation for statement-$1$.

3. Statement-$1$ is true but statement-$2$ is false.

4. Both statement-$1$ and statement-$2$ are true.

Q.

Prove the following by using the principle of mathematical induction for all n ∈ N:

Q. For every natural number n, n(n+1) is always

1. odd
2. even
3. divisible by 3
4. divisible by 4