Proof by mathematical induction

Q.

Two finite sets have $m$ and $n$elements. The total number of subsets of the first is $56$ more than the total number of subsets of the second set. The values of $m$ and $n$are

1. $7,6$

2. $6,3$

3. $5,1$

4. $8,7$

Q.

Show that the function $f:R\to R$ defined by $f\left(x\right)=\frac{x}{x^2+1},\forall xϵR$ is neither one-one nor onto. Also, if $g:R\to R$ is defined as $g\left(x\right)=2x-1$, find $fog\left(x\right)$.

Q. If in the English alphabet, every alternate letter starting from $B$ is written in small letter while rest are written in capital, then how will the $3^{\text{rd}}$ day from Tuesday be coded?

1. FRiDay
2. frIdAY
3. thUrSdAY
4. ThUrSdAY

Q.

The number of elements in the set {$(a,b):2a^2+3b^2=35$, $a,b$ belongs to $Z$}, where $Z$ is the set of all integers, is

1. $2$

2. $4$

3. $8$

4. $12$

5. $16$

Q.

Let $A$and $B$ be two sets containing $2$ elements and $4$ elements, respectively. The number of subsets of $AxB$ having $3$ or more elements, is equal to?

1. $256$

2. $220$

3. $219$

4. $211$

Q.

Prove that: 1 + 2 + 3 + ......... + n = $\frac{n(n+1)}{2}$ i.e., the sum of the first n natural numbers is $\frac{n(n+1)}{2}$.

Q.

If A and B are square matrices of the same order such that AB = BA, then prove by induction that. Further, prove that for all n ∈ N

Q. The value of the determinant
$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$ will be

1. $(a-b-c)(a^2+b^2+c^c)$
2. $(a+b+c{)}^3$
3. $(a+b+c)(ab+bc+ca)$
4. $(a+b+c{)}^2$

Q.

$a+ar+a{r}^2+......+a{r}^{n-1}=a\left(\frac{r^n-1}{r-1}\right)$ $r\ne 1$.

Q. Let $y={\sin }^{-1}\left(\sin 8\right)-{\tan }^{-1}\left(\tan 10\right)+{\cos }^{-1}\left(\cos 12\right)-{\sec }^{-1}\left(\sec 9\right)+{\cot }^{-1}\left(\cot 6\right)-{\text{cosec}}^{–1}\left(\text{cosec}7\right).$ If $y$ simplifies to $a\pi +b,$ then the value of $a-b$ is

Q.

The following statement $(p\to q)\to \left[\right(~p\to q)\to q]$ is

1. equivalent to $~p\to q$

2. equivalent to $p\to ~q$

3. a fallacy

4. a tautology

Q.

1.2.3 + 2.3.4 + … + n(n + 1) (n + 2) =

Q.

$1+3+5+.....+(2n-1)=n^2$ i.e., the sum of first n odd natural numbers is $n^2$.

Q.

Prove that $7+77+777+......+777\underset{n-digits}{.............}7=\frac{7}{81}({10}^{n+1}-9n-10)$

Q. The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but det $\left(A\right)$ is divisible by $p$ is
[Note : The trace of a matrix is the sum of its diagonal entries]

1. $(p-1{)}^2$
2. $p^3-(p-1{)}^2$
3. $(p-1)(p^2-2)$
4. $(p–1)(p^2–p+1)$

Q.

Let $A$ be the sum of the first $20$ terms and $B$ be the sum of the first $40$ terms of the series.$1^2+2.2^2+3^2+2.4^2+5^2+2.6^2+...$If $B-2A=100\lambda$, then $\lambda$ is equal to :

1. $464$

2. $496$

3. $232$

4. $248$

Q.

Prove that the number of subsets of a set containing n distinct elements is $2^n$ for all $nϵN$.

Q.

If $C_r={}^{25}C_r$ and $C_0+5\cdot C_1+9\cdot C_2+\cdots +101·C_{25}=2^{25}·k$ then $k$ is equal to

Q.

If ${10}^n+3\times 4^{n+2}+\lambda$ is divisible by 9 for all $nepsilonN$, then the least positive integral value of $\lambda$ is

1. 5

2. 3

3. 7

4. 1

Q.

How do find the square of a number ending with $5$?

Q.

$7^{2n}+2^{3n-3}{.3}^{n-1}$ is divisible by 25 for all $nϵ$ N.

Q.

$n(n+1)(n+5)$ is a multiple of 3 for all $nϵN$.

Q.

If $a,b,c,d\&e$ are prime integers, then the number of divisors of $a\times b^2\times c^2\times d\times e$ excluding $1$ as a factor, is?

1. $94$

2. $72$

3. $36$

4. $71$

Q.

$\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+......+\frac{1}{(2n+1)(2n+3)}=\frac{n}{3(2n+3)}$

Q.

$1.2+{2.2}^2+{3.2}^3+....+n{.2}^n=(n-1)2^{n+1}+2$

Q.

If $a-b=3$ and $a^3-b^3=117$, then find the value of $a+b$?

Q.

Prove that $\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{2n}>\frac{13}{24}$, for all natural numbers n > 1.

Q.

$1^2+2^2+3^2+......+n^2=\frac{n(n+1)(2n+1)}{6}$

Q. Can the number $6^n$, where n is a natural number, end with digit 5? Give reasons.

Q.

Sum of $n$ terms of the following series $1^3+3^3+5^3+7^3+............$ is