nCr Definitions and Properties

Q.

If $5^{40}$ is divided by 11 then the remainder is

1. 2

2. 3

3. 5

4. 1

Q. Find square root of 2+2√3i

Q. If $f:A\to B$ such that n(A) = 3, n(B) = 4 then total number of functions that can be defined from A to B is

1. 3!
2. 4!
3. 
4. 

Q.

An examination paper containing 12 questions consists of two parts, A and B. Part A contains 7 questions and part B contains 5 questions. A candidate is required to attempt 8 questions, selecting at least 3 from each part. In how many ways can the candidate select the questions?

Q.

There are 7 chocolates be distributed among 10 children subject to condition that child can take any number of chocolates. Find the required number of ways to do this if chocolates are identical.

1. 

2. 

3. 

4. 

Q.

The least positive integer n which will reduce ${\frac{i-1}{i+1}}^n$to a real number , is

1. 2

2. 3

3. 5

4. 4

Q.

If f(x) = $\{\begin{array}{cc}m{x}^2+n,& x<0\\ nx+m,& 0\le x\le 1\\ n{x}^3+m,& x>1\end{array}$ for what intergers m and n does both $\underset{x\to 0}{\text{lim}}$ f(x) and $\underset{x\to 1}{\text{lim}}$ f(x) exist?

Q.

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}$ = 30800:1, then r =

1. 51

2. 40

3. 31

4. 41

Q.

If ${}^n{P}_r=336,{}^n{C}_r=56$, then find n and r and hence find ${}^{n-1}{C}_{r-1}$

Q. If ${}^n{C}_{r-1}=36,{}^n{C}_r=84and{}^n{C}_{r+1}=126$ then r is

1. 1
2. 2
3. 3
4. none of these

Q.

The least positive integer n which will reduce ${\left(\frac{i-1}{i+1}\right)}^n$to a real number , is

1. 2

2. 3

3. 4

4. 5

Q. Evalaute : ${[i^{18}+{\left(\frac{1}{i}\right)}^{25}]}^3$

Q. If ${\left(\frac{1-i}{1+i}\right)}^{50}=x+iy$ , then find the value of $(x,y)$.

1. $(1,0)$
2. $(0,1)$
3. $(0,i)$
4. $(-1,0)$

Q.

If ${}^n{C}_4={}^n{C}_5$, find ${}^n{C}_3$

__

Q. If ${}^n{\text{C}}_4,{}^n{\text{C}}_5$ and ${}^n{\text{C}}_6$ are in A.P., then $n$ can be :

1. $9$
2. $11$
3. $12$
4. $14$

Q.

If ${}^{56}{P}_{r+6}.{}^{54}{P}_{r+3}$ = 30800:1, then r =

1. 31

2. 41

3. 51

4. 40

Q. The number of ways of selecting two squares from a chess board so that they have exactly one common corner is

1. $36$
2. $72$
3. $98$
4. $112$

Q.

If ${}^{56}{P}_{r+6}:{}^{54}{P}_{r+3}$ = 30800:1, then r =

1. 31

2. 51

3. 40

4. 41

Q. The equation whose roots are the values of $r$ satisfying the equation ${}^{69}{C}_{3r-1}-{}^{69}{C}_{r^2}={}^{69}{C}_{r^2-1}-{}^{69}{C}_{3r}$ is

1. $x^3-10x^2+21x=0$
2. $x^2+10x+21=0$
3. $x^2-10x+21=0$
4. $x^2-21x+10=0$

Q.

Prove that ${}^{2n}{C}_n=\frac{2^n\times \left[\mathrm{1.3.5...}\right(2n-1\left)\right]}{n!}.$

Q.

if f(x) = $\frac{1}{x}$ and g(x) = $x^2$ find f(g(x))

1. x²

2. 

3. x

4. 

Q. If P (10, r) = 5040, then r =

1. 2
2. 4
3. 6
4. 8

Q.

If $a_1,a_2,a_3$, ................., $a_n$ are in H.P., then $a_1{a}_2+a_2{a}_3+$...........+$a_{n-1}{a}_n$ will be equal to

1. 

2. 

3. 

4. 

Q. The value of ${\sum }_{r=1}^{10}r{.}^r{P}_r$ =

1. 11!
2. 11! - 1
3. 11! + 1
4. 11! - 11

Q. The digit at units place of the sum $(1!{)}^2+(2!{)}^2+(3!{)}^2+.....+(20009{)}^2$

1. 5
2. 0
3. 1
4. 7

Q. The value of the expression ${}^{47}{C}_4+{\sum }_{j=1}^5{}^{52-j}{C}_3$ is

1. $Noneofthese$
2. ${}^{47}{C}_5$
3. ${}^{52}{C}_5$
4. ${}^{52}{C}_4$

Q. If P(n, 5) = 20P (n, 3), then n =

1. 4
2. 6
3. 8
4. 2

Q. Square root of 4i - 3

Q. If ${}^n{C}_{r-1}=36,{}^n{C}_r=84and{}^n{C}_{r+1}=126$ then r is

1. 1
2. 2
3. 3
4. none of these

Q. The total number of ways in which three
distinct numbers in $A.P$ can be selected from the set $\{1,2,3,\cdots ,24\}$ is equal to :

1. $66$
2. $132$
3. $198$
4. none of these