Question 2 :
If A, B and C are three finite sets then what is $\displaystyle \left [ \left ( A\cup B \right )\cap C \right ]'$ equal to?
Question 5 :
Let $S=\{2,4,6,8,......20\}$. What is the maximum number of subsets does $S$ have ?
Question 6 :
If $A$ and $B$ are subsets of $U$ such that $n(U) = 700, n(A) = 200, n(B) = 300, n$ <br> $\displaystyle \left ( A\cap B \right )$ $= 100$, then find $n\displaystyle \left ( A'\cap B' \right )$
Question 7 :
Given $\displaystyle A= \left \{ 1,2,3 \right \}, B= \left \{ 3,4 \right \}, C= \left \{ 4,5,6 \right \}$ find:$\displaystyle A\cup \left ( B\cup C \right )$
Question 8 :
If $A = \left \{1, 2, 3, 4\right \}$, what is the number of subsets of A with at least three elements?
Question 9 :
If X and Y are two sets such that $n(X)=45, n(X \cup Y)=76, n(X \cap Y)=12,$ find $n(Y)$.<i></i>
Question 10 :
If X $=$ (multiples of 2), Y $=$ (multiples of 5), Z $=$ (multiples of 10), then $X \cap ( Y \cap Z )$ is equal to<br>
Question 11 :
While preparing the progress reports of the students, the class teacher found that $70$% of the students passed in Hindi, $80$% passed in English and only $65$% passed in both the subjects. Find out the percentage of students who failed in both the subjects
Question 12 :
If universal set $\xi = \{a, b, c, d, e, f, g, h\}, A = \{b, c, d, e, f\}, B =\{a, b, c, g, h\}$ and $C = \{c, d, e, f, g\}$, then find $B - A$
Question 13 :
In a town of 10000 families, it was found that 40% families buy a newspaper A, 20% families buy newspaper B and 10% families buy newspaper C. 5% families buy both A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers, then the number of families which buy A only.
Question 14 :
Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey ; 80 played cricket and basketball and 40 played cricket and hockey 24 player all the three games. The number of boys who did not play any game is
Question 15 :
If $A$ and $B$ have some elements in common, then $n(A \cup B)$ is:<br/>
Question 16 : <div>Identify the type of Set</div>$A= \{ x| x \epsilon N, 2 < x <3 \} $
Question 17 :
Given that the universal set,$ U =$ {x : 1 < x < 12 and x is an integer} and the sets P = {x : x is a prime number}, Q = {x : x is a multiple of 4} and R = {2, 3, 8, 9} the elements of the set $(Q \cup R)' \cap P$ are:<br/>
Question 18 :
If n(U) = 50, n(A) = 20, $n((A \cup B)')$ = 18 then n(B - A) is
Question 19 :
Let $P_1$ be the set of all prime numbers, i.e., $P_1=\left \{2, 3, 5, 7, 11, ....\right \}$, Let $Pn=\left \{np|p\epsilon P_1|\right \}$, i.e., the set of all prime multiples of n. Then which of the following sets is non empty?<br>
Question 20 :
The set $\left( A\cup B\cup C \right) \cap \left( A\cap B'\cap C' \right) '\cap C'$ is equal to
Question 21 :
If $A=[x:x$ is a multiple of $3]$ and $B=[x:x$ is a multiple of $5]$, then $A-B$ is $(\bar { A }$ means complements of $A)$
Question 22 :
Suman is given an aptitude test containing 80 problems, each carrying I mark to be tackled in 60 minutes. The problems are of 2 types; the easy ones and the difficult ones. Suman can solve the easy problems in half a minute each and the difficult ones in 2 minutes each. (The two type of problems alternate in the test). Before solving a problem, Suman must spend one-fourth of a minute for reading it. What is the maximum score that Suman can get if he solves all the problems that he attempts?
Question 23 :
The value of $'c'$ for which the set $[(x, y)|x^{2} + y^{2} + 2x \leq 1]\cap [(x, y)|x - y + c\geq 0]$ contains only one point in common is :
Question 24 :
$A = \{1,2,4\}, B = \{2, 4,5\}, C = \{2, 5\}, $then $(A - B) \cup (B- C)$ is
Question 25 :
Let $A_{1}, A_{2}, ........., A_{m}$ be m sets such that $O(A_{i}) = p \forall i = 1, 2, ......... m$ and $B_{1}, B_{2}, .........., B_{n}$ be n sets such that $O(B_{j}) = q \forall j = 1, 2, ........., n$. If $\bigcup_{i=1}^{m} A_{i}$ = $\bigcup_{j=1}^{n} B_{j} = S$ and each element of S belongs to exactly $\alpha$ number of $A_{i}'s$ and $\beta$ number of $B_{j}'s$, then
Question 26 :
In how many ways can we distribute $5$ different balls in $4$ different boxes when order is not consider inside the boxes and empty boxes are not allowed
Question 27 :
If the letters of word ' IMPORTANCE ' are arranged from left to right in alphabetic order , then which letter will be the fifth from left <br>
Question 29 :
The number of integers x for which the number $\displaystyle \sqrt{x^{2}+x+1}$ is rational is :
Question 31 :
Let the coefficient of $6^{th}$ term of an expansion be $a$ and $b$ be the power.<br/>Expansion:$\left (\dfrac{4x}{5}- \dfrac 5{2x}\right)^{9}$<br/>Find $a \times b$.<br/>
Question 32 :
A man has 9 friends, 4 boys and 5 girls. In how many ways can be invite them, if there have to be exactly three girls in the invites?
Question 34 :
A batsman can score $0,1,2,3,4$ or $6$ runs from a ball. The number of different sequences in which he can score exactly $30$ runs in an over of six balls is:
Question 35 :
When we realize a specific implementation of a pancake algorithm, every move when we find the greatest of the sized array and flipping can be modeled through ____________.<span><br/></span>
Question 37 :
Out of $5$ men and $2$ women, a committee of $3$ is to be formed. In how many ways can it be formed if at least one woman is included in each committee?
Question 38 :
The number of five-letter words that are formed out of the letters of the word INFINITESIMAL
Question 39 :
${ _{ }^{ 2n }{ C } }_{ 2 }-2({ _{ }^{ n }{ C } }_{ 2 })=$
Question 40 :
Six persons A, B, C, D, E and Fare to be seated at a circular table. The number of ways this can be done if A must have either B or C on his right and B must have either C or D on his right, is ______.<br/>
Question 41 :
If $^{n-1}C_3 + {^{n-1}C_4} > {^nC_3}$, then the least value of $n$ is
Question 42 :
There are $10$ seats in the first row of a theatre of which 4 are to be occupied. The number of ways of arranging 4 persons so that no two person sit side by side is :
Question 43 :
$^{n }{ C }_{ r }+^{ n }{ C }_{ r+1 }$ is equal to______________.
Question 44 :
How many numbers greater than 50000 can be formed using all the digits 1, 1, 5, 9, 0?
Question 45 :
For $2 \geq \, r \, \geq \, n , \left ( _{n}^{r} \right) \, + \, 2 \left ( _{n}^{r - 1} \right) \, + \, \left ( _{n}^{r - 2} \right) $
Question 46 :
How many different words can be formed using all the letters of the word $'\text{ALLAHABAD}' $ when both $L's$ are not together
Question 47 :
Find the number of ways in which $5$ persons $A,B,C,D,E$ can be seated at a round table such that $C$ and $D$ must not sit together?
Question 48 :
The number of signals that can be given using any number of flags of 5 different colors, is
Question 49 :
Total number of four-digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are<br>
Question 50 : Match the column<br><table class="wysiwyg-table"><tbody><tr><td></td><td><b>List 1</b></td><td></td><td><b>List 2</b></td></tr><tr><td>A.</td><td>Given $^nC_7=^nC_4$<br><br>$n=?$</td><td>1.</td><td>$3$</td></tr><tr><td>B.</td><td>$^nP_r=720\ and\ ^nC_r=120$<br><br>$r=?$</td><td>2.</td><td>$4$</td></tr><tr><td>C.</td><td>$P(22, r+1):P(20, r+2)=11:52 r=?$<br></td><td>3.</td><td>$11$</td></tr><tr><td>D.</td><td>Given $^{10}P_r=5040$<br><br>$r=?$<br></td><td>4.</td><td>$7$</td></tr></tbody></table>
Question 51 :
The domain of the function $f(x) = \dfrac{1}{\sqrt{|x| - x}}$ is:
Question 52 :
If y is a function of x and $\log (x + y) = 2xy$, then the value of y'(0) = .....
Question 54 :
If $A$ and $B$ are two sets containing four and two elements, respectively. Then the number of subsets of the set $A\times B$ each having at least three elements is
Question 55 :
If $A=\left \{x:x^2-3x+2=0\right \}$ and $B=\left \{x:x^2+4x-5=0\right \}$ then the value of A-B is
Question 56 :
If $y$ is second entry and $x$ is first entry then its ordered pair will be ............
Question 57 :
If $2\leq a < 3$, then the value of $\cos^{-1} \cos [a] + \text{cosec}^{-1} \text{cosec }[a] + \cot^{-1} \cot [a]$. where [.] denotes greater integer less then equal to $x$) is equal to:
Question 58 :
$A$ and $B$ are two sets having $3$ and $4$ elements respectively and having $2$ elements in common. The number of relations which can be defined from $A$ to $B$ is:
Question 59 :
If domain of $f(x)=\sqrt{{x}^{2}+bx+4}$ is $R$, then maximum possible integral value of $b$ is
Question 60 :
<span>If $\displaystyle a + b = 1$ and $\displaystyle a - b = 7$; find </span>$\displaystyle ab$
Question 61 :
$A = \left \{1, 2, 3, 4\right \}$ and $B = \left \{a, b, c\right \}$. The relations from $A$ to $B$ is
Question 62 :
If $R$ be a relation defined from $\displaystyle A=\left \{ 1,2,3,4 \right \}$ to $\displaystyle B=\left \{ 1,3,5 \right \},i.e.\left ( a,b \right )\in R$ iff $a<b$ then $\displaystyle R o R^{-1}$ is
Question 63 :
Find all real values of $x$ such that $f(x)=0$ given that $f$ is a function defined by<br>$f(x)=({x}^{2}+2x-3)/(x-1)$
Question 64 :
Let $\displaystyle f\left ( x \right ) = x^{3} + 2x^{2} + 3x + 4$, then the equation $\displaystyle \frac{1}{x - f\left ( 1 \right )} + \frac{2}{x - f\left ( 2 \right )} + \frac{3}{x - f\left ( 3 \right )} = 0$, has<br>
Question 65 :
If $x={ \left( \sqrt { 3 } +1 \right) }^{ n }$ such that $n \in N$ and is an odd number then $[x]$ is (where $[x]$ denotes the greatest integers less than or equal to $x$)
Question 66 :
If $\left |x^2-2x-8 \right | + \left |x^2+x-2 \right | = 3 \left |x +2 \right |$, then the set of all real values of x is
Question 68 :
Let the number of elements of the sets $A$ and $B$ be $p$ and $q$ respectively. Then the number of relations from the set $A$ to the set $B$ is
Question 70 :
The domain of the function f(x)= log |x - 1| is ........
Question 71 : Match the statements of Column $ I $ with values of Column $II$<br><table class="wysiwyg-table"><tbody><tr><td></td><td><b>Column I</b></td><td></td><td><b>Column II</b></td></tr><tr><td>A.</td><td>If $f(x)=x+1$, when $x<0$,<br>$f(x)=x^{2}-1$,for $x\geq 0,$<br>then $f(f(x))$, for $-1\leq x\leq 0$ is</td><td>1.</td><td>$\displaystyle \frac{x-3}{2}$</td></tr><tr><td>B.</td><td>If $\displaystyle f \left(\frac{2\tan x}{1+\tan^{2}x}\right)=$ <br> $\dfrac{(\cos 2x+1)(\sec^{2}x+2\tan x)}{2}$ then $f(x)$ is</td><td>2.</td><td>$x^{2}+2x$</td></tr><tr><td>C.</td><td>If $f(x+y+1)=(\sqrt{f(x)}+\sqrt{f(y)})^{2}$ for all $x,y \in R$ and $f(0)=1,$ then $f(x)$ is</td><td>3.</td><td>$1+x$</td></tr><tr><td>D.</td><td>If $4 < x < 5 $ and $\displaystyle f(x)=\left[\frac{x}{4}\right] +2x+2$, where $[y]$ is the greatest integer $\leq y,$ then $f^{-1}(x)$ is</td><td>4.</td><td>$(x+1)^{2}$</td></tr></tbody></table>
Question 73 :
The solution of ${x}[{x}]-{x}^{2}-5[{x}]+5{x}\ge0$ (where $[{x}]$ stands for the greatest integer less than or equal to $x$) lies in the interval<br/>
Question 75 :
The solution set of $(x)^{2}+(x+1)^{2}=25,$ where $(x)$ denotes the nearest integer greater than or equal to $x$ is<br/>
