Logarithmic Function

Q.

Let $N$ denote the set of all-natural numbers and $R$ be the relation on $N\times N$ defined by $(a,b)R(c,d)$ if $ad(b+c)=bc(a+d)$, then $R$ is

1. Symmetric only

2. Reflexive only

3. Transitive only

4. An equivalence relation

Q. The domain of the function $\log |x^2-9|,$ is

1. $R$
2. $R-[-3,3]$
3. $R-\{-3,3\}$
4. $(-3,3)$

Q.

Let $A=\left\{n\in N:nisa3-digitnumber\right\}$, $B=\left\{9k+2:k\in N\right\}$ and $C=\left\{9k+l:k\in N\right\}$ for some $l(0<l<9)$. If the sum of all the elements of the set $A\cap (B\cup C)$ is $274\times 400$, then $l$ is equal to

Q.

What is the main screen of windows is called?

Q.

The domain of the function $f\left(x\right)={\log }_2\left[{\log }_3\left({\log }_{4}x\right)\right]$ is

1. $\left(-\infty ,4\right)$

2. $\left(4,\infty \right)$

3. $\left(0,4\right)$

4. $\left(1,\infty \right)$

5. $\left(-\infty ,1\right)$

Q. Let $f\left(x\right)$ be a function defined from $A$ to $B$ where number of elements in $A$ and $B$ are $3$ and $7$ respectively. Then which among the following is/are correct.

1. Total number of functions that can be defined from $A$ to $B$ is $7^3$
2. Total number of functions that can be defined from $B$ to $A$ is $3^7$
3. Number of one-one functions from $A$ to $B$ is $210$
4. Number of many-one functions from $B$ to $A$ is $3^7$

Q.

In a bivariate data, $\sum _{}x=30,\sum _{}y=400,\sum _{}{x}^2=196,\sum _{}xy=850,$ and $n=10$. The regression coefficient of $y$ on $x$ is

1. $-3.1$

2. $-3.2$

3. $-3.3$

4. $-3.4$

Q.

Which one of the following relations on R is an equivalence relation?

1. $a{R}_{1}b\iff \left|a\right|=\left|b\right|$

2. $a{R}_{2}b\iff a\ge b$

3. ${\mathrm{aR}}_{3}b\iff \text{a divides}b$

4. $a{R}_{4}b\iff a<b$

Q.

What is the value of ${248}^2-{249}^2$?

Q.

$\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+..........\infty$equals

1. $e$

2. $-e$

3. $\frac{1}{e}$

4. none of these

Q. For a real number $x,$ let $\left[x\right]$ denote the largest integer less than or equal to $x,$ and let $\left\{x\right\}=x–\left[x\right].$ The number of solutions $x$ to the equation $\left[x\right]\left\{x\right\}=5$ with $0\le x\le 2015$ is

1. $0$
2. $3$
3. $2009$
4. $2008$

Q. If $f:A\to B,g:B\to A$ be two functions such that $gof=I_A$, then which among following is/are correct?

1. $f$ is an injective function.
2. $g$ is a surjective function.
3. $f$ is many-one function.
4. $g$ is an into function.

Q.

$\frac{\left({\displaystyle \frac{1}{2!}}+{\displaystyle \frac{1}{4!}}+{\displaystyle \frac{1}{6!}}+...\right)}{\left(1+\frac{1}{3!}+\frac{1}{5!}+...\right)}$equals

1. $e+1$

2. $\frac{(e-1)}{(e+1)}$

3. $e-1$

4. None of these

Q.

In order that a relation $R$ defined on a non–empty set $A$ is an equivalence relation, it is sufficient, if $R$:

1. Is reflexive

2. Is symmetric

3. Is transitive

4. Possesses all the above three properties

Q. For a function $f:A\to B,$ let the number of elements in set $A$ and $B$ be $4$ and $5$ respectively. If the number of many one functions are $N,$ then value of $\frac{N}{101}$ is

Q. Find the domain of the following function :
$f\left(x\right)={\log }_7{\log }_5{\log }_3{\log }_2(2x^3+5x^2-14x)$

1. $\left[-2,-\frac{1}{2}\right]\cup [4,\infty )$
2. $\left(-4,-\frac{1}{2}\right)\cup (2,\infty )$
3. $\left(-2,-\frac{1}{2}\right)\cup (4,\infty )$
4. none of these

Q.

Say true or false and justify your answer: $2^3>5^2$

1. True
2. False

Q.

$lo{g}_{b}a=?$

1. $\frac{lo{g}_{10}a}{lo{g}_{10}b}$

2. $\frac{lo{g}_{e}a}{lo{g}_{e}b}$

3. $\frac{lo{g}_{10}b}{lo{g}_{10}a}$

4. $\frac{lo{g}_{e}a}{lo{g}_{e}b}$

Q. Let $H$ be the set of all houses in a village where each house is faced in one of the directions, East, West, North, South.
Let $R=\left\{\right(x,y\left)|\right(x,y)\in H\times H$ and $x,y$ are faced in same direction$\}$. Then the relation '$R$' is

1. Only symmetric and transitive
2. Only reflexive and symmetric
3. Only reflexive and transitive
4. An equivalence relation

Q. Let $f:R\to R$ given by $f\left(x\right)={(3-x^3)}^{1/3}$. Then $fof\left(x\right)$ is

1. $x$
2. $x^{1/3}$
3. $x^3$
4. $\frac{1}{x^3}$

Q.

Let $n$ be a fixed positive integer. Let a relation $R$ defined on $I$ (the set of all integers) as follows: $aRb$ iff $n/(a-b)$, that is, iff $a-b$ is divisible by $n$, then, the relation $R$ is

1. Reflexive only
2. Symmetric only
3. An equivalence relation
4. Transitive only

Q.

Let $R$ be the relation over the set $N\times N$ and is defined by $\left(a,b\right)R\left(c,d\right)\Rightarrow a+d=b+c$, Then $R$ is

1. Reflexive only

2. Symmetric only

3. Transitive only

4. An equivalence relation

Q.

If $R$ is a relation defined on the set of natural numbers $N$ such that $(a,b)R(c,d)$ if and only if $a+d=b+c$, then $R$ is

1. Symmetric and transitive but not reflexive

2. Reflexive and transitive but not symmetric

3. Reflexive and symmetric but not transitive

4. An equivalence relation

Q.

If $A=\{1,2,3,4\}$ & $B=\{2,4,6\}$. Then, the number of set ‘C’ such that $A\cap B\subseteq C\subseteq A\cup B$ is

1. $6$

2. $9$

3. $8$

4. $10$

Q. If $3^{x-y}=27,3^{x+y}=243$, what is the value of $x$?

1. $1$
2. $3$
3. $4$
4. $2$

Q.

$a^{{\log }_{c}b}=b^{{\log }_{c}a}\{a,b,c>0andc\ne 1\}$

1. True

2. False

Q. The value of
$\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+\frac{85}{5!}+...\infty$ is $xe-y$ Find $x-y$

Q.

$a^{{\log }_{c}b}=b^{{\log }_{c}a}\{a,b,c>0andc\ne 1\}$

1. True

2. False

Q. If $f\left(x\right)={\left(\frac{12}{13}\right)}^x+{\left(\frac{5}{13}\right)}^x-1,x\in R$, then the equation $f\left(x\right)=0$ has

1. one solution
2. no solution
3. two solutions
4. more than two solutions

Q. If $a={99}^{50}+{100}^{50}$ and $b={101}^{50}$, then :

1. $a<b$
2. $a=b$
3. $a>b$
4. $a-b={100}^{49}$