Match List-I with List-II and select the correct answer using the code given below the lists :

$\mathrm{If} \mathrm{x} + {\log }_{10}\left(1 + 2^{\mathrm{x}}\right) = {\mathrm{xlog}}_{10}\left(5\right) + {\log }_{10}\left(6\right) \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 2, - 3

• 2 only

• 1

• 3

$\mathrm{The} \mathrm{remainder} \mathrm{and} \mathrm{the} \mathrm{quotient} \mathrm{of} \mathrm{the} \mathrm{binary} \mathrm{division} (101110{)}_2 : (110{)}_2, \mathrm{are} \mathrm{respectively} :$

• ${\left(111\right)}_2 \mathrm{and} {\left(100\right)}_2$

• ${\left(100\right)}_2 \mathrm{and} {\left(111\right)}_2$

• ${\left(101\right)}_2 \mathrm{and} {\left(101\right)}_2$

• ${\left(100\right)}_2 \mathrm{and} {\left(100\right)}_2$

The matrix A has x rows and x + 5 columns. The matrix B has y rows and 11 - y columns. Both AB and BA exist. What are the values of x and y respectively ?

• 8 and 3

• 3 and 4

• 3 and 8

• 8 and 8

$$\begin{gathered} \mathrm{If} {\mathrm{S}}_{\mathrm{n}} = \mathrm{nP} + \frac{\mathrm{n}\left(\mathrm{n} - 1\right)\mathrm{Q}}{2}, \mathrm{where} {\mathrm{S}}_{\mathrm{n}} \mathrm{denotes} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{first} \mathrm{n} \mathrm{terms} \mathrm{of} \mathrm{an} \mathrm{AP}, \\ \mathrm{then} \mathrm{the} \mathrm{common} \mathrm{difference} \mathrm{is} : \end{gathered}$$

•  P + Q

• 2P + 3Q

• 2Q`

• Q

The roots of the equation :

$\left(\mathrm{q} - \mathrm{r}\right){\mathrm{x}}^2 + \left(\mathrm{r} - \mathrm{p}\right)\mathrm{x} + \left(\mathrm{p} - \mathrm{q}\right) = 0 \mathrm{are}$

• $\frac{\left(\mathrm{r} - \mathrm{p}\right)}{\left(\mathrm{q} - \mathrm{r}\right)}, \frac{1}{2}$

• $\frac{\left(\mathrm{p} - \mathrm{q}\right)}{\left(\mathrm{q} - \mathrm{r}\right)}, 1$

• $\frac{\left(\mathrm{q} - \mathrm{r}\right)}{\left(\mathrm{p} - \mathrm{q}\right)}, 1$

• $\frac{\left(\mathrm{r} - \mathrm{p}\right)}{\left(\mathrm{p} - \mathrm{q}\right)}, \frac{1}{2}$

If E is the universal set and A = B $\cup$ C, then the set E - (E - (E - (E - (E - A)))) is same as the set :

• $\mathrm{B}\text{'} \cup \mathrm{C}\text{'}$

• $\mathrm{B} \cup \mathrm{C}$

• $\mathrm{B}\text{'} \cap \mathrm{C}\text{'}$

• $\mathrm{B} \cap \mathrm{C}$

If A = {x : x is a multiple of 2}, B = {x : x is a multiple of 5} and C = {x : x is a multiple of 10}, then $\mathrm{A} \cap \left(\mathrm{B} \cap \mathrm{C}\right)$ is equal to :

• A

• B

• C

• {x : x is a multiple of 100}

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are the roots of the equation 1 + x + x² = 0, then the matrix product $\left[\begin{array}{cc}1& \mathrm{\beta }\\ \mathrm{\alpha }& \mathrm{\alpha }\end{array}\right]\left[\begin{array}{cc}\mathrm{\alpha }& \mathrm{\beta }\\ 1& \mathrm{\beta }\end{array}\right]$ = ?

• $\left[\begin{array}{cc}1& 1\\ 1& 2\end{array}\right]$

• $\left[\begin{array}{cc} - 1& - 1\\ - 1& 2\end{array}\right]$

• $\left[\begin{array}{cc}1& - 1\\ - 1& 2\end{array}\right]$

• $\left[\begin{array}{cc} - 1& - 1\\ - 1& - 2\end{array}\right]$

$\mathrm{If} \left|\mathrm{a}\right|$ denotes the absolute value of an integer, then which of the following are correct ?

$$\begin{gathered} 1. \left|\mathrm{ab}\right| = \left|\mathrm{a}\right|\left|\mathrm{b}\right| \\ 2. \left|\mathrm{a} + \mathrm{b}\right| \le \left|\mathrm{a}\right| + \left|\mathrm{b}\right| \\ 3. \left|\mathrm{a} - \mathrm{b}\right| \ge \left|\left|\mathrm{a}\right| - \left|\mathrm{b}\right|\right| \end{gathered}$$

Select the correct answer using the code given below :

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3