Number System - Aptitude MCQ Questions and Solutions with Explanations

$$\eqalign{ & \left( {{3^{25}} + {3^{26}} + {3^{27}} + {3^{28}}} \right)\, \cr & = {3^{25}}\left( {{3^0} + {3^1} + {3^2} + {3^3}} \right)\, \cr & = {3^{25}} \times 40 \cr & = {3^{24}} \times 120 \cr & {\text{now check with option }} \cr & {\text{Only 30 can divide this}}{\text{.}} \cr} $$

$$\eqalign{ & \Rightarrow {\text{999}}\frac{{998}}{{999}} \times 999 \cr & \Rightarrow \left( {999 + \frac{{998}}{{999}}} \right) \times 999 \cr & \Rightarrow \left[ {\left( {1000 - 1} \right) + \frac{{998}}{{999}}} \right] \times 999 \cr & \Rightarrow 999000 - 999 + 998 \cr & \Rightarrow 998999 \cr} $$

$$0.\overline {001} \,\, \Rightarrow \frac{1}{{999}}$$

$$\eqalign{ & {\text{a + b = 3 }}.......{\text{ (i)}} \cr & {\text{(a and b are two numbers)}} \cr & {{\text{a}}^2}{\text{ + }}{{\text{b}}^2} = 12 \cr & {\text{On squaring}}\,\,{\text{equation (i)}} \cr & {\left( {{\text{a + b}}} \right)^2} = \,{3^2} \cr & {{\text{a}}^2}{\text{ + }}{{\text{b}}^2} + 2{\text{ab}} = 9 \cr & 12 + 2{\text{ab = 9}} \cr & {\text{2ab = - 3}} \cr & {\text{ab = }}-\frac{{3}}{2} \cr} $$

Let total income is 15x (L.C.M. of 3, 5)
$$\eqalign{ & {\text{Spend on food}} \cr & {\text{ = }}\frac{1}{3} \times 15x \cr & = 5x \cr & {\text{Spend on rent}} \cr & {\text{ = }}\frac{2}{5} \times 15x \cr & = 6x \cr & {\text{Spend on clothes }} \cr & {\text{ = }}\frac{1}{5} \times 15x \cr & = 3x \cr & {\text{Income left }} \cr & = 15x - \left( {5 + 6 + 3} \right)x \cr & = 15x - 14x = x \cr & x = 400 \cr & 15x = 6000 \cr & {\text{Income = 6000 }} \cr} $$

$$\eqalign{ & {\text{Let the number multiplied by }}x \cr & x \times 0.022 = 66 \cr & x = \frac{{66}}{{22}} \times 1000 = \,3000 \cr} $$

$$\eqalign{ & = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{{14}} + \frac{1}{{28}} \cr & = \frac{{28 + 14 + 7 + 4 + 2 + 1}}{{28}} \cr & = \frac{{56}}{{28}} \cr & = 2 \cr} $$

Sum first n natural numbers
$${\text{ = }}\left[ {\frac{{{\text{n(n + 1)}}}}{2}} \right] = 25 \times 13$$
Sum of first 25 natural numbers
$${\text{ = }}\frac{{25(26)}}{2} = 25 \times 13$$
Now check option
Clearly, we can see 13 is factor
So, '13' is answer

$$\frac{{3.157 \times 4126 \times 3.198}}{{63.972 \times 2835.121}}$$     = 0.22
$$\eqalign{ & \Rightarrow {\text{Closest value = 0}}{\text{.2 ( approx}}{\text{.)}} \cr & {\text{ }} \cr} $$

$$\eqalign{ & {\text{Let }}x{\text{ is the number}} \cr & {\text{According to question,}} \cr & 5x = 2{x^2} - 3 \cr & 2{x^2} - 5x - 3 = 0 \cr & 2{x^2} - 6x + x - 3 = 0 \cr & 2x\left( {x - 3} \right) + 1\left( {x - 3} \right) = 0 \cr & \left( {2x + 1} \right)\left( {x - 3} \right) = 0 \cr & x = 3 \cr} $$