Simplification-Test 8 (Based on Algebraic formulae/Statement simplifications)

Answer: Option B

Explanation

$\begin{array}{l}\text{Here, }\\ \Rightarrow \text{56\% of 450 + ? = 300}\\ \\ \text{It can be simplified as}\\ \\ \Rightarrow \frac{56}{100}\times 450+?=300\\ \\ \Rightarrow \text{56}\times \text{4.50 + ? = 300}\\ \\ \Rightarrow \text{252 + ? = 300}\\ \\ \Rightarrow \text{? = 300}-\text{252 = 48}\end{array}$

Hence, option B is correct.

Answer: Option E

Explanation

$\begin{array}{l}\text{Here, }\\ \text{Given expression is}\\ \frac{{\text{(1111)}}^{\text{3}}{\text{+ (2212)}}^{\text{3}}-{\text{(3.323)}}^{\text{3}}}{\text{(1.111)(2212)(- 3.323)}}\\ \\ \text{Here, put a= 1.111, b = 2.112}\\ \text{and c = 3.323}\\ \\ \text{So, given expression can also be }\\ \text{written as}\\ \\ \Rightarrow \frac{{\text{a}}^{\text{3}}{\text{+ b}}^{\text{3}}{\text{+(-c)}}^{\text{3}}}{-abc}\\ \\ \text{Now, here, a + b + c = 0}\\ \\ {\text{Hence, a}}^{\text{3}}{\text{+ b}}^{\text{3}}{\text{+ c}}^{\text{3}}\text{= 3abc}\\ \\ \text{So,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{3abc}{abc}=3\\ \\ \text{So, its value lies between 2.5 and 3.5.}\end{array}$

Hence, option E is correct.

Answer: Option D

Explanation

$\begin{array}{l}\text{Here, it can also be written as}\\ \sqrt{{\text{(6)}}^{\text{2}}\text{+ (}\sqrt{\text{7}}{\text{)}}^{\text{2}}-\text{2.6}\sqrt{\text{7}}}\\ -\frac{2}{\sqrt{\text{(}\sqrt{\text{7}}{\text{)}}^{\text{2}}{\text{+ (3)}}^{\text{2}}\text{+2}\sqrt{\text{7.3}}}}\\ \\ \text{Apply the formula}\\ [{\text{ a}}^{\text{2}}{\text{+ b}}^{\text{2}}-{\text{2ab =(a- b)}}^{\text{2}}]\\ \text{and }\left[{\text{a}}^{\text{2}}{\text{+ b}}^{\text{2}}{\text{+ 2ab = (a + b)}}^{\text{2}}\right]\\ \\ \text{so, }\sqrt{\text{(6}-\sqrt{\text{7}}{\text{)}}^{\text{2}}}-\frac{2}{\sqrt{\text{(3+}\sqrt{\text{7}}{\text{)}}^{\text{2}}}}\\ \\ =\text{(6}-\sqrt{\text{7}}\text{)}-\frac{2}{3+\sqrt{7}}\\ =\text{(6}-\sqrt{\text{7}}\text{)}-\frac{2}{3+\sqrt{7}}\frac{(3-\sqrt{7})}{(3-\sqrt{7})}\\ \\ =\text{(6}-\sqrt{\text{7}}\text{)}-\frac{2(3-\sqrt{7})}{(9-7)}\\ \\ \text{[}\because {\text{a}}^{\text{2}}-{\text{b}}^{\text{2}}\text{=(a+b)(a}-\text{b)]}\\ \\ \text{=(6}-\sqrt{\text{7}}\text{)}-\text{(3}-\sqrt{\text{7}}\text{)}\\ \\ \text{= 6}-\sqrt{\text{7}}-\text{3 +}\sqrt{\text{7}}\text{=3}\end{array}$

Hence, option D is correct.

Answer: Option B

Explanation

Here, this expression is based on

$$\begin{gathered} \begin{array}{l}[{\text{a}}^{\text{2}}-{\text{b}}^{\text{2}}\text{= (a+ b)(a}-\text{b)}]\\ \\ \text{So, }\\ {\text{= 66}}^{\text{2}}-{\text{34}}^{\text{2}}\end{array} \\ \text{= (66 + 34)×(66}-\text{34)} \\ \text{= 100}\times \text{32 = 3200} \end{gathered}$$

Hence, Option B is correct.

Answer: Option A

Explanation

Here,

$\begin{array}{l}\Rightarrow {(78.95)}^{\text{2}}-{(43.35)}^{\text{2}}\text{= (78.95 + 43.35)}\\ \\ \Rightarrow \text{(78.95}-\text{43.35)}\\ \text{[}\because {\text{a}}^{\text{2}}-{\text{b}}^{\text{2}}\text{=(a + b)(a}-\text{b)]}\\ \\ \text{= (122.3)}\times \text{(35.6) = 4353.88}\end{array}$

Hence, option A is correct.