## 6th SSC CPO Solved question set, 1st on Surds and indices

This is the 6th set of 10 solved questions for SSC CPO, SSC CAPF, SSC CISF, Delhi Police ASI SI and 1st on Surds and indices. All problems are solved by specially quick methods based on discovered patterns. The solved question set contains,

**Question set on Surds and indices for SSC CPO**to be answered in 15 minutes (10 chosen questions)**Answers to the questions**, and,**Detailed conceptual solutions**to the questions.

For maximum gains, the test should be taken first, and then the solutions are to be read.

The answers and detailed solutions follow the questions. First try to solve the questions yourself and then go through the solutions. All problems are solved by specially quick methods based on discovered patterns.

**IMPORTANT:** To absorb the concepts, techniques and reasoning explained in the solutions fully and apply those in solving problems on Surds and indices quickly, one must solve many problems in a systematic manner using the conceptual analytical approach.

Learning by doing is the best learning. There is no other alternative towards achieving excellence.

#### 6th Question set - 10 problems for SSC CPO exam: 1st on topic Surds and indices - answering time 15 mins

**Q1.** If $\sqrt{5}=2.236$, then what is the value of $\displaystyle\frac{\sqrt{5}}{2}+\displaystyle\frac{5}{3\sqrt{5}}-\sqrt{45}$?

- $-2.987$
- $-6.261$
- $-4.845$
- $-8.571$

**Q2.** If $P=\displaystyle\frac{(\sqrt{7}-\sqrt{6})}{(\sqrt{7}+\sqrt{6})}$, then what is the value of $P+\displaystyle\frac{1}{P}$?

- $26$
- $13$
- $24$
- $12$

**Q3.** What is the value of $3^2+7^2+11^2+13^2+17^2-1^2-5^2-9^2-11^2-15^2$?

- $5$
- $184$
- $72$
- $92$

**Q4.** $\displaystyle\frac{\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}}{\sqrt[3]{8}}=$?

- $4$
- $\displaystyle\frac{1}{2}$
- $8$
- $2$

**Q5.** What is the simplified value of $(x^{128}+1)(x^{32}+1)(x^{64}+1)(x^{16}+1)(x^8+1)(x^4+1)(x^2+1)(x+1)$?

- $\displaystyle\frac{x^{256}-1}{x-1}$
- $\displaystyle\frac{x^{128}-1}{x-1}$
- $\displaystyle\frac{x^{64}-1}{x-1}$
- $x^{256}-1$

**Q6.** The simplified value of $(\sqrt{6}+\sqrt{10}-\sqrt{21}-\sqrt{35})(\sqrt{6}-\sqrt{10}+\sqrt{21}-\sqrt{35})$ is,

- $11$
- $10$
- $12$
- $13$

**Q7.** If $p=5+2\sqrt{6}$, then $\displaystyle\frac{\sqrt{p}-1}{\sqrt{p}}$ is,

- $1-\sqrt{2}-\sqrt{3}$
- $1-\sqrt{2}+\sqrt{3}$
- $1+\sqrt{2}-\sqrt{3}$
- $-1+\sqrt{2}-\sqrt{3}$

**Q8.** What is the value of $\sqrt{1+\displaystyle\frac{1}{2^2}+\displaystyle\frac{1}{3^2}}+\sqrt{1+\displaystyle\frac{1}{3^2}+\displaystyle\frac{1}{4^2}}+\sqrt{1+\displaystyle\frac{1}{4^2}+\displaystyle\frac{1}{5^2}}$?

- $\displaystyle\frac{7}{3}$
- $\displaystyle\frac{4}{3}$
- $\displaystyle\frac{33}{10}$
- $\displaystyle\frac{18}{5}$

**Q9.** Find the simplified value of the following expression,

$\displaystyle\frac{1}{\sqrt{11-2\sqrt{30}}}-\displaystyle\frac{3}{\sqrt{7-2\sqrt{10}}}-\displaystyle\frac{4}{\sqrt{8+4\sqrt{3}}}$.

- $0$
- $1$
- $\sqrt{3}$
- $\sqrt{2}$

**Q10.** The value of $(3+2\sqrt{2})^{-3}+(3-2\sqrt{2})^{-3}$ is,

- $128$
- $189$
- $198$
- $180$

### Answers to the questions

**Q1. Answer:** Option c: $-4.845$.

**Q2. Answer:** Option a: $26$.

**Q3. Answer:** Option b: $184$.

**Q4. Answer:** Option d: $2$.

**Q5. Answer:** Option a: $\displaystyle\frac{x^{256}-1}{x-1}$.

**Q6. Answer:** Option b: $10$.

**Q7. Answer:** Option c: $1+\sqrt{2}-\sqrt{3}$.

**Q8. Answer:** Option c: $\displaystyle\frac{33}{10}$.

**Q9. Answer:** Option a: $0$.

**Q10. Answer:** Option c: $198$.

### 6th solution set - 10 problems for SSC CPO exam: 1st on topic Surds and indices - answering time 15 mins

**Q1.** If $\sqrt{5}=2.236$, then what is the value of $\displaystyle\frac{\sqrt{5}}{2}+\displaystyle\frac{5}{3\sqrt{5}}-\sqrt{45}$?

- $-2.987$
- $-6.261$
- $-4.845$
- $-8.571$

#### Solution 1: Quick solution by target expression simplification first by surd term factoring

Adopting target expression simplification first strategy and surd term factoring you get a simplified target expression as,

$\displaystyle\frac{\sqrt{5}}{2}+\displaystyle\frac{5}{3\sqrt{5}}-\sqrt{45}$

$=\displaystyle\frac{\sqrt{5}}{2}+\displaystyle\frac{\sqrt{5}}{3}-3\sqrt{5}$

$=\sqrt{5}\left(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}-3\right)$

$=-\displaystyle\frac{13}{6}\times{2.236}=-13\times{0.3727}=-4.8451$.

**Answer:** Option c: $-4.845$.

**Key concepts used:** **Target expression simplification first -- Surd term factoring.**

For accuracy it is better to carry out the simple calculation on paper.

**Q2.** If $P=\displaystyle\frac{(\sqrt{7}-\sqrt{6})}{(\sqrt{7}+\sqrt{6})}$, then what is the value of $P+\displaystyle\frac{1}{P}$?

- $26$
- $13$
- $24$
- $12$

#### Solution 2: Immediate solution by Key pattern identification of simple result of direct addition of two mutually inverse terms

By direct addition of the two mutually inverse terms, you get a simple result,

$P+\displaystyle\frac{1}{P}=\displaystyle\frac{(\sqrt{7}-\sqrt{6})}{(\sqrt{7}+\sqrt{6})}+\displaystyle\frac{(\sqrt{7}+\sqrt{6})}{(\sqrt{7}-\sqrt{6})}$

$=\displaystyle\frac{(\sqrt{7}-\sqrt{6})^2+(\sqrt{7}+\sqrt{6})^2}{(\sqrt{7}+\sqrt{6})(\sqrt{7}-\sqrt{6})}$.

Denominator simplifies to 1, and the numerator to,

$(a-b)^2+(a+b)^2=2(a^2+b^2)=2(7+6)=26$.

**Answer:** Option a: 26.

**Key concepts used:** **Key pattern identification -- Surd arithmetic -- Basic algebraic concepts -- Solving in mind.**

**Q3.** What is the value of $3^2+7^2+11^2+13^2+17^2-1^2-5^2-9^2-11^2-15^2$?

- $5$
- $184$
- $72$
- $92$

#### Solution 3: Quick solution by Key pattern identification of pairing $(a^2-b^2)$

Examine the target expression to discover the key pattern that the terms can be paired as $(a^2-b^2)=(a+b)(a-b)$,

$E=(17^2-15^2)+(13^2-11^2)+(11^2-9^2)+(7^2-5^2)+(9-1)$

$=2(32+24+20+12+4)=2\times{92}=184$.

Without tedious calculation of squares, the target expression could easily be evaluated.

**Answer:** Option b: $184$.

**Key concepts used:** **Key pattern identification -- Basic algebraic concept -- Solving in mind.**

Q4. $\displaystyle\frac{\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}}{\sqrt[3]{8}}=$?

- $4$
- $\displaystyle\frac{1}{2}$
- $8$
- $2$

#### Solution 4: Quick solution by simplifying from the innermost square root, nested square root evaluation

The numerator is a nested square root that can be simplified only by first evaluating the innermost square root of $\sqrt{225}=15$.

The numerator is thus simplified stage by stage as,

$N=\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+15}}}}$

$=\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{169}}}}$

$=\sqrt{10+\sqrt{25+\sqrt{108+13}}}$

$=\sqrt{10+\sqrt{25+\sqrt{121}}}$

$=\sqrt{10+\sqrt{25+11}}$

$=\sqrt{10+\sqrt{36}}$

$=\sqrt{10+6}$

$=4$.

With denominator simplified as $\sqrt[3]{8}=2$, the target expression value is $2$.

**Answer:** Option d: $2$.

**Key concepts used:** **Nested square root evaluation -- Solving in mind.**

**Q5.** What is the simplified value of $(x^{128}+1)(x^{32}+1)(x^{64}+1)(x^{16}+1)(x^8+1)(x^4+1)(x^2+1)(x+1)$?

- $\displaystyle\frac{x^{256}-1}{x-1}$
- $\displaystyle\frac{x^{128}-1}{x-1}$
- $\displaystyle\frac{x^{64}-1}{x-1}$
- $x^{256}-1$

#### Solution 5: Quick solution by missing element introduction

The key pattern identified is the identification of missing element $(x-1)$.

If only this factor were there, the target expression would have been transformed in stages to $(2^{256}-1)$ by basic algebraic identity relatioon,

$(a+b)(a-b)=a^2-b^2$.

So introduce this missing element by multiplying and dividing it to transform the target expression to simply,

$\displaystyle\frac{x^{256}-1}{x-1}$.

**Answer:** Option a: $\displaystyle\frac{x^{256}-1}{x-1}$.

**Key concepts used:** **Key pattern identification -- Missing element introduction technique -- Basic algebraic concepts -- Solving in mind.**

**Q6.** The simplified value of $(\sqrt{6}+\sqrt{10}-\sqrt{21}-\sqrt{35})(\sqrt{6}-\sqrt{10}+\sqrt{21}-\sqrt{35})$ is,

- $11$
- $10$
- $12$
- $13$

#### Solution 6: Quick solution by key pattern identification of converting target expression in the form of $(a-b)(a+b)=a^2-b^2$

Identify the key pattern that imagining $a=\sqrt{6}-\sqrt{35}$ and $b=\sqrt{21}-\sqrt{10}$, the target expression is simplified to,

$E=(a-b)(a+b)=a^2-b^2$

$=(\sqrt{6}-\sqrt{35})^2-(\sqrt{21}-\sqrt{10})^2$

$=41-\sqrt{420}-31+\sqrt{420}=10$.

**Answer:** Option b: $10$.

**Key concepts used:** **Key pattern identification -- Abstraction -- Basic algebraic relation of $(a+b)(a-b)=a^2-b^2$ -- Surd arithmetic -- Solving in mind.**

**Q7.** If $p=5+2\sqrt{6}$, then $\displaystyle\frac{\sqrt{p}-1}{\sqrt{p}}$ is,

- $1-\sqrt{2}-\sqrt{3}$
- $1-\sqrt{2}+\sqrt{3}$
- $1+\sqrt{2}-\sqrt{3}$
- $-1+\sqrt{2}-\sqrt{3}$

#### Solution 7: Solve quickly by double square root surd simplification and target expression simplification first strategy

As $\sqrt{p}$ is a double square root surd, first we simplify it as,

$\sqrt{p}=\sqrt{5+2\sqrt{6}}$

$=\sqrt{(\sqrt{3}+\sqrt{2})^2}$, as sum of 3 and 2 is 5 and product 6,

$=(\sqrt{3}+\sqrt{2})$.

Instead of substituting this value straightaway in the target let us first simplify the target and then substitute,

$E=\displaystyle\frac{\sqrt{p}-1}{\sqrt{p}}$

$=1-\displaystyle\frac{1}{\sqrt{p}}=1-\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}$.

Just rationalize the denominator of the fraction surd term,

$E=1+\sqrt{2}-\sqrt{3}$.

**Answer:** Option c: $1+\sqrt{2}-\sqrt{3}$.

**Key concepts used:** **Double square root surd simplification -- Target expression simplification first -- Surd rationalization -- Solving in mind.**

**Q8.** What is the value of $\sqrt{1+\displaystyle\frac{1}{2^2}+\displaystyle\frac{1}{3^2}}+\sqrt{1+\displaystyle\frac{1}{3^2}+\displaystyle\frac{1}{4^2}}+\sqrt{1+\displaystyle\frac{1}{4^2}+\displaystyle\frac{1}{5^2}}$?

- $\displaystyle\frac{7}{3}$
- $\displaystyle\frac{4}{3}$
- $\displaystyle\frac{33}{10}$
- $\displaystyle\frac{18}{5}$

#### Solution 8: Quick solution by key pattern identification of simple values of each fraction term under square root

The fraction terms under square roots are simplified to just,

$1+\displaystyle\frac{1}{2^2}+\displaystyle\frac{1}{3^2}=1+\displaystyle\frac{1}{4}+\displaystyle\frac{1}{9}=\displaystyle\frac{49}{36}$,

$1+\displaystyle\frac{1}{3^2}+\displaystyle\frac{1}{4^2}=1+\displaystyle\frac{1}{9}+\displaystyle\frac{1}{16}=\displaystyle\frac{169}{144}$, and,

$1+\displaystyle\frac{1}{4^2}+\displaystyle\frac{1}{5^2}=1+\displaystyle\frac{1}{16}+\displaystyle\frac{1}{25}=\displaystyle\frac{441}{400}$.

Each of the six integers involved are squares of integers.

The target expression evaluates to,

$\displaystyle\frac{7}{6}+\displaystyle\frac{13}{12}+\displaystyle\frac{21}{20}$

$=3+\displaystyle\frac{1}{6}+\displaystyle\frac{1}{12}+\displaystyle\frac{1}{20}=3+\displaystyle\frac{18}{60}=\displaystyle\frac{33}{10}$.

Integer parts were separately added together to reduce computational load. This is part of **efficient fraction simplification techniques.**

**Answer:** Option c: $\displaystyle\frac{33}{10}$.

**Key concepts used:** **Fraction arithmetic -- ***Efficient fraction simplification techniques* -- **Solving in mind.**

**Q9.** Find the simplified value of the following expression,

$\displaystyle\frac{1}{\sqrt{11-2\sqrt{30}}}-\displaystyle\frac{3}{\sqrt{7-2\sqrt{10}}}-\displaystyle\frac{4}{\sqrt{8+4\sqrt{3}}}$.

- $0$
- $1$
- $\sqrt{3}$
- $\sqrt{2}$

#### Solution 9: Quick solution by Double square root surd simplification and surd rationalization

The three double square root surd expressions involved are simplified by expressing these as whole square expressions,

$\sqrt{11-2\sqrt{30}}=(\sqrt{6}-\sqrt{5})$, sum of 6 and 5 is 11, and product 30,

$\sqrt{7-2\sqrt{10}}=(\sqrt{5}-\sqrt{2})$, sum of 5 and 2 is 7, and product 10, and,

$\sqrt{8+4\sqrt{3}}=(\sqrt{6}+\sqrt{2})$, sum of 6 and 2 is 8, and product $12$, as, $(4\sqrt{3}=2\sqrt{12})$.

The target expression is simplified to,

$E=\displaystyle\frac{1}{(\sqrt{6}-\sqrt{5})}-\displaystyle\frac{3}{(\sqrt{5}-\sqrt{2})}-\displaystyle\frac{4}{(\sqrt{6}+\sqrt{2})}$.

Rationalize the denominators by multiplying and dividing the fraction terms by $(\sqrt{6}+\sqrt{5})$, $(\sqrt{5}+\sqrt{2})$ and, $(\sqrt{6}-\sqrt{2})$ respectively.

The second term numerator 3 and third term numerator 4 cancel out. Result is,

$(\sqrt{6}+\sqrt{5})-(\sqrt{5}+\sqrt{2})-(\sqrt{6}-\sqrt{2})$

$=0$.

**Answer:** Option a: $0$.

**Key concepts used:** **Double square root surd simplification -- Surd rationalization -- Solving in mind.**

**Q10.** The value of $(3+2\sqrt{2})^{-3}+(3-2\sqrt{2})^{-3}$ is,

- $128$
- $189$
- $198$
- $180$

#### Solution 10: Quick solution by key pattern identification, simplification by surd rationalization, abstraction and sum of cubes factorization

Identify the first key pattern,

$(3+2\sqrt{2})=\displaystyle\frac{1}{(3-2\sqrt{2})}$, as the product of the two is 1.

By this, the target expression is simplified to a sum of cubes of compound variables,

$E=(3+2\sqrt{2})^{-3}+(3-2\sqrt{2})^{-3}$

$=(3-2\sqrt{2})^3+(3+2\sqrt{2})^3$.

By abstraction, you may take this expression as a sum of cubes of compound variables,

$x=(3-2\sqrt{2})$, and $y=(3+2\sqrt{2})$.

By sum of cubes factorization,

$x^3+y^3=(x+y)(x^2-xy+y^2)$.

With the values of $x$ and $y$,

$(x+y)=6$,

$x^2+y^2=2[3^2+(2\sqrt{2})^2]=34$, and,

$xy=3^2-(2\sqrt{2})^2=1$.

So, target expression value,

$E=6(34-1)=198$.

**Answer:** Option c: $198$.

**Key concepts used:** **Key pattern identification -- Simplification by surd rationalization -- Abstraction in imagining the two surd expressions as single compound variables -- Sum of cubes factorization -- Solving in mind.**

With clear concepts, you can easily solve the problem in mind.

### End note

Observe that, each of the problems could be quickly and cleanly solved in minimum number of steps using special key patterns and methods in each case.

This is the hallmark of **quick problem solving:**

**Concept based pattern and method formation**, and,**Identification of the key pattern and use of the method associated with it.**Every special pattern has its own method, and not many such patterns are there.

Important is the *concept based pattern identification and use of quick problem solving method.*

### SSC CPO level Question and Solution sets

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**SSC CPO level Solved Question set 1 on Algebra 1**

**SSC CPO level Solved Question set 2 on Algebra 2**

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**SSC CPO level Solved Question set 3 on Trigonometry 1**

**SSC CPO level Solved Question set 4 on Trigonometry 2**

#### Number system

**SSC CPO level Solved Question set 5 on Number system 1**

#### Surds and Indices

**SSC CPO level Solved Question set 6 on Surds and indices 1**