## 7th SSC CHSL Solved Question Set, 1st on Surds and indices

This is the 7th solved question set of 10 practice problem exercise for SSC CHSL exam and the 1st on topic Surds and indices. It contains,

**Question set on Surds and Indices**for SSC CHSL 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 referred to. But more importantly, to absorb the concepts, techniques and reasoning explained in the solutions, 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.

### 7th Question set - 10 problems for SSC CHSL exam: 1st on topic Surds and Indices - answering time 15 mins

**Q1. **The total number of prime factors in $4^{10}\times{7^3}\times{16^2}\times{11}\times{10^2}$ is,

- 36
- 34.
- 37
- 35

**Q2.** If $x^{\frac{1}{4}}+x^{-\frac{1}{4}}=2$ then what is the value of $x^{\frac{1}{81}}+x^{-\frac{1}{81}}$?

- $-2$
- $0$
- $2$
- $1$

**Q3. **If $x$, $y$ are rational numbers and $\displaystyle\frac{5+\sqrt{11}}{3-2\sqrt{11}}=x+y\sqrt{11}$, the values of $x$ and $y$ are,

- $x=\displaystyle\frac{4}{13}$, $y=\displaystyle\frac{11}{17}$
- $x=-\displaystyle\frac{14}{17}$, $y=-\displaystyle\frac{13}{26}$
- $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$
- $x=-\displaystyle\frac{27}{25}$, $y=-\displaystyle\frac{11}{37}$

**Q4. **The greatest among the numbers, $0.16$, $\sqrt{0.16}$, $(0.16)^2$, $0.04$ is,

- $0.16$
- $\sqrt{0.16}$
- $(0.16)^2$
- $0.04$

**Q5.** Out of the numbers 0.3, 0.03, 0.9 and 0.09, the number that is nearest to the value of $\sqrt{0.9}$ is,

- $0.09$
- $0.3$
- $0.03$
- $0.9$

**Q6.** Which of the following statement(s) is/are TRUE?

- $\sqrt{144}\times{\sqrt{36}} \lt \sqrt[3]{125}\times{\sqrt{121}}$
- $\sqrt{324}+\sqrt{49} \lt \sqrt[3]{216}\times{\sqrt{9}}$

- Only I
- Only II
- Both I and II
- Neither I nor II

**Q7.** If $\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})}{\sqrt{75}-\sqrt{50}}=1$, then the value of $x$ is,

- $3\sqrt{5}$
- $5$
- $\sqrt{5}$
- $2\sqrt{5}$

**Q8.** The value of $\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\displaystyle\frac{1}{\sqrt{100}+\sqrt{99}}$ is,

- $1$
- $\sqrt{99}$
- $9$
- $\sqrt{99}-1$

**Q9.** If $1^3+2^3+3^3....+10^3=3025$, then the value of $2^3+4^3+....+20^3$ is,

- 7590
- 5060
- 12100
- 24200

**Q10.** $\left(\displaystyle\frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}}+\displaystyle\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}\right)$ simplifies to,

- $\sqrt{5}-\sqrt{6}$
- $\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}-3\sqrt{6}$

### Answers to the questions

**Q1. Answer:** Option a: 36.

**Q2. Answer:** Option c: $2$.

**Q3. Answer:** Option c: $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$

**Q4. Answer:** Option b: $\sqrt{0.16}$.

**Q5. Answer:** Option d: $0.9$.

**Q6. Answer:** Option d : Neither I nor II.

**Q7. Answer:** Option b: 5.

**Q8. Answer:** Option c: 9.

**Q9. Answer:** Option d: 24200.

**Q10. Answer:** Option a: $\sqrt{5}-\sqrt{6}$.

### 7th solution set - 10 problems for SSC CHSL exam: 1st on topic Surds and Indices - answering time 15 mins

**Q1. **The total number of prime factors in $4^{10}\times{7^3}\times{16^2}\times{11}\times{10^2}$ is,

- 36
- 34.
- 37
- 35

** Solution 1: Problem analysis and solution by prime factorization of each term of the product**

Let's first take care of base factor 2.

$4^{10}=2^{20}$ contributes to 20 number of 2s.

$16^2=2^8$ contributes to 8 number of 2s, and

$10^2=2^2.5^2$ contributes to 2 number of 2s.

Total number of 2s is 30.

Add to this 2 number of 5s from $10^2$, 1 number of 11 and 3 number of 7s from $7^3$.

The total number of prime factors in the product is,

$30+2+1+3=36$.

**Answer:** Option a: 36.

**Key concepts used: Counting prime factors in a product -- Indices -- Prime factorization** --

*.*

**Solving in mind****Q2.** If $x^{\frac{1}{4}}+x^{-\frac{1}{4}}=2$ then what is the value of $x^{\frac{1}{81}}+x^{-\frac{1}{81}}$?

- $-2$
- $0$
- $2$
- $1$

**Solution 2: Problem solving using principle of interaction of inverses**

You cannot derive the value of the target expresion with large inverse powers in $x$ from the given expression directly. The only possibility is to get a simple value of $x$ from the given expression, which should be 1, and use the value in the target expression.

The LHS of the given equation is a sum of inverses that can be expressed as a square of inverses in powers of $x$ as 2 easily because the middle term will be neutralized to an integer,

$x^{\frac{1}{4}}+x^{-\frac{1}{4}}=2$,

Or, $(x^{\frac{1}{2}})^2-2.x^{\frac{1}{2}}.x^{-\frac{1}{2}}+(x^{-\frac{1}{2}})^2=0$,

Or, $(x^{\frac{1}{2}}-x^{-\frac{1}{2}})^2=0$,

So, $x^{\frac{1}{2}}=x^{-\frac{1}{2}}$,

Or, $x^{\frac{1}{4}}=1$,

So, $x=1$.

Answer is,

$x^{\frac{1}{81}}+x^{-\frac{1}{81}}=1+1=2$

**Answer:** Option c: $2$.

**Key concepts used:** **Principle of interaction of sum of inverses -- Mathematical reasoning -- Indices-- Solving in mind****.**

**Q3. **If $x$, $y$ are rational numbers and $\displaystyle\frac{5+\sqrt{11}}{3-2\sqrt{11}}=x+y\sqrt{11}$, the values of $x$ and $y$ are,

- $x=\displaystyle\frac{4}{13}$, $y=\displaystyle\frac{11}{17}$
- $x=-\displaystyle\frac{14}{17}$, $y=-\displaystyle\frac{13}{26}$
- $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$
- $x=-\displaystyle\frac{27}{25}$, $y=-\displaystyle\frac{11}{37}$

**Solution 3: Problem analysis and solution by pattern identification, remainder concept and HCF concept**

You have to first simplify the LHS of the given equation by eliminating the surd denominator to get an expression of the form, $a+b\sqrt{11}$ so that comparing the coefficients of the surd terms and equating the rational terms of two sides of the equation you would get the values of $x$ and $y$.

To eliminate the surd expression in the denominator, rationalize it by multiplying both numerator and denominator by $(3+2\sqrt{11})$ getting,

$\displaystyle\frac{5+\sqrt{11}}{3-2\sqrt{11}}\times{\displaystyle\frac{3+2\sqrt{11}}{3+2\sqrt{11}}}=x+y\sqrt{11}$,

Or, $-\displaystyle\frac{15+22+13\sqrt{11}}{35}=x+y\sqrt{11}$,

Or, $-\displaystyle\frac{37}{35}-\displaystyle\frac{13}{35}.\sqrt{11}=x+y\sqrt{11}$.

By the surd priperties then values of $x$ and $y$, both rational numbers, must be,

$x=-\displaystyle\frac{37}{35}$, and

$y=-\displaystyle\frac{13}{35}$.

**Answer:** Option c: $x=-\displaystyle\frac{37}{35}$, $y=-\displaystyle\frac{13}{35}$.

**Key concepts used: Surd denominator rationalization -- Comparing rational coefficients of surd terms and equating rational terms on both sides of a surd equation.**

**Q4. **The greatest among the numbers, $0.16$, $\sqrt{0.16}$, $(0.16)^2$, $0.04$ is,

- $0.16$
- $\sqrt{0.16}$
- $(0.16)^2$
- $0.04$

**Solution 4: Problem solving using concept of powers of a decimal number**

The basic principle of raising a decimal number to a power is,

If a decimal that is less than 1, is raised to a positive integer power, the value of the decimal with power gets smaller than the original decimal number. For example, $(0.4)^2=0.16 \lt 0.4$.

Similarly, if a decimal that is less than 1, is raised to a positive power less than 1, the value of the decimal in power increases. For example, $(0.25)^{\frac{1}{2}}=0.5 \gt 0.25$.

Applying this pair of concepts, identify $\sqrt{0.16}$ as the largest among the first three given nimbers, $0.16$, $\sqrt{0.16}$ and $(0.16)^2$.

Now actually take the square root of $0.16$ to compare it with the fourth number $0.04$,

$\sqrt{0.16}=0.4 \gt 0.04$.

So, $\sqrt{0.16}$ is the largest among the four given numbers.

**Answer:** Option b: $\sqrt{0.16}$.

**Key concepts used: Principle of powers of decimal numbers -- Square root of a decimal number -- Solving in mind.**

**Q5.** Out of the numbers 0.3, 0.03, 0.9 and 0.09, the number that is nearest to the value of $\sqrt{0.9}$ is,

- $0.09$
- $0.3$
- $0.03$
- $0.9$

**Solution 5: Problem analysis and solving by Approximation of square root of a decimal number**

First find the approximate value of,

$\sqrt{0.9}=0.94$

We have formed $(0.9)^2=0.81$ and then mentally evalauted $(0.95)^2=0.9025$, just exceeding 0.9.

Among the four choice values then $0.9$ will be closest to $\sqrt{0.9}$.

**Answer:** Option d: $0.9$.

*Key concepts used:* **Approximate square root estimation of a decimal in square root -- Solving in mind****.**

**Q6.** Which of the following statement(s) is/are TRUE?

- $\sqrt{144}\times{\sqrt{36}} \lt \sqrt[3]{125}\times{\sqrt{121}}$
- $\sqrt{324}+\sqrt{49} \lt \sqrt[3]{216}\times{\sqrt{9}}$

- Only I
- Only II
- Both I and II
- Neither I nor II

**Solution 6: Problem analysis and solution by squares and cubes of common integers and inequality analysis**

First evaluate the four roots of the first inequality,

$\sqrt{144}=12$,

$\sqrt{36}=6$,

$\sqrt[3]{125}=5$, and

$\sqrt{121}=11$.

$\text{LHS of the first inequality}= 72$, and

$\text{RHS of the first inequality}=55$.

So the first inequality is FALSE. So only option b or d may be the answer.

Now evaluate the four roots of the second inequality,

$\sqrt{324}=18$,

$\sqrt{49}=7$,

$\sqrt[3]{216}=6$, and,

$\sqrt{9}=3$.

$\text{LHS of the second inequality}= 18+7=25$, and

$\text{RHS of the second inequality}=6\times{3}=18$.

Again this inequality is FALSE.

Neither I nor II is TRUE.

**Answer:** Option d : Neither I nor II.

**Key concepts used:** *Squares and cubes of common integers -- Inequality analysis -- Solving in mind.*

**Q7.** If $\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})}{\sqrt{75}-\sqrt{50}}=1$, then the value of $x$ is,

- $3\sqrt{5}$
- $5$
- $\sqrt{5}$
- $2\sqrt{5}$

**Solution 7: Problem analysis and Solving by rationalization and surd arithmetic**

We'll not carry out the product in the numerator. The first step is to simplify the LHS expression by rationalization of the denominator.

Multiply and divide the LHS by $(\sqrt{75}+\sqrt{50})$,

$\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})}{\sqrt{75}-\sqrt{50}}=1$

Or, $\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})}{\sqrt{75}-\sqrt{50}}\times{\displaystyle\frac{\sqrt{75}+\sqrt{50}}{\sqrt{75}+\sqrt{50}}}=1$

Or, $\displaystyle\frac{(x-\sqrt{24})(\sqrt{75}+\sqrt{50})^2}{25}=1$

Or, $\displaystyle\frac{(x-\sqrt{24})(125+25.2\sqrt{6})}{25}=1$, factoring $\sqrt{25}$ out of $\sqrt{75}$ and $\sqrt{50}$,

Or, $(x-\sqrt{24})(5+2\sqrt{6})=1$

Though this is a much simplified expression, even now we'll not expand the product.

Instead we'll take $(5+2\sqrt{6})$ in the denominator of the RHS and then rationalize. It is easy to see that numerator of RHS will simply become $(5-2\sqrt{6})$ as, $[5^2-(2\sqrt{6})^2=1]$.

Accordingly you get in the next step,

$x-2\sqrt{6}=\displaystyle\frac{1}{5+2\sqrt{6}}\times{\displaystyle\frac{5-2\sqrt{6}}{5-2\sqrt{6}}}=5-2\sqrt{6}$,

Or, $x=5$, a simple result.

**Answer:** Option b: 5.

** Key concepts used:** ** Surd rationalization in stages **--

**Surd term factoring -- Convenient pattern of difference of square of two surd terms as 1 -- Surd arithmetic***--*Solving in mind.If you are comfortable in Surd arithmetic and rationalization, you can solve this problem easily in mind.

**Q8.** The value of $\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\displaystyle\frac{1}{\sqrt{100}+\sqrt{99}}$ is,

- $1$
- $\sqrt{99}$
- $9$
- $\sqrt{99}-1$

** Solution 8: Problem analysis and solving by identifying and using Simplified surd rationalization result pattern**

Identify the pattern that the difference of squares of the two terms in each denominator of the terms in the sum is 1.

When you rationalize each term, the denominator is cleanly eliminated.

Rationalizing all the terms in the sum you get,

$\displaystyle\frac{1}{\sqrt{2}+1}+\displaystyle\frac{1}{\sqrt{3}+\sqrt{2}}+\displaystyle\frac{1}{\sqrt{4}+\sqrt{3}}+.....+\displaystyle\frac{1}{\sqrt{100}+\sqrt{99}}$

$=(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})+.....+(\sqrt{100}-\sqrt{99})$.

Now identify the second pattern that the $(-\sqrt{2})$ of the second term cancels out $(\sqrt{2})$ of the first term leaving 1.

The $(-\sqrt{3})$ of the third term cancel out $(\sqrt{3})$ of the second term, thus eliminating both the elements of the second term.

This pattern of cancellation goes on to finally $(-\sqrt{99})$ of the last term cancelling out $(\sqrt{99})$ of the last but one term.

Finally, only $(-1)$ of the first term and, $\sqrt{100}=10$ of the last term are left.

Result is, $10-1=9$.

**Answer:** Option c: 9.

**Key concepts used:** **Simplified surd rationalization pattern identification -- Surd rationalization -- Identification of Pattern of term cancellation in the whole series -- Solving in mind.**

**Q9.** If $1^3+2^3+3^3....+10^3=3025$, then the value of $2^3+4^3+....+20^3$ is,

- 7590
- 5060
- 12100
- 24200

**Solution 9: Problem analysis and Solving by term factoring and product rule of equal power terms in a series**

Identify the key pattern that if you take $2^3$ out of each term in the series it is converted simply to,

$2^3+4^3+....+20^3$

$=2^3\times{(1^3+2^3+....+10^3)}$

$=2^3\times{3025}$

$=24200$

This happens because of the product rule,

$(xy)^3=x^3\times{y^3}$.

**Answer:** Option d: 24200.

**Key concepts used:** * Product rule for variables of equal power *--

**Solving in mind.****Q10.** $\left(\displaystyle\frac{1+\sqrt{2}}{\sqrt{5}+\sqrt{3}}+\displaystyle\frac{1-\sqrt{2}}{\sqrt{5}-\sqrt{3}}\right)$ simplifies to,

- $\sqrt{5}-\sqrt{6}$
- $\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}+\sqrt{6}$
- $2\sqrt{5}-3\sqrt{6}$

**Solution 10: Problem analysis and Solution by simple cross-multiplication forming numerator**

Denominator simplifies to,

$(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})=(\sqrt{5})^2-(\sqrt{3})^2=2$.

In the numerator you have eight terms,

$(\sqrt{5}-\sqrt{3})(1+\sqrt{2})+(\sqrt{5}+\sqrt{3})(1-\sqrt{2})$

$=(\sqrt{10}-\sqrt{10})+2\sqrt{5}+(-\sqrt{3}+\sqrt{3})-2\sqrt{6}$

$=2(\sqrt{5}-\sqrt{6})$.

As denominator already is evaluated as 2, the final result is,

$\sqrt{5}-\sqrt{6}$.

**Answer: **Option a: $\sqrt{5}-\sqrt{6}$.

Observing the similarities in the numerator and denominator terms and the opposite signs of the terms, with a little care you can easily solve the problem in mind.

**Key concepts used:** **Surd arithmetic -- Simplifying elimination by cross-multiplication -- Solving in mind.**

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