## 81st SSC CGL level Solution Set, 17th on Algebra

This is the 81st solution set of 10 practice problem exercise for SSC CGL exam and the 17th on topic Algebra. For maximum gains, the test should be taken first and then this solution set should be referred to.

If you have not yet taken the corresponding test yet you may take it by referring to the * SSC CGL level question set 81 on Algebra 17* before going through the solution.

You may also watch the **video solutions** in the two-part video.

**Part 1: Q1 to Q5**

**Part 2: Q6 to Q10**

### 81st solution set - 10 problems for SSC CGL exam: 17th on topic Algebra - answering time 12 mins

**Q1. **If $x+\displaystyle\frac{1}{x}=3$, then value of $x^8+\displaystyle\frac{1}{x^8}$ is,

- 2213
- 2207
- 2201
- 2203

** Solution 1: Problem analysis and solution by applying principle of inverses: Solving in mind**

Given,

$x+\displaystyle\frac{1}{x}=3$.

Raising the equation to its square will eliminate $x$ in the middle term 2 because of interaction of inverses $x$ and $\displaystyle\frac{1}{x}$. We will then easily take the 2 from LHS to RHS reducing $3^2=9$ in RHS to $7$.

This gives at the first squaring,

$x^2+\displaystyle\frac{1}{x^2}=7$.

Without writing any more, we can raise this equation again to its square to get,

$x^4+\displaystyle\frac{1}{x^4}=7^2-2=47$.

And lastly squaring for the third time,

$x^8+\displaystyle\frac{1}{x^8}=47^2-2=2207$.

**Answer:** Option b: 2207.

**Key concepts used:** * Problem analysis* --

*Key pattern identification**.*

**-- Principle of interaction of inverses -- Solving in mind**Knowing the power and use of the principle of interaction of inverses, we could evaluate the final result easily in mind without writing the steps. For accuracy we calculated $47^2$ by hand.

**Q2.** If $(x-4)(x^2+4x+16)=x^3-p$, then value of $p$ is,

- 27
- 0
- 8
- 64

**Solution 2: Problem analysis and solving in mind by coefficient comparison and mathematical reasoning**

$-p$ on the RHS will be equal to the product of the numeric terms in the two factors on the LHS and will not depend on anything else. So,

$-p=-4\times{16}=-64$,

Or, $p=64$.

**Answer:** Option d : 64.

**Key concepts used:** **Coefficient comparison technique -- Solving in mind****.**

**Q3. **If $a+b+c=15$ and $\displaystyle\frac{1}{a}+\displaystyle\frac{1}{b}+\displaystyle\frac{1}{c}=\displaystyle\frac{71}{abc}$, then the value of $a^3+b^3+c^3-3abc$ is,

- 180
- 160
- 220
- 200

**Solution 3: Problem analysis and solution using three-variable sum of cubes expression**

By the three-variable sum of cubes expression we have,

$a^3+b^3+c^3-3abc$

$=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

$=15(a^2+b^2+c^2-ab-bc-ca)$.

We will get the second set of three terms of the above expression from the second given expression,

$\displaystyle\frac{1}{a}+\displaystyle\frac{1}{b}+\displaystyle\frac{1}{c}=\displaystyle\frac{71}{abc}$.

Multiplying both sides by $abc$,

$ab+bc+ca=71$.

And with values of $a+b+c$ and $ab+bc+ca$ known it is easy to get value of $a^2+b^2+c^2$,

$(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)=225$,

Or, $a^2+b^2+c^2=225-142=83$.

Finally going back to our previous result,

$a^3+b^3+c^3-3abc$

$=15(a^2+b^2+c^2-ab-bc-ca)$

$=15\times(83-71)$

$=180$

**Answer:** Option a: 180.

**Key concepts used: Problem analysis -- Three-variable **

**Sum of cubes -- Three-variable square of sum -- Solving in mind.**

This problem also could also be solved in mind as the steps are clear and calculations easy. Due to two stages of solution—first, getting the value of $ab+bc+ca$ and second the value of $a^2+b^2+c^2$—the time taken was a bit more, but it was still within a minute.

**Q4. **If $x=12$, and $y=4$, then the value of $(x+y)^{\displaystyle\frac{x}{y}}$ is,

- 570
- 48
- 1792
- 4096

**Solution 4: Problem analysis and solving in mind by unit's digit behavior analysis**

Simplifying the target expression to $(x+y)=16$ raised to the power of 3 by $\displaystyle\frac{x}{y}=\frac{12}{4}=3$ is easy.

At the next step, we didn't do any multiplication at all. We have used just the result of unit's digit behavior for powers of 6,

$\text{Unit's digit of } 6^n=6$.

The only choice value 4096 having unit's digit of 6, we knew, $16^3$ must be equal to 4096.

**Answer:** Option d: 4096.

**Key concepts used: Indices -- Unit's digit behavior analysis -- **

**Free resource use principle—we have used the free resource of choice values****--**

*S*

**olving in mind**

*.*#### Unit's digit behavior analysis—two principles define unit's digit behavior

The first principle of unit's digit behavior is,

Unit's digit of the product of two integers is the unit's digit of the product of the unit's digits of the integers.

For example, unit's digit of $4398\times{79}$ will be 2, the unit's digit of $8\times{9}=72$.

The second principle is,

Unit's digit for powers of the digits 2 through 9 will follow a fixed pattern specific to the digit.

For example, unit's digit of any power of 5 and 6 will remain always locked on 5 and 6 respectively, whereas unit's digit for powers of 3 will cycle through the 4 digits 3, 9, 7, 1 and then back to 3.

You can easily form the pattern specific to a digit and save valuable calculation time using these two principles.

Unit's digit for a few powers of 6 are,

$6^2=36$=>$\text{unit's digit } 6$.

$6^3=216$=>$\text{unit's digit } 6$.

$6^4=1296$=>$\text{unit's digit } 6$, and so on.

However many times you multiply 6 with itself, the unit's digit remains locked at the value of 6.

**Q5. **If $xy+yz+zx=1$, then the value of $\displaystyle\frac{1+y^2}{(x+y)(y+z)}$ is,

- 1
- 4
- 2
- 3

**Solution 5: Problem analysis, key pattern identification and solving in mind by targeted substitution**

We use the given expression value that can be used in the only way—by expanding $(x+y)(y+z)=xy+yz+zx+y^2$, and substituting given value 1 of $xy+yz+zx$ to get the denominator same as the numerator.

**Answer:** Option a: 1.

**Key concepts used:** **Key pattern identification -- Substitution -- solving in mind.**

**Q6.** If $x+y+z=0$, then the value of $\displaystyle\frac{x^2}{3z}+\displaystyle\frac{y^3}{3xz}+\displaystyle\frac{z^2}{3x}$ is,

- $0$
- $3y$
- $y$
- $xz$

**Solution 6: Problem analysis, key pattern identification and solving in mind**

The key pattern identified as—multiplying the target expression by $xz$ transforms the numerator to $(x^3+y^3+z^3)$, and we know by three variable sum of cubes expression,

If $a+b+c=0$, then $a^3+b^3+c^3=3abc$.

Thus the numerator is simplified to $3xyz$ and the denominator is $3xz$. The answer turns out to $y$.

Let us show the deductions.

Target expression,

$\displaystyle\frac{x^2}{3z}+\displaystyle\frac{y^3}{3xz}+\displaystyle\frac{z^2}{3x}$

$=\displaystyle\frac{1}{3xz}(x^3+y^3+z^3)$,

$=\displaystyle\frac{3xyz}{3xz}$, as with $x+y+z=0$, $x^3+y^3+z^3=3xyz$,

$=y$.

**Answer:** Option c : $y$.

**Key concepts used: **

*.*

**Key pattern identification -- Target expression transformation -- Three-variable sum of cubes expression -- Solving in mind**** Q7.** If the expression $(px^3-2x^2-qx+18)$ is completely divisible by $(x^2-9)$, then the ratio of $p$ to $q$ is,

- 9 : 1
- 1 : 9
- 3 : 1
- 1 : 3

**Solution 7: Problem analysis, mathematical reasoning and solving in mind by coefficient comparison**

By examining the two given expressions we can form the quotient factor $(ax-2)$,

$px^3-2x^2-qx+18=(x^2-9)(ax-2)$.

Here $a$ is an additional dummy variable and the numeric term of $(ax-2)$ must be $-2$ to make the product $(-9)\times{(-2)}=18$.

Analyzing the two expressions on the LHS and RHS while comparing the coefficients of $x$ and $x^3$ we can form the relations,

$p=a$, and

$-q=-9a$,

Or, $q=9$.

The ratio is, $p:q=1:9$.

**Answer:** Option b: 1 : 9.

** Key concepts used:** * Key pattern identification* --

**Quotient expression formation -- Coefficient comparison -- Basic ratio concepts.**

** Q8.** If $\displaystyle\frac{x}{3}+\displaystyle\frac{3}{x}=1$, then the value of $x^3$ is,

- $0$
- $27$
- $-27$
- $1$

** Solution 8: Problem analysis, key pattern identification and solving in mind by factor extraction technique**

Simplifying the given expression,

$\displaystyle\frac{x}{3}+\displaystyle\frac{3}{x}=1$

Or, $x^2-3x+9=0$.

Multiplying by $x$ to form term $x^3$,

$x^3-3x^2+9x=0$,

Or, $x^3-3(x^2-3x+9) + 27=0$, extracting zero valued factor,

Or, $x^3=-27$.

**Answer:** Option c: $-27$.

**Key concepts used:** * Problem analysis* --

**Key pattern identification -- Factor extraction technique -- Solving in mind****.**The problem is formed intending to test your problem skill and not math skills. In fact, the values of $x$ are imaginary from the quadratic equation, whereas $x=-3$ from $x^3$. Nevertheless, we are not to look more closely into the value of $x$, we are to find the value of $x^3$.

**Q9.** If $\displaystyle\frac{1}{x}+\displaystyle\frac{1}{y}+\displaystyle\frac{1}{z}=0$ and $x+y+z=9$, then the value of $x^3+y^3+z^3-3xyz$ is,

- 81
- 6561
- 729
- 361

**Solution 9: Problem analysis, key pattern identification and solving in mind by three-variable sum of cubes relation**

The first given equation is transformed to,

$xy+yz+zx=0$.

The three-variable sum of cubes relation is,

$x^3+y^3+z^3-3xyz$

$=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$

$=9(x^2+y^2+z^2)$.

Again by three-variable square of sum expansion,

$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$,

Or, $x^2+y^2+z^2=9^2=81$.

So the value of target expression is,

$9^3=729$.

**Answer:** Option c: 729.

**Key concepts used:** **Key pattern identification -- ****Input transformation -- Three-variable sum of cubes -- Three-variable square of sum -- Solving in mind.**

** Q10.** If $a(a+b+c)=45$, $b(a+b+c)=75$ and $c(a+b+c)=105$, then the value of $a^2+b^2+c^2$ is,

- 75
- 83
- 225
- 217

**Solution 10: Problem analysis, key pattern identification and Collection of friendly terms: Solving in mind**

Identifying the pattern of $(a+b+c)$ common to the three equations we sum up the three to get,

$(a+b+c)^2=225$,

Or, $a+b+c=15$.

This immediately gives values of $a$, $b$ and $c$ as,

$a=3$, from first equation,

$b=5$, from second equation and,

$c=7$, from third equation.

So target expression,

$a^2+b^2+c^2=9+25+49=83$.

**Answer: **Option b: 83.

**Key concepts used:** * Key pattern identification* --

*--*

**Principle of collection of friendly terms**

**Solving in mind.**### Additional help on SSC CGL Algebra

Apart from a **large number of question and solution sets** and a valuable article on "* 7 Steps for sure success on Tier 1 and Tier 2 of SSC CGL*" rich with concepts and links, you may refer to our other articles specifically on Algebra listed on latest shown first basis,

#### First to read tutorials on Basic and rich Algebra concepts and other related tutorials

**Basic and rich algebraic concepts for elegant Solutions of SSC CGL problems **

**More rich algebraic concepts and techniques for elegant solutions of SSC CGL problems**

**SSC CGL level difficult Algebra problem solving by Componendo dividendo**

**Proof of least value of sum of reciprocals for any number of positive variables**

**How to factorize 25 selected quadratic equations quickly by factor analysis**

#### SSC CGL Tier II level Questions and Solutions on Algebra

**SSC CGL Tier II level Solved Question Set 30, Algebra 7**

**SSC CGL Tier II level Question Set 17, Algebra 6**

**SSC CGL Tier II level Solution Set 17, Algebra 6**

**SSC CGL Tier II level Question Set 14, Algebra 5**

**SSC CGL Tier II level Solution Set 14, Algebra 5**

**SSC CGL Tier II level Question Set 9, Algebra 4**

**SSC CGL Tier II level Solution Set 9, Algebra 4**

**SSC CGL Tier II level Question Set 3, Algebra 3**

**SSC CGL Tier II level Solution Set 3, Algebra 3**

**SSC CGL Tier II level Question Set 2, Algebra 2**

**SSC CGL Tier II level Solution Set 2, Algebra 2**

**SSC CGL Tier II level Question Set 1, Algebra 1**

**SSC CGL Tier II level Solution Set 1, Algebra 1**

#### Efficient solutions for difficult SSC CGL problems on Algebra in a few steps

**How to solve a difficult surd algebra question by repeated componendo dividendo in a few steps 17**

**How to solve difficult SSC CGL level problem mentally using patterns and methods 16**

**How to solve a difficult SSC CGL Algebra problem mentally in quick time 15**

**How to solve difficult SSC CGL Algebra problems in a few steps 14**

**How to solve difficult SSC CGL Algebra problems in a few steps 13**

**How to solve difficult SSC CGL Algebra problems in a few steps 12**

**How to solve difficult SSC CGL Algebra problems in a few steps 11**

**How to solve difficult SSC CGL Algebra problems in a few assured steps 10**

**How to solve difficult SSC CGL Algebra problems in a few steps 9**

**How to solve difficult SSC CGL Algebra problems in a few steps 8**

**How to solve difficult SSC CGL Algebra problems in a few steps 7**

**How to solve difficult Algebra problems in a few simple steps 6**

**How to solve difficult Algebra problems in a few simple steps 5**

**How to solve difficult surd Algebra problems in a few simple steps 4**

**How to solve difficult Algebra problems in a few simple steps 3**

**How to solve difficult Algebra problems in a few simple steps 2**

**How to solve difficult Algebra problems in a few simple steps 1**

#### SSC CGL level Question and Solution Sets on Algebra

**SSC CGL level Solved Question Set 90, Algebra 18**

**SSC CGL level Question Set 81, Algebra 17**

**SSC CGL level Solution Set 81, Algebra 17**

**SSC CGL level Question Set 74, Algebra 16**

**SSC CGL level Solution Set 74, Algebra 16**

**SSC CGL level Question Set 64, Algebra 15**

**SSC CGL level Solution Set 64, Algebra 15**

**SSC CGL level Question Set 58, Algebra 14**

**SSC CGL level Solution Set 58, Algebra 14**

**SSC CGL level Question Set 57, Algebra 13**

**SSC CGL level Solution Set 57, Algebra 13**

**SSC CGL level Question Set 51, Algebra 12**

**SSC CGL level Solution Set 51, Algebra 12**

**SSC CGL level Question Set 45 Algebra 11**

**SSC CGL level Solution Set 45, Algebra 11**

**SSC CGL level Solution Set 35 on Algebra 10**

**SSC CGL level Question Set 35 on Algebra 10**

**SSC CGL level Solution Set 33 on Algebra 9**

**SSC CGL level Question Set 33 on Algebra 9**

**SSC CGL level Solution Set 23 on Algebra 8**

**SSC CGL level Question Set 23 on Algebra 8**

**SSC CGL level Solution Set 22 on Algebra 7**

**SSC CGL level Question Set 22 on Algebra 7**

**SSC CGL level Solution Set 13 on Algebra 6**

**SSC CGL level Question Set 13 on Algebra 6**

**SSC CGL level Question Set 11 on Algebra 5**

**SSC CGL level Solution Set 11 on Algebra 5**

**SSC CGL level Question Set 10 on Algebra 4**

**SSC CGL level Solution Set 10 on Algebra 4**

**SSC CGL level Question Set 9 on Algebra 3**

**SSC CGL level Solution Set 9 on Algebra 3**

**SSC CGL level Question Set 8 on Algebra 2**

**SSC CGL level Solution Set 8 on Algebra 2**

**SSC CGL level Question Set 1 on Algebra 1**

**SSC CGL level Solution Set 1 on Algebra 1**

#### SSC CPO CAPF CISF level Solved Question sets on Algebra

**SSC CPO level Solved Question Set 1 Algebra 1**

**SSC CPO level Solved Question Set 2 Algebra 2**

#### SSC CHSL level Solved Question Sets on Algebra

**SSC CHSL level Solved Question set 11 on Algebra 1**

**SSC CHSL level Solved Question set 12 on Algebra 2**

**SSC CHSL level Solved Question set 13 on Algebra 3**

**SSC CHSL level Solved Question set 14 on Algebra 4**

### Getting content links in your mail

#### You may get link of any content published

- from this site by
or,**site subscription** - on competitive exams by
.**exams subscription**