## 90th SSC CGL Solved Question Set, 18th on Algebra

This is the 90th solved question set of 10 practice problem exercise for SSC CGL exam and the 18th on topic Algebra. It contains,

**Question set on Algebra**for SSC CGL 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.

**IMPORTANT:** To absorb the concepts, techniques and reasoning explained in the solutions fully and apply those in solving problems on Algebra 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.

### 90th Question set - 10 problems for SSC CGL exam: 18th on topic Algebra - answering time 15 mins

**Q1. **If $x=a+\displaystyle\frac{1}{a}$ and $y=a-\displaystyle\frac{1}{a}$, then $\sqrt{x^4+y^4-2x^2y^2}$ is equal to,

- $4$
- $8$
- $\displaystyle\frac{8}{a^2}$
- $16a^2$

**Q2.** If $x+y+z=19$, $x^2+y^2+z^2=133$, and $xz=y^2$, then the difference between $z$ and $x$ is,

- $3$
- $4$
- $5$
- $6$

**Q3. **If $x^4+x^{-4}=194$ with $x \gt 0$, then the value of $(x-2)^2$ is,

- $1$
- $2$
- $3$
- $6$

**Q4. **If $\displaystyle\frac{5x-y}{5x+y}=\frac{3}{7}$, then what is the value of $\displaystyle\frac{(4x^2+y^2-4xy)}{(9x^2+16y^2+24xy)}$?

- $\displaystyle\frac{1}{6}$
- $0$
- $\displaystyle\frac{3}{7}$
- $\displaystyle\frac{18}{49}$

**Q5.** If the expression $x^2+x+1$ is written in the form of $\left(x+\displaystyle\frac{1}{2}\right)^2+q^2$, then the possible values of $q$ are,

- $\pm\displaystyle\frac{1}{3}$
- $\pm\displaystyle\frac{1}{2}$
- $\pm\displaystyle\frac{2}{\sqrt{3}}$
- $\pm\displaystyle\frac{\sqrt{3}}{2}$

**Q6.** If $a+b+c=0$, then the values of $\left(\displaystyle\frac{a+b}{c}+\displaystyle\frac{b+c}{a}+\displaystyle\frac{c+a}{b}\right)\left(\displaystyle\frac{a}{b+c}+\displaystyle\frac{b}{c+a}+\displaystyle\frac{c}{a+b}\right)$ is,

- $-3$
- $8$
- $9$
- $0$

**Q7.** If $\displaystyle\frac{5x}{2}-\displaystyle\frac{[7(6x-\frac{3}{2}]}{4}=\displaystyle\frac{5}{8}$, then what is the value of $x$?

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

**Q8.** If $2apq=(p+q)^2-(p-q)^2$, then value of $a$ is,

- $2$
- $8$
- $1$
- $4$

**Q9.** Find the value of $\sqrt{(x^2+y^2+z)(x+y-3z)} \div {\sqrt[3]{xy^3z^2}}$ when $x=+1$, $y=-3$ and $z=-1$.

- $3$
- $-1$
- $-3$
- $0$

**Q10.** If $(5\sqrt{5}x^3-81\sqrt{3}y^3) \div (\sqrt{5}x-3\sqrt{3}y)=(Ax^2+By^2+Cxy)$, then the value of $(6A+B-\sqrt{15}C)$ is,

- 10
- 15
- 12
- 9

### Answers to the questions

**Q1. Answer:** Option a: $4$.

**Q2. Answer:** Option c: $5$.

**Q3. Answer:** Option c: $3$.

**Q4. ****Answer:** Option b: $0$.

**Q5. Answer:** Option d: $\pm\displaystyle\frac{\sqrt{3}}{2}$.

**Q6. Answer:** Option c : $9$.

**Q7. Answer:** Option a: $\displaystyle\frac{1}{4}$.

**Q8. Answer:** Option a: $2$.

**Q9. Answer:** Option b: $-1$.

**Q10. Answer:** Option c: 12.

### 90th solution set - 10 problems for SSC CGL exam: 18th on topic Algebra - answering time 15 mins

**Q1. **If $x=a+\displaystyle\frac{1}{a}$ and $y=a-\displaystyle\frac{1}{a}$, then $\sqrt{x^4+y^4-2x^2y^2}$ is equal to,

- $4$
- $8$
- $\displaystyle\frac{8}{a^2}$
- $16a^2$

** Solution 1: Quick solution by Key pattern identification and use of basic algebraic relations**

Focusing on the target expression, the key pattern is immediately identified as,

$E=\sqrt{x^4+y^4-2x^2y^2}$

$=\sqrt{(x^2-y^2)^2}$

$=(x^2-y^2)$

$=(x+y)(x-y)$.

Turning attention to the two given expression,

$x+y=2a$, and

$x-y=\displaystyle\frac{2}{a}$.

When multiplied together, $a$ cancels out leaving just 4,

$E=(x+y)(x-y)=2a\times{\displaystyle\frac{2}{a}}=4$.

**Answer:** Option a: $4$.

**Key concepts used: Key pattern identification -- Basic algebraic relations -- **

*.*

**Solving in mind****Q2.** If $x+y+z=19$, $x^2+y^2+z^2=133$, and $xz=y^2$, then the difference between $z$ and $x$ is,

- $3$
- $4$
- $5$
- $6$

**Solution 2: Solution by square of three variable sum, key pattern identification and substitution**

To use the sum of squares of three variables equal to 133, first step is to raise sum of three variables, $(x+y+z)$ to its square,

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

Or, $19\times{19}=133+2y(x+y+z)$, substituting $zx=y^2$,

Or, $19-7=2y$, as $(x+y+z)=19$ and $133=7\times{19}$, the factor 19 cancels out,

Or, $y=6$,

So, $y^2=zx=36$.

You have reached very near to the solution.

Taking up sum of squares again for utilizing $z^2+x^2$, as our target is $(z-x)^2$,

$x^2+y^2+z^2=133$.

Subtract $3y^2$ from both sides of the equation to get finally $(z-x)^2$,

$x^2-2y^2+z^2=133-3\times{36}=25$,

Or, $(x-z)^2=25$.

So, difference between $x$ and $z$ is 5, which one is greater we don't have to find out.

**Answer:** Option c: $5$.

**Key concepts used:** **Three variable square of sum -- Key pattern identification -- Substitution.**

**Q3. **If $x^4+x^{-4}=194$ with $x \gt 0$, then the value of $(x-2)^2$ is,

- $1$
- $2$
- $3$
- $6$

**Solution 3: Solution by Property of sum of inverses, Power down of sum of inverses in higher power of $x$ and key pattern identification**

There is no other option than to power down the sum of inverses in 4th power of $x$.

The first step is to add 2 to both sides of the given equation and get the value of sum of inverses in squares of $x$,

$x^4+2+\displaystyle\frac{1}{x^4}=194+2=196$,

Or, $\left(x^2+\displaystyle\frac{1}{x^2}\right)^2=14^2$.

Or, $x^2+\displaystyle\frac{1}{x^2}=14$, as the LHS is a sum of squares, the square root must be +ve.

In the second step similarly adding 2 to both sides of this resultant equation you get,

$x^2+2+\displaystyle\frac{1}{x^2}=14+2=16$

Or, $\left(x+\displaystyle\frac{1}{x}\right)^2=4^2$,

Or, $x+\displaystyle\frac{1}{x}=4$, as $x$ is positive, sum of its inverses must also be positive.

Now we will simplify this equation to,

$x^2-4x+1=0$.

Add 3 to both sides. Result is,

$(x^2-4x+4)=(x-2)^2=3$

**Answer:** Option c: $3$.

**Key concepts used: Principle of interaction of inverses -- Property of sum of inverses -- Power down of sum of inverses in higher powers of $x$ -- Key pattern identification -- Solving in mind.**

With clear concepts you can easily solve the problem in mind if you can see the final simplifying step of the sum of inverses in $x$.

**Q4. **If $\displaystyle\frac{5x-y}{5x+y}=\frac{3}{7}$, then what is the value of $\displaystyle\frac{(4x^2+y^2-4xy)}{(9x^2+16y^2+24xy)}$?

- $\displaystyle\frac{1}{6}$
- $0$
- $\displaystyle\frac{3}{7}$
- $\displaystyle\frac{18}{49}$

**Solution 4: Quick problem solving by Componendo dividendo and substitution**

The given equation is perfectly ready for applying componendo dividendo with **signature of componendo dividendo** of all terms of numerator and denominator same except one term differing in sign.

Applying the first of the 3 step componendo dividendo method on the given expression, add 1 to both sides of the equation,

$\displaystyle\frac{10x}{5x+y}=\frac{10}{7}$.

In the second step, **subtract** both sides of the equation **from 1**,

$\displaystyle\frac{2y}{5x+y}=\frac{4}{7}$.

In the third step, divide the first result by the second,

$\displaystyle\frac{5x}{y}=\frac{5}{2}$,

Or, $y=2x$.

Turning attention to the target expression, identify the key pattern that the numerator is $(2x-y)^2=0$,

$4x^2+y^2-4xy=(2x-y)^2=0$.

Result is 0.

**Answer:** Option b: $0$.

**Key concepts used: Componendo dividendo -- Signature pattern of componendo dividendo -- Key pattern identification -- Solving in mind.**

**Q5.** If the expression $x^2+x+1$ is written in the form of $\left(x+\displaystyle\frac{1}{2}\right)^2+q^2$, then the possible values of $q$ are,

- $\pm\displaystyle\frac{1}{3}$
- $\pm\displaystyle\frac{1}{2}$
- $\pm\displaystyle\frac{2}{\sqrt{3}}$
- $\pm\displaystyle\frac{\sqrt{3}}{2}$

#### Solution 5: Problem analysis and solution by Expression matching

Concentrating on the target expression first you get,

$\left(x+\displaystyle\frac{1}{2}\right)^2+q^2$

$=x^2+x+\displaystyle\frac{1}{4}+q^2$

$=x^2+x+1$, by reverse substitution of given expression.

So, $q^2+\displaystyle\frac{1}{4}=1$,

Or, $q=\pm\displaystyle\frac{\sqrt{3}}{2}$.

**Answer:** Option d: $\pm\displaystyle\frac{\sqrt{3}}{2}$.

*Key concepts used:* Expression matching**—we have matched the target expression directly with the given expression**** -- Solving in mind****.**

**Q6.** If $a+b+c=0$, then the values of $\left(\displaystyle\frac{a+b}{c}+\displaystyle\frac{b+c}{a}+\displaystyle\frac{c+a}{b}\right)\left(\displaystyle\frac{a}{b+c}+\displaystyle\frac{b}{c+a}+\displaystyle\frac{c}{a+b}\right)$ is,

- $-3$
- $8$
- $9$
- $0$

**Solution 6: Problem analysis and solution by Key pattern identification and most favorable substitution**

As $a+b+c=0$, identify the pattern that if you substitute each of $(a+b)$ by $-c$, $(b+c)$ by $-a$ and $(c+a)$ by $-b$ in the target expression, all six terms reduce to $-1$. Final result is thus,

$E=(-1-1-1)(-1-1-1)=9$.

We have done the most favorable substitution and solved the problem in mind practically instantly.

**Answer:** Option c : $9$.

**Key concepts used:** *Key pattern identification -- Most favorable substitution -- Solving in mind.*

**Q7.** If $\displaystyle\frac{5x}{2}-\displaystyle\frac{[7(6x-\frac{3}{2}]}{4}=\displaystyle\frac{5}{8}$, then what is the value of $x$?

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

**Solution 7: Problem analysis and Solving by Efficient algebraic simplification using fraction elimination strategy**

First simplify the fraction in the numerator of the 2nd term in the LHS. Target expression becomes,

$\displaystyle\frac{5x}{2}-\displaystyle\frac{21(4x-1)}{8}=\displaystyle\frac{5}{8}$

Now multiply both sides by 8 to elimnate fractions altogether. Fraction handling is troublesome. The result becomes,

$20x-84x+21=5$,

Or, $64x=16$,

Or, $x=\displaystyle\frac{1}{4}$.

It's straightforward efficient simplification and the problem can easily be solved in mind if you are a bit careful.

**Answer:** Option a: $\displaystyle\frac{1}{4}$.

** Key concepts used:** *Efficient algebraic simplification -- Fraction elimination -- ***Solving in mind.**

**Q8.** If $2apq=(p+q)^2-(p-q)^2$, then value of $a$ is,

- $2$
- $8$
- $1$
- $4$

** Solution 8: Problem analysis and solution by pattern identification and basic algebraic relations**

Expand the RHS of the given expression and only $4pq$ remains in the RHS, and value of $a$ becomes 2,

$2apq=(p+q)^2-(p-q)^2=4pq$, the terms $p^2$ and $q^2$ cancel out,

Or, $a=2$.

It's an easy problem solved instantly using basic algebraic relations.

**Answer:** Option a: $2$.

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

**Q9.** Find the value of $\sqrt{(x^2+y^2+z)(x+y-3z)} \div {\sqrt[3]{xy^3z^2}}$ when $x=+1$, $y=-3$ and $z=-1$.

- $3$
- $-1$
- $-3$
- $0$

**Solution 9: Problem Solution by careful substitution and efficient simplification**

Whenever you face an awkward algebraic simplification involving substution, you need to be a bit more careful. For example, *do not make the mistake of taking the division symbol as a plus symbol in a hurry.*

Starting simplification, first evaluate the first factor of the numerator as,

$(x^2+y^2+z)=1+9-1=9$, and the second factor as,

$(x+y-3z)=1-3+3=1$.

So, $\text{Numerator}=\sqrt{9}=3$.

In the denominator, first and third factor producing $+1$,

$\text{Denominator}=\sqrt[3]{y^3}=y=-3$.

Final result is, $3 \div (-3)=-1$.

**Answer:** Option b: $-1$.

**Key concepts used:** **Efficient simplification -- Solving in mind****.**

**Q10.** If $(5\sqrt{5}x^3-81\sqrt{3}y^3) \div (\sqrt{5}x-3\sqrt{3}y)=(Ax^2+By^2+Cxy)$, then the value of $(6A+B-\sqrt{15}C)$ is,

- 10
- 15
- 12
- 9

**Solution 10: Problem analysis and Solution by identifying key pattern, two factor expansion of subtractive sum of cubes and comparison of coefficients of like terms**

The very first thing you do is to identify the key pattern that,

$(\sqrt{5}x)^3=5\sqrt{5}x^3=a^3$, say, the first term of the numerator of the LHS of the given expression, and,

$(3\sqrt{3}y)^3=81\sqrt{3}y^3=b^3$, say, the second term of the numerator of the LHS of the given expression.

It follows immediately by two factor expansion of subtractive sum of cubes, that, numerator is

$(5\sqrt{5}x^3-81\sqrt{3}y^3)=(a^3-b^3)=(a-b)(a^2+ab+b^2)$, and denominator is just,

$(a-b)$.

So the given equation is simplified to,

$(a^2+ab+b^2)=5x^2+3\sqrt{15}xy+27y^2$

$=(Ax^2+By^2+Cxy)$

Comparing the coefficients of like terms $x^2$, $y^2$ and $xy$ on both sides of the equation,

$A=5$,

$B=27$, and,

$C=3\sqrt{15}$.

With these values of $A$, $B$ and $C$, the value of the target expression becomes,

Or, $(6A+B-\sqrt{15}C)=30+27-45=57-45=12$.

**Answer:** Option c: 12.

**Summary of the techniques and concepts used** for solving this not so easy problem:

- We have formed the key pattern of surd cube terms in the numerator of the LHS as well as in the denominator in single power.
- Then expanded the subtractive sum of cubes in two factor form with first factor eliminated by the denominator leaving an expression in $x^2$, $y^2$ and $xy$ in the LHS.
- By comparison of coefficients of like terms in both sides of the equation, values of $A$, $B$ and $C$ found out and substituted in the target expression.

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