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## 90th SSC CGL Set of Algebra questions answers and solution

Algebra Questions Answers and Solutions: Take the 10 question mini-mock test, verify answers and then learn how to solve quickly in 15 mins.

Contents are,

**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.

### 10 Algebra Questions for SSC CGL Set 90 - 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 Algebra questions for SSC CGL Set 90

**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.

### Quick conceptual solution to the algebra questions for SSC CGL Set 90 - answering time was 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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