## 51st SSC CGL level Solution Set, 12th on Algebra

This is the 51st solution set of 10 practice problem exercise for SSC CGL exam and 12th on topic Algebra.

For maximum gains, the test should be taken first, that is obvious. But more importantly, to absorb the concepts, techniques and deductive reasoning elaborated through these solutions, one must solve many problems in a systematic manner using this conceptual analytical approach.

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

If you have not yet taken this test you may take it by referring to the * 51st SSC CGL question set and 12th on Algebra* before going through the solution.

Watch **quick algebra solutions on two-part video**.

**Part I: Q1 to Q5**

**Part II: Q6 to Q10**

### 51st solution set - 10 problems for SSC CGL exam: 12th on topic Algebra - answering time 15 mins

**Problem 1.**

If $x^2=y+z$, $y^2=z+x$ and $z^2=x+y$, the value of $\displaystyle\frac{1}{x+1}+\displaystyle\frac{1}{y+1}+\displaystyle\frac{1}{z+1}$ is,

- $1$
- $4$
- $-1$
- $-4$

** Solution 1 - Problem analysis**

Analyzing the target expression we find no apparent similarity between the offending denominators and the given expressions. On a second look though, we just added a $x$ to both sides of the first given equation to have,

$x^2+x=x+y+z$,

Or, $x(x+1)=x+y+z$,

Or, $\displaystyle\frac{1}{x+1}=\frac{x}{x+y+z}$.

We know we have the solution.

We have * equalized the denominators* by

*. And to transform the input expressions with the target of arriving at the target terms, we have used*

**transforming the input expressions***Time and again we have infused or introduced an additional element in an expression to transform it in a very desirable manner.*

**Extra element infusion technique.**#### Solution 1 - Problem solving execution

Similarly,

$\displaystyle\frac{1}{y+1}=\frac{y}{x+y+z}$, and

$\displaystyle\frac{1}{z+1}=\frac{z}{x+y+z}$.

Adding the three,

$\displaystyle\frac{1}{x+1}+\displaystyle\frac{1}{y+1}+\displaystyle\frac{1}{z+1}=\displaystyle\frac{x+y+z}{x+y+z}=1$.

**Answer:** Option a: $1$.

**Key concepts used:** * End state anlysis approach* --

*--*

**Extra element infusion technique***--*

**given expression transformation to create the denominator expressions**

*Pattern identification --*

*Denominator equalization.***Problem 2.**

If $a^2+b^2+c^2+3=2(a+b+c)$ then the value of $(a+b+c)$ is,

- 2
- 3
- 5
- 4

**Solution 2 - Problem analysis and solving**

As there are no multiplicative terms like $ab$ or $bc$ in the given expression there is no possibility of expressing the given expression as a square of $(a+b+c)$. So the only way out is to find the values of $a$, $b$ and $c$ individually.

Analysis of the given expression gave us the clue of rearranging the terms by applying * principle of collection of friendly terms*, to transform the given expression to a sum of squares,

$a^2+b^2+c^2+3=2(a+b+c)$,

Or, $(a-1)^2+(b-1)^2+(c-1)^2=0$

By the * principle of zero sum of square terms* then,

$a-1=b-1=c-1=0$.

By the * principle of zero sum of square terms*, we know for sure that if there is an expression, $x^2+y^2+z^2=0$, where $x$, $y$ and $z$ are expressions involving real variables (variables evaluating to real numbers), first squaring each of $x$, $y$ and $z$ will transform the negative signs in the terms if any, and summing the square terms and equating the sum to zeo will result in the truth that, each term and hence each of the variables individually must be zero, $x=y=z=0$.

So,

$a=b=c=1$,

and $(a+b+c)=3$

**Answer:** Option b : 3.

**Key concepts used:** **I*** nput transformation technique* --

**Principle of collection of friendly terms -- Principle of zero sum of square terms.****Problem 3.**

If $a^2-4a-1=0$, then the value of $a^2+\displaystyle\frac{1}{a^2}+3a-\displaystyle\frac{3}{a}$ is,

- 40
- 35
- 30
- 25

**Solution 3 - Problem analysis**

As the target expression has two part expressions that are sum of inverses (and sum of squares of inverses) we look forward to transform the given expression in the form of a sum or subtraction of inverses. If we can do that, we know by the ** principle of interaction of inverses** that we should easily be able to evaluate the parts of target expression.

**Solution 3 - Problem solving execution**

The given expression is,

$a^2-4a-1=0$,

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

Squaring both sides,

$a^2-2+\displaystyle\frac{1}{a^2}=16$,

Or, $a^2+\displaystyle\frac{1}{a^2}=18$.

Now we are ready to evaluate the target expression,

$E=a^2+\displaystyle\frac{1}{a^2}+3a-\displaystyle\frac{3}{a}$

$=18+3\times{4}=30$.

**Answer:** Option c: 30.

**Key concepts used: ***Comparison of target expression with the given expression following End state analysis -- input expression transformation so that it becomes similar to portions of the target expression in the form of sum of inverses -- application of principle of interaction of inverses to evaluate both the parts of the target expression.*

**Problem 4.**

If $x+\displaystyle\frac{1}{x}=99$, find the value of $\displaystyle\frac{100x}{2x^2+102x+2}$.

- $\displaystyle\frac{1}{6}$
- $\displaystyle\frac{1}{3}$
- $\displaystyle\frac{1}{2}$
- $\displaystyle\frac{1}{4}$

**Solution 4 - Problem analysis**

By bringing down the $x$ from numerator to the denominator and dividing the three terms in thre denominator by $x$, the middle term reduces to a number and the first and third term form the sum of inverses, value of which is given in the input. It is **transformation of target expression to match the input expression.**

**Solution 4 - Problem solving execution**

The target expression,

$E=\displaystyle\frac{100x}{2x^2+102x+2}$

$=\displaystyle\frac{100}{2x+102+\displaystyle\frac{2}{x}}$

$=\displaystyle\frac{100}{2\left(x+\displaystyle\frac{1}{x}\right)+102}$

$=\displaystyle\frac{100}{2\times{99}+102}$

$=\displaystyle\frac{100}{300}$

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

**Answer:** Option b: $\displaystyle\frac{1}{3}$.

**Key concepts used: Target expression transformation -- pattern recognition -- direct use of input expression value -- substitution.**

**Problem 5.**

If $\sqrt{1+\displaystyle\frac{x}{961}}=\displaystyle\frac{32}{31}$, then the value of $x$ is,

- 63
- 64
- 61
- 65

**Solution 5 - Problem analysis and execution**

Early on we identified $961$ as $31^2$ and accordingly while squaring the LHS, kept 961 in square form to speed up the solution. Let us see how.

The given expression is,

$\sqrt{1+\displaystyle\frac{x}{31^2}}=\displaystyle\frac{32}{31}$

Squaring both sides,

$1+\displaystyle\frac{x}{31^2}=\displaystyle\frac{32^2}{31^2}$,

Or, $\displaystyle\frac{x}{31^2}=\displaystyle\frac{32^2}{31^2}-1=\displaystyle\frac{(32+31)(32-31)}{31^2}$,

Or, $x=63$.

It is a very quick solution and must be an **efficient simplification.**

**Answer:** Option a: 63.

**Key concepts used:** * Pattern recognition* --

*--*

**delayed evaluation***--*

**basic algebraic concepts**

**efficient simplification.****Problem 6.**

If $1.5a=0.04b$ then the value of $\displaystyle\frac{b-a}{b+a}$ will be equal to,

- $\displaystyle\frac{73}{77}$
- $\displaystyle\frac{75}{2}$
- $\displaystyle\frac{2}{75}$
- $\displaystyle\frac{77}{33}$

**Solution 6 - Problem analysis and solving**

The form of target expression is suitable for use of * componendo dividendo technique* and accordingly it is needed to first find the value of $\displaystyle\frac{b}{a}$.

The given expression is,

$1.5a=0.04b$,

Or, $\displaystyle\frac{b}{a}=\frac{15}{0.04}=\frac{150}{4}=\frac{75}{2}$.

We have used * Safe decimal division technique* by multiplying the numerator and denominator by 100 and eliminating the decimals.

Applying componendo dividendo technique then,

$\displaystyle\frac{b-a}{b+a}=\frac{75-2}{75+2}=\frac{73}{77}$.

**Answer:** Option a : $\displaystyle\frac{73}{77}$.

**Key concepts used: **

*--*

**Key pattern identification***.*

**Safe decimal division technique -- Componendo dividendo technique****Problem 7.**

The value of the expression, $\displaystyle\frac{(a-b)^2}{(b-c)(c-a)}+\displaystyle\frac{(b-c)^2}{(a-b)(c-a)}+\displaystyle\frac{(c-a)^2}{(a-b)(b-c)}$ is,

- $2$
- $3$
- $0$
- $\displaystyle\frac{1}{3}$

**Solution 7 - Problem analysis and solving**

Though the given target expression is quite daunting, it has the same three component expressions $(a-b)$, $(b-c)$ and $(c-a)$ each used in unchanged form throughout the expression. This is the pattern of **Use of unchanged component expressions.**

Under such a condition * without losing any information we can replace the three expressions by three simple variables*, $p=(a-b)$, $q=(b-c)$ and $r=(c-a)$ so that additionally we get a helper equation, $p+q+r=0$. The last expression always helps in simplifying complex algebraic expressions and so we classify it as a

**helper equation.**This is use of * abstraction* and

*. This simplfies the complex expression greatly to,*

**component expression substitution**$E=\displaystyle\frac{p^2}{qr}+\displaystyle\frac{q^2}{rp}+\displaystyle\frac{r^2}{pq}$

$=\displaystyle\frac{p^3+q^3+r^3}{pqr}$, where $p+q+r=0$.

This is * target expression transformation* as well as

*. We do not need to use the variables $a$, $b$, and $c$ any more.*

**problem transformation**As a next step to simplification of form, we have made the * denominators equal* so that the transformed numerators can directly be summed up.

The problem now is reduced to evaluating $p^3+q^3+r^3$, where $p+q+r=0$.

Let us now look back to the helper equation,

$p+q+r=0$,

Or, $p+q=-r$,

Or, $p^3+q^3+3pq(p+q)=-r^3$,

Or, $p^3+q^3+r^3=3pqr$.

Thus the target expression is simply evaluated to,

$E=\displaystyle\frac{3pqr}{pqr}=3$.

**Answer:** Option b: $3$.

** Key concepts used:** * Pattern identification* --

*Abstraction -- component expression substitution -- problem transformation -- denominator equalization -- helper equation -- efficient simplification.***Problem 8.**

If $9\sqrt{x}=\sqrt{12}+\sqrt{147}$, then $x$ is,

- 5
- 2
- 3
- 4

** Solution 8 - Problem analysis and solving execution**

The RHS containing two surd terms unless these are combined to a single term, squaring both sides will lead us nowhere near the solution integer choice values. This is * deductive reasoning*. So we look into the numbers under the square roots more closely and then could see the way to combining the two terms to a single term,

$9\sqrt{x}=\sqrt{12}+\sqrt{147}$,

Or, $9\sqrt{x}=2\sqrt{3}+7\sqrt{3}$,

Or, $9\sqrt{x}=9\sqrt{3}$,

Or, $x=3$.

Once you discover the key pattern, solution comes quickly.

Mark that while examining the individual terms on the RHS for possibility of combining them to a single term we knew that it is a necessity to combine, as the choice values are perfect integers, and not surds. This is an indirect **use of the free resource of the choice values.**

**Answer:** Option c: 3.

**Key concepts used:** * Deductive reasoning* --

**Key pattern discovery -- Principle of free resource use.****Problem 9.**

If $p:q=r:s=t:u=2:3$ then $(mp+nr+ot):(mq+ns+ou)$ is equal to,

- 2 : 3
- 3 : 2
- 2 : 1
- 1 : 2

**Solution 9 - Problem analysis**

The target expression uses three new variables * that are not present in the given relations*. This is

*But as it is a ratio problem, we felt that the new variables*

**an oddity at first thought.***s of the ratio*

**in both the term***and be eliminated by ratio operation. With this hope we proceeded with solving the problem.*

**should form a common expression**This is * deductive reasoning* based on

*and properties.*

**basic ratio concepts**#### Solution 9 - Problem solving execution

We have the given expression in combined form as,

$p:q=r:s=t:u=2:3$.

This gives rise to three expressions in tune with the target expression form,

$p=\displaystyle\frac{2}{3}q$,

$r=\displaystyle\frac{2}{3}s$, and

$t=\displaystyle\frac{2}{3}u$,

Substituting these values of $p$, $r$ and $t$ in the first term (numerator) of the target expression ratio,

$E=(mp+nr+ot):(mq+ns+ou)$,

$=\displaystyle\frac{2}{3}[(mq+ns+ou):(mq+ns+ou)]$

$=2:3$.

As expected the new variables were eliminated by the ratio operation.

**Answer:** Option a: 2 : 3.

**Key concepts used:** * Deductive reasoning -- Basic ratio concepts *--

**Substitution.****Problem 10.**

If $\displaystyle\frac{1}{a+1}+\displaystyle\frac{1}{b+1}+\displaystyle\frac{1}{c+1}=2$ then $a^2+b^2+c^2$ is,

- $\displaystyle\frac{3}{4}$
- $\displaystyle\frac{1}{3}$
- $\displaystyle\frac{27}{16}$
- $\displaystyle\frac{4}{3}$

**Solution 10: Problem analysis and intuitive solution**

After a few moments, you decide straightway that deriving the value of target expression from the given expression by deductive steps won't be feasible. But then how to proceed?

At least you know, because of the perfectly symmetric nature of the LHS in variables $a$, $b$ and $c$, these values must be same.

Trying out equal values of $a$, $b$ and $c$ that would result in RHS value of 2, you may intuitively identify $\displaystyle\frac{1}{2}$ to be the value for each of the variables, that would give 2 on the RHS, and so the answer would be, $\displaystyle\frac{3}{4}$.

Like,

$\displaystyle\frac{1}{\displaystyle\frac{1}{2}+1}+\displaystyle\frac{1}{\displaystyle\frac{1}{2}+1}+\displaystyle\frac{1}{\displaystyle\frac{1}{2}+1}$

$=\displaystyle\frac{2}{3}+\displaystyle\frac{2}{3}+\displaystyle\frac{2}{3}=2$, and,

$a^2+b^2+c^2=\displaystyle\frac{1}{4}+\displaystyle\frac{1}{4}+\displaystyle\frac{1}{4}=\displaystyle\frac{3}{4}$.

But that is intuitive which is uncertain.

#### Solution 10: Alternate assured solution by working backwards and mathematical reasoning

You may also proceed in a more certain way without depending on any chances.

This would be a new approach of trying out which choice value can satisfy the perfectly balanced and symmetric sum of squares, $a^2+b^2+c^2$, as well as satisfy the given expression.

To satisfy the given equation, the values of $a$, $b$ and $c$ cannot be surds, and so the values $\displaystyle\frac{1}{3}$ for option b and $\displaystyle\frac{4}{3}$ for option d are immediately considered as invalid for $a^2+b^2+c^2$.

Because of symmetry of both LHS of given expression and the target expression, you have already decided that the values of $a$, $b$ and $c$ must be equal.

$\displaystyle\frac{27}{16}$ can be a candidate for answer with $a=b=c=\displaystyle\frac{3}{4}$. But this would violate the given equation,

$\displaystyle\frac{1}{\displaystyle\frac{3}{4}+1}+\displaystyle\frac{1}{\displaystyle\frac{3}{4}+1}+\displaystyle\frac{1}{\displaystyle\frac{3}{4}+1} \neq 2$.

The last choice $\displaystyle\frac{3}{4}$ satisfies requirements of both target and the given expressions.

You have used working backwards approach with verification from the given conditions, a mathematically sound and assured path to the solution.

**Answer: **Option a: $\displaystyle\frac{3}{4}$.

**Key concepts used:** * Deductive reasoning* --

*--*

**Symmetric balanced expression***--*

**key pattern recognition***.*

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