## 33rd SSC CGL level Solution Set, 9th on Algebra

This is the 33rd solution set of 10 practice problem exercise for SSC CGL exam and 9th 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 * 33rd SSC CGL question set and 9th on Algebra* before going through the solution.

### 33rd solution set - 10 problems for SSC CGL exam: 9th on topic Algebra - answering time 15 mins

**Q1. **If $x = 2.361$, $y=3.263$, and $z=5.624$, then the value of $x^3 + y^3 - z^3 + 3xyz$ is,

- 35.621
- 1
- 0
- 19.277

** Solution - Problem analysis**

The target expression being a well-known cube expression in three variables, we remember vaguely an expression similar to $x+y+z = 0$ associated with it so that the value of the target expression is very much simplified.

With this goal in view we try to discover an additive-subtractive relationship between $x$, $y$ and $z$ and quickly find that,

$x + y = z$.

#### Solution - Simplifying actions

We have $x + y = z$

Or, $x^3 + y^3 + 3xy(x+y) = z^3$

Or, $x^3 + y^3 - z^3 + 3xyz = 0$.

**Answer:** Option c: 0.

**Key concepts used:** First establishing simple relationship between $x$, $y$, and $z$ -- using cube of sums expression in two variables -- * basic algebra concepts*.

**Q2.** If $6 + \displaystyle\frac{1}{x}=x$, then the values of $x^4 + \displaystyle\frac{1}{x^4}$ is,

- 1444
- 1442
- 1448
- 1446

**Solution - Problem analysis:**

The target expression being in the form of sum of inverses the given expression also needs to be transformed into a sum of inverses in single power of $x$.

$6 + \displaystyle\frac{1}{x}=x$,

Or, $x - \displaystyle\frac{1}{x} = 6$.

We will now derive higher powers of sum of inverses in two stages to arrive at the sum of inverses in fourth power of $x$.

#### Solution - Simplifying actions

$x - \displaystyle\frac{1}{x} = 6$,

Or squaring both sides,

$x^2 - 2 + \displaystyle\frac{1}{x^2} = 36$

Or, $x^2 + \displaystyle\frac{1}{x^2} = 38$.

Squaring both sides again,

$x^4 + 2 + \displaystyle\frac{1}{x^4} = 38^2=1444$,

Or, $x^4 + \displaystyle\frac{1}{x^4} = 1444 - 2=1442$.

**Answer:** Option b : 1442.

**Key concepts used:** Observing that the target expression is in the form of sum of inverses, the given expression is also transformed to a sum (subtraction) of inverses and in two stages of squaring the value of the target expression is reached -- **sum of inverses concept.**

You may refer to the * article on principle of inverses* for more details on how effectively the concept on sum of inverses can be used.

**Q3.** If $x^2 + \displaystyle\frac{1}{x^2} = 66$, then the value of $\displaystyle\frac{x^2 - 1 + 2x}{x}$ is,

- $\pm{8}$
- $6, -10$
- $10, -6$
- $\pm{4}$

**Solution - Problem analysis:**

Usually we have to derive value of higher powers of sum of inverses when lower power of sum of inverses is given. In this problem, the situation is opposite and from higher powers, we have to derive lower powers of sum of inverses. But first we need to verify the nature of the target expression to see whether it is really a sum of inverses or not.

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

So the target expression is really in the form of sum of inverses.

#### Solution - Simplifying actions:

Given expression,

$x^2 + \displaystyle\frac{1}{x^2} = 66$,

Or, $x^2 - 2 + \displaystyle\frac{1}{x^2} = 64$,

Or, $\left(x - \displaystyle\frac{1}{x}\right)^2 = 8^2$

Or, $x - \displaystyle\frac{1}{x} = \pm{8}$

So the target expression when evaluated is,

$x - \displaystyle\frac{1}{x} + 2 = \pm{8} + 2 = 10, -6$.

**Answer:** Option c: $10, -6$.

**Key concepts used:** From higher powers of sum of inverses value of lower power of sum of inverses derived after confirming that the target expression is really a sum of inverses -- * principle of inverses* --

**working backwards approach.****Q4. **Find the minimum value of $2x^2 - (x - 3)(x + 5)$, where $x$ is real,

- 20
- 14
- -12
- 8

**Solution - Problem analysis**

As this is a minimization of quadratic expression problem, the sign of $x^2$ term should be positive. But before any significant action can be taken on the target expression, it needs to be converted in proper form consisting of a square term in $x$ where all terms of $x$ are absorbed leaving only a numeric term outside the square term.

Let us transform the given expression in that specific form and then apply the reasoning of minimization.

**Solution - Problem simplification**

$2x^2 - (x - 3)(x + 5)$

$=2x^2 -(x^2 + 5x - 3x - 15)$

$=2x^2 - x^2 - 2x + 15$

$=x^2 -2x +15$

$=(x-1)^2 + 14$.

We can now apply our reasoning.

As $x$ is real, the square in $x$, that is, $(x-1)^2$ will always have a positive or zero value and the target expression will have the minimum value only when the square expression is zero at $x=1$.

In that case the minimum value of the target expression is 14.

**Answer:** Option b: 14.

**Key concepts used: Maxima minima technique** -- Transforming the given expression in terms of an expression containing a square term in $x$ absorbing all terms in $x$ so that the remaining numeric term can be identified as the minimum value of the expression.

**Q5. **If $x+y=7$ then the value of $x^3 + y^3 + 21xy$ is,

- 243
- 143
- 443
- 343

**Solution - Problem analysis:**

To use the value of the given expression, the target expression needs to be expressed as a cube of sum expression.

#### Solution - Simplifying actions

$(x+y)^3=x^3+y^3+3xy(x+y)$

$=x^3 + y^3 + 3xy\times{7}$

$=x^3 + y^3 + 21xy$,

Or, $7^3 = x^3 + y^3 + 21xy$,

Or, $x^3 + y^3 + 21xy = 343$.

**Answer:** Option d: 343.

**Key concepts used:** Using the cube of sum expression and the value of given expression to transform the target expression suitably -- * basic algebra concepts*.

**Q6.** If $x=\displaystyle\frac{\sqrt{3}}{2}$ then the value of $\displaystyle\frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}$ will be,

- $-\sqrt{3}$
- $1$
- $\sqrt{3}$
- $-1$

**Solution - Problem analysis:**

As the value of $x$ contains a square root as also the target expression is in terms of square root of $x$, it is a case of double square root. Under these circumstances, the target expression can't be evaluated if the values of square root of $(1 +x)$ and $(1 - x)$ are not freed of one square root.

To do this we need to transform each of the expressions under the square roots in terms of a square of surd expression.

With this need identified let us try to transform the square root expressions to the desired form.

#### Solution - Simplifying steps

$\sqrt{1 +x} = \sqrt{1 + \displaystyle\frac{\sqrt{3}}{2}} $

$= \sqrt{\displaystyle\frac{2 + \sqrt{3}}{2}} $

$= \sqrt{\displaystyle\frac{4+2\sqrt{3}}{4}}$

$=\displaystyle\frac{1}{2}\sqrt{(\sqrt{3} + 1)^2}$

$=\displaystyle\frac{1}{2}(\sqrt{3} + 1)$.

Similarly,

$\sqrt{1 - x} = \displaystyle\frac{1}{2}(\sqrt{3} - 1)$.

So we have the the target expressoion as,

$\displaystyle\frac{\sqrt{1+x} + \sqrt{1-x}}{\sqrt{1+x} - \sqrt{1-x}}$

$=\displaystyle\frac{\sqrt{3} + 1 + \sqrt{3} - 1}{\sqrt{3} + 1 - \sqrt{3} + 1}$, the $\frac{1}{2}$ canceled out.

$=\displaystyle\frac{2\sqrt{3}}{2}$

$=\sqrt{3}$

**Answer:** Option c : $\sqrt{3}$.

**Key concepts used: ** Freeing a surd expression out of a square root by expressing a two term surd expression as a three term expanded square of surd expression --

**surd simplification technique.**** Q7.** If $p^3 + 3p^2 + 3p = 7$ then the value of $p^2 + 2p$ is,

- 3
- 4
- 5
- 6

**Solution - Problem analysis: **

Identifying the given expression as a part of a cube of sum expression we examine the target expression to find it also as a part of square expression in a sum of $x$.

#### Solution - Simplifying actions

$p^3 + 3p^2 + 3p = 7$,

Or, $p^3 + 3p^2 + 3p + 1 = 8$,

Or, $(p+1)^3 = 2^3$,

Or, $p+1 = 2$,

Or, $(p+1)^2 = p^2 + 2p + 1 = 4$.

So finally,

$p^2 + 2p = 3$.

**Answer:** Option a: 3.

** Key concepts used:** Identifying the given expression as a part of cube of sum expression as well as the target expression as a part of square of sum expression -- **basic algebra concepts.**

** Q8.** If $\left(x + \displaystyle\frac{1}{x}\right)^2 = 3$ then the value of $(x^{72} + x^{66} + x^{54} + x^{36} + x^{24} + x^6 + 1)$ is,

- 4
- 2
- 3
- 1

** Solution - Problem analysis:**

We can visualize transforming the given expression first into a sum of square of inverses with a value of 1 and then proceed to try for further simplification. We must try for maximum simplification of the given expression, as the target expression is in very high powers of $x$.

#### Solution - Simplifying actions

$\left(x + \displaystyle\frac{1}{x}\right)^2 = 3$,

Or, $x^2 + 2 + \displaystyle\frac{1}{x^2} =3$

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

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

Now we remember the sum of inverses in cube form as,

$x^3 + \displaystyle\frac{1}{x^3} =\left(x + \displaystyle\frac{1}{x}\right)\left(x^2 - 1 + \displaystyle\frac{1}{x^2}\right) = 0$.

This is a very significant simplification of the given expression and we know this will lead us to the solution.

The target expression,

$E=(x^{72} + x^{66} + x^{54} + x^{36} + +x^{24} + x^6 + 1)$

$=x^{69}\left(x^3 + \displaystyle\frac{1}{x^3}\right) + x^{54} + x^{36} + x^{24} + x^6 + 1$

$=x^{54} + x^{36} + x^{24} + x^6 + 1$.

It is clear to us now, if we can form a pair of terms in powers of $x$ with difference in powers as 6, we can reduce the sum of the two terms to zero using simplified result of given expression.

We have achieved this simplification with the first two terms in highest powers of $x$. But as the difference between next two powers of $x$ is 18, we need to introduce two pairs of dummy terms with suitable powers. This is use of introduction of **dummy terms technique.**

$E=x^{54} + x^{36} + x^{24} + x^6 + 1$

$=x^{54} + x^{48} - x^{48} - x^{42} + x^{42} + x^{36} + x^{24} + x^6 + 1$

$=x^{51}\left(x^3 + \displaystyle\frac{1}{x^3}\right) - x^{45}\left(x^3 + \displaystyle\frac{1}{x^3}\right) + x^{42} + x^{36} + x^{24} + x^6 + 1$

$=x^{42} + x^{36} + x^{24} + x^6 + 1$.

The first two terms results in to 0 and repeating the same process of introducing two pairs of dummy terms between $x^{24}$ and $x^6$,

$E=x^{24} + x^{18} - x^{18} - x^{12} + x^{12} + x^6 + 1$

$=1$, as three pairs of terms combine to 0.

**Answer:** Option d: 1.

**Key concepts used:** * Simplifying* the given

*to the maximum extent possible --*

**sum of inverses***from the given expression -- use of*

**continued extraction of the simplified factor**

**dummy terms technique -- rich algebra concepts.**

**Q9.** If $a^2 + b^2 + c^2 = 2(a -b -c) -3$, then $4a - 3b + 5c$ is,

- 3
- 2
- 5
- 6

**Solution - Problem analysis:**

As the target expression is asymmetric and not easily relatable to the given expresson, it is expected that actual values need to be found out for $a$, $b$ and $c$ for substitution and final evaluation.

#### Solution - Simplifying actions

We analyze the given expression and gather friendly terms on the LHS,

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

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

As the sum of squares is 0, each of the squares must be 0.

So, $a = 1$, $b=-1$ and $c=-1$.

Thus the target expression is,

$4a - 3b + 5c = 4 + 3 - 5 = 2$.

**Answer:** Option b: 2.

**Key concepts used:** Using * principle of collection of friendly terms* for expressing the given

*-- resulting each of the square to become 0 -- evaluation of the variables and the target expression value.*

**expression as a sum of squares equal to 0**** Q10.** If $3x + \displaystyle\frac{1}{2x} = 5$, then the value of $8x^3 + \displaystyle\frac{1}{27x^3}$ is,

- $118\frac{1}{2}$
- $0$
- $30\frac{10}{27}$
- $1$

**Solution - Problem analysis:**

The given expression though is a sum of inverse its coefficients do not exactly correspond with those of the target inverse expression. We need to first make the coefficients of the corresponding terms of the given and the target expression conform to each other. Effectively we have to apply **coefficient conformation technique between the given and the target expressions.**

#### Solution - Simplifying actions

The target expression is a sum of inverses in power of cube of terms, $2x$ direct and $3x$ inverse. We will transform the given expression in that form.

$3x + \displaystyle\frac{1}{2x} = 5$,

Multiplying both sides by $\displaystyle\frac{2}{3}$ for making the coefficients between the given and the target expressions conform we have,

$2x + \displaystyle\frac{1}{3x}=\displaystyle\frac{10}{3}$.

So by the sum of cubes expression,

$(2x)^3 + \left(\displaystyle\frac{1}{3x}\right)^3 $

$= \left(2x + \displaystyle\frac{1}{3x}\right)\left(\left(2x + \displaystyle\frac{1}{3x}\right)^2 - 3\times{2x}\times{\displaystyle\frac{1}{3x}}\right)$

$=\displaystyle\frac{10}{3}\left(\left(\displaystyle\frac{10}{3}\right)^2 - 2\right)$

$=\displaystyle\frac{10}{3}\left(\displaystyle\frac{82}{9}\right)$

$=\displaystyle\frac{820}{27}$

$=30\frac{10}{27}$.

**Answer:** Option c: $30\frac{10}{27}$.

**Key concepts used:** * Input transformation* to conform the given coefficients to the coefficients of similar terms of the target expression --

*-- cube of sum expression --*

**principle of inverses**

**coefficient conformation technique.**### Additional help on SSC CGL Algebra

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