## Solution to hard algebra questions for Competitive exams SSC CGL 22: Quick solutions

Solution to hard algebra questions for Competitive exams SSC CGL 22.

Questions are all solved by concepts and techniques.

Hope you have already taken the related test.

If not, do take the test at * SSC CGL Question Set 22 on Algebra* before going through the solutions. Otherwise you may not be able to appreciate the finer points in getting over the difficulties in the solutions.

Each question is solved quick and how to solve it quick also explained as clearly as possible.

**Tip:** After reading the solutions, take the test again. That will help you to absorb the techniques you learned into your reflexes.

### Solution to hard algebra questions for Competitive exams, SSC CGL test set 22 - answering time was 12 mins

**Q1. **The value of $a = \displaystyle\frac{b^2}{b-a}$, then the value of $a^3 + b^3$ is,

- 1
- 6ab
- 2
- 0

** Solution - Problem analysis**

The target expression is a sum of cubes that always needs two-factor expansion,

$a^3+b^3=(a+b)(a^2-ab+b^2)$.

Examine the given equation.

Can we express it as a quadratic equation?

Of course we can, and that will give immediate solution.

#### Solution - Simplifying actions

$a = \displaystyle\frac{b^2}{b-a}$,

Or, $ab - a^2 = b^2$,

Or, $a^2 -ab + b^2 = 0$.

So,

$a^3 + b^3 = (a + b)(a^2 -ab + b^2)=0$.

**Answer:** d: 0

**Key concepts and techniques used:**

- Two-factor expansion of sum of cubes.
- Examining possibility to transform given equation to the quadratic factor.
- Question solved in mind.

**Q2.** If $x = \sqrt[3]{5} + 2$, then the value of $x^3 - 6x^2 + 12x -13$, is,

- $0$
- $-1$
- $2$
- $1$

**Solution - Problem analysis:**

To get the target expression in cube of $x$, it is obvious that the given equation has to be raised to its cube.

But should we raise the equation as it is given?

No, naturally not.

The most used rule to raise a given unity power equation to its cube is,

Form a sum of two terms in the LHS leaving a numeric value on the RHS. In this form it is easiest to raise the LHS to powers of 3.

That is the only way to go,

$x = \sqrt[3]{5} + 2$,

Or, $x-2=\sqrt[3]{5}$,

Or, $(x - 2)^3 = 5$

Or, $x^3 - 6x^2 + 12x - 8 = 5$,

Or, $x^3 - 6x^2 + 12x - 13 = 0$.

**Answer:** Option a : $0$.

**Key concepts and techniques used:**

- Decision to raise given equation to its cube by comparison of powers of $x$ between target and given expression.
- Forming sum of two terms in LHS of given equation to make raising to its cube easiest.
- Cube of sum of two terms.
- Target matching.
- Question solved in mind.

**Q3.** If $p=124$, then the value of $\sqrt[3]{p(p^2 + 3p + 3) + 1}$, is,

- 123
- 5
- 7
- 125

**Solution - Problem analysis:**

If we substitute the value of $p$ in the target expression, it will be a long drawn out process of calculations.

In this type of problems, it is always necessary to simplify the target expression first and then substitute the value of the variable.

This is the golden rule of **Target expression simplification first.**

Before using any given expression, simplify the target expression as much as possible by itself.

#### Solution - Simplifying the target expression:

For simplifying a polynomial usual approach is to factorize.

But in this case a **key pattern is identified** in the target expression when examined closely.

The four terms formed by expanding the first product under cube root represent simply $(p+1)^3$,

$\sqrt[3]{p(p^2 + 3p + 3) + 1}$

$=\sqrt[3]{p^3 + 3p^2 + 3p + 1}$

$=\sqrt[3]{(p + 1)^3}$

$=p + 1$

$=125$.

**Answer:** Option d: 125.

**Key concepts and techniques used:**

- Strategy of target expression simplification first.
- Key pattern identification of lightly hidden cube of sum under cube root.
- Cube of sum expansion.
- Question solved in mind.

**Q4. **The factors of $(a^2 + 4b^2 + 4b - 4ab - 2a - 8)$ are,

- $(a + 2b - 4)(a + 2b + 2)$
- $(a - 2b - 4)(a - 2b + 2)$
- $(a + 2b - 1)(a - 2b + 1)$
- $(a - b + 2)(a - 4b - 4)$

**Solution 1: Quick solution by Strategic comparison of coefficients of like terms between choice expressions and given expression**

At first the problem seems to be difficult.

Factorizing the given expression is out of question.

**Reasoning:** Suppose the first option is the answer. When this product is expanded into indivoidual terms, all the terms must be same as the given expression, isn't it?

It is obvious.

But what would it mean?

It would simply mean that coefficients of $a^2$, $b^2$, $b$, $ab$, $a$ and the numeric term will all be same in both.

That's the key. Strategy decided,

We will quickly eliminate an option if its product differs from the given expression in its coefficients of

any ONE of the like terms.

Forming coefficients of $a^2$ or $b^2$ is easy, but these two are same for the expressions in all options.

Do we have to continue and compare all six sets of coefficients?

The experience based rule to follow in such a case is,

Compare the coefficients of

mixed products of variables, the more the number of variables in the product, better it is.

Let's make it clear and specific,

Instead of comparing coefficients of terms $a$ or $a^2$ in only one variable, COMPARE COEFFICIENT OF A TERM IN TWO OR MORE VARIABLES FIRST.

Hiding the **coefficient mismatch** in only one variable makes it easy to discover. Possibly that's why it is generally hidden in a product of two variables.

By this rule, first the coefficients of $ab$ in the four options are compared with $-4$ in given expression,

**Option a:** $2 + 2 = 4 \Rightarrow$ invalid choice.

**Option b:** $-2 -2 = -4 \Rightarrow$ a possible answer.

**Option c:** $-2+2=0 \Rightarrow$ invalid choice.

**Option d:** $-4-1=-5 \Rightarrow$ invalid choice.

So answer is Option:b.

This solution is quick if you can identify the pattern.

**Answer:** Option b: $(a - 2b - 4)(a - 2b + 2)$.

**Key concepts and techniques used:**

- Strategic elimination of the option of factorization.
- Analytical reasoning to identify easiest path to solution as
**Coefficient comparison of like terms**between each choice and the given expression. - Following the rule of comparing coefficient of a term with
**maximum number of variables first**, in this case, of $ab$. - Actual comparison of coefficients of $ab$.
- Choice by elimination.
- Question solved in mind.

**Q5. **When the expression $12x^3 - 13x^2 -5x + 7$ is divided by $3x + 2$ the remainder is,

- $0$
- $2$
- $1$
- $-1$

**Solution - Problem analysis:**

The easiest approach is to take out the factor from the larger expression step by step eliminating at each step the highest power of $x$ and compensating for the second term.

The final leftover will be the remainder.

This is * continued factor extraction technique* for simplifying an expression in higher power of variable using an expression in smaller power of the variable.

Essentially, this method is exactly same as dividing a larger number by a smaller number.

Let us show how it is done.

#### Solution - Algebraic expression division by Continued factor extraction

In this case, the factor of $3x+2$ will be extracted from the first given expression step by step.

In each step, the term with highest power in $x$ will be consumed in the factor and a compensating term will be introduced (added or subtracted depending on the situaion).

In the first step, $12x^3$ is consumed in the factor of $(3x+2)$,

$12x^3 - 13x^2 -5x + 7$

$=4x^2(3x + 2) - 8x^2 - 13x^2 -5x + 7$, $-8x^2$ compensates $+2$ inside the brackets of the factor.

In the second step, the terms other than the factor are to be considered for factor extraction. $-21x^2$ is consumed in this step.

$=4x^2(3x + 2) - 7x(3x + 2) + 14x -5x + 7$.

Third step is still easier,

$=4x^2(3x + 2) - 7x(3x + 2) + 3(3x + 2) -6 + 7$

$=4x^2(3x + 2) - 7x(3x + 2) + 3(3x + 2) + 1$.

Thus remainder will be 1.

**Answer:** Option c: $1$.

**Key concepts and techniques used:**

Taking out the factor of step by step by **Continued factor extraction technique.**

**Method:**

- At each step, term with highest power of $x$ is absorbed in the factored expression.
- A new term equal to the second term equivalent inside the factor and of opposite sign is generated in each step
- This extra term compensates for the second term.
- At each step, only one factor is taken out, reducing the size of the expression systematically and eliminating any chances of error.

With experience of using this method, answering the question in mind is easy.

**Q6.** If $x(x-3)=-1$ then the value of $x^3(x^3 - 18)$ will be,

- $2$
- $0$
- $1$
- $-1$

**Solution: Quick solution by Mathematical reasoning, Pattern based strategy and reverse substitution in cube of sum**

By comparing the target expression with the given expression it is clear that the given expression is to be raised to its cube in some form.

As $x^3$ is one of the two factors in the target, while raising to its cube, the given expression factors are cubed in same form as given. In this way separate identity of $x^3$ as factor is maintained.

The result is,

$x^3(x-3)^3 =-1$

Or, $x^3[x^3-9x(x-3)-27]=-1$, $(a+b)^3=a^3+b^3+3ab(a+b)$ is the compact form of expansion of cube of sum,

Or, $x^3(x^3+9-27)=-1$, as $x(x-3)=-1$, it is reverse substitution,

or, $x^3(x^3-18)=-1$.

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

**Key concepts and techniques used:**

- By comparing given and target expression, decision taken to raise given expression to its cube in the given form itself.
- Using compact form of expansion of cube of sum $(a+b)^3=a^3+b^3+3ab(a+b)$. This form is frequently used and in this form reverse substitution is easy to see.
- Reverse substitution to simplify quick.
- Question solved in mind.

** Q7.** If $x = \sqrt{5} + 2$, then the value of $\displaystyle\frac{2x^2 - 3x - 2}{3x^2 - 4x - 3}$ is,

- 0.525
- 0.625
- 0.785
- 0.185

**Solution - Problem analysis: **

Our first task is to eliminate the square root by transposing and squaring the given expression and then only start analyzing further,

$x = \sqrt{5} + 2$,

Or. $(x - 2)^2 = 5$,

Or, $x^2 - 4x +4 - 5 = 0$

Or, $x^2- 4x -1 = 0$.

What is the similarity between this expression and the two quadratic expressions in the target numerator and denominator?

By close examination the **only similarity identified** between the three is,

In all three quadratic expressions, the absolute value of the coefficient of $x^2$ and numeric term are equal.

Any quadratic expression like $ax^2+bx+a$ can be converted in the form of sum of inverses $\left(x+\displaystyle\frac{1}{x}\right)$ by dividing with $a$,

$ax^2+bx+a=\displaystyle\frac{1}{a}\left(x+\displaystyle\frac{1}{x}+\displaystyle\frac{b}{a}\right)$.

With this knowledge convert the result of given expression to,

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

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

In the same way, both the numerator and denominator can be converted in terms of $\left(x-\displaystyle\frac{1}{x}\right)$.

Substituting 4 for the subtractive sum of inverses solution is straightforward.

#### Solution - Simplying actions on the target expression

Working on the target expression now,

$\displaystyle\frac{2x^2 - 3x - 2}{3x^2 - 4x - 3}$

$=\displaystyle\frac{2x\left(x - \displaystyle\frac{1}{x} - \displaystyle\frac{3}{2}\right)}{3x\left(x - \displaystyle\frac{1}{x} - \displaystyle\frac{4}{3}\right)}$

$=\displaystyle\frac{2\left(4 - \displaystyle\frac{3}{2}\right)}{3\left(4 - \displaystyle\frac{4}{3}\right)}$

$=\displaystyle\frac{5}{8}$

$=0.625$

**Answer:** Option b: 0.625.

** Key concepts used:**

- First tactical action of eliminating the square root surd by transposing and squaring.
- Identifying condition of sum of inverses in the transformed given expression and numerator and denominator of target expression.
- Converting all three in terms of subtractive sum of inverses.
- Substitution and simplification to get the answer.
- For accuracy, it is better to write a few intermediate results.

**Note:** Again, if you get the opportunity to use sum of inverses form, it will lead you in most cases to the quickest solution.

** Q8.** If $\displaystyle\frac{p}{a} + \displaystyle\frac{q}{b} + \displaystyle\frac{r}{c} = 1$, and $\displaystyle\frac{a}{p} + \displaystyle\frac{b}{q} + \displaystyle\frac{c}{r} = 0$, where $p$, $q$, $r$, $a$, $b$ and $c$ are non-zero, the value of $\displaystyle\frac{p^2}{a^2} + \displaystyle\frac{q^2}{b^2} + \displaystyle\frac{r^2}{c^2}$ is,

- $2$
- $-1$
- $0$
- $1$

** Solution - Problem analysis:**

The key pattern identified is,

The algebraic fractions $\displaystyle\frac{p}{a}$, $\displaystyle\frac{q}{b}$ and $\displaystyle\frac{r}{c}$

appear unchanged throughout the problem. This is the condition for simplification by abstraction substituting single dummy variables for the compound variables.

This is the condition for abstraction by substituting all three compound variables by single dummy variables.

Substitute dummy variables,

$x=\displaystyle\frac{p}{a}$,

$y= \displaystyle\frac{q}{b}$ and,

$z= \displaystyle\frac{r}{c}$.

The given expressions are simplified to,

$x + y + z = 1$ and ,

$\displaystyle\frac{1}{x} + \displaystyle\frac{1}{y} + \displaystyle\frac{1}{z}=0$.

And the target expression is transformed to,

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

There is no sign of $p$, $q$, $r$ or $a$, $b$ and $c$.

**Solution - Simplifying actions:**

By raising sum of three variables to its square,

$(x+y+z)^2=1$,

Or, $x^2+y^2+z^2+2(xy+yz+zx)=1$.

At this point it is easy to identify that if the three terms of the second tranformed given expression are combined you will get $(xy+yz+zx=0)$,

$\displaystyle\frac{1}{x} + \displaystyle\frac{1}{y} + \displaystyle\frac{1}{z}=0$,

Or, $\displaystyle\frac{xy+yz+zx}{xyz}=0$,

Or, $xy+yz+zx=0$.

So target expression value is,

$x^2 + y^2 + z^2 + 2(xy + yz + zx) = 1$

Or, $x^2 + y^2 + z^2 = 1$.

**Answer:** Option d: $1$.

**Key concepts and techniques used:**

- Key pattern identification in the form of three expressions $\displaystyle\frac{p}{a}$, $\displaystyle\frac{q}{b}$ and $\displaystyle\frac{r}{c}$ appearing unchanged throughout the question.
- Abstraction of these three compound variables by substitution of dummy variables.
- Square of three variable sum.
- Second pattern identification in the form of $xy+yz+zx=0$.
- Question solved in mind.

**Q9.** If equation $2x^2 -7x + 12 = 0$ has two roots $\alpha$ and $\beta$, then the value of $\displaystyle\frac{\alpha}{\beta} + \displaystyle\frac{\beta}{\alpha}$ is,

- $\displaystyle\frac{1}{24}$
- $\displaystyle\frac{7}{24}$
- $\displaystyle\frac{7}{2}$
- $\displaystyle\frac{97}{24}$

**Solution by sum of zeroes and product of zeros of a quadratic expression**

In a general quadratic expression,

$(x-p)(x-q)=x^2-(p+q)x+pq$, where the zeroes of the quadratic expression are $p$ and $q$.

So sum of zeroes $(p+q)$ equals the coefficient of the middle term and product of zeroes $pq$ equals the numeric term.

As $\alpha$ and $\beta$ are the two roots we can write,

$2x^2 -7x + 12 = 0$,

Or, $x^2 - \displaystyle\frac{7}{2}x + 6 $

$= (x - \alpha)(x - \beta) $

$= x^2 -x(\alpha + \beta) + \alpha\beta $

$= 0$.

Equating coefficients of like powers (terms),

$\alpha + \beta = \displaystyle\frac{7}{2}$, and,

$\alpha\beta = 6$.

Taking up the target expression now,

$\displaystyle\frac{\alpha}{\beta} + \displaystyle\frac{\beta}{\alpha} = \displaystyle\frac{{\alpha}^2 + {\beta}^2}{\alpha\beta}$

$=\displaystyle\frac{(\alpha + \beta)^2 - 2\alpha\beta}{\alpha\beta}$

$=\displaystyle\frac{\displaystyle\frac{49}{4} - 12}{6}$

$=\displaystyle\frac{1}{24}$.

**Answer:** Option a: $\displaystyle\frac{1}{24}$.

**Key concepts used:**

- Concepts of sum of zeroes and product of zeroes of a quadratic expression.
- Getting
**values of sum and product of zeroes**or roots by comparing coefficients of like terms. - Expressing target in terms of sum and product of roots.
- Question solved in mind.

**Note:** Values of $\alpha$ and $beta$ are not needed, their sum and product only are needed.

** Q10.** If $a - b= 3$, and $a^3 - b^3 = 117$, then absolute value of $a + b$ is,

- 5
- 7
- 3
- 9

**Solution - Problem analysis and execution:**

From the two given expressions it is easy to get the value of $ab$,

$(a - b)^3 = a^3 -b^3 - 3ab(a - b)$,

Or, $9ab = 117 - 27 = 90$,

So $ab = 10$, and

$(a + b)^2 = (a - b)^2 + 4ab = 49$,

Or, absolute value of $a + b = 7$

**Answer:** Option b: 7.

**Key concepts and techniques used:**

- Use of cube of sum expression.
- $(a+b)^2$ in terms of $(a-b)^2$.
- Question solved in mind.

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