Solutions to quantitative algebra questions SSC CGL Set 33
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33rd SSC CGL question set and 9th on Algebra.
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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 1: Quick solution by principle of target expression simplification first and pattern discovery
Following the golden principle of algebraic expression simplification, we first try to see whether the target expression can be simplified by itself before substitution of the surd value of $x$.
We find it easily possible by multiplying both numerator and denominator with $\sqrt{1+x}+\sqrt{1-x}$. This is the key pattern discovery.
Denominator reduces to just $2x$,
$(1+x)-(1-x)=2x=\sqrt{3}$.
And the numerator reduces to a numeric value,
$(\sqrt{1+x}+\sqrt{1-x})^2=2+2\sqrt{(1-x^2)}=2+1=3$.
Result is,
$\displaystyle\frac{3}{\sqrt{3}}=\sqrt{3}$.
Answer: Option c : $\sqrt{3}$.
This is a quick solution all in mind without the complexity of double square root surd simplification.
Solution 2: By double square root surd simplification: Problem analysis:
As the value of $x$ contains a square root and the target expression is in terms of square root of $x$, it is apparently a case of double square root surd simplification.
To free $\sqrt{1+x}$ and $\sqrt{1-x}$ of the double square roots, we need to transform each of the expressions under the square roots in terms of a square of surd expression.
Let us try to transform the square root expressions to the desired form.
Solution 2: 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 -- simplification.
Before going into the complexity of double square root surd simplification, you should try to look for any feasible way of avoiding the complexity. This would save valuable solving time.
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: Quick solution by pattern discovery based on difference in powers of consecutive terms and mathematical reasoning
The target expression has 7 terms and highest power of $x$ is 72. Seems to be a complex problem. There must be some pattern in the expression, and this usually is a relation between two consecutive terms. We are using mathematical reasoning.
And when we look for it, we find an interesting pattern between each of the three pairs of terms—the power difference between the first two terms is 6 and for the next two pairs of terms it is 18.
For two terms in $x$ with power difference of 6 you can get a factor $x^3+\displaystyle\frac{1}{x^3}$ out of the sum of two terms.
Let's show you,
$x^{72}+x^{66}=x^{69}\left(x^3+\displaystyle\frac{1}{x^3}\right)$.
In the same way when the power difference between two terms in $x$ is 18, you get a factor of $x^9+\displaystyle\frac{1}{x^9}$ out of the sum of two terms.
$x^{54}+x^{36}=x^{45}\left(x^9+\displaystyle\frac{1}{x^9}\right)$, and
$x^{24}+x^6=x^{15}\left(x^9+\displaystyle\frac{1}{x^9}\right)$.
The job is now to get the value of $x^3+\displaystyle\frac{1}{x^3}$ and then of $x^9+\displaystyle\frac{1}{x^9}$ from the given expression.
At this point itself we are sure from the simple numeric choice values that both $x^3+\displaystyle\frac{1}{x^3}$ and $x^9+\displaystyle\frac{1}{x^9}$ are zero and answer will be option d: 1. There cannot be any term in $x$ left over.
Still let us check whether our reasoning is true.
Given expression is,
$\left(x+\displaystyle\frac{1}{x}\right)^2=3$,
Or, $x^2-1+\displaystyle\frac{1}{x^2}=0$.
This is the second factor in the two-factor expansion of sum of inverses of cubes,
$x^3+\displaystyle\frac{1}{x^3}=\left(x+\displaystyle\frac{1}{x}\right)\left(x^2−1+\displaystyle\frac{1}{x^2}\right)=0$.
And so, $x^9+\displaystyle\frac{1}{x^9}$ also is 0 as it has the first factor in its two-factor expansion of sum of cubes as $x^3+\displaystyle\frac{1}{x^3}$, which is 0.
The first six terms then add to 0 leaving 1 as the final result.
Answer: Option d: 1.
Key concepts used: Mathematical reasoning -- Pattern discovery based on difference in power of two consecutive terms -- Two-factor expansion of sum of cubes.
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 expression, 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 expression as a sum of squares equal to 0 -- resulting each of the square to become 0 -- evaluation of the variables and the target expression value.
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 -- principle of inverses -- cube of sum expression -- coefficient conformation technique.
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