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SSC CGL level Solution Set 10, Algebra

SSC CGL algebra questions with solutions set 4

10 MCQ Algebra Questions with Solutions 4th set for SSC CGL and other exams: Learn to solve algebra quick

Fourth set of SSC CGL Algebra questions with solutions shows how to solve MCQ algebra questions easy and quick using algebra problem solving techniques.

For best results, take the timed test first at SSC CGL Question set 10 algebra 4 and then go through the solutions.

The 10 MCQ Algebra questions are carefully selected to test you. These are of six different types that need appropriate special problem solving techniques to solve quick.

Solutions to 4th set of 10 MCQ SSC CGL Algebra questions for SSC CGL: Learn to solve algebra quick - time to solve was 12 mins

Q1. If $2a + \displaystyle\frac{1}{3a} = 6$, then the value of the expression $3a + \displaystyle\frac{1}{2a}$ is,

  1. 12
  2. 9
  3. 4
  4. 8

Solution:

Examine similarities between the expression to be evaluated and given expression. This is applying powerful and natural problem solving End state analysis approach in action.

As a result form first conclusion:

The product of the direct and inverse terms of the given expression and the target expression are equal, that is, $2a\times{3a}=3a\times{2a}=6a^2$.

So this is a case of matching coefficients. 

Transform first dividing the equation by 2 and then multiplying by 3,

$2a + \displaystyle\frac{1}{3a} = 6$

Or, $a + \displaystyle\frac{1}{6a} = 3$

Or, $3a + \displaystyle\frac{1}{2a} = 9$. 

Directly reached the answer.

Answer: Option b: 9.

Key concepts used:

End state analysis to detect similarities between expression to be evaluated and the given expression -- Key pattern identified as product of coefficients of mutually inverse variable $a$ to be equal in both given and target expressions -- Matching coefficients of the inverse terms.

Q2. If $x^2 + y^2 - 2x + 6y + 10 = 0$, then $(x^2 + y^2)$ is,

  1. 6
  2. 4
  3. 10
  4. 8

Solution:

It being impossible to deduce the target exapression value from the given expression by deductive steps, the only possibility to solution is to evaluate $x$ and $y$ directly from the gfiven equation.

With intent as you examine the given equation way forward becomes clear.

Collect like terms together to express the given expression as a sum of square expressions equal to zero,

$x^2 + y^2 - 2x + 6y + 10 = 0$,

Or, $(x^2 - 2x + 1) + (y^2 + 6y + 9) = 0$,

Or, $(x - 1)^2 + (y + 3)^2 = 0$,

For real $x$ and $y$,

The sum of squares of two expressions to be 0 both the squares individually must be 0.

This is the principle of zero sum of square terms.

So,

$x - 1  = 0$, and $y + 3$ = 0,

Or, $x = 1$, and $y = -3$,

Or, $x^2 + y^2 = 10$

Answer: Option c: 10.

Key concepts used:

Collection of like terms together -- Principle of zero sum of square terms.

Q3. If $x^2 = 2$, then $x + 1$ is,

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

Solution:

The given and the target expression being very simple we decide to use the free resource of the choice values and test each one by one.

Without actual deduction we can see the first choice expression to be mathematically invalid. In case of second choice, $x^2 - 1 = 2$, or $x^2 = 3$, an invalid result.

For the third choice, $2x = 3$, again an invalid choice.

For the fourth expression we get,

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

Or, $(x + 1)(3 - 2x) = x - 1$

Or, $3x - 2x^2 + 3 - 2x = x - 1$

Or, $2x^2 = 4$,

$x^2 = 2$, a confirmation of given expression.

Answer: Option d: $\displaystyle\frac{x - 1}{3 - 2x}$.

Key concepts used:

Deciding to use the choice expressions as a resource and testing each choice expression to see whether it satisfies the given value of  $x$.

Q4. If $a^2 + b^2 + \displaystyle\frac{1}{a^2} + \displaystyle\frac{1}{b^2} = 4$  then $a^2 + b^2$ is,

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

Solution:

This again requires a collection of like terms terms towards forming sum of two squares, but this time we use the resource of the value on the RHS also. 

$a^2 + b^2 + \displaystyle\frac{1}{a^2} + \displaystyle\frac{1}{b^2} = 4$,

Or, $\left(a^2 - 2 + \displaystyle\frac{1}{a^2}\right) + \left(b^2 - 2 + \displaystyle\frac{1}{b^2}\right)= 0$,

Or, $\left(a - \displaystyle\frac{1}{a}\right)^2 + \left(b - \displaystyle\frac{1}{b}\right)^2=0$.

And so, $a = \displaystyle\frac{1}{a}$,  or, $a^2 = 1$, and,

$b = \displaystyle\frac{1}{b}$,  or, $b^2 = 1$,

Or, $a^2 + b^2 = 2$

Answer: Option d: $2$.

Key concepts used:

Again collection of like terms to form a sum of two squares. It is a powerful technique if you can apply it.

Q5. If $a + b + c = 6$, $a^2 + b^2 + c^2 = 14$ and $a^3 + b^3 + c^3 = 36$, then the value of $abc$ is,

  1. 3
  2. 6
  3. 9
  4. 12

Solution:

The three given expressions with values can be related together by the single expanded form of three variable sum of cubes,

$a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) + 3abc$.

It is to some extent similar to,

$x^3 + y^3 = (x + y)(x^2 + y^2 - xy)$

In our problem case,

$a^3 + b^3 + c^3 = 36=(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) + 3abc$,

Or, $36=6[14 - (ab + bc + ca)] + 3abc$.

Lastly get the sum of three products of two variables from square of three variable sum,

$(a + b + c)^2 = 36= a^2 + b^2 + c^2 + 2(ab + bc + ca)$,

Or, $ab + bc + ca = 11$.

So, $36=6[14 - (ab + bc + ca)] + 3abc$,

Or, $36 = 6(14 - 11) + 3abc$,

Or, $3abc = 18$,

Or, $abc = 6$.

Answer: Option b: 6.

Key concepts used:

Expanded form of three variable sum of cubes -- Square of three variable sum.

Q6. If $x + \displaystyle\frac{1}{16x} = 1$, then the value of $64x^3 + \displaystyle\frac{1}{64x^3}$ is,

  1. 64
  2. 76
  3. 52
  4. 4

Solution:

The terms of the target expression are cubes of $4x$ and $\displaystyle\frac{1}{4x}$.

Though the given expression looks different, the coefficient and power of $x$ in the product of $x\times{16x}=16x^2$ is same as $4x\times{4x}$.

So transform the coefficients of the given expression terms by multiplying the equation by 4 and then take the square,

$x + \displaystyle\frac{1}{16x} = 1$,

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

Or, $\left(4x + \displaystyle\frac{1}{4x}\right)^2 = 16$,

Or, $\left(16x^2 + \displaystyle\frac{1}{16x^2}\right) = 14$.

Express target in two factor expanded form for sum of cubes,

$64x^3 + \displaystyle\frac{1}{64x^3} $

$\hspace{5mm} = \left(4x + \displaystyle\frac{1}{4x}\right)\left(16x^2 + \displaystyle\frac{1}{16x^2} - 1\right) $

$\hspace{5mm} = 4\times{13} $

$\hspace{5mm} = 52$

Answer: Option c: 52.

Key concepts used:

Coefficient matching -- Two factor expansion of sum of cubes.

Q7. If $a^4 + a^2b^2 + b^4 = 8$ and $a^2 + ab + b^2 = 4$, then the value of $ab$ is,

  1. $0$
  2. $2$
  3. $-1$
  4. $1$

Solution:

The main question is, what is the similarity between the given expression and the target expression?

Examine the two to identify the key pattern of the identity,

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

Or, $4(a^2+b^2-ab)=8$

$a^2+b^2-ab=2$.

Subtract this result from $a^2+b^2+ab=4$,

$2ab=1$,

Or, $ab=1$.

Alternate solution by direct expression matching:

As a first step square up the second expression to reach a form similar to the first expression.

$a^4 + a^2b^2 + b^4 + 2ab(a^2 + ab + b^2) = 16$,

Or, $8 + 2ab\times{4} = 16$,

Or, $8ab = 8$,

Or, $ab = 1$.

Answer: Option d: 1.

Key concepts used:

Key pattern identification of identity relating two given expressions -- Or Expression matching.

Q8. If $x^2 + y^2 + z^2 = xy + yz + zx$, then the value of, $\displaystyle\frac{4x +2y -3z}{2x}$ is,

  1. $1$
  2. $0$
  3. $\displaystyle\frac{1}{2}$
  4. $\displaystyle\frac{3}{2}$

Solution:

There being no simple way to deduce the target expression that is not at all similar to the given expression decide to simplify the given expression first. With intent clear, identify now the key pattern of zero sum of square terms in the given expression,

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

Or, $(x - y)^2 + (y - z)^2 + (z - x)^2 = 0$

Again the use of Principle of zero sum of squares.

So, $(x - y) = (y - z) = (z - x) = 0$

Or, $x = y = z$

So the target expression is simplied as,

$\displaystyle\frac{4x +2y -3z}{2x} = \frac{3x}{2x} = \frac{3}{2}$

Answer: Option d: $\displaystyle\frac{3}{2}$.

Key concepts used:

Mathematical reasoning -- Collection of like terms - Principle of zero sum of square terms.

Q9. If $x\left(3 - \displaystyle\frac{2}{x}\right) = \displaystyle\frac{3}{x}$, and $x\neq{0}$ then $x^2 + \displaystyle\frac{1}{x^2}$ is,

  1. $2\displaystyle\frac{5}{9}$
  2. $2\displaystyle\frac{4}{9}$
  3. $2\displaystyle\frac{1}{3}$
  4. $2\displaystyle\frac{2}{3}$

Solution:

How to evaluate the target sum of inverses of squares from the given expression?

It would be easy if a sum of inverses in $x$ is obtained from the given expression.

Now with focused intent it is easy to transform the given expression to a subtractive sum of inverses.

The key pattern is easy to see as the coefficient of $x$ in LHS and coefficient of $\displaystyle\frac{1}{x}$ in RHS both are equal as 3,

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

Or, $3x - \displaystyle\frac{3}{x} = 2$

Or, $x - \displaystyle\frac{1}{x} = \frac{2}{3}$.

Squaring up this resultant equation,

$x^2 + \displaystyle\frac{1}{x^2} - 2 = \displaystyle\frac{4}{9}$,

Or, $x^2 + \displaystyle\frac{1}{x^2} = 2+\displaystyle\frac{4}{9}=2\displaystyle\frac{4}{9}$.

Answer: Option b: $2\displaystyle\frac{4}{9}$.

Key concepts used:

Target matching to transform given expression in suitable form - Principle of interaction of inverses.

Q10. If $(x -a)(x - b) = 1$ and $(a - b) + 5 = 0$, then $(x - a)^3 - \displaystyle\frac{1}{(x - a)^3}$ is

  1. 140
  2. 125
  3. -125
  4. 1

Solution:

Identify the key pattern that, $x$ is eliminated by subtracting $x - b$ from $x - a$ to get $b - a$ which equals $5$.

In other words,

$(x - a) - (x - b) = b-a=5$.

Now substitute inverse of $(x-a)$ for $(x-b)$ from first given equation,

$(x - a) - (x - b) = 5$,

Or, $(x - a) - \displaystyle\frac{1}{(x - a)} = 5$,

The target is in form of sum of inverse of cubes as well as the given expression is a sum of inverses.

To visualize this clearly, substitute dummy variable $p$ for $(x - a)$. The given expression is transformed to,

$p - \displaystyle\frac{1}{p} = 5$,

Squaring,

$p^2 + \displaystyle\frac{1}{p^2} = 25 + 2 = 27$

Express the target sum of cubes in two factor expanded form,

$E = \left(p - \displaystyle\frac{1}{p}\right)\left(p^2 + \displaystyle\frac{1}{p^2} + 1\right)$

$\hspace{5mm} = 5\times{\left(27 + 1\right)} = 140$.

Answer: Option a: 140.

Key concepts used:

Key pattern identification of transforming both the second given expression and the target expression in terms of inverses of $(x-a)$.

First formation of  $(x - a) - (x - b) = 5$ from $b-a=5$ has been crucial.


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