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SSC CGL level Solution Set 60, Fractions indices and surds 4

Surds indices square root questions SSC CGL solution set 60

Surds indices and square root questions solved SSC CGL set 60

Learn to solve 10 questions on surds indices and square roots for SSC CGL set 60 in 12 minutes using surds and indices problem solving techniques.

Contents are,

  1. Surds and indices questions solved.
  2. Square root questions solved.
  3. Cube root question solved.
  4. Concepts and techniques for quick solution.

For best results take the test first at,

SSC CGL question set 60 on surds indices square roots.

Solution to 10 questions on surds indices and square roots SSC CGL set 60 - time to solve was 12 mins

Problem 1.

The sum of cubes of the numbers $22$, $-15$ and $-7$ is equal to,

  1. 3
  2. 9630
  3. 0
  4. 6930

Solution 1: Problem analysis and solving

The first cube is of 22 which is a sum of other two cube base values, 15 and 7, $22=15+7$. Moreover, the second two cubes will evaluate to negative values. So we decide to use cube of sum expression,

$(22)^3=(15)^3+7^3+3\times{15}\times{7}\times{22}$

$=(15)^3+7^3+105\times{66}$

$=(15)^3+7^3+6930$.

So,

$(22)^3+(-15)^3+(-7)^3=6930$.

Answer: Option d: 6930.

Key concepts used: Key pattern identification -- basic algebraic concepts -- Efficient simplification.

Problem 2.

Which of the following is the correct relation?

  1. $\sqrt{5}+\sqrt{3} \lt \sqrt{6}+\sqrt{2}$
  2. $\sqrt{5}+\sqrt{3} = \sqrt{6}+\sqrt{2}$
  3. $\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$
  4. $\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{6}+\sqrt{2}\right)=1$

Solution 2: Problem analysis

In surd expression comparison, one of the most important concepts that we use frequently is what we call, the Equal difference surd comparison concept.

Let us briefly state and explain the mechanism of this important concept.

Equal difference surd comparison concept

Let the two surd expressions we need to compare be,

$\sqrt{a}-\sqrt{b}$, and

$\sqrt{c}-\sqrt{d}$

where $a-b=c-d$.

If $a \gt c$, by equal difference surd comparison concept,

$\sqrt{a}-\sqrt{b} \lt \sqrt{c}-\sqrt{d}$.

Proof of equal difference surd comparison concept

$a-b=c-d$,

Or, $a-c=b-d$.

As $a \gt c$, $b \gt d$.

Taking up the surd expressions,

$\sqrt{a}-\sqrt{b}=\displaystyle\frac{a-b}{\sqrt{a}+\sqrt{b}}$, by rationalization.

Similarly,

$\sqrt{c}-\sqrt{d}=\displaystyle\frac{c-d}{\sqrt{c}+\sqrt{d}}$.

Taking the ratio,

$\displaystyle\frac{\sqrt{a}-\sqrt{b}}{\sqrt{c}-\sqrt{d}}=\frac{\sqrt{c}+\sqrt{d}}{\sqrt{a}+\sqrt{b}}$.

As $a \gt c$ and $b \gt d$,

$\sqrt{a}+\sqrt{b} \gt \sqrt{c}+\sqrt{d}$.

So,

$\sqrt{a} - \sqrt{b} \lt \sqrt{c}-\sqrt{d}$.

Solution 2: Problem solving execution

In all four options the same two surd expressions appear. Let us evaluate the comparative relation between these two expressions.

By equal difference surd comparison concept,

$\sqrt{6}-\sqrt{5} \lt \sqrt{3}-\sqrt{2}$,

Or, $\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$.

So only Option: $c$ is true.

Answer: Option c: $\sqrt{5}+\sqrt{3} \gt \sqrt{6}+\sqrt{2}$.

Key concepts used: Surd rationalization -- inequality concepts -- equal difference surd comparison concept.

Problem 3.

$\sqrt[3]{(333)^3+(333)^3+(334)^3-3\times{333}\times{333}\times{334}}$ is closest to,

  1. 11
  2. 10
  3. 12
  4. 15

Solution 3: Problem analysis and solving

As direct calculation will be inordinately lengthy and as the numeric cubed terms have a patttern, we will use sum of cubes expression, $x^3-y^3=(x-y)(x^2+xy+y^2)$.

$E=\sqrt[3]{(333)^3+(333)^3+(334)^3-3\times{333}\times{333}\times{334}}$

$=\sqrt[3]{(334)^3-(333)^3-3\times{(333)^2}(334-333)}$

$=\sqrt[3]{(334)^2+333\times{334}+(333)^2-3(333)^2}$

$=\sqrt[3]{(334)^2-(333)^2+333}$, the 334 in product is split as $334=(333+1)$ cancelling two 333 squares

$=\sqrt[3]{667+333}$, using $a^2-b^2=(a-b)(a+b)$ where $334+333=667$

$=\sqrt[3]{1000}$

$=10$.

Answer: Option b: 10.

Key concepts used: Basic algebra concepts -- efficient simplification -- abstraction, we have used the numeric constant terms of $333$ and $334$ as variables in algebraic relations -- delayed evaluation, we have carried out numeric calculation only at the last step.

Problem 4.

The value of $\sqrt{0.00060516}$ is equal to,

  1. 0.246
  2. 0.0246
  3. 0.00246
  4. 0.000246

Solution 4: Problem analysis and solving execution

It can be assumed from the values that,

$(24.6)^2=605.16$.

So squaring only the value in the second choice, $0.0246$ will shift the decimal point in comparison to the square of $24.6$ six places left and thus will generate three leading zeros before $605.16$.

Answer: Option b: 0.0246.

Key concepts used: Basic decimal concepts -- place value mechanism -- free resource use -- choice value test.

Problem 5.

The value of $(3+2\sqrt{2})^{-3}+(3-2\sqrt{2})^{-3}$ is,

  1. 108
  2. 189
  3. 180
  4. 198

Solution 5: Problem analysis and solving execution

The problem involves cube of surd terms, and so we need to use the surd rationalization technique after expressing the negative powers of terms in fraction form.

The target expression,

$E=(3+2\sqrt{2})^{-3}+(3-2\sqrt{2})^{-3}$

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

$=\left(3-2\sqrt{2}\right)^3+\left(3+2\sqrt{2}\right)^3$

This is in the form of $a^3+b^3$, where $a=3-2\sqrt{2}$ and $b=3+2\sqrt{2}$.

So the target expression is,

$E=(a+b)(a^2-ab+b^2)$

$=6[(a+b)^2-3ab]$

$=6\times{[(6)^2-3(9-8)]}$

$=6\times{33}$

$=198$.

Answer: Option d: 198.

Key concepts used: Basic algebraic concepts -- sum of cubes -- abstraction -- delayed evaluation -- surd rationalization -- basic indices concepts -- efficient simplification.

Problem 6.

$4^{61}+4^{62}+4^{63}+4^{64}$ is always divisible by,

  1. 3
  2. 11
  3. 13
  4. 17

Solution 6: Problem analysis and solving execution

We need to take the largest common factor $4^{61}$ out of the four terms to simplify the expression.

The target expression is thus,

$E=4^{61}+4^{62}+4^{63}+4^{64}$

$=4^{61}(1+4+16+64)$

$=4^{61}(85)$.

Out of the given choice values 17 as a factor satisfies the target expression.

Answer: Option d: 17.

Key concepts used: Simplification by taking common factor out of each term -- basic factors and multiples concepts.

Problem 7.

The value of $\sqrt{30+\sqrt{30+\sqrt{30+...}}}$ is,

  1. 6
  2. 5
  3. 7
  4. $3\sqrt{10}$

Solution 7: Problem analysis and solving execution

With such an additive non-terminating square root series, we need to assume the target as $p$ and square it to take out one 30 from the target leaving $p$,

$p=\sqrt{30+\sqrt{30+\sqrt{30+...}}}$,

Or, $p^2=30+p$,

Or, $p^2-p-30=0$,

Or, $(p-6)(p+5)=0$.

As $p$ cannot be negative, $p=6$.

Answer: Option a: 6.

Key concepts used: Basic number system concepts -- Non-terminating expression conversion technique -- basic algebraic concepts -- factorization of quadratic equations.

Problem 8.

The value of $\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4...}}}}$ is,

  1. $2$
  2. $2^3$
  3. $2^5$
  4. $2^2$

Solution 8: Problem analysis and solving execution

To access the repeating sequence within the square roots we need to raise the target expression first to square and then to cube. Assuming the target expression as $p$,

$p^2=2\sqrt[3]{4\sqrt{2\sqrt[3]{4...}}}$.

Again cubing,

$p^6=32\sqrt{2\sqrt[3]{4...}}$

Or, $p^6=32p$,

Or, $p^5=32$,

Or, $p=2$.

Here the expression has been multiplicative that aided final evaluation.

Answer: Option a: 2.

Key concepts used: Basic indices concepts -- basic algebraic concepts -- non-terminating expression conversion technique.

Problem 9.

The number which when multiplied with $(\sqrt{3}+\sqrt{2})$ gives $(\sqrt{12}+\sqrt{18})$ is,

  1. $3\sqrt{2}-2\sqrt{3}$
  2. $\sqrt{6}$
  3. $2\sqrt{3}-3\sqrt{2}$
  4. $3\sqrt{2}+2\sqrt{3}$

Problem analysis and solving execution

We need to factorize the second expression so that we get one factor as the first expression. So let us factorize the second expression,

$(\sqrt{12}+\sqrt{18})$

$=2\sqrt{3}+3\sqrt{2}$,

$=\sqrt{3}(2+\sqrt{6})$

$=\sqrt{6}(\sqrt{3}+\sqrt{2})$.

So it is simply $\sqrt{6}$.

Answer. Option b: $\sqrt{6}$.

Key concepts used: Surd factorization concepts.

Problem 10.

When it is given that $\sqrt{3}=1.732$, the value of $\displaystyle\frac{3+\sqrt{6}}{5\sqrt{3}-2\sqrt{12}-\sqrt{32}+\sqrt{50}}$ is,

  1. 1.414
  2. 1.732
  3. 2.551
  4. 4.899

Solution 10: Problem analysis and solving

We need to simplify the target expression first.

The target expression,

$E=\displaystyle\frac{3+\sqrt{6}}{5\sqrt{3}-2\sqrt{12}-\sqrt{32}+\sqrt{50}}$

$=\displaystyle\frac{3+\sqrt{6}}{5\sqrt{3}-4\sqrt{3}-4\sqrt{2}+5\sqrt{2}}$

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

$=\sqrt{3}$

$=1.732$.

Answer: Option b: 1.732.

Key concepts used: Surd simplification -- Surd factorization.


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