## 58th SSC CGL level Question Set, 14th on Algebra

This is the 58th question set of 10 practice problem exercise for SSC CGL exam and 14th on topic Algebra.

In solving the problems in this session, efforts should be to solve the problems using analytical conceptual strategies and powerful techniques with least amount of number crunching. To absorb the concepts though the student must first understand and then apply the concepts and techniques for actual problem solving during managed practice sessions.

Learning by doing is the best learning. There is no other alternative towards achieving excellence.

After taking the test, you need to refer to the companion solution set (link at the end) for answers and detailed conceptual solutions.

Before taking the test you may like to refer to our **concept tutorials** on Algebra and other related topics,

**Basic and rich Algebraic concepts for elegant solutions of SSC CGL problems,**

**More Basic and rich Algebraic concepts for elegant solutions of SSC CGL problems **

* SSC CGL level difficult Algebra problem solving by Componendo dividendo*.

### 58th question set - 10 problems for SSC CGL exam: 14th on topic Algebra - answering time 12 mins

**Problem 1.**

If $x=5-\sqrt{21}$, then the value of $\displaystyle\frac{\sqrt{x}}{\sqrt{32-2x}-\sqrt{21}}$ is,

- $\displaystyle\frac{1}{\sqrt{2}}\left(\sqrt{3}-\sqrt{7}\right)$
- $\displaystyle\frac{1}{\sqrt{2}}\left(\sqrt{7}+\sqrt{3}\right)$
- $\displaystyle\frac{1}{\sqrt{2}}\left(\sqrt{7}-\sqrt{3}\right)$
- $\displaystyle\frac{1}{\sqrt{2}}\left(7-\sqrt{3}\right)$

**Problem 2.**

If $a+b+c+d=4$, then find the value of $\displaystyle\frac{1}{(1-a)(1-b)(1-c)}+\displaystyle\frac{1}{(1-b)(1-c)(1-d)}+$

$\hspace{30mm}\displaystyle\frac{1}{(1-c)(1-d)(1-a)}+\displaystyle\frac{1}{(1-d)(1-a)(1-b)}$ is,

- 1
- 4
- 5
- 0

**Problem 3.**

If $a(2+\sqrt{3})=b(2-\sqrt{3})=1$, then the positive value of $\displaystyle\frac{1}{a^2+1}+\displaystyle\frac{1}{b^2+1}$ is,

- $1$
- $4$
- $9$
- $-5$

**Problem 4.**

If $\displaystyle\frac{x}{xa+yb+zc}=\displaystyle\frac{y}{ya+zb+xc}=\displaystyle\frac{z}{za+xb+yc}$, and $x+y+z \neq 0$ then each ratio can be expressed as,

- $\displaystyle\frac{1}{a+b-c}$
- $\displaystyle\frac{1}{a+b+c}$
- $\displaystyle\frac{1}{a-b-c}$
- $\displaystyle\frac{1}{a-b+c}$

**Problem 5.**

If $3(a^2+b^2+c^2)=(a+b+c)^2$, then the relation between $a$, $b$ and $c$ is,

- $a=b=c$
- $a \neq b=c$
- $a=b \neq c$
- $a \neq b \neq c$

**Problem 6.**

If $x=\sqrt{5} + \sqrt{3}$ and $y=\sqrt{5} - \sqrt{3}$, then the value of $(x^4-y^4)$ is,

- $16$
- $544$
- $64\sqrt{15}$
- $32\sqrt{15}$

**Problem 7.**

Let $p=\displaystyle\frac{5}{18}$, then $27p^3-\displaystyle\frac{1}{216} - \displaystyle\frac{9}{2}p^2 + \displaystyle\frac{1}{4}p$ is equal to,

- $\displaystyle\frac{4}{27}$
- $\displaystyle\frac{10}{27}$
- $\displaystyle\frac{5}{27}$
- $\displaystyle\frac{8}{27}$

**Problem 8.**

For real $x+y+z=6$, then the value of $(x-1)^3+(y-2)^3+(z-3)^3$ is,

- $3xyz$
- $3(x-1)(y-2)(z-3)$
- $2(x-1)(y-2)(z-3)$
- $(x-1)(y-2)(z-3)$

**Problem 9.**

If $x+\displaystyle\frac{1}{x}=\sqrt{3}$, then the value of $x^{30}+x^{24}+x^{18}+x^{12}+x^6+1$ is,

- $1$
- $\sqrt{3}$
- $-\sqrt{3}$
- $0$

**Problem 10.**

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 $a$, $b$, $c$ and $p$, $q$, $r$ 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,

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

### Solutions to the problems

For detailed conceptual solutions with answers you should refer to the companion * SSC CGL level Solution Set 58 on Algebra* where you will get detailed explanations on easiest path to the solutions.

You may watch the **video solutions** in the **two-part video set**.

**Part 1: Q1 to Q5**

**Part 2: Q6 to Q10**

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