PRACTICE QUESTIONS ON ALGEBRA

To Take a Practice Test on Quantitative Aptitude‌ Click Here

This section contains a practice test of 15 questions on ALGEBRA and about 20 model questions based on previous examinations on algebra. Practice in a time bound manner and evaluate yourself or just go through model questions and their solutions to improve your understanding of problems on algebra.

PRACTICE TEST ON ALGEBRA

SCROLL DOWN FOR MODEL QUESTIONS ON ALGEBRA ON MOBILE SCREENS

Directions: In the questions given below, four options have been given. Click on the option which you think is the RIGHT ANSWER to the question.

Press Start to enable the test.

Stop when you have finished.

Reset to erase answers.

You have 15 minutes to complete the test.

If you are facing any technical issues with this page, please mail to us at admin@leadthecompetition.in

Score 0

Timer 00:00

Question
1. The value of ${1.66}^{2} + {0.66}^{2} + 1.66 + 0.66$ is equal to
$2. When 2 x + \frac{2}{3} = 3,$ then the value of $x^{3} + \frac{1}{x^{3}} + 2$ is
3. If $x y (x + y) = 1$ then the value of $\frac{1}{x^{3} y^{3}} - x^{3} - y^{3}$ is
4. $If a + \frac{1}{a - 2} = 4, then the value of$ ${(a - 2)}^{2} + {(\frac{1}{a - 2})}^{2} is$
5. If $\frac{4 x - 3}{x} + \frac{4 y - 3}{y} + \frac{4 z - 3}{z} = 0$ Then the value of: $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is
6. $If x^{2} - 3 x + 1 = 0, then the value of$ $x^{2} + x + \frac{1}{x} + \frac{1}{x^{2}} is?$
7. If (a + b + c) = 0, then $(\frac{a^{2}}{bc} + \frac{b^{2}}{ca} + \frac{c^{2}}{ab}) is equal to$
8. If $x = 1 - \sqrt2,$ $the value of {(x - \frac{1}{x})}^{3} is$
9. If $\frac{a}{b} + \frac{b}{a} - 1 = 0,$ Find the value of a³ + b³
10. If $x + \frac{1}{x} = 99 then what is the value of \frac{100 x}{2 x^{2} + 102 x + 2} ?$
11. If a = 2 + √3, then the value of $a^{2} + \frac{1}{a^{2}}$ is
12. If $a + b + c = 0$ what is the value of $(\frac{a + b}{c} + \frac{b + c}{a} + \frac{c + a}{b}) (\frac{c}{a + b} + \frac{a}{b + c} + \frac{b}{c + a})$
13. If $x = \sqrt 3 + \sqrt 2,$ then what is the value of $x^{3} - \frac{1}{x^{3}}$
14. If $x - \frac{1}{x} = 1, then the value of \frac{x^{4} - \frac{1}{x^{2}}}{3 x^{2} + 5 x - 3} is$
15. If $x + \frac{1}{x} = 5, then x^{6} + \frac{1}{x^{6}}$ is

Question

1. The value of ${1.66}^{2} + {0.66}^{2} + 1.66 + 0.66$ is equal to

$2. When 2 x + \frac{2}{3} = 3,$

then the value of $x^{3} + \frac{1}{x^{3}} + 2$ is

3. If $x y (x + y) = 1$ then the value of

$\frac{1}{x^{3} y^{3}} - x^{3} - y^{3}$ is

4. $If a + \frac{1}{a - 2} = 4, then the value of$

${(a - 2)}^{2} + {(\frac{1}{a - 2})}^{2} is$

5. If $\frac{4 x - 3}{x} + \frac{4 y - 3}{y} + \frac{4 z - 3}{z} = 0$

Then the value of:

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is

6. $If x^{2} - 3 x + 1 = 0, then the value of$

$x^{2} + x + \frac{1}{x} + \frac{1}{x^{2}} is?$

7. If (a + b + c) = 0, then

$(\frac{a^{2}}{bc} + \frac{b^{2}}{ca} + \frac{c^{2}}{ab}) is equal to$

8. If $x = 1 - \sqrt2,$

$the value of {(x - \frac{1}{x})}^{3} is$

9. If $\frac{a}{b} + \frac{b}{a} - 1 = 0,$

Find the value of a³ + b³

10. If $x + \frac{1}{x} = 99 then what is the value of \frac{100 x}{2 x^{2} + 102 x + 2} ?$

11. If a = 2 + √3, then the value of

$a^{2} + \frac{1}{a^{2}}$ is

12. If $a + b + c = 0$ what is the value of

$(\frac{a + b}{c} + \frac{b + c}{a} + \frac{c + a}{b}) (\frac{c}{a + b} + \frac{a}{b + c} + \frac{b}{c + a})$

13. If $x = \sqrt 3 + \sqrt 2,$ then what is the value of

$x^{3} - \frac{1}{x^{3}}$

14. If $x - \frac{1}{x} = 1, then the value of \frac{x^{4} - \frac{1}{x^{2}}}{3 x^{2} + 5 x - 3} is$

15. If $x + \frac{1}{x} = 5, then x^{6} + \frac{1}{x^{6}}$ is

EXAMINATION QUESTIONS ON ALGEBRA

Click on the Show Answer button to see the answer.

1. $If x + \frac{1}{x} = 1, then the value of$

$\frac{x^{2} + 3 x + 1}{x^{2} + 7 x + 1} is$

a. 2	b. 0
c. ½	d. 1

Answer (c).

Dividing both numerator and denominator by x we get $\frac{x^{2} + 3 x + 1}{x^{2} + 7 x + 1} = \frac{\frac{x^{2} + 3 x + 1}{x}}{\frac{x^{2} + 7 x + 1}{x}} = \frac{x + \frac{1}{x} + 3}{x + \frac{1}{x} + 7} = \frac{1 + 3}{1 + 7} = \frac{1}{2}$

2. If a + b + c = 0, find the value of

$\frac{1}{a^{3}} + \frac{1}{b^{3}} > + \frac{1}{c^{3}} + \frac{1}{{(a+b)}^{3}} + \frac{1}{{(b+c)}^{3}} + \frac{1}{{(c+a)}^{3}}$

a. 3abc	b. 1
c. 0	d. – 1

Answer (c).

Since a + b + c = 0, a + b = - c, b + c = - a and a + c = - b
The above equation can be written as
$\frac{1}{a^{3}} + \frac{1}{b^{3}} > + \frac{1}{c^{3}} + \frac{1}{{(-c)}^{3}} + \frac{1}{{(-a)}^{3}} + \frac{1}{{(-b)}^{3}}$

3. If x + y = 4 and $\frac{1}{x} + \frac{1}{y} = 4$ , then the value of $x^{3} + y^{3} is$

a. 64	b. 4
c. 25	d. 52

Answer (d).

x + y = 4 ⇒ (x + y)² = 16 ⇒ x² + y ² + 2xy = 16 ⇒ x² + y² = 14 ......(1)
$\frac{1}{x} + \frac{1}{y} = 4$

$⇒ \frac{y + x}{xy} = 4$

⇒xy=1(since x+ y = 4).......(2)

We know that x³ + y³ = (x + y)(x² + y² + xy)

Substituting the values of (1) and (2) in the above equation we get (4) x (14 - 1) = 52

4. Given that a + b + c = 2 and ab + bc + ca = 1, then the value of (a + b)² + (b + c)² + (c + a)² is

a. 16	b. 6
c. 8	d. 10

5. (a² + 2a)² + 12(a² + 2a) – 45 can be expressed as

(a – 1)(a + 3)(a² + 2a + 15)
(a + 1)(a + 3)(a² + 2a + 15)
(a + 1)(a – 3)(a² + 2a + 15)
(a – 1)(a – 3)(a² + 2a + 15)

6. If 3(a² + b² + c²) = (a + b + c)², then the relation between a, b, c is

a. a = b = c	b. a = b ≠ c
c. a < b < c	d. a > b > c

7. If a = 2.234, b = 3.121 and c = -5.355, then the value of a³ + b³ + c³ – 3abc is

a. – 1	b. 0
c. 1	d. 2

8. The equations 3x + 4y = 10 and – x + 2y = 0 have the solution (a, b). The value of a + b is

a. 1	b. 2
c. 3	d. 4

9. If $x^{2} + 4x + 3 = 0, then the value of$

$\frac{x^{3}}{x^{6} + 27 x^{3} + 27}$ is

a. – $\frac{1}{2}$	b. 1
c. $\frac{1}{2}$	d. – 1

Answer (d).

$x^{2} + 4 x + 3 = 0 \Rightarrow x^{2} + 3 = – 4x$

$⇒ x + \frac{3}{x} = –4 ....... (1)$

$\Rightarrow (x + \frac{3}{x})^{3} = (- 4)^{3}$

$\Rightarrow x^{3} + \frac{27}{x^{3}} + 3 (x) \frac{3}{x} (x + \frac{3}{x}) = - 64$

$\Rightarrow x^{3} + \frac{27}{x^{3}} + 9 x - 4 = - 64$

[see (1) above]

$\Rightarrow x^{3} + \frac{27}{x^{3}} - 36 = - 64$

$\Rightarrow x^{3} + \frac{27}{x^{3}} = - 28 ....(2)$

We now divide the numerator and denominator of equation by $x^{3} to obtain$

$\frac{1}{x^{3} + 27 x^{3} + \frac{27}{x^{3}}}$

Substituting the value of (2) in the above we get

$\frac{1}{27 - 28} = - 1$

10. If 999x + 888y = 1332 and 888x + 999y = 555, then x² – y² is equal to

a. 7	b. 8
c. 9	d. 5

11. If a – b = 4, and a² + b² = 40 where a and b are positive integers, then a³ + b⁶ is equal to

a. 264	b. 280
c. 300	d. 324

12. If $(x^{2} + \frac{1}{x^{2}}) = \frac{17}{4}$

then what is $(x^{3} - \frac{1}{x^{3}}) equal to$

a.	75
	16
b.	63
	8
c.	95
	8
d.	None	of these

13. If x + y = 5, y + 3 = 10 and z + x = 15, then which of the following is correct?

a. z > x > y	b. z > y > x
c. x > y > z	d. x > z > y

14. What is $\frac{1}{a - b} - \frac{1}{a + b} - \frac{2b}{a^{2} + b^{2}} - \frac{{4b}^{3}}{a^{4} + b^{4}} - \frac{{8b}^{7}}{a^{8} - b^{8}}$

equal to

a. a + b	b. a – b
c. 1	d. 0

Answer (d).

Taking first 2 expressions

$\frac{1}{a - b} - \frac{1}{a + b} = \frac{a + b - a + b}{a^{2} - b^{2}} = \frac{2b}{a^{2} - b^{2}}$

Taking the above and the next expression

$\frac{2b}{a^{2} - b^{2}} - \frac{2b}{a^{2} + b^{2}} = \frac{4 b^{3}}{a^{4} - b^{4}}$

Once again taking the above and the next expression

$\frac{4 b^{3}}{a^{4} - b^{4}} - \frac{4 b^{3}}{a^{4} + b^{4}} = \frac{8 b^{7}}{a^{8} - b^{8}}$

Finally $\frac{8 b^{7}}{a^{8} - b^{8}} - \frac{8 b^{7}}{a^{8} - b^{8}} = 0$

15. If x + y – 7 = 0 and 3x + y – 13 = 0, then what is 4x² + y² + 4xy equal to

a. 75	b. – 85
c. 91	d. 100

16. If $x + \frac{1}{x} = 99,$

find the value of $\frac{100 x}{2 x^{2} + 102 x + 2}$

a.	1
	3
b.	1
	4
c.	1
	6
d.	1
	2

17. If	a	+	b	– 1 = 0, then the value of a³ + b³ is
	b		a

a. 1	b. – 1
c. 3	d. 0

18. If x² – 3x + 1 = 0, then the value of

$x^{2} + x + \frac{1}{x} + \frac{1}{x^{2}} is$

a. 6	b. 8
c. 10	d. 2

19. If $\frac{4 x - 3}{x} + \frac{4 y - 3}{y} + \frac{4 z - 3}{z} = 0,$

then what is the value of

$\frac{1}{x} + \frac{1}{y} + \frac{1}{x}$

a. 4	b. 6
c. 9	d. 3

$20. If a + b + c = 2s, then$

$\frac{{(s - a)}^{2} + {(s - a)}^{2} + {(s - a)}^{2} + s^{2}}{a^{2} + b^{2} + c^{2}} is equal to$

a. a² + b² + c²

b. 0

c. 1

d. 2

100x	=	100x
2x² + 102x + 2	=	2(x² + 51x + 1)
100x	=	100x	since x² + x + 1 = 100x
2(x² + 50x + x + 1)	=	x(100x + 50x)	since x² + x + 1 = 100x

PRACTICE QUESTIONS ON ALGEBRA

To Take a Practice Test on Quantitative Aptitude‌ Click Here

PRACTICE TEST ON ALGEBRA

SCROLL DOWN FOR MODEL QUESTIONS ON ALGEBRA ON MOBILE SCREENS

Score 0

Timer 00:00

EXAMINATION QUESTIONS ON ALGEBRA

Click on the Show Answer button to see the answer.

QA Practice Tests

Geometry

Others

Arithmetic

⇒	4s² – 2s x 2s + a² + b² + c²	( a + b + c = 2s)
	a² + b² + c²

(x –	1	)² =	9
	x		4

100x	=	1
300x	=	3

x +	1	= 3 .......(1)
	x

(x +	1	)² = 9
	x