# Algebra | Free Online Test and Practice

**“Algebra”** is a common chapter in many competitive exams such as **SSC**, **Bank,** **UPSC** and other exams. Many students practice a lot of **Algebra Formula and equations **to make his and her calculation strong. Sometimes, A student gets confused at the time of choosing the right answer among the given options.

## Basic Algebra formula to improve accuracy.

- a
^{2}+ b^{2}= (a + b)^{2}– 2ab - a
^{2}– b^{2}= (a – b)(a + b) - (a+b)
^{2}= a^{2}+ 2ab + b^{2} - (a – b)
^{2}= a^{2}– 2ab + b^{2} - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2bc + 2ca - (a – b – c)
^{2}= a^{2}+ b^{2}+ c^{2}– 2ab + 2bc – 2ca - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2}) - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}; (a + b)^{3}= a^{3}+ b^{3}+ 3ab(a + b) - (a – b)
^{3}= a^{3}– 3a^{2}b + 3ab^{2}– b^{3} - a
^{4}– b^{4}= (a – b)(a + b)(a^{2}+ b^{2}) - (a + b)
^{4}= a^{4}+ 4a^{3}b + 6a^{2}b^{2}+ 4ab^{3}+ b^{4} - (a – b)
^{4}= a^{4}– 4a^{3}b + 6a^{2}b^{2}– 4ab^{3}+ b^{4} - a
^{5}– b^{5}= (a – b)(a^{4}+ a^{3}b + a^{2}b^{2}+ ab^{3}+ b^{4})

**Click Here to Download Algebra Formula PDF**

**Let’s have some warm-up questions before taking the test…**

**1. If x = 2 then the value of x ^{3} + 27x^{2} + 243x + 631 is…..?**

A. 1213

B. 1233

C. 1336

D. 1356

Ans: B

**Solution**

Given equation,

f(x) = x

^{3}+ 27x

^{2}+ 243x + 631

⇒ x(x

^{2}+ 27x+ 243) + 631

Now, put the value of x = 2

⇒ 2(2

^{2}+ 27 × 2 + 243) + 631 ⇒ 2 (4 + 54 + 243) + 631

⇒ 2(301) + 631 = 602 + 631 = 1233.

**2. The expression x ^{4} – 2x^{2} + k will be a perfect square when the value of k is….?**

A. 1

B. 0

C. -1

D. 2

Ans: A

**Solution**

Let’s assume x

^{2}= m, therefore the given eq. will be:

m

^{2}– 2m + k which is a quadratic equation (ax

^{2}+ bx + c).

Now we know that a quadratic eqn. is a perfect square if its discriminant (b

^{2}– 4ac) is equal to zero.

In the eq. a = 1, b = – 2, c = k

**∴**(– 2)

^{2}– 4 (1).k = 0

–4k = –4

**∴**k = 1

**3. If x ^{2} + y^{2} + 1 = 2x, then the value of x^{3} + y^{5} is…..?**

A. 1

B. -2

C. 2

D. -1

Ans: A

**Solution**

x

^{2}+ y

^{2}+ 1 = 2x

⇒ x

^{2}+ y

^{2}+ 1 – 2x = 0 ⇒ x

^{2}– 2x + 1 + y

^{2}= 0

⇒ (x – 1)

^{2}+ y

^{2}= 0

In the above eq. the L.H.S. only becomes zero when the base of terms; (x – 1)

^{ }and y becomes zero because for any other value the sum of their squares shall always be a positive integer.

Taking (x – 1) = 0

and y = 0

Therefore, x = 1 and y = 0

**∴**x

^{3}+ y

^{5}= 1 + 0 = 1.

### You are doing well, Let’s have a few more question…

**4. If x = 2015, y = 2014 and z = 2013, then value of x ^{2} + y^{2} + z^{2} – xy – yz – zx is…?**

A. 12

B. 9

C. 3

D. 6

Ans: C

**Solution**

x – y = 2015 – 2014 = 1

y – z = 2014 – 2013 = 1

z – x = 2013 – 2015 = –2

∴ x

^{2}+ y

^{2}+ z

^{2}– xy – yz – zx

Numerator & denominator multiplied by 2

= 1/2 (2x

^{2}+ 2y

^{2}+ 2z

^{2}– 2xy – 2yz – 2zx) ⇒ 1/2 (x

^{2}+ y

^{2}– 2xy + y

^{2}+ z

^{2}– 2yz + z

^{2}+ x

^{2}– 2zx)

= 1/2 [(x– y)

^{2}+ (y – z)

^{2}+ (z – x)

^{2}] ⇒ 1/2 [1 + 1 + 4] = 1/2 × 6 = 3

**5. Consider the following statements I. x + 3 is the factor of x ^{3} + 2x^{2} + 3x + 8. II. x – 2 is the factor of x^{3} + 2x^{2} + 3x + 8. Which of the statements given above is/are correct?**

A. Only I

B. Only II

C. Neither I nor II

D. Both I and II

Ans:C

**Solution**

Put x = –3 in equation x

^{3}+ 2x

^{2}+ 3x + 8

= (–3)

^{3}+ 2(–3)

^{2}+ 3(–3) + 8

= –10 ≠ 0

So, (x + 3) is not the factor of x

^{3}+ 2x

^{2}+ 3x + 8

Similarly, put x = 2 in above equation

= (2)

^{3}+ 2(2)

^{2}+ 3(2) + 8

= 30 ≠ 0

So, (x – 2) is also not the factor of x

^{3}+ 2x

^{2}+ 3x + 8.

## Algebra | Free Online Test Details

Exam Practice Name | Algebra |

Number of Sets | 2 |

Number of Questions | 20 X 2 = 40 |

Type of Questions | MULTIPLE CHOICE |

Languages | English / Hindi |

#### Are you ready for this exam practice?

### Free Online Test Set-1

Instructions:

**Each question carry 1 mark, no negative marks.**

** Click the ‘Finish quiz’ button to Submit your answers.**

** Don’t refresh the page or click back button of browser. your work will be lost.**

** If you want to give the answer to the question later, click the Review Question button**.

### Free Online Test Set-2

Instructions:

**Each question carry 1 mark, no negative marks.**

** Click the ‘Finish quiz’ button to Submit your answers.**

** Don’t refresh the page or click back button of browser. your work will be lost.**

** If you want to give the answer to the question later, click the Review Question button**.

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