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- Online Algebra Exercise with Correct Answer Key and Solutions
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- Question 1 of 23
##### 1. Question

If 2x + 3y = 78 and 3x + 2y = 72, what is the value of x + y?

##### Hint

2x + 3y = 78 …(i)

3x + 2y = 72 …(ii)

From equations (i) and (ii),

x = 12, y = 18

∴ x + y = 12 + 18 = 30

- Question 2 of 23
##### 2. Question

If x + y =18 and xy = 72, what is the value of (x)² + (y)²

##### Hint

x + y = 18

⇒ (x + y)² = 18² = 324

⇒ x² + y² + 2xy = 324

⇒ x² + y² = 324 – 2xy

⇒ x² + y² = 324 – 2(72)

⇒ x² + y² = 324 – 144 = 180

- Question 3 of 23
##### 3. Question

If 3x + 5y = 44 and 10x–2y=16, what is the value of x?

##### Hint

3x + 5y = 44 …(i)

10x – 2y = 16 …(ii)

By equation (i) × 2 + equation (ii) × 5

we have,

6x + 10y + 50x – 10y = 88 + 80

or 56x = 168

∴ x = = 3

- Question 4 of 23
##### 4. Question

If 3

^{4x-2}= 729, then the value of X is##### Hint

729 = 9³ = 3⁶, Now 4X – 2 = 6 ⇒ X = 2.

- Question 5 of 23
##### 5. Question

If 2

^{x+4}– 2^{x+2}= 3 then x is equal to##### Hint

2

^{x + 4}– 2^{x + 2}= 3⇒ 2

^{x + 2}(2² – 1) = 3⇒ 2

^{x + 2}= 2⁰∴ x + 2 = 0 ⇒ x = –2

- Question 6 of 23
##### 6. Question

If 4

^{x}= 5^{y}= 20^{z}then z is (x, y, z ≠ 0)##### Hint

4 × 5 = 20z/x × 20z/y = 20z/x + z/y = 20

i.e. 1 = z/x + z/y

i.e. z = xy/(x + y)

- Question 7 of 23
##### 7. Question

If , then x is

##### Hint

We have,

⇒

⇒

⇒

⇒ x =

- Question 8 of 23
##### 8. Question

If x

^{abc}= x^{a}x^{b}x^{c}and x are all positive integers, then a² + b² + c² =?##### Hint

x

^{abc}= x^{a}x^{b}x^{c}⇒ x

^{abc}= x^{a + b + c}⇒ abc = a + b + c.

This happens for positive integer value of a, b, c, only when a, b, c take value of 1, 2 & 3.

Hence, a² + b² + c² = 1² + 2² + 3² = 1 + 4 + 9 = 14.

- Question 9 of 23
##### 9. Question

The pure form of the surd is

##### Hint

- Question 10 of 23
##### 10. Question

A student was asked to solve the fraction and his answer was ¼. By how much was his answer wrong?

##### Hint

∴ Required answer

- Question 11 of 23
##### 11. Question

A man plants 15376 apple trees in his garden and arranges them so that there are as many rows as there are apples trees in each row. The number of rows is:

##### Hint

∴ Number of rows = 124.

- Question 12 of 23
##### 12. Question

Number of prime factors in is

##### Hint

=

=

=

=

Hence, the number of prime factors = (2 + 3 + 2) = 7

- Question 13 of 23
##### 13. Question

The difference between two numbers is 3 and the difference of their squares is 63. Find the greater number.

##### Hint

Let the larger and smaller numbers be x and y respectively.

Then, x – y = 3 …(i)

and, x² – y² = 63

⇒ (x + y) (x – y) = 63

…(ii)

From equation (i) and (ii), x = 12

- Question 14 of 23
##### 14. Question

How much less is ⅘ of 1150 from ⅚ of 1248?

##### Hint

= 1040 – 920 = 120

- Question 15 of 23
##### 15. Question

The fourth root of 28 + 16 is:

##### Hint

=

= =

= =

= = + 1

- Question 16 of 23
##### 16. Question

Find the value of x

^{a}+ y^{b}, if a^{y}= 19683, where y is a multiple of a and b^{x}= 1024, where b is a factor of x. a, b, x and y being positive integers:##### Hint

a

^{y}= 19683⇒ 3⁹ = 27³ = (19683)¹ = 19683

As y is a multiple of a, a = 3, y = 9, b

^{x}= 1024⇒ 2¹⁰ = 4⁵ = 32² = (1024)¹ = 1024

As b is a factor of x, b = 2, x = 10

Now, x

^{a}+ y^{b}⇒ 10³ + 9² = 1000 + 81 = 1081.

- Question 17 of 23
##### 17. Question

What is the difference in the sum of squares and the difference of squares of n and a whenand the sum the of a and n is 12.

##### Hint

Given, a + n = 12 ⇒ n = 12 – a

∴ Given expression becomes

= 2

^{12 – a}. 3^{12 – a}By comparing, power of 3, we get a + 2 = 12 – a ⇒ a = 5 and n = 7

∴ Required answer = (a² + n²) – (n² – a²) = (49 + 25) – (49 – 25) = 50

- Question 18 of 23
##### 18. Question

If mx

^{m}– nx^{n}= 0, then what is the value of + terms of x^{n}?##### Hint

mx

^{m}= nx^{n}∴ x

^{m}=∴ Given

=

= = .

- Question 19 of 23
##### 19. Question

Three of the following four parts are exactly equal. Which parts is not equal to the other three?

##### Hint

Check all options one by one.

(a)

(b) + (15.2 × 4) = 100

(c) (15 × 8) – (5 × 4) = 100

(d) (76 × 1.5) – (5.5 × 2.6) = 99.7

- Question 20 of 23
##### 20. Question

Three of the following four parts are exactly equal. Which parts is not equal to the other three?

##### Hint

All others are equal to 100. But option (c) gives a value equal to 98.

- Question 21 of 23
##### 21. Question

Three of the following four parts are exactly equal. Which parts is not equal to the other three?

##### Hint

Others equal 5650.

- Question 22 of 23
##### 22. Question

Three of the following four parts are exactly equal. Which parts is not equal to the other three?

##### Hint

Except it, others are equal to 550.

- Question 23 of 23
##### 23. Question

Three of the following four parts are exactly equal. Which parts is not equal to the other three?

##### Hint

Other parts are equal to 300.