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QUESTION: 1

What should be subtracted to x^{2}+2x+2 to make it a perfect square?

Solution:

By subtracting 1 from 2,you will get 1 that makes the whole square of (x+1)=1

QUESTION: 2

If a + b = 12 and a^{2} + b^{2} = 60, the value of a^{3} + b^{3} is:

Solution:

**The correct option is Option C.**

**(a + b)² = a² + b² + 2ab**

**(12)² = 60 + 2ab**

**ab = (144 - 60)/2**

**ab = 84/2**

**ab = 42**

**Now, **

**(a + b)³ = a³ + b³ + 3ab(a+b)**

**(12)³ = a³ + b³ + (3 × 42 × 12)**

**a³ + b³ = 1728 - 1512**

**Therefore, a³ + b³ = 216**

QUESTION: 3

One of the factors of (16y^{2} – 1) + (1 -4y)^{2} is

Solution:

QUESTION: 4

(101)^{2} is equal to

Solution:

(100-1)²= (100)²+(1)²+2×100×1 10000+1+200 =10201

QUESTION: 5

Find the value of (9a + 10b)^{2}

Solution:

QUESTION: 6

a^{2} + b^{2} + c^{2} – ab – bc – ca equals:

Solution:

= ( a² + b² + c² - ab - bc - ca )

By multiplying it by 2 and dividing it by 2.

= 2 ( a² + b² + c² - ab - bc - ca ) ÷ 2

= ( 2a² + 2b² + 2c² - 2ab - 2bc - 2ca ) ÷ 2

= ( a² + a² + b² + b² + c² + c² - 2ab - 2bc - 2ca ) ÷ 2

= ( a² + b² - 2ab + b² + c² - 2bc + a² + c² - 2ca ) ÷ 2

= [ ( a - b )² + ( b - c )² + ( a - c )² ] ÷ 2

Now , whatever is the value of ( a - b ) , ( b - c ) and ( a - c ) but its square will be always positive, and 2 is also a positive number.

So, the sum of ( a - b )², ( b - c )² and ( a - c )² will be a positive number and if a positive number is divided by a positive number then the result is also a positive number.

So, it is a positive number,hence it can't be a negative number.

QUESTION: 7

Evaluate 98 x 102.

Solution:

(100+2)×(100-2). [(a+b)(a-b)=a^2-b^2

100^2 - 2^2

10000 - 4

9996

QUESTION: 8

a^{2} + b^{2} is equal to

Solution:

QUESTION: 9

Solve the equation x^{2} + 6x - 7 = 0

Solution:

QUESTION: 10

If *a* - *b* = 3 and *a*^{2} + *b*^{2} = 29, find the value of *ab.*

Solution:

2*ab* = (*a*^{2} + *b*^{2}) - (*a* - *b*)^{2}

= 29 - 9 = 20

⇒ *ab* = 10.

QUESTION: 11

What should be added to x^{2}+2x+0.5 to make it a perfect square?

Solution:

QUESTION: 12

The missing figure in the expression

(5p +…)^{2} = ….+ 40 pq + 16q^{2} are:

Solution:

QUESTION: 13

Factorise: x^{4}-1

Solution:

QUESTION: 14

Factorise: x^{2}– 81

Solution:

QUESTION: 15

What is the product of (x+a) and (x+b)?

Solution:

QUESTION: 16

The value of 101 x 99 is

Solution:

(100+1)×(100 - 1) it is the formula of a²-b² so, (100)² -(1)² = 10000-1 = 9999.

QUESTION: 17

If 2x + 3y = 12 and xy = 5, find the value of 4x^{2} + 9y^{2}.

Solution:

QUESTION: 18

Find the value of (109)^{2}

Solution:

QUESTION: 19

(99)^{2} is equivalent to

Solution:

QUESTION: 20

The expansion of (2a – 3b + 5c)^{2}.

Solution:

The expansion of (2a - 3b + 5c)² -: (2a)² + (- 3b)² + (5c)² + 2× 2a × (-3b) + 2× (-3b)× 5c + 2× 5c + 2a. = 4a² + 9b² + 25c² - 12ab - 30bc + 20ac

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