Factor the given expression completely.

• Step 1:We have given area of one face of the cube.

Area of face = 81

• Step 2: For total surface area, find out the product of the square of side length by 6.

Surface area =  6 (area of face)
                      =  6 (81)
                      =   486

• Step 4: Therefore, the surface of the given cube is 486.
• Therefore, the correct answer is option A) 486 .

Hint:- Slant height L = 
where h is height
r is radius of base of cone
Solution:- We have given the dimensions of funnel in the shape of cone
Radius, r = 3 cm
Height, h = 4 cm
Let us find the slant height
L = 
L = 
= 
= 
L = 5 cm
Therefore, the correct option is b) 5 cm.

Ans :-   is the required product of binomials.
Given , 
Write
As  
Here a = x and b = 

∴  is the required product of binomials.

We have given the dimensions of triangle having sides equal to 7cm, 24cm and 25cm
It is revolved about 24 cm side
Therefore, the cone formed will have dimensions as
Height , h = 24 cm
Radius , r = 7 cm
So, the volume of cone = ()πr²h

= ()()(7 x 7)(24)

= 22 x 7 x 8

= 1232 cm³
Therefore, the correct option is b) 1232 cm³

Hint :- factorize the given expression by taking out common elements and also using the required formulas .
Ans:- 4x(x+3)(x+3) is the factorized form of the given expression.
Explanation :-
Given, 
Taking out common factor 4 we get  
Taking out common element x we get  
Splitting out 6x into 3x+3x we get 
Taking out x+3 common out we get 
As we get 
∴ 4x (x + 3)(x + 3) is the factorized form of the given expression.

• Step 1:We have given area of one face of the cube.

Area of face = 81

• Step 2: For total surface area, find out the product of the square of side length by 6.

Surface area =  6 (area of face)
                      =  6 (81)
                      =   486

• Step 4: Therefore, the surface of the given cube is 486.
• Therefore, the correct answer is option A) 486 .

• Step 1:We have given the length of the side of the cube.

Side = 4 mm

• Step 2: Find the square of the length of the side of the cube.

(Side)²= (4)²= 16 mm²

• Step 3: For total surface area, find out the product of the square of side length by 6.

Surface area = 6(side)²

= 6(16)

= 96 mm²

• Step 4: Therefore, the surface area of the given cube is 96 mm².
• Therefore, the correct answer is option B) 96 mm².

Solution :- Here is the activity for finding the volume of cone
Let us take a cylinder  of height "h", base radius "r", and take 3 cones of height "h". Fill the cones with water and empty out one cone at a time


Each cone fills the cylinder to one-third quantity. Hence, such three cones will fill the cylinder. Thus, the volume of a cone is one-third of the volume of the cylinder.
Volume of cone = (1/3) × Volume of cylinder
= () × πr²h
= ()πr²h
So the ratio of volume of cone to the volume of cylinder is 1:3
Therefore , the correct option is b) 1:3

Hint:- We have given three different dimensions ,
Total Surface Area(TSA) of cuboid = 2[ lb + bh + hl ]
where
l → length of the cuboid
b → breadth of the cuboid
h → height of the cuboid
Solution:-
We will calculate the total surface area of a cuboid by using the following formula:
The dimensions of the given cuboid:
Length, l = 4 cm
Breadth, b = 5 cm
Height, h = 6 cm
By using the above formula of the total surface area of the cuboid, we get
The total surface area of the given cuboid is,
= 
= 
= 
= 
= 148 cm²
Thus, the total surface area of a cuboid of dimensions 6 cm, 4 cm & 5 cm is 148 cm².
The correct option is C)148 cm².
Note:- In some examples it may be given the surface area and any two dimensions , then we have to adjust the formula such a that we will be able to find out the required value . For eg- If area , length and breadth is given the formula for height becomes

ANS :-  (3x-10)(3x+10) is the factorized form of the given expression.
Given , 
Using square root and squaring on 9 and 100 . we get 
using the formula 
Here 
Then 
∴ (3x - 10) (3x + 10) is the factorized form of the given expression.

Ans :-  is the required product of binomials.
Given , 
Write then we get 
As 
Here a = x and b =  
x² + x + = (x +)² = (x + ) (x +)
∴ (x+)(x+) is the required product of binomials.

ANS :-  (x-8)(x+8) is the factorized form of the given expression.
Given 
Using square root and squaring on 64 . we get 
using the formula 
Here 

Then
∴ ( x - 8) (x + 8) is the factorized form of the given expression.

We have given that
Slanted height L = 5 cm
Height, h = 4cm
From these given values we can find the radius of base of cone
We know that

h² + r² = L²

r² = L² - h²

= 5² – 4²

= 25 – 16

r² = 9

r= 3
So, the volume of cone = ()πr²h

= ()(3.14)(3 x 3)(4)

= 3.14 x 3 x 4

= 37.68
Therefore, the correct option is d) 37.6

Ans :-  is the required product of binomials.
Given , 
Write  then we get 
As 
Here a = p and b = 

∴  is the required product of binomials.

Hint:- Volume of a cone = (1/3)πr²h
Solution :- We have given the dimensions of a cone
Radius , r = 21 cm
Height , h = 5 cm

            We have to find the volume of the given cone

We know that

Volume of a cone = (1/3)πr²h

= ()()(21 x 21) (5)

= ()(22)(3 x 21) (5)

= (22 x 21 x 5)

= 2310 cm³
Therefore correct option is a) 2310cm³