It is given that, the curved surface area of the cylinder = 2\(\pi \)(y²+ 7y + 12).....(i)
We know that, the formula of curved surface area of the cylinder = 2\(\pi \)rh.....(ii)
where, r = Radius of the cylinder
h = Height of the cylinder
On comparing Eqs. (l) and (ii), we get
2\(\pi \)rh = 2\(\pi \)(y²+ 7y + 12)
r\(\times\)h=(y²+7y+12).....(iii)
But radius of the cylinder = R = (y + 3)
Putting this value in Eq.(iii), we get
(y+ 3)\(\times\)h =(y² + 7y +12)
\(h=\frac { \left( { y }^{ 2 }+7y+12 \right) }{ \left( y+3 \right) } \)
We have to factorise (y²+ 7y + 12)
Since, 4 \(\times\) 3 = 12 and 4 + 3 = 7
Putting this value in (y² + 7y + 12), we get
y² + (4 + 3)y+ 4 \(\times\) 3
= y²+4y+3y+4x³
= y(y + 4)+ 3(y + 4)=(y + 4)(y + 3)
\(h=\frac { \left( { y }^{ 2 }+7y+12 \right) }{ \left( y+3 \right) } =\frac { \left( y+4 \right) \left( y+3 \right) }{ \left( y+3 \right) } \)
= (y + 4) units
Hence, height of the cylinder = (y + 4) units.

Side of the square =(4x + 5) units
Area of the square = Side \(\times\) Side
= (4x+5)\(\times\)(4x+5)
= (4x + 5)² sq units
It is given that the area = 900 sq units
So, (4x + 5)² = 900
\(⇒\) (4x+5)²=30\(\times\)30
\(⇒\) (4x+ 5)² =(30)²
\(⇒\) (4x+5)=30
\(⇒\) 4x = 30 - 5 \(⇒\) 4x = 25
\(\Rightarrow\ x=\frac{25}{4}\)
Side of the square = 4x + 5
\(=4\times\frac{25}{4}+=25+5\)
= 30 units.

Number of chocolates Abhinav bought = (x + 4)
Cost of one chocolate = Rs.(x + 4)
Total amount paid by him = Number of chocolates \(\times\) Cost of one chocolate
= Rs. ( x + 4) x (x + 4) = Rs.(x + 4)²
= Rs.{(x² + (4)² + 2 \(\times\) x \(\times\) 4))
[using identity, (a + b)² = (a² + b² + 2ab)]
= Rs.(x² + 4\(\times\)4+ 8x)
= Rs.(x² + 16 + 8x)
It is given that, if x = 10 then,
the amount paid by him
= Rs. {(10)² + 16 + 8 \(\times\)10}
= Rs.(10 x10 + 16+ 8 \(\times\)10)
= Rs.(100+16+ 80)=Rs.196
(ii) The value depicted here is that Abhinav is helpful and friendly by nature to his friend.

\(\left( 9R+\frac { 37 }{ 4 } \right) \left( 9R-\frac { 35 }{ 4 } \right) \)

Area of square x⁴=18x² + 81 sq unit 
Area if x = 2 is 25 sq units.

a.b=19800

Number of observations = 4x²+9 

3\(\pi \) unit 

3860

10609