Area, General Mathematics

Question: The area of a circle inscircle in an equilateral triangle of side 24 cm, is :
A
`18 π cm^2`

B
`24 π cm^2`

C
`36 π cm^2`

D
`48 π cm^2`

Note: `1/2 xx 24 xx h = sqrt(3)/4 xx 24 xx 24 or h` `= 12sqrt(3)` `:. 3r = 12sqrt(3) or r = 4sqrt(3)` :. Area of circle` = π xx (4sqrt(3))^2` `= (48 π) cm^2`.
1. Report
Question: A circular wire of radius 42 cm is cut and bent into the from of a rectangle whose sides are in the ratio of 6 : 5. The smaller side of the rectangle is :
A
`30 cm`

B
`60 cm`

C
`72 cm`

D
`132 cm`

Note: Length of wire = circumference of circle of radius 42 cm `= (2 xx 22/7 xx 42) cm.` `= 264 cm` :. perimeter of rectangle = 264 cm. Let, length = 6x cm & breadth = 5x cm. `:. 2 (6x + 5x) = 264 or x = 12.` :. Smaller side = 60 cm
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Question: Four circular cardboard pieces, each of radius 7 cm are placed in such a way that each piece touches two other pieces, The area of the space enclosed by the four pieces is :
A
`21 cm^2`

B
`42 cm^2`

C
`84 cm^2`

D
`168 cm^2`

Note: Required area` = (14 xx 14 - 4 xx 1/4 xx 22/7 xx 7 xx 7) cm^2` `= (196 - 154) cm^2` `= 42 cm^2`
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Question: Four house are tethered at four corners of a square plot of side 63 metres so that they just can not reach one another. The area left ungrazed is :
A
`675.5 m^2`

B
`780.6 m^2`

C
`785.8 m^2`

D
`850.5 m^2`

Note: Required area` = (63 xx 63 - 4 xx 1/4 xx 22/7 xx 63/2 xx 63/2) m^2` `= 850.5 m^2`
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Question: `1/(1 + x^(b - a) + x^(c - a)) + 1/(1 + x^(a - b) + x^(c - b)) + 1/(1 + x^(b - c) + x^(a - c))=?`
A
`x^(a - b - c)`

B
`1`

C
`0`

D
`3`

Note: Given Exp. `= 1/(1 + x^b/x^a + x^c/x^a) + 1/(1 + x^a/x^b + x^c/x^b) + 1/(1 + x^b/x^c + x^a/x^c) `= (x^a/(x^a + x^b + x^c) + x^b/(x^b + x^a + x^c) + x^c/x^c + x^b + x^a)` `= ((x^a + x^b + x^c)/(x^a + x^b + x^c)) = 1`
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Question: `(x^b/x^c)^((b + c + a)) .(x^c/x^a)^((c + a + b)) .(x^a/x^b)^((a + b + c)) = ?`
A
`x^abc`

B
`x^(a + b + c)`

C
`x^(ab + bc + ca))

D
1

Note: Given Exp.` x = x^(b - c)(b + c - a) .x^(c - a)(c + a - b) .x^(a - b)(a + b - c)` `= x^((b^2 - c^2) + (c^2 - a^2) + (a^2 - b^2)) . x^(-a (b - c) - b (c - a) - c(a - b))` `= x^0 .x^0 = 1.
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Question: If pqr = 1, then` (1/(1 + p + q^-1) + 1/(1 + q + r^-1) + 1/(1 + r + p^-1) )= ?`
A
0

B
`1/pq`

C
pq

D
1

Note: Not available
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Question: `(x^a/x^b)^(a + b) xx (x^b/x^c)^(b + c) xx (x^c/x^a)^(c + a) = ?`
A
0

B
`x^abc`

C
`x^a + b + c`

D
1

Note: Not available
1. Report

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Question: The area of a circle inscircle in an equilateral triangle of side 24 cm, is :

Question: A circular wire of radius 42 cm is cut and bent into the from of a rectangle whose sides are in the ratio of 6 : 5. The smaller side of the rectangle is :

Question: Four circular cardboard pieces, each of radius 7 cm are placed in such a way that each piece touches two other pieces, The area of the space enclosed by the four pieces is :

Question: Four house are tethered at four corners of a square plot of side 63 metres so that they just can not reach one another. The area left ungrazed is :

Question: `1/(1 + x^(b - a) + x^(c - a)) + 1/(1 + x^(a - b) + x^(c - b)) + 1/(1 + x^(b - c) + x^(a - c))=?`

Question: `(x^b/x^c)^((b + c + a)) .(x^c/x^a)^((c + a + b)) .(x^a/x^b)^((a + b + c)) = ?`

Question: If pqr = 1, then` (1/(1 + p + q^-1) + 1/(1 + q + r^-1) + 1/(1 + r + p^-1) )= ?`

Question: `(x^a/x^b)^(a + b) xx (x^b/x^c)^(b + c) xx (x^c/x^a)^(c + a) = ?`