Geometry Test

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Geometry Test - 2

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Question 1
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A cyclic quadrilateral is such that two of its adjacent angles are divisible by 6 and 10 respectively. One of the remaining angles will necessarily be divisible by:

A
3

B
4

C
8

D
None of these

Solution

We know that the sum of the opposite angles of a cyclic quadrilateral is 180 degrees. Let the four angles be A, B, C, and D, with A and B being the angles divisible by 6 and 10, respectively.

Since A is divisible by 6 and B is divisible by 10, we know that A = 6m and B = 10n for some integers m and n.

Now, consider the opposite angles. Since the sum of opposite angles is 180 degrees, we have:

C = 180 - B = 180 - 10n
D = 180 - A = 180 - 6m

We want to find which of the given options the angles C or D are necessarily divisible by. Let 's examine each option:

1. 3: Since B is divisible by 10, it is possible that B is divisible by 5 but not 3 (e.g. B = 10). In this case, C = 180 - B would not be divisible by 3. Also, A is divisible by 6, so A is always divisible by 3, which means D = 180 - A would never be divisible by 3. So, this option is incorrect.

2. 4: Since A is divisible by 6, it is possible that A is divisible by 2 but not 4 (e.g. A = 6). In this case, D = 180 - A would not be divisible by 4. Also, B is divisible by 10, so B is always divisible by 2, which means C = 180 - B would never be divisible by 4. So, this option is also incorrect.

3. 8: If A is divisible by 6, then it can be even or odd multiples of 6 (e.g. A = 6, 12, 18, ...). D will be 180 - A, which means D can be both even and odd (e.g. D = 180 - 6 = 174, D = 180 - 12 = 168, D = 180 - 18 = 162, ...). Since D can be both even and odd, it is not necessarily divisible by 8. Similarly, C can also be both even and odd, so it is not necessarily divisible by 8. Thus, this option is also incorrect.

4. None of these: Since none of the previous options work, the correct answer is None of these.

So, the correct answer is option 4: None of these.
Question 2
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The volume of two spheres are in the ratio 27 : 125. The ratio of their surface area is?

A
25 : 9

B
27 : 11

C
11 : 27

D
9 : 25

Solution

Option 4 : 9 : 25
Given
Ratio of volume of two spheres = 27 : 125
Formula used
surface area of sphere =4 πr²
Volume of sphere = (4/3)πr³
Where r is radius respectively
Calculation
[(4/3)πR3/(4/3)πr3] = 27/125
Where R and r are radius of two sphere
⇒R³ /r³ = 27/125
⇒R = 3r/5
Surface area of 1st sphere/Surface area of 2nd sphere =[4 π(3r/5)2]/4 πr2
⇒9 : 25
∴Ratio of surface area of two spheres is 9 : 25.
Question 3
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Sides of a triangle are 6, 10 and x for what value of x is the area of the △the maximum?

A
8 cm²

B
9 cm²

C
12 cm²

D
None of these

Solution
Question 4
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A

B

C

D

Solution
Question 5
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A square is inscribed in a semi circle of radius 10 cm. What is the area of the inscribed square? (Given that the side of the square is along the diameter of the semicircle.)

A
70 cm²

B
50 cm²

C
25 cm²

D
80 cm²

Solution
Question 6
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Two circles of an equal radii are drawn, without any overlap, in a semicircle of radius 2 cm. If these are the largest possible circles that the semicircle can accommodate, what is the radius (in cm) of each of the circles?

A
0.828

B
0.414

C
0.172

D
0.586

Solution
Question 7
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PQRS is a Trapezzium, in which PQ is Parralel to RS, and PQ = 3 (RS) . The diagnol of the Trapezzium intersect each other at X, then the ratio of, ar ( ∆PXQ) : ar ( ∆RXS) is?

A
6:1

B
3:1

C
9:1

D
7:1

Solution

In ∆PXQ and ∆RXS
=>angle P = angle R
    angle Q = angle S
:- ∆PXQ ~ ∆RXS ( AA similarity rule)
ar ( ∆PXQ) / ar ( ∆RXS) = ( PQ / RS) ^ 2
= ( 3 / 1 ) ^ 2
=   9 / 1
Therefore, ar ( ∆PXQ) : ar      ( ∆RXS)

= 9:1
Question 8
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Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?

A
1/3

B
1/2

C
2/3

D
5/6

Solution
Question 9
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A pond 100 m in diameter is surrounded by a circular grass walk-way 2 m wide. How many square metres of grass is the on the walk-way?

A
98 π

B
100 π

C
204 π

D
202 π

Solution
Question 10
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The dimensions of a rectangular box are in the ratio of 1:2:4 and the difference between the costs of covering it with the cloth and a sheet at the rate of Rs 20 and Rs 20.5 per sq m respectively is Rs 126. Find the dimensions of the box.

A
2 m, 6 m, 9 m

B
6 m, 12 m, 24 m

C
1 m, 2 m, 4 m

D
None of these

Solution
Question 11
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The ratio of the area of a square inscribed in a semicircle to that of the area of a square inscribed in the circle of the same radius is

A
2:1

B
2:3

C
2:5

D
2 : 7

Solution
Question 12
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The ratio of the area of a square to that of the square drawn on the its the diagonal is

A
1:4

B
2:1

C
1:2

D
1:3

Solution
Question 13
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What is the area of the triangle in which two of its medians 9 cm and 12 cm long intersect at the right angles?

A
72

B
60

C
56

D
48

Solution
Question 14
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Four horses are tethered at four comers of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m2 at the centre. Find the ungrazed area.

A
42 m²

B
22 m²

C
84 m²

D
168 m²

Solution

Total area of plot = 14 * 14 = 196m²
Horses can graze in quarter circle of radius = 7m
Grazed area = 4 * (pie r² )/4 = 154 m²
Area of plot when horses cannot reach = (196 - 154) = 42m²
Ungrazed area = 42 - 20 = 22m²
Question 15
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Two sides of a triangle are 4 and 5. Then, for the area of the triangle, which one of the following bounds is the sharpest?

A
<10.

B
≤ 10.

C
<8.

D
>5.

Solution

Let AB = 4 and BC = 5 and AB is perpendicular to BC
then Area = 1/2 AB . AC = 1/2 . 4.5 = 10