Geometry- 2 - UPSC Free MCQ Test with solutions

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.

 Option 4 : 9 : 25

Given

Ratio of volume of two spheres = 27 : 125

Formula used

surface area of sphere =4πr²

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 =[4π(3r/5)2]/4πr2

⇒ 9 : 25

∴ Ratio of surface area of two spheres is 9 : 25.

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

Step 1: Understand the Problem
In a triangle ABC, medians are line segments from a vertex to the midpoint of the opposite side. Here, two medians (say from vertices A and B) are 9 cm and 12 cm long and intersect at the centroid G at 90°. The centroid divides each median in a 2:1 ratio (vertex to centroid is twice the length from centroid to midpoint).

Step 2: Assign the Medians and Use the Centroid Property
Assume:

• Median AD (from A to midpoint D of BC) = 9 cm.

• Median BE (from B to midpoint E of CA) = 12 cm.

At the centroid G:

• AG : GD = 2 : 1, so AG = (2/3) × 9 = 6 cm, GD = 3 cm.

• BG : GE = 2 : 1, so BG = (2/3) × 12 = 8 cm, GE = 4 cm.

Since the medians intersect at G at 90°, triangle AGB has a right angle at G.

Step 3: Use the Perpendicular Medians Property
A key geometric result states that if two medians of a triangle intersect at right angles at the centroid, the area of the triangle can be found using the formula:
Area of triangle ABC = (2/3) × m1 × m2,
where m1 and m2 are the lengths of the perpendicular medians.

Substitute the given lengths:
Area = (2/3) × 9 × 12 = (2/3) × 108 = 72 cm².

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

Let AB = 4 and BC = 5 and AB is perpendicular to BC

then Area = 1/2 AB . AC = 1/2 . 4.5 = 10