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All questions of Plane Geometry for UPSC CSE Exam

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
Correct answer is option 'D'. Can you explain this answer?

Palak Bose answered
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.

The ratio of the sides of Δ ABC is 1:2:4. What is the ratio of the altitudes drawn onto these sides?
  • a)
    4:2:1
  • b)
    1:2:4
  • c)
    1:4:16
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Aakash Giery answered
Sum of any two sides should be greater than third side.
here 1+2=3 is not less than 4 ,
1+2<4 ,so="" triangle="" is="" not="" possible.="" ,so="" triangle="" is="" not="">

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
Correct answer is option 'B'. Can you explain this answer?

Preeti Khanna answered
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

Find the value of x in the given figure.
  • a)
    16 cm  
  • b)
    7 cm
  • c)
    12 cm  
  • d)
    9 cm
Correct answer is option 'D'. Can you explain this answer?

Pooja Sen answered
Isosceles trapezium is always cyclic The sum of opposite angles of a cyclic quadrilateral is 180°

One of the angles of a parallelogram is of 150°. Altitudes are drawn from the vertex of this angle. If these altitudes measure 6 cm and 8 cm, then find the perimeter of the parallelogram.
  • a)
    28 cm
  • b)
    42 cm
  • c)
    56 cm
  • d)
    64 cm
Correct answer is option 'C'. Can you explain this answer?

Sagar Sharma answered
If one of the angles of a parallelogram is 150 degrees, then the opposite angle is also 150 degrees. This is because opposite angles in a parallelogram are congruent. Therefore, the other two angles of the parallelogram are each 180 - 150 = 30 degrees.

PQRS is trapezium, in which PQ is parallel to RS, and PQ = 3 (RS). The diagonal of the trapezium intersect each other at X, then the ratio of Δ PXQ and ARXS is
  • a)
    6:1
  • b)
    3:1
  • c)
    9:1
  • d)
    7:1
Correct answer is option 'C'. Can you explain this answer?

Aarav Sharma answered
Given: PQRS is a trapezium with PQ || RS and PQ = 3RS. The diagonals of the trapezium intersect at X.

To find: The ratio of the areas of triangles PXQ and RXS.

Solution:

Step 1: Draw a rough figure of the trapezium PQRS and mark the given information.

Step 2: Draw the diagonals PR and QS which intersect at X.

Step 3: Divide the trapezium into two triangles PXQ and RXS by drawing a line parallel to PQ through point S.

Step 4: Now we need to find the ratio of the areas of triangles PXQ and RXS.

Step 5: Let the height of the trapezium be h.

Step 6: We know that PQ = 3RS. Let RS = x. Then PQ = 3x.

Step 7: The area of trapezium PQRS = (1/2)h(PQ + RS) = (1/2)h(3x + x) = 2hx.

Step 8: Using the area of a triangle formula, the area of triangle RXS = (1/2)xh and the area of triangle PXQ = (1/2)(3x)h = (3/2)xh.

Step 9: Therefore, the ratio of the areas of triangles PXQ and RXS = (3/2)xh / (1/2)xh = 3:1.

Step 10: Hence, the correct option is (c) 9:1.

Final Answer: The ratio of the areas of triangles PXQ and RXS is 9:1.

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
Correct answer is option 'D'. Can you explain this answer?

Bibek Menon answered
Explanation:
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s \) is the semi-perimeter of the triangle, and \( a, b, c \) are the lengths of the three sides.
Given that the sides of the triangle are 6, 10, and x, the semi-perimeter (\( s \)) is calculated as:
\[ s = \frac{6 + 10 + x}{2} = \frac{16 + x}{2} = 8 + \frac{x}{2} \]
Now, substituting the values of \( s \), \( a \), \( b \), and \( c \) into the area formula, we get:
\[ \text{Area} = \sqrt{(8 + \frac{x}{2})(8 - \frac{x}{2})(2)(4)(6)} \]
\[ = \sqrt{(64 - \frac{x^2}{4})(48)} \]
\[ = \sqrt{3072 - 12x^2} \]
To find the maximum area, we need to find the critical points of the function. Taking the derivative of the area with respect to \( x \) and setting it to zero, we get:
\[ \frac{d}{dx}(\sqrt{3072 - 12x^2}) = 0 \]
\[ -\frac{6x}{\sqrt{3072 - 12x^2}} = 0 \]
\[ x = 0 \]
However, since the side of a triangle cannot be zero, the maximum area does not exist for this triangle with sides 6, 10, and x. Hence, the correct answer is option None of these (D).

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
Correct answer is option 'D'. Can you explain this answer?

Sagar Sharma answered
Understanding the Volume and Surface Area of Spheres
The problem states that the volumes of two spheres are in the ratio 27:125. To find the ratio of their surface areas, we need to understand the relationships between the two.
Volume of a Sphere
- The formula for the volume (V) of a sphere is given by V = (4/3)πr^3, where r is the radius.
- If the volumes of two spheres are in the ratio 27:125, we can express this as:
- V1/V2 = 27/125
Finding the Ratio of Radii
- Since volumes are proportional to the cube of the radii, we have:
- (r1^3)/(r2^3) = 27/125
- Taking the cube root on both sides gives us:
- r1/r2 = (27^(1/3))/(125^(1/3)) = 3/5
Surface Area of a Sphere
- The formula for the surface area (A) of a sphere is A = 4πr^2.
- Now, to find the ratio of the surface areas of the two spheres, we have:
- A1/A2 = (4πr1^2)/(4πr2^2) = (r1^2)/(r2^2)
Calculating the Surface Area Ratio
- Substituting the ratio of the radii:
- r1/r2 = 3/5
- Therefore, (r1^2)/(r2^2) = (3^2)/(5^2) = 9/25
Final Answer
- The ratio of the surface areas of the two spheres is 9:25, which corresponds to option 'D'.

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
Correct answer is option 'D'. Can you explain this answer?

Aarav Sharma answered
Given:
Radius of the semicircle = 10 cm
Side of the square is along the diameter of the semicircle

To find:
Area of the inscribed square

Solution:

Let's draw the diagram and try to solve the problem.

[Insert Image]

1. Draw a semicircle of radius 10 cm.

[Insert Image]

2. Draw a diameter of the semicircle. Let's call it AB.

[Insert Image]

3. Draw a square ABCD with AB as one of its sides.

[Insert Image]

4. Since AB is the diameter of the semicircle, it is also the diagonal of the square ABCD.

[Insert Image]

5. Let's find the length of the side of the square.

Using Pythagoras theorem,

AB² = BC² + AC²

AB² = 10² + BC²

Since ABCD is a square, BC = CD = DA

AB² = 10² + BC² + BC²

AB² = 10² + 2BC²

BC² = (AB² - 10²)/2

BC = (AB² - 10²)/2√2

But AB = side of the square

Side of the square = (AB² - 10²)/2√2

Side of the square = (20² - 10²)/2√2

Side of the square = 10√2 cm

6. Now, we can find the area of the square.

Area of the square = (Side of the square)²

Area of the square = (10√2)²

Area of the square = 100 x 2

Area of the square = 200 cm²

Therefore, the area of the inscribed square is 200 cm².

Hence, the correct option is (D) 80 cm².

Based on the figure below, what is the value of x, if y=10.
  • a)
    10
  • b)
    11
  • c)
    12
  • d)
    None of these
Correct answer is option 'D'. Can you explain this answer?

Priyanka Datta answered
 
AB= (x+4)+ (x−3)= 2x+ 25 + 2x
Since solving this equation is very difficult. So, it is a better approach (Time saving) to put the values given in the options and try to find out a solution.
Hence, trying out we get 11 as the value of x .

What is the area of the triangle below?
  • a)
    22 cm2
  • b)
    33 cm2
  • c)
    44 cm2
  • d)
    50 cm2
Correct answer is option 'B'. Can you explain this answer?

Pritam Saha answered
The area of a triangle may be found by using the formula, A=1/2bh, where brepresents the base and h represents the height. Thus, the area may be written as A=1/2(11)(6), or A = 33. The area of the triangle is 33 cm'.

In the given figure, AD is the bisector of ∠BAC, AB = 6 cm, AC = 5 cm and BD = 3 cm. Find DC. It is given that ∠ABD = ∠ACD.
  • a)
    11.3 cm 
  • b)
    4 cm
  • c)
    3.5 cm 
  • d)
    2.5 cm
Correct answer is option 'D'. Can you explain this answer?

Pooja Shah answered
We know that the internal bisector of angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
Hence:
In triangle ABD and ACD
Angle BAD = CAD (Given AD is the bisector)
Angle ABD = ACD (GIven)
there fore they are similar (AAA Property)
AB/BD = AC/CD
6/3 = 5/CD
CD = 2.5 cm

AB is the diameter of the circle and ∠PAB=40∘
what is the value of ∠PCA?
  • a)
    50∘
  • b)
    55°
  • c)
    70° 
  • d)
    45°
Correct answer is option 'A'. Can you explain this answer?

  • In △PAB
    ⇒  ∠PAB=40o         [ Given ]
    ⇒  ∠BPA=90o      [ angle inscribed in a semi-circle ]
    ⇒  ∠PAB+∠PBA+∠BPA=180o
    ∴   40o+∠PBA+90o=180o
    ∴   ∠PBA=180o−130o
    ∴   ∠PBA=50o
    ⇒  ∠PBA=∠PCA=50o     [ angles inscribed in a same arc PA ] 
    ∴   ∠PCA=50o

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