Olympiad Test: Mensuration - CTET & State TET MCQ

Area of trapezium= 1/2*d*(a+b) sq. Units.

 

Where d= perpendicular distance between parellel sides which are a & b respectively, 25 &13 cm.

 

25–13=12, 12/2= 6 cm is base of RT. Angled ∆.

 

Non parallel sides=10cm, this is hypotenuese of this ∆.

 

Find ‘d ‘ to solve.

 

d=√{10²–6²}=8 cm.

 

Area of trapezium= 1/2*8*(25+13)=152 cm²

Quad ABCD is made of two triangles ADC and ABC 

so, area(ABCD) = area(ABC) + area(ADC)

= 4 x 12/2 + 3.5 x12/2

= 24 + 21

= 45 cm²

Problem:
Find the altitude of a trapezium, the sum of the lengths of whose bases is 6.5 cm and whose area is 26 cm².

To find the altitude of the trapezium, we can use the formula for the area of a trapezium:
Area = (1/2) * (sum of bases) * (altitude)
Given that the sum of the lengths of the bases is 6.5 cm and the area is 26 cm², we can set up the equation:
26 = (1/2) * 6.5 * altitude
Simplifying the equation:
52 = 6.5 * altitude
Dividing both sides by 6.5:
altitude = 52 / 6.5
altitude = 8 cm
The altitude of the trapezium is 8 cm.
Answer: B. 8 cm

Given:
- Capacity of the tank = 5632 m^3
- Diameter of the base = 16 m
We need to find the depth of the tank.
To find the depth, we can use the formula for the volume of a cylinder:
Volume = π * r^2 * h
where:
- π is a constant approximately equal to 3.14
- r is the radius of the base
- h is the height (or depth) of the cylinder
Step 1: Find the radius of the base:
The diameter of the base is given as 16 m. We can use the formula for the radius of a circle to find the radius:
Radius = Diameter / 2 = 16 / 2 = 8 m
Step 2: Substitute the values into the volume formula:
5632 = 3.14 * 8^2 * h
Simplifying the equation:
5632 = 3.14 * 64 * h
5632 = 200.96 * h
Step 3: Solve for h:
h = 5632 / 200.96
h ≈ 28 m
Therefore, the depth of the cylindrical tank is approximately 28 m.
Answer: D. 28 m

Area of rhombus =1/2 ×d1×d2
Length of the one diagonal= 20 cm
Length of the other diagonal =16 cm
Area of the rhombus =1/2×20×16
= 1×10×16
= 160
The area of the rhombus is 160cm^2.

Total Surface Area of Cuboid: 2(lb + bh + hl)
L = length
B = Breadth
H = Height

Length of the cuboid = 8 cm

Breadth of the cuboid = 6 cm

Height of the cuboid = 3.5 cm

Volume of the cuboid = length × breadth × height

= 8 x 6 x 3.5 = 168cm³

Therefore,volume of  the cuboid = 168cm³

Given:
Dimensions of the godown: 60 m × 40 m × 30 m
Volume of one box: 0.8 m³
To find:
Number of cuboidal boxes that can be stored in the godown
Formula:
Volume of a cuboid = length × width × height
Solution Steps:
1. Calculate the volume of the godown using the given dimensions:
Volume of godown = 60 m × 40 m × 30 m = 72000 m³
2. Divide the volume of the godown by the volume of one box to find the number of boxes that can be stored:
Number of boxes = Volume of godown / Volume of one box
Number of boxes = 72000 m³ / 0.8 m³ = 90000
Answer:
The number of cuboidal boxes that can be stored in the godown is 90000.
So, the correct answer is option D.

Perimeter of a Trapezium=sum of all sides
52= sum of parallel sides +20
Sum of parallel sides = 32
Area of Trapezium=
1/2(Sum of parallel sides) x h
                              =1/2(32)*8 = 128cm²

Given:
- Area of the trapezium = 384 cm^2
- Ratio of the parallel sides = 3:5
- Distance between the parallel sides = 12 cm
Let the lengths of the parallel sides be 3x and 5x, where x is a constant.
Step 1: Write the formula for the area of a trapezium:
Area = (sum of parallel sides) * (distance between them) / 2
Step 2: Substitute the given values into the formula:
384 = (3x + 5x) * 12 / 2
Step 3: Simplify the equation:
384 = 8x * 12 / 2
384 = 96x
Step 4: Solve for x:
x = 384 / 96
x = 4
Step 5: Find the lengths of the parallel sides:
3x = 3 * 4 = 12 cm
5x = 5 * 4 = 20 cm
Therefore, the lengths of the parallel sides are 12 cm and 20 cm.
Hence, the correct answer is option C: 12 cm and 20 cm.

Area of Rectangle ABCD = 110*150 = 16500cm²
Area of rectangle EFGH = (150-10)(110-10) = 14000cm²
Area of path = 16500-14000 = 2500cm²

Given:
- The area of the rhombus is 28 cm^2.
- One of its diagonals is 4 cm.
To find:
- The perimeter of the rhombus.
Step 1: Find the length of the other diagonal.
- We know that the area of a rhombus can be calculated using the formula: Area = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals.
- Since the area is given as 28 cm^2 and one diagonal is given as 4 cm, we can rearrange the formula to find the length of the other diagonal.
- 28 = (4 * d2) / 2
- 56 = 4 * d2
- d2 = 56 / 4
- d2 = 14 cm
Step 2: Find the side length of the rhombus.
- The side length of a rhombus can be found using the formula: s = √(Area / sinθ), where θ is the angle between the diagonals.
- Since the diagonals of a rhombus are perpendicular bisectors of each other, the angle between them is 90 degrees.
- sin90 = 1, so the formula simplifies to s = √Area.
- Therefore, the side length of the rhombus is √28 cm.
Step 3: Find the perimeter of the rhombus.
- The perimeter of a rhombus is simply 4 times the length of its sides.
- Therefore, the perimeter of the rhombus is 4 * √28 cm.
Step 4: Calculate the perimeter.
- Substituting the value of √28, we get the perimeter as 4 * √28 cm.
- Simplifying further, the perimeter is equal to 4 * 2√7 cm.
- This gives us a final answer of 8√7 cm.
Therefore, the correct answer is A: 8√7 cm.

To find the side of a cube given its surface area, we can use the formula:
Surface Area = 6 * (side)^2
Given that the surface area is 2400 cm^2, we can set up the equation:
2400 = 6 * (side)^2
Now, let's solve for the side:
1. Divide both sides of the equation by 6:
2400/6 = (side)^2
400 = (side)^2

2. Take the square root of both sides to solve for the side:
√400 = √(side)^2
20 = side

Therefore, the side of the cube is 20 cm.
So, the correct answer is option B: 20 cm.

Area of the base = Product of length and breadth

Volume of cuboid = Length x breadth x Height
= Area of its base x Height
275 = 25 x Height
Height = 275 / 25 cm
=11 cm

Volume of cuboid = base area × height
490 = 35 × height
height = 14cm.

To find the surface area of a cube, we need to calculate the area of each face and then sum them up.
Step 1: Identify the formula for the surface area of a cube:
The formula for the surface area of a cube is given by 6 times the square of the length of one side (s):
Surface Area = 6s^2
Step 2: Break down the formula and calculate:
- Each face of a cube has the same length, so we can substitute s for l in the formula.
- Calculate the square of the length:
l^2
- Multiply the square of the length by 6:
6 * l^2
Step 3: Write the final answer:
The surface area of a cube is 6l^2.
Answer: A: 6 l^2
Note: The surface area of a cube is always equal to 6 times the square of one side length.

Circumference of semi circle = 
perimeter = sum of two sides + circumference = 4 + 5.5 = 9.5cm