GEOMETRY
Note: Question numbers are numbers from the actual exam in the respective years mentioned below
- In a plane, line X is perpendicular to line Y and parallel to line Z; line U is perpendicular to both lines V and W; line X is perpendicular to line V. Which one of the following statements is correct?(2015)
(A) Z, U and W are parallel.
(B) X, V and Y are parallel.
(C) Z, V and U are all perpendicular to W.
(D) Y, V and W are parallel.
Answer-D
The correct statement is (d) Y, V, and W are parallel.
Let's analyze the given information step by step:
Line X is perpendicular to line Y and parallel to line Z: This means that X and Z are parallel, and X and Y are perpendicular.
Line U is perpendicular to both lines V and W: This implies that U is perpendicular to both V and W.
Line X is perpendicular to line V: This means that X and V are perpendicular.
From the above information, we can conclude the following:
X and Y are perpendicular.
X and V are perpendicular.
Since X is perpendicular to both Y and V, we can infer that Y and V are parallel to each other.
X is parallel to Z.
Now, since X and Z are parallel, and Y and V are parallel, and both X and Y are perpendicular to V, we can conclude that Y and Z are also parallel.
Finally, we know that U is perpendicular to both V and W. Since V and Y are parallel, and Y is perpendicular to U, we can conclude that U is also perpendicular to W.
Hence, the correct statement is (d) Y, V, and W are parallel.
- A piece of tin is in the form of a rectangle having length 12 cm and width 8 cm.
This is used to construct a closed cube. The side of the cube is : (2016)
(A) 2 cm
(B) 3 cm
(C) 4 cm
(D) 6 cm
Answer-C
The surface area of the rectangular piece of tin is 12 cm x 8 cm = 96 cm². This surface area will be covered by the surface area of the cube formed from it.
Let the side of the cube be "x" cm. Each face of the cube will be a square of side "x" cm. The total surface area of the cube is 6 times the area of one face of the cube, which is 6x².
We know that the surface area of the cube (6x²) is equal to the surface area of the rectangular piece of tin (96 cm²).
6x² = 96
x² = 16
x = 4 cm
Therefore, the side of the cube is 4 cm. So, the answer is (c) 4 cm.
- An agricultural field is in the form of a rectangle having length X1 meters and breadth X2 meters (X1 and X2 are variable).
If X1+ X2=40 meters, then the area of the agricultural field will not exceed which one of the following values?(2016)
(A) 400 sq m
(B) 300 sq m
(C) 200 sq m
(D) 80 sq m
Answer-A
To find the maximum possible area of the agricultural field, we need to consider the scenario where X1 and X2 are equal, which would give us a square shape. In this case, each side would be 20 meters (half of the total perimeter).
The area of a square is calculated by multiplying the length of one side by itself. Therefore, the maximum possible area of the agricultural field is:
Area = 20 meters × 20 meters = 400 square meters
So, the area of the agricultural field will not exceed 400 square meters.
Among the given options, the correct answer is (a) 400 sq m.
- AB is a vertical trunk of a huge tree with A being the point where the base of the trunk touches the ground. Due to a cyclone, the trunk has been broken at C which is at a height of 12 meters, broken part is partially attached to the vertical portion of the trunk at C. If the end of the broken part B touches the ground at D which is at a distance of 5 meters from A, then the original height of the trunk is(2016)
(A) 20 m
(B) 25 m
(C) 30 m
(D) 35 m
Answer-B
Let's solve the problem using the given information.
We have a right triangle ADC, where AD is the base of the trunk touching the ground, CD is the broken part at a height of 12 meters, and BD is the remaining part of the trunk.
We are given that BD = 5 meters and CD = 12 meters.
Using the Pythagorean theorem, we can find the length of AD:
AD^2 = BD^2 + CD^2
AD^2 = 5^2 + 12^2
AD^2 = 25 + 144
AD^2 = 169
Taking the square root of both sides, we find:
AD = √169
AD = 13 meters
So, the original height of the trunk, AB, is equal to AD + CD:
AB = AD + CD
AB = 13 + 12
AB = 25 meters
Therefore, the original height of the trunk is 25 meters.
The correct answer is (b) 25 m.
- A cylindrical overhead tank of radius 2 m and height 7 m is to be filled from an underground tank of size 5.5 m × 4 m × 6 m. How much portion of the underground tank is still filled with water after filling the overhead tank completely ?(2016)
(A) 1/3
(B) 1/2
(C) 1/4
(D) 1/6
Answer-A
To determine the portion of the underground tank that is still filled with water after filling the overhead tank completely, we need to calculate the volume of the cylindrical overhead tank and compare it with the volume of the underground tank.
Volume of the cylindrical overhead tank:
V1 = πr^2h
V1 = π(2^2)(7)
V1 = 28π
Volume of the underground tank:
V2 = lwh
V2 = 5.5 × 4 × 6
V2 = 132
The portion of the underground tank that is still filled with water can be calculated using the formula:
Portion filled = (V2 - V1) / V2
Portion filled = (132 - 28π) / 132
To simplify the expression, we can divide both the numerator and denominator by 4:
Portion filled = (33 - 7π) / 33
Since the value of π is approximately 3.14, we can substitute it into the equation:
Portion filled = (33 - 7(3.14)) / 33
Portion filled = (33 - 21.98) / 33
Portion filled = 11.02 / 33
Portion filled ≈ 0.334
So, the portion of the underground tank that is still filled with water after filling the overhead tank completely is approximately 0.334, which is equivalent to 1/3.The correct answer is (a) 1/3.
- If for a sample data Mean < Median < Mode then the distribution is(2017)
(A) symmetric
(B) skewed to the right
(C) neither symmetric nor skewed
(D) skewed to the left
Answer-D
Mean < Median < Mode for a sample data, it means that the distribution is negatively skewed or skewed to the left.
The mode represents the value that occurs most frequently in the data, and it is the peak of the distribution. The median is the middle value of the data when it is arranged in ascending or descending order. The mean is the average value of the data.
When the mean is less than the median, it indicates that the data has some low values that pull the mean down, and these low values are not present in the median. When the median is less than the mode, it indicates that the distribution is skewed to the left, with a tail extending towards the lower values.
Therefore, the correct answer is (d) skewed to the left.
- Two walls and a ceiling of a room meet at right angles at a point P. A fly is in the air 1m from one wall, 8 m from the other wall and 9 m from the point P. How many meters is the fly from the ceiling?(2017)
(A) 4
(B) 6
(C) 12
(D) 15
Answer-A
Let's consider the fly's position in the room.
We have two walls and a ceiling meeting at right angles at point P. Let's label the walls as Wall 1 and Wall 2.
Given that the fly is 1 meter from Wall 1, 8 meters from Wall 2, and 9 meters from point P, we can create a right-angled triangle with these distances as the sides.
Let's label the distance from the fly to the ceiling as x.
Using the Pythagorean theorem, we can set up the following equation:
(1)^2 + x^2 = (9)^2
1 + x^2 = 81
x^2 = 80
x = √80
x = √(16 × 5)
x = √16 × √5
x = 4√5
Therefore, the fly is located at a distance of 4√5 meters from the ceiling.The correct answer is (a) 4.
- Twelve equal squares are placed to fit in a rectangle of diagonal 5 cm. There are three rows containing four squares each. No gaps are left between adjacent squares. What is the area of each square?(2018)
(A) 5/7 sq cm
(B)7/5 sq cm
(C) 1 sq cm
(D) 25/12 sq cm
Answer-C
Let's solve the problem step by step.
We are given that twelve equal squares are placed to fit in a rectangle, with three rows containing four squares each. No gaps are left between adjacent squares.
Let's assume that the side length of each square is "x" cm.
In each row, four squares are placed side by side. Therefore, the length of the rectangle is 4x cm.
Since there are three rows, the height of the rectangle is 3x cm.
Using the Pythagorean theorem, we can relate the diagonal of the rectangle to its length and height:
Diagonal^2 = Length^2 + Height^2
(5)^2 = (4x)^2 + (3x)^2
25 = 16x^2 + 9x^2
25 = 25x^2
Dividing both sides by 25, we get:
1 = x^2
x = 1 cm
Therefore, the side length of each square is 1 cm.
The area of each square is given by the formula:
Area = side length^2
Area = (1 cm)^2
Area = 1 sq cm
Therefore, the area of each square is 1 sq cm.
The correct answer is (c) 1 sq cm.
- How many diagonals can be drawn by joining the vertices of an octagon?(2018)
(A) 20
(B) 24
(C) 28
(D) 64
Answer-A
To find the number of diagonals in an octagon, we need to consider that each vertex can be connected to every other non-adjacent vertex.
An octagon has 8 vertices, and each vertex can be connected to 5 other non-adjacent vertices.
However, when we count these diagonals, we count each diagonal twice (once for each vertex it is connected to), so we need to divide the total count by 2 to avoid double-counting.
Therefore, the number of diagonals in an octagon can be calculated as:
Number of diagonals = (8 * 5) / 2
Number of diagonals = 40 / 2
Number of diagonals = 20
So, the correct answer is (a) 20 diagonals.
- The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is (2019)
(A) 18
(B) 24
(C) 32
(D) 36
Answer-D
To count the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, we need to consider the possible combinations of these lines.
From each set of parallel lines, we can choose two lines to form the opposite sides of a parallelogram. Therefore, the number of parallelograms can be calculated by choosing 2 lines from each set of 4 lines.
For the first set of parallel lines, we can choose 2 lines out of 4 in (4 choose 2) ways, which is equal to 6.
Similarly, for the second set of parallel lines, we can also choose 2 lines out of 4 in 6 ways.
To find the total number of parallelograms, we multiply these two choices together:
Total number of parallelograms = 6 * 6 = 36
Therefore, the correct answer is (d) 36 parallelograms.
- If you have two straight sticks of length 7.5 feet and 3.25 feet, what is the minimum length can you measure?(2020)
(A) 0.05 foot
(B) 0.25 foot
(C) 1 foot
(D) 3.25 feet
Answer-B
To determine the minimum length that can be measured using two straight sticks of lengths 7.5 feet and 3.25 feet, we need to consider the concept of greatest common divisor (GCD) or highest common factor (HCF).
The minimum length that can be measured using the two sticks is equal to the GCD of their lengths.
The GCD of 7.5 feet and 3.25 feet can be found by converting both lengths to a common unit, such as inches.
Since 1 foot is equal to 12 inches, we have:
7.5 feet = 7.5 * 12 inches = 90 inches
3.25 feet = 3.25 * 12 inches = 39 inches
Now, let's calculate the GCD of 90 inches and 39 inches.
Using the Euclidean algorithm, we can find the GCD as follows:
90 = 2 * 39 + 12
39 = 3 * 12 + 3
12 = 4 * 3 + 0
The remainder becomes zero at this point, so the GCD is 3.
Therefore, the minimum length that can be measured using the two sticks is 3 inches.
Converting back to feet, 3 inches is equal to 3/12 = 0.25 feet.
Therefore, the correct answer is (b) 0.25 foot.
- If 1 litre of water weighs 1 kg, then how many cubic millimetres of water will weigh
0.1 gm?(2020)
(A) 1
(B) 10
(C) 100
(D) 1000
Answer-C
To solve this problem, we need to convert the given units of volume and weight into a common unit
Given that 1 liter of water weighs 1 kg, we can convert the weight of 0.1 g to kilograms:
0.1 g = 0.1/1000 kg = 0.0001 kg
Since the density of water is 1 kg/L, we can use this information to find the volume of water that corresponds to 0.0001 kg:
1 kg of water = 1 L
0.0001 kg of water = (0.0001 L) = (0.0001 * 1000 mL) = 0.1 mL
Therefore, 0.1 mL of water will weigh 0.1 g.
The correct answer is (c) 100.
- Consider the following statements:
- The minimum number of points of intersection of a square and a circle is 2.
- The maximum number of points of intersection of a square and a circle is 8.
Which of the above statements is/are correct?(2020)
(A) 1 only
(B) 2 only
(C) Both 1 and 2
(D) Neither 1 nor 2
Answer-B
Let's analyze each statement separately:
Statement 1: The minimum number of points of intersection of a square and a circle is 2.
This statement is incorrect. A square and a circle can have 0, 2, 4, or 8 points of intersection. If the circle lies entirely outside the square or entirely inside the square, there will be no points of intersection. However, if the circle intersects the square at any two points, there will be a minimum of 2 points of intersection.
Statement 2: The maximum number of points of intersection of a square and a circle is 8.
This statement is correct. The maximum number of points of intersection between a square and a circle is 8. This occurs when the circle passes through the four vertices of the square, and the center of the circle lies on the diagonals of the square. In this case, each side of the square intersects the circle at two points, resulting in a total of 8 points of intersection.
Based on the analysis above, the correct answer is (b) 2 only.
- A cubical vessel of side 1m is filled completely with water. How many millilitres of water is contained in it (neglect thickness of the vessel ?(2021)
(A) 1000
(B) 10000
(C) 100000
(D) 1000000
Answer-D
The volume of a cube of side 1 meter is given by V = s^3 = 1^3 = 1 cubic meter.
1 cubic meter is equal to 1,000,000 milliliters (ml).
Therefore, the vessel contains 1,000,000 ml of water.
So, the answer is (d) 1000000.
- Consider the following statements in respect of a rectangular sheet of length 20 cm and breadth 8 cm :
- It is possible to cut the sheet exactly into 4 square sheets.
- It is possible to cut the sheet into 10 triangular sheets of equal area.
Which of the above statements is/are correct?(2022)
(A) 1 only
(B) 2 only
(C) Both 1 and 2
(D) Neither 1 nor 2
Answer-C
1.To determine if it is possible to cut the rectangular sheet into 4 square sheets, we need to check if the area of the sheet can be divided equally into 4 parts.
The area of the rectangular sheet is 20 cm * 8 cm = 160 square cm.
It is possible to divide the sheet into 4 squares. For example, we can have one square with an area of 100 square cm, two squares with an area of 25 square cm each, and one square with an area of 10 square cm.
Therefore, statement 1 is correct.
2.It is possible to cut the sheet into 10 triangular sheets of equal area.
The total area of the rectangular sheet is 20*8 = 160 square cm. To cut it into 10 triangular sheets of equal area, each triangle should have an area of 16 square cm. We can draw a line from one corner of the rectangle to the opposite corner, dividing it into two triangles of equal area. Then, we can draw another line parallel to one of the sides of the rectangle, dividing each of the two triangles into two smaller triangles of equal area. We repeat this process for each of the resulting triangles until we have a total of 10 triangles, each with an area of 16 square cm. Therefore, statement 2 is correct.
So, both the statements are true.
