Geometry PYQ

The angle of elevation of the top of a tower from a certain point is 60°. If the observer moves 10 m away from the tower, the angle of elevation of the top of the tower decreases by 15°. The height of the tower is:

Find the value of x in the adjoining figure, if it is given that PR and QS are diameters of the circle. 

The area of the right-angled isosceles triangle having hypotenuse A cm is:

A cylindrical jar having a base of radius 15 cm, is filled with water up to a height of 20 cm. If a solid iron spherical ball of radius 10 cm is dropped in the jar to submerge completely in the water, then the increase in the level of water is:

If the distance between the points (2, -2) and (5, k) is 5 units, then one of the possible values of k shall be which of the following?

If two vertices of a triangle are (5, 4) and (-2, 4) and centroid is (5,6), then third vertex is:

If the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the vertices of a square then what is the length of the diagonal of square?

The area of a rectangle whose length is 5 more than twice its width is 75 square unit. What is the perimeter of the rectangle?

(A). The angles of depression of two ships from the top of a lighthouse are $60^\circ$ and $45^\circ$ towards the east. If the ships are $300$ meter apart, the height of the lighthouse is $150(3+\sqrt{3})$ $$\newline$ (B). If the surface area of a cube is $726 \text{ m}^2$, then its volume shall be $1331 \text{ m}^3$ $$\newline$ (C) If the ratio of diameters of two spheres is $3:5$, then the ratio of their surface area shall be $9:25$ $$\newline$$ Determine as to which of the statements given above are correct :

Match the given areas of various solid objects with their respective formulae: | List I | List II | |---|---| | Area of solid object | Formula | | (A) Lateral surface area of a cylinder | (I) $\pi r\sqrt{r^2 + h^2}$ | | (B) Total surface area of a hemisphere | (II) $4\pi r^2$ | | (C) Lateral surface area of a cone | (III) $2\pi rh$ | | (D) Surface area of a sphere | (IV) $3\pi r^2$ | (Where, r = radius of the solid shape (or base) and h = height of the solid shape.) Choose the correct answer from the options given below:

A solid, metallic right circular cone of radius 7 cm and height 3 cm is melted into three cubes. If the sides of two cubes are 3 cm and 5 cm, then the side of the third cube will be how much?

If area of a square is 44 cm², find the area of the circle formed by the same perimeter.

From A point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45°, respectively. If the bridge is at a height of 9 m from the surface of the river, then find the width of the river.

The area of a sector of a circle of radius 36 cm is $72\pi$ cm². Find the length of the corresponding arc of the sector.

If the areas of adjacent faces of a cuboid (rectangular prism) are in the ratio of 2: 3: 5 and its volume is 900 cm³, then the length of the longest side is

Consider the volumes of the following: (A) A cylinder with a radius of base 7 cm and height 10 cm. (B) A cone of radius 7 cm and height 9 cm. (C) A sphere of radius 6 cm. The volumes of these figures in decreasing order shall be: Choose the correct answer from the options given below:

In the adjoining figure, a rectangle has dimensions of 18 cm x 12 cm. The area of the parallelogram (shaded portion) is 

The angle of elevation of the top of a building from the foot of a tower is 30°. The angle of elevation of the top of the tower when seen from the foot of the building is 60°. If the tower is 60 m high, then the height of the building is:

If the slope of a line passing through the points A (2, 1) and B (3, y) is 4, then the value of y will be:

At the foot of a mountain, the elevation of the summit is 45°. After ascending 2 kilometers towards the mountain, at an incline of 30°, the elevation changes to 60°. Determine the height of the mountain?

Which of the following statements are true? (A) If the diagonal of a cube is $6\sqrt{3}$ cm, then its volume and surface area are equal. (B) If a rectangular block having dimensions 6 cm x 12 cm x 15 cm is cut into an exact number of equal cubes, then the least possible number of such cubes is 27. Choose the correct answer from the options given below:

From a point exactly midway between the foot of two towers P and Q, the angle of elevation of their tops are 30° and 60°, respectively. The ratio of the heights of tower P to that of Q is:

In what ratio are the volumes of a cylinder, a cone and a sphere, if each has the same diameter and the same height?

The diagonal of a square is $4\sqrt{2}$ cm. The diagonal of another square whose area is double that of the first square, is: