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:

If the length of a cuboid is increased by 20% and its breadth is decreased by 20%, then the volume of cuboid

If the points A(11, y), B(13, 5), C(14, 7) and D(12, 6) are the vertices of a parallelogram, taken in order, find the value of y:

Two men are on opposite side of tower. They measure the angles of elevation of the top of the tower as 30° and 45° respectively. If the height of the tower is 50 meters, find the distance between the two men.

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?

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?

Consider the surface area of the following: (A) A cube having each side as 6 cm. (B) A cylinder with a diameter of base 14 cm and length 80 cm. (C) A cone of diameter 14 cm and a slant height of 10 cm. (D) A sphere of radius 10.5 cm. The surface area of these in decreasing order are: Choose the correct answer from the options given below:

A 2 m tall boy is 30 m away from a tower. The angle of elevation of the top of the tower from his eye is 45°. What is the height of the tower?

(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:

Find the value of k for which the points A (-1,3), B (2, k) and C (5, -1) are collinear.

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?

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.

A wheel makes 2000 rounds when it covers a distance of 88 km. Find the radius of the wheel?

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

$V_1, V_2, V_3$ and $V_4$ are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4x cm respectively. The following statements regarding these volumes are given below. (A) $V_1 + V_2 + 2V_3 < V_4$ (B) $V_1 + 3V_2 > V_3 + V_4$ (C) $2(V_1 + V_3) + V_2 = V_4$ (D) $V_1 + 4V_2 + V_3 < V_4$ Which of these statements is/are correct? Choose the correct answer from the options given below:

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:

Find the coordinates of the point which divides the line segment joining the points (4, -3) and (8,5) in the ratio 3:1 internally?

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 slant height of a cone is decreased by 15 percent and the radius of its base is increased by 20 percent, then by what percent will its curved surface area change?

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:

How many meters of cloth 6 metre wide will be required to make a conical tent, the radius of whose base is 21m and height is 20m?

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: