Galileo, in his book Two new sciences, stated that "for elevations which exceed or fall short of `45^(@)` by equal amounts, the ranges are equal. Prove this statement.

If the angle of projection of a projector with same initial velocity exceed or fall short of 45^(@) by equal amount alpha , then the ratio of horizontal rages is

State whether the statements are true (T) or false (F). The areas of any two faces of a cuboid are equal.

State whether the statements are true (T) or false (F). The areas of any two faces of a cube are equal.

Which of the following statements are true (T) and which are false (F): Side opposite to equal angles of a triangle may be unequal. Angle opposite to equal sides of a triangle are equal. The measure of each angle of an equilateral triangle is 60^0 If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles. The bisectors of two equal angles of a triangle are equal. If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles. The two altitudes corresponding to two equal sides of a triangle need not be equal. If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent. Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

State whether the statements are true (T) or false (F). Two cuboids with equal volumes will always have equal surface areas.

(When one of the Old Partners Gifts his Share). A and B are Partners sharing profits in the ration of which half share was gifted by A and the remaining share was taken by C equally from A and B. Calculate new profit-sharing ratio.

Which of the following statements are true and which are false; All squares are congruent.If two squares have equal areas,they are congruent.If two rectangles have equal areas, they are congruent.If two triangles have equal areas,they are congruent.

In two concentric circles prove that all chords of the outer circle which touch the inner circle are of equal length.

Consider the following statements: 1. If both the terms of a ratio are multiplied or divided by the same natural number, then the ratio remains unaltered. 2. A statements which states that two ratios are equivalent is called proportion. 3. If 4 quantities are in proportion, the product of extremes is not equal to the means. 4. The mean proportional between any two numbers is equal to the square root of their product. The wrong one statements is//are :

State whether the statements are true (T) or false (F). Two cylinders with equal volume will always have equal surface areas.