(English Medium) ICSE Class 9 - CISCE Question Bank Solutions

Given log₂ x = m. Express 2^{m - 3}  in terms of x.

Given log₅ y = n. Express 5^{3n + 2} in terms of y.

If log₂x = a  and log₃ y = a, write 72^a in terms of x and y.

Solve for x:

log(x - 1) + log (x + 1) = log₂1

If log (x² - 21) = 2, show that x = ± 11.

In triangle ABC, D is a point in AB such that AC = CD = DB. If ∠B = 28°, find the angle ACD.

ABC is an equilateral triangle. Its side BC is produced up to point E such that C is mid-point of BE. Calculate the measure of angles ACE and AEC.

In triangle ABC; angle ABC = 90^o and P is a point on AC such that ∠PBC = ∠PCB.
Show that: PA = PB.

In triangle ABC; ∠A = 60^o, ∠C = 40^o, and the bisector of angle ABC meets side AC at point P. Show that BP = CP.

Using the information given of the following figure, find the values of a and b. [Given: CE = AC] 

In triangle ABC, AB = AC; BE ⊥ AC and CF ⊥ AB.

Prove that:

1. BE = CF
2. AF = AE

Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.

The sides AB and AC of a triangle ABC are produced; and the bisectors of the external angles at B and C meet at P. Prove that if AB > AC, then PC > PB.

Prove that the straight line joining the vertex of an isosceles triangle to any point in the base is smaller than either of the equal sides of the triangle.

In the following figure, ABC is an equilateral triangle and P is any point in AC;
prove that: BP > PA

From the following figure;

prove that:
(i) AB > BD
(ii) AC > CD
(iii) AB + AC > BC.

Adjacent sides of a parallelogram are equal and one of the diagonals is equal to any one of the sides of this parallelogram. Show that its diagonals are in ratio √3:1.

A ladder 13 m long rests against a vertical wall. If the foot of the ladder is 5 m from the foot of the wall, find the distance of the other end of the ladder from the ground.

A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.

In ΔABC,  Find the sides of the triangle, if:

1. AB =  ( x - 3 ) cm, BC = ( x + 4 ) cm and AC = ( x + 6 ) cm
2. AB = x cm, BC = ( 4x + 4 ) cm and AC = ( 4x + 5) cm