number, divisibility

Prove that if x and y are both odd positive integers, 

then x² + y² is even but not divisible by 4.

number, divisibility

Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q

numbers, divisibility

Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

number, divisibility

For any positive integer n, prove that n³ – n is divisible by 6.

numbers , divisibility

If n is an odd integer, then show that n² – 1 is divisible by 8.

Irrational numbers

Prove that √2 + √5 is irrational.

Division , HCF and LCM

Use Euclid’s division algorithm to find the HCF. Also find which of the following pairs of numbers are co-prime: 

NCERT Exemplary Question

(i)                 231, 396

(ii)               847, 2160

ICSE Mock TEST 2014

IPM/ MTSE test paper STD IV

trigonometry NCERT

A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and β, respectively. Prove that the height of the other house is h (1 + tan α cot β) metres

trigonometry NCERT --- ↑↑↑

The angle of elevation of the top of a vertical tower from a point on the ground is 60^o. From another point 10 m vertically above the first, its angle of elevation is 45^o. Find the height of the tower.

trigonometry NCERT ↑↑↑↑

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal.

Show that:

trigonometry - NCERT ↑↑

The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower.

trigonometry -- NCERT ↑↑↑↑

 

trigonometry NCERT ↑↑↑↑

trigonometry -- NCERT ↑↑

Simplify:

(1+ tan²A)(1-sinA)(1+ sinA)

trigonometry - NCERT ↑↑

Find the value of A:

2sin²A – cos²A = 2

trigonometry ---- NCERT ↑↑↑

trigonometry - NCERT ↑↑↑

Show that:

tan⁴A +tan²A = sec⁴A –sec²A

NCERT

trigonometry NCERT ↑↑↑

If √3 tan A = 1

Find the value of sin²A - cos²A

NCERT