Contents

- 1 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions
- 1.1 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 1:
- 1.2 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 2 (a):
- 1.3 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 2 (b):
- 1.4 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 3:
- 1.5 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 4:
- 1.6 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 5:
- 1.7 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 6:
- 1.8 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 7 (a):
- 1.9 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 7 (b):
- 1.10 NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 8:

# NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 1:

ABC is a right triangle with BC as a hypotenuse. The medians are drawn to BC and AC are perpendicular to each other. if AB has length 1 cm, find the area of the triangle ABC.

**ANSWER: ** cm^{2}.

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 2 (a):

Find the smallest positive integer such that it has exactly 100 different positive integer divisors including 1 and itself.

**ANSWER:** 45360

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 2 (b):

A rectangle can be divided into n equal squares. The same rectangle can also be divided into (n + 76) equal squares. Find the value of n.

**ANSWER:** 324

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 3:

Prove that 1^{n} + 2^{n} + 3^{n} + … + 15^{n} is divisible by 480 for all odd n >= 5.

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 4:

Is it possible to have 19 lines in a plane such that no three lines have a common point and they have exactly 95 points of intersections.

**ANSWER:** NO

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 5:

In a trapezium ABCD with AB parallel to CD, the diagonals intersect at P. The area of triangle ABP is 72 cm^{2} and the area of triangle CDP is 50 cm^{2}. Find the area of trapezium.

**ANSWER:** 242

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 6:

Let a < b < c be three positive integers. Prove that among any 2c consecutive positive integers there exists three different numbers x, y, z such that abc divides xyz.

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 7 (a):

Let m and n be two positive integers. If m^{3} + n^{3} is the square of an integer, then prove that (m + n) is not a product of two different prime numbers.

**Solution:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 7 (b):

a, b, c are three real numbers such that ab + bc + ca = -1. Prove that a^{2} + 5b^{2} + 8c^{2} >= 4.

**Solution****:**

## NMTC 2018 Final Stage Class 9 and 10 Question Paper With Solutions Question No 8:

ABCD is a quadrilateral in a circle whose diagonals intersect at right angles. Through O the center of the circle GOG’ and HOH’ are drawn parallel to AC and BD respectively, meeting AB and CD in G, H and DC, AB produced in G’ and H’. Prove that GH. G’H’ are parallel to BC and AD respectively.

**Solution****:**