# NMTC 2010 Question Papers with Solutions Junior Level

NMTC is an acronym for * National Mathematics Talent Contest*. NMTC is conducted by AMTI (

*) to identify the talent of the students in Mathematics. In this post,*

**Association of Mathematics Teachers of India****NMTC 2010 Question Papers with Solutions Junior Level**is published.

**Part A: Instruction:**

**Only One of the choices A, B, C, D is correct. NMTC 2010 Question Papers with Solutions Junior Level For each correct response, you get 1 mark. For each incorrect response, you lose 1/2 mark.**

**NMTC 2010 Paper For ****Junior**** Level Ques No 1:**

The numbers which when subtracted from the terms of the ratio a : b makes it equal to c : d is

**Options:**

A. (ab – cd)/(ab + cd)

B. (bc – ad)/(c + d)

C. (ab + cd)/(c + d)

D. (ab – cd)/(b – c)

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 2**

In a kilometer race, Ram beats Shyam by 25 meters or 5 seconds. The time taken by Ram to complete the race is

**Options:**

A. 1 minute

B. 5 minute 30 second

C. 3 minute 15 second

D. 4 minute 10 second

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 3**

Through D, the mid-point of the side BC of a triangle ABC, a straight line is drawn to meet AC at E and AB produced at F so that AE = AF Then, the ratio of BF : CE is

**Options:**

A. 1 : 2

B. 2 : 1

C. 1 : 3

D. None of These

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 4**

In the bigger of two concentric circles, two chords AB and AC are drawn to touch line is drawn to touch the smaller circle at D and E, then BC is equal to

**Options:**

A. 3DE

B. 4DE

C. 2DE

D. (3/2)DE

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 5**

The number of solution of the equation

**Options:**

A. 0

B. 1

C. 2

D. 3

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 6**

The internal bisector of AE of the angle A of triangle ABC is

**Options:**

A. Not greater than the median through A for all triangles.

B. Not greater than the median through A for only acute angled triangles.

C. Not greater than the median through A for only obtuse angled triangles.

D. Not less than the median through A for all triangles.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 7**

In the adjoining diagram, ABC is an equilateral triangle BCDE is a square. The side of the equilateral triangle is 2010. The radius of the circle is

**Options:**

A. 2010

B. 4020

C. 6030

D. 8040

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 8**

Given a and b are integers the expression (a^{2} + a + 2011)(2b + 1) is

**Options:**

A. Odd for exactly 2010 values of a and b.

B. Odd for exactly 2010 values of a and b.

C. Even for exactly one value of a and two values of b.

D. Odd for exactly one value of a and one values of b.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 9**

A real sequence of real numbers x_{n} is defined recursively as follows x_{0}, x_{1}, are arbitrary positive real numbers and x_{n+2} = (1 + x_{n +1})/x_{n}, n = 0,1,2,3,…. Then the value of x_{2011} is

**Options:**

A. 1

B. x_{0}

C. x_{1}

D. x_{2}

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 10**

If xy = 6 and x^{2}y + y^{2}x + x + y = 63, then the value of x^{2} + y^{2} is

**Options:**

A. 81

B. 18

C. 2010

D. 78

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 11**

If p is the perpendicular drawn from the vertex of a regular tetrahedron to the opposite face and if each edge is equal to 2 units, the p is

**Options:**

A.

B. /2

C. /3

D. /5

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 12**

The remainder when the polynomial x + x^{3} + x^{9} + x^{27} + x^{81} + x^{243} is divided by x^{2} – 1.

**Options:**

A. 6x

B. 2x

C. 3x

D. 1

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 13**

Consider the sequence 4, 4, 8, 2, 0, 2, 4, 6, 0, …where the nth term is the unit place of the sum of the sum of previous two terms for n ≥ 3. If Sn is the sum to n terms of this sequence, then the smallest n for which S_{n} > 2011 is.

**Options:**

A. 253

B. 502

C. 503

D. 504

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 14**

P is the point inside an equilateral triangle of side 2010 units. The sum of the lengths of the perpendiculars drawn from P to the sides is equal to

**Options:**

A. 2010

B. 2010

C. 1005

D. 2010/

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 15**

The equation

has

**Options:**

A. A root less than 1

B. Has only root greater than 1

C. Two irrational roots

D. No real roots

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 16**

The value of

is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 17**

If a, b are positive and a + b = 1 then the minimum value of a^{4} + b^{4} is______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 18**

The whole surface area of rectangular block is 1332 cm^{2}. The length, breadth and height are in the ratio of 6:5:4. The sum of the length, breadth and height is _______ cms.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 19**

If |x| + x + y = 10 and x + |y| – y = 12, then x + y is _____.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 20**

Two parallel sides of a trapezoid are 3 and 9, the non-parallel sides are 4 and 6. A line parallel to the bases (parallel sides) divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 21**

Triangle ABC has AB = 17 and AC = 25 and altitude to BC has length is 15. The sum of possible values of BC is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 22**

If

where 0 ≤ a_{i} < i, i = 1,2,3,4,5,6. Then, a_{2} + a_{3} + a_{4} + a_{5} + a_{6} is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 23**

A circle is circumscribed about a triangle with sides 30, 34, 16. It divides the circle into four regions with the non-triangular regions being A, B, C. C being the largest. Then the value of (C – A – B) is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 24**

If a number n is divisible by 8 and 30, then the smallest number of divisors that n has is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 25**

Both the quadratic equation x^{2} – 12x + K = 0 are prime numbers. The sum of all such value of K is _______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 26**

In a convex polygon of sides 16 sides the maximum number of angles which can all be equal to 10 degrees is _______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 27**

If an arc of circle 1 subtending 60 degree at the centre, has double the length as the arc subtending 75 degrees at the centre in the circle 2, then the ratio of (Area of Circle 1)/ (Area of Circle 2) is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 28**

A two-digit number is equal to the sum of the product of its digits and the sum of its digits. Then the units place of the number is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 29**

Let f(x) be a polynomial of degree 1. If f(10) – f(5) = 15 and, then f(20) – f(5) is ______.

**Solution:**

**NMTC 2010 Paper For ****Junior**** Level Ques No 30**

The number of perfect square divisors of the number 12! is _______.

**Solution:**