## NMTC 2016 JUNIOR LEVEL IX & X ANSWER KEYS WITH SOLUTION

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**Ques No 1:**

The sum of the values of x and y that satisfy the equation (x + y)2^{(y – x)} = 1 and (x + y)2^{(x – y)} = 2 simultaneously is

**Options:**

A. 2

B. 3/2

C. 5/2

D. 7/2

**Solution:**

**Ques No 2:**

If

the value of a is

**Options:**

A. 4/1007

B. 3/2016

C. 4/2017

D. None of these

**Solution:**

**Ques No 3:**

ABCD is a square inscribed in a circle of radius of 1 unit. The tangent to the circle at C meets AB produced at P. The length of PD is

**Options:**

A. 2

B. 3

C.

D.

**Solution:**

**Ques No 4:**

Quadrilateral ABCD is inscribed in a circle with radius 1 unit. AC is diameter of the circle and BD = AB. The diagonals cut at P. If PC = 2/5 then the length of CD is equal to

**Options:**

A. 2/3

B. 2/7

C. 1/8

D. 3/4

**Solution: **

Read : Number Theory Syllabus for NMTC

**Ques No 5:**

The number of natural numbers n for which the expression (23n2 + 18n + 4)/n is also a natural number is

**Options:**

A. 3

B. 2

C. 1

D. 0

**Solution:**

**Ques No 6:**

The cost price of 16 oranges is equal to the selling price of 12 oranges. Then there is a

**Options:**

A. 40% Profit

B. 20% Loss

C. 33 1/3% Profit

D. 23 1/3% Profit

**Solution:**

**Ques No 7:**

The number of positive integer pairs (a, b) such that ab – 24 = 2b

**Options:**

A. 6

B. 7

C. 8

D. 9

**Solution:**

**Ques No 8:**

If A = (2 + 1)(2^{2} + 1)(2^{4} + 1)…(2^{2016} + 1), then the value of (A + 1)^{1/2016} is

**Options:**

A. 4

B. 2016

C. 2^{4032}

D. 2

**Solution:**

**Ques No 9:**

The sum of two numbers a, b where a < b is 1215 and their HCF is 81. The number of such pairs (a, b) is

**Options:**

A. 1

B. 2

C. 3

D. 4

**Solution:**

**Ques No 10:**

The first Republic Day of India was celebrated on 26th January 1950. What was the day of the week on that date?

**Options:**

A. Tuesday

B. Wednesday

C. Thursday

D. Friday

**Solution:**

**Ques No 11:**

The 12 numbers a_{1}, a_{2}, …, a_{12} are in arithmetic progression. The sum of all these numbers is 354. Let P = a_{2} + a_{4} + … + a_{12} and Q = a_{1} + a_{3} + … + a_{11}. If the ratio of P : Q is 32 : 27, the common ratio of the progression is

**Options:**

A. 2

B. 3

C. 4

D. 5

**Solution:**

**Ques No 12:**

A shopkeepers marks the price of his goods at 20% higher than the original price. There is an increase in demand of the goods and he further increases the price by 20%. The total profit % is

**Options:**

A. 40

B. 38

C. 42

D. 44

**Solution:**

**Ques No 13:**

A circle passes through the vertices A and D and touches the side BC of a square. The side of the square 2 cm. The radius of the circle (in cm) is

**Options:**

A. 5/4

B. 4/5

C. 1

D. 5/2

**Solution:**

**Ques No 14:**

There are 4 balls – one green, one red, one blue and one yellow and there are four boxes – one green, one red, one blue and one yellow. A child playing with the balls decides to put the balls in the boxes, one ball in each box. The number of ways in which the child can put the balls in the boxes such that no ball is in a box of its own color is

**Options:**

A. 12

B. 9

C. 24

D. 6

**Solution:**

**Ques No 15:**

The 5 x 5 array of dots represents trees an orchard. If you were standing at the central spot marked C, you would not be able to see of 8 of the 24 trees (shown as x). if you were standing at the center of 9 x 9 arrays of trees, how many of the 80 trees would be hidden?

**Options:**

A. 40

B. 32

C. 36

D. 44

**Solution:**

**Ques No 16:**

Fill in blank:

a and b are positive integers such that a^{2} + 2b = b^{2} + 2a + 5. The value of b is ______ .

**Solution:**

**Ques No 17:**

Fill in blank:

After full simplification, the value of the product

is ____________ .

**Solution:**

**Ques No 18:**

Fill in blank:

ABCD is rectangle with AD = 1 and AB = 2. DFEB is also a rectangle. The area of DFEB is ____ .

**Solution:**

**Ques No 19:**

Fill in blank:

The two digit number whose unit digit is exceeds the tens digit by 2 and such that the product of the number and the sum of the digits is 144 is ______ .

**Solution:**

**Ques No 20:**

Fill in blank:

If x = p/q where p, q are integers having no common divisors other than 1 satisfies

then x is ____ .

**Solution:**

**ANSWER:**

**1____ 5/2**

**2____ 4/1007**

**3____ **

**4____ 2/3**

**5____ 3**

**6____ 33 1/3%**

**7____ 8**

**8____ 4**

**9____ 4**

**10___ Thursday**

**11___ 5**

**12___ 44%**

**13___ 5/4**

**14___ 9**

**15___ 32 **

**16___ 3**

**17___ 1**

**18___ 2**

**19___ 24**

**20___ 25/16**

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