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# Bhaskara Contest (NMTC JUNIOR LEVEL-IX and X Grades) 2023 - Problems and Solutions

###### Problem 1

If are real numbers such that the polynomial is the cube of then
a) is divisible by 13
b)
c) and .
(4) is divisible by 11 .

###### Problem 2

In the adjoining figure, , is the median and . Then measure of (in degrees) is

a) 100
b) 140
c) 45
d) 120

###### Problem 3

If and then the value of is

a) 68
b) 72
c) 65
d) 70

###### Problem 4

If , then the value of is

a)
b)
c)
d)

###### Problem 5

AB = BC = CD

is the midpoint of . If , then is equal to

a)
b)
c)
d)

###### Problem 6

In the adjoining figure, is the midpoint of the are .
Given that and .

Then the value of is

a) 22
b) 23
c) 22.5
d) 23.5

###### Problem 7

The number of real numbers which satisfy the equation is

a) 1
b) 2
c) 0
d) 4

###### Problem 8

are real numbers such that . Then the maximum value of is

a) 256
b) 1024
c) 1262
d) 16

###### Problem 9

The number of ordered pairs of integers such that and is

a) 4
b) 3
c) 2
d) 1

###### Problem 10

In the adjoining figure, three equal squares are placed. The squares are unit squares. The area of the shaded region is

a)
b)
c)
d)

###### Problem 11

In the adjoining figure, is a diameter of the circle. Given , Then the measure (in degrees) of is

a) 12
b) 10
c) 14
d) 16

###### Problem 12

The number of ordered pairs of integers such that and leaves a remainder 1 when divided by 4 is

a) 2250
b) 1000
c) 1125
d) 1250

###### Problem 13

The number of ordered pairs of positive integers satisfying the equation is

a) 1
b) 2
c) 3
d) 4

###### Problem 14

The algebraic expression reduces to

a)
b)
c)
d)

###### Problem 15

The sum of terms is equal to

a) 122500
b) 116800
c) 11800
d) 117600

###### Problem 16

If the equations and have a common root, then the value of is .

###### Problem 17

If are positive reals such that abcd=1 then the maximum value of is .

###### Problem 18

The sum of all natural numbers ' ' for which is a perfect square is .

###### Problem 19

is a point inside the square such that Distance of from .
The ratio of the areas of the triangle to the area of the square is where are relatively prime integers. Then the value of .

###### Problem 20

The sum of roots of the simultuneous equations
is .

###### Problem 21

If where are natural numbers, then the value of is .

###### Problem 22

Then the measure (in degrees) of angle is .

###### Problem 23

If (where are all not zero), then the numerical value of is .

###### Problem 24

The geometric and arithmetic means of two positive numbers are respectively 8 and 17 . The larger among the two numbers is .

###### Problem 25

The number of two-digit numbers in which the tens and the units digit are different and odd is .

###### Problem 26

The value of is equal to .

###### Problem 27

If , then the numerical value of is .

###### Problem 28

The sum of all natural numbers which satisfy the simultaneous inequations and is .

###### Problem 29

In an increasing geometric progression (with term and term ). the difference between the fourth and the first term is 52 and the sum of the first three terms is 26. Then the numerical value of is .

###### Problem 30

The base of a triangle is 4 units less than the altitude drawn to it. The area of the triangle is unit . The ratio of the base to height is where are relatively prime to each other. Then the value of is .

###### Problem 1

If are real numbers such that the polynomial is the cube of then
a) is divisible by 13
b)
c) and .
(4) is divisible by 11 .

###### Problem 2

In the adjoining figure, , is the median and . Then measure of (in degrees) is

a) 100
b) 140
c) 45
d) 120

###### Problem 3

If and then the value of is

a) 68
b) 72
c) 65
d) 70

###### Problem 4

If , then the value of is

a)
b)
c)
d)

###### Problem 5

AB = BC = CD

is the midpoint of . If , then is equal to

a)
b)
c)
d)

###### Problem 6

In the adjoining figure, is the midpoint of the are .
Given that and .

Then the value of is

a) 22
b) 23
c) 22.5
d) 23.5

###### Problem 7

The number of real numbers which satisfy the equation is

a) 1
b) 2
c) 0
d) 4

###### Problem 8

are real numbers such that . Then the maximum value of is

a) 256
b) 1024
c) 1262
d) 16

###### Problem 9

The number of ordered pairs of integers such that and is

a) 4
b) 3
c) 2
d) 1

###### Problem 10

In the adjoining figure, three equal squares are placed. The squares are unit squares. The area of the shaded region is

a)
b)
c)
d)

###### Problem 11

In the adjoining figure, is a diameter of the circle. Given , Then the measure (in degrees) of is

a) 12
b) 10
c) 14
d) 16

###### Problem 12

The number of ordered pairs of integers such that and leaves a remainder 1 when divided by 4 is

a) 2250
b) 1000
c) 1125
d) 1250

###### Problem 13

The number of ordered pairs of positive integers satisfying the equation is

a) 1
b) 2
c) 3
d) 4

###### Problem 14

The algebraic expression reduces to

a)
b)
c)
d)

###### Problem 15

The sum of terms is equal to

a) 122500
b) 116800
c) 11800
d) 117600

###### Problem 16

If the equations and have a common root, then the value of is .

###### Problem 17

If are positive reals such that abcd=1 then the maximum value of is .

###### Problem 18

The sum of all natural numbers ' ' for which is a perfect square is .

###### Problem 19

is a point inside the square such that Distance of from .
The ratio of the areas of the triangle to the area of the square is where are relatively prime integers. Then the value of .

###### Problem 20

The sum of roots of the simultuneous equations
is .

###### Problem 21

If where are natural numbers, then the value of is .

###### Problem 22

Then the measure (in degrees) of angle is .

###### Problem 23

If (where are all not zero), then the numerical value of is .

###### Problem 24

The geometric and arithmetic means of two positive numbers are respectively 8 and 17 . The larger among the two numbers is .

###### Problem 25

The number of two-digit numbers in which the tens and the units digit are different and odd is .

###### Problem 26

The value of is equal to .

###### Problem 27

If , then the numerical value of is .

###### Problem 28

The sum of all natural numbers which satisfy the simultaneous inequations and is .

###### Problem 29

In an increasing geometric progression (with term and term ). the difference between the fourth and the first term is 52 and the sum of the first three terms is 26. Then the numerical value of is .

###### Problem 30

The base of a triangle is 4 units less than the altitude drawn to it. The area of the triangle is unit . The ratio of the base to height is where are relatively prime to each other. Then the value of is .

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