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Math Olympiad Test: Linear Equations in Two Variables- 3 - Class 10 MCQ


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10 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Linear Equations in Two Variables- 3

Math Olympiad Test: Linear Equations in Two Variables- 3 for Class 10 2024 is part of Olympiad Preparation for Class 10 preparation. The Math Olympiad Test: Linear Equations in Two Variables- 3 questions and answers have been prepared according to the Class 10 exam syllabus.The Math Olympiad Test: Linear Equations in Two Variables- 3 MCQs are made for Class 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Math Olympiad Test: Linear Equations in Two Variables- 3 below.
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Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 1

The sum of the digits of a two-digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 1

Let the unit’s and ten’s digits in the number be y and x respectively
So, the number be 10x + y.
According to the question, x + y = 12
Also, 10x + y + 18 = 10y + x
⇒ 9x – 9y = –18 ⇒ x – y = –2
Solving (1) and (2), we get x = 5 and y = 7
∴ Required number is 57.

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 2

The sum of two numbers is 8 and the sum of their reciprocals is 8/15. Find the numbers.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 2

Let the two numbers be x and y
According to the given conditions, x + y = 8...(1)
 and  ...(2)
Putting value of x = 8 – y in (2), we get

⇒ y2 – 8y + 15 = 0 ⇒ y2 – 5y – 3y + 15 = 0
⇒ (y – 5) (y – 3) = 0 ⇒ y = 5 or y = 3
From (1), x = 3 or x = 5
Thus, the numbers are 5 and 3.

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Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 3

If am = bl and bn ≠ cm, then the system of equations
ax + by = c
lx + my = n

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 3

We have, ax + by = c and lx + my = n
Now, 
∴ The given system of equations has no solution.

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 4

The denominator of a rational number is greater than its numerator by 3. If 3 is subtracted from  the numerator and 2 is added to the denominator, the new number becomes 1/5. Then the original number was _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 4

Let the number be x/y
According to the question,
y = x + 3 ⇒ x = y − 3 ... (1)
Also,  ⇒ 5x – 15 = y + 2
⇒ 5x = y + 17 ...(2)
Solving (1) and (2), we get x = 5 and y = 8
∴ Required number = x/y = 5/8.

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 5

If the system of equations
2x + 3y = 7
2ax + (a + b)y = 28
has infinitely many solutions, then the values of a and b respectively are _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 5

Given equations are
2x + 3y = 7 and 2ax + (a + b)y = 28
For infinitely many solutions, we have

Taking first two members, we get 2a + 2b = 6a
⇒ 4a = 2b ⇒ 2a = b ...(1)
Also,  ...(2)
From (1) and (2), we have
2(4) = b ⇒ b = 8

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 6

The value of k, for which the system of equations kx – 3y + 6 = 0, 4x – 6y + 15 = 0 represent parallel lines, is _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 6

It is given that, kx – 3y + 6 = 0 and 4x – 6y + 15 = 0 are two parallel lines.
i.e., The given lines has no solution or

⇒ 

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 7

ax + by + c = 0 does not represent an equation of line when _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 7

ax + by + c = 0
When a and b are equal to zero, then the above linear equation does not represent an equation of line.

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 8

In a ΔABC, ∠C = 3 ∠B = 2 (∠A + ∠B). The three angles will be _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 8

In a triangle, sum of angles is 180°
∴ ∠A + ∠B + ∠C = 180° ...(1)
∠C = 3∠B ...(2)
and 3∠B = 2∠A + 2∠B
∴ ∠A = ∠B/2 ...(3)
From (1), (2) and (3), we get
∠B/2 + ∠B + 3∠B = 180°
⇒ (9/2)∠B = 180° ⇒ ∠B = 40°
∴ ∠C = 120° and ∠A = 20°

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 9

How many values of c, for which the system of equations 6x + 3y = c – 3, 12x + cy = c has infinitely many solutions?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 9

Given equations are
6x + 3y = c – 3 and 12x + cy = c
For infinitely many solutions, 
Taking first two members, we have 
Thus, only one value of c exists.

Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 10

The ratio of a 2-digit number to the sum of digits of that number is 4 : 1. If the digit in the unit place is 3 more than the digit in the tens place, what is the number?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 3 - Question 10

Let the digit at unit place be x and the digit at tens place be y, then the number = 10y + x
Now, according to the question,
 
⇒ 10y + x = 4y + 4x
⇒ 6y = 3x ⇒ x = 2y ...(1)
Also, x = 3 + y ⇒ 2y = 3 + y [From (1)]
⇒ y = 3 and x = 6
∴ Number = 36

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