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Math Olympiad Test: Linear Equations in Two Variables- 1 - Class 10 MCQ


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10 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Linear Equations in Two Variables- 1

Math Olympiad Test: Linear Equations in Two Variables- 1 for Class 10 2024 is part of Olympiad Preparation for Class 10 preparation. The Math Olympiad Test: Linear Equations in Two Variables- 1 questions and answers have been prepared according to the Class 10 exam syllabus.The Math Olympiad Test: Linear Equations in Two Variables- 1 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- 1 below.
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Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 1

What are the values of x and y if

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

Let

then 57a + 6b = 5 ...(1)
38a + 21b = 9 ...(2)
Solving (1) and (2) we get a = 1/19, b = 1/3
Hence
x + y = 19 ...(4)
x - y = 3 ...(5)
Solving (4) and (5), we get x = 11, y = 8

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 2

The sum of two numbers is 16 and the sum of their reciprocals is 1/3. What are the numbers?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 2

Let the numbers be x and y
x + y = 16 ...(1)
and

Now


⇒ x - y = 8 ...(2)
From (1) and (2), we get x = 12, y = 4.

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

In a triangle ΔABC, 3∠B = ∠C = 2 (∠A + ∠B). Which angle is the largest and what is its value?

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

L et ∠A = x°, ∠B = y°, ∠C = 3∠B = 3y°
In ΔABC, ∠A + ∠B + ∠C = 180°
x° + y° +3y° = 180°
x° + 4y° = 180 ...(1)
∠C = 2(∠A + ∠B) ⇒ 3y = 2 (x° + y°)
2x° - y° = 0 ...(2)
From (1) and (2)
x = 20 , y = 40
∴ ∠A = 20°, ∠B = 40°, ∠C = 3 × 40 = 120°

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 4

What is the value of k for which the system of equations 3x + y = 1 and (2k - 1) x + (k - 1) y = 2k + 1 has no solution.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 4

For no solution, it must have

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 5

What are the values of x and y, if 2(ax - by) + (a + 4b) = 0 and 2 (bx + ay) + (b - 4a) = 0

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 5

We have
2ax - 2by = - a - 4b ...(1)
and 2bx + 2ay = 4a - b ...(2)
2a2x - 2aby = -a2 - 4ab ...(3)
Now 2b2x + 2aby = 4ab - b2
2x (a2 + b2) = - (a2 + b2) ⇒ x = -1/2
Now putting the value of 2aby from eq. (3), we get

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 6

The length of a field exceeds its breadth by 3 meters. If the length is increased by 3 meters and the breadth is decreased by 2 meters. The area remains the same. What are the length and breadth respectively of the field?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 6

Let the breadth of the field be x m and length be y m.
Then, area = xy m2
Now y = 3 + x
⇒ y - x = 3 ...(1)
New length = y + 3
breadth = x -2
∴ area =  (x - 2) (y + 3) = xy
⇒ xy + 3x - 2y - 6 = xy
⇒ 3x -2y = 6 ...(2)
From (1) and (2), we get
x = 12 m, y = 15 m

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 7

What is the value of k except which the given system of equations has a unique solution?
2x + 3y - 5 = 0 and kx - 6y - 8 = 0

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 7

The given system of equations are 
2x + 3y – 5 = 0 
kx - 6y - 8 = 0 
This system is of the form: 
a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 
where, a1 = 2, b1 = 3, c1 = -5 and a2 = k, b2 = -6, c2 = -8 
Now, for the given system of equations to have a unique solution, we must have:

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 8

Five years ago, Ravi was thrice as old as Shashi. Ten years later Ravi will be twice as old as Shashi. What is the age of Shashi?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 8

Let Shashi’s present age be x and Ravi’s present age be y
y - 5 = 3 (x - 5) ⇒ 3x - y = 10 ...(1)
y + 10 = 2 (x + 10) ⇒ 2x - y = 10 ...(2)
Solving eqns (1) and (2), we get
x = 20, y = 50

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 9

In a cyclic quadrilateral ABCD, ∠A = 2x - 1, ∠B = y + 5, ∠C = 2y + 15, ∠D = 4x - 7. Which is the greatest angle of quadrilateral.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 9

In a cyclic quadrilateral,
∠A + ∠C = 180°, ∠B + ∠D = 180°
Here 2x - 1 + 2y + 15 = 180°
⇒ 2x + 2y = 166
x + y = 83 ...(1)
and 4x - 7 + y + 5 = 180°
⇒ 4x + y = 182 ...(2)
Solving eq. (1) and (2), we get x = 33, y = 50
∠A = 2x - 1 = 2 × 33 - 1 = 65°
∠B = y + 5 = 50 + 5 = 55°
∠C = 2y + 15 = 2 × 50 + 15 = 115°
∠D = 4x - 7 = 4 × 33 - 7 = 125°

Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 10

Rajesh scored 40 marks on a test getting 3 marks for each right answer and losing one mark for each wrong answer. If 4 marks have been awarded for each correct answer and 2 marks have been deducted for each incorrect answer then Rajesh will score 50 marks. What is the number of questions in the test?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 1 - Question 10

Let the number of right answer be x and the number of wrong answer be y
3x - y = 40 ...(1)
4x - 2y = 50 ⇒ 2x - y = 25 ...(2)
Solving (1) and (2) x = 15, y = 5
No. of questions = 15 + 5 = 20

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