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Math Olympiad Test: Linear Equations in Two Variables- 4 - Class 10 MCQ


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10 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Linear Equations in Two Variables- 4

Math Olympiad Test: Linear Equations in Two Variables- 4 for Class 10 2024 is part of Olympiad Preparation for Class 10 preparation. The Math Olympiad Test: Linear Equations in Two Variables- 4 questions and answers have been prepared according to the Class 10 exam syllabus.The Math Olympiad Test: Linear Equations in Two Variables- 4 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- 4 below.
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Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 1

If a pair of linear equations in two variables is consistent, then the lines represented by two equations are

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

Two lines can have solution in two different cases,
Case 1: When they intersect at one point (unique solution)
Case 2: When the lines are coincident (infinite solutions)
Thus, if a pair of linear equations is consistent, then the lines will be intersecting or coincident.

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 2

Sum of two numbers is 64 and their difference is 22. Find the numbers.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 2

Let the two numbers be x and y
Then, x + y = 64 ...(1)
Also, x – y = 22 ...(2)
or y – x = 22 ...(3)
(i) If x – y = 22, from (1) and (2), we have 2x = 86
⇒ x = 43 and y = 21
(ii) If y – x = 22, from (1) and (3), we have 2y = 86 ⇒ y = 43 and x = 21
∴ Numbers are 43 and 21.

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

Value of x in pair of linear equations 36x + 24y = 702 and 24x + 36y = 558 is

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

We have, 36x + 24y = 702
and 24x + 36y = 558
Simplifying above equations, we get
6x + 4y = 117 ...(1) and 4x + 6y = 93 ...(2)
Multiplying eq. (1) by 3, eq. (2) by –2 and then adding, we get 18x + 12y – 8x – 12y = 351 – 186
⇒ 10x = 165 ⇒ x = 165/10 = 33/2 

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

The solution of the system of equations  (where x ≠ 0, y ≠ 0), is _______.

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

We have,
  ......(1)
Also,
  ...(2)
Let 1/y = a and 1/x = b
So, (1) and (2) become
2a + 5b = 6 ...(3)
and, 4a – 5b = –3 ...(4)
Adding (3) and (4), we get
a = 1/2 and b = 1 [From (3)]
∴ x = 1 and y = 2

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 5

Arun and Prabhat have some books with them. Once Prabhat said to Arun that, if Arun gives 3 books to Prabhat then Arun will have only 1/2 of the books that Prabhat will have with him. Then Arun asked frankly that if Prabhat gives him only two books (to Arun), then Prabhat will have as many books as Arun will have. The total number of books that Arun and Prabhat have with them is _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 5

Let Arun has x books and Prabhat has y books
According to the question,
x − 3 = (1/2)(y + 3)⇒ 2x = y + 9 ... (1)
Also, x + 2 = y – 2 ⇒ x + 4 = y ... (2)
Solving (1) and (2), we get 
x = 13 and y = 17 ⇒ x + y = 30

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 6

Which of the following linear equation coincide with the line 4x + 5y = 15?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 6

For coincident lines, 
i.e., 

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 7

A boat goes 32 km upstream and 36 km downstream in 7 hours. In 9 hours, it can go 40 km upstream and 48 km downstream. If x represents the speed of the boat in still water in km/hr and y represents the speed of the stream in km/hr, then_______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 7

Let speed of boat in still water = x km/h
Speed of stream current = y km/h
According to question,

In these type of questions, make factor of 24 and 36 and choose the common values which satisfy the above equations.
24 = 2, 3, 4, 6, 8, 12
36 = 3, 4, 9, 12
Choose the common factor i.e. Put this value in equation (i)

x − y = 8
∴ x + y = 12
∴ x = 10, y = 2
Speed of the current,
y = 2 km/h.

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 8

Places M and N are 90 km apart from each other on a national highway. A truck starts from M and another from N at the same time. If they go in the same direction, then they meet in 9 hours and if they go in opposite directions they meet in 9/7 hours. The speed of the trucks are _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 8

Let x km/hr be the speed of truck starting from M and y km/hr be the speed of truck starting from N.
Case I : If they meet at P.


⇒ 10 + y = x ...(1)
Case II : If they meet at P.

For M, 
⇒ 70 = x + y ...(2)
Solving (1) and (2), we get
y = 30 km/hr and x = 40 km/hr

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 9

On selling a tea-set at 5% loss and lemonset at 15% gain, a crockery seller gains ₹ 7. If he sells the tea-set at 5% gain and the lemon-set at 10% gain, he gains ₹ 13. Find the actual prices of the tea-set and the lemon-set.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 9

Let cost price of tea-set = ₹ x
and cost price of lemon-set = ₹ y
(i) Loss on tea-set = 
Gain on lemon-set = 
Also,  ...(1)
(ii) Gain on tea-set = 
Gain on lemon-set = 
Also,  ..(2)
Solving (1) and (2), we get ⇒ y = ₹ 80 and x = ₹ 100

Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 10

At the end of the year 2002, Ram was half  as old as his grandpa. The sum of the years in which they were born is 3854. Age of Ram at the end of year 2003 is _______.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 4 - Question 10

Let the year in which Ram was born be x and the year in which Ram’s grandpa was born be y
Then, according to question, 
⇒ 2x – y = 2002 ...(1)
and x + y = 3854 ...(2)
Solving (1) and (2), we get ⇒ x = 1952
Thus in 2003, Ram’s age would be 2003 – 1952 = 51 yrs.

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