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Math Olympiad Test: Linear Equations in Two Variables- 2 - Class 10 MCQ


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15 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Linear Equations in Two Variables- 2

Math Olympiad Test: Linear Equations in Two Variables- 2 for Class 10 2024 is part of Olympiad Preparation for Class 10 preparation. The Math Olympiad Test: Linear Equations in Two Variables- 2 questions and answers have been prepared according to the Class 10 exam syllabus.The Math Olympiad Test: Linear Equations in Two Variables- 2 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- 2 below.
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Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 1

The sum of the digits of a two digits number is 12. The number obtained by interchanging the digits exceeds the given number by 18. What is that number?

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

Let unit’s digit be x and ten’s digit be y
Then number = 10y + x
x + y = 12 ...(1)
10x + y - (10y + x) = 18
9x - 9y = 18 ⇒ x - y = 2 ...(2)
Solving (1) and (2), we get x = 7, y = 5
∴ Number =  10 × 5 + 7 = 50 + 7 = 57

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

Five years ago, Ram was thrice as old as Mohan. Ten years later Ram shall be twice as old as Mohan. What is the present age of Ram and Mohan?

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

Let the present age of Ram be x years and the present age of Mohan be y years
x - 5 = 3 (y - 5)
⇒ x - 3y = -10 ...(1)
and x + 10 = 2(y + 10)
x - 2y = 10 ...(2)
x - 3y = -10
x - 2y = 10
Subtracting (1) and (2)
-y = - 20 ⇒ y = 20
Now x = 10 + 2y = 10 + 2 × 20 = 50

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

What is the value of k for which the system of equations 3x + 5y = 0 and kx + 10y = 0, has a non-zero solution?

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

For a non-zero solution,

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

What are the values of x and y, if

where x ≠ 0, y ≠ 0?

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

 Let
1/x = m, 1/y = n
Then am - bn = 0 ...(1)
ab2m + a2bn = a2 + b2 ...(2)
Now am - bn + 0 = 0
ab2m + a2bn - (a2 + b2) = 0
By cross multiplication,

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 5

What are the values of m and n for which the system of linear equations has infinitely many solutions, 3x + 4y = 12, (m + n)x + 2(m - n)y = (5m - 1)?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 5

For infinitely many solutions,

On solving these equations, we get
m = 5, n = +1

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 6

What are the values of x and y if x + y = a + b, ax - by = a2 - b2?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 6

Here
bx + by = ab + b2 ...(1)
and ax - by = a2 - b2 ...(2)
Adding, x (a + b) = a (b + a) ⇒ x = a
Putting x = a in eqn (2), we have
a2 - by = a2 - b2
⇒ by = - b2 ⇒ y = b
Hence x = a, y = b

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 7

What is the value of K for which the system has no solution?
2x - ky + 3 = 0; 3x + 2y - 1 = 0

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 7

Given
2x - ky + 3 = 0
3x + 2y - 1 = 0
For no solution, we have

Now

So,

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 8

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. What is the fraction?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 8

Let the fraction be x/y
then x + y = 12 .....(1)
and

By solving eqn (1) and (2), we get x = 5, y = 7
x/y = 5/7

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 9

The larger of the two supplementary angles exceeds the smaller by 18º. What is the value of a larger angle?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 9

Let x be the larger angle and y be the smaller angle
then x + y = 180° ...(1)
and x - y = 18°
Solving eq. (1) and (2), we get
x = 99°, y = 81°

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 10

A railway half ticket cost half the full fare and the reservation charge is the same on a half ticket as on a full ticket. one reserved first-class ticket from Delhi to Patna costs ₹ 216. One full and one half reserved first-class ticket cost ₹ 328. What is the reservation charge?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 10

Let ₹ x be the charge of full first-class ticket and ₹ y be the reservation charge
x + y = 216 ...(1)
x + y + x/2 + y = 327
⇒ 3x + 4y = 654 ...(2)
By solving eq. (1) and (2), we get
x = 210, y = 6 

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 11

The student of a class are made to stand in rows. If 4 students are extra in each row, there would be 2 rows less. If 4 students are less to each row, there would be 4 rows more. What is the number of students in the class?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 11

Let the number of rows be x and the number of students in each row be y.
Total number of students = xy
(x - 2) (y + 4) = xy
⇒ xy + 4x - 2y - 8 = xy
⇒ 4x - 2y = 8 ...(1)
and (x + 4) (y - 4) = xy
xy - 4x + 4y - 16 = xy
⇒ -4x + 4y = 16 ...(2)
From eqn. (1) and (2)
4x - 2y - 4x + 4y = 24
2y = 24 ⇒ y = 12
and 4x - 2 × 12 = 8
⇒ 4x = 8 + 24 = 32 ⇒ x = 8
Total no. of students = 12 × 8 = 96.

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 12

If three times the larger of two numbers is divided by the smaller one, we get 4 as quotient and 3 as remainder. If seven times the smaller number is divided by the larger one, we get 5 as the quotient and 1 as the remainder. What is the smaller number?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 12

Let the larger number be x and the smaller number be y.
Now according to question, 3x = 4y + 3
3x - 4y = 3 ...(1)

and 3 × 25 - 4y = 3
⇒ 4y = 72 ⇒ y = 18
∴ Smaller number = 18

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 13

A chemist has one solution containing 50% acid and a second one containing 25% acid. How much of each should be used respectively to make 10 litres of a 40% acid solution?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 13

Let x litres of the 40% solution be mixed with y litres of 25% solution.
then x + y = 10 ...(1)
50% of x + 25% of y = 40% of 10

⇒ 50x + 25y = 400
⇒ 2x + y = 16 ...(2)
From (1) and (2)

∴ x =  10 - 4 = 6

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 14

A three digit number abc is 459 more than the sum of its digits. What is the sum of the 2 digit number ab and the 1-digit number a?

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 14

Let abc be the three-digit number.
Given abc = a + b + c + 459 .......(i)
Again abc = 100a + 10b + c...........(ii)
From equation(i) and (ii), we get
100a + 10 b + c = a + b + c + 459
100a - a + 10b - b + c - c = 459
99a + 9b = 459
9(11a + b) = 459
11a + b = 51 .....(iii)
To find: ab + a = 10a + b + a
= 11a + b
∴ ab + b = 51 [From equation(iii)]

Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 15

Swathi starts her job with a certain monthly salary and earns a fixed increment every year. If her salary was ₹ 22500 per month after 6 years of service and ₹ 30000 per month after 11 years of service. Find her salary in rupees after 8 years of service.

Detailed Solution for Math Olympiad Test: Linear Equations in Two Variables- 2 - Question 15

Let Swathi's first salary is Rs. x & fixed increment is Rs. y
∴ Swathi's salary after 6 years = Rs. (x + 6y)
Swathi's salary after 11 years = Rs. (x + 11y)
Swathi's salary after 8 years = Rs. (x + 8y)
∴ x + 6y = 22500 ......(1)
& x + 11y = 30000 .......(2)
From (1) & (2), we get
22500 - 6y + 11y = 30000
⇒ 5y = 30000 - 22500
= 7500
⇒ y = 7500/5 = 1500
∴∴ From (1), we get
x = 22500 - 6y
= 22500 - 6 × 1500
= 22500 - 9000
= 13500
∴∴ Swathi's salary after 8 years of service = Rs. (x + 8y)
= Rs. (13500 + 8 × 1500)
= Rs. (13500 + 12000)
= Rs. 25500

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