IMO Practice Test — Pair of Linear Equations in Two Variables
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
medium
For what value of k will the system 2x+3y=7 and (k−1)x+(k+2)y=3k have infinitely many solutions?
4
5
6
7
Explanation: Coincident condition: 2/(k-1)=3/(k+2)=7/(3k). From first two: 2(k+2)=3(k-1)→2k+4=3k-3→k=7. Check with third: 2/6=1/3, 7/21=1/3 ✓
Question 2 of 6
medium
If x = a, y = b is the solution of x+y=5 and 2x−3y=5, find the value of \(a^{2} + b^{2}\).
10
13
17
25
Explanation: Solve $x+y=5$ and $2x-3y=5$. From $3(x+y)=15$, adding $2x-3y=5$ gives $5x=20 \Rightarrow x=4$, so $y=1$. Then $a^{2}+b^{2}=4^{2}+1^{2}=17$.
Question 3 of 6
medium
A fraction becomes 1/3 when 1 is subtracted from numerator and becomes 1/4 when 8 is added to denominator. Find the fraction.
3/8
5/12
7/20
2/9
Explanation: Let fraction=x/y. (x-1)/y=1/3 → 3x-3=y …(1); x/(y+8)=1/4 → 4x=y+8 …(2). From (1) y=3x-3, sub in (2):4x=3x-3+8→x=5, y=12 → 5/12
Question 4 of 6
medium
Solve for x and y: (x+y−8)/2 = (x+2y−14)/3 = (3x+y−12)/11
x=4,y=4
x=5,y=3
x=2,y=6
x=3,y=5
Explanation: Equating the first two ratios: $3(x+y-8)=2(x+2y-14) \Rightarrow x-y=-4$. Equating the first and third: $11(x+y-8)=2(3x+y-12) \Rightarrow 5x+9y=64$. Substituting $x=y-4$ gives $14y=84 \Rightarrow y=6,\ x=2$. Each ratio then equals $0$, so $(x,y)=(2,6)$.
Question 5 of 6
medium
A two-digit number is 4 times the sum of its digits. If 18 is added, the digits reverse. Find the number.
24
36
48
72
Explanation: Let the tens digit be $x$ and the units digit $y$, so the number is $10x+y$. Since it is four times the digit sum: $10x+y=4(x+y) \Rightarrow y=2x$. Adding $18$ reverses the digits: $10x+y+18=10y+x \Rightarrow y-x=2$. Solving $y=2x$ and $y-x=2$ gives $x=2,\ y=4$, so the number is $24$ (check: $4 \times 6 = 24$ and $24+18=42$).
Question 6 of 6
medium
A man travels 600 km partly by train and partly by car. If he covers 400 km by train and rest by car, it takes 6 hours 30 minutes. If he travels 200 km by train and rest by car, it takes 7 hours. Find speed of train.
80 km/h
100 km/h
120 km/h
140 km/h
Explanation: Let train speed $=t$, car speed $=c$, with $u=\frac{1}{t},\ v=\frac{1}{c}$. Then $400u+200v=6.5$ and $200u+400v=7$. Doubling the first: $800u+400v=13$; subtracting the second: $600u=6 \Rightarrow u=\frac{1}{100}$, so $t=100$. Then $4+200v=6.5 \Rightarrow v=\frac{1}{80}$, so $c=80$. The train's speed is $100$ km/h.