Olympiad Mathematics - Linear Equations in Two Variables
Exam Duration: 45 Mins Total Questions : 30
What does an equation of the form ax + by + C = 0, where a and b are nonzero numbers represent?
- (a)
A straight line
- (b)
A circle
- (c)
A triangle
- (d)
A quadrilateral
Which of the following is correct with respect to the line x + 1 = 0?
- (a)
It is parallel to y-axis.
- (b)
It passes through (0, -1).
- (c)
It is parallel to x-axis.
- (d)
It passes through (0,0).
Which is the equation of a line passing through the origin?
- (a)
y = 2
- (b)
x = 4
- (c)
x = 5x
- (d)
x = -7
The equation of the line passing through the origin is y = mx.
Hence y = 5x is the required equation.
Identify the correct statement from the following with respect to the line y - 2 = 0.
- (a)
It is parallel to X-axis.
- (b)
It is parallel to Y-axis.
- (c)
It passes through the origin.
- (d)
It passes through x = 2.
y - 2 = a \(\Rightarrow\) y = 2
It represents a line parallel to X-axis.
If (3, 4) is a solution of the equation 5x - 2y = k, find the value of k.
- (a)
7
- (b)
6
- (c)
5
- (d)
4
Given (3, 4) is a solution of 5x - 2y = k
\(\Rightarrow\) 5(3) - 2(4) = k \(\Rightarrow\) k = 7
If the point (2,-3) lies on the graph of the equation ay = 7x - 26, what is the value of 'a'?
- (a)
37
- (b)
16
- (c)
-5
- (d)
4
Given (2, -3) lies on ay = 7x - 26
\(\Rightarrow\) a (-3) = 7 (2) - 26 \(\Rightarrow\) \(a=\frac{-12}{-3}=4\)
What is the solution set of \(\sqrt {k+64}-8=-2?\)
- (a)
{-28}
- (b)
{-124}
- (c)
{4}
- (d)
{}
Given equation is \(\sqrt {k+64}-8=-2\)
\(\Rightarrow\) k = - 28
Identify the point at which the graph of the equation 7x - 9y - 21 = 0 cuts x - axis.
- (a)
(21, 0)
- (b)
(0, 9)
- (c)
(3,0)
- (d)
(3,1)
Which of the following equations represents a straight line passing through the points (1,6), (0,4) and (-2,0)?
- (a)
2x - y = - 4
- (b)
x - 2y = - 4
- (c)
2x + y = 4
- (d)
x + 2y = - 4
If the point (2p, p - 3) lies on the graph of the equation 3x + 2y + 12 = 0, find the value of p.
- (a)
\(\frac{-3}{4}\)
- (b)
\(\frac{7}{15}\)
- (c)
\(\frac{-1}{6}\)
- (d)
\(\frac{6}{7}\)
A man is five times as old as his son. After 2 years the man will be four times as old as his son. What is the present age of the man?
- (a)
35 years
- (b)
30 years
- (c)
6 years
- (d)
31 years
Megacity High School earned Rs 5100 on tickets sales for a play. The cost per ticket was Rs 12. If t represents the number of tickets sold to the play, which of the following equations could be used to determine the number of tickets sold for the play?
- (a)
12 = 5100t
- (b)
12t = 5100
- (c)
t=5100-12
- (d)
t=5100.12
Given. amount earned by school by selling tickets for a play = Rs 5100
Cost of each ticket = Rs 12
No. of tickets sold = t
Cost of t tickets = 12 t
\(\therefore\) The equation used to determine the number tickets were sold is 12t = 5100.
The function f(x) = 35 + 15x represents the amount of money, in Rupees, Mr. Ramesh earns for working X hours. How much money does Mr. Ramesh earn for working 25 hours?
- (a)
Rs 75
- (b)
Rs 375
- (c)
Rs 410
- (d)
Rs 1250
Amount = 35 + 15(25) = Rs 410
Two planes start from a city and fly in opposite directions, one averaging a speed of 40 km/hour greater than the second. If they are 3400 km apart from 5 hours, find the sum of their average speeds.
- (a)
680 km/h
- (b)
360 km/h
- (c)
320 km/h
- (d)
640 km/h
Let the speed of one plane be x km/hour.
Then the speed of other plane is (x + 40) km/hour.
Distance travelled by first plane in 5 hours = Speed x Time = x\(\times\) 5 = 5x
Distance travelled by second plane in 5 hours = (x + 40)5.
Distance travelled by first plane + Distance travelled by the other plane = 3400 km
5x + 5(x + 40) = 3400
\(\Rightarrow x=\frac{3200}{10}=320km/hour\)
Sum of speeds = (320 + 360) km/h
= 680 km/hour
Which equation represents the relationship between time, t1 and distance, d?
| Time (hours) | Distance (km) |
| 2 | 90 |
| 3 | 135 |
| 4 | 180 |
| 5 | 225 |
- (a)
d = t + 45
- (b)
d = 45t
- (c)
t = 45d
- (d)
\(t=\frac{45}{d}\)
Straight lines represented by linear equations x + y = 2 and 5x - 3y = 2 intersect at which of the given points?
- (a)
(1, 2)
- (b)
(1, 1)
- (c)
(2, 1)
- (d)
(3, 2)
A student was asked to divide a number by 17/8. Instead, he actually multiplied it by 17/8 and hence got 225 more than the expected answer. What was the expected answer?
- (a)
126
- (b)
136
- (c)
64
- (d)
84
In a class, \(\frac{3}{5}\) of the students are girls and rest are bays. If \(\frac{2}{9}\) of the girls and \(\frac{1}{4}\) of the bays are absent, what part of the total number of students are present?
- (a)
\(\frac{23}{30}\)
- (b)
\(\frac{23}{36}\)
- (c)
\(\frac{18}{49}\)
- (d)
\(\frac{17}{25}\)
How many solutions does a linear equation in two variable have?
- (a)
1
- (b)
Infinite
- (c)
2
- (d)
0
The breadth of a rectangular room is 2 m less than its length(l).lf the perimeter of the room is 14 m, find the length(l) and breadth(b) of the room.
- (a)
l= 2.5 m, b = 4.5 m
- (b)
l= 3.5 m, b = 3.5 m
- (c)
l= 4.5 m, b = 2.5 m
- (d)
l= 2.5 m, b = 5.5 m
b = l- 2......(i)
2(l + b) = 14
\(\Rightarrow\) l + b = 7 (or) b = 7 -l...(ii)
From (i) & (ii)
l - 2 = 7 -l(Since b = l - 2)
\(\Rightarrow\) l = 4.5 m and b = 2.5 m
x = 3 and y = -1 is a solution of which of the linear equations given?
- (a)
x + y = 3
- (b)
2x + y = 3
- (c)
x + 2y = 1
- (d)
2x - y = 1
Substituting x = 3 and y = -1 in
x + 2y = 1
\(\Rightarrow\) 3 + 2(-1) = 1
\(\Rightarrow\) 3 - 2 = 1 \(\Rightarrow\) 1 = 1 (True)
Hence (3. -1) is the solution of x + 2y = 1.
Find the equation of the line that passes through the points (5, 15) and (10, 20).
- (a)
y = x + 10
- (b)
y = x - 30
- (c)
y = x + 30
- (d)
y = x + 15
y = x + 10 put x = 5, y = 15
\(\Rightarrow\) 15=5+10
15 = 15
Similarly. x = 10. Y = 20 also satisfies the line y = x + 10
Hence. y = x + 10 is the required line.
The graph of y = 6 is a line
- (a)
Parallel to x-axis at a distance of 6 units from the origin.
- (b)
Parallel to y-axis at a distance of 6 units from the origin.
- (c)
Making an intercept of 6 units on the x-axis.
- (d)
Making an intercept of 6 units on both the axes.
How many linear equations in x and y can be satisfied by x = 2, Y = 3?
- (a)
Only one
- (b)
Only two
- (c)
Infinitely many
- (d)
None of these
Infinitely many equations in x and y can be satisfied by x = 2 and y = 3.
A straight line parallel to the y-axis has equation _________.
- (a)
x = a
- (b)
Y = a
- (c)
y = x
- (d)
y =-x
If (2, 0) is a solution of the linear equation 2x + 3y = k, then the value of k is________.
- (a)
4
- (b)
6
- (c)
5
- (d)
2
Since, (2, 0) is the solution of 2x + 3y = k. So (2, 0) satisfies it.
\(\therefore\) 2 x 2 + 3 x 0 = k \(\Rightarrow\) k = 4
Point (4, 1) lies on the line________
- (a)
x + 2y = 5
- (b)
2x + Y = - 6
- (c)
x+2y=6
- (d)
x+y=16
The equation of the line whose graph passes through the origin is_______.
- (a)
4x + 2y = - 1
- (b)
x + y = 1
- (c)
8x + 7y = 0
- (d)
8x - 1 = 4y
(A) The point (0, 0) satisfy the equation 8x+7y=0
So, 8x + 7y = 0 is the equation of line whose graph passes through the origin.
If \(\angle\)A and \(\angle\)B are complementary angles and m\(\angle\)A is x, which equation can be used to find m \(\angle\)B which is denoted by y?
- (a)
Y = (90° + x)
- (b)
Y = (90° - x)
- (c)
y = (180° - x)
- (d)
y = (x + 180°)
We have given, \(\angle\)A + \(\angle\)B = 90°
\(\Rightarrow\) x + Y = 90° \(\Rightarrow\) y = (90° - x)
A and B are friends. A is elder to B by 5 years. B's sister C is half the age of B while A's father D is 8 years older than twice the age of B. If the present age of D is 48 respectively.
- (a)
50years, 25years, 20years
- (b)
40 years, 20 years, 15 years
- (c)
20 years, 15 years, 10 years
- (d)
25 years, 20 years, 10 years
Let the present ages of A, B, C and 0 are x, y, z
and t respectively.
Since, present age of 0 = t = 48 years.
According to question,
x = y+ 5
z=\(\frac{1}{2}\)y
t= 2y+ 8 (iii)
From (iii), 48 = 2y + 8
\(\Rightarrow\)2y = 40 \(\Rightarrow\) y = 20 years
From (ii), z = \(\frac{1}{2}\)x 20 = 10 years
From (i), x = 20 + 5 = 25 years
So, present ages of A, Band Care 25 years,
20 years and 10 years respectively.
