📝 Pair of linear equation in two variables
Multiple Choice questions
1. Graphically, the pair of equations
`6x – 3y + 10 = 0`
`2x – y + 9 = 0`
represents two lines which are
(A) intersecting at exactly one point.
(B) intersecting at exactly two points.
(C) coincident.
(D) parallel.
2. The pair of equations `x + 2y + 5 = 0` and `–3x – 6y + 1 = 0`
have
(A) a unique solution
(B) exactly two solutions
(C) infinitely many solutions
(D) no solution
3. If a pair of linear equations is consistent, then the lines will be
(A) parallel
(B) always coincident
(C) intersecting or coincident
(D) always intersecting
4. The pair of equations `y = 0` and `y = –7` has
(A) one solution
(B) two solutions
(C) infinitely many solutions
(D) no solution
5. The pair of equations `x = a` and `y = b` graphically represents lines
that are
(A) parallel
(B) intersecting at `(b, a)`
(C) coincident
(D) intersecting at `(a, b)`
6. For what value of k, do the equations `3x – y + 8 = 0` and
`6x – ky = –16` represent coincident lines?
(A) `1/2`
(B) `1/2`
(C) `2`
(D) `–2`
7. If the lines given by `3x + 2ky = 2` and `2x + 5y + 1 = 0` are parallel,
then the value of `k` is
(A) `–5/4`
(B) `2/5`
(C) `15/4`
(D) `3/2`
8. The value of c for which the pair of equations `cx – y = 2` and `6x –
2y = 3` will have infinitely many solutions is
(A) 3
(B) – 3
(C) –12
(D) no value
9. One equation of a pair of dependent linear equations is `–5x + 7y =
2`. The second equation can be
(A) `10x + 14y + 4 = 0`
(B) `–10x – 14y + 4 = 0`
(C) `–10x + 14y + 4 = 0`
(D) `10x – 14y = –4`
10. A pair of linear equations that has a unique solution `x = 2, y = –3`
is
(A) `x + y = –1 , 2x – 3y = –5`
(B) `2x + 5y = –11, 4 x + 10y = –22`
(C) `2x – y = 1` , `3x + 2y = 0`
(D) `x – 4y –14 = 0`, `5x – y – 13 = 0`
11. If x = a, y = b is the solution of the equations `x – y = 2` and `x + y
= 4`, then the values of a and b are, respectively
(A) 3 and 5
(B) 5 and 3
(C) 3 and 1
(D) –1 and –3
12. Aruna has only ₹1 and ₹2 coins with her. If the total number of coins that she has is 50 and the
amount of money with her is ₹75, then the number of ₹1 and ₹2 coins are, respectively
(A) 35 and 15
(B) 35 and 20
(C) 15 and 35
(D) 25 and 25
13. The father’s age is six times his son’s age. Four years hence, the age
of the father will be four times his son’s age. The present ages, in years,
of the son and the father, are, respectively
(A) 4 and 24
(B) 5 and 30
(C) 6 and 36
(D) 3 and 24
14. Which of these linear equations have a unique solution?
(a)
(b)
(c)
(d)
(a) These lines have a unique solution as they are intersecting at a point.
(b) These lines have infinitely many solutions as they lie in the same
quadrant.
(c) These lines have a unique solution as the coefficient of x in both the
equations is one.
(d) These lines have infinitely many solutions as they lie in the same
quadrant.
16. Consider the equations as shown:
`9x + 6y = 5`
`3x + 2y = 7`
★★★★★★
Case Studies
Case Study 1
Amit is planning to buy a house and the layout is given below. The design and
the measurement have been made such that areas of two bedrooms and kitchen
together is 95 sq.m.
1. Form the pair of linear equations in two variables from this situation.
2. Find the length of the outer boundary of the layout.
3. Find the area of each bedroom and kitchen in the layout.
4. Find the area of the living room in the layout.
5. Find the cost of laying tiles in the kitchen at the rate of ₹50 per
sq.m
Case Study 2
A test consists of ‘True’ or ‘False’ questions. One mark is awarded for
every correct answer while ¼ mark is deducted for every wrong answer. A
student knew the answers to some of the questions. The rest of the questions
he attempted by guessing. He answered 120 questions and got 90 marks.
1. If answers to all questions he attempted by guessing were wrong, then how
many questions did he answer correctly?
2. How many questions did he guess?
3. If answers to all questions he attempted by guessing were wrong and
answered 80 correctly, then how many marks he got?
4. If answers to all questions he attempted by guessing were wrong, then how
many questions were answered correctly to score 95 marks?
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📝 Coordinate Geometry (Question Bank)
Multiple Choice Questions
1. The distance of the point P (2, 3) from the x-axis is
(A) 2
(A) 2
(B) 3
(C) 1
(D) 5
2. The distance between points A (0, 6) and B (0, –2) is
(A)6
(B) 8
(C) 4
(D) 2
3. The distance of the point P (–6, 8) from the origin is
(A) 8
(B) `2 sqrt7`
(C) 10
(D) 6
4. The distance between the points (0, 5) and (–5, 0) is
(A) 5
(B) `5 sqrt2`
(C) `2 sqrt5`
(D) 10
5. AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0,
0) and B (5, 0). The length of its diagonal is
(A) 5
(B) 3
(C) `sqrt34`
(D) 4
6. The perimeter of a triangle with vertices (0, 4), (0, 0) and
(3, 0) is
(A) 5
(B) 12
(C) 11
(D) `7+ sqrt5`
7. The area of a triangle with vertices A (3, 0), B (7, 0) and C (8,
4) is
(A) 14
(B) 28
(C) 8
(D) 6
8. The points (–4, 0), (4, 0), (0, 3) are the vertices of a
(A) right triangle
(B) isosceles triangle
(C) equilateral triangle
(D) scalene triangle
9. The point which divides the line segment joining the points (7,
–6) and (3, 4) in ratio 1 : 2 internally lies in the
(A) I quadrant
(B) II quadrant
(C) III quadrant
(D) IV quadrant
10. The point which lies on the perpendicular bisector of the line
segment joining the points A (–2, –5) and B (2, 5) is
(A) (0, 0)
(B) (0, 2)
(C) (2, 0)
(D) (–2, 0)
11. The fourth vertex D of a parallelogram ABCD whose three
vertices are A (–2, 3), B (6, 7) and C (8, 3) is
(A) (0, 1)
(B) (0, –1)
(C) (–1, 0)
(D) (1, 0)
12. If the point P (2, 1) lies on the line segment joining points
A (4, 2) and B (8, 4), then
(A) AP `=1/3`AB
(B) AP = PB
( C) PB `=1/3`AB
(D) AP `=1/2`AB
13. If `P (a/3, 4)` is the mid-point of the line segment joining the points Q (– 6, 5)
and R (– 2, 3), then the value of a is
(A) – 4
(B) – 12
(C) 12
(D) – 6
14. The perpendicular bisector of the line segment joining the
points A (1, 5) and B (4, 6) cuts the y-axis at
(A) (0, 13)
(B) (0, –13)
(C) (0, 12)
(D) (13, 0)
15. The coordinates of the point which is equidistant from the three vertices of the `triangleAOB` as shown in Fig. is
(A) `(x, y)`
(B) `(y, x)`
(C) `x/2, y/2`
(D) `y/2, x/2`
16. A circle is drawn with origin as the centre passes through `(13/2,0)`. The point which does not lie in the interior of the circle is
(A) `-3/4, 1`
(B) `2, 7/3`
(C) `5, -1/2`
(D) `-6, 5/2`
17. A-line intersects the y-axis and x-axis at the points P and Q,
respectively. If (2, –5) is the mid-point of PQ, then the coordinates of
P and Q are, respectively
(A) (0, – 5) and (2, 0)
(B) (0, 10) and (– 4, 0)
(C) (0, 4) and (– 10, 0)
(D) (0, – 10) and (4, 0)
18. The area of a triangle with vertices `(a, b + c), (b, c + a)`,
and `(c, a + b)` is
(A) `(a + b + c)^2`
(B) 0
(C) `a + b + c`
(D) `abc`
19. If the distance between the points (4, p) and (1, 0) is 5,
then the value of p is
(A) 4 only
(B) ± 4
(C) – 4 only
(D) 0
20. If the points A (1, 2), O (0, 0), and C (a, b) are collinear,
then
(A) `a = b`
(B) `a = 2b`
(C) `2a = b`
(D) `a = –b`
★★★★★★
Case Studies
CASE STUDY 1:
In order to conduct Sports Day activities in your School, lines have been
drawn with chalk powder at a distance of 1 m each, in a rectangular-shaped
ground ABCD, 100 flower pots have been placed at a distance of 1 m from
each other along with AD, as shown in the given figure below. Niharika
runs `1/4` th the distance AD on the 2nd line and posts a green flag.
Preet runs `1/5` th distance AD on the eighth line and posts a red flag.
1. Find the position of the green flag
a) (2, 25)
b) (2, 0.25)
c) (25, 2)
d) (0, -25)
2. Find the position of the red flag
a) (8, 0)
b) (20, 8)
c) (8, 20)
d) (8, 0.2)
3. What is the distance between both flags?
a) √41
b) √11
c) √61
d) √51
4. If Rashmi has to post a blue flag exactly halfway between the line
segment joining the two flags, where should she post her flag?
a) (5, 22.5)
b) (10, 22)
c) (2, 8.5)
d) (2.5 ,20)
5. If Joy has to post a flag at one-fourth distance from the green
flag, in the line segment joining the green and red flags, then where
should he post his flag?
a) (3.5, 24)
b) (0.5, 12.5)
c) (2.25, 8.5)
d) (25, 20)
ANSWERS
1. a) (2, 25)
2. c) (8, 20)
3. c) √61
4. a) (5, 22.5)
5. a) (3.5, 24)
Case Study 2:
The class X students school in Krishnagar have been allotted a rectangular
plot of land for their gardening activity. Saplings of Gulmohar are
planted on the boundary at a distance of 1 m from each other. There is a
triangular grassy lawn in the plot as shown in the figure. The students
are to sow seeds of flowering plants on the remaining area of the plot.
1. Taking A as origin, find the coordinates of P
a) (4, 6)
b) (6, 4)
c) (0, 6)
d) (4, 0)
2. What will be the coordinates of R, if C is the origin?
a) (8, 6)
b) (3, 10)
c) (10, 3)
d) (0, 6)
3. What will be the coordinates of Q, if C is the origin?
a) (6, 13)
b) (-6,13)
c) (-13, 6)
d) (13, 6)
4. Calculate the area of the triangles if A is the origin
a) 4.5
b) 6
c) 8
d) 6.25
5. Calculate the area of the triangles if C is the origin
a) 8
b) 5
c) 6.25
d) 4.5
ANSWERS:
1. a) (4, 6)
2. c) (10, 3)
3. d) (13, 6)
4. a) 4.5
5. d) 4.5
Case Study 3
A hockey field is the playing surface for the game of hockey. Historically,
the game was played on natural turf (grass) but nowadays it is predominantly
played on an artificial turf. It is rectangular in shape - 100 yards by 60
yards. Goals consist of two upright posts placed equidistant from the centre
of the backline, joined at the top by a horizontal crossbar. The inner edges
of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the
crossbar must be 2.14 metres (7 feet) above the ground. Each team plays with
11 players on the field during the game including the goalie. Positions you
might play include-
◘ Forward: As shown by players A, B, C, and D.
◘ Midfielders: As shown by players E, F, and G.
◘ Fullbacks: As shown by players H, I, and J.
◘ Goalie: As shown by player K
Using the picture of a hockey field below, answer the questions that follow:
1. The coordinates of the centroid of ΔEHJ are
(a) `(-2/3, 1)`
(b) `(1,-2/3)`
(c) `(2/3,1)`
(d) `( -2/3,-1)`
2. If a player P needs to be at equal distances from A and G, such that A,
P and G are in straight line, then position of P will be given by
(a) `(-3/2, 2)`
(b) `(2,-3/2)`
(c) `(2, 3/2)`
(d) `( -2,-3)`
3. The point on x-axis equidistant from I and E is
(a) `(1/2, 0)`
(b) `(0,-1/2)`
(c) `(-1/2,0)`
(d) `( 0,1/2)`
4. What are the coordinates of the position of a player Q such that his
distance from K is twice his distance from E and K, Q, and E are
collinear?
(a) `(1, 0)`
(b) `(0,1)`
(c) `(-2,1)`
(d) `( -1,0)`
5. The point on y-axis equidistant from B and C is
(a) `(-1, 0)`
(b) `(0,-1)`
(c) `(1,0)`
(d) `( 0,1)`
Ans:
1 (d)
2 (c)
3 (a)
4 (b)
5 (d)
.
More questions will be updated soon.
Very helpful and many questions you are a very nice person
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