RowQ
The Vault
RowQ
The Vault
CBSE Class 9 Maths · 11 questions · 26 marks
Descartes' great idea was that a pair of numbers can pin down a point exactly, turning pictures into arithmetic. Here you learn the axes, the four quadrants and their sign rules, and how to plot and read off points with confidence. It is a short chapter, but every graph you draw for the rest of school depends on getting these conventions right.
The point (-8, 3) lies in which quadrant?
Answer
It lies in Quadrant II. The x-coordinate is negative and the y-coordinate is positive, which is the sign pattern (-, +). That combination belongs to the second quadrant, the region above the x-axis and to the left of the y-axis.
A point whose ordinate is 0 and whose abscissa is -6 lies:
Answer
It lies on the x-axis. The ordinate is the y-coordinate, so the point is (-6, 0). Any point with y = 0 has zero perpendicular distance from the x-axis, so it lies on the x-axis itself — specifically 6 units to the left of the origin.
The perpendicular distance of the point P(5, -9) from the y-axis is:
Answer
The distance is 5 units. Distance from the y-axis is measured horizontally, so it equals the absolute value of the x-coordinate: |5| = 5. The value 9 would be the distance from the x-axis instead. Distance can never be negative, which rules out -5.
If the points A(3, k) and B(3, 7) are joined, the line segment AB (for k ≠ 7) will be:
Answer
It will be parallel to the y-axis. Both points share the same x-coordinate, 3, so every point of the segment is 3 units to the right of the y-axis. A set of points at a constant horizontal distance forms a vertical line, which is parallel to the y-axis.
Assertion (A): The points (2, 9) and (9, 2) represent the same location in the Cartesian plane. Reason (R): In an ordered pair, the first entry is the abscissa and the second is the ordinate.
Answer
A is false but R is true. Reason R correctly describes an ordered pair. Precisely because the order matters, (2, 9) sits 2 units right and 9 units up, while (9, 2) sits 9 units right and 2 units up — two clearly different points. So Assertion A is false, though the reason given is a true statement.
Write the coordinates of a point that lies on the y-axis, 7 units below the origin, and state the sign of each coordinate.
Answer
A point on the y-axis has abscissa 0, so its coordinates are of the form (0, b). Moving 7 units below the origin means b = -7. The coordinates are (0, -7). The x-coordinate is 0, which is neither positive nor negative, and the y-coordinate is negative.
Plot the points P(-4, 2), Q(-4, -3) and R(2, -3) on a Cartesian plane, name the quadrant of each, and state what kind of angle is formed at Q.
Answer
P(-4, 2) has signs (-, +), so it lies in Quadrant II. Q(-4, -3) has signs (-, -), so it lies in Quadrant III. R(2, -3) has signs (+, -), so it lies in Quadrant IV. PQ is a vertical segment because P and Q share the abscissa -4, while QR is a horizontal segment because Q and R share the ordinate -3. A vertical line meets a horizontal line at a right angle, so the angle at Q is 90°.
In which quadrant or on which axis does each of the following lie: A(0, 0), B(11, -6), C(-2, -2), D(0, 5)?
Answer
A(0, 0) is the origin, the point where the two axes intersect; it lies on both axes. B(11, -6) has signs (+, -), so it lies in Quadrant IV. C(-2, -2) has signs (-, -), so it lies in Quadrant III. D(0, 5) has abscissa 0, so it lies on the y-axis, 5 units above the origin.
Plot the points A(1, 1), B(7, 1), C(7, 5) and D(1, 5) and join them in order. (a) Identify the figure ABCD. (b) Find the lengths AB and BC. (c) Find its area and perimeter. (d) State the quadrant in which the whole figure lies.
Answer
(a) A and B share the ordinate 1, so AB is horizontal; B and C share the abscissa 7, so BC is vertical; similarly CD is horizontal and DA is vertical. Adjacent sides therefore meet at right angles, and opposite sides are equal, so ABCD is a rectangle. (b) AB is the difference of the x-coordinates of A and B: 7 - 1 = 6 units. BC is the difference of the y-coordinates of B and C: 5 - 1 = 4 units. (c) Area = length × breadth = 6 × 4 = 24 square units. Perimeter = 2(6 + 4) = 20 units. (d) Every one of the four points has both coordinates positive, so the entire rectangle lies in Quadrant I.
(a) Explain, with examples, how the signs of the coordinates change as a point moves from Quadrant I to Quadrant IV. (b) A point M(a, b) satisfies a < 0 and b > 0. If the point is reflected so that its ordinate keeps its value but its abscissa changes sign, name the quadrant of the new point and illustrate with a = -5, b = 4.
Answer
(a) In Quadrant I both coordinates are positive, e.g. (3, 6). Moving anticlockwise into Quadrant II the abscissa turns negative while the ordinate stays positive, e.g. (-3, 6). In Quadrant III both are negative, e.g. (-3, -6). In Quadrant IV the abscissa is positive again while the ordinate is negative, e.g. (3, -6). So going from Quadrant I to Quadrant IV, the x-coordinate keeps its positive sign while the y-coordinate becomes negative, because the point has crossed below the x-axis without crossing the y-axis. (b) M(a, b) has a negative abscissa and positive ordinate, so M lies in Quadrant II. Changing only the sign of the abscissa makes it positive while the ordinate remains positive, giving the sign pattern (+, +), which is Quadrant I. With a = -5 and b = 4: M(-5, 4) lies in Quadrant II, and the new point (5, 4) lies in Quadrant I. The two points are mirror images of each other in the y-axis.
Read the following and answer the questions that follow: A town planner draws a map of a park on graph paper, taking the central fountain as the origin, with 1 unit representing 10 metres. The gate is marked at G(6, 0), the pond at P(-4, 3), the cafe at C(-2, -5) and the bandstand at B(0, 8). (a) Which of these landmarks lie on an axis, and on which axis? (b) State the quadrant of the pond and of the cafe. (c) How far is the cafe from the x-axis, in metres? (d) The planner adds a bench at the point whose abscissa equals that of the pond and whose ordinate equals that of the gate. Write its coordinates and describe its position.
Answer
(a) G(6, 0) has ordinate 0, so it lies on the x-axis, 6 units to the right of the fountain. B(0, 8) has abscissa 0, so it lies on the y-axis, 8 units above the fountain. (b) P(-4, 3) has signs (-, +) and so lies in Quadrant II. C(-2, -5) has signs (-, -) and so lies in Quadrant III. (c) The distance from the x-axis is the absolute value of the ordinate, |-5| = 5 units. Since 1 unit = 10 metres, the cafe is 50 metres from the x-axis. (d) The abscissa of the pond is -4 and the ordinate of the gate is 0, so the bench is at (-4, 0). Because its ordinate is 0, the bench lies on the x-axis, 4 units (that is, 40 metres) to the left of the fountain.
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