maths class 9

Key Concept: Distance calculation, Concept of quadrants

a) First Quadrant
[Solution Description]
To find the midpoint $C(x, y)$ of a line segment connecting two points $A(x_1, y_1)$ and $B(x_2, y_2)$, we use the formula:
$C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Substituting the given coordinates of $A$ and $B$:
$C\left(\frac{3 + (-1)}{2}, \frac{4 + (-2)}{2}\right) = C\left(\frac{2}{2}, \frac{2}{2}\right) = C(1, 1)$
The coordinates of point $C$ are $(1, 1)$. In the Cartesian plane, this corresponds to the first quadrant where both $x$ and $y$ are positive.

Key Concept: Quadrant Identification

b) Second Quadrant
[Solution Description] A point with coordinates $(-4, 5)$ has a negative x-coordinate and a positive y-coordinate. According to the quadrant rules:
– First Quadrant: $(+, +)$
– Second Quadrant: $(-, +)$
– Third Quadrant: $(-, -)$
– Fourth Quadrant: $(+, -)$
The point $(-4, 5)$ corresponds to the second quadrant.

Key Concept: Complex Coordinates, Analytical Reasoning

b) $\sqrt{29}$
[Solution Description] Treating complex components as separate dimensions, the points are represented as $(0, 2)$ and $(5, 0)$ in the real numbers corresponding to imaginary parts and vice versa for real parts. Applying the distance formula:
$d = \sqrt{(5 – 0)^2 + (0 – 2)^2}$
Simplify further:
$d = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4}$
Continue calculation:
$d = \sqrt{29}$
Hence, the distance between $K$ and $L$ in the complex plane is $\sqrt{29}$ units.

Key Concept: Integration of Sensory Experience and Mathematical Concepts

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion is accurate, as textures provide additional cues to identify and differentiate between various elements on the grid. The reason supports this idea by explaining how textures make abstract ideas more tangible through sensory perception, thus improving understanding.

Key Concept: Midpoint Formula, System Analysis

a) (7, 2)
[Solution Description]
Applying the midpoint relationship:
$J = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
For $K(3, 9)$ and $L(x, -5)$:
\begin{align*}
\frac{3 + x}{2} &= 5
\frac{9 – 5}{2} &= y
\end{align*}
Solve for $x$:
\begin{align*}
3 + x &= 10
x &= 10 – 3
x &= 7
\end{align*}
For $y$:
\begin{align*}
y &= \frac{9 – 5}{2}
y &= \frac{4}{2}
y &= 2
\end{align*}
The solution yields $(x, y) = (7, 2)$.

Key Concept: Coordinate Geometry: Slope of a Line

a) -$\frac{1}{2}$
[Solution Description] The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as:
$m = \frac{y_2 – y_1}{x_2 – x_1}$
Substituting the coordinates for $G(3, 7)$ and $H(9, 4)$:
$m = \frac{4 – 7}{9 – 3}$
$m = \frac{-3}{6} = -\frac{1}{2}$
Therefore, the slope of the line is $-\frac{1}{2}$.

Key Concept: Using Midpoint to Find Another Point

a) $(-5, 0)$
[Solution Description] Let midpoint $M = (-1, 2)$ and point $A = (3, 4)$. So,
\begin{align*}
\frac{3 + x}{2} &= -1
3 + x &= -2
x &= -5
\end{align*}
Similarly for $y$-coordinates,
\begin{align*}
\frac{4 + y}{2} &= 2
4 + y &= 4
y &= 0
\end{align*}
Thus, $B(x, y) = (-5, 0)$.

Key Concept: Midpoint Formula, Midpoint Verification

a) Yes, E is the midpoint
[Solution Description]
The formula to find the midpoint $M$ is:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Compute the midpoint of $C(-2, 4)$ and $D(6, -8)$:
\begin{align*}
M_x &= \frac{-2 + 6}{2} = \frac{4}{2} = 2
M_y &= \frac{4 – 8}{2} = \frac{-4}{2} = -2
\end{align*}
Midpoint $M = (2, -2)$ which matches point $E(2, -2)$, thus point $E$ is indeed the midpoint.

Key Concept: Distance Calculation

a) 5 units
[Solution Description] The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.$
Substituting the values, we get:
\begin{align*}
d &= \sqrt{(5 – 2)^2 + (7 – 3)^2}
&= \sqrt{3^2 + 4^2}
&= \sqrt{9 + 16}
&= \sqrt{25}
&= 5.
\end{align*}
Therefore, the distance is 5 units.

Key Concept: Coordinates in Ancient Civilizations

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Both the assertion and reason are true, and the reason correctly explains why a grid-based layout allows precise locational descriptions by providing a structured framework for coordinates.

Key Concept: Algebraic Entities and Coordinate Planes

b) (-1, -1)
[Solution Description]
The formula for finding the midpoint $M(x,y)$ of segment $AB$ with endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$ is:
$M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Substituting the given values:
$M(x, y) = \left( \frac{3 + (-5)}{2}, \frac{-4 + 2}{2} \right) = \left( \frac{-2}{2}, \frac{-2}{2} \right) = (-1, -1)$
Thus, the midpoint $M(x,y)$ is $(-1, -1)$.

Key Concept: Quadrants, Coordinate Sign

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion is true as per the definition, points in the third quadrant have both negative x and y coordinates. The reason is false; only the fourth quadrant meets this condition of having one positive coordinate (positive x), as the second quadrant has (-, +).

Key Concept: Distance formula, midpoint concept

d) Assertion is false, but Reason is true.
[Solution Description]
Using the midpoint formula for B:
$M_x = \frac{0 + x_B}{2} = -2 \\ \\ 0 + x_B = 2 \times (-2) \\ \\ x_B = -4$
Solve for $y_B$:
$M_y = \frac{6 + y_B}{2} = 3 \\ \\ 6 + y_B = 6 \\ \\ y_B = 0$
Coordinates of B are (-4, 0).
Calculate distance MB using:
$MB = \sqrt{((-4) – (-2))^2 + (0 – 3)^2} \\ \\ MB = \sqrt{(-2)^2 + (-3)^2} \\ \\ MB = \sqrt{4 + 9} \\ \\ MB = \sqrt{13}$
Since $\sqrt{13} \neq 5$, Assertion is false. The Reason statement explains how distances are found correctly.

Key Concept: Distance between points, Baudhāyana-Pythagoras Theorem

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the distance between two points in a 2-D plane, the formula derived from the Pythagorean theorem is used:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
This is because if you consider a triangle formed by these points, the line segment joining them serves as the hypotenuse of a right-angled triangle whose legs are parallel to the x-axis and y-axis respectively.
Since the assertion uses this correct formula for calculating the distance and it correctly relates to the reason provided which describes the basic principle behind the theorem, both assertion and reason are true and related.

Key Concept: Point on Axes

b) $(0, -4)$
[Solution Description] A point on the y-axis has the form $(0, y)$. Given that $y = -4$, the coordinates are $(0, -4)$.

Key Concept: Finding midpoint from endpoints

b) (6, 3)
[Solution Description]
To find the midpoint of a line segment with endpoints G(x_1, y_1) and H(x_2, y_2), we apply the formula:
$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Substituting the given values,
$x = \frac{10 + 2}{2} = 6$
$y = \frac{-5 + 11}{2} = 3$
Hence, the midpoint of GH is (6, 3).

Key Concept: Symmetry and Distance, Quadrant Understanding

b) $2\sqrt{a^2 + b^2}$
[Solution Description] Using the distance formula for points $H(a, b)$ and $J(-a, -b)$, we have:
$d = \sqrt{((-a) – a)^2 + ((-b) – b)^2}$
Simplifying, we get:
$d = \sqrt{(-2a)^2 + (-2b)^2} = \sqrt{4a^2 + 4b^2}$
Factor out the common term:
$d = \sqrt{4(a^2 + b^2)} = 2\sqrt{a^2 + b^2}$
So, the distance between $H(a, b)$ and $J(-a, -b)$ is $2\sqrt{a^2 + b^2}$.

Key Concept: Spatial understanding through scaling

c) Assertion is true, but Reason is false.
[Solution Description]
While scaling down objects is intended to simplify complex environments into more comprehensible parts, doing so primarily enhances the ability to perceive broader patterns, contrary to the suggestion that it allows focus on specifics. The assertion speaks about simplification, whereas the stated reason implies an unrelated advantage of scaling.

Key Concept: Quadrant Identification

d) Assertion is false, but Reason is true.
[Solution Description]
The point $(-2, 3)$ has a negative $x$-coordinate and a positive $y$-coordinate, placing it in the second quadrant, not the third. The reason correctly describes the third quadrant, but the assertion is false.

Key Concept: Identifying Coordinates

a) $(-6, -3)$
[Solution Description] Since the point is 6 units to the left, its x-coordinate is $-6$. Being 3 units below means the y-coordinate is $-3$. Therefore, the coordinates of the point are $(-6, -3)$.

Key Concept: Quadrants, Distance Formula application

c) Assertion is true, but Reason is false.
[Solution Description]
For points $(-3, 4)$ and $(5, -2)$:
Calculate differences:
$x_{\text{difference}} = 5 – (-3) = 5 + 3 = 8$
$y_{\text{difference}} = -2 – 4 = -6$
Applying the distance formula:
$d = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10$
The reason erroneously implies that absolute difference directly affects calculation, rather than being handled within the Pythagorean approach. Thus, the assertion is true, but the reason is false.

Key Concept: Origin, Coordinates

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion is true as the intersection of the x-axis and y-axis at the origin creates four quadrants in the Cartesian plane. The reason is true since the origin is defined as having coordinates $(0, 0)$. However, knowing the coordinates of the origin does not explain how it leads to the division into quadrants.

Key Concept: Rectangular Grid

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states the use of a specific scale for mapping. The reason explains how scales convert real-world measurements into smaller representations, making it relevant but not directly explaining why that scale was chosen.

Key Concept: Quadrants, Distance Formula

Correct Answer option Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] To find the distance from the origin $(0, 0)$ to the point $(3, 4)$, apply the distance formula: $\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. Therefore, the assertion is true. The reasoning provides the correct formula for calculating distances using the Baudhāyana–Pythagoras Theorem. While both statements are true, the reason does not specifically address the distance from the origin; hence, they are related but not directly explanatory.

Key Concept: Navigation and Mapping, Role of Āryabhaṭa

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Āryabhaṭa’s introduction of sines indeed simplified calculating celestial coordinates since sines provided more straightforward computation than chords. This directly impacted how positions were determined in astronomy and navigation using spherical trigonometry. Hence, both assertions and reasons are true, and Reason is the correct explanation of Assertion.

Key Concept: Use of Mathematical Concepts in Mapping

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Āryabhaṭa’s replacement of ‘chords’ with ‘sines’ indeed simplified calculations and aided in celestial mapping by providing a more straightforward trigonometric function for measurement.

Key Concept: Coordinate Signs

c) Both x and y are negative
[Solution Description] In Quadrant III, both the x- and y-coordinates are negative. So, the correct statement is: “Both x and y are negative.”

Key Concept: Midpoint extension, triangle properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Given U is the midpoint of JK, let J be (x, y); K is (a, b) so:
$3 = \frac{x + a}{2}, \quad 6 = \frac{y + b}{2} \\ \\ x + a = 6, \quad y + b = 12$
Similarly, consider V being the midpoint of KL, and calculate backward:
$6 = \frac{a + c}{2}, \quad 9 = \frac{b + d}{2} \\ \\ a + c = 12, \quad b + d = 18$
Finally, examine W as the midpoint of LJ:
$9 = \frac{c + x}{2}, \quad 12 = \frac{d + y}{2} \\ \\ c + x = 18, \quad d + y = 24$
Solving this system confirms J = (0, 3). Thus, assertion holds true, and reason supports it.
Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Development of Coordinate Geometry

Correct Answer option b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] While both statements are true about Baudhāyana’s contributions and the theorem’s role in geometry, the theorem itself did not specifically introduce grids for navigation; it was more fundamental to geometric constructions.

Key Concept: Trisection, geometric applications

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For E dividing GH in the ratio 1:2:
$x_E = \frac{1 \cdot 8 + 2 \cdot 2}{1+2} = \frac{8 + 4}{3} = 4 \\ \\ y_E = \frac{1 \cdot 9 + 2 \cdot 3}{1+2} = \frac{9 + 6}{3} = 5$
Hence, E(4, 5) accurately trisects segment GH as specified.
Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Use of Coordinates, History of Coordinate System

Correct Answer option b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Baudhāyana–Pythagoras Theorem contributed to the mathematical basis needed for establishing coordinate geometry, as it involves right-angled triangles used in grid-based systems. However, Baudhāyana’s use of directional lines also had practical applications beyond religion, such as city planning, which implies his work was not solely religious. Therefore, both statements are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Exploration, significance, accuracy

c) 20
[Solution Description] Starting from J(0, 0) to I(3, 3), Reiaan needs to make three rightward moves and three upward moves. This is a combinatorial path problem involving arranging six steps (three ‘R’ for right, three ‘U’ for up).
Number of distinct paths:
$\binom{6}{3} = 20$
Thus, there are 20 different sequences leading to point I(3, 3).

Key Concept: Baudhāyana-Pythagoras Theorem

c) Assertion is true, but Reason is false.
[Solution Description] The assertion is true as it gives the correct formula for finding the distance between two points in a 2-D plane. However, the reason is false because the Baudhāyana-Pythagoras theorem is applied broadly across many geometric contexts, including but not limited to right-angled triangles.

Key Concept: Application of Distance Formula

b) 5 units
[Solution Description] The diagonal can be found using any two opposite corners of the rectangle. Let’s use the points $(2, 3)$ and $(5, 7)$. Using the distance formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Here,
\begin{align*}
x_1 &= 2, \quad y_1 = 3,
x_2 &= 5, \quad y_2 = 7
\end{align*}
Substituting these values:
\begin{align*}
d &= \sqrt{(5 – 2)^2 + (7 – 3)^2}
&= \sqrt{3^2 + 4^2}
&= \sqrt{9 + 16}
&= \sqrt{25}
&= 5
\end{align*}
Thus, the length of the diagonal is 5 units.

Key Concept: Grid Coordinate Understanding

a) (5, 0)
[Solution Description] The midpoint formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ Applying this to the points (0, 0) and (10, 0):
$M = \left(\frac{0+10}{2}, \frac{0+0}{2}\right) = (5, 0)$

Key Concept: Quadrants Identification

c) Quadrant III
[Solution Description] A point with both coordinates negative, such as $(-x, -y)$, is found in Quadrant III.
(i) Quadrant I: $(+, +)$
(ii) Quadrant II: $(-, +)$
(iii) Quadrant III: $(-, -)$
(iv) Quadrant IV: $(+, -)$

Key Concept: Concept of Origin and Axes, Reflection across axes

a) $(-a, -b)$
[Solution Description]
When a point $(a, b)$ is reflected across the x-axis, its new coordinates become $(a, -b)$. Reflecting this point again across the y-axis results in $(-a, -b)$.
Thus, the final coordinates of $D’$ after reflecting across both axes are $(-a, -b)$.

Key Concept: Quadrants in Cartesian plane

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion states that the point $(-3, -4)$ is in the third quadrant. This is true since both the x and y-coordinates are negative. Reason provides a valid explanation by stating that in the third quadrant, both coordinates are negative. Therefore, the reason correctly explains the assertion.

Key Concept: Concept of scale

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A scale like 1 cm : 1 foot provides a means of translating real-world dimensions into a manageable form for modeling. This enables accurate representation of spatial relationships within the model, reflecting real-world proportions exactly. Since the assertion involves utilizing this principle effectively, both the Assertion and Reason are true and Reason explains why using such a scale is important.

Key Concept: Rectangular Grid

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion mentions using pins for marking points for tactile assistance. The reason incorrectly links the purpose of a pin solely to contact points between objects and grid surfaces.

Key Concept: Using Midpoints to Find Vertices

a) (-6, 8)
[Solution Description]
The coordinates of vertex opposite to any side in a triangle when midpoints are known require doubling the midpoint and subtracting one corresponding vertex twice.
Consider J as midpoint of a side segment opposite the third vertex; we have:
$x_{\text{vertex}} = 2 \times 2 \times x_J – (x_K + x_L) = 4 – (6 + 4) = -6$
$y_{\text{vertex}} = 2 \times 2 \times y_J – (y_K + y_L) = 10 – (3 – 1) = 8$
Thus, the coordinates of the vertex opposite to J are (-6, 8).

Key Concept: Coordinates and Quadrants

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both the Assertion and Reason are true. Points in the second quadrant have negative x-coordinates and positive y-coordinates, so they fit the form $(-, +)$. Hence, the Reason correctly explains why the Assertion is true.

Key Concept: Quadrants, Midpoint

c) Quadrant III
[Solution Description] To find the midpoint $M(x_m, y_m)$:
\begin{align*}
x_m &= \frac{-4 + 2}{2} = -1
y_m &= \frac{3 + (-7)}{2} = -2
\end{align*}
Therefore, the midpoint $M$ is $(-1, -2)$.
Both coordinates are negative, so the midpoint lies in Quadrant III.

Key Concept: Quadrants, Coordinate Plane

c) II, III, and IV
[Solution Description] The quadrants can be determined by checking the signs of the coordinates of the endpoints.
Point $A(3, -4)$ is in Quadrant IV because $3 > 0$ and $-4 < 0$.
Point $B(-5, 2)$ is in Quadrant II because $-5 0$.
Since the line connects these two points, it will pass through Quadrants II, III, and IV as it moves from $B$ to $A$.

Key Concept: Geometric Constructions and Historical Context

b) The diameter is twice the radius
[Solution Description]
Baudhāyana conceptualized geometric constructions with precision. In a circle, the radius is half of the diameter. Therefore, if the length of the rope used as the radius is $r$, then the diameter $d$ of the circle is given by:
$d = 2 \times r$
This is a foundational concept used in understanding circles within coordinate geometry, where every point on the circumference is equidistant from the center.

Key Concept: Use of tactile learning aids

c) Assertion is true, but Reason is false.
[Solution Description]
Tactile learning aids like pins and threads can indeed facilitate understanding of spatial concepts for visually impaired individuals by allowing them to ‘feel’ the layout. However, while these aids engage the sense of touch, they do not inherently provide auditory feedback unless specifically designed to do so. As a result, the assertion is valid, but the reason provided does not apply to this specific case.

Key Concept: Midpoint in 2D Plane

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion correctly states the formula used to find the midpoint of a segment connecting two points $A(x_1, y_1)$ and $B(x_2, y_2)$. However, the reason provides the formula for calculating the distance between two points, which is unrelated to finding the midpoint. Therefore, while both statements are true, the reason does not explain the assertion.

Key Concept: Point locations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion is true because the point $(0, -7)$ lies on the y-axis with an x-coordinate of zero. Reason also holds true since all points on the y-axis have their x-coordinates as zero. Furthermore, Reason correctly justifies this Assertion.

Key Concept: Navigating with coordinate geometry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
By assigning numerical values to positions on a grid via coordinates, one can systematically plan and describe routes. This numerical representation makes path calculation straightforward using formulas from coordinate geometry, thus supporting the assertion directly. Consequently, both statements are true and the Reason supports the Assertion perfectly.

Key Concept: Historical Influence on Modern Systems

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] Descartes’s coordination system indeed enabled the visualization of algebraic equations in geometric terms, unifying algebra and geometry and enhancing mathematical analysis.

Key Concept: Integration of Concepts

c) By integrating algebra with geometry through coordinates
[Solution Description] René Descartes introduced the idea that any point in a two-dimensional plane can be represented by a pair of numerical coordinates. This was a critical step in merging algebraic equations with geometric shapes, forming the basis of analytic geometry.

Key Concept: Application of Coordinate Geometry in Real-Life Navigation

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion correctly states that coordinates enable exact positioning on the grid, enhancing navigational accuracy for Reiaan. The reason accurately describes how a coordinate system functions, detailing the role of numerical pairs in defining location. Additionally, the reason thoroughly explains the working principle of using coordinates.

Key Concept: Right triangles, Coordinate geometry

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Let’s calculate distances for AB and BC:
AB:
$AB = \sqrt{(4 – 1)^2 + (6 – 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
BC:
$BC = \sqrt{(7 – 4)^2 + (2 – 6)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Although AB = BC, reason provided does not relate to assertion as it deals with all orientations. Therefore, both assertion and reason are true, but reason does not explain the assertion.

Key Concept: Quadrant Location

b) x is positive and y is negative
[Solution Description] In Quadrant IV, the x-coordinate is positive and the y-coordinate is negative. Hence, the statement “x is positive and y is negative” is true for Quadrant IV.

Key Concept: Midpoint formula, finding unknown coordinates

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the coordinates of T, use the midpoint formula:
$M_x = \frac{3 + x_T}{2} = 1$
Solve for $x_T$:
$3 + x_T = 2 \times 1 \\ \\ x_T = 2 – 3 \\ \\ x_T = -1$
Now solve for $y_T$:
$M_y = \frac{4 + y_T}{2} = 2 \\ \\ 4 + y_T = 2 \times 2 \\ \\ y_T = 4 – 4 \\ \\ y_T = 0$
Therefore, the coordinates of T are (-1, 0).
Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Quadrants

b) Second Quadrant
[Solution Description]
Points in the first quadrant have positive x and y coordinates $(+, +)$, whereas points in the second quadrant have negative x and positive y coordinates $(-, +)$. Here, $(-3, 4)$ fits the description for the second quadrant.

Key Concept: Distance Calculation on Grid

a) 5 units
[Solution Description] To calculate the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the distance formula:
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Substituting the given values:
$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Key Concept: Distance Between Two Points

c) 5 units
[Solution Description] Using the distance formula:
$\text{Distance} = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \\ \\ = \sqrt{(4 – 1)^2 + (6 – 2)^2} \\ \\ = \sqrt{3^2 + 4^2} \\ \\ = \sqrt{9 + 16} \\ \\ = \sqrt{25} \\ \\ = 5$
Therefore, the distance is 5 units.

Key Concept: Coordinate Conversion and Celestial Mapping

c) $x = \cos(\delta) \cdot \cos(\alpha), y = \cos(\delta) \cdot \sin(\alpha), z = \sin(\delta)$
[Solution Description]
To transform celestial coordinates $(\alpha, \delta)$ into Cartesian coordinates on a unit sphere, use the following transformations:
$x = \cos(\delta) \cdot \cos(\alpha) \\ \\ \\ y = \cos(\delta) \cdot \sin(\alpha) \\ \\ \\ z = \sin(\delta)$
These equations project the spherical coordinates onto a Cartesian plane, representing positions on a celestial globe.

Key Concept: Trigonometry Integration in Coordinate Systems

a) $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$
[Solution Description]
For a unit circle, the coordinates of a point on the circle are given by the trigonometric functions cosine and sine. Specifically, for an angle $\theta$, the point’s coordinates $(x, y)$ are found using:
$x = \cos(\theta), \quad y = \sin(\theta)$
At $\theta = 45^\circ$, known angles provide standard trigonometric values:
$x = \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad y = \sin(45^\circ) = \frac{\sqrt{2}}{2}$
Therefore, the point on the unit circle is $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Key Concept: Quadrant Coordinates

c) Assertion is true, but Reason is false.
[Solution Description]
In the second quadrant, the x-coordinate is negative and the y-coordinate is positive, so the Assertion is true. However, the Reason incorrectly states that both x and y are negative, which is false.

Key Concept: Coordinates in Quadrants

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The given coordinates $(3, 5)$ have both $x = 3$ and $y = 5$ positive, which matches the condition for a point to be in the first quadrant. The reason correctly explains why the assertion is true.

Key Concept: Evolution of Navigation Techniques

Correct Answer option c) Assertion is true, but Reason is false.
[Solution Description] Al-Bīrūnī improved the astrolabe using trigonometry for better navigation, not just time-keeping, making the reason incorrect concerning its primary usage.

Key Concept: Distance Between Points, Quadrants

c) $\sqrt{85}$
[Solution Description] To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
For points $A(-3, 4)$ and $B(4, -2)$, substitute the coordinates into the formula:
$d = \sqrt{(4 – (-3))^2 + (-2 – 4)^2}$
Simplify the expression:
$d = \sqrt{(4 + 3)^2 + (-6)^2} = \sqrt{7^2 + 6^2}$
Calculate further:
$d = \sqrt{49 + 36} = \sqrt{85}$
Therefore, the distance between the points is $\sqrt{85}$ units.

Key Concept: Midpoint Calculation

b) $(6, 0)$
[Solution Description] Using the midpoint formula $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$:
$M = \left(\frac{8 + 4}{2}, \frac{2 + (-2)}{2}\right) = (6, 0)$

Key Concept: Point on X-axis

a) It is located on the x-axis.
[Solution Description] A point with coordinates $(x, 0)$ implies it lies directly on the x-axis since the y-coordinate is zero. The exact position on the x-axis depends on the value of $x$, whether positive or negative.

Key Concept: Coordinates of points on x-axis

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion states that a point with coordinates $(5, 0)$ lies on the x-axis, which is true because any point on the x-axis has its y-coordinate as zero. Reason correctly explains why the assertion is true: the y-coordinate for points on the x-axis is always zero.

Key Concept: Distance Formula

d) Assertion is false, but Reason is true.
[Solution Description]
Using the distance formula $\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$, we plug in the values:
$\text{Distance} = \sqrt{(7 – 3)^2 + (2 – (-4))^2} = \sqrt{4^2 + 6^2}$
$= \sqrt{16 + 36} = \sqrt{52}$
$\neq 10$
So, the assertion is incorrect in stating that the distance is 10.

Key Concept: Coordinates, Quadrants

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion is true because by definition, the first quadrant contains points where both coordinates $(x, y)$ are positive. The reason is also true since in the second quadrant the x-coordinate is negative and the y-coordinate is positive. However, the reason does not explain why points in the first quadrant have positive coordinates; thus, it is not a correct explanation for the assertion.

Key Concept: Coordinates on Axes

c) Assertion is true, but Reason is false.
[Solution Description]
The coordinates $(0, -7)$ mean $x = 0$, thus it lies on the y-axis as per the definition. However, the reason incorrectly states that points on the y-axis have the form $(x, 0)$, which actually describes points on the x-axis.

Key Concept: Historical Influence and Coordinate Systems

b) 56
[Solution Description]
To solve this problem, recognize that moving 5 units north and 3 units east requires a total of $5 + 3 = 8$ steps. The number of distinct paths is determined by choosing which 3 of these 8 steps are eastward movements (or equivalently, which 5 are northward). This can be calculated using combinations:
$\text{Number of distinct paths} = \binom{8}{3} = \frac{8!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$
Hence, there are 56 distinct paths.

Key Concept: Rectangular Grid

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion is about using a rectangular grid for navigation by visually impaired individuals. Reason explains the method of connecting points with wool threads for tactile understanding, linking both statements conceptually.

Key Concept: Distance between points

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion is true as the distance from $(3, 4)$ to the origin $(0, 0)$ using the formula $\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25}$. The Reason provides the correct formula for calculating this distance; hence both statements are true, with Reason explaining the Assertion.

Key Concept: Understanding Rectangular Grid

a) Square
[Solution Description] The coordinates provided represent the vertices of a rectangle in a two-dimensional plane. The difference between x-coordinates (5 – 2 = 3) gives the width, and the difference between y-coordinates (6 – 3 = 3) gives the height. Since both differences are equal, it forms a square.

Key Concept: Practical Implications of Using Grids in Spatial Orientation

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion highlights the advantages of tactile models over verbal descriptions for spatial awareness, which is valid given their interactive nature. The reason expands on this by stating that physical models lead to the formation of lasting mental images, supporting the effectiveness of such approaches.

Key Concept: Using Baudhāyana-Pythagoras Theorem

c) 5 units
[Solution Description] From the distance formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Given $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$,
$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Thus, the distance is 5 units.

Key Concept: Reflection About Origin

a) $(-2, 3)$
[Solution Description] Reflecting a point $(a, b)$ about the origin results in $(-a, -b)$. Thus, reflecting $(2, -3)$ about the origin gives us $(-2, 3)$.

Key Concept: Axes, Coordinates

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion is true because by definition, if the x-coordinate is zero, the point lies on the y-axis. The reason is false; for a point on the y-axis, its x-coordinate is zero, but its y-coordinate (and hence the vertical distance from the x-axis) can be non-zero unless it’s at the origin.

Key Concept: Using Rectangular Grid and Coordinate Geometry

Correct Answer option b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] The assertion that Shalini uses a rectangular grid to help Reiaan navigate is correct; it provides tactile feedback for orientation. The reason states that a grid system transforms 3D spaces into 2D planes, which is indeed true as grids typically represent floors or surfaces in two dimensions. However, this transformation isn’t why Shalini’s method helps Reiaan; her method helps due to the tactile nature of the setup, not specifically because of the dimensional conversion.

Key Concept: Coordinate Geometry

c) Assertion is true, but Reason is false.
[Solution Description]
In coordinate geometry, different quadrants have specific sign conventions for their coordinates. In the first quadrant, both coordinates are positive: $(+, +)$. Thus, Assertion is true. However, in the fourth quadrant, the coordinates are actually $(+, -)$—where x is positive and y is negative. Hence, the Reason is false.

Key Concept: Distance Formula

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
When calculating the distance between two points with the same y-coordinate, the formula simplifies to the absolute difference of their x-coordinates, i.e., $|x_2-x_1|$, so the assertion is true. The reason mentions the Baudhāyana–Pythagoras Theorem, which is generally used for finding the distance between any two points $(x_1, y_1)$ and $(x_2, y_2)$ in the Cartesian plane; however, it is not specifically required when the y-coordinates are identical.

Key Concept: Calculation of Distance Using Formula

b) 5 units
[Solution Description] Utilizing the distance formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
With $(x_1, y_1) = (-2, 1)$ and $(x_2, y_2) = (1, -3)$,
$d = \sqrt{(1-(-2))^2 + ((-3)-1)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Consequently, the distance is 5 units.

Key Concept: Midpoint of Triangle Sides

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Knowing the midpoints D, E, and F helps calculate the vertices A, B, and C through the use of the midpoint formula. Each midpoint divides its respective side into two equal segments, thus helping develop equations to solve for the individual coordinates of A, B, and C.

Key Concept: Astronomical Tools

b) Astrolabe
[Solution Description] Al-Bīrūnī developed and improved the astrolabe, a device that enabled navigators to find their position on Earth by making measurements of celestial bodies, such as stars.

Key Concept: Relative position of points, Concept of Origin

c) 15 units
[Solution Description]
The shortest path along the coordinate axes from $K(7, 0)$ to $L(0, -8)$ involves moving horizontally from $K$ to the origin $(0, 0)$ and then vertically to $L$. This path consists of two segments:
– From $K(7, 0)$ to the origin $(0, 0)$: horizontal distance is $|7 – 0| = 7$ units.
– From the origin $(0, 0)$ to $L(0, -8)$: vertical distance is $|0 – (-8)| = 8$ units.
The total distance traveled is $7 + 8 = 15$ units.

Key Concept: Distance from Origin

c) 15 units
[Solution Description] The distance from the origin $(0, 0)$ to a point $(x, y)$ is given by:
$d = \sqrt{x^2 + y^2}$
For the point $(9, -12)$:
\begin{align*}
x &= 9,
y &= -12
\end{align*}
Therefore,
\begin{align*}
d &= \sqrt{9^2 + (-12)^2}
&= \sqrt{81 + 144}
&= \sqrt{225}
&= 15
\end{align*}
So, the distance is 15 units.

Key Concept: Quadrants, Coordinates on Axes

Correct Answer option Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] The assertion that a point $(0, 6)$ lies on the y-axis is true because it satisfies the condition for points on the y-axis: having an x-coordinate of $0$. The reason given states this property directly and accurately describes why the assertion is true; hence, both statements are true, and the reason is the correct explanation.

Key Concept: Rectangular grid, coordinate geometry, navigation

b) -1
[Solution Description] The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a plane can be found using the distance formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Given that $d = 5$, $A = (3, 4)$, and $B = (x, 7)$, we substitute into the formula:
$5 = \sqrt{(x – 3)^2 + (7 – 4)^2}$
Simplifying inside the square root:
$5 = \sqrt{(x – 3)^2 + 3^2} = \sqrt{(x – 3)^2 + 9}$
Squaring both sides:
$25 = (x – 3)^2 + 9$
Subtracting 9 from both sides:
$16 = (x – 3)^2$
Taking the square root of both sides gives:
$x – 3 = \pm 4$
Solving for $x$:
$x = 3 + 4 = 7 \quad \text{or} \quad x = 3 – 4 = -1$
Thus, the possible values of $x$ are $-1$ or $7$.

Key Concept: Rectangular Grid

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion correctly highlights how tactile exploration aids familiarity. The reason, however, inaccurately suggests no visual aid is required, ignoring the possibility of mixed methods.

Key Concept: Point Representation

c) Assertion is true, but Reason is false.
[Solution Description]
The origin is a special point distinct from the axes and quadrants with coordinates $(0, 0)$. Thus, the assertion about the origin’s coordinates being $(0, 0)$ is true. However, the reason indicating every point is either on an axis or within a quadrant neglects the origin, which is neither part of an axis nor a quadrant. Hence, the reason is false.

Key Concept: Quadrants, Distance between points

Correct Answer option Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To calculate the distance between two points $(3, 5)$ and $(7, 5)$ where both points have the same y-coordinate, we use the formula for horizontal distance: $|x_2 – x_1|$. Here, $x_2 = 7$ and $x_1 = 3$. Therefore, the distance is $|7 – 3| = 4$. Both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Distance between Points

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The points $(1, 4)$ and $(1, 8)$ have the same $x$-coordinate. Therefore, the distance is $|8 – 4| = 4$. The reason correctly explains this calculation method.

Key Concept: Coordinate Connection

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For M(-7, 1) to be the midpoint, it holds:
$-7 = \frac{3 + x}{2}, \quad 1 = \frac{-4 + y}{2}$
Solving:
From the first equation:
$-14 = 3 + x \Rightarrow x = -17$
From the second equation:
$2 = -4 + y \Rightarrow y = 6$
Thus, B’s coordinates are (-17, 6). The assertion and reason both reflect the correct mathematical principle accurately.

Key Concept: Coordinate Systems

c) To facilitate navigation and location finding within cities
[Solution Description] The Sindhu-Sarasvatī civilization used a grid-based coordinate system for urban planning. The streets were laid out with precision in North–South and East–West directions, allowing inhabitants to locate places like shops or warehouses by counting units of distance from the city center.

Key Concept: Quadrants, Reflection

c) $(x, -y)$
[Solution Description] For reflection across the y-axis, the x-coordinate changes sign while the y-coordinate remains unchanged.
Suppose $C(x, y) = (-x_1, -y_1)$, then its reflection would be $(-(-x_1), -y_1) = (x_1, -y_1)$.
Hence, the reflected coordinates are $(|x|, -|y|)$.

Key Concept: Distance from origin

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To find the distance from the origin $(0, 0)$ to the point $(3, 4)$, we use the formula for the distance between two points, where $x_1 = 0$, $y_1 = 0$, $x_2 = 3$, and $y_2 = 4$:
$\text{Distance} = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Hence, the distance is indeed 5 units. However, while the Reason describes an accurate method for calculating distance, it doesn’t directly explain why the specific assertion about the distance being 5 units is true, so Reason is not the correct explanation of Assertion.

Key Concept: Concept of Origin

d) $(0, 0)$
[Solution Description] The origin is defined as the intersection point of the x-axis and the y-axis, and its coordinates are $(0, 0)$.

Key Concept: Distance using Pythagorean theorem, Coordinate system

c) 13 units
[Solution Description]
Using the Baudhāyana–Pythagoras theorem, the distance $d$ between the origin $(0, 0)$ and a point $(x, y)$ is:
$d = \sqrt{(x-0)^2 + (y-0)^2} = \sqrt{x^2 + y^2}$
Substitute $x = 12$ and $y = 5$:
$d = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$
The distance is 13 units.

Key Concept: Quadrants Identification

d) Quadrant IV
[Solution Description] The coordinates $(4, -7)$ have a positive x-coordinate and a negative y-coordinate.
Based on the given definitions:
(i) Quadrant I: $(+, +)$
(ii) Quadrant II: $(-, +)$
(iii) Quadrant III: $(-, -)$
(iv) Quadrant IV: $(+, -)$
Since the point has a positive x-value and a negative y-value, it lies in Quadrant IV.

Key Concept: Midpoint Formula, Trisection Points

a) (5.33, 38)
[Solution Description]
Since P and Q divide AB into three equal segments, P will be one-third from A, giving us:
\begin{align*}
P_x &= \frac{2(8) + 8(2)}{3}
P_y &= \frac{2(-3) + 8(15)}{3}
\end{align*}
Calculating $P_x$:
\begin{align*}
P_x &= \frac{2 \times 8 + 8 \times 2}{3}
&= \frac{16 + 16}{3}
&= \frac{32}{3}
&\approx 5.33
\end{align*}
Calculating $P_y$:
\begin{align*}
P_y &= \frac{2 \times (-3) + 8 \times 15}{3}
&= \frac{-6 + 120}{3}
&= \frac{114}{3}
&= 38
\end{align*}
Therefore, $P$ is approximately at $\left(\frac{32}{3}, 38\right)$ or $(5.33, 38)$.

Key Concept: Rectangular grid, orientation, exact physical locations

b) 11 meters
[Solution Description] To find the shortest path moving only along the grid lines, calculate the horizontal and vertical distances separately:
Horizontal movement: $|x_2 – x_1| = |6 – 2| = 4$ meters
Vertical movement: $|y_2 – y_1| = |8 – 1| = 7$ meters
Total walking distance along the grid lines is:
$|x_2 – x_1| + |y_2 – y_1| = 4 + 7 = 11 \text{ meters}$

Key Concept: Coordinate Systems and Historical Development

Correct answer option b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that the concept of coordinate systems began with the Sindhu-Sarasvatī Civilization’s use of grids, which is historically accurate as they used a systematic layout in city-planning. The reason connects the idea of coordinate systems to their current applications in cartography and navigation. Both statements are true; however, while the reason provides context on the significance of coordinate systems today, it doesn’t directly explain the historical origin mentioned in the assertion. Thus, both are true, but the reason is not the direct explanation of the assertion.

Key Concept: Points on x-axis

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The coordinates $(5, 0)$ indicate a point where $y = 0$, confirming its location on the x-axis. This matches the correct description of points on the x-axis provided by the reason.

Key Concept: Coordinate Geometry: Equation of a Line in Point-Slope Form

a) $y = -2x + 7$
[Solution Description] The equation of a line in point-slope form is given by:
$y – y_1 = m(x – x_1)$
Substitute the given point $I(2, 3)$ and slope $m = -2$:
$y – 3 = -2(x – 2)$
Simplifying,
$y – 3 = -2x + 4$
Adding 3 to both sides,
$y = -2x + 7$
This is the equation of the line.

Key Concept: Midpoint of a Segment

a) (–1, –2)
[Solution Description]
To find the coordinates of B, let’s use the midpoint formula: if M(x, y) is the midpoint of segment AB where A(x_1, y_1) and B(x_2, y_2), then:
$x = \frac{x_1 + x_2}{2}, \quad y = \frac{y_1 + y_2}{2}$
Given that M(2, 3) and A(5, 8), we can set up the equations:
$2 = \frac{5 + x_2}{2} \quad \Rightarrow \quad 4 = 5 + x_2 \quad \Rightarrow \quad x_2 = -1$
$3 = \frac{8 + y_2}{2} \quad \Rightarrow \quad 6 = 8 + y_2 \quad \Rightarrow \quad y_2 = -2$
Therefore, the coordinates of B are (-1, -2).

Key Concept: Coordinate Geometry: Midpoint of a Line Segment

b) (5, 2.5)
[Solution Description] The midpoint $M$ of the line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is found by using the formula:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Substituting the given coordinates for $A(3, 4)$ and $B(7, 1)$:
$M = \left( \frac{3 + 7}{2}, \frac{4 + 1}{2} \right)$
$M = \left( \frac{10}{2}, \frac{5}{2} \right) = (5, 2.5)$
Hence, the coordinates of point $C$ are $(5, 2.5)$.

Key Concept: Practical example, orientation, coordinate geometry

b) 5
[Solution Description] Since the sounds are heard equally, Reiaan is equidistant from both beacons. Thus, $H$ lies on the perpendicular bisector of line $FG$.
The midpoint of $F(0, 0)$ and $G(10, 0)$ is $M(5, 0)$.
The line through $M$ and perpendicular to $FG$ (horizontal line) has equation $x = 5$.
Therefore, the coordinate $H(x, 5)$ must satisfy $x = 5$.

Key Concept: Distance Between Two Points

d) 5 units
[Solution Description] To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Here, $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (5, 7)$. So,
$d = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
Hence, the distance is 5 units.

Key Concept: Quadrants, Coordinate System

b) $(p, -q)$
[Solution Description] In Quadrant I, both coordinates are positive: $p > 0$ and $q > 0$.
When shifting to Quadrant IV, the x-coordinate remains positive, but the y-coordinate becomes negative: $p > 0$ and $q < 0$.
The transformation maintains $p$'s positivity while changing the sign of $q$.

Key Concept: Origin and Axes

b) $(0, 0)$
[Solution Description]
The origin is the point where the x-axis and y-axis intersect in a 2-D Cartesian coordinate system. By definition, the coordinates of the origin are $(0, 0)$.

Key Concept: Quadrants and their Coordinates

c) Assertion is true, but Reason is false.
[Solution Description]
In the Cartesian coordinate system, each quadrant has defined coordinates:
– Quadrant I: $(+,+)$
– Quadrant II: $(-,+)$
– Quadrant III: $(-,-)$
– Quadrant IV: $(+,-)$
The assertion is true because points in the second quadrant do have positive y-coordinates. However, the reason provided states that both coordinates are negative in the second quadrant, which is incorrect since only the x-coordinate is negative while the y-coordinate is positive in the second quadrant.

Key Concept: Conceptual Understanding of Scale and Distance Measurement

Correct Answer option c) Assertion is true, but Reason is false.
[Solution Description] The assertion is valid as the scale allows a tangible experience relative to real-world distances. The reason incorrectly explains proportional scaling; different units do affect perceived sizes unless scaled accordingly. Thus, while scaling aids understanding, the claim about size consistency across units is false.

Key Concept: Development of Coordinate Systems

d) Zero and negative numbers
[Solution Description] Brahmagupta formalized the notion of zero and negative numbers which are essential in defining the origin and axes in modern coordinate systems.

Key Concept: Symmetry in Coordinates

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The statement $(x, y) = (y, x)$ implies equality between corresponding components of the ordered pairs. This means $x=y$ for the equality to hold true, thus making the Assertion true. The Reason provides the geometrical interpretation of this condition, showing symmetry about the line $y=x$. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Quadrants, Identifying Quadrant based on Sign of Coordinates

Correct Answer option Assertion is true, but Reason is false.
[Solution Description] Points in the third quadrant have both x- and y-coordinates negative. For a point with coordinates $(-4, -10)$, both coordinates are indeed negative, so it lies in the third quadrant. However, the reason incorrectly describes the characteristics of points in the third quadrant—they should have both coordinates negative, not a positive y-coordinate. Hence, the assertion is true, but the reason is false.

Key Concept: Coordinate Geometry: Distance on a Grid

b) 5 feet
[Solution Description] The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a coordinate plane is given by the formula:
$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Substituting the given coordinates for $A(3, 4)$ and $B(7, 1)$:
$d = \sqrt{(7 – 3)^2 + (1 – 4)^2}$
$d = \sqrt{4^2 + (-3)^2}$
$d = \sqrt{16 + 9} = \sqrt{25} = 5$
Thus, the distance is 5 feet.

Key Concept: Midpoint Formula

c) $(4, 5)$
[Solution Description] The formula for midpoint $M(x, y)$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. For points $A(2, 3)$ and $B(6, 7)$,
$M = \left(\frac{2 + 6}{2}, \frac{3 + 7}{2}\right) = (4, 5)$

Key Concept: Verification of Points on Circle

a) Yes
[Solution Description] A point $(x, y)$ lies on a circle centered at the origin with radius $r$ if:
$\sqrt{x^2 + y^2} = r$
For point $(6, 8)$ and radius $r = 10$:
\begin{align*}
x &= 6,
y &= 8
\end{align*}
Calculating the distance:
\begin{align*}
\sqrt{6^2 + 8^2} &= \sqrt{36 + 64}
&= \sqrt{100}
&= 10
\end{align*}
Since the calculated distance equals the radius, the point $(6, 8)$ does lie on the circle.

Key Concept: Pioneers in Geometry and Trigonometry

Correct answer option b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion talks about Āryabhaṭa’s innovation which indeed made trigonometric calculations more straightforward by simplifying the method from chords to sines. The reason discusses Brahmagupta’s introduction of zero, which was crucial for mathematical advancements including the development of the Cartesian plane. While both statements are independently correct, the reason does not directly explain the change from chords to sines mentioned in the assertion. Hence, both are true, but the reason is not the explanation for the assertion.

Key Concept: Distance Formula – Baudhāyana-Pythagoras Theorem

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ using the Baudhāyana–Pythagoras theorem, we use the formula:
$AD = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Here, $x_1 = 3$, $y_1 = 4$, $x_2 = 7$, and $y_2 = 1$. Substitute these values into the formula:
$AD = \sqrt{(7-3)^2 + (1-4)^2} \\ = \sqrt{4^2 + (-3)^2} \\ = \sqrt{16 + 9} \\ = \sqrt{25} \\ = 5 \text{ units}$
Therefore, both Assertion and Reason are true, and Reason explains the Assertion correctly.

Key Concept: Historical Developments

d) Āryabhaṭa
[Solution Description] Āryabhaṭa is credited with replacing the use of Greek ‘chords’ with ‘sines’. This replacement made it easier to perform calculations involving trigonometry, crucial for determining celestial or terrestrial coordinates.

Key Concept: Properties of coordinates

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion is true since the origin is defined as having coordinates $(0, 0)$. However, the Reason provided does not explain the Assertion; rather, it’s additional information about the role of the origin in dividing the plane into quadrants. So, while both statements are true, they are unrelated in terms of cause and effect.

Key Concept: Midpoint using known midpoint

a) (2, -4)
[Solution Description]
Using the midpoint formula for C(4, -3) being the midpoint of DE, where D has coordinates (6, -2) and E has coordinates (x_2, y_2):
$4 = \frac{6 + x_2}{2} \quad \Rightarrow \quad 8 = 6 + x_2 \quad \Rightarrow \quad x_2 = 2$
$-3 = \frac{-2 + y_2}{2} \quad \Rightarrow \quad -6 = -2 + y_2 \quad \Rightarrow \quad y_2 = -4$
Thus, the coordinates of E are (2, -4).

Key Concept: Midpoint in 2D Plane

c) Assertion is true, but Reason is false.
[Solution Description]
Calculate the midpoint for the segment joining $(5, 7)$ and $(9, 11)$ using the midpoint formula:
$M = \left( \frac{5 + 9}{2}, \frac{7 + 11}{2} \right) = \left( \frac{14}{2}, \frac{18}{2} \right) = (7, 9)$
Hence, the assertion is true. However, the reason incorrectly describes a midpoint as the centroid, which is false since a centroid divides a triangle into three median segments.

Key Concept: Conceptual Properties, Distance Expression

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When two points have the same y-coordinate, they indeed lie horizontally relative to each other. Given points $(a, b)$ and $(c, b)$:
The distance formula becomes:
$d = \sqrt{(c – a)^2 + (b – b)^2} = \sqrt{(c – a)^2} = |c-a|$
Hence, both statements accurately describe the situation and the relationship; therefore, both assertions are true and tied together through the reason.

Key Concept: Quadrants Identification

b) $(0, y)$
[Solution Description] Points on the y-axis have the form $(0, y)$ where the x-coordinate is always zero.

Key Concept: Coordinate Analysis, Pythagorean Application

b) $5$
[Solution Description] In a right-angled triangle, the hypotenuse is the longest side which can be found using the distance formula. Here, $E(1, 1)$ and $F(5, 1)$ lie on the same horizontal line, while $F(5, 1)$ and $G(5, 4)$ on the same vertical line. The horizontal leg $EF$ is:
$EF = |5 – 1| = 4$
The vertical leg $FG$ is:
$FG = |4 – 1| = 3$
By the Baudhāyana–Pythagoras theorem, the hypotenuse $EG$ is:
$EG = \sqrt{EF^2 + FG^2} = \sqrt{4^2 + 3^2}$
Calculate further:
$EG = \sqrt{16 + 9} = \sqrt{25} = 5$
Thus, the length of the hypotenuse is $5$ units.

Key Concept: Coordinate geometry, navigation, real-life application

c) 10
[Solution Description] Since Reiaan moves parallel to the axes, his path consists of horizontal and vertical segments. Given that the total distance is 20 units, if $E(a, a)$, then the sum of movements in both directions equals twice the coordinate value because:
$2a = 20$
Solving for $a$:
$a = \frac{20}{2} = 10$

Key Concept: Midpoint Formula, Finding Unknown Coordinates

a) (-1, 1)
[Solution Description]
To find the coordinates $x$ and $y$, we use the midpoint formula:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Given that $M = (1, 5)$, $S = (3, y)$, and $T = (x, 9)$, these equations can be established:
\begin{align*}
\frac{3 + x}{2} &= 1
\frac{y + 9}{2} &= 5
\end{align*}
Solving for $x$:
\begin{align*}
3 + x &= 2 \cdot 1
3 + x &= 2
x &= 2 – 3
x &= -1
\end{align*}
Solving for $y$:
\begin{align*}
y + 9 &= 2 \cdot 5
y + 9 &= 10
y &= 10 – 9
y &= 1
\end{align*}
Thus, the coordinates $(x, y)$ are $(-1, 1)$.

Key Concept: Quadrants Identification

b) Quadrant II
[Solution Description] The coordinates of the point are $(-3, 5)$. According to the rules for quadrants:
(i) Quadrant I: $(+, +)$
(ii) Quadrant II: $(-, +)$
(iii) Quadrant III: $(-, -)$
(iv) Quadrant IV: $(+, -)$
The first value is negative and the second is positive, so it lies in Quadrant II.

Key Concept: Quadrant Coordinates Understanding

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description] The assertion is true as points in the first quadrant have both coordinates positive. The reason is also true because if $x = y$, then it satisfies the equation of the line $y = x$. However, the reason does not explain why a point with positive coordinates lies in the first quadrant.

Key Concept: Impact of Negative Numbers and Zero

Correct answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion accurately claims that Brahmagupta’s contributions were essential for creating the four-quadrant system since zero and negative numbers allow us to define all four quadrants in a Cartesian plane. The reason given supports this by explaining how the Cartesian plane utilizes these concepts to depict both positive and negative values. In this case, the reason directly supports and explains the validity of the assertion.

Key Concept: Introduction of Coordinate Systems in Europe, Influence of Al-Bīrūnī

Correct Answer option c) Assertion is true, but Reason is false.
[Solution Description]
While Al-Bīrūnī did advance the design of the astrolabe, these advancements primarily affected navigational and astronomical practices rather than directly influencing Descartes. Descartes’ formalisation of coordinate geometry arose from other developments in European mathematics. Therefore, while the Assertion is true, the Reason is false.

Key Concept: General Understanding of Pythagorean Application

d) Assertion is false, but Reason is true.
[Solution Description] The assertion is false since the statement holds true only for right-angled triangles, not any triangle. The reason is true because the Baudhāyana-Pythagoras theorem specifically deals with right-angled triangles.

Key Concept: Tools for Navigation

c) Astrolabe
[Solution Description] Al-Bīrūnī perfected the ‘astrolabe’, a navigational instrument that allowed sailors to determine their location by observing celestial objects.

Key Concept: Midpoint Verification

a) Yes
[Solution Description] First, calculate the midpoint $(M_x, M_y)$ using:
$M_x = \frac{x_1 + x_2}{2}, \quad M_y = \frac{y_1 + y_2}{2}$
For points $(3, 4)$ and $(11, 10)$:
\begin{align*}
M_x &= \frac{3 + 11}{2} = 7,
M_y &= \frac{4 + 10}{2} = 7
\end{align*}
Midpoint $M$ is $(7, 7)$. Now calculate the distances $AM$ and $MB$ where $A = (3, 4)$ and $B = (11, 10)$:
\begin{align*}
AM &= \sqrt{(7 – 3)^2 + (7 – 4)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5
MB &= \sqrt{(11 – 7)^2 + (10 – 7)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5
\end{align*}
As both distances are equal, the midpoint $(7, 7)$ is equidistant from both endpoints.

Key Concept: Historical Reference Points

c) Central longitude meridian
[Solution Description] Ujjayinī was described as marking the central longitude meridian from which all other locations were measured in ancient geographical texts.

Key Concept: Role of Brahmagupta, Use of Negative Numbers

Correct Answer option a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Brahmagupta’s contributions made it possible to include both positive and negative values systematically in mathematics, which is crucial for understanding the Cartesian plane where quadrants are defined around an origin point at zero. Thus, both assertion and reason are true, and Reason explains why Assertion is significant to coordinate planes.

Key Concept: Midpoint Concept, Distance Formula

b) $\sqrt{53}$
[Solution Description] First, determine the midpoint $M(x, y)$ of points $C(-5, 6)$ and $D(9, 2)$. The formula for the midpoint is:
$M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Substitute the given values:
$M(x, y) = \left(\frac{-5 + 9}{2}, \frac{6 + 2}{2}\right) = \left(2, 4\right)$
Now, find the distance from midpoint $M(2, 4)$ to point $C(-5, 6)$:
$d = \sqrt{(2 – (-5))^2 + (4 – 6)^2}$
Simplify the expression:
$d = \sqrt{(2 + 5)^2 + (-2)^2} = \sqrt{7^2 + 2^2}$
Calculate further:
$d = \sqrt{49 + 4} = \sqrt{53}$
Therefore, the distance from midpoint to point $C$ is $\sqrt{53}$ units.

Key Concept: Midpoint in 2D Plane

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Using the formula for the midpoint, calculate the midpoint for $(2, -3)$ and $(10, 5)$:
$M = \left( \frac{2 + 10}{2}, \frac{-3 + 5}{2} \right) = \left( \frac{12}{2}, \frac{2}{2} \right) = (6, 1)$
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Point on Axes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For any point on the x-axis, the y-coordinate must indeed be zero, fitting the general form $(x, 0)$. The assertion is true as for any point lying on the x-axis its y-coordinate will be zero. The reason correctly explains this fact by mentioning the representation of such points.

Key Concept: Quadrants Identification

b) Quadrant II
[Solution Description] The point $(-5, 7)$ has a negative x-coordinate and a positive y-coordinate. According to the Cartesian coordinate system, points with a negative x-coordinate and a positive y-coordinate are located in Quadrant II.

Key Concept: Coordinate Geometry: Area of a Triangle

b) 32.5 square units
[Solution Description] The area $(A)$ of a triangle formed by three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by:
$A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|$
By substituting the given points $D(2, 3)$, $E(5, 11)$, and $F(12, 8)$:
$A = \frac{1}{2} \left| 2(11-8) + 5(8-3) + 12(3-11) \right|$
$A = \frac{1}{2} \left| 2(3) + 5(5) + 12(-8) \right|$
$A = \frac{1}{2} \left| 6 + 25 – 96 \right|$
$A = \frac{1}{2} \times 65 = 32.5$
Therefore, the area of the triangle is $32.5$ square units.

Key Concept: Distance Formula Application

c) Assertion is true, but Reason is false.
[Solution Description] The assertion is true because any point on the x-axis will have its y-coordinate as zero and vice versa for the y-axis. However, the reason is false; the distance from the origin to a point $(x, y)$ is correctly given by $\sqrt{x^2 + y^2}$ according to the Baudhāyana-Pythagoras theorem.

Key Concept: Finding midpoints, trisecting segments

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a section point dividing a segment in the ratio $m:n$, use:
$x_P = \frac{mx_2 + nx_1}{m+n}, \quad y_P = \frac{my_2 + ny_1}{m+n}$
Here, $m = 1$, $n = 2$, coordinates of C(8, 5), D(14, 11):
$x_P = \frac{1 \cdot 14 + 2 \cdot 8}{1+2} = \frac{14 + 16}{3} = 10 \\ \\ y_P = \frac{1 \cdot 11 + 2 \cdot 5}{1+2} = \frac{11 + 10}{3} = 7$
Therefore, P(10, 7) indeed divides CD in the ratio 1:2.
Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Coordinate Trisection

a) (7, 10)
[Solution Description]
Since P is closer to X, it divides the distance between X and Y into $1/3$ and $2/3$. The coordinate of P is determined as follows:
$P_x = \frac{2 \cdot 9 + 1 \cdot 3}{3} = \frac{18 + 3}{3} = \frac{21}{3} = 7$
$P_y = \frac{2 \cdot 12 + 1 \cdot 6}{3} = \frac{24 + 6}{3} = \frac{30}{3} = 10$
Therefore, the coordinates of P are (7, 10).

Key Concept: Identifying Points on Grid

a) Yes, they form a rectangle
[Solution Description] An axis-aligned rectangle has parallel sides to the axes. Checking the given points: (1, 1) to (1, 4) forms a vertical line segment; (1, 4) to (3, 4) forms a horizontal line segment; (3, 4) to (3, 1) forms another vertical line segment; and (3, 1) to (1, 1) completes the horizontal base. This confirms an axis-aligned rectangle.