Assertion-Reason MCQs for Sub-Topics of Topic 6: Coordinate Geometry

Question 1. Assertion (A): A point lying on the Y-axis has its abscissa equal to $0$.

Reason (R): The set of points on the Y-axis are those with an x-coordinate of $0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The point $(5, -1)$ lies in the fourth quadrant.

Reason (R): In the fourth quadrant, the x-coordinate is positive and the y-coordinate is negative.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The distance of the point $(a, b)$ from the X-axis is $|b|$.

Reason (R): The absolute value of the y-coordinate gives the perpendicular distance from the X-axis.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The origin $(0, 0)$ is not considered to be in any of the four quadrants.

Reason (R): A point is in a quadrant only if both its coordinates are non-zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The point $(-3, -5)$ lies in the third quadrant.

Reason (R): Points with negative x and negative y coordinates lie in the third quadrant.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): To plot the point $(-2, 4)$, we move $2$ units left from the origin along the X-axis and then $4$ units up parallel to the Y-axis.

Reason (R): In the ordered pair $(x, y)$, the first value $x$ determines horizontal position and the second value $y$ determines vertical position relative to the origin.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): A point located $3$ units right of the Y-axis and $5$ units below the X-axis has coordinates $(3, -5)$.

Reason (R): 'Right of Y-axis' corresponds to positive x-coordinate, and 'below X-axis' corresponds to negative y-coordinate.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The point $(a, a)$ for any non-zero real number $a$ lies in the first or third quadrant.

Reason (R): If $a \neq 0$, then either $a>0$ (both positive, Q1) or $a<0$ (both negative, Q3).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The point $(0, -2)$ lies on the negative Y-axis.

Reason (R): A point on the Y-axis has its x-coordinate as $0$, and if the y-coordinate is negative, it is on the negative part of the Y-axis.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): Plotting points helps visualize algebraic relationships and geometric shapes.

Reason (R): The Cartesian system establishes a one-to-one correspondence between points in a plane and ordered pairs of real numbers.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The distance between points $(2, 3)$ and $(5, 7)$ is $5$ units.

Reason (R): The distance formula between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The points $(1, 2), (3, 4), (5, 6)$ are collinear.

Reason (R): The area of the triangle formed by three collinear points is zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The distance of a point $(x, y)$ from the origin is $\sqrt{x^2 + y^2}$.

Reason (R): The origin has coordinates $(0, 0)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): If the distance between $(k, 0)$ and $(0, 3)$ is $5$, then $k = \pm 4$.

Reason (R): The square of the distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $(x_2-x_1)^2 + (y_2-y_1)^2$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The distance formula is derived directly from the Pythagorean theorem applied in the coordinate plane.

Reason (R): The difference in x-coordinates and the difference in y-coordinates form the legs of a right triangle, with the distance between the points as the hypotenuse.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The midpoint of the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$.

Reason (R): The midpoint divides the line segment internally in the ratio $1:1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The point that divides the line segment joining A and B externally in the ratio $m:n$ (with $m>n$) lies between A and B.

Reason (R): For external division, the point lies on the line containing the segment but outside the segment.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$.

Reason (R): The centroid is the point of intersection of the medians and divides each median in the ratio $2:1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The point $(0, 0)$ divides the line segment joining $(1, 2)$ and $(-1, -2)$ in the ratio $1:1$ internally.

Reason (R): The midpoint of the segment joining $(1, 2)$ and $(-1, -2)$ is $(\frac{1+(-1)}{2}, \frac{2+(-2)}{2}) = (0, 0)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The Y-axis divides the line segment joining $(5, -6)$ and $(-1, -4)$ in the ratio $5:1$ internally.

Reason (R): A point on the Y-axis has its x-coordinate equal to $0$. Using the section formula with a ratio $k:1$, the x-coordinate of the division point is $\frac{k(-1) + 1(5)}{k+1}$. Setting this to $0$ gives $5-k=0$, so $k=5$. The ratio is $5:1$. Since the ratio is positive, the division is internal.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The area of the triangle with vertices $(0, 0), (a, 0), (0, b)$ is $\frac{1}{2}|ab|$ square units.

Reason (R): For a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, the area is $\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): Three points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ are collinear if $x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) = 0$.

Reason (R): The area of the triangle formed by three collinear points is always zero.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The area of a triangle can be negative if the vertices are taken in a clockwise order.

Reason (R): The formula for the area of a triangle using coordinates involves a determinant-like expression whose sign depends on the order of vertices.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The points $(1, 1), (2, 3), (3, 5)$ are collinear.

Reason (R): The slope of the line segment joining $(1, 1)$ and $(2, 3)$ is $\frac{3-1}{2-1} = 2$, and the slope of the line segment joining $(2, 3)$ and $(3, 5)$ is $\frac{5-3}{3-2} = 2$. Since the slopes are equal, the points are collinear.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): If the area of a triangle with vertices $(k, 0), (1, 1), (0, 2)$ is non-zero, then the points are not collinear.

Reason (R): Collinear points lie on the same straight line and do not form a triangle with a positive area.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The centroid of a triangle is the point of intersection of its medians.

Reason (R): A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The incenter of a triangle is equidistant from the vertices of the triangle.

Reason (R): The incenter is the intersection of angle bisectors and is the center of the inscribed circle (incircle) which is tangent to all sides.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): For a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse.

Reason (R): The circumcenter is the intersection of the perpendicular bisectors of the sides and is equidistant from the vertices.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The orthocenter of a triangle always lies inside the triangle.

Reason (R): The orthocenter is the point of intersection of the altitudes of the triangle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): For an equilateral triangle, the centroid, incenter, circumcenter, and orthocenter all coincide at the same point.

Reason (R): In an equilateral triangle, the medians, angle bisectors, altitudes, and perpendicular bisectors are all the same lines.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The locus of a point P$(x, y)$ such that its distance from the origin is constant is a circle.

Reason (R): A circle is defined as the set of all points in a plane that are at a fixed distance from a fixed point (the center).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The locus of a point equidistant from two fixed points A and B is the perpendicular bisector of the segment AB.

Reason (R): Any point on the perpendicular bisector is by definition equidistant from the two endpoints of the segment.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The equation of the locus of a point P$(x, y)$ such that its distance from the line $x=a$ is equal to its distance from the point $(-a, 0)$ is a parabola.

Reason (R): A parabola is the locus of a point equidistant from a fixed point (focus) and a fixed line (directrix).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): To find the equation of a locus, we assume the moving point has coordinates $(x, y)$ and express the given geometric condition as an equation in terms of $x$ and $y$.

Reason (R): The set of all points satisfying the geometric condition constitutes the locus, and the equation satisfied by their coordinates is the equation of the locus.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The locus of a point P$(x, y)$ such that $x = 3$ is a vertical line.

Reason (R): The equation $x=3$ means that the x-coordinate of any point on the locus is always 3, while the y-coordinate can be any real number, which describes a vertical line.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): If the origin is shifted to $(h, k)$, the new coordinates $(X, Y)$ of a point with original coordinates $(x, y)$ are given by $X = x-h$ and $Y = y-k$.

Reason (R): This transformation involves translating the coordinate axes without rotation, meaning the new coordinates are the original coordinates relative to the new origin.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): Shifting the origin changes the equation of a curve.

Reason (R): The geometric properties of the curve, such as its shape or size, are altered by shifting the origin.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): To simplify the equation $x^2 + y^2 - 4x - 6y + 12 = 0$, we can shift the origin to $(2, 3)$.

Reason (R): The center of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is $(-g, -f)$, and shifting the origin to the center eliminates the linear terms.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): If the original coordinates of a point are $(5, -1)$ and the origin is shifted to $(2, 3)$, the new coordinates are $(3, -4)$.

Reason (R): New coordinates are found by subtracting the shift values from the original coordinates.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): Shifting the origin affects the slope of a straight line.

Reason (R): The slope of a line is determined by the angle it makes with the positive X-axis, and shifting the origin does not change the orientation of the axes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The slope of the line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is $\frac{y_2-y_1}{x_2-x_1}$ for $x_1 \neq x_2$.

Reason (R): The slope is the tangent of the angle the line makes with the positive X-axis, which is the ratio of the vertical change to the horizontal change between any two points on the line.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): A line parallel to the Y-axis has an undefined slope.

Reason (R): For a vertical line, the change in x-coordinates between any two distinct points is $0$, resulting in division by zero in the slope formula.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): Two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1 m_2 = -1$.

Reason (R): If two lines are perpendicular, the angle between them is $90^\circ$, and $\tan 90^\circ$ is undefined. The formula for the angle between lines involves $\tan \theta$, leading to the condition $1 + m_1 m_2 = 0$ if the angle is $90^\circ$ (and $m_1, m_2$ are finite).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The slope of the line $2x + 3y - 6 = 0$ is $-2/3$.

Reason (R): The slope of a line with the general equation $Ax + By + C = 0$ is given by $-A/B$, provided $B \neq 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): Two lines are parallel if and only if their slopes are equal (provided they are non-vertical).

Reason (R): Parallel lines make the same angle with the positive X-axis, and the slope is the tangent of this angle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The equation of a line parallel to the X-axis is of the form $x = k$, where $k$ is a constant.

Reason (R): Lines parallel to the X-axis have a slope of $0$ and can be written as $y = c$ for some constant $c$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The equation of the line passing through $(x_1, y_1)$ with slope $m$ is given by $y - y_1 = m(x - x_1)$.

Reason (R): This form, the point-slope form, directly incorporates a specific point on the line and its slope.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The equation of a line with x-intercept $a$ and y-intercept $b$ is $\frac{x}{a} + \frac{y}{b} = 1$, provided $a \neq 0$ and $b \neq 0$.

Reason (R): This form, the intercept form, highlights where the line crosses the coordinate axes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The equation of the line passing through points $(1, 2)$ and $(3, 6)$ can be found using the two-point form.

Reason (R): The two-point form is derived from the point-slope form by calculating the slope using the two given points.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The normal form of the line $x + y = \sqrt{2}$ is $x \cos 45^\circ + y \sin 45^\circ = 1$.

Reason (R): The general equation $Ax + By + C = 0$ is converted to normal form by dividing by $\pm\sqrt{A^2+B^2}$, where the sign is chosen such that the constant term becomes positive. Here $A=1, B=1, C=-\sqrt{2}$. $\sqrt{A^2+B^2} = \sqrt{1^2+1^2} = \sqrt{2}$. Dividing by $\sqrt{2}$: $\frac{1}{\sqrt{2}}x + \frac{1}{\sqrt{2}}y - 1 = 0 \implies x \cos 45^\circ + y \sin 45^\circ = 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The equation $3x + 4y - 7 = 0$ represents a straight line.

Reason (R): Any equation of the form $Ax + By + C = 0$, where A and B are not both zero, is the general equation of a straight line.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The slope of the line $5x - 2y + 10 = 0$ is $5/2$.

Reason (R): The slope of a line $Ax + By + C = 0$ is $-A/B$ (when $B \neq 0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The point of intersection of the lines $x+y=3$ and $x-y=1$ is $(2, 1)$.

Reason (R): A point of intersection of two lines satisfies both equations simultaneously. Adding the equations gives $2x=4$, $x=2$. Substituting into $x+y=3$ gives $2+y=3$, $y=1$. So $(2, 1)$ is the intersection.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2}$.

Reason (R): Two non-vertical lines are parallel if their slopes are equal. The slopes are $-a_1/b_1$ and $-a_2/b_2$. Equality implies $a_1/b_1 = a_2/b_2$, or $a_1b_2 = a_2b_1$, which is $\frac{a_1}{a_2} = \frac{b_1}{b_2}$ (if $a_2, b_2 \neq 0$). This condition also covers vertical lines ($b_1=b_2=0$) and horizontal lines ($a_1=a_2=0$).

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The general equation of a line is always unique for a given line.

Reason (R): If $Ax + By + C = 0$ is the equation of a line, then $k(Ax + By + C) = 0$ (for any non-zero constant $k$) represents the same line.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The distance of the point $(1, -1)$ from the line $3x + 4y - 5 = 0$ is $6/5$ units.

Reason (R): The distance of a point $(x_0, y_0)$ from the line $Ax + By + C = 0$ is $\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$. Using this, the distance is $\frac{|3(1) + 4(-1) - 5|}{\sqrt{3^2+4^2}} = \frac{|3 - 4 - 5|}{\sqrt{9+16}} = \frac{|-6|}{\sqrt{25}} = \frac{6}{5}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The distance between the parallel lines $2x + 3y = 6$ and $2x + 3y = 12$ is $\frac{6}{\sqrt{13}}$.

Reason (R): The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is $\frac{|C_1 - C_2|}{\sqrt{A^2+B^2}}$. Here $A=2, B=3, C_1=-6, C_2=-12$. Distance = $\frac{|-6 - (-12)|}{\sqrt{2^2+3^2}} = \frac{|-6+12|}{\sqrt{4+9}} = \frac{6}{\sqrt{13}}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The equation $(x + y - 1) + \lambda(x - y - 1) = 0$ represents a family of lines passing through the point of intersection of $x+y-1=0$ and $x-y-1=0$.

Reason (R): Any equation of the form $L_1 + \lambda L_2 = 0$ represents a line passing through the intersection of lines $L_1 = 0$ and $L_2 = 0$, where $\lambda$ is a parameter.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The distance of the origin from the line $y = mx + c$ is $\frac{|c|}{\sqrt{1+m^2}}$.

Reason (R): The distance of a point $(x_0, y_0)$ from $Ax + By + C = 0$ is $\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$. For the origin $(0, 0)$ and line $mx - y + c = 0$, $A=m, B=-1, C=c, x_0=0, y_0=0$. Distance = $\frac{|m(0) - 0 + c|}{\sqrt{m^2+(-1)^2}} = \frac{|c|}{\sqrt{m^2+1}}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The family of lines parallel to $Ax + By + C = 0$ is given by $Ax + By + K = 0$, where $K$ is a parameter.

Reason (R): Parallel lines have the same slope, and changing only the constant term in the general equation results in lines with the same slope but different intercepts.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The equation of the YZ-plane in 3D Cartesian coordinates is $x = 0$.

Reason (R): Any point on the YZ-plane has its x-coordinate equal to zero, while y and z can be any real numbers.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The three coordinate planes divide the 3D space into eight regions called octants.

Reason (R): Each octant is characterized by the signs of the x, y, and z coordinates of the points within it.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The distance of the point $(2, 3, 4)$ from the X-axis is $\sqrt{3^2 + 4^2} = 5$ units.

Reason (R): The distance of a point $(x, y, z)$ from the X-axis is $\sqrt{y^2 + z^2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The coordinates of the origin in three dimensions are $(0, 0, 0)$.

Reason (R): The origin is the point where the three coordinate axes intersect.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): A point $(x, y, z)$ lies on the XY-plane if $z=0$.

Reason (R): The XY-plane is the plane containing the X and Y axes.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The distance between points $(1, 2, 3)$ and $(4, 6, 15)$ is $\sqrt{169} = 13$ units.

Reason (R): The distance formula in 3D is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. Calculating, $\sqrt{(4-1)^2 + (6-2)^2 + (15-3)^2} = \sqrt{3^2+4^2+12^2} = \sqrt{9+16+144} = \sqrt{169} = 13$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The distance of the point $(x, y, z)$ from the origin is $\sqrt{x^2+y^2+z^2}$.

Reason (R): The origin is the point $(0, 0, 0)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The distance of the point $(a, b, c)$ from the YZ-plane is $|a|$.

Reason (R): The distance of a point from a coordinate plane is the absolute value of the coordinate perpendicular to that plane.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The distance formula in 3D is derived using the concept of vectors.

Reason (R): The distance between two points is the magnitude of the vector connecting them.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): If the distance between $(k, 0, 0)$ and $(0, 4, 0)$ is $5$, then $k = \pm 3$.

Reason (R): The distance between $(k, 0, 0)$ and $(0, 4, 0)$ is $\sqrt{(0-k)^2 + (4-0)^2 + (0-0)^2} = \sqrt{k^2 + 16}$. Setting this equal to $5$, $k^2+16=25 \implies k^2=9 \implies k=\pm 3$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The midpoint of the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$.

Reason (R): The midpoint divides the segment internally in the ratio $1:1$, and the section formula for internal division is a linear combination of the coordinates.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The point that divides the line segment joining A and B externally in the ratio $m:n$ (with $m>n$) lies between A and B.

Reason (R): External division implies the point lies on the line containing the segment but outside the segment.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The centroid of a triangle in 3D space with vertices $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$ is $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3})$.

Reason (R): The centroid is the average of the coordinates of the vertices.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): If the point P divides the line segment joining A and B in the ratio $m:n$, the ratio of directed segments AP/PB is equal to $m/n$ for both internal and external division.

Reason (R): The section formula for internal division is $\frac{mx_2+nx_1}{m+n}$, and for external division is $\frac{mx_2-nx_1}{m-n}$. The ratio $m:n$ in the formula refers to the ratio of lengths of the segments AP and PB.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The Z-axis divides the line segment joining $(2, 3, 4)$ and $(-4, -6, -8)$ in the ratio $1:2$ internally.

Reason (R): A point on the Z-axis has its x and y coordinates equal to zero. Let the ratio be $k:1$. The x-coordinate of the division point is $\frac{k(-4) + 1(2)}{k+1}$. Setting this to $0$ gives $2-4k=0$, $k=1/2$. The ratio is $1:2$. Since $k$ is positive, the division is internal.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): A parabola is a conic section with eccentricity $e=1$.

Reason (R): The eccentricity is the ratio of the distance of a point on the conic from the focus to its distance from the directrix.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): A circle is a special type of ellipse.

Reason (R): A circle is obtained when the cutting plane is perpendicular to the axis of the cone and does not pass through the vertex, which is a specific case of the conditions for an ellipse.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): Degenerate conic sections are formed when the cutting plane passes through the vertex of the cone.

Reason (R): When the plane passes through the vertex, the intersection is typically a point, a line, or a pair of lines, rather than the usual curves.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): If the eccentricity of a conic section is $e < 1$, it is an ellipse.

Reason (R): The eccentricity is a key parameter that classifies the type of conic section based on its value relative to 1.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The latus rectum of a conic section is a chord passing through the focus and parallel to the directrix.

Reason (R): The latus rectum is defined as the chord passing through the focus and perpendicular to the axis of the conic.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The equation $x^2 + y^2 - 6x + 4y - 3 = 0$ represents a circle with center $(3, -2)$ and radius $4$.

Reason (R): The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, with center $(-g, -f)$ and radius $\sqrt{g^2+f^2-c}$. For the given equation, $2g=-6 \implies g=-3$, $2f=4 \implies f=2$, $c=-3$. Center is $(-(-3), -2) = (3, -2)$. Radius is $\sqrt{(-3)^2+2^2-(-3)} = \sqrt{9+4+3} = \sqrt{16}=4$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): A line intersects a circle at two distinct points if the distance from the center of the circle to the line is greater than the radius.

Reason (R): The number of intersection points between a line and a circle depends on the comparison between the distance from the center to the line and the radius.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The equation $(x-h)^2 + (y-k)^2 = 0$ represents a point circle at $(h, k)$.

Reason (R): For a real circle, the radius $r$ in $(x-h)^2 + (y-k)^2 = r^2$ must be a positive real number. If $r=0$, the equation is only satisfied by the point $(h, k)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): Two circles intersect externally if the distance between their centers is equal to the sum of their radii.

Reason (R): If the distance between centers is greater than the sum of radii, the circles do not intersect and are outside each other.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The equation of a circle with diameter endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$.

Reason (R): The angle subtended by a diameter at any point on the circumference is $90^\circ$. If P$(x, y)$ is a point on the circle, the product of slopes of AP and BP is $-1$ (unless the diameter is vertical or horizontal). The slope of AP is $\frac{y-y_1}{x-x_1}$ and slope of BP is $\frac{y-y_2}{x-x_2}$. Product is $-1 \implies (y-y_1)(y-y_2) = -(x-x_1)(x-x_2) \implies (x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The vertex of the parabola $y^2 = 4ax$ is at the origin $(0, 0)$.

Reason (R): The vertex of a parabola is the point on the parabola closest to the directrix and is the midpoint of the segment joining the focus and the point of intersection of the axis and the directrix.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): For the parabola $x^2 = 8y$, the focus is at $(0, 2)$.

Reason (R): The equation is of the form $x^2 = 4ay$. Comparing, $4a = 8 \implies a = 2$. Since the axis is the Y-axis and it opens upwards, the focus is $(0, a) = (0, 2)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The length of the latus rectum of the parabola $y^2 = 12x$ is $12$.

Reason (R): The length of the latus rectum of a parabola of the form $y^2 = 4ax$ or $x^2 = 4ay$ is $|4a|$. Here $4a = 12$, so the length is $|12| = 12$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The directrix of the parabola $y^2 = 4ax$ is the line $x = -a$.

Reason (R): The directrix is a fixed line such that any point on the parabola is equidistant from the focus and the directrix.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The parametric equations $x = t, y = t^2/4a$ represent the parabola $y^2 = 4ax$.

Reason (R): Substituting $y = t^2/4a$ into $y^2 = 4ax$ gives $(t^2/4a)^2 = 4ax \implies t^4/16a^2 = 4ax$. This does not match $x=t$. The standard parametric form $(at^2, 2at)$ works because $(2at)^2 = 4a(at^2)$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): For an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant.

Reason (R): This constant sum is equal to the length of the major axis of the ellipse.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The eccentricity of an ellipse is always greater than $1$.

Reason (R): The eccentricity $e$ of an ellipse is given by $c/a$, where $c$ is the distance from the center to the focus and $a$ is the length of the semi-major axis. For an ellipse, the foci are inside the ellipse, so $c < a$, which means $e < 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$ represents an ellipse with major axis along the X-axis.

Reason (R): In the standard equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, if $a^2 > b^2$, the major axis is along the X-axis. Here $a^2=25, b^2=16$, and $25>16$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The length of the latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (with $a>b$) is $\frac{2b^2}{a}$.

Reason (R): The latus rectum is a chord through the focus perpendicular to the major axis.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The parametric equations $x = 5\cos\theta, y = 4\sin\theta$ represent an ellipse.

Reason (R): By eliminating the parameter $\theta$, we get $\frac{x}{5} = \cos\theta, \frac{y}{4} = \sin\theta$. Squaring and adding gives $(\frac{x}{5})^2 + (\frac{y}{4})^2 = \cos^2\theta + \sin^2\theta = 1$, which is $\frac{x^2}{25} + \frac{y^2}{16} = 1$, the equation of an ellipse.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): For a hyperbola, the absolute difference of the distances from any point on the hyperbola to the two foci is constant.

Reason (R): This constant difference is equal to the length of the transverse axis of the hyperbola.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The eccentricity of a hyperbola is always greater than $1$.

Reason (R): For a hyperbola, the distance from the center to the focus ($c$) is always greater than the distance from the center to the vertex ($a$), so $c > a$, and $e = c/a > 1$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The equations of the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are $y = \pm \frac{b}{a}x$.

Reason (R): Asymptotes are lines that the hyperbola approaches infinitely closely but never touches, and they pass through the center of the hyperbola.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The equation $\frac{x^2}{16} - \frac{y^2}{25} = 1$ represents a hyperbola with transverse axis along the Y-axis.

Reason (R): In the standard equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, the transverse axis lies along the axis corresponding to the term with the positive sign.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): A rectangular hyperbola has an eccentricity of $\sqrt{2}$.

Reason (R): A rectangular hyperbola is one whose asymptotes are perpendicular, which occurs when the lengths of the semi-transverse axis and semi-conjugate axis are equal ($a=b$). The eccentricity is then $e = \sqrt{1 + b^2/a^2} = \sqrt{1 + a^2/a^2} = \sqrt{2}$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): The parametric equations $x = r\cos t, y = r\sin t$ represent a circle centered at the origin with radius $r$.

Reason (R): Squaring and adding the equations gives $x^2 + y^2 = r^2(\cos^2 t + \sin^2 t) = r^2$, which is the Cartesian equation of a circle centered at the origin with radius $r$.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The parametric equations $x = at^2, y = 2at$ represent the parabola $x^2 = 4ay$.

Reason (R): Eliminating $t$ from $x = at^2, y = 2at$ gives $t = y/2a$. Substituting this into $x = at^2$ gives $x = a(y/2a)^2 = a(y^2/4a^2) = y^2/4a$, which simplifies to $y^2 = 4ax$. So the assertion is false.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): Parametric equations of a conic section are useful for representing points on the curve using a single parameter.

Reason (R): Some curves, like a circle, cannot be represented as a function $y = f(x)$ or $x = g(y)$ over their entire domain, but can be represented parametrically.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): The parametric equations $x = 3\cos\theta, y = 2\sin\theta$ represent a circle with radius $3$.

Reason (R): These equations represent an ellipse centered at the origin with semi-major axis $3$ and semi-minor axis $2$, since $\frac{x^2}{9} + \frac{y^2}{4} = \cos^2\theta + \sin^2\theta = 1$. A circle is a special case of an ellipse where the semi-axes are equal.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): The parametric equations $x = \sec t, y = \tan t$ represent the hyperbola $x^2 - y^2 = 1$.

Reason (R): The trigonometric identity $\sec^2 t - \tan^2 t = 1$ can be used to eliminate the parameter $t$ and obtain the Cartesian equation.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 1. Assertion (A): Coordinate geometry can be used to prove that the diagonals of a parallelogram bisect each other.

Reason (R): By assigning coordinates to the vertices of the parallelogram, we can use the midpoint formula to show that the midpoints of the two diagonals coincide.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 2. Assertion (A): The path of a projectile, neglecting air resistance, is a parabola.

Reason (R): The horizontal motion is uniform velocity, and the vertical motion is uniformly accelerated (due to gravity). Combining these movements algebraically in a coordinate system results in a parabolic trajectory.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 3. Assertion (A): The distance formula in 3D can be used to determine if four given points are coplanar.

Reason (R): While the distance formula is used for lengths, checking coplanarity typically involves verifying if one point lies on the plane formed by the other three, which requires concepts beyond just distances, such as vector methods or plane equations.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 4. Assertion (A): Coordinate geometry is a powerful tool for solving geometric problems because it allows us to translate geometry into algebra and vice versa.

Reason (R): Points, lines, and figures are represented by coordinates and equations, which can then be manipulated using algebraic techniques.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.

Question 5. Assertion (A): Using coordinate geometry, we can easily determine the type of triangle (e.g., isosceles, right-angled) given the coordinates of its vertices.

Reason (R): By calculating the lengths of the sides using the distance formula and checking for conditions like equality of sides or satisfying the Pythagorean theorem, we can classify the triangle.

(A) Both A and R are true and R is the correct explanation of A.

(B) Both A and R are true but R is not the correct explanation of A.

(C) A is true but R is false.

(D) A is false but R is true.