Completing Statements MCQs for Sub-Topics of Topic 6: Coordinate Geometry

Question 1. The point where the X-axis and the Y-axis intersect is known as the _______.

(A) Quadrant

(B) Abscissa

(C) Ordinate

(D) Origin

Question 2. In the Cartesian plane, the horizontal axis is called the _______.

(A) Y-axis

(B) Z-axis

(C) X-axis

(D) Origin

Question 3. The four regions into which the coordinate axes divide the plane are called _______.

(A) Octants

(B) Halves

(C) Quadrants

(D) Sectors

Question 4. The x-coordinate of a point is also known as its _______.

(A) Ordinate

(B) Abscissa

(C) Origin

(D) Quadrant

Question 5. A point in the first quadrant has both its x-coordinate and y-coordinate as _______.

(A) Negative

(B) Zero

(C) Positive

(D) Equal

Question 6. If a point lies on the Y-axis, its _______ is zero.

(A) Ordinate

(B) Abscissa

(C) Distance from origin

(D) Quadrant number

Question 7. The coordinates of the origin are _______.

(A) $(1, 1)$

(B) $(0, 1)$

(C) $(1, 0)$

(D) $(0, 0)$

Question 8. A point in the fourth quadrant has a positive x-coordinate and a negative _______.

(A) Abscissa

(B) Origin

(C) Quadrant

(D) Ordinate

Question 9. The Cartesian plane is also known as the _______.

(A) Number line

(B) Coordinate plane

(C) Complex plane

(D) Vector space

Question 10. The y-coordinate of a point is its distance from the _______.

(A) Origin

(B) Y-axis

(C) X-axis

(D) Quadrant

Question 1. To plot the point $(4, 2)$, you move $4$ units right from the origin and then _______.

(A) 2 units left

(B) 2 units down

(C) 2 units up

(D) 4 units up

Question 2. A point located $3$ units to the left of the Y-axis and $5$ units below the X-axis has coordinates _______.

(A) $(3, -5)$

(B) $(-3, 5)$

(C) $(-3, -5)$

(D) $(3, 5)$

Question 3. A point lying on the positive X-axis at a distance of $6$ units from the origin has coordinates _______.

(A) $(6, 6)$

(B) $(0, 6)$

(C) $(-6, 0)$

(D) $(6, 0)$

Question 4. If a point is on the negative Y-axis at a distance of $4$ units from the origin, its coordinates are _______.

(A) $(-4, 0)$

(B) $(0, 4)$

(C) $(0, -4)$

(D) $(4, 0)$

Question 5. A point with a negative abscissa and a positive ordinate is located in the _______ quadrant.

(A) First

(B) Second

(C) Third

(D) Fourth

Question 6. The ordered pair $(x, y)$ represents a point where $x$ indicates movement along the X-axis and $y$ indicates movement along the _______ axis.

(A) Z

(B) Y

(C) W

(D) Vertical

Question 7. If a point lies on the X-axis to the left of the origin, its coordinates are of the form _______ where $a > 0$.

(A) $(a, 0)$

(B) $(0, a)$

(C) $(-a, 0)$

(D) $(0, -a)$

Question 8. To identify the coordinates of a plotted point, we drop perpendiculars from the point to the _______ axes.

(A) Parallel

(B) Coordinate

(C) Diagonal

(D) Inclined

Question 9. A point $(x, y)$ is located such that $x > 0$ and $y < 0$. This point is in the _______ quadrant.

(A) First

(B) Second

(C) Third

(D) Fourth

Question 10. The starting point for plotting coordinates is typically the _______.

(A) X-axis

(B) Y-axis

(C) Origin

(D) First quadrant

Question 1. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$, which is derived from the _______ theorem.

(A) Thales'

(B) Euclidean

(C) Pythagorean

(D) Fermat's

Question 2. The distance of a point $(x, y)$ from the origin is _______.

(A) $x+y$

(B) $x^2+y^2$

(C) $\sqrt{x^2+y^2}$

(D) $|x|+|y|$

Question 3. If three points A, B, and C are collinear, then the sum of the distances between any two pairs of points is equal to the _______ between the remaining pair.

(A) Product of distances

(B) Difference of distances

(C) Distance

(D) Average distance

Question 4. The distance between the points $(2, 3)$ and $(2, 7)$ is _______ units.

(A) $4$

(B) 3

(C) 4

(D) 5

Question 5. To check if a triangle is a right triangle using the distance formula, we verify if the lengths of the sides satisfy the _______ theorem.

(A) Similarity

(B) Congruence

(C) Pythagorean

(D) Sine Rule

Question 6. The distance of a point $(p, q)$ from the X-axis is _______.

(A) $|p|$

(B) $|q|$

(C) $\sqrt{p^2+q^2}$

(D) $p$

Question 7. If the distance between $(k, 0)$ and $(0, 4)$ is $5$ units, then $k^2$ is equal to _______.

(A) 9

(B) 16

(C) 25

(D) 41

Question 8. The points $(1, 5), (2, 3), (-2, -1)$ are collinear if the area of the triangle formed by them is _______.

(A) Positive

(B) Negative

(C) Zero

(D) Undefined

Question 9. The distance formula can be used to verify if a quadrilateral is a rhombus by checking if all four _______ are equal.

(A) Angles

(B) Diagonals

(C) Vertices

(D) Sides

Question 10. The distance formula in two dimensions measures the length of a _______ line segment between two points.

(A) Curved

(B) Broken

(C) Straight

(D) Dotted

Question 1. The coordinates of the midpoint of the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ are given by the formula _______.

(A) $(\frac{x_1+y_1}{2}, \frac{x_2+y_2}{2})$

(B) $(\frac{x_1x_2}{2}, \frac{y_1y_2}{2})$

(C) $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

(D) $(\frac{x_2-x_1}{2}, \frac{y_2-y_1}{2})$

Question 2. The point that divides the line segment joining points A and B internally in the ratio $m:n$ lies _______ A and B.

(A) Outside

(B) Beyond

(C) Between

(D) Next to

Question 3. The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is found using the formula that averages the corresponding _______.

(A) Distances

(B) Slopes

(C) Ratios

(D) Coordinates

Question 4. If a point P divides the line segment AB externally in the ratio $m:n$, the point P lies on the line containing A and B but _______ the segment AB.

(A) Inside

(B) On

(C) Outside

(D) Parallel to

Question 5. The midpoint formula is a special case of the section formula where the ratio of division is _______.

(A) $2:1$

(B) $1:2$

(C) $1:1$

(D) $0:1$

Question 6. To find the coordinates of a point that trisects a line segment, we use the section formula with ratios $1:2$ and _______.

(A) $1:1$

(B) $2:1$

(C) $3:1$

(D) $1:3$

Question 7. The section formula for internal division in the ratio $m:n$ is $(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n})$. If $m=n$, this simplifies to the _______ formula.

(A) Distance

(B) Slope

(C) Midpoint

(D) External division

Question 8. The ratio in which the X-axis divides the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ can be found by setting the _______ coordinate of the division point to zero and solving for the ratio.

(A) x

(B) y

(C) z

(D) horizontal

Question 9. The centroid of a triangle divides each median in the ratio _______, from the vertex to the midpoint of the opposite side.

(A) $1:1$

(B) $1:2$

(C) $2:1$

(D) $3:1$

Question 10. If a point divides a line segment externally in the ratio $m:n$, the ratio $\frac{m}{n}$ (considering lengths) must be _______.

(A) Equal to 1

(B) Not equal to 1

(C) Less than 1

(D) Equal to 0

Question 1. The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. The vertical bars $|...|$ indicate taking the _______ value.

(A) Negative

(B) Square

(C) Absolute

(D) Cube

Question 2. Three points are collinear if and only if the area of the triangle formed by them is _______.

(A) Greater than zero

(B) Less than zero

(C) Equal to zero

(D) Undefined

Question 3. The area of the triangle with vertices $(0, 0)$, $(3, 0)$, and $(0, 4)$ is _______ square units.

(A) $6$

(B) 6

(C) 12

(D) 7

Question 4. If points $(2, 3), (4, k), (6, -3)$ are collinear, the value of $k$ is found by setting the area of the triangle formed by them to _______.

(A) 1

(B) -1

(C) 0

(D) 2

Question 5. The formula for the area of a triangle using vertex coordinates is derived using concepts from _______ geometry.

(A) Euclidean

(B) Synthetic

(C) Coordinate

(D) Projective

Question 6. If the area calculation using the formula results in a negative value, it indicates the _______ of traversing the vertices.

(A) Length

(B) Orientation

(C) Center

(D) Size

Question 7. A degenerate triangle is one where the vertices are _______.

(A) Concurrent

(B) Collinear

(C) Equidistant

(D) Perpendicular

Question 8. The area of a triangle formed by three points on the same straight line is always equal to the area of a _______.

(A) Square

(B) Point

(C) Line segment

(D) Circle

Question 9. The condition for collinearity of three points using the area formula is that the value of the expression inside the absolute value bars must be _______.

(A) Non-zero

(B) Positive

(C) Negative

(D) Zero

Question 10. If the area of a triangle is non-zero, the vertices are not _______.

(A) Concurrent

(B) Collinear

(C) Equidistant

(D) Parallel

Question 1. The point of intersection of the medians of a triangle is called the _______.

(A) Incenter

(B) Orthocenter

(C) Circumcenter

(D) Centroid

Question 2. The incenter of a triangle is the point of intersection of its _______.

(A) Altitudes

(B) Medians

(C) Angle bisectors

(D) Perpendicular bisectors

Question 3. The circumcenter of a triangle is equidistant from the _______ of the triangle.

(A) Sides

(B) Midpoints of sides

(C) Vertices

(D) Feet of altitudes

Question 4. For a right-angled triangle, the orthocenter lies at the _______.

(A) Midpoint of hypotenuse

(B) Vertex with the right angle

(C) Centroid

(D) Outside the triangle

Question 5. For an equilateral triangle, the centroid, incenter, circumcenter, and orthocenter _______.

(A) Are distinct points

(B) Form a square

(C) Coincide

(D) Form a line

Question 6. The circumcenter is the center of the _______ circle, which passes through the vertices of the triangle.

(A) Inscribed

(B) Excentral

(C) Circumscribed

(D) Medial

Question 7. The incenter is the center of the _______ circle, which is tangent to the sides of the triangle.

(A) Circumscribed

(B) Inscribed

(C) Excentral

(D) Nine-point

Question 8. The orthocenter of a triangle is the point of intersection of its _______.

(A) Medians

(B) Angle bisectors

(C) Altitudes

(D) Perpendicular bisectors

Question 9. For an obtuse-angled triangle, the orthocenter lies _______ the triangle.

(A) Inside

(B) Outside

(C) On a vertex

(D) On a side

Question 10. The centroid divides each median in the ratio $2:1$, with the longer segment being from the _______.

(A) Midpoint of the side

(B) Centroid

(C) Vertex

(D) Foot of the altitude

Question 1. The path traced by a point moving under a given condition is called its _______.

(A) Area

(B) Volume

(C) Locus

(D) Perimeter

Question 2. The equation of the locus of a point P$(x, y)$ such that its distance from the origin is always $7$ units is _______.

(A) $x^2 + y^2 = 7$

(B) $x+y=7$

(C) $\sqrt{x^2+y^2} = 7$

(D) $x^2-y^2=49$

Question 3. The locus of a point equidistant from two fixed points is a _______.

(A) Circle

(B) Parabola

(C) Perpendicular bisector

(D) Ellipse

Question 4. The equation of the locus of a point P$(x, y)$ which is equidistant from the point $A(1, 0)$ and the line $x = -1$ is a _______.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 5. The equation of a locus represents the _______ condition algebraically.

(A) Geometric

(B) Arithmetic

(C) Trigonometric

(D) Vector

Question 6. To find the equation of a locus, we let the coordinates of the moving point be _______.

(A) $(a, b)$

(B) $(h, k)$

(C) $(x, y)$

(D) $(x_0, y_0)$

Question 7. The locus of a point such that the sum of its distances from two fixed points is constant is an _______.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 8. The locus of a point such that the difference of its distances from two fixed points is constant is a _______.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 9. Every point on the locus must _______ the given geometric condition.

(A) Contradict

(B) Not satisfy

(C) Satisfy

(D) Ignore

Question 10. The process of finding the equation of a locus involves translating a geometric problem into an _______ problem.

(A) Arithmetic

(B) Algebraic

(C) Trigonometric

(D) Vector

Question 1. If the origin is shifted to $(h, k)$, and $(x, y)$ are the original coordinates, the new x-coordinate $(X)$ is given by _______.

(A) $x+h$

(B) $x-h$

(C) $h-x$

(D) $h+x$

Question 2. If the origin is shifted to $(2, 3)$, the new coordinates of the point $(5, 7)$ are _______.

(A) $(7, 10)$

(B) $(3, 4)$

(C) $(-3, -4)$

(D) $(2, 4)$

Question 3. Shifting the origin changes the _______ of points but preserves distances between them.

(A) Slopes

(B) Coordinates

(C) Intercepts

(D) Orientation

Question 4. If the equation of a line is $y = mx + c$, after shifting the origin, its slope in the new coordinate system is _______.

(A) $m$

(B) $m+h$

(C) $m-k$

(D) $m(h+k)$

Question 5. To eliminate the linear terms from $x^2 + y^2 + 2gx + 2fy + c = 0$, the origin is shifted to the point _______.

(A) $(g, f)$

(B) $(-g, -f)$

(C) $(c, 0)$

(D) $(0, c)$

Question 6. If the original coordinates are $(x, y)$ and the new origin is $(h, k)$, then $x$ is equal to _______ in the new system.

(A) $X-h$

(B) $X+h$

(C) $x-h$

(D) $x+h$

Question 7. Shifting of origin is a type of coordinate transformation called _______.

(A) Rotation

(B) Scaling

(C) Reflection

(D) Translation

Question 8. If the original coordinates are $(x, y)$ and the new coordinates are $(X, Y)$ after shifting the origin, then $Y = _______$.

(A) $y-k$

(B) $y+k$

(C) $k-y$

(D) $y/k$

Question 9. If the equation $x^2 + y^2 = r^2$ is transformed by shifting the origin to $(1, 1)$, the new equation is _______.

(A) $(X+1)^2 + (Y+1)^2 = r^2$

(B) $(X-1)^2 + (Y-1)^2 = r^2$

(C) $X^2 + Y^2 = r^2 - 2$

(D) $X^2 + Y^2 = r^2$

Question 10. The process of changing the coordinate system by moving the origin to a new point without rotating the axes is called _______.

(A) Rotation of axes

(B) Shifting of origin

(C) Scaling of axes

(D) Reflection of axes

Question 1. The slope of a straight line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by _______ (assuming $x_1 \neq x_2$).

(A) $\frac{x_2-x_1}{y_2-y_1}$

(B) $\frac{y_2-y_1}{x_2-x_1}$

(C) $\frac{x_1+x_2}{y_1+y_2}$

(D) $\frac{y_1-x_1}{y_2-x_2}$

Question 2. The slope of a horizontal line is _______.

(A) 1

(B) -1

(C) 0

(D) Undefined

Question 3. Two non-vertical lines are parallel if and only if their slopes are _______.

(A) Negative reciprocals

(B) Equal

(C) Proportional

(D) Add up to zero

Question 4. Two non-vertical lines are perpendicular if and only if the product of their slopes is _______.

(A) 1

(B) -1

(C) 0

(D) Undefined

Question 5. The tangent of the angle between two lines with slopes $m_1$ and $m_2$ is given by $|\frac{m_1-m_2}{1+m_1m_2}|$, provided $1+m_1m_2 \neq$ _______.

(A) 1

(B) -1

(C) 0

(D) Undefined

Question 6. The slope of the line joining $(2, 5)$ and $(2, 10)$ is NOT:

(A) Undefined.

(B) The slope of a vertical line.

(C) Calculated using $\frac{10-5}{2-2}$, which involves division by zero.

(D) $5/0$, which is $0$.

Question 7. If a line makes an angle of $45^\circ$ with the positive X-axis, its slope is _______.

(A) 0

(B) 1

(C) -1

(D) $\sqrt{2}$

Question 8. The slope of a vertical line is _______.

(A) 1

(B) -1

(C) 0

(D) Undefined

Question 9. If two lines have slopes $m_1$ and $m_2$ and $m_1 = m_2$, then the lines are _______ (assuming non-vertical).

(A) Perpendicular

(B) Intersecting

(C) Parallel

(D) Coincident (unless their intercepts are different)

Question 10. The slope of the line joining $(5, 2)$ and $(8, 2)$ is _______.

(A) $0$

(B) Undefined

(C) 1

(D) 0

Question 1. The equation of a line parallel to the X-axis at a distance of $d$ units from it is _______.

(A) $x = d$

(B) $y = d$ or $y = -d$

(C) $x^2 = d^2$

(D) $y^2 = d^2$

Question 2. The point-slope form of the equation of a line with slope $m$ passing through $(x_1, y_1)$ is _______.

(A) $y = mx + c$

(B) $\frac{x}{a} + \frac{y}{b} = 1$

(C) $y - y_1 = m(x - x_1)$

(D) $Ax + By + C = 0$

Question 3. The slope-intercept form of the equation of a line is _______, where $m$ is the slope and $c$ is the y-intercept.

(A) $Ax + By + C = 0$

(B) $y = mx + c$

(C) $y - y_1 = m(x - x_1)$

(D) $\frac{x}{a} + \frac{y}{b} = 1$

Question 4. The intercept form of the equation of a line with x-intercept $a$ and y-intercept $b$ is _______ (assuming $a \neq 0, b \neq 0$).

(A) $y = mx + c$

(B) $\frac{x}{a} + \frac{y}{b} = 1$

(C) $y - b = m(x - a)$

(D) $x \cos\alpha + y \sin\alpha = p$

Question 5. The normal form of the equation of a line is $x \cos\alpha + y \sin\alpha = p$, where $p$ is the _______ from the origin to the line.

(A) Slope

(B) Intercept

(C) Perpendicular distance

(D) Angle

Question 6. The equation of the X-axis is _______.

(A) $x = 0$

(B) $y = 0$

(C) $x = y$

(D) $x + y = 0$

Question 7. The two-point form of the equation of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is $\frac{y-y_1}{x-x_1} = _______$ (assuming $x_1 \neq x_2$).

(A) $\frac{x_2-x_1}{y_2-y_1}$

(B) $\frac{y_2-y_1}{x_2-x_1}$

(C) $m$

(D) 1

Question 8. The equation of a line parallel to the Y-axis passing through the point $(a, b)$ is _______.

(A) $x = a$

(B) $y = b$

(C) $x = b$

(D) $y = a$

Question 9. The equation of a line passing through the origin is always of the form $y = mx$, where $m$ is the _______.

(A) X-intercept

(B) Y-intercept

(C) Slope

(D) Distance from origin

Question 10. To convert the general equation $Ax + By + C = 0$ into intercept form, we divide by $_______$ (assuming $C \neq 0$).

(A) A

(B) B

(C) C

(D) -C

Question 1. The general equation of a straight line is $Ax + By + C = 0$, where A and B are not both equal to _______.

(A) 1

(B) -1

(C) 0

(D) C

Question 2. The slope of the line $Ax + By + C = 0$ (where $B \neq 0$) is _______.

(A) $A/B$

(B) $-A/B$

(C) $C/B$

(D) $-C/A$

Question 3. The y-intercept of the line $Ax + By + C = 0$ (where $B \neq 0$) is _______.

(A) $-C/A$

(B) $-A/B$

(C) $-C/B$

(D) $C/B$

Question 4. To convert the general equation $Ax + By + C = 0$ to slope-intercept form, we need to isolate the _______ term.

(A) x

(B) y

(C) C

(D) AB

Question 5. The point of intersection of two distinct lines is found by _______ their equations simultaneously.

(A) Adding

(B) Subtracting

(C) Multiplying

(D) Solving

Question 6. If two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are parallel, then $\frac{a_1}{a_2} = _______$ (assuming $a_2, b_2 \neq 0$).

(A) $\frac{b_2}{b_1}$

(B) $\frac{b_1}{b_2}$

(C) $\frac{c_1}{c_2}$

(D) $-\frac{a_2}{b_2}$

Question 7. If two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are perpendicular, then $a_1a_2 + b_1b_2 = _______$ (assuming $b_1, b_2 \neq 0$).

(A) 1

(B) -1

(C) 0

(D) $c_1c_2$

Question 8. The equation of a line passing through the origin is one where the constant term C in $Ax + By + C = 0$ is _______.

(A) Positive

(B) Negative

(C) Zero

(D) Non-zero

Question 9. If the equations of two lines are proportional, the lines are _______.

(A) Parallel and distinct

(B) Perpendicular

(C) Coincident

(D) Intersecting

Question 10. To convert the general equation $Ax + By + C = 0$ to intercept form $\frac{x}{a} + \frac{y}{b} = 1$, we divide by $_______$ (assuming $C \neq 0$).

(A) $A$

(B) $B$

(C) $C$

(D) $-C$

Question 1. The distance of a point $(x_0, y_0)$ from the line $Ax + By + C = 0$ is given by the formula $_______$.

(A) $\frac{Ax_0 + By_0 + C}{\sqrt{A^2+B^2}}$

(B) $\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$

(C) $|Ax_0 + By_0 + C|$

(D) $\sqrt{(x_0-A)^2 + (y_0-B)^2}$

Question 2. The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is $_______$.

(A) $\frac{|C_1 + C_2|}{\sqrt{A^2+B^2}}$

(B) $\frac{|C_1 - C_2|}{\sqrt{A^2+B^2}}$

(C) $|C_1 - C_2|$

(D) $\sqrt{(C_1-C_2)^2}$

Question 3. The equation of any line passing through the intersection of two given lines $L_1 = 0$ and $L_2 = 0$ is of the form $_______$, where $\lambda$ is a parameter.

(A) $L_1 + L_2 = 0$

(B) $L_1 - L_2 = 0$

(C) $L_1 \times L_2 = 0$

(D) $L_1 + \lambda L_2 = 0$

Question 4. The distance of the origin $(0, 0)$ from the line $Ax + By + C = 0$ is $_______$.

(A) $|C|$

(B) $C/\sqrt{A^2+B^2}$

(C) $|C|/\sqrt{A^2+B^2}$

(D) $A+B+C$

Question 5. The distance between the parallel lines $y = mx + c_1$ and $y = mx + c_2$ is $_______$.

(A) $|c_1 - c_2|$

(B) $|c_1 - c_2| / \sqrt{1+m^2}$

(C) $|c_1 + c_2| / \sqrt{1+m^2}$

(D) $|c_1 - c_2| / (1+m^2)$

Question 6. The shortest distance from a point to a line is the length of the _______ from the point to the line.

(A) Segment

(B) Parallel line

(C) Perpendicular

(D) Angle bisector

Question 7. The family of lines passing through the origin can be represented by $y = mx$ (for finite slope) or $_______$ (for infinite slope).

(A) $y = c$

(B) $x = k$

(C) $x = 0$

(D) $y = 0$

Question 8. If a point lies on a line, the distance of the point from the line is _______.

(A) Positive

(B) Negative

(C) Zero

(D) Undefined

Question 9. The concept of a family of lines is useful for finding the equation of a line that satisfies an additional _______.

(A) Slope

(B) Distance

(C) Point

(D) Condition

Question 10. The equation $Ax + By + C = 0$ can represent a family of parallel lines if A and B are fixed and _______ is varied.

(A) A

(B) B

(C) C

(D) x and y

Question 1. The Cartesian coordinate system in three dimensions consists of three mutually _______ axes.

(A) Parallel

(B) Intersecting

(C) Perpendicular

(D) Skew

Question 2. The point where the three coordinate axes intersect is called the _______ in 3D.

(A) Center

(B) Apex

(C) Node

(D) Origin

Question 3. The three coordinate planes in 3D space divide the space into _______ octants.

(A) Four

(B) Six

(C) Eight

(D) Twelve

Question 4. The equation of the XY-plane in 3D is _______.

(A) $x = 0$

(B) $y = 0$

(C) $z = 0$

(D) $x + y = 0$

Question 5. The coordinates of a point in 3D space are represented by an ordered _______ of real numbers.

(A) Pair

(B) Triple

(C) Quadruple

(D) Single

Question 6. The distance of a point $(x, y, z)$ from the YZ-plane is _______.

(A) $|x|$

(B) $|y|$

(C) $|z|$

(D) $\sqrt{y^2+z^2}$

Question 7. A point $(a, b, c)$ lies on the X-axis if $b=0$ and $c=_______$.

(A) 1

(B) -1

(C) a

(D) 0

Question 8. The coordinates of the origin in 3D are _______.

(A) $(1, 1, 1)$

(B) $(0, 0, 0)$

(C) $(0, 1, 0)$

(D) $(1, 0, 0)$

Question 9. The distance of a point $(x, y, z)$ from the XZ-plane is _______.

(A) $|x|$

(B) $|y|$

(C) $|z|$

(D) $\sqrt{x^2+z^2}$

Question 10. The three coordinate planes are mutually _______.

(A) Parallel

(B) Perpendicular

(C) Inclined

(D) Skew

Question 1. The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by the formula $_______$.

(A) $(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2$

(B) $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$

(C) $\sqrt{|x_2-x_1| + |y_2-y_1| + |z_2-z_1|}$

(D) $|x_2-x_1| + |y_2-y_1| + |z_2-z_1|$

Question 2. The distance of a point $(x, y, z)$ from the origin is $_______$.

(A) $x^2+y^2+z^2$

(B) $\sqrt{x^2+y^2+z^2}$

(C) $|x|+|y|+|z|$

(D) $x+y+z$

Question 3. The distance of a point $(a, b, c)$ from the X-axis is $_______$.

(A) $|a|$

(B) $\sqrt{a^2+b^2+c^2}$

(C) $\sqrt{b^2+c^2}$

(D) $\sqrt{a^2+b^2}$

Question 4. The distance formula in three dimensions is derived using the _______ theorem extended to 3D.

(A) Thales'

(B) Euclidean

(C) Pythagorean

(D) Fermat's

Question 5. If the distance between $(k, 0, 0)$ and $(0, 0, 4)$ is $5$ units, then $|k|$ is equal to _______.

(A) 1

(B) 3

(C) 4

(D) 5

Question 6. The distance between two points in 3D space is always a _______ value.

(A) Negative

(B) Complex

(C) Zero or positive

(D) Undefined

Question 7. The distance from a point to a coordinate plane is the _______ value of the coordinate perpendicular to that plane.

(A) Negative

(B) Square

(C) Absolute

(D) Inverse

Question 8. To check if three points in 3D are collinear using the distance formula, we verify if the sum of the distances between any two pairs equals the _______ between the remaining pair.

(A) Product of distances

(B) Difference of distances

(C) Distance

(D) Average distance

Question 9. The distance between the points $(x, y, z)$ and $(x, y, 0)$ is _______.

(A) $|x|$

(B) $|y|$

(C) $|z|$

(D) $\sqrt{x^2+y^2}$

Question 10. The formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ represents the length of the _______ line segment in 3D.

(A) Curved

(B) Broken

(C) Straight

(D) Dotted

Question 1. The coordinates of the midpoint of the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are given by _______.

(A) $(\frac{x_1x_2}{2}, \frac{y_1y_2}{2}, \frac{z_1z_2}{2})$

(B) $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$

(C) $(\frac{x_2-x_1}{2}, \frac{y_2-y_1}{2}, \frac{z_2-z_1}{2})$

(D) $(\frac{x_1+y_1+z_1}{3}, \frac{x_2+y_2+z_2}{3})$

Question 2. The point that divides the line segment joining A and B internally in the ratio $m:n$ lies _______ the segment AB.

(A) Outside

(B) Beyond

(C) Between

(D) Parallel to

Question 3. The centroid of a triangle with vertices in 3D is found by averaging the corresponding _______ of the vertices.

(A) Distances

(B) Slopes

(C) Ratios

(D) Coordinates

Question 4. If a point P divides the line segment AB externally in the ratio $m:n$, the point P lies on the line containing A and B but _______ the segment AB.

(A) Inside

(B) On

(C) Outside

(D) Parallel to

Question 5. The section formula for internal division in the ratio $m:n$ is $(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n})$. If $m=n$, this simplifies to the _______ formula in 3D.

(A) Distance

(B) Slope

(C) Midpoint

(D) External division

Question 6. The ratio in which a coordinate plane (e.g., XY-plane) divides the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ can be found by setting the _______ coordinate of the division point to zero.

(A) x

(B) y

(C) z

(D) horizontal

Question 7. The centroid of a tetrahedron with vertices $(x_1, y_1, z_1), \dots, (x_4, y_4, z_4)$ is found using the formula that averages the corresponding _______ of the vertices.

(A) Distances

(B) Slopes

(C) Ratios

(D) Coordinates

Question 8. The section formula in 3D is a direct extension of the section formula in _______ dimensions.

(A) One

(B) Two

(C) Four

(D) Zero

Question 9. If a point divides a line segment internally, the ratio $m:n$ (considering positive lengths) must be _______.

(A) Negative

(B) Positive

(C) Zero

(D) Undefined

Question 10. The coordinates of the point dividing the line segment joining $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ externally in the ratio $m:n$ are $_______$, assuming $m \neq n$.

(A) $(\frac{mx_2 + nx_1}{m+n}, \dots)$

(B) $(\frac{mx_2 - nx_1}{m-n}, \dots)$

(C) $(\frac{nx_1 - mx_2}{m-n}, \dots)$

(D) $(\frac{x_1-x_2}{m-n}, \dots)$

Question 1. Conic sections are curves obtained by the intersection of a plane with a double _______ cone.

(A) Square

(B) Circular

(C) Rectangular

(D) Polygonal

Question 2. The ratio of the distance from a point on a conic to the focus and its distance from the directrix is called the _______.

(A) Vertex

(B) Axis

(C) Eccentricity

(D) Latus Rectum

Question 3. For a parabola, the eccentricity is equal to _______.

(A) 0

(B) 1

(C) Less than 1

(D) Greater than 1

Question 4. For an ellipse, the eccentricity is _______ 1.

(A) Equal to

(B) Greater than

(C) Less than

(D) Not related to

Question 5. For a hyperbola, the eccentricity is _______ 1.

(A) Equal to

(B) Greater than

(C) Less than

(D) Not related to

Question 6. A circle is a special case of an ellipse where the eccentricity is _______.

(A) 1

(B) -1

(C) 0

(D) Undefined

Question 7. When the cutting plane passes through the vertex of the cone, the resulting intersection is a _______ conic section.

(A) Non-degenerate

(B) Degenerate

(C) Standard

(D) Regular

Question 8. A single line can be formed as a degenerate conic section when the plane is tangent to the cone along a _______ and passes through the vertex.

(A) Circle

(B) Focus

(C) Generator

(D) Axis

Question 9. The fixed point used in the definition of a conic section is called the _______.

(A) Directrix

(B) Vertex

(C) Center

(D) Focus

Question 10. A pair of intersecting lines is a degenerate conic section formed when the plane passes through the vertex and contains the _______ of the cone.

(A) Base

(B) Circumference

(C) Axis

(D) Directrix

Question 1. The equation of a circle with center $(h, k)$ and radius $r$ is _______.

(A) $x^2 + y^2 = r^2$

(B) $(x+h)^2 + (y+k)^2 = r^2$

(C) $(x-h)^2 + (y-k)^2 = r^2$

(D) $(x-h)^2 - (y-k)^2 = r^2$

Question 2. The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. The coordinates of the center are _______.

(A) $(g, f)$

(B) $(-g, -f)$

(C) $(g, -f)$

(D) $(-g, f)$

Question 3. For the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ to represent a real circle, the condition on $g, f, c$ is _______.

(A) $g^2 + f^2 - c < 0$

(B) $g^2 + f^2 - c = 0$

(C) $g^2 + f^2 - c > 0$

(D) $c > 0$

Question 4. A line is tangent to a circle if the distance from the center of the circle to the line is equal to the _______.

(A) Diameter

(B) Circumference

(C) Radius

(D) Chord length

Question 5. Two circles intersect at two distinct points if the distance between their centers is _______ the sum of their radii and _______ the absolute difference of their radii.

(A) Greater than, less than

(B) Less than, greater than

(C) Equal to, equal to

(D) Greater than, greater than

Question 6. The equation of a circle with diameter endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = _______$.

(A) $r^2$

(B) 1

(C) 0

(D) $-1$

Question 7. For a point circle, the radius is equal to _______.

(A) A positive value

(B) A negative value

(C) Zero

(D) Undefined

Question 8. The equation $x^2 + y^2 = r^2$ represents a circle centered at the _______.

(A) Vertex

(B) Focus

(C) Origin

(D) Directrix

Question 9. If $g^2 + f^2 - c < 0$ in the general equation of a circle, it represents an _______ circle.

(A) Real

(B) Point

(C) Imaginary

(D) Unit

Question 10. Two circles touch internally if the distance between their centers is equal to the _______ of their radii.

(A) Sum

(B) Product

(C) Absolute difference

(D) Average

Question 1. A parabola is the locus of a point equidistant from a fixed point (focus) and a fixed line (_______).

(A) Vertex

(B) Axis

(C) Directrix

(D) Latus Rectum

Question 2. The standard equation of a parabola symmetric about the X-axis, opening to the right, with vertex at origin is _______.

(A) $x^2 = 4ay$

(B) $y^2 = 4ax$

(C) $x^2 = -4ay$

(D) $y^2 = -4ax$

Question 3. For the parabola $y^2 = 8x$, the coordinates of the focus are _______.

(A) $(2, 0)$

(B) $(-2, 0)$

(C) $(0, 2)$

(D) $(0, -2)$

Question 4. For the parabola $y^2 = 4ax$, the equation of the directrix is _______.

(A) $x = a$

(B) $x = -a$

(C) $y = a$

(D) $y = -a$

Question 5. The length of the latus rectum of the parabola $x^2 = 16y$ is _______.

(A) 4

(B) 8

(C) 12

(D) 16

Question 6. The vertex of a parabola is the point on the parabola closest to the _______.

(A) Focus

(B) Axis

(C) Latus Rectum

(D) Directrix

Question 7. The axis of symmetry for the parabola $y^2 = -12x$ is the _______.

(A) Y-axis

(B) X-axis

(C) Line $x=3$

(D) Line $y=3$

Question 8. The parametric equations $x = at^2, y = 2at$ represent a _______.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 9. For the parabola $x^2 = 4ay$, the axis is the _______.

(A) X-axis

(B) Y-axis

(C) Line $x=a$

(D) Line $y=a$

Question 10. The equation of the parabola with vertex at origin and focus at $(5, 0)$ is _______.

(A) $y^2 = 5x$

(B) $y^2 = 10x$

(C) $y^2 = 20x$

(D) $x^2 = 20y$

Question 1. For an ellipse, the sum of the distances from any point on the ellipse to the two foci is _______.

(A) Variable

(B) Zero

(C) Constant

(D) Infinity

Question 2. The standard equation of an ellipse centered at the origin is _______.

(A) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

(B) $x^2 + y^2 = r^2$

(C) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

(D) $y^2 = 4ax$

Question 3. For the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, the length of the major axis is _______.

(A) 5

(B) 10

(C) 3

(D) 6

Question 4. For the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1$, the major axis is along the _______ axis.

(A) X

(B) Y

(C) Z

(D) Neither

Question 5. The length of the latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (with $a>b$) is _______.

(A) $2a^2/b$

(B) $2b^2/a$

(C) $a^2/b$

(D) $b^2/a$

Question 6. The parametric equations $x = a\cos\theta, y = b\sin\theta$ represent an _______ centered at the origin.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 7. The eccentricity of an ellipse is _______ than 1.

(A) Equal to

(B) Greater

(C) Less

(D) Not comparable

Question 8. The vertices of an ellipse are the endpoints of the _______ axis.

(A) Minor

(B) Conjugate

(C) Transverse

(D) Major

Question 9. The equation of an ellipse with center $(h, k)$ and major axis parallel to the X-axis is $_______$, where $a>b$.

(A) $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

(B) $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

(C) $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

(D) $(x-h)^2 + (y-k)^2 = a^2$

Question 10. The directrices of an ellipse are lines perpendicular to the major axis, located _______ the ellipse.

(A) Inside

(B) On the boundary of

(C) Outside

(D) At the foci

Question 1. For a hyperbola, the absolute difference of the distances from any point on the hyperbola to the two foci is _______.

(A) Variable

(B) Zero

(C) Constant

(D) Infinity

Question 2. The standard equation of a hyperbola centered at the origin with transverse axis along the X-axis is _______.

(A) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

(B) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

(C) $x^2 + y^2 = r^2$

(D) $y^2 = 4ax$

Question 3. For the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$, the length of the transverse axis is _______.

(A) 4

(B) 8

(C) 3

(D) 6

Question 4. The eccentricity of a hyperbola is _______ 1.

(A) Equal to

(B) Greater than

(C) Less than

(D) Not related to

Question 5. The equations of the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ are $_______$.

(A) $y = \pm \frac{a}{b}x$

(B) $y = \pm \frac{b}{a}x$

(C) $x = \pm \frac{a}{b}y$

(D) $x = \pm \frac{b}{a}y$

Question 6. The parametric equations $x = a\sec\theta, y = b\tan\theta$ represent a _______.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 7. The vertices of a hyperbola are the endpoints of the _______ axis.

(A) Minor

(B) Conjugate

(C) Transverse

(D) Major

Question 8. The equation of a hyperbola with center $(h, k)$ and transverse axis parallel to the Y-axis is $_______$, where $a$ is related to the positive term.

(A) $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

(B) $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$

(C) $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$

(D) $(x-h)^2 + (y-k)^2 = a^2$

Question 9. The directrices of a hyperbola are lines perpendicular to the transverse axis, located _______ the two branches of the hyperbola.

(A) Inside

(B) Outside

(C) At the foci

(D) Between

Question 10. A rectangular hyperbola is a hyperbola whose asymptotes are _______.

(A) Parallel

(B) Perpendicular

(C) Coincident

(D) Tangent

Question 1. Parametric equations represent the coordinates of a point on a curve as functions of a single variable called the _______.

(A) Variable

(B) Parameter

(C) Constant

(D) Coordinate

Question 2. The parametric equations $x = a\cos t, y = a\sin t$ represent a _______ centered at the origin.

(A) Parabola

(B) Ellipse

(C) Circle

(D) Hyperbola

Question 3. For the parabola $y^2 = 4ax$, the standard parametric coordinates are often taken as _______.

(A) $(a\cos t, a\sin t)$

(B) $(at^2, 2at)$

(C) $(a\sec t, b\tan t)$

(D) $(a\cos t, b\sin t)$

Question 4. By eliminating the parameter from parametric equations, we obtain the _______ equation of the curve.

(A) Vector

(B) Polar

(C) Cartesian

(D) Normal

Question 5. The parametric equations $x = a\cos \theta, y = b\sin \theta$ represent an _______ centered at the origin.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 6. The parametric equations $x = a\sec \theta, y = b\tan \theta$ represent a _______ centered at the origin.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 7. Parametric representation is useful for describing curves that are not functions of _______ or _______.

(A) Parameters, constants

(B) X, Y

(C) Slope, intercept

(D) Focus, directrix

Question 8. The parametric equations for a circle centered at $(h, k)$ with radius $r$ are $x = h + r\cos t$ and $y = _______$.

(A) $k + r\cos t$

(B) $k + r\sin t$

(C) $h + r\sin t$

(D) $k + h\sin t$

Question 9. The Cartesian equation corresponding to $x = 2t, y = t^2$ is _______.

(A) $y = x$

(B) $y = x^2$

(C) $y = x^2/4$

(D) $x = y^2/4$

Question 10. Parametric equations are commonly used in physics to describe the _______ of a particle over time.

(A) Mass

(B) Trajectory

(C) Energy

(D) Charge

Question 1. Coordinate geometry allows us to solve geometric problems by translating them into _______ problems.

(A) Physical

(B) Chemical

(C) Algebraic

(D) Biological

Question 2. Using coordinate geometry, we can verify if three points are collinear by checking if the area of the triangle formed by them is _______.

(A) Positive

(B) Negative

(C) Zero

(D) Large

Question 3. The distance formula is used in coordinate geometry to calculate the _______ between two points.

(A) Slope

(B) Area

(C) Distance

(D) Angle

Question 4. Coordinate geometry is used in _______ systems like GPS to determine location.

(A) Communication

(B) Navigation

(C) Imaging

(D) Security

Question 5. The section formula in coordinate geometry is used to find the coordinates of a point that divides a line segment in a given _______.

(A) Distance

(B) Angle

(C) Area

(D) Ratio

Question 6. Coordinate methods can be used to prove that the diagonals of a parallelogram _______ each other.

(A) Are perpendicular to

(B) Are equal to

(C) Bisect

(D) Are parallel to

Question 7. In physics, the trajectory of a projectile (neglecting air resistance) is described by an equation of a _______.

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 8. Coordinate geometry is fundamental in _______ graphics for representing and manipulating objects.

(A) Sound

(B) Text

(C) Computer

(D) Data

Question 9. The equation of a locus is the _______ representation of a geometric condition.

(A) Visual

(B) Verbal

(C) Algebraic

(D) Physical

Question 10. In 3D geometry, coordinate methods are used to describe points, lines, planes, and _______ shapes.

(A) Curved

(B) Plane

(C) Solid

(D) Flat