Negative Questions MCQs for Sub-Topics of Topic 6: Coordinate Geometry

Question 1. Which of the following statements about the Cartesian coordinate system in two dimensions is FALSE?

(A) The X-axis and Y-axis are perpendicular.

(B) The point of intersection of the axes is called the origin.

(C) The system divides the plane into four quadrants.

(D) Points on the X-axis have non-zero ordinates.

Question 2. Regarding the coordinates of a point $(x, y)$, which statement is INCORRECT?

(A) $x$ is called the abscissa.

(B) $y$ is called the ordinate.

(C) The coordinates uniquely identify the position of the point in the plane.

(D) The order of coordinates does not matter (i.e., $(x, y)$ is the same as $(y, x)$).

Question 3. Which of the following points does NOT lie in the second quadrant?

(A) $(-1, 5)$

(B) $(-3, 2)$

(C) $(4, -6)$

(D) $(-5, 10)$

Question 4. If a point $(a, b)$ is in the third quadrant, which of the following must NOT be true?

(A) $a < 0$

(B) $b < 0$

(C) $ab > 0$

(D) $a > 0$

Question 5. Which statement is NOT true about a point on the Y-axis?

(A) Its abscissa is $0$.

(B) Its coordinates are of the form $(0, y)$.

(C) Its distance from the Y-axis is non-zero.

(D) If it is not the origin, it is in either the positive or negative part of the Y-axis.

Question 6. Which of the following points does NOT lie on either the X-axis or the Y-axis?

(A) $(5, 0)$

(B) $(0, -8)$

(C) $(2, 7)$

(D) $(0, 0)$

Question 7. The Cartesian plane is NOT also known as the:

(A) Coordinate plane

(B) XY-plane

(C) Rectangular coordinate system (plane)

(D) Complex plane

Question 8. Which of the following statements is INCORRECT regarding the perpendicular distances of a point $(x, y)$ from the axes?

(A) Distance from the X-axis is $|y|$.

(B) Distance from the Y-axis is $|x|$.

(C) The distance is always non-negative.

(D) The distance from the X-axis is $y$ (without absolute value).

Question 9. If a point $(p, q)$ is in the fourth quadrant, which condition must NOT hold true?

(A) $p > 0$

(B) $q < 0$

(C) $p/q < 0$

(D) $p+q > 0$ (This might be true, e.g. (5, -1), but not necessary)

Question 10. The coordinate axes do NOT divide the plane into:

(A) Four quadrants

(B) Two axes

(C) A region where $x>0, y>0$ (1st quadrant)

(D) Five distinct regions.

Question 1. To plot the point $(5, -2)$ starting from the origin, which movement is NOT correct?

(A) Move 5 units right along the X-axis.

(B) Move 2 units down parallel to the Y-axis.

(C) Move 5 units up along the Y-axis.

(D) The final point is in the fourth quadrant.

Question 2. A point is located $3$ units to the left of the Y-axis and $1$ unit above the X-axis. Which of the following is NOT true about its coordinates?

(A) The x-coordinate is $-3$.

(B) The y-coordinate is $1$.

(C) The coordinates are $(-3, 1)$.

(D) The point is in the third quadrant.

Question 3. Which statement is FALSE about plotting the point $(-4, 0)$?

(A) The point lies on the X-axis.

(B) The point is $4$ units to the left of the origin.

(C) The point is in the second quadrant.

(D) The ordinate of the point is $0$.

Question 4. If a point $(p, q)$ is plotted and $p < 0$ while $q > 0$, which of the following is NOT true about the location of the point?

(A) It is in the second quadrant.

(B) It is to the left of the Y-axis.

(C) It is below the X-axis.

(D) Its distance from the Y-axis is $|p|$.

Question 5. Which of the following points is NOT equidistant from the X-axis and the Y-axis?

(A) $(2, 2)$

(B) $(-5, 5)$

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

(D) $(4, -5)$

Question 6. A point is on the positive Y-axis at a distance of $5$ units from the origin. Which of the following is NOT true about this point?

(A) Its coordinates are $(0, 5)$.

(B) Its abscissa is $0$.

(C) It is in the first quadrant.

(D) Its distance from the X-axis is $5$ units.

Question 7. Which statement is INCORRECT about identifying the coordinates of a plotted point?

(A) Draw a perpendicular line from the point to the X-axis to find the x-coordinate.

(B) Draw a perpendicular line from the point to the Y-axis to find the y-coordinate.

(C) The distance from the point to the X-axis is the x-coordinate.

(D) The ordered pair $(x, y)$ gives the coordinates.

Question 8. If a point is located in the first quadrant, which of the following must NOT be true about its coordinates $(x, y)$?

(A) $x > 0$

(B) $y > 0$

(C) $x$ is positive and $y$ is positive.

(D) $x < 0$

Question 9. A point has coordinates $(a, b)$. If $ab = 0$, which of the following must NOT be true?

(A) The point lies on the X-axis.

(B) The point lies on the Y-axis.

(C) The point is the origin.

(D) The point lies in a quadrant.

Question 10. To plot the point $(0, -5)$, which of the following steps is NOT correct?

(A) Start at the origin.

(B) Move $5$ units down parallel to the Y-axis.

(C) Move $5$ units right along the X-axis.

(D) The point lies on the negative Y-axis.

Question 1. Which of the following is NOT the correct formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$?

(A) $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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

(C) $\sqrt{(x_2+x_1)^2 + (y_2+y_1)^2}$

(D) The length of the hypotenuse of a right triangle with vertices at $(x_1, y_1), (x_2, y_1), (x_2, y_2)$.

Question 2. Which of the following is NOT a correct application of the distance formula?

(A) Finding the length of a line segment.

(B) Checking if three points are collinear.

(C) Finding the slope of a line.

(D) Verifying if a triangle is a right triangle.

Question 3. The distance between the points $(-3, 5)$ and $(1, -1)$ is NOT equal to:

(A) $\sqrt{(1-(-3))^2 + (-1-5)^2}$

(B) $\sqrt{4^2 + (-6)^2}$

(C) $\sqrt{16 + 36} = \sqrt{52}$

(D) $\sqrt{16+9} = 5$

Question 4. If points A, B, and C are collinear, which of the following is NOT necessarily true?

(A) The area of triangle ABC is $0$.

(B) $AB + BC = AC$ (assuming B is between A and C).

(C) $AB^2 + BC^2 = AC^2$ (Pythagoras theorem).

(D) The slope of AB is equal to the slope of BC (if non-vertical).

Question 5. Which statement about the distance from a point to the origin is FALSE?

(A) The distance from $(x, y)$ to $(0, 0)$ is $\sqrt{x^2 + y^2}$.

(B) The distance is always positive for any point other than the origin.

(C) The distance from $(a, b)$ is the same as the distance from $(-a, -b)$.

(D) The distance from $(a, b)$ is $|a| + |b|$.

Question 6. If the distance between $(4, k)$ and $(1, 0)$ is $5$, which of the following is NOT a possible value for $k$?

(A) $\sqrt{(4-1)^2 + (k-0)^2} = 5 \implies \sqrt{3^2 + k^2} = 5 \implies 9 + k^2 = 25 \implies k^2 = 16 \implies k = \pm 4$.

(B) $4$

(C) $-4$

(D) $3$

Question 7. Which of the following quadrilaterals' properties can NOT be solely verified using the distance formula?

(A) All sides are equal (Rhombus).

(B) Opposite sides are equal and diagonals are equal (Rectangle).

(C) Opposite angles are equal (Parallelogram).

(D) All sides are equal and diagonals are equal (Square).

Question 8. The vertices of a triangle are $(0, 0)$, $(5, 0)$, and $(0, 12)$. Which statement about this triangle is INCORRECT based on the distance formula?

(A) The sides have lengths $5$, $12$, and $\sqrt{5^2+12^2} = 13$.

(B) It is a right-angled triangle at the origin ($5^2 + 12^2 = 13^2$).

(C) It is an equilateral triangle.

(D) It is a scalene triangle (since all sides are different lengths: 5, 12, 13).

Question 9. If the distance between $(x, 2)$ and $(3, 4)$ is $\sqrt{8}$, which of the following is NOT a correct step to find $x$?

(A) $(3-x)^2 = 8 - 4$

(B) $(3-x)^2 = 4$

(C) $3-x = \pm 2$

(D) $x=0$

Question 10. Which statement about the distance formula is FALSE?

(A) It gives the length of the straight line segment between two points.

(B) It is based on the coordinates of the points.

(C) It can be used to find the midpoint of a line segment directly.

(D) It is always a non-negative value.

Question 1. Which of the following is NOT a direct application of the section formula for internal division?

(A) Finding the coordinates of a point that divides a segment in a given ratio.

(B) Finding the midpoint of a segment.

(C) Finding the length of a segment.

(D) Finding the centroid of a triangle (as a point dividing a median in $2:1$).

Question 2. The midpoint formula is a special case of the section formula. Which statement is FALSE?

(A) The midpoint formula is obtained by setting the ratio $m:n$ to $1:1$ in the internal section formula.

(B) The midpoint formula is obtained by setting the ratio $m:n$ to $1:1$ in the external section formula.

(C) The midpoint divides the segment into two equal parts.

(D) The midpoint is equidistant from the two endpoints.

Question 3. If point P divides the segment AB internally in the ratio $3:2$, which statement is NOT true?

(A) AP/PB = 3/2.

(B) P is closer to A than to B.

(C) P lies between A and B.

(D) AP = $\frac{3}{5}AB$.

Question 4. The coordinates of the point that divides the line segment joining $(1, 7)$ and $(4, -3)$ internally in the ratio $2:1$ is NOT:

(A) $(\frac{2(4) + 1(1)}{2+1}, \frac{2(-3) + 1(7)}{2+1})$

(B) $(\frac{8+1}{3}, \frac{-6+7}{3})$

(C) $(3, 1/3)$

(D) $(3, 1)$

Question 5. Which statement about external division of a line segment AB by a point P is FALSE?

(A) P lies on the line containing A and B.

(B) If P divides AB externally in ratio $m:n$ (where $m,n$ are positive), the ratio of distances $AP/PB$ is $m/n$.

(C) The point of external division lies within the segment AB.

(D) The external section formula is $(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n})$.

Question 6. In what ratio does the Y-axis NOT divide the line segment joining points $(5, -6)$ and $(-1, -4)$?

(A) $5:1$ internally.

(B) $1:5$ internally.

(C) This is an internal division.

(D) The ratio is $5:1$.

Question 7. If the midpoint of the segment joining $(a, 5)$ and $(-2, b)$ is $(1, 4)$, which equation is NOT correct?

(A) $a-2=1$

(B) $5+b=8$

(C) $a=4$

(D) $b=3$

Question 8. Which statement is FALSE regarding the centroid of a triangle?

(A) It is the point of intersection of the medians.

(B) It divides each median in the ratio $2:1$ (vertex to midpoint).

(C) Its coordinates are the average of the coordinates of the vertices.

(D) It is equidistant from the vertices of the triangle.

Question 9. The points P, Q, R trisect the line segment AB. Which statement is INCORRECT?

(A) P divides AB in the ratio $1:2$ internally.

(B) Q divides AB in the ratio $1:1$ internally (Q is the midpoint).

(C) R divides AB in the ratio $2:1$ internally.

(D) P divides AQ in the ratio $1:1$ internally.

Question 10. Which application is NOT primarily solved using the section formula?

(A) Finding the coordinates of a point on a line segment given a ratio.

(B) Determining if three points are collinear.

(C) Finding the coordinates of the centroid of a triangle.

(D) Finding the midpoint of a line segment.

Question 1. Which statement about the area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is FALSE?

(A) The area is given by $\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$.

(B) The area is always a non-negative value.

(C) If the vertices are collinear, the area is $1$.

(D) The formula is derived using determinants or coordinate geometry techniques.

Question 2. Which condition does NOT imply that three points are collinear?

(A) The area of the triangle formed by the points is zero.

(B) The slope of the line segment joining the first two points is equal to the slope of the segment joining the last two points (if non-vertical).

(C) The sum of the squares of the distances between any two pairs of points equals the square of the distance between the remaining pair ($AB^2 + BC^2 = AC^2$).

(D) The sum of the distances between any two pairs of points equals the distance between the remaining pair ($AB + BC = AC$).

Question 3. The area of the triangle with vertices $(0, 0), (4, 0), (0, 6)$ is NOT equal to:

(A) $\frac{1}{2} \times 4 \times 6$ (using base and height).

(B) $\frac{1}{2} |0(0-6) + 4(6-0) + 0(0-0)|$ (using coordinate formula).

(C) $12$ sq units.

(D) $24$ sq units.

Question 4. If the area of the triangle formed by $(2, 3), (4, k), (6, -3)$ is zero, which statement is FALSE?

(A) $2(k+3) + 4(-6) + 6(3-k) = 0$

(B) $2k+6 - 24 + 18 - 6k = 0$

(C) $-4k = 0$

(D) $k = 5$

Question 5. Which set of points is NOT collinear?

(A) $(1, 1), (0, 0), (-1, -1)$.

(B) $(1, 0), (2, 0), (3, 0)$.

(C) $(0, 1), (0, 2), (0, 3)$.

(D) $(1, 2), (2, 5), (3, 6)$

Question 6. The vertices of a triangle are $(t, t), (t+1, t), (t+2, t+1)$. Which statement about the area of this triangle is FALSE?

(A) Area $= \frac{1}{2} |(t-t-1)(t+1-t) - (t+1-t-2)(t-t)| = \frac{1}{2} |(-1)(1) - (-1)(0)| = \frac{1}{2} |-1| = 1/2$.

(B) The area is $1/2$ sq unit.

(C) The area is independent of $t$.

(D) The vertices are collinear for some value of $t$.

Question 7. Which of the following values for the area of a triangle indicates that its vertices are collinear?

(A) 5 sq units

(B) -10 sq units (assuming absolute value is intended for area)

(C) 0 sq units

(D) $1/\sqrt{2}$ sq units

Question 8. If the area of a triangle is known, which of the following can NOT be uniquely determined without more information?

(A) The lengths of its sides.

(B) The length of an altitude if the corresponding base is known.

(C) Whether the triangle is degenerate.

(D) Whether the triangle has a positive area.

Question 9. Which method is NOT directly used to calculate the area of a triangle given its vertices in coordinate geometry?

(A) Using the distance formula to find side lengths and then Heron's formula.

(B) Using the formula $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$.

(C) Using the concept of slopes to find a right angle and then $1/2 \times base \times height$.

(D) Using the section formula to find the centroid.

Question 10. Which statement regarding collinearity is FALSE?

(A) Three points are collinear if they lie on the same straight line.

(B) If three points are collinear, the sum of the distances between any two pairs is equal to the distance between the remaining pair.

(C) If three points are collinear, their slopes calculated pairwise are different.

(D) Collinearity can be checked using area formula or slope formula.

Question 1. Which statement about the centroid of a triangle is FALSE?

(A) It is the point of intersection of the medians.

(B) It divides each median in the ratio $1:2$ (vertex to midpoint).

(C) Its coordinates are the average of the vertex coordinates.

(D) It is the center of gravity of the triangle.

Question 2. The incenter is the center of the inscribed circle. Which property does the incenter NOT have?

(A) It is the intersection of the angle bisectors.

(B) It is equidistant from the sides of the triangle.

(C) It lies inside the triangle for any triangle.

(D) It is equidistant from the vertices of the triangle.

Question 3. The circumcenter is the center of the circumscribed circle. Which statement about the circumcenter is INCORRECT?

(A) It is the intersection of the perpendicular bisectors of the sides.

(B) It is equidistant from the vertices of the triangle.

(C) It lies inside the triangle for an obtuse-angled triangle.

(D) For a right-angled triangle, it lies at the midpoint of the hypotenuse.

Question 4. The orthocenter is the intersection of the altitudes. Which statement is FALSE about the orthocenter?

(A) For an acute triangle, it lies inside the triangle.

(B) For a right triangle, it lies at the vertex containing the right angle.

(C) For an obtuse triangle, it lies outside the triangle.

(D) It is equidistant from the sides of the triangle.

Question 5. Which statement about triangle centers is NOT true for an equilateral triangle?

(A) The centroid and incenter coincide.

(B) The circumcenter and orthocenter coincide.

(C) All four major centers coincide at a single point.

(D) The circumcenter lies outside the triangle.

Question 6. Which statement is FALSE about the Euler line?

(A) It passes through the centroid.

(B) It passes through the incenter (unless the triangle is equilateral).

(C) It passes through the circumcenter.

(D) It passes through the orthocenter.

Question 7. The coordinates of the vertices of a triangle are $A(0, 0), B(4, 0), C(2, 6)$. Which statement about its centers is INCORRECT?

(A) Centroid G $= (\frac{0+4+2}{3}, \frac{0+0+6}{3}) = (2, 2)$.

(B) Midpoint of AB is $(2, 0)$. Perpendicular bisector of AB is $x=2$. Midpoint of BC is $(3, 3)$. Slope of BC is $(6-0)/(2-4) = 6/-2 = -3$. Perpendicular bisector of BC has slope $1/3$, equation $y-3 = \frac{1}{3}(x-3) \implies 3y-9 = x-3 \implies x-3y+6=0$. Circumcenter is intersection of $x=2$ and $x-3y+6=0 \implies 2-3y+6=0 \implies 3y=8 \implies y=8/3$. Circumcenter is $(2, 8/3)$.

(C) Slope of AC is $(6-0)/(2-0) = 3$. Altitude from B to AC has slope $-1/3$, equation $y-0 = -\frac{1}{3}(x-4) \implies 3y = -x+4 \implies x+3y-4=0$. Altitude from C to AB (x-axis) is vertical line $x=2$. Orthocenter is intersection of $x=2$ and $x+3y-4=0 \implies 2+3y-4=0 \implies 3y=2 \implies y=2/3$. Orthocenter is $(2, 2/3)$.

(D) The incenter is the average of the coordinates of the vertices.

Question 8. Which triangle center's calculation formula does NOT directly use the coordinates of the vertices and possibly side lengths?

(A) Centroid

(B) Incenter

(C) Circumcenter

(D) Orthocenter

Question 9. Which of the following points is NOT necessarily a triangle center?

(A) The intersection of angle bisectors.

(B) The point equidistant from the vertices.

(C) The point dividing a median in the ratio $1:1$ (midpoint of median).

(D) The intersection of altitudes.

Question 10. Which statement about the location of triangle centers is FALSE?

(A) The incenter is always inside the triangle.

(B) The centroid is always inside the triangle.

(C) The orthocenter is always inside the triangle.

(D) The circumcenter can be inside, outside, or on a vertex.

Question 1. Which of the following is NOT a correct definition of a locus?

(A) The path traced by a point under given conditions.

(B) The set of all points satisfying a specific geometric condition.

(C) A single fixed point in a plane.

(D) A geometric figure formed by a moving point.

Question 2. The equation of the locus of a point P$(x, y)$ such that its distance from the origin is always $k$ units (where $k$ is a positive constant) is $x^2 + y^2 = k^2$. Which statement is FALSE?

(A) The locus is a circle.

(B) The center of the circle is the origin.

(C) The radius of the circle is $k^2$.

(D) Every point $(x, y)$ on the circle satisfies the given condition.

Question 3. The locus of a point that is equidistant from two fixed points A and B is the perpendicular bisector of AB. Which statement is FALSE?

(A) The locus is a straight line.

(B) Every point on the locus is the same distance from A as from B.

(C) The locus passes through the midpoint of AB.

(D) The locus is parallel to the segment AB.

Question 4. Which of the following is NOT a type of locus commonly studied in coordinate geometry?

(A) Straight line

(B) Circle

(C) Quadrant

(D) Parabola

Question 5. To find the equation of a locus, which step is NOT typically involved?

(A) Let the coordinates of the moving point be $(x, y)$.

(B) Express the given geometric condition as an equation in terms of $x$ and $y$.

(C) Solve the resulting equation for a specific value of $x$ or $y$.

(D) Simplify the equation to get the relationship between $x$ and $y$ that holds for all points on the locus.

Question 6. The equation of the locus of a point P$(x, y)$ such that its distance from the Y-axis is always equal to its distance from the point $(2, 0)$ is:

(A) Distance from P to Y-axis ($x=0$) is $|x|$. Distance from P to $(2, 0)$ is $\sqrt{(x-2)^2 + y^2}$.

(B) $|x| = \sqrt{(x-2)^2 + y^2}$. Squaring: $x^2 = (x-2)^2 + y^2 = x^2 - 4x + 4 + y^2$. $0 = -4x + 4 + y^2$. $y^2 = 4x - 4 = 4(x-1)$.

(C) $y^2 = 4(x-1)$, which is a parabola with vertex $(1, 0)$.

(D) $x^2 = 4(y-1)$, which is a parabola with vertex $(0, 1)$.

Question 7. Which of the following is NOT an example of a locus?

(A) The set of points forming a straight line segment.

(B) The set of all points inside a circle.

(C) The set of points forming an ellipse.

(D) The set of points forming a hyperbola.

Question 8. The equation of the locus of a point P$(x, y)$ such that the sum of the squares of its distances from A$(0, 0)$ and B$(2, 0)$ is $8$ is:

(A) $PA^2 + PB^2 = 8$. $(x-0)^2 + (y-0)^2 + (x-2)^2 + (y-0)^2 = 8$. $x^2 + y^2 + x^2 - 4x + 4 + y^2 = 8$. $2x^2 + 2y^2 - 4x + 4 = 8$. $2x^2 + 2y^2 - 4x - 4 = 0$. $x^2 + y^2 - 2x - 2 = 0$.

(B) $x^2 + y^2 - 2x - 2 = 0$. This is a circle.

(C) $x^2 + y^2 - 2x - 2 = 0$.

(D) $x^2 + y^2 + 2x - 2 = 0$.

Question 9. Which statement about the equation of a locus is FALSE?

(A) It is an algebraic equation involving the coordinates $(x, y)$.

(B) Every point on the locus satisfies the equation.

(C) Points not on the locus also satisfy the equation.

(D) The equation represents the geometric condition algebraically.

Question 10. The locus of a point that is always at a fixed distance from a fixed line is:

(A) A single straight line parallel to the fixed line (if the distance is positive).

(B) Two parallel straight lines, one on each side of the fixed line, at the given distance.

(C) The fixed line itself (if the distance is zero).

(D) A circle.

Question 1. If the origin is shifted to the point $(h, k)$, and $(x, y)$ are the original coordinates and $(X, Y)$ are the new coordinates, which transformation formula is INCORRECT?

(A) $x = X + h$

(B) $y = Y + k$

(C) $X = x - h$

(D) $Y = y + k$

Question 2. The original coordinates of a point are $(3, 4)$. If the origin is shifted to $(1, 1)$, which statement about the new coordinates is FALSE?

(A) The new x-coordinate is $3-1=2$.

(B) The new y-coordinate is $4-1=3$.

(C) The new coordinates are $(2, 3)$.

(D) The new coordinates are $(3+1, 4+1) = (4, 5)$.

Question 3. Shifting the origin (translation of axes) does NOT change which of the following?

(A) The equation of a straight line.

(B) The coordinates of a point.

(C) The distance between two points.

(D) The intercepts of a line on the axes.

Question 4. If the equation of a curve is $x^2 + y^2 = r^2$, and the origin is shifted to $(h, k)$, the new equation is $(X-h)^2 + (Y-k)^2 = r^2$. Which statement is FALSE?

(A) The original curve is a circle centered at the origin.

(B) The new equation represents a circle centered at the new origin $(h, k)$.

(C) The radius of the circle changes after shifting the origin.

(D) The shape of the curve remains the same (a circle).

Question 5. To eliminate the linear terms in $x$ and $y$ from the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ by shifting the origin, the new origin should be shifted to $(-g, -f)$. Which statement is FALSE?

(A) $(-g, -f)$ is the center of the circle represented by the equation.

(B) After the shift, the new equation will be of the form $X^2 + Y^2 = R^2$ for some constant $R^2$.

(C) The shift affects the quadratic terms ($x^2, y^2$).

(D) The shift simplifies the equation by making the center the new origin.

Question 6. The original coordinates of a point are $(-2, 5)$. If the new coordinates after shifting the origin are $(0, 0)$, which statement is FALSE?

(A) The original point $(-2, 5)$ is the new origin.

(B) The shift in origin was from $(0, 0)$ to $(-2, 5)$.

(C) Using $X=x-h, Y=y-k$: $0 = -2-h \implies h = -2$. $0 = 5-k \implies k = 5$. The shift was to $(-2, 5)$.

(D) The shift in origin was from $(0, 0)$ to $(2, -5)$.

Question 7. Which of the following is NOT a property of shifting the origin?

(A) It is a type of translation of the coordinate system.

(B) The direction of the axes remains unchanged.

(C) The distance between points is preserved.

(D) The orientation of geometric figures changes.

Question 8. If the origin is shifted to $(2, 0)$, the equation $y^2 = 4x$ becomes $(Y+0)^2 = 4(X+2) \implies Y^2 = 4X + 8$. Which statement is FALSE?

(A) The original equation is a parabola with vertex at $(0, 0)$.

(B) The new equation is $Y^2 = 4(X+2)$.

(C) The new equation is $Y^2 = 4X+8$.

(D) The vertex of the parabola in the new system is at $(-2, 0)$.

Question 9. Which statement is INCORRECT regarding the effect of shifting the origin on the equation of a line $Ax + By + C = 0$?

(A) The equation changes to $A(X+h) + B(Y+k) + C = 0$ if the shift is to $(h, k)$.

(B) The slope of the line remains unchanged.

(C) The intercepts on the axes remain unchanged.

(D) The distance of a point from the line remains unchanged.

Question 10. If the original coordinates of a point are $(x, y)$ and the new coordinates are $(X, Y)$ after a shift of origin, which relationship is NOT always true?

(A) The distance of the point from the origin in the original system is $\sqrt{x^2+y^2}$.

(B) The distance of the point from the new origin is $\sqrt{X^2+Y^2}$.

(C) $\sqrt{x^2+y^2} = \sqrt{X^2+Y^2}$.

(D) $(x-h)^2 + (y-k)^2 = X^2+Y^2$, where $(h,k)$ is the new origin in the old system.

Question 1. Which statement about the slope of a straight line is FALSE?

(A) Slope is defined as the tangent of the angle the line makes with the positive X-axis.

(B) A horizontal line has an undefined slope.

(C) A vertical line has an undefined slope.

(D) Lines with positive slope rise from left to right.

Question 2. The slope of the line $3x - 4y = 12$ is NOT equal to:

(A) $3/4$ (from $4y = 3x - 12 \implies y = 3/4 x - 3$).

(B) $-(-3)/4$ (from $-A/B$).

(C) The tangent of the angle the line makes with the X-axis.

(D) $4/3$

Question 3. Which statement about parallel lines is FALSE?

(A) They have the same slope (if non-vertical).

(B) They make the same angle with the X-axis.

(C) They never intersect unless they are coincident.

(D) If their slopes are $m_1$ and $m_2$, then $m_1 m_2 = -1$.

Question 4. Two lines are perpendicular. Which condition is NOT necessarily true?

(A) The angle between them is $90^\circ$.

(B) The product of their slopes is $-1$ (if both are non-vertical).

(C) If one line is horizontal, the other is vertical.

(D) Their slopes are equal.

Question 5. The angle between two lines with slopes $m_1$ and $m_2$ is $\theta$. Which formula for $\tan \theta$ is INCORRECT?

(A) $|\frac{m_1-m_2}{1+m_1m_2}|$

(B) $|\frac{m_2-m_1}{1+m_1m_2}|$

(C) $\frac{m_1-m_2}{1+m_1m_2}$ (without absolute value - gives directed angle).

(D) $\frac{m_1+m_2}{1+m_1m_2}$

Question 6. The slope of the line joining points $(2, 5)$ 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 the angle made by a line with the positive X-axis is obtuse, which statement about its slope is FALSE?

(A) The slope is negative.

(B) The tangent of the angle is negative.

(C) The line falls from left to right.

(D) The slope is positive.

Question 8. Which condition does NOT guarantee that two lines $L_1$ and $L_2$ are perpendicular?

(A) $L_1$ is horizontal, and $L_2$ is vertical.

(B) $L_1$ has slope $m_1$, $L_2$ has slope $m_2$, and $m_1 = -1/m_2$ (for $m_2 \neq 0$).

(C) The angle between $L_1$ and $L_2$ is $90^\circ$.

(D) $L_1$ has slope $m_1$, $L_2$ has slope $m_2$, and $m_1 + m_2 = 0$.

Question 9. The slope of the line $y = 5x - 2$ is NOT:

(A) 5

(B) The coefficient of $x$

(C) Equal to the slope of $y = 5x + 3$

(D) $-2$ (This is the y-intercept)

Question 10. Which statement about the angle between lines is FALSE?

(A) The angle between a horizontal line and a vertical line is $90^\circ$.

(B) The angle between two parallel lines is $0^\circ$ or $180^\circ$.

(C) The formula $\tan\theta = |\frac{m_1-m_2}{1+m_1m_2}|$ gives the obtuse angle between the lines.

(D) If $1+m_1m_2 = 0$, the lines are perpendicular (assuming finite slopes).

Question 1. Which of the following is NOT a standard form for the equation of a straight line in two dimensions?

(A) Slope-intercept form

(B) Point-slope form

(C) Area form

(D) Intercept form

Question 2. The equation of a line parallel to the Y-axis passing through the point $(3, 5)$ is NOT:

(A) $x = 3$

(B) A line with undefined slope.

(C) A vertical line.

(D) $y = 5$

Question 3. The equation of the line with slope $-2$ and y-intercept $3$ is NOT:

(A) $y = -2x + 3$

(B) $\frac{x}{3/2} + \frac{y}{3} = 1$

(C) Passes through $(0, 3)$.

(D) Passes through $(3, 0)$ with slope $-2$.

Question 4. The equation of the line passing through $(2, 5)$ and $(4, 9)$ is NOT:

(A) $y - 5 = 2(x - 2)$

(B) $y = 2x + 1$

(C) $\frac{x}{1/2} + \frac{y}{1} = 1$

(D) $\frac{x}{-1/2} + \frac{y}{1} = 1$

Question 5. Which statement about the normal form $x \cos\alpha + y \sin\alpha = p$ is FALSE?

(A) $p$ represents the perpendicular distance from the origin to the line, and $p \geq 0$.

(B) $\alpha$ is the angle made by the perpendicular from the origin to the line with the positive X-axis.

(C) If the line passes through the origin, then $p > 0$.

(D) It is a unique form for a given line (assuming $p \ge 0$ and $0 \le \alpha < 2\pi$).

Question 6. The equation of the X-axis is NOT:

(A) $y = 0$

(B) A line with slope $0$.

(C) A horizontal line.

(D) $x = 0$

Question 7. Which of the following forms of equation cannot represent a line parallel to the Y-axis?

(A) $x = k$

(B) A line with undefined slope.

(C) A vertical line.

(D) $y = mx + c$

Question 8. The equation of the line passing through the origin is NOT necessarily of the form:

(A) $y = mx$

(B) $Ax + By = 0$

(C) $y = mx + c$ where $c=0$.

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

Question 9. The equation of the line passing through $(a, b)$ and parallel to the Y-axis is NOT:

(A) $x = a$

(B) A vertical line.

(C) A line with undefined slope.

(D) $y = b$

Question 10. Given the equation $2x - 3y = 6$. Which statement about converting it to different forms is FALSE?

(A) Slope-intercept form is $y = \frac{2}{3}x - 2$.

(B) Intercept form is $\frac{x}{3} + \frac{y}{-2} = 1$.

(C) Normal form requires dividing by $\sqrt{2^2+(-3)^2} = \sqrt{13}$.

(D) The perpendicular distance from the origin is $6$.

Question 1. The general equation of a straight line is $Ax + By + C = 0$. Which condition must NOT be true for this to represent a line?

(A) A, B, C are real constants.

(B) $A = 0$ and $B = 0$ simultaneously.

(C) A and B are not both zero.

(D) C can be zero.

Question 2. Which statement about the slope of the line $Ax + By + C = 0$ is FALSE?

(A) If $B \neq 0$, the slope is $-A/B$.

(B) If $A = 0$ and $B \neq 0$, the slope is $0$.

(C) If $A \neq 0$ and $B = 0$, the slope is $-A/0$, which is undefined.

(D) The slope is always a finite real number.

Question 3. The point of intersection of two lines can be found by solving the system of their equations. Which statement is FALSE?

(A) If the lines are parallel and distinct, there is no point of intersection.

(B) If the lines are coincident, there is exactly one point of intersection.

(C) If the lines intersect at a single point, the system of equations has a unique solution.

(D) If the lines are perpendicular, they intersect at a single point (unless one or both are axes and the intersection is the origin).

Question 4. Consider the lines $2x + 3y = 5$ and $4x + 6y = 10$. Which statement is FALSE?

(A) The first equation is $L_1: 2x+3y-5=0$. The second equation is $L_2: 4x+6y-10=0$.

(B) $a_1/a_2 = 2/4 = 1/2$. $b_1/b_2 = 3/6 = 1/2$. $c_1/c_2 = -5/-10 = 1/2$.

(C) Since $a_1/a_2 = b_1/b_2 = c_1/c_2$, the lines are parallel and distinct.

(D) The lines are coincident (represent the same line).

Question 5. The general equation $Ax + By + C = 0$ can be converted to other forms. Which conversion statement is FALSE?

(A) To slope-intercept form $y=mx+c$: If $B \ne 0$, $y = -\frac{A}{B}x - \frac{C}{B}$.

(B) To intercept form $\frac{x}{a}+\frac{y}{b}=1$: If $C \ne 0$, divide by $-C$: $\frac{Ax}{-C} + \frac{By}{-C} = 1 \implies \frac{x}{-C/A} + \frac{y}{-C/B} = 1$. So $a=-C/A, b=-C/B$ (if $A, B \ne 0$).

(C) To normal form $x\cos\alpha+y\sin\alpha=p$: Divide by $\sqrt{A^2+B^2}$. For example, $3x+4y-5=0$, divide by 5: $\frac{3}{5}x + \frac{4}{5}y - 1 = 0 \implies \frac{3}{5}x + \frac{4}{5}y = 1$. Here $\cos\alpha=3/5, \sin\alpha=4/5, p=1$.

(D) The general equation can always be converted to intercept form, even if it passes through the origin.

Question 6. Which statement about the general equation $Ax + By + C = 0$ is INCORRECT?

(A) If $C=0$, the line passes through the origin.

(B) If $A=0, B \ne 0$, it's a horizontal line $y=-C/B$.

(C) If $A \ne 0, B = 0$, it's a vertical line $x=-C/A$.

(D) If $A=B=0$, it represents a point.

Question 7. Consider the system of equations $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. Which statement is FALSE?

(A) If $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, there is a unique solution (intersection point).

(B) If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, there is no solution (parallel and distinct lines).

(C) If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, there is no solution (parallel and distinct lines).

(D) If $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, there are infinitely many solutions (coincident lines).

Question 8. To convert the general equation $Ax + By + C = 0$ (with $A, B, C \neq 0$) to intercept form $\frac{x}{a} + \frac{y}{b} = 1$, which step is INCORRECT?

(A) Move the constant term to the right side: $Ax + By = -C$.

(B) Divide the entire equation by $-C$: $\frac{Ax}{-C} + \frac{By}{-C} = \frac{-C}{-C}$.

(C) Rewrite the terms as $\frac{x}{(-C/A)} + \frac{y}{(-C/B)} = 1$.

(D) The x-intercept is $-C/B$ and the y-intercept is $-C/A$.

Question 9. Which statement is FALSE about the point of intersection of two lines?

(A) It is the point that lies on both lines.

(B) Its coordinates satisfy the equations of both lines.

(C) There is always exactly one point of intersection for any two lines.

(D) It can be found by solving the system of linear equations.

Question 10. Which statement is INCORRECT regarding the general equation $Ax + By + C = 0$?

(A) If A and B are both zero, the equation represents a line.

(B) If A=0, B=1, C=-5, the equation is $y-5=0$, a horizontal line.

(C) If A=1, B=0, C=3, the equation is $x+3=0$, a vertical line.

(D) If A=1, B=1, C=0, the equation is $x+y=0$, a line through the origin.

Question 1. Which statement about the distance of a point from a line is FALSE?

(A) The distance is the length of the perpendicular from the point to the line.

(B) The formula for the distance of $(x_0, y_0)$ from $Ax + By + C = 0$ is $\frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}$.

(C) The distance is always a non-negative value.

(D) If the point lies on the line, the distance is positive.

Question 2. Which statement about the distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is FALSE?

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

(B) The distance is defined only if the lines are parallel and distinct.

(C) If $C_1 = C_2$, the distance is non-zero.

(D) The distance can be found by taking any point on one line and finding its distance from the other line.

Question 3. The equation $L_1 + \lambda L_2 = 0$ represents a family of lines passing through the intersection of $L_1=0$ and $L_2=0$. Which statement is FALSE?

(A) $L_1$ and $L_2$ must be intersecting lines for this family to be defined.

(B) For different values of $\lambda$, we get different lines passing through the common point.

(C) The line $L_1=0$ is part of this family (when $\lambda=0$).

(D) The line $L_2=0$ is not part of this family (unless $\lambda \to \infty$ is considered or the family is written as $\mu L_1 + L_2 = 0$).

Question 4. The distance of the point $(2, 3)$ from the line $x = -1$ is NOT:

(A) 3 units.

(B) The horizontal distance between the point and the line.

(C) $|2-(-1)|$.

(D) 0 units.

Question 5. Which statement is FALSE about the distance between two parallel lines?

(A) It is the constant shortest distance between any point on one line and the other line.

(B) It is always positive for distinct parallel lines.

(C) It is zero if the lines are coincident.

(D) It is calculated using the formula involving the slopes of the lines.

Question 6. The family of lines parallel to the X-axis is represented by $y = k$. Which statement is FALSE?

(A) This family includes the X-axis itself (when $k=0$).

(B) All lines in this family have a slope of $0$.

(C) All lines in this family pass through the origin.

(D) For each value of $k$, we get a distinct horizontal line.

Question 7. The distance of the origin from the line $Ax + By + C = 0$ is NOT:

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

(B) The length of the perpendicular from $(0, 0)$ to the line.

(C) $C$ (unless $A^2+B^2=1$).

(D) $0$ if the line passes through the origin.

Question 8. The equation of the family of lines passing through the intersection of $x-y=0$ and $x+y=2$ is $(x-y) + \lambda(x+y-2) = 0$. Which line is NOT part of this family?

(A) $x-y=0$

(B) $x+y-2=0$

(C) A line that does not pass through the point $(1,1)$ (which is the intersection of $x-y=0$ and $x+y=2$).

(D) A line passing through the point $(1,1)$.

Question 9. Which is NOT a correct application of the distance formula for a point from a line?

(A) Finding the shortest distance from a point to a line.

(B) Finding the distance between two parallel lines.

(C) Checking if a point lies on a line (distance = $0$).

(D) Finding the length of a line segment.

Question 10. Which statement about a family of lines is FALSE?

(A) A family of parallel lines share the same slope.

(B) A family of lines passing through a point share the same point.

(C) $Ax+By+K=0$ represents a family of lines perpendicular to $Ax+By+C=0$.

(D) $L_1 + \lambda L_2 = 0$ represents a family of lines through the intersection of $L_1=0, L_2=0$.

Question 1. Which statement about the 3D Cartesian coordinate system is FALSE?

(A) There are three mutually perpendicular axes.

(B) The axes intersect at the origin.

(C) The system defines two coordinate planes.

(D) The space is divided into eight octants by the coordinate planes.

Question 2. A point in 3D space is given by coordinates $(x, y, z)$. Which statement is INCORRECT?

(A) $|x|$ is the distance from the YZ-plane.

(B) $|y|$ is the distance from the XZ-plane.

(C) $|z|$ is the distance from the XY-plane.

(D) $|x|+|y|+|z|$ is the distance from the origin.

Question 3. Which statement about points on the coordinate axes in 3D is FALSE?

(A) A point on the X-axis has $y=0$ and $z=0$.

(B) A point on the Y-axis has $x=0$ and $z=0$.

(C) A point on the Z-axis has $x=0$ and $y=0$.

(D) A point on the X-axis has only its x-coordinate as non-zero.

Question 4. Which of the following is NOT an equation of a coordinate plane?

(A) $x = 0$

(B) $y = 0$

(C) $z = 0$

(D) $x + y + z = 0$

Question 5. Which statement about octants is FALSE?

(A) Octants are the eight regions into which 3D space is divided by the coordinate planes.

(B) Each octant is characterized by the signs of the coordinates $(x, y, z)$.

(C) The first octant is where $x>0, y>0, z>0$.

(D) The origin lies in the first octant.

Question 6. The coordinates of a point in 3D space are $(a, b, c)$. Which statement is FALSE?

(A) The perpendicular distance to the YZ-plane is $|a|$.

(B) The perpendicular distance to the X-axis is $\sqrt{b^2+c^2}$.

(C) The distance from the origin is $\sqrt{a^2+b^2+c^2}$.

(D) The perpendicular distance to the XZ-plane is $b$ (without absolute value).

Question 7. Which of the following is NOT a coordinate axis?

(A) X-axis

(B) Y-axis

(C) Z-axis

(D) XY-axis

Question 8. A point $(x, y, z)$ lies on the YZ-plane. Which condition is NOT true?

(A) $x = 0$

(B) The point is of the form $(0, y, z)$.

(C) The distance from the YZ-plane is non-zero.

(D) The point is equidistant from the XZ and XY planes if $|y|=|z|$.

Question 9. Which statement about the 3D Cartesian system is INCORRECT?

(A) It provides a framework for locating points in space using ordered triples of real numbers.

(B) The axes are usually labelled as X, Y, and Z, following the right-hand rule convention.

(C) The coordinate planes are perpendicular to the coordinate axes.

(D) Any point in space lies on at least one coordinate plane.

Question 10. Which statement about a point located at $(a, b, c)$ is FALSE?

(A) Its projection onto the XY-plane is $(a, b, 0)$.

(B) Its distance from the XY-plane is $|c|$.

(C) Its projection onto the X-axis is $(0, b, c)$.

(D) Its distance from the origin is $\sqrt{a^2+b^2+c^2}$.

Question 1. Which is NOT a correct formula for the distance between $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$?

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

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

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

(D) The magnitude of the vector $(x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}$.

Question 2. The distance between points $(1, 2, 3)$ and $(1, 2, 5)$ is NOT equal to:

(A) $\sqrt{(1-1)^2 + (2-2)^2 + (5-3)^2}$

(B) $\sqrt{0^2 + 0^2 + 2^2} = \sqrt{4} = 2$.

(C) 2 units.

(D) 8 units.

Question 3. Which statement about the distance from a point $(x, y, z)$ to the coordinate planes is FALSE?

(A) Distance to XY-plane is $|z|$.

(B) Distance to YZ-plane is $|x|$.

(C) Distance to XZ-plane is $|y|$.

(D) Distance to XY-plane is $z$ (without absolute value).

Question 4. Which is NOT a valid application of the distance formula in 3D geometry?

(A) Finding the length of a line segment in space.

(B) Checking if three points in space are collinear.

(C) Finding the equation of a plane.

(D) Verifying properties of geometric shapes like triangles or quadrilaterals in space.

Question 5. If the distance between $(k, k, k)$ and $(0, 0, 0)$ is $\sqrt{27}$, which statement is FALSE?

(A) $\sqrt{(k-0)^2 + (k-0)^2 + (k-0)^2} = \sqrt{27}$.

(B) $\sqrt{k^2 + k^2 + k^2} = \sqrt{3k^2} = \sqrt{27}$.

(C) $3k^2 = 27 \implies k^2 = 9 \implies k = \pm 3$.

(D) $k=0$ is a possible value for $k$.

Question 6. Which statement about the distance from a point $(x, y, z)$ to the coordinate axes is INCORRECT?

(A) Distance to X-axis is $\sqrt{y^2+z^2}$.

(B) Distance to Y-axis is $\sqrt{x^2+z^2}$.

(C) Distance to Z-axis is $\sqrt{x^2+y^2}$.

(D) Distance to X-axis is $|x|$.

Question 7. The vertices of a triangle are $A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)$. Which statement about this triangle is FALSE?

(A) $AB = \sqrt{(-1)^2+1^2+0^2} = \sqrt{2}$.

(B) $BC = \sqrt{0^2+(-1)^2+1^2} = \sqrt{2}$.

(C) $AC = \sqrt{(-1)^2+0^2+1^2} = \sqrt{2}$.

(D) It is a right-angled triangle.

Question 8. The distance formula in 3D is an extension of the Pythagorean theorem. Which dimension is NOT included in the generalization from 2D to 3D?

(A) x-difference

(B) y-difference

(C) z-difference

(D) Time difference

Question 9. Which statement about the distance between two points in 3D is FALSE?

(A) The distance is always non-negative.

(B) The distance is $0$ if and only if the two points are identical.

(C) The order of the points in the formula matters for the final distance value.

(D) It represents the length of the straight line segment connecting the points.

Question 10. Which of the following is NOT an equation derived from the distance formula in 3D?

(A) Equation of a sphere (locus of points equidistant from a center).

(B) Equation of a plane (locus of points equidistant from two fixed points).

(C) Condition for collinearity of three points ($AB+BC=AC$).

(D) Equation of a cylinder.

Question 1. Which statement about the midpoint formula in 3D is FALSE?

(A) It finds the point that divides a segment in the ratio $1:1$ internally.

(B) The coordinates of the midpoint are the average of the corresponding coordinates of the endpoints.

(C) It is a special case of the external section formula.

(D) It is equidistant from the two endpoints.

Question 2. If point P divides the line segment joining $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ internally in the ratio $m:n$, which formula for the x-coordinate of P is INCORRECT?

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

(B) $\frac{nx_1 + mx_2}{n+m}$

(C) $\frac{mx_1 + nx_2}{m+n}$

(D) $\frac{mx_2 + nx_1}{n+m}$

Question 3. Which statement about external division of a line segment AB by a point P in ratio $m:n$ is FALSE?

(A) The point P lies outside the segment AB.

(B) If $m=n$, the point of external division is undefined (at infinity).

(C) The formula for the x-coordinate of P is $\frac{mx_2 - nx_1}{m-n}$ (assuming $m \neq n$).

(D) The ratio $m:n$ (of lengths) must be equal to $1$ for external division.

Question 4. The centroid of a triangle with vertices in 3D is the intersection of its medians. Which statement is FALSE about the centroid G?

(A) Its coordinates are the average of the coordinates of the vertices.

(B) It divides each median in the ratio $2:1$ (vertex to midpoint).

(C) It is equidistant from the sides of the triangle.

(D) It is always inside the triangle.

Question 5. Which statement is FALSE regarding the section formula in 3D?

(A) It is a direct extension of the 2D section formula.

(B) It applies to points dividing a line segment internally.

(C) It applies to points dividing a line segment externally.

(D) It can be used to find the angle between two lines.

Question 6. If the centroid of a triangle is at the origin $(0, 0, 0)$ and two vertices are $(1, 2, 3)$ and $(-1, -2, -3)$, which statement is FALSE?

(A) Let the third vertex be $(x, y, z)$. $(\frac{1-1+x}{3}, \frac{2-2+y}{3}, \frac{3-3+z}{3}) = (0, 0, 0)$.

(B) $\frac{x}{3}=0 \implies x=0$. $\frac{y}{3}=0 \implies y=0$. $\frac{z}{3}=0 \implies z=0$.

(C) The third vertex is $(0, 0, 0)$.

(D) The three vertices are collinear.

Question 7. The coordinates of the points which trisect the line segment joining $A(3, 6, 9)$ and $B(12, 15, 18)$ are found using section formula. Which point is NOT one of the trisection points?

(A) The points divide AB in ratios $1:2$ and $2:1$ internally.

(B) Point 1 (1:2) $= (\frac{1(12)+2(3)}{3}, \frac{1(15)+2(6)}{3}, \frac{1(18)+2(9)}{3}) = (\frac{18}{3}, \frac{27}{3}, \frac{36}{3}) = (6, 9, 12)$.

(C) Point 2 (2:1) $= (\frac{2(12)+1(3)}{3}, \frac{2(15)+1(6)}{3}, \frac{2(18)+1(9)}{3}) = (\frac{27}{3}, \frac{36}{3}, \frac{45}{3}) = (9, 12, 15)$.

(D) $(5, 8, 11)$

Question 8. Which statement about the centroid of a tetrahedron with vertices $(x_1, y_1, z_1), \dots, (x_4, y_4, z_4)$ is FALSE?

(A) Its coordinates are the average of the corresponding coordinates of the four vertices.

(B) It is the point of intersection of the lines joining each vertex to the centroid of the opposite face.

(C) It divides the lines mentioned in (B) in the ratio $3:1$ (vertex to face centroid).

(D) It is equidistant from the four vertices.

Question 9. The point $(0, 0, 0)$ divides the segment joining $(a, b, c)$ and $(-a, -b, -c)$. Which statement is FALSE?

(A) The origin is the midpoint of the segment.

(B) The origin divides the segment in the ratio $1:1$ internally.

(C) The origin divides the segment in the ratio $-1:-1$ internally.

(D) The origin divides the segment in the ratio $1:-1$ externally.

Question 10. Which application is NOT typically solved using the section formula in 3D?

(A) Finding the coordinates of a point that divides a segment in a given ratio.

(B) Finding the midpoint of a segment.

(C) Finding the centroid of a triangle or tetrahedron.

(D) Finding the angle between two lines in 3D.

Question 1. Which of the following is NOT a conic section obtained by intersecting a plane with a double cone?

(A) Circle

(B) Square

(C) Parabola

(D) Hyperbola

Question 2. Which statement about the eccentricity ($e$) of conic sections is FALSE?

(A) $e=0$ for a circle.

(B) $e<1$ for an ellipse.

(C) $e=1$ for a parabola.

(D) $e>1$ for an ellipse.

Question 3. Which of the following is NOT a degenerate conic section?

(A) A single point.

(B) A pair of intersecting lines.

(C) A single line.

(D) A rectangle.

Question 4. The definition of a conic section involves a fixed point (focus) and a fixed line (directrix). Which statement is FALSE?

(A) For a point P on the conic, the ratio of its distance from the focus (PF) to its distance from the directrix (PD) is constant.

(B) The constant ratio PF/PD is called the eccentricity.

(C) The directrix is always perpendicular to the axis of the conic (except for a circle where directrix is at infinity).

(D) The focus always lies on the directrix.

Question 5. Which statement about the formation of conic sections from a double cone is FALSE?

(A) A circle is formed when the plane is parallel to the base.

(B) An ellipse is formed when the plane cuts one cone and is not parallel to a generator or the base.

(C) A parabola is formed when the plane is parallel to a generator.

(D) A hyperbola is formed when the plane is parallel to the axis of the cone and passes through the vertex.

Question 6. The axis of a conic section is a line of symmetry. Which statement is FALSE?

(A) For a parabola, there is one axis of symmetry.

(B) For an ellipse, there is one axis of symmetry (the major axis).

(C) For a hyperbola, there are two axes of symmetry (transverse and conjugate axes).

(D) For a circle, any line passing through the center is an axis of symmetry.

Question 7. Which statement about the latus rectum of a conic section is FALSE?

(A) It is a chord passing through the focus.

(B) It is perpendicular to the axis of the conic.

(C) Its length is the same for all types of conic sections.

(D) Its endpoints lie on the conic.

Question 8. Which type of conic section has two foci?

(A) Parabola

(B) Ellipse

(C) Hyperbola

(D) Circle

Question 9. Which statement is FALSE about the definition of a conic section as a locus?

(A) It is the locus of a point moving in a plane.

(B) The distance from the moving point to a fixed point is always equal to the distance from the moving point to a fixed line.

(C) The fixed point is called the focus.

(D) The fixed line is called the directrix.

Question 10. Which type of conic section has asymptotes?

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 1. Which of the following is NOT the standard equation of a circle?

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

(B) $x^2 + y^2 = r^2$ (centered at origin)

(C) $x^2 + y^2 + 2gx + 2fy + c = 0$ (general form)

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

Question 2. Which statement about the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is FALSE?

(A) The coefficients of $x^2$ and $y^2$ are equal and non-zero.

(B) There is an $xy$ term in the equation.

(C) The center of the circle is $(-g, -f)$.

(D) The radius is $\sqrt{g^2 + f^2 - c}$.

Question 3. Which statement about the equation $x^2 + y^2 - 4x + 6y + 13 = 0$ is FALSE?

(A) $g = -2, f = 3, c = 13$.

(B) $g^2 + f^2 - c = (-2)^2 + 3^2 - 13 = 4 + 9 - 13 = 0$.

(C) The radius is $\sqrt{0} = 0$.

(D) The equation represents a real circle with positive radius.

Question 4. Which statement about the relative position of a line and a circle is FALSE?

(A) If the distance from the center to the line is less than the radius, the line intersects the circle at two distinct points.

(B) If the distance from the center to the line is equal to the radius, the line is tangent to the circle.

(C) If the distance from the center to the line is greater than the radius, the line intersects the circle at two distinct points.

(D) If the distance from the center to the line is $0$, the line passes through the center.

Question 5. Two circles $C_1$ (center $O_1$, radius $r_1$) and $C_2$ (center $O_2$, radius $r_2$). Let $d = O_1O_2$. Which statement about their relative position is FALSE?

(A) If $d > r_1 + r_2$, they do not intersect and are outside each other.

(B) If $d = r_1 + r_2$, they touch externally.

(C) If $d < |r_1 - r_2|$, they intersect at two distinct points.

(D) If $d = |r_1 - r_2|$, they touch internally.

Question 6. The equation of a circle whose diameter has endpoints $(1, -1)$ and $(3, 5)$ is NOT:

(A) $(x-2)^2 + (y-2)^2 = 10$

(B) $(x-1)(x-3) + (y+1)(y-5) = 0$

(C) Center $(2, 2)$

(D) Radius $\sqrt{40}$

Question 7. Which statement about a point circle is FALSE?

(A) It is a degenerate circle with radius $0$.

(B) Its equation is $(x-h)^2 + (y-k)^2 = 0$.

(C) It consists of only one point $(h, k)$.

(D) It is a real circle with a very small positive radius.

Question 8. The equation of a circle is $x^2 + y^2 + 8x - 10y - 8 = 0$. Which statement is FALSE?

(A) Center is $(-4, 5)$.

(B) Radius is $\sqrt{(-4)^2 + 5^2 - (-8)} = \sqrt{16+25+8} = \sqrt{49} = 7$.

(C) Radius is $7$.

(D) The circle passes through the origin (since $c=-8 \ne 0$).

Question 9. Which statement about the equation of a circle $(x-h)^2 + (y-k)^2 = r^2$ is FALSE?

(A) $(h, k)$ are the coordinates of the center.

(B) $r$ is the radius, and $r > 0$ for a real circle.

(C) The coefficients of $x^2$ and $y^2$ are always equal (and usually 1).

(D) The equation always has an $xy$ term.

Question 10. Which statement about the relative position of two circles is FALSE?

(A) They can intersect at two distinct points.

(B) They can touch externally at one point.

(C) They can touch internally at two points.

(D) One can be completely inside the other without touching.

Question 1. Which statement about the definition of a parabola is FALSE?

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

(B) The fixed point is called the vertex.

(C) The fixed line is called the directrix.

(D) The ratio of the distance from a point on the parabola to the focus and the distance to the directrix is the eccentricity, which is equal to $1$ for a parabola.

Question 2. Which of the following is NOT a standard equation of a parabola with vertex at the origin?

(A) $y^2 = 4ax$

(B) $x^2 = 4ay$

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

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

Question 3. For the parabola $y^2 = 16x$, which statement is FALSE?

(A) $4a = 16 \implies a = 4$.

(B) The vertex is at $(0, 0)$.

(C) The focus is at $(0, 4)$.

(D) The directrix is $x = -4$.

Question 4. Which statement about the latus rectum of the parabola $x^2 = 4ay$ is FALSE?

(A) It is a chord passing through the focus $(0, a)$.

(B) It is perpendicular to the axis (Y-axis, $x=0$).

(C) Its length is $|4a|$.

(D) Its endpoints are at $(a, a)$ and $(-a, a)$.

Question 5. Which statement about the vertex of a parabola is FALSE?

(A) It is the point on the parabola closest to the directrix.

(B) It lies on the axis of the parabola.

(C) It is the same point as the focus.

(D) It is the midpoint of the segment joining the focus and the foot of the perpendicular from the focus to the directrix.

Question 6. The equation of a parabola with vertex at $(h, k)$ and axis parallel to the Y-axis is $(x-h)^2 = 4a(y-k)$ or $(x-h)^2 = -4a(y-k)$. Which statement is FALSE?

(A) If $a>0$, $(x-h)^2 = 4a(y-k)$ opens upwards.

(B) If $a>0$, $(x-h)^2 = -4a(y-k)$ opens downwards.

(C) The vertex is at $(h, k)$.

(D) The axis of symmetry is $y=k$.

Question 7. The parametric equations $x = at^2, y = 2at$ represent the parabola $y^2 = 4ax$. Which statement about these parametric equations is FALSE?

(A) $t$ is the parameter.

(B) For each real value of $t$, there is a unique point $(x, y)$ on the parabola.

(C) Eliminating $t$ gives the Cartesian equation $y^2 = 4ax$.

(D) These equations represent only the part of the parabola where $x \ge 0$.

Question 8. The reflective property of a parabola states that rays parallel to the axis are reflected towards the focus. Which statement is FALSE?

(A) This property is used in satellite dishes (incoming parallel rays to focus).

(B) This property is used in searchlights (light source at focus to produce parallel beam).

(C) This property is related to the definition of the parabola.

(D) This property applies to all conic sections.

Question 9. For the parabola $x^2 = -8y$, which statement is FALSE?

(A) $4a = 8 \implies a = 2$.

(B) It opens downwards.

(C) The focus is at $(0, 2)$.

(D) The directrix is $y = 2$.

Question 10. Which of the following properties is NOT characteristic of a parabola?

(A) Has one focus.

(B) Has one directrix.

(C) Has two vertices.

(D) Has an axis of symmetry.

Question 1. Which statement about the definition of an ellipse is FALSE?

(A) An ellipse is the locus of a point such that the sum of its distances from two fixed points is constant.

(B) The two fixed points are called the foci.

(C) The constant sum is equal to the length of the minor axis.

(D) The distance between the two foci is less than the constant sum.

Question 2. Which of the following is NOT a standard equation of an ellipse centered at the origin?

(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) $Ax^2 + By^2 = C$ (where $A, B, C > 0, A \ne B$).

Question 3. For the ellipse $\frac{x^2}{100} + \frac{y^2}{36} = 1$, which statement is FALSE?

(A) $a^2 = 100 \implies a = 10$. $b^2 = 36 \implies b = 6$. Since $a > b$, the major axis is along the X-axis.

(B) Length of major axis is $2a = 20$.

(C) Length of minor axis is $2b = 12$.

(D) $c^2 = a^2 + b^2 = 100 + 36 = 136$. Foci are at $(\pm \sqrt{136}, 0)$.

Question 4. The eccentricity of an ellipse is $e$. Which statement is FALSE?

(A) $0 \le e < 1$.

(B) $e = c/a$ (where a is semi-major axis length).

(C) If $e=0$, the ellipse is a circle.

(D) If $e=1$, the ellipse is a parabola.

Question 5. Which statement about the latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (with $a>b$) is FALSE?

(A) It passes through a focus.

(B) It is perpendicular to the major axis.

(C) Its length is $2b^2/a$.

(D) Its length is $2a^2/b$.

Question 6. Which statement about the vertices of an ellipse is FALSE?

(A) They are the endpoints of the major axis.

(B) The distance from the center to a vertex is the length of the semi-major axis.

(C) For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the vertices are $(\pm a, 0)$ if $a>b$, and $(0, \pm b)$ if $b>a$.

(D) The vertices are the same as the foci.

Question 7. The parametric equations $x = a\cos\theta, y = b\sin\theta$ represent an ellipse centered at the origin. Which statement is FALSE?

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

(B) $a$ is always the semi-major axis length.

(C) $b$ is always the semi-minor axis length.

(D) These equations cover the entire ellipse for $\theta \in [0, 2\pi)$.

Question 8. Which statement about the reflective property of an ellipse is FALSE?

(A) A ray from one focus reflects off the ellipse and passes through the other focus.

(B) This property is used in whispering galleries.

(C) This property means that $PF_1 + PF_2 = \text{constant}$ for any point P on the ellipse and foci $F_1, F_2$.

(D) This property is unique to ellipses and does not apply to parabolas or hyperbolas.

Question 9. For the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$, which statement is FALSE?

(A) Major axis is along the Y-axis.

(B) Foci are at $(0, \pm 4)$.

(C) Vertices are at $(\pm 3, 0)$.

(D) Length of latus rectum is $18/5$.

Question 10. Which statement about the directrices of an ellipse is FALSE?

(A) There are two directrices.

(B) They are lines perpendicular to the major axis.

(C) They are located outside the ellipse.

(D) They pass through the foci.

Question 1. Which statement about the definition of a hyperbola is FALSE?

(A) A hyperbola is the locus of a point such that the absolute difference of its distances from two fixed points is constant.

(B) The two fixed points are called the foci.

(C) The constant difference is equal to the length of the conjugate axis.

(D) The distance between the two foci is greater than the constant difference.

Question 2. Which of the following is NOT a standard equation of a hyperbola centered at the origin?

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

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

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

(D) $Ax^2 + By^2 = C$ (where A, B have opposite signs, C!=0).

Question 3. For the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, which statement is FALSE?

(A) $a^2 = 9 \implies a = 3$. $b^2 = 16 \implies b = 4$. Transverse axis is along X-axis.

(B) Vertices are at $(\pm 3, 0)$.

(C) $c^2 = a^2 + b^2 = 9 + 16 = 25 \implies c = 5$. Foci are at $(0, \pm 5)$.

(D) Asymptotes are $y = \pm \frac{b}{a}x = \pm \frac{4}{3}x$.

Question 4. The eccentricity of a hyperbola is $e$. Which statement is FALSE?

(A) $e > 1$.

(B) $e = c/a$.

(C) Asymptotes become flatter as $e$ increases. (Incorrect, steeper).

(D) $e = \sqrt{1 + (b/a)^2}$.

Question 5. Which statement about the asymptotes of a hyperbola is FALSE?

(A) They are lines that the hyperbola approaches as it extends infinitely.

(B) They intersect at the center of the hyperbola.

(C) They are parallel to the axes of the hyperbola.

(D) For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the equations are $y = \pm \frac{b}{a}x$.

Question 6. Which statement about the latus rectum of the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ is FALSE?

(A) It passes through a focus.

(B) It is perpendicular to the transverse axis.

(C) Its length is $2b^2/a$ (if $y^2/a^2 - x^2/b^2 = 1$).

(D) Its length is $4a$ (like a parabola).

Question 7. Which statement about the directrices of a hyperbola is FALSE?

(A) There are two directrices.

(B) They are lines perpendicular to the transverse axis.

(C) They are located between the two branches of the hyperbola.

(D) For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the equations are $x = \pm a/e$.

Question 8. Which type of conic section does NOT have two real foci?

(A) Ellipse

(B) Hyperbola

(C) Circle (foci coincide)

(D) Parabola (has one real focus, the other is at infinity)

Question 9. The parametric equations $x = a\sec\theta, y = b\tan\theta$ represent the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Which statement is FALSE?

(A) $\frac{x}{a} = \sec\theta, \frac{y}{b} = \tan\theta$.

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

(C) The range of $\theta$ from $0$ to $2\pi$ covers the entire hyperbola.

(D) The vertices are at $(\pm a, 0)$.

Question 10. Which statement is FALSE about the relationship $c^2 = a^2 + b^2$ for a hyperbola centered at the origin?

(A) $c$ is the distance from the center to a focus.

(B) $a$ is the length of the semi-transverse axis.

(C) $b$ is the length of the semi-conjugate axis.

(D) This relationship holds for the hyperbola $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Question 1. Which statement about parametric equations of conics is FALSE?

(A) They represent the x and y coordinates as functions of a single parameter.

(B) Eliminating the parameter yields the Cartesian equation.

(C) The parameter always represents time.

(D) Different choices of parameter can represent the same curve.

Question 2. Which of the following is NOT a correct standard set of parametric equations for a conic centered at the origin?

(A) Circle: $x = r\cos t, y = r\sin t$

(B) Ellipse: $x = a\cos \theta, y = b\sin \theta$

(C) Parabola ($y^2 = 4ax$): $x = t, y = t^2$

(D) Hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$): $x = a\sec\theta, y = b\tan\theta$

Question 3. Which statement about the parametric representation of an ellipse is FALSE?

(A) $x = h + a\cos\theta, y = k + b\sin\theta$ represents an ellipse centered at $(h, k)$.

(B) $a$ and $b$ represent the semi-major and semi-minor axis lengths (not necessarily in that order).

(C) The parameter $\theta$ is always an angle in standard geometric sense (0 to $2\pi$).

(D) Eliminating $\theta$ gives the Cartesian equation of the ellipse.

Question 4. Which of the following parametric equations does NOT represent a hyperbola centered at the origin?

(A) $x = a\cosh t, y = b\sinh t$ ($\cosh^2t - \sinh^2t = 1 \implies x^2/a^2 - y^2/b^2 = 1$)

(B) $x = a\sec\theta, y = b\tan\theta$ ($\sec^2\theta - \tan^2\theta = 1 \implies x^2/a^2 - y^2/b^2 = 1$)

(C) $x = b\tan\theta, y = a\sec\theta$ ($\sec^2\theta - \tan^2\theta = 1 \implies y^2/a^2 - x^2/b^2 = 1$)

(D) $x = a\cos t, y = b\sin t$

Question 5. Which statement about the utility of parametric equations is FALSE?

(A) They can simplify calculations of tangent slopes and areas under curves.

(B) They provide a unique representation for each conic section (i.e., only one set of parametric equations exists for a given conic).

(C) They can represent curves that are not functions of x or y.

(D) They provide a way to describe the position of a point on a curve based on the value of a single parameter.

Question 6. Which of the following steps is NOT a correct way to eliminate the parameter $t$ from the parametric equations $x = t+1, y = t^2$?

(A) From the first equation, $t = x-1$.

(B) Substitute $t = x-1$ into the second equation: $y = (x-1)^2$.

(C) The Cartesian equation is $y = x^2 - 2x + 1$.

(D) Substitute $t = \sqrt{y}$ into the first equation: $x = \sqrt{y} + 1$.

Question 7. Which statement is FALSE about the parameter in parametric equations?

(A) It is typically denoted by $t$ or $\theta$ or $u$.

(B) It is an independent variable.

(C) Its value determines the coordinates of a point on the curve.

(D) It is a constant value for all points on the curve.

Question 8. Which of the following is NOT a correct set of parametric equations for a parabola $x^2 = 4ay$?

(A) $x = 2at, y = at^2$

(B) $x = t, y = t^2/(4a)$

(C) $x = t, y = \sqrt{4at}$

(D) $x = 4at^2, y = 8at$

Question 9. Which of the following is NOT a standard parameter used in the parametric equations of conics?

(A) $t$

(B) $\theta$

(C) $\lambda$ (often used in family of lines)

(D) $\phi$

Question 10. Which statement about the parametric equation of a circle centered at $(h, k)$ is FALSE?

(A) $x = h + r\cos t, y = k + r\sin t$.

(B) $r$ is the radius.

(C) $(x-h)^2 + (y-k)^2 = r^2$ is the resulting Cartesian equation.

(D) The parameter $t$ must be restricted to $[0, \pi]$ to get the whole circle.

Question 1. Which of the following geometric problems can NOT be effectively solved using coordinate geometry?

(A) Finding the area of a polygon given its vertices.

(B) Proving geometric theorems algebraically.

(C) Finding the exact value of $\pi$.

(D) Determining the type of triangle formed by three points.

Question 2. In physics, coordinate geometry is used to describe motion. Which description is FALSE?

(A) Position of an object is given by coordinates $(x, y)$ or $(x, y, z)$.

(B) Velocity is the rate of change of position coordinates with respect to time.

(C) Trajectories of objects can be represented by equations of curves.

(D) Coordinate geometry can calculate the mass of an object directly from its position.

Question 3. Which geometric property can NOT be directly verified using the distance formula?

(A) Whether all sides of a quadrilateral are equal (rhombus).

(B) Whether two line segments are parallel.

(C) Whether a triangle is isosceles.

(D) Whether the lengths of the diagonals of a rectangle are equal.

Question 4. Coordinate geometry is applied in various fields. Which application is LEAST related to coordinate geometry concepts?

(A) Navigation systems (like GPS).

(B) Computer graphics.

(C) Stock market trend analysis.

(D) Engineering design (e.g., structural analysis).

Question 5. Which statement about using coordinate geometry to check collinearity is FALSE?

(A) Calculate the slope between two pairs of points; if they are equal, the points are collinear (if non-vertical).

(B) Calculate the distances between the three points; if the sum of the two smaller distances equals the largest distance, the points are collinear.

(C) Calculate the area of the triangle formed by the three points; if the area is non-zero, the points are collinear.

(D) All three points lie on the same straight line if they are collinear.

Question 6. Which geometric concept's equation is NOT derived or represented using coordinate geometry?

(A) A straight line.

(B) A circle.

(C) A specific angle value (like $30^\circ$).

(D) An ellipse.

Question 7. In 3D geometry, coordinate methods are used. Which is NOT a typical application?

(A) Finding the distance between two points in space.

(B) Finding the volume of a simple solid like a cube given its vertices.

(C) Determining the texture of a surface.

(D) Finding the midpoint of a line segment in space.

Question 8. Which statement about using coordinate geometry for proofs is FALSE?

(A) Choosing convenient coordinates (like placing a vertex at the origin or a side on an axis) can simplify the algebra.

(B) Algebraic manipulation of equations derived from coordinates replaces traditional geometric arguments.

(C) Coordinate geometry can prove every geometric theorem without exception.

(D) Properties like slopes, distances, and midpoints are used in coordinate proofs.

Question 9. Which application area does NOT heavily rely on coordinate geometry?

(A) Meteorology (weather forecasting).

(B) Robotics.

(C) Architecture.

(D) Film production (cinematography).

Question 10. Which of the following geometric problems can NOT be effectively solved using coordinate geometry?

(A) Finding the area of a polygon given its vertices.

(B) Proving geometric theorems algebraically.

(C) Finding the exact value of $\pi$.

(D) Determining the type of triangle formed by three points.

Question 11. In physics, coordinate geometry is used to describe motion. Which description is FALSE?

(A) Position of an object is given by coordinates $(x, y)$ or $(x, y, z)$.

(B) Velocity is the rate of change of position coordinates with respect to time.

(C) Trajectories of objects can be represented by equations of curves.

(D) Coordinate geometry can calculate the mass of an object directly from its position.

Question 12. Which geometric property can NOT be directly verified using the distance formula?

(A) Whether all sides of a quadrilateral are equal (rhombus).

(B) Whether two line segments are parallel.

(C) Whether a triangle is isosceles.

(D) Whether the lengths of the diagonals of a rectangle are equal.

Question 13. Coordinate geometry is applied in various fields. Which application is LEAST related to coordinate geometry concepts?

(A) Navigation systems (like GPS).

(B) Computer graphics.

(C) Stock market trend analysis.

(D) Engineering design (e.g., structural analysis).

Question 14. Which statement about using coordinate geometry to check collinearity is FALSE?

(A) Calculate the slope between two pairs of points; if they are equal, the points are collinear (if non-vertical).

(B) Calculate the distances between the three points; if the sum of the two smaller distances equals the largest distance, the points are collinear.

(C) Calculate the area of the triangle formed by the three points; if the area is non-zero, the points are collinear.

(D) All three points lie on the same straight line if they are collinear.

Question 15. Which geometric concept's equation is NOT derived or represented using coordinate geometry?

(A) A straight line.

(B) A circle.

(C) A specific angle value (like $30^\circ$).

(D) An ellipse.

Question 16. In 3D geometry, coordinate methods are used. Which is NOT a typical application?

(A) Finding the distance between two points in space.

(B) Finding the volume of a simple solid like a cube given its vertices.

(C) Determining the texture of a surface.

(D) Finding the midpoint of a line segment in space.

Question 17. Which statement about using coordinate geometry for proofs is FALSE?

(A) Choosing convenient coordinates (like placing a vertex at the origin or a side on an axis) can simplify the algebra.

(B) Algebraic manipulation of equations derived from coordinates replaces traditional geometric arguments.

(C) Coordinate geometry can prove every geometric theorem without exception.

(D) Properties like slopes, distances, and midpoints are used in coordinate proofs.

Question 18. Which application area does NOT heavily rely on coordinate geometry?

(A) Meteorology (weather forecasting).

(B) Robotics.

(C) Architecture.

(D) Film production (cinematography).

Question 19. Which statement about solving geometric problems using coordinate methods is FALSE?

(A) Geometric figures are represented by algebraic equations.

(B) Algebraic manipulations are used to solve the problem.

(C) The solution is interpreted back into geometric terms.

(D) Coordinate methods are always more complicated than synthetic geometry methods.