Multiple Correct Answers MCQs for Sub-Topics of Topic 6: Coordinate Geometry

Question 1. Which of the following statements about the Cartesian coordinate system are true?

(A) The X-axis is also called the axis of abscissae.

(B) The Y-axis is also called the axis of ordinates.

(C) The point where the axes intersect is called the origin, and its coordinates are $(0, 0)$.

(D) The system divides the plane into three quadrants.

Question 2. A point lies on the Y-axis. Which of the following must be true about its coordinates $(x, y)$?

(A) $x = 0$

(B) $y = 0$

(C) The point is the origin.

(D) The abscissa is zero.

Question 3. Which of the following points lie in the fourth quadrant?

(A) $(3, -5)$

(B) $(-2, -1)$

(C) $(6, -2)$

(D) $(1, 4)$

Question 4. A point $(x, y)$ lies in the second quadrant. Which of the following inequalities must be true?

(A) $x > 0$

(B) $y > 0$

(C) $x < 0$

(D) $y < 0$

Question 5. The distance of a point $(x, y)$ from the X-axis is:

(A) $|x|$

(B) $x$

(C) $|y|$

(D) $y$

Question 6. Which of the following points lie on the coordinate axes?

(A) $(5, 0)$

(B) $(0, -3)$

(C) $(2, 4)$

(D) $(0, 0)$

Question 7. If a point $(a, b)$ is in the third quadrant, which of the following is true?

(A) $a > 0$

(B) $b < 0$

(C) $a < 0$

(D) $ab > 0$

Question 8. The abscissa of a point is its distance from which axis?

(A) X-axis

(B) Y-axis

(C) Z-axis

(D) Origin

Question 9. Which of the following describes the origin in the Cartesian plane?

(A) It is the point of intersection of the X and Y axes.

(B) Its coordinates are $(0, 0)$.

(C) It lies in all four quadrants.

(D) It is the reference point for measuring distances.

Question 10. A point $(p, q)$ lies in the first quadrant. Which of the following must be true?

(A) $p > 0$

(B) $q > 0$

(C) $p$ is the distance from the Y-axis.

(D) $q$ is the distance from the X-axis.

Question 11. If a point lies on the X-axis to the left of the origin, which of the following is true about its coordinates $(x, y)$?

(A) $x > 0$

(B) $x < 0$

(C) $y = 0$

(D) The point is in the second or third quadrant.

Question 1. To plot the point $(-4, 5)$ starting from the origin, you would move:

(A) 4 units to the left along the X-axis.

(B) 4 units to the right along the X-axis.

(C) 5 units up parallel to the Y-axis.

(D) 5 units down parallel to the Y-axis.

Question 2. A point is located 3 units below the X-axis and 2 units to the right of the Y-axis. The coordinates of the point are:

(A) $(2, -3)$

(B) $(-2, 3)$

(C) In the fourth quadrant.

(D) Abscissa is 2 and ordinate is -3.

Question 3. Which of the following statements correctly describe the location of the point $(0, -6)$?

(A) It lies on the X-axis.

(B) It lies on the Y-axis.

(C) It is 6 units below the origin.

(D) It is 6 units to the left of the origin.

Question 4. A point $(x, y)$ is plotted such that its distance from the Y-axis is 5 units and it is in the second quadrant. Which of the following could be the coordinates of the point?

(A) $(5, 2)$

(B) $(-5, 2)$

(C) $(-5, 7)$

(D) $(5, -2)$

Question 5. To plot a point $(a, b)$, where $a < 0$ and $b < 0$, starting from the origin, you would move:

(A) Left along the X-axis.

(B) Right along the X-axis.

(C) Up parallel to the Y-axis.

(D) Down parallel to the Y-axis.

Question 6. Which of the following points are equidistant from the X and Y axes?

(A) $(3, 3)$

(B) $(-4, -4)$

(C) $(5, -5)$

(D) $(2, -3)$

Question 7. A point $(p, q)$ is plotted. If $pq > 0$, in which quadrants could the point lie?

(A) First Quadrant

(B) Second Quadrant

(C) Third Quadrant

(D) Fourth Quadrant

Question 8. Which of the following statements are true about plotting the point $(0, 5)$?

(A) Start at the origin.

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

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

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

Question 9. If a point lies on the negative X-axis, which of the following is true about its coordinates $(x, y)$?

(A) $x < 0$

(B) $y = 0$

(C) The point is $(-a, 0)$ for some $a > 0$.

(D) The point is in the third quadrant.

Question 10. Which of the following describes the location of a point whose abscissa is $-2$ and ordinate is $-3$?

(A) $(-2, -3)$

(B) In the third quadrant.

(C) 2 units left of Y-axis and 3 units below X-axis.

(D) 3 units left of Y-axis and 2 units below X-axis.

Question 11. Identify the correct statements about plotting points.

(A) The first coordinate tells you how far to move horizontally from the origin.

(B) The second coordinate tells you how far to move vertically from the origin.

(C) Positive $x$ means moving right, negative $x$ means moving left.

(D) Positive $y$ means moving down, negative $y$ means moving up.

Question 1. Which of the following represents the distance between points $A(x_1, y_1)$ and $B(x_2, y_2)$?

(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) The length of the hypotenuse of a right triangle with legs $|x_2-x_1|$ and $|y_2-y_1|$.

(D) $(x_2-x_1)^2 + (y_2-y_1)^2$

Question 2. The distance between the points $(-1, -3)$ and $(2, 1)$ is:

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

(B) $\sqrt{(2+1)^2 + (1+3)^2}$

(C) $\sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$ units.

(D) 25 units.

Question 3. Which of the following can be used to determine if three points A, B, and C are collinear?

(A) Check if $AB + BC = AC$ (assuming B is between A and C, or any permutation).

(B) Check if the area of triangle ABC is zero.

(C) Check if the slope of AB is equal to the slope of BC.

(D) Calculate the midpoints of AB and BC.

Question 4. Consider the points A$(1, 2)$, B$(5, 2)$, C$(3, 4)$. Which of the following statements are true?

(A) AB has length $|5-1| = 4$ units.

(B) BC has length $\sqrt{(3-5)^2 + (4-2)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}$ units.

(C) AC has length $\sqrt{(3-1)^2 + (4-2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4+4} = \sqrt{8} = 2\sqrt{2}$ units.

(D) Triangle ABC is an isosceles triangle.

Question 5. The distance of the point $(p, q)$ from the origin is:

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

(B) The radius of a circle centered at the origin passing through $(p, q)$.

(C) Equal to the distance of $(-p, -q)$ from the origin.

(D) $|p| + |q|$

Question 6. If the distance between points $(x, 0)$ and $(0, 5)$ is 13 units, which of the following is/are possible value(s) for $x$?

(A) $12$

(B) $-12$

(C) $\pm \sqrt{144}$

(D) $13^2 - 5^2$

Question 7. To prove that a quadrilateral is a rhombus using the distance formula, you need to show that:

(A) All four sides are equal in length.

(B) Opposite sides are equal in length.

(C) The diagonals are perpendicular bisectors of each other (implies parallelogram). To prove it's a rhombus, check if diagonals are perpendicular AND bisect (or just check all sides are equal and diagonals bisect). Just equal sides is enough for rhombus if you know it's a parallelogram. If starting with 4 points, checking all sides equal is sufficient.

(D) The diagonals are equal in length.

Question 8. The distance of a point $(a, b)$ from the Y-axis is:

(A) The magnitude of its abscissa.

(B) $|a|$

(C) The magnitude of its ordinate.

(D) $|b|$

Question 9. Which of the following sets of points are collinear?

(A) $(1, 1), (2, 3), (3, 5)$ (Slope $(3-1)/(2-1) = 2$, Slope $(5-3)/(3-2) = 2$)

(B) $(0, 0), (1, 2), (2, 4)$ (Slope $(2-0)/(1-0) = 2$, Slope $(4-2)/(2-1) = 2$)

(C) $(1, 0), (2, 0), (3, 0)$ (All on X-axis)

(D) $(1, 2), (3, 4), (5, 6)$ (Slope $(4-2)/(3-1) = 2/2 = 1$, Slope $(6-4)/(5-3) = 2/2 = 1$)

Question 10. If the points $(0, 0)$, $(a, 0)$, and $(0, b)$ form a right-angled triangle, which is true?

(A) The angle at the origin is $90^\circ$.

(B) The legs are along the coordinate axes.

(C) The lengths of the legs are $|a|$ and $|b|$.

(D) The hypotenuse has length $\sqrt{a^2+b^2}$.

Question 11. The distance formula in two dimensions is based on:

(A) The concept of slope.

(B) The Pythagorean theorem.

(C) Finding the difference in x and y coordinates.

(D) The section formula.

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

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

(B) The point that divides the segment AB internally in the ratio $1:1$.

(C) The point that is equidistant from A and B.

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

Question 2. Point P divides the line segment AB internally in the ratio $m:n$. Which of the following statements are true?

(A) P lies between A and B.

(B) The coordinates of P are $(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n})$ where A=$(x_1, y_1)$ and B=$(x_2, y_2)$.

(C) The ratio $m:n$ is positive.

(D) $AP/PB = m/n$.

Question 3. Point Q divides the line segment AB externally in the ratio $m:n$. Which of the following statements are true?

(A) Q lies outside the segment AB on the line containing A and B.

(B) The ratio $m:n$ is positive.

(C) The coordinates of Q are $(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n})$ where A=$(x_1, y_1)$ and B=$(x_2, y_2)$, assuming $m \neq n$.

(D) $AQ/QB = m/n$.

Question 4. The vertices of a triangle are $A(1, 2)$, $B(3, -1)$, and $C(-2, 4)$. Which of the following are correct about its centroid G?

(A) The coordinates of G are $(\frac{1+3+(-2)}{3}, \frac{2+(-1)+4}{3})$.

(B) The coordinates of G are $(\frac{2}{3}, \frac{5}{3})$.

(C) G is the intersection of the medians of the triangle.

(D) G divides each median in the ratio $2:1$.

Question 5. In what ratio does the X-axis divide the line segment joining $A(2, 3)$ and $B(-1, -6)$?

(A) The point of intersection with the X-axis has a y-coordinate of 0.

(B) Let the ratio be $k:1$. The y-coordinate of the division point is $\frac{k(-6) + 1(3)}{k+1} = 0$.

(C) $-6k + 3 = 0 \implies 6k = 3 \implies k = 1/2$.

(D) The X-axis divides the segment internally in the ratio $1:2$.

Question 6. If the point P divides the line segment AB such that $AP/PB = 3/2$, which of the following is true about the division?

(A) It is internal division.

(B) It is external division.

(C) P is closer to B than A.

(D) P is closer to A than B.

Question 7. The points $P, Q, R$ trisect the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$. Which of the following is true?

(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) The coordinates of Q are $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$.

Question 8. If the midpoint of the line segment joining $(a, 5)$ and $(-2, b)$ is $(1, 3)$, then the values of $a$ and $b$ are:

(A) $\frac{a-2}{2} = 1 \implies a-2 = 2 \implies a = 4$.

(B) $\frac{5+b}{2} = 3 \implies 5+b = 6 \implies b = 1$.

(C) $a=4$

(D) $b=1$

Question 9. The section formula can be used to prove that the medians of a triangle are concurrent (intersect at a single point). This point is the centroid.

(A) True

(B) False

(C) The centroid divides each median in the ratio $2:1$.

(D) The section formula is not related to the concurrency of medians.

Question 10. Which of the following are applications of the section formula?

(A) Finding the midpoint of a line segment.

(B) Finding the centroid of a triangle.

(C) Determining the ratio in which a point divides a line segment.

(D) Calculating the distance between two points.

Question 11. If point P divides the line segment joining A and B externally in the ratio $m:n$, with $m > n$, where does P lie?

(A) On the extension of AB beyond B.

(B) On the extension of AB beyond A.

(C) Between A and B.

(D) At a distance from A and B proportional to $m$ and $n$.

Question 1. The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be calculated using:

(A) $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$

(B) $\frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|$

(C) Base $\times$ Height, if base and corresponding height are known.

(D) Heron's formula, if side lengths are known.

Question 2. Three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear if:

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

(B) $x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) = 0$

(C) The slope of the line segment joining the first two points is equal to the slope of the line segment joining the last two points.

(D) They lie on the same straight line.

Question 3. Consider the points $A(1, 2)$, $B(3, 4)$, and $C(5, 6)$. Which of the following statements are true?

(A) Slope of AB = $\frac{4-2}{3-1} = \frac{2}{2} = 1$.

(B) Slope of BC = $\frac{6-4}{5-3} = \frac{2}{2} = 1$.

(C) Points A, B, and C are collinear.

(D) The area of triangle ABC is non-zero.

Question 4. The area of the triangle with vertices $(0, 0)$, $(5, 0)$, and $(0, 8)$ is:

(A) $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 8 = 20$ sq units.

(B) $\frac{1}{2} |0(0-8) + 5(8-0) + 0(0-0)| = \frac{1}{2} |40| = 20$ sq units.

(C) $20$ sq units.

(D) $\sqrt{5^2+8^2}$ sq units.

Question 5. If the area of the triangle formed by points $(k, 0), (1, 1), (0, 2)$ is 4 sq units, then possible values of $k$ satisfy:

(A) $\frac{1}{2} |k(1-2) + 1(2-0) + 0(0-1)| = 4$

(B) $\frac{1}{2} |-k + 2| = 4$

(C) $|-k + 2| = 8 \implies -k + 2 = 8$ or $-k + 2 = -8$.

(D) $k = -6$ or $k = 10$.

Question 6. Which of the following areas are non-zero?

(A) Area of a line segment.

(B) Area of a degenerate triangle (collinear vertices).

(C) Area of a triangle with distinct vertices not lying on a single line.

(D) Area of a point.

Question 7. The vertices of a triangle are $(0, 0)$, $(a, b)$, and $(c, d)$. Its area is given by:

(A) $\frac{1}{2} |ad - bc|$

(B) $\frac{1}{2} |a(b-d) + c(d-b)|$

(C) $\frac{1}{2} |0(b-d) + a(d-0) + c(0-b)|$

(D) $\frac{1}{2} |ad - bc|$ (calculated using formula)

Question 8. The condition for three points to be collinear using the area formula is that the area of the triangle formed by them must be zero. This implies that the points do not form a triangle.

(A) True

(B) False

(C) Collinear points lie on a single straight line.

(D) Non-collinear points form a triangle with positive area.

Question 9. The area of a triangle can be negative according to the formula if the order of vertices is taken in a certain way. However, area is a physical quantity and must be non-negative. Therefore, we take the absolute value of the result.

(A) True

(B) False

(C) The sign of the area indicates the orientation (clockwise or counterclockwise) of the vertices.

(D) The formula always gives a non-negative value.

Question 10. If the area of a triangle with vertices $(1, 2)$, $(3, 4)$, and $(5, k)$ is zero, which of the following are possible values of $k$?

(A) Area $= \frac{1}{2} |1(4-k) + 3(k-2) + 5(2-4)| = 0$

(B) $|4-k + 3k-6 - 10| = 0$

(C) $|2k - 12| = 0 \implies 2k - 12 = 0 \implies k = 6$.

(D) $k=6$ is the only possible value.

Question 11. Consider the points $(a, b+c), (b, c+a), (c, a+b)$. Which of the following statements are true?

(A) The sum of the y-coordinates is $(b+c) + (c+a) + (a+b) = 2(a+b+c)$.

(B) The area of the triangle formed by these points is $\frac{1}{2} |a(c+a-(a+b)) + b(a+b-(b+c)) + c(b+c-(c+a))|$.

(C) Area $= \frac{1}{2} |a(c-b) + b(a-c) + c(b-a)| = \frac{1}{2} |ac-ab + ab-bc + bc-ac| = \frac{1}{2} |0| = 0$.

(D) The points are collinear.

Question 12. If the area of a triangle is given as A and the length of the base is $b$, the corresponding height $h$ is given by $h = 2A/b$. This is an application of the area formula.

(A) True

(B) False

(C) This allows finding the distance from a vertex to the opposite side (altitude length).

(D) This formula is only valid for right triangles.

Question 1. Which of the following are correct definitions of triangle centers?

(A) Centroid: Intersection of medians.

(B) Incenter: Intersection of angle bisectors.

(C) Circumcenter: Intersection of altitudes.

(D) Orthocenter: Intersection of perpendicular bisectors of sides.

Question 2. The centroid of a triangle with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ is the point G such that:

(A) $G = (\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$

(B) G divides each median in the ratio $2:1$ (vertex to midpoint of opposite side).

(C) It is the center of gravity of the triangle (if mass is uniformly distributed).

(D) It is the intersection of perpendicular bisectors.

Question 3. The incenter of a triangle is the center of the incircle. Which of the following is true about the incircle?

(A) It passes through the vertices of the triangle.

(B) It is tangent to all three sides of the triangle internally.

(C) Its radius is the perpendicular distance from the incenter to any side.

(D) The incenter is equidistant from the sides of the triangle.

Question 4. The circumcenter of a triangle is the center of the circumcircle. Which of the following is true about the circumcircle?

(A) It is tangent to all three sides of the triangle.

(B) It passes through all three vertices of the triangle.

(C) Its radius is the distance from the circumcenter to any vertex.

(D) The circumcenter is equidistant from the vertices of the triangle.

Question 5. For a right-angled triangle, where does the orthocenter lie?

(A) Inside the triangle.

(B) At the vertex where the right angle is formed.

(C) On the hypotenuse.

(D) Outside the triangle.

Question 6. For an obtuse-angled triangle, where do the orthocenter and circumcenter lie?

(A) Both lie inside the triangle.

(B) Both lie outside the triangle.

(C) Orthocenter is outside, Circumcenter is inside.

(D) Orthocenter is inside, Circumcenter is outside.

Question 7. The incenter of a triangle always lies inside the triangle, regardless of the type of triangle (acute, obtuse, right-angled).

(A) True

(B) False

(C) This is because the angle bisectors always intersect inside the triangle.

(D) The incenter can lie on a side for a degenerate triangle.

Question 8. For an equilateral triangle, which of the following statements are true?

(A) The centroid, incenter, circumcenter, and orthocenter all coincide at a single point.

(B) The medians, angle bisectors, altitudes, and perpendicular bisectors are all the same lines.

(C) The distance from the center to any vertex is equal to the distance from the center to any side (circumradius = inradius).

(D) It is a special case where all four centers coincide.

Question 9. The Euler line passes through which of the following triangle centers (for a non-equilateral triangle)?

(A) Centroid

(B) Incenter

(C) Circumcenter

(D) Orthocenter

Question 10. If the vertices of a triangle are $(0, 0)$, $(a, 0)$, and $(a, a)$ where $a \neq 0$, which of the following statements are true?

(A) It is a right-angled triangle at $(a, 0)$.

(B) It is an isosceles triangle.

(C) The circumcenter lies at $(a/2, a/2)$ (midpoint of hypotenuse).

(D) The orthocenter lies at $(a, 0)$.

Question 11. Which center's coordinates require the side lengths of the triangle for calculation using a direct formula?

(A) Centroid

(B) Incenter

(C) Circumcenter

(D) Orthocenter

Question 1. Which of the following can represent the locus of a point?

(A) A straight line.

(B) A circle.

(C) A single point.

(D) A curve satisfying a specific geometric condition.

Question 2. The locus of a point P$(x, y)$ that is equidistant from the points $A(1, 0)$ and $B(5, 0)$ is:

(A) A line parallel to the Y-axis.

(B) The perpendicular bisector of the segment AB.

(C) The line $x = 3$.

(D) A circle with center $(3, 0)$.

Question 3. The equation of the locus of a point P$(x, y)$ such that $PA^2 + PB^2 = k$ (constant), where A and B are fixed points, is generally a:

(A) Straight line.

(B) Circle.

(C) Point (for specific k).

(D) Parabola.

Question 4. The locus of a point P$(x, y)$ such that its distance from the X-axis is twice its distance from the Y-axis is:

(A) $|y| = 2|x|$.

(B) $y = 2x$ or $y = -2x$.

(C) A pair of straight lines passing through the origin.

(D) A parabola.

Question 5. The locus of a point P$(x, y)$ such that the area of the triangle formed by P, A$(0, 0)$ and B$(2, 0)$ is 5 sq units is:

(A) $\frac{1}{2} |x(0-0) + 0(0-y) + 2(y-0)| = 5$

(B) $\frac{1}{2} |2y| = 5$

(C) $|y| = 5 \implies y = 5$ or $y = -5$.

(D) A pair of parallel lines.

Question 6. Which of the following are examples of loci?

(A) A circle (locus of points equidistant from a fixed point).

(B) A parabola (locus of points equidistant from a fixed point and a fixed line).

(C) The perpendicular bisector of a line segment (locus of points equidistant from two fixed points).

(D) A point (trivial locus).

Question 7. To find the equation of a locus, we typically:

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

(B) Write the given geometric condition in terms of $x$ and $y$.

(C) Simplify the resulting algebraic equation.

(D) Plot a few points that satisfy the condition.

Question 8. The locus of the center of a circle that touches the X-axis is:

(A) A horizontal line.

(B) A vertical line.

(C) A pair of horizontal lines parallel to the X-axis.

(D) The equation $|y| = r$, where r is the radius (if radius is constant) or $|y|=$ distance from center to x-axis.

Question 9. The locus of a point P such that the sum of its distances from two fixed points $F_1$ and $F_2$ is constant ($2a$) is:

(A) A circle (if $F_1=F_2$).

(B) An ellipse (if $2a >$ distance $F_1F_2$).

(C) A line segment (if $2a =$ distance $F_1F_2$).

(D) A hyperbola.

Question 10. The locus of a point P such that the difference of its distances from two fixed points $F_1$ and $F_2$ is constant ($2a$) is:

(A) A hyperbola (if $0 < 2a <$ distance $F_1F_2$).

(B) A pair of rays (if $2a =$ distance $F_1F_2$).

(C) A circle.

(D) An ellipse.

Question 11. The equation of the locus of a point P$(x, y)$ such that it is equidistant from the point $A(2, 0)$ and the line $x = -2$ is:

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

(B) The distance from P to the line $x=-2$ is $|x - (-2)| = |x+2|$.

(C) $\sqrt{(x-2)^2 + y^2} = |x+2| \implies (x-2)^2 + y^2 = (x+2)^2$.

(D) $x^2 - 4x + 4 + y^2 = x^2 + 4x + 4 \implies y^2 = 8x$, which is a parabola.

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

(A) $x = X + h$

(B) $y = Y + k$

(C) $X = x - h$

(D) $Y = y - k$

Question 2. If the original coordinates of a point are $(7, -5)$ and the new coordinates after shifting the origin are $(2, 3)$, then the origin was shifted to the point:

(A) $(7-2, -5-3) = (5, -8)$.

(B) $(2-7, 3-(-5)) = (-5, 8)$.

(C) $(h, k)$ such that $7 = 2+h$ and $-5 = 3+k$.

(D) $(5, -8)$.

Question 3. If the equation of a curve is $y = x^2$, and the origin is shifted to $(1, 2)$, the new equation in terms of new coordinates $(X, Y)$ is:

(A) Replace $x$ with $X+1$ and $y$ with $Y+2$.

(B) $Y+2 = (X+1)^2$.

(C) $Y = (X+1)^2 - 2$.

(D) $Y = X^2 + 2X - 1$.

Question 4. Shifting the origin (translation of axes) changes which of the following?

(A) The equation of a geometric locus.

(B) The distance between two points.

(C) The coordinates of a point.

(D) The orientation of the axes.

Question 5. To eliminate the first-degree terms from the equation $x^2 + y^2 + 2gx + 2fy + c = 0$ by shifting the origin, the new origin should be shifted to:

(A) $(-g, -f)$

(B) $(g, f)$

(C) The center of the circle.

(D) A point $(h, k)$ such that $x = X+h, y = Y+k$ results in no linear terms in X and Y.

Question 6. If the equation of a line is $y = mx + c$, and the origin is shifted to $(h, k)$, the new equation is $Y+k = m(X+h) + c$. Which of the following is true?

(A) The slope of the line remains $m$ in the new system.

(B) The y-intercept changes.

(C) The equation of the line changes.

(D) The line itself moves relative to the new axes.

Question 7. The effect of shifting the origin is equivalent to:

(A) Rotating the coordinate axes.

(B) Translating the entire plane (including the curve and axes) so the old origin moves to a new position, while the axes directions remain the same.

(C) Translating the curve or point relative to fixed axes.

(D) Changing the scale of the axes.

Question 8. If the equation $ax^2 + by^2 + 2hxy + 2gx + 2fy + c = 0$ is transformed by shifting the origin to $(h, k)$, the terms $ax^2, by^2, 2hxy$ remain unchanged in form (i.e., the quadratic terms are invariant under translation).

(A) True

(B) False

(C) The linear terms $2gx + 2fy$ are generally affected.

(D) The constant term $c$ is generally affected.

Question 9. The original coordinates $(x, y)$ of a point are $(3, 5)$. The origin is shifted to $(1, 1)$. The new coordinates $(X, Y)$ are:

(A) $X = 3-1 = 2$

(B) $Y = 5-1 = 4$

(C) $(2, 4)$

(D) $(4, 6)$

Question 10. The equation $x^2 - 4x + y^2 = 0$ represents a circle. If the origin is shifted to $(2, 0)$, the new equation is:

(A) $(X+2)^2 - 4(X+2) + Y^2 = 0$

(B) $X^2 + 4X + 4 - 4X - 8 + Y^2 = 0$

(C) $X^2 + Y^2 - 4 = 0$

(D) $X^2 + Y^2 = 4$, which is a circle centered at the new origin with radius 2.

Question 1. The slope of a straight line represents:

(A) The steepness of the line.

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

(C) The ratio of the change in y-coordinates to the change in x-coordinates between any two distinct points on the line.

(D) The y-intercept.

Question 2. Which of the following statements about slope are true?

(A) A line parallel to the X-axis has a slope of 0.

(B) A line parallel to the Y-axis has an undefined slope.

(C) The slope of the line $y = 5$ is 5.

(D) The slope of the line $x = -3$ is undefined.

Question 3. Two non-vertical lines with slopes $m_1$ and $m_2$ are perpendicular if:

(A) $m_1 = m_2$

(B) $m_1 m_2 = -1$

(C) $m_2 = -1/m_1$ (if $m_1 \neq 0$)

(D) One slope is positive and the other is negative.

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

(A) Convert to slope-intercept form: $3y = -2x + 6 \implies y = -\frac{2}{3}x + 2$.

(B) The slope is $-\frac{2}{3}$.

(C) Using the formula $-A/B = -2/3$.

(D) $2/3$

Question 5. If two lines are parallel, which of the following must be true?

(A) Their slopes are equal (if both are non-vertical).

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

(C) They never intersect unless they are coincident.

(D) Their equations are of the form $y = mx + c_1$ and $y = mx + c_2$ (for non-vertical lines).

Question 6. The angle between the lines $y = x$ and $y = -x$ is:

(A) The slope of $y=x$ is $m_1 = 1$, angle $\theta_1 = 45^\circ$.

(B) The slope of $y=-x$ is $m_2 = -1$, angle $\theta_2 = 135^\circ$.

(C) The angle between them is $|\theta_2 - \theta_1| = |135^\circ - 45^\circ| = 90^\circ$.

(D) The lines are perpendicular since $m_1 m_2 = 1 \times (-1) = -1$.

Question 7. The slope of the line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is:

(A) $\frac{y_2-y_1}{x_2-x_1}$, provided $x_1 \neq x_2$.

(B) $\frac{\Delta y}{\Delta x}$ where $\Delta y = y_2-y_1$ and $\Delta x = x_2-x_1$.

(C) Undefined if $x_1 = x_2$ and $y_1 \neq y_2$.

(D) 0 if $y_1 = y_2$ and $x_1 \neq x_2$.

Question 8. If the angle between two lines is $45^\circ$, and the slope of one line is $1/2$, the slope(s) of the other line can be found using $\tan 45^\circ = |\frac{m_2 - m_1}{1 + m_1 m_2}|$. Which of the following are possible values for the slope of the other line?

(A) $1 = |\frac{m_2 - 1/2}{1 + m_2/2}| = |\frac{2m_2 - 1}{2 + m_2}|$

(B) $\frac{2m_2 - 1}{2 + m_2} = 1 \implies 2m_2 - 1 = 2 + m_2 \implies m_2 = 3$.

(C) $\frac{2m_2 - 1}{2 + m_2} = -1 \implies 2m_2 - 1 = -2 - m_2 \implies 3m_2 = -1 \implies m_2 = -1/3$.

(D) $3$

(E) $-1/3$

Question 9. The condition for three points $A, B, C$ to be collinear based on slope is that the slope of AB equals the slope of BC. This holds true unless the line is vertical.

(A) True, provided the line is not vertical.

(B) If the line is vertical, all points have the same x-coordinate.

(C) This method is an alternative to the area method for checking collinearity.

(D) This method requires calculating distances.

Question 10. The slope of a line with equation $Ax + By + C = 0$ is:

(A) $-A/B$, if $B \neq 0$.

(B) $A/B$, if $B \neq 0$.

(C) Undefined, if $B = 0$ (and $A \neq 0$).

(D) 0, if $A = 0$ (and $B \neq 0$).

Question 11. If a line makes an angle $\theta$ with the positive X-axis, its slope is $m = \tan \theta$. Which of the following are true?

(A) If $0^\circ < \theta < 90^\circ$, the slope is positive.

(B) If $90^\circ < \theta < 180^\circ$, the slope is negative.

(C) If $\theta = 90^\circ$, the slope is undefined.

(D) If $\theta = 0^\circ$ or $180^\circ$, the slope is 0.

Question 12. Two lines are perpendicular. Which of the following must be true?

(A) The product of their slopes is -1 (if both non-vertical).

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

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

(D) Their equations are of the form $y=m_1x+c_1$ and $y=m_2x+c_2$ with $m_1m_2 = -1$.

Question 13. The slope of the line $y = -4x + 7$ is:

(A) $-4$

(B) The coefficient of $x$ when the equation is in slope-intercept form.

(C) $7$ (this is the y-intercept)

(D) Positive.

Question 1. Which of the following are valid forms for the equation of a straight line?

(A) Point-slope form: $y - y_1 = m(x - x_1)$

(B) Slope-intercept form: $y = mx + c$

(C) Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$

(D) General form: $Ax + By + C = 0$

Question 2. The equation of a line parallel to the X-axis can be written as:

(A) $y = \text{constant}$

(B) $Ax + C = 0$ for some constants A, C with $A \neq 0$. (This is a vertical line)

(C) $By + C = 0$ for some constants B, C with $B \neq 0$. (This is a horizontal line)

(D) $y = k$ for some real number $k$.

Question 3. The equation of a line passing through the point $(3, -2)$ with slope $1/2$ is:

(A) $y - (-2) = \frac{1}{2}(x - 3)$

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

(C) $2(y + 2) = x - 3 \implies 2y + 4 = x - 3 \implies x - 2y - 7 = 0$.

(D) $x - 2y = 7$.

Question 4. The equation of the line passing through $(1, 5)$ and $(3, 7)$ is:

(A) Slope $m = \frac{7-5}{3-1} = \frac{2}{2} = 1$.

(B) Using point $(1, 5)$: $y - 5 = 1(x - 1) \implies y - 5 = x - 1 \implies y = x + 4$.

(C) Using point $(3, 7)$: $y - 7 = 1(x - 3) \implies y - 7 = x - 3 \implies y = x + 4$.

(D) $y = x + 4$.

Question 5. The line $y = -3x + 5$ has:

(A) Slope $-3$.

(B) Y-intercept $5$.

(C) Is in slope-intercept form.

(D) Is parallel to the line $y = -3x + 10$.

Question 6. A line has x-intercept 4 and y-intercept -2. Which of the following represents its equation?

(A) $\frac{x}{4} + \frac{y}{-2} = 1$

(B) $\frac{x}{4} - \frac{y}{2} = 1$

(C) $2x - 4y = 8 \implies x - 2y = 4$.

(D) Passes through $(4, 0)$ and $(0, -2)$.

Question 7. The equation of the Y-axis is:

(A) $x = 0$

(B) A vertical line.

(C) Has undefined slope.

(D) Passes through the origin.

Question 8. The normal form of a line's equation is $x \cos\alpha + y \sin\alpha = p$. Which of the following is true?

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

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

(C) It is a unique representation for a given line (if $p>0$ and $0 \leq \alpha < 2\pi$).

(D) If the line passes through the origin, $p=0$.

Question 9. The equation of a line passing through the origin is always of the form:

(A) $y = mx + c$ with $c=0$, i.e., $y = mx$.

(B) $Ax + By = 0$ for some constants A, B not both zero.

(C) Has a y-intercept of 0.

(D) Can be represented in intercept form $\frac{x}{a} + \frac{y}{b} = 1$ unless it is a coordinate axis.

Question 10. Consider the equation $5x - 12y + 26 = 0$. To convert it to normal form $x \cos\alpha + y \sin\alpha = p$, we divide by $\sqrt{A^2+B^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25+144} = \sqrt{169} = 13$. Which of the following forms are correct?

(A) $\frac{5}{13}x - \frac{12}{13}y + \frac{26}{13} = 0 \implies \frac{5}{13}x - \frac{12}{13}y + 2 = 0$.

(B) $\frac{5}{13}x - \frac{12}{13}y = -2$. Since p must be non-negative, multiply by -1.

(C) $-\frac{5}{13}x + \frac{12}{13}y = 2$.

(D) $\cos\alpha = -5/13, \sin\alpha = 12/13, p = 2$.

Question 11. The equation of a line with undefined slope passing through the point $(-5, 2)$ is:

(A) A vertical line.

(B) Parallel to the Y-axis.

(C) $x = -5$.

(D) $y = 2$.

Question 1. The equation $Ax + By + C = 0$ is the general equation of a straight line. Which of the following conditions apply to A, B, and C?

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

(B) At least one of A or B must be non-zero.

(C) If $A=0$ and $B \neq 0$, it represents a horizontal line.

(D) If $A \neq 0$ and $B = 0$, it represents a vertical line.

Question 2. Consider the lines $L_1: x + 2y = 5$ and $L_2: 2x - y = 0$. Which of the following are true about their intersection?

(A) Solving the system of equations gives the point of intersection.

(B) From $L_2$, $y = 2x$. Substituting into $L_1$: $x + 2(2x) = 5 \implies 5x = 5 \implies x=1$. Then $y = 2(1) = 2$.

(C) The point of intersection is $(1, 2)$.

(D) The lines are parallel.

Question 3. To convert the general equation $Ax + By + C = 0$ into slope-intercept form ($y = mx + c$), assuming $B \neq 0$, we:

(A) Isolate the y term: $By = -Ax - C$.

(B) Divide by B: $y = -\frac{A}{B}x - \frac{C}{B}$.

(C) Identify the slope $m = -A/B$.

(D) Identify the y-intercept $c = -C/B$.

Question 4. The intercept form of the line $6x - 4y = 12$ is:

(A) Divide the equation by 12: $\frac{6x}{12} - \frac{4y}{12} = \frac{12}{12}$.

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

(C) $\frac{x}{2} + \frac{y}{-3} = 1$.

(D) X-intercept is 2, Y-intercept is -3.

Question 5. If two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are parallel, then:

(A) $\frac{a_1}{a_2} = \frac{b_1}{b_2}$

(B) $\frac{a_1}{a_2} \neq \frac{c_1}{c_2}$ (unless they are coincident)

(C) $a_1b_2 - a_2b_1 = 0$

(D) Their slopes are equal (if non-vertical).

Question 6. If two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are coincident, then:

(A) They represent the same line.

(B) They have infinitely many points of intersection.

(C) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (provided denominators are non-zero).

(D) Their slopes and y-intercepts are equal (if non-vertical).

Question 7. The general equation $Ax + By + C = 0$ can represent which types of lines?

(A) Horizontal lines (when A=0, B!=0)

(B) Vertical lines (when B=0, A!=0)

(C) Lines passing through the origin (when C=0)

(D) Lines not passing through the origin (when C!=0)

Question 8. The point of intersection of two distinct lines can be found by:

(A) Solving the system of two linear equations.

(B) Using substitution method.

(C) Using elimination method.

(D) Using graphical method by plotting both lines.

Question 9. If two lines are perpendicular, $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$, then:

(A) $a_1a_2 + b_1b_2 = 0$ (derived from $m_1m_2 = -1$ where $m_1 = -a_1/b_1, m_2 = -a_2/b_2$)

(B) Their slopes are negative reciprocals (if non-vertical/horizontal).

(C) $a_1b_2 - a_2b_1 = 0$ (This is for parallel lines)

(D) $a_1/b_1 = a_2/b_2$ (This is for parallel lines)

Question 10. Consider the lines $y = 2x + 1$ and $y = 2x - 3$. Which of the following are true?

(A) Their slopes are equal.

(B) They are parallel lines.

(C) They have different y-intercepts.

(D) They intersect at a single point.

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

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

(B) The shortest distance from the point to any point on the line.

(C) The length of the perpendicular dropped from the point to the line.

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

(C) The distance between any point on one line and the other line.

(D) Undefined if the lines are coincident ($C_1=C_2$).

Question 3. The equation of a family of lines passing through the intersection of two lines $L_1 = 0$ and $L_2 = 0$ is $L_1 + \lambda L_2 = 0$, where $\lambda$ is a parameter. Which of the following are true?

(A) Every line in the family passes through the common point of $L_1=0$ and $L_2=0$ (if they intersect).

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

(C) This family includes the line $L_1=0$ (when $\lambda=0$).

(D) This family includes the line $L_2=0$ (except when we consider $\lambda \to \infty$ or write it as $\mu L_1 + L_2 = 0$).

Question 4. The distance of the origin from the line $3x - 4y + 10 = 0$ is:

(A) $\frac{|3(0) - 4(0) + 10|}{\sqrt{3^2 + (-4)^2}}$

(B) $\frac{|10|}{\sqrt{9 + 16}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2$ units.

(C) $2$ units.

(D) The value of $C$ divided by $\sqrt{A^2+B^2}$.

Question 5. The lines $2x + 3y - 5 = 0$ and $2x + 3y + 10 = 0$ are parallel. Their distance apart is:

(A) $|-5 - 10| / \sqrt{2^2 + 3^2} = |-15| / \sqrt{4+9} = 15 / \sqrt{13}$.

(B) $\frac{15}{\sqrt{13}}$ units.

(C) $\frac{|C_1 - C_2|}{\sqrt{A^2+B^2}}$ with $C_1=-5, C_2=10$.

(D) $\frac{|5 - (-10)|}{\sqrt{13}} = \frac{15}{\sqrt{13}}$.

Question 6. The equation of a line passing through the intersection of $x + y - 2 = 0$ and $x - y = 0$, and through the point $(3, 1)$ is $L_1 + \lambda L_2 = 0$. Which of the following are true?

(A) $x + y - 2 + \lambda(x - y) = 0$.

(B) Substitute $(3, 1)$: $3 + 1 - 2 + \lambda(3 - 1) = 0 \implies 2 + 2\lambda = 0 \implies \lambda = -1$.

(C) The equation is $(x + y - 2) - 1(x - y) = 0 \implies x + y - 2 - x + y = 0 \implies 2y - 2 = 0 \implies y = 1$.

(D) The line is $y=1$. (Check: Intersection of $x+y=2, x-y=0 \implies 2x=2 \implies x=1, y=1$. Point is $(1,1)$. The line $y=1$ passes through $(1,1)$ and $(3,1)$).

Question 7. The shortest distance between a point and a line is the length of the perpendicular from the point to the line.

(A) True

(B) False

(C) This is a fundamental concept used in the distance formula for a point from a line.

(D) Any other distance is greater than the perpendicular distance.

Question 8. The family of lines passing through the origin is given by:

(A) $y = mx$ (for finite slope)

(B) $x = 0$ (for infinite slope)

(C) $Ax + By = 0$

(D) $L_1 + \lambda L_2 = 0$ where $L_1=x, L_2=y$. (This gives $x+\lambda y = 0$, which is $y = (-1/\lambda)x$ or $x=0$ if $\lambda \to \infty$)

Question 9. If two parallel lines have equations $y = mx + c_1$ and $y = mx + c_2$, their distance is derived by taking a point on one line (say $(0, c_1)$ on the first line) and finding its distance from the other line $y = mx + c_2$ or $mx - y + c_2 = 0$. Which of the following is true?

(A) The distance is $\frac{|m(0) - c_1 + c_2|}{\sqrt{m^2 + (-1)^2}} = \frac{|c_2 - c_1|}{\sqrt{m^2 + 1}}$.

(B) The distance is $\frac{|c_1 - c_2|}{\sqrt{1+m^2}}$.

(C) This method works for non-vertical parallel lines.

(D) This formula is only for lines passing through the origin.

Question 10. The concept of a family of lines is useful in finding the equation of a line that satisfies an additional condition, such as passing through a specific point or being parallel/perpendicular to another line.

(A) True

(B) False

(C) It simplifies finding lines through intersection points without explicitly finding the intersection point first.

(D) It can represent all lines in the plane.

Question 1. In the Cartesian coordinate system for three dimensions, which of the following are true?

(A) There are three mutually perpendicular axes: X, Y, and Z axes.

(B) The axes intersect at the origin $(0, 0, 0)$.

(C) The axes define three coordinate planes: XY, YZ, and ZX planes.

(D) The planes divide the space into four octants.

Question 2. A point in three-dimensional space is uniquely identified by its coordinates $(x, y, z)$. What do these coordinates represent?

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

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

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

(D) The perpendicular distances from the point to the respective coordinate planes.

Question 3. A point lies on the Z-axis. Which of the following must be true about its coordinates $(x, y, z)$?

(A) $x = 0$

(B) $y = 0$

(C) $z = 0$

(D) The point is of the form $(0, 0, k)$ for some real number $k$.

Question 4. The equation of the XY-plane is:

(A) $x = 0$

(B) $y = 0$

(C) $z = 0$

(D) Represents all points $(x, y, 0)$.

Question 5. The distance of the point $(a, b, c)$ from the X-axis is:

(A) $|a|$

(B) $\sqrt{b^2 + c^2}$ (Distance from point to its projection on X-axis, which is $(a, 0, 0)$)

(C) $\sqrt{(a-a)^2 + (b-0)^2 + (c-0)^2} = \sqrt{b^2+c^2}$.

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

Question 6. A point $(x, y, z)$ lies in Octant I. Which of the following must be true about its coordinates?

(A) $x > 0$

(B) $y > 0$

(C) $z > 0$

(D) All coordinates are positive.

Question 7. The distance of a point $(5, 6, 7)$ from the YZ-plane is:

(A) $5$ units.

(B) The absolute value of the x-coordinate.

(C) $\sqrt{6^2+7^2}$.

(D) $\sqrt{5^2+6^2+7^2}$.

Question 8. The point $(-2, -3, 4)$ lies in which octant?

(A) Octant II ($-,+,+$)

(B) Octant III ($-, -,+$)

(C) Octant VI ($-,-,-$)

(D) Octant VII ($-,-,+$)

Question 9. The coordinates of the projection of the point $(a, b, c)$ onto the XY-plane are:

(A) $(a, b, 0)$

(B) $(0, b, c)$

(C) $(a, 0, c)$

(D) $(a, b, c)$

Question 10. Which of the following statements about distances in 3D are true?

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

(B) Distance from $(x,y,z)$ to XY-plane is $|z|$.

(C) Distance from $(x,y,z)$ to origin is $\sqrt{x^2+y^2+z^2}$.

(D) Distance from $(x,y,z)$ to the point $(x_0, y_0, z_0)$ is $\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$.

Question 11. The three coordinate planes are mutually perpendicular. Their intersection lines are the coordinate axes.

(A) True

(B) False

(C) The intersection of the XY and XZ planes is the X-axis.

(D) The intersection of all three planes is the origin.

Question 1. The distance formula in three dimensions is a generalization of the 2D distance formula and is derived using the Pythagorean theorem in 3D space.

(A) True

(B) False

(C) It involves the square root of the sum of squared differences of corresponding coordinates.

(D) It requires vector operations for derivation.

Question 2. The distance between the points $(2, -3, 4)$ and $(-1, 5, -2)$ is:

(A) $\sqrt{(-1-2)^2 + (5-(-3))^2 + (-2-4)^2}$

(B) $\sqrt{(-3)^2 + 8^2 + (-6)^2}$

(C) $\sqrt{9 + 64 + 36} = \sqrt{109}$ units.

(D) $\sqrt{109}$ units.

Question 3. The distance of a point $(a, b, c)$ from the origin $(0, 0, 0)$ is:

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

(B) The magnitude of the position vector of the point.

(C) Equal to the distance of $(-a, -b, -c)$ from the origin.

(D) Always positive unless the point is the origin itself.

Question 4. Consider the points $A(1, 2, 3)$, $B(4, 5, 6)$, $C(7, 8, 9)$. Which of the following are true?

(A) AB = $\sqrt{(4-1)^2 + (5-2)^2 + (6-3)^2} = \sqrt{3^2+3^2+3^2} = \sqrt{27} = 3\sqrt{3}$.

(B) BC = $\sqrt{(7-4)^2 + (8-5)^2 + (9-6)^2} = \sqrt{3^2+3^2+3^2} = \sqrt{27} = 3\sqrt{3}$.

(C) AC = $\sqrt{(7-1)^2 + (8-2)^2 + (9-3)^2} = \sqrt{6^2+6^2+6^2} = \sqrt{3 \times 36} = 6\sqrt{3}$.

(D) A, B, and C are collinear because AB + BC = AC ($3\sqrt{3} + 3\sqrt{3} = 6\sqrt{3}$).

Question 5. The distance of a point $(x, y, z)$ from the Y-axis is:

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

(B) Distance to the projection on Y-axis, which is $(0, y, 0)$.

(C) $\sqrt{(x-0)^2 + (y-y)^2 + (z-0)^2} = \sqrt{x^2+z^2}$.

(D) $|y|$.

Question 6. If the distance between $(k, 2, 3)$ and $(1, 4, 5)$ is $\sqrt{17}$, what are the possible values of $k$?

(A) $\sqrt{(k-1)^2 + (4-2)^2 + (5-3)^2} = \sqrt{17}$

(B) $(k-1)^2 + 2^2 + 2^2 = 17$

(C) $(k-1)^2 + 4 + 4 = 17 \implies (k-1)^2 = 9$.

(D) $k-1 = 3$ or $k-1 = -3 \implies k = 4$ or $k = -2$.

Question 7. The vertices of a tetrahedron are $A(0,0,0)$, $B(a,0,0)$, $C(0,b,0)$, $D(0,0,c)$, where $a,b,c > 0$. The length of the edges connecting the origin are:

(A) OA = $a$

(B) OB = $b$

(C) OC = $c$

(D) These edges are mutually perpendicular (along axes).

Question 8. A point P is equidistant from two fixed points A and B in 3D space. The locus of P is the perpendicular bisector plane of the segment AB.

(A) True

(B) False

(C) This is an extension of the 2D locus concept to 3D.

(D) The equation of the locus is a sphere.

Question 9. Which of the following distances are equal for a point $(x, y, z)$?

(A) Distance from XY-plane and $|z|$.

(B) Distance from XZ-plane and $|y|$.

(C) Distance from YZ-plane and $|x|$.

(D) Distance from X-axis and $\sqrt{x^2+y^2}$.

Question 10. The distance formula in 3D can be used to verify geometric properties of figures in space, such as checking if vertices form a specific type of triangle or quadrilateral/parallelepiped.

(A) True

(B) False

(C) For example, checking if all edges of a cube are equal length.

(D) For example, checking if opposite edges of a parallelepiped are parallel (using vectors or direction cosines).

Question 1. The coordinates of the midpoint of the line segment joining $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ are:

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

(B) The point that divides AB internally in the ratio $1:1$.

(C) The center of the segment AB.

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

Question 2. The 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 of the following are true about the coordinates of P?

(A) $P = (\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n})$

(B) The division is internal, so $m/n > 0$.

(C) P lies on the line segment AB.

(D) If $m=n$, P is the midpoint.

Question 3. The point Q divides the line segment joining $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ externally in the ratio $m:n$. Which of the following are true about the coordinates of Q?

(A) $Q = (\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}, \frac{mz_2 - nz_1}{m-n})$, provided $m \neq n$.

(B) The division is external, so $m/n < 0$ is sometimes used in internal formula with negative ratio.

(C) Q lies outside the segment AB.

(D) If $m=n$, the external division point is at infinity, and the formula is undefined.

Question 4. The vertices of a triangle in 3D are $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, and $C(x_3, y_3, z_3)$. The centroid G is the point that divides the line segment joining a vertex to the centroid of the opposite face in the ratio $3:1$ is incorrect. Centroid divides the median in 2:1. The centroid is $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3})$.

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

(B) G divides each median in the ratio $2:1$ (vertex to midpoint of opposite side).

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

(D) G lies inside the triangle (or on edge/vertex for degenerate cases).

Question 5. The centroid of a tetrahedron with vertices $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, $C(x_3, y_3, z_3)$, $D(x_4, y_4, z_4)$ is given by:

(A) $G = (\frac{x_1+x_2+x_3+x_4}{4}, \frac{y_1+y_2+y_3+y_4}{4}, \frac{z_1+z_2+z_3+z_4}{4})$

(B) G is the point of intersection of the lines joining each vertex to the centroid of the opposite face, and it divides these lines in the ratio $3:1$ (vertex to face centroid).

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

(D) G lies inside the tetrahedron (for a non-degenerate tetrahedron).

Question 6. The point $(1, -2, 0)$ divides the line segment joining $A(3, 2, -4)$ and $B(x, y, z)$ in the ratio $1:2$. Which of the following are true?

(A) It is internal division if the ratio is positive (like $1:2$).

(B) $1 = \frac{1(x) + 2(3)}{1+2} = \frac{x+6}{3} \implies 3 = x+6 \implies x = -3$.

(C) $-2 = \frac{1(y) + 2(2)}{1+2} = \frac{y+4}{3} \implies -6 = y+4 \implies y = -10$.

(D) $0 = \frac{1(z) + 2(-4)}{1+2} = \frac{z-8}{3} \implies 0 = z-8 \implies z = 8$. The coordinates of B are $(-3, -10, 8)$.

Question 7. The section formula in 3D is used for:

(A) Finding a point that divides a line segment in a given ratio.

(B) Finding the midpoint of a line segment.

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

(D) Calculating the distance between two points.

Question 8. If the centroid of a triangle with vertices $(1, 2, 3)$, $(a, b, c)$, and $(-1, -2, -3)$ is the origin $(0, 0, 0)$, which of the following are true?

(A) $\frac{1+a-1}{3} = 0 \implies a = 0$.

(B) $\frac{2+b-2}{3} = 0 \implies b = 0$.

(C) $\frac{3+c-3}{3} = 0 \implies c = 0$.

(D) The second vertex is $(0, 0, 0)$.

Question 9. The points $P, Q$ divide the line segment joining $A(1, 2, 3)$ and $B(7, 8, 9)$ into three equal parts. Which of the following are true?

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

(B) Q divides AB in the ratio $2:1$ internally.

(C) P = $(\frac{1(7)+2(1)}{1+2}, \frac{1(8)+2(2)}{1+2}, \frac{1(9)+2(3)}{1+2}) = (\frac{9}{3}, \frac{12}{3}, \frac{15}{3}) = (3, 4, 5)$.

(D) Q = $(\frac{2(7)+1(1)}{2+1}, \frac{2(8)+1(2)}{2+1}, \frac{2(9)+1(3)}{2+1}) = (\frac{15}{3}, \frac{18}{3}, \frac{21}{3}) = (5, 6, 7)$.

Question 10. If the ratio of division $m:n$ is negative in the section formula for internal division, it represents external division.

(A) True

(B) False

(C) The formula $\frac{mx_2 + nx_1}{m+n}$ can be used for external division by taking the ratio as $m:(-n)$.

(D) The external division formula is a special case of the internal division formula.

Question 1. Conic sections are formed by the intersection of a plane with a double-napped right circular cone. Which of the following conic sections are obtained when the cutting plane does *not* pass through the vertex of the cone?

(A) Circle

(B) Ellipse

(C) Parabola

(D) Hyperbola

Question 2. A conic section is defined by the locus of a point P such that its distance from a fixed point (Focus, F) is $e$ times its distance from a fixed line (Directrix, L). Which of the following are true?

(A) $PF = e \cdot PD$, where D is the foot of the perpendicular from P to L.

(B) $e$ is called the eccentricity.

(C) The value of $e$ determines the type of conic section.

(D) The directrix is perpendicular to the axis of the conic.

Question 3. Match the eccentricity value to the corresponding conic section:

(A) (i) - (c)

(B) (ii) - (d)

(C) (iii) - (a)

(D) (iv) - (b)

Question 4. Degenerate conic sections are formed when the cutting plane passes through the vertex of the cone. Which of the following can be degenerate conic sections?

(A) A pair of intersecting lines (plane contains the axis of the cone).

(B) A single point (plane is perpendicular to the axis).

(C) A single line (plane is tangent to the cone along a generator).

(D) An ellipse.

Question 5. The axis of a conic section is a line passing through the focus and perpendicular to the directrix.

(A) True

(B) False

(C) For a central conic (ellipse/hyperbola), the axis passes through both foci.

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

Question 6. The latus rectum of a conic section is a line segment passing through the focus, perpendicular to the axis, with endpoints on the conic. Its length depends on the specific type of conic and its parameters.

(A) True

(B) False

(C) Its length is $2 \times$ distance from focus to directrix multiplied by eccentricity $e$. (This might not be standard). Length is $2b^2/a$ or $4a$ etc depending on conic.

(D) It's a chord of the conic.

Question 7. The angle of the cutting plane relative to the axis of the cone determines the type of conic section (excluding degenerate cases). Let $\beta$ be the angle between the plane and the axis of the cone, and $\alpha$ be the semi-vertical angle of the cone. Which of the following is true?

(A) If $\beta = 90^\circ$, the intersection is a circle.

(B) If $\alpha < \beta < 90^\circ$, the intersection is an ellipse.

(C) If $\beta = \alpha$, the intersection is a parabola.

(D) If $0 \leq \beta < \alpha$, the intersection is a hyperbola.

Question 8. The directrix of a conic section is:

(A) A fixed point.

(B) A fixed line.

(C) Perpendicular to the axis of the conic (if it exists and is unique).

(D) Involved in the locus definition $PF = e \cdot PD$.

Question 9. A circle is a special case of:

(A) A parabola (e=1)

(B) An ellipse (e=0)

(C) A conic section formed when the cutting plane is perpendicular to the axis of the cone (and does not pass through the vertex).

(D) An ellipse where the two foci coincide (at the center).

Question 10. Conic sections are important in physics and astronomy. For example, the orbits of planets around the sun are approximately ellipses.

(A) True

(B) False

(C) The path of a projectile is a parabola.

(D) The shape of some telescope mirrors is parabolic.

Question 1. The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$. Which of the following are true?

(A) If the center is at the origin, the equation is $x^2 + y^2 = r^2$.

(B) $r$ must be a positive real number for a real circle.

(C) The equation represents the locus of all points $(x, y)$ that are at a fixed distance $r$ from the point $(h, k)$.

(D) $(x-h)^2 + (y-k)^2 = 0$ represents a single point $(h, k)$.

Question 2. The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. Which of the following are true?

(A) The coefficients of $x^2$ and $y^2$ are equal and non-zero (usually taken as 1 by dividing).

(B) There is no $xy$ term.

(C) The center is at $(-g, -f)$.

(D) The radius is $\sqrt{g^2 + f^2 - c}$, provided $g^2 + f^2 - c > 0$.

Question 3. Consider the equation $x^2 + y^2 + 4x - 6y + 13 = 0$. Which of the following are true?

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

(B) The center is $(-2, 3)$.

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

(D) The equation represents a point circle at $(-2, 3)$.

Question 4. The equation of a circle with endpoints of a diameter at $(x_1, y_1)$ and $(x_2, y_2)$ is $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$. Which of the following are true?

(A) The center of the circle is the midpoint of the diameter: $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$.

(B) The radius is half the distance between the endpoints of the diameter.

(C) The angle subtended by the diameter at any point on the circumference is $90^\circ$. This property is used to derive the diameter form.

(D) If $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (a, b)$, the equation is $x(x-a) + y(y-b) = 0$.

Question 5. A line $y = mx + c$ intersects a circle $x^2 + y^2 = r^2$. Which of the following statements regarding their intersection are true?

(A) If the distance from the center $(0, 0)$ to the line is less than the radius $r$, the line intersects the circle at two distinct points.

(B) The distance from $(0, 0)$ to $mx - y + c = 0$ is $\frac{|c|}{\sqrt{m^2 + (-1)^2}} = \frac{|c|}{\sqrt{m^2+1}}$.

(C) If $\frac{|c|}{\sqrt{m^2+1}} = r$, the line is tangent to the circle.

(D) If $\frac{|c|}{\sqrt{m^2+1}} > r$, the line does not intersect the circle.

Question 6. Consider two circles $C_1$ with center $O_1$ and radius $r_1$, and $C_2$ with center $O_2$ and radius $r_2$. Let $d$ be the distance between their centers $O_1O_2$. Which of the following are true about their relative position?

(A) If $d > r_1 + r_2$, the circles do not intersect and lie outside each other.

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

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

(D) If $d < |r_1 - r_2|$, one circle lies completely inside the other without touching.

Question 7. The equation $x^2 + y^2 - 2x - 2y + 1 = 0$ represents:

(A) A circle with center $(1, 1)$.

(B) $g = -1, f = -1, c = 1$. $g^2+f^2-c = (-1)^2+(-1)^2-1 = 1+1-1 = 1 > 0$. So it's a real circle.

(C) Radius is $\sqrt{1} = 1$.

(D) A circle tangent to both X and Y axes (since radius = $|h| = |k|$ and center is $(1,1)$).

Question 8. Which of the following can be equations of circles?

(A) $x^2 + y^2 + 6x - 4y + 10 = 0$. ($g=3, f=-2, c=10$. $g^2+f^2-c = 9+4-10=3>0$)

(B) $2x^2 + 2y^2 - 8x + 12y - 6 = 0$. (Divide by 2: $x^2+y^2-4x+6y-3=0$. $g=-2, f=3, c=-3$. $g^2+f^2-c = 4+9-(-3)=16>0$)

(C) $x^2 + y^2 + 2xy + 5 = 0$. (Has xy term)

(D) $x^2 - y^2 = 4$. (Coefficients of $x^2$ and $y^2$ are not equal)

Question 9. The equation of the circle concentric with $x^2 + y^2 - 4x + 6y - 3 = 0$ and passing through the point $(2, 3)$ is:

(A) Concentric means same center. Center of given circle is $(-g, -f) = (-(-2), -(3)) = (2, -3)$.

(B) The new circle has center $(2, -3)$.

(C) The radius of the new circle is the distance from $(2, -3)$ to $(2, 3)$: $\sqrt{(2-2)^2 + (3-(-3))^2} = \sqrt{0^2 + 6^2} = 6$.

(D) The equation is $(x-2)^2 + (y-(-3))^2 = 6^2 \implies (x-2)^2 + (y+3)^2 = 36$.

Question 10. A line intersects a circle at exactly one point. This means the line is tangent to the circle. This happens when the distance from the center to the line equals the radius.

(A) True

(B) False

(C) The discriminant of the quadratic equation formed by substituting the line into the circle equation is zero.

(D) The line could pass through the center, resulting in two intersection points (unless it's a point circle).

Question 1. The definition of a parabola involves a fixed point (focus) and a fixed line (directrix). For any point P on the parabola, its distance from the focus is equal to its distance from the directrix.

(A) True

(B) False

(C) This means the eccentricity $e=1$.

(D) The focus lies on the directrix.

Question 2. The standard equation of a parabola with vertex at the origin and axis along the Y-axis, opening upwards, is $x^2 = 4ay$. Which of the following are true about this parabola (assuming $a>0$)?

(A) The focus is at $(0, a)$.

(B) The directrix is the line $y = -a$.

(C) The axis is the Y-axis (equation $x=0$).

(D) The length of the latus rectum is $4a$.

Question 3. Consider the parabola $y^2 = 16x$. Which of the following are true?

(A) It is of the form $y^2 = 4ax$, with $4a = 16 \implies a = 4$.

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

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

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

Question 4. The length of the latus rectum of a parabola is defined as the length of the chord passing through the focus and perpendicular to the axis. For any standard parabola ($y^2=4ax$ or $x^2=4ay$), the length of the latus rectum is $4|a|$.

(A) True

(B) False

(C) It represents the width of the parabola at the focus.

(D) For $y^2 = -8x$, the length of the latus rectum is $|-8| = 8$.

Question 5. The equation of a parabola with vertex at $(h, k)$ and axis parallel to the X-axis is $(y-k)^2 = 4a(x-h)$ or $(y-k)^2 = -4a(x-h)$. Which of the following are true?

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

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

(C) The focus is at $(h+a, k)$ for $(y-k)^2 = 4a(x-h)$.

(D) The directrix is $x = h-a$ for $(y-k)^2 = 4a(x-h)$.

Question 6. The parametric equations of the parabola $y^2 = 4ax$ are $x = at^2, y = 2at$. Which of the following are true?

(A) $t$ is the parameter.

(B) By eliminating $t$, we get the Cartesian equation: $t = y/(2a) \implies x = a(y/(2a))^2 = a(y^2/(4a^2)) = y^2/(4a) \implies y^2 = 4ax$.

(C) Any point on the parabola can be represented by a value of $t$.

(D) These specific parametric equations only represent the upper half of the parabola ($y \ge 0$).

Question 7. The vertex of a parabola is the midpoint of the segment joining the focus and the point of intersection of the axis and the directrix.

(A) True

(B) False

(C) The vertex lies on the axis of the parabola.

(D) The vertex is the point on the parabola closest to the directrix.

Question 8. For the parabola $x^2 = -12y$, which of the following are true?

(A) It opens downwards.

(B) $4a = 12 \implies a = 3$.

(C) The focus is at $(0, -a) = (0, -3)$.

(D) The directrix is $y = a = 3$.

Question 9. The equation of the parabola with vertex at $(1, 2)$ and focus at $(3, 2)$ is:

(A) The axis is horizontal ($y=2$).

(B) The distance from vertex to focus is $a = |3-1| = 2$.

(C) Since focus is to the right of the vertex, it opens right. Equation is $(y-k)^2 = 4a(x-h)$.

(D) $(y-2)^2 = 4(2)(x-1) \implies (y-2)^2 = 8(x-1)$.

Question 10. A property of parabolas is that any ray parallel to the axis, upon reflection from the inner surface of the parabola, passes through the focus. This property is used in satellite dishes and headlights.

(A) True

(B) False

(C) This is the reflective property of the parabola.

(D) This property is unique to parabolas among conic sections.

Question 1. The definition of an ellipse involves two fixed points (foci, $F_1, F_2$) and a constant length ($2a$). For any point P on the ellipse, $PF_1 + PF_2 = 2a$. Which of the following are true?

(A) $2a$ is the length of the major axis.

(B) The distance between the foci is $2c$. The relationship is $a^2 = b^2 + c^2$ for $a>b$, or $b^2 = a^2 + c^2$ for $b>a$. Eccentricity $e = c/a$ or $c/b$.

(C) $2a$ must be greater than the distance between the foci ($2c$).

(D) If $F_1$ and $F_2$ coincide, the ellipse becomes a circle.

Question 2. The standard equation of an ellipse centered at the origin with major axis along the Y-axis ($b>a$) is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Which of the following are true about this ellipse?

(A) The vertices are at $(0, \pm b)$.

(B) The foci are at $(0, \pm c)$, where $c^2 = b^2 - a^2$.

(C) The length of the minor axis is $2a$.

(D) The directrices are $y = \pm b/e$.

Question 3. Consider the ellipse $\frac{x^2}{100} + \frac{y^2}{64} = 1$. Which of the following are true?

(A) $a^2 = 100 \implies a = 10$ (semi-major axis).

(B) $b^2 = 64 \implies b = 8$ (semi-minor axis).

(C) $c^2 = a^2 - b^2 = 100 - 64 = 36 \implies c = 6$. The foci are at $(\pm 6, 0)$.

(D) The eccentricity $e = c/a = 6/10 = 3/5$.

Question 4. The length of the latus rectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $2b^2/a$ (if $a>b$) or $2a^2/b$ (if $b>a$). Which of the following are true?

(A) The latus rectum passes through a focus.

(B) The endpoints of the latus rectum lie on the ellipse.

(C) It is perpendicular to the major axis.

(D) For $\frac{x^2}{25} + \frac{y^2}{9} = 1$, $a=5, b=3$, the length of the latus rectum is $2(3^2)/5 = 18/5$.

Question 5. The parametric equations $x = a\cos\theta, y = b\sin\theta$ represent an ellipse centered at the origin. Which of the following are true?

(A) $a$ and $b$ are the semi-major and semi-minor axis lengths (in some order).

(B) By eliminating $\theta$, we get $(x/a)^2 + (y/b)^2 = \cos^2\theta + \sin^2\theta = 1$, which is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

(C) $\theta$ is the eccentric angle.

(D) This representation covers all points on the ellipse.

Question 6. The vertices of an ellipse are the endpoints of the major axis. The co-vertices are the endpoints of the minor axis.

(A) True

(B) False

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

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

Question 7. The eccentricity of an ellipse is $e = c/a$ (where $a$ is semi-major axis length and $c$ is distance from center to focus). Which of the following are true about the eccentricity of an ellipse?

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

(B) If $e=0$, it's a circle.

(C) As $e$ approaches 1, the ellipse becomes more elongated.

(D) As $e$ approaches 0, the ellipse becomes more circular.

Question 8. The directrices of an ellipse are lines perpendicular to the major axis, located outside the ellipse. The equations are $x = \pm a/e$ (if major axis is X-axis) or $y = \pm b/e$ (if major axis is Y-axis).

(A) True

(B) False

(C) There are two directrices for an ellipse.

(D) The distance from any point P on the ellipse to a focus F is $e$ times the distance from P to the corresponding directrix L ($PF = e \cdot PD$).

Question 9. A property of ellipses is the reflective property: a ray originating from one focus and reflecting off the ellipse passes through the other focus. This property is used in designing whispering galleries.

(A) True

(B) False

(C) This property is related to the definition $PF_1 + PF_2 = 2a$.

(D) This property is also true for parabolas and hyperbolas.

Question 10. The equation of an ellipse centered at $(h, k)$ with major axis parallel to the X-axis is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (where $a>b$). Which of the following are true?

(A) The center is $(h, k)$.

(B) The foci are at $(h \pm c, k)$, where $c^2 = a^2 - b^2$.

(C) The vertices are at $(h \pm a, k)$.

(D) The directrices are $x = h \pm a/e$.

Question 1. The definition of a hyperbola involves two fixed points (foci, $F_1, F_2$) and a constant length ($2a$). For any point P on the hyperbola, $|PF_1 - PF_2| = 2a$. Which of the following are true?

(A) $2a$ is the length of the transverse axis.

(B) $2a$ must be less than the distance between the foci ($2c$).

(C) The distance between the foci is $2c$. The relationship is $c^2 = a^2 + b^2$.

(D) The eccentricity $e = c/a$.

Question 2. The standard equation of a hyperbola centered at the origin with transverse axis along the Y-axis is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$. Which of the following are true about this hyperbola?

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

(B) The foci are at $(0, \pm c)$, where $c^2 = a^2 + b^2$.

(C) The length of the transverse axis is $2a$.

(D) The equations of the asymptotes are $y = \pm (a/b)x$.

Question 3. Consider the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$. Which of the following are true?

(A) $a^2 = 9 \implies a = 3$.

(B) $b^2 = 16 \implies b = 4$.

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

(D) The eccentricity $e = c/a = 5/3$.

Question 4. The asymptotes of a hyperbola are lines that the hyperbola approaches as $|x|$ or $|y|$ tends to infinity. Which of the following are true?

(A) The asymptotes pass through the center of the hyperbola.

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

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

(D) The pair of asymptotes is a degenerate hyperbola.

Question 5. The length of the latus rectum of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $2b^2/a$. Which of the following are true?

(A) It passes through a focus.

(B) It is perpendicular to the transverse axis.

(C) Its endpoints lie on the hyperbola.

(D) For $\frac{y^2}{4} - \frac{x^2}{5} = 1$, $a=2, b=\sqrt{5}$, the length of the latus rectum is $2(\sqrt{5})^2/2 = 5$.

Question 6. The parametric equations $x = a\cosh t, y = b\sinh t$ or $x = a\sec\theta, y = b\tan\theta$ can represent a hyperbola. Which of the following are true?

(A) For $x = a\sec\theta, y = b\tan\theta$, $\sec^2\theta - \tan^2\theta = 1 \implies (x/a)^2 - (y/b)^2 = 1 \implies \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

(B) For $x = a\cosh t, y = b\sinh t$, $\cosh^2 t - \sinh^2 t = 1 \implies (x/a)^2 - (y/b)^2 = 1 \implies \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

(C) Parametric equations provide a way to represent points on the hyperbola using a single parameter.

(D) The choice of parameterization (trigonometric or hyperbolic) depends on convenience.

Question 7. The eccentricity of a hyperbola is $e = c/a$. Which of the following are true about the eccentricity of a hyperbola?

(A) $e > 1$.

(B) As $e$ increases, the hyperbola becomes 'wider' (asymptotes are steeper). (The angle between asymptotes is $2 \arctan(b/a)$. As $e=c/a=\sqrt{a^2+b^2}/a = \sqrt{1+(b/a)^2}$ increases, $b/a$ increases, so angle increases. So asymptotes become steeper, hyperbola becomes wider.)

(C) There is no upper bound on the value of $e$ for a hyperbola.

(D) If $e=1$, it's a parabola.

Question 8. The directrices of a hyperbola are lines perpendicular to the transverse axis, located between the vertices and the foci. The equations are $x = \pm a/e$ (if transverse axis is X-axis) or $y = \pm a/e$ (if transverse axis is Y-axis).

(A) True

(B) False (located between vertices and foci is incorrect, $a/e < a < c$)

(C) There are two directrices for a hyperbola.

(D) The distance from any point P on the hyperbola to a focus F is $e$ times the distance from P to the corresponding directrix L ($PF = e \cdot PD$).

Question 9. A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular. This occurs when $b=a$.

(A) True

(B) False

(C) The equation of a rectangular hyperbola centered at the origin is $x^2 - y^2 = a^2$ or $y^2 - x^2 = a^2$.

(D) The eccentricity of a rectangular hyperbola is $\sqrt{a^2+a^2}/a = \sqrt{2a^2}/a = \sqrt{2}$.

Question 10. The equation of a hyperbola centered at $(h, k)$ with transverse axis parallel to the Y-axis is $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$. Which of the following are true?

(A) The center is $(h, k)$.

(B) The foci are at $(h, k \pm c)$, where $c^2 = a^2 + b^2$.

(C) The vertices are at $(h, k \pm a)$.

(D) The equations of the asymptotes are $(y-k) = \pm (a/b)(x-h)$.

Question 1. Parametric equations allow us to describe a curve by expressing the coordinates of points on the curve as functions of a single independent variable (parameter). Which of the following are advantages of using parametric equations?

(A) Simplifies finding the slope of the tangent at a point.

(B) Can easily represent curves that are not functions of $x$ (e.g., a circle).

(C) Describes the movement along the curve over time if the parameter is time.

(D) Eliminating the parameter gives the Cartesian equation.

Question 2. Match the parametric equations to the corresponding conic section centered at the origin:

(A) (i) - (b)

(B) (ii) - (d)

(C) (iii) - (c)

(D) (iv) - (a)

Question 3. Which of the following are parametric equations for a circle centered at $(h, k)$ with radius $r$?

(A) $x = r\cos t, y = r\sin t$ (This is for origin center)

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

(C) $(x-h)^2 + (y-k)^2 = r^2$ (This is Cartesian form)

(D) $x = h + r\cos t, y = k + r\sin t$ allows recovering Cartesian form: $(x-h) = r\cos t, (y-k) = r\sin t \implies (x-h)^2 + (y-k)^2 = r^2(\cos^2 t + \sin^2 t) = r^2$.

Question 4. The parametric equations $x = 2t, y = t^2$ represent:

(A) Eliminate t: $t = x/2 \implies y = (x/2)^2 = x^2/4 \implies x^2 = 4y$.

(B) A parabola.

(C) A parabola with vertex at origin and axis along Y-axis, opening upwards.

(D) A parabola with latus rectum length 4.

Question 5. The parametric equations of an ellipse centered at the origin with major axis along the Y-axis ($b>a$) are $x = a\cos\theta, y = b\sin\theta$. Which of the following are true?

(A) $a$ is the semi-minor axis length.

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

(C) The domain of $\theta$ to cover the entire ellipse can be $[0, 2\pi)$.

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

Question 6. The parametric equations for the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ can be:

(A) $x = b\tan\theta, y = a\sec\theta$. (Since $a^2 \sec^2\theta - b^2 \tan^2\theta = a^2 (y/a)^2 - b^2 (x/b)^2 = y^2 - x^2$. Need $y^2/a^2 - x^2/b^2 = 1 \implies (a\sec\theta/a)^2 - (b\tan\theta/b)^2 = \sec^2\theta - \tan^2\theta = 1$. So $y=a\sec\theta, x=b\tan\theta$ is correct).

(B) $x = a\cosh t, y = b\sinh t$. (This gives $x^2/a^2 - y^2/b^2 = 1$)

(C) $x = b\sinh t, y = a\cosh t$. (Since $a^2\cosh^2 t - b^2\sinh^2 t = a^2 (y/a)^2 - b^2 (x/b)^2 = y^2 - x^2$. Need $y^2/a^2 - x^2/b^2=1 \implies (a\cosh t/a)^2 - (b\sinh t/b)^2 = \cosh^2 t - \sinh^2 t = 1$. So $y=a\cosh t, x=b\sinh t$ is correct).

(D) $x = a\cos t, y = b\sin t$. (This is for ellipse)

Question 7. The parametric representation is unique for a given conic section. (i.e. there is only one set of parametric equations for a specific conic).

(A) True

(B) False

(C) Multiple different parameterizations can represent the same curve (e.g., $x = t^2, y = 2t$ and $x = (u+1)^2, y = 2(u+1)$ represent the same parabola segment).

(D) Standard parametric forms are chosen for convenience and simplicity.

Question 8. Consider the parametric equations $x = 2 + 3\cos t, y = 1 + 3\sin t$. Which of the following are true?

(A) $(x-2) = 3\cos t, (y-1) = 3\sin t$.

(B) $(x-2)^2 + (y-1)^2 = (3\cos t)^2 + (3\sin t)^2 = 9(\cos^2 t + \sin^2 t) = 9$.

(C) The Cartesian equation is $(x-2)^2 + (y-1)^2 = 9$.

(D) This represents a circle centered at $(2, 1)$ with radius 3.

Question 9. The parametric equations $x = t, y = t^2$ represent a parabola. Which standard form does it match?

(A) $y = x^2$.

(B) $x^2 = 4ay$ where $4a=1 \implies a=1/4$.

(C) Vertex at origin, axis along Y-axis, opens upwards.

(D) Focus at $(0, 1/4)$, directrix $y = -1/4$.

Question 10. Which of the following parameters are commonly used in parametric equations of conics?

(A) $t$ (a general real parameter)

(B) $\theta$ (angle)

(C) Eccentric angle (for ellipse)

(D) Slope ($m$ in $y=mx+c$ or $y=mx+\sqrt{a^2m^2+b^2}$ for tangents)

Question 1. Coordinate geometry provides a powerful method to solve geometric problems by translating them into algebraic problems. Which of the following are examples of geometric properties that can be easily verified using coordinate methods?

(A) Collinearity of points.

(B) Perpendicularity of lines.

(C) Properties of triangles (e.g., isosceles, right-angled, equilateral).

(D) Properties of quadrilaterals (e.g., parallelogram, rhombus, rectangle, square).

Question 2. In 2D geometry, you can prove that the diagonals of a parallelogram bisect each other by showing that the midpoint of one diagonal coincides with the midpoint of the other diagonal, using the midpoint formula.

(A) True

(B) False

(C) This method uses coordinates of the vertices of the parallelogram.

(D) This property is unique to parallelograms among quadrilaterals.

Question 3. Which of the following geometric concepts can be effectively represented and manipulated using coordinate equations?

(A) Straight lines.

(B) Circles and other conic sections.

(C) Planes in 3D geometry.

(D) Spheres in 3D geometry.

Question 4. Coordinate geometry is widely used in various fields. Which of the following are real-world applications?

(A) GPS navigation (locating positions on Earth).

(B) Computer graphics and animation.

(C) Engineering design (e.g., bridge structures, car design).

(D) Physics (e.g., describing motion, trajectories).

Question 5. In 3D geometry, determining the equation of a plane passing through three non-collinear points is a common problem with applications in graphics and design.

(A) True

(B) False

(C) This often involves using vectors and coordinate information.

(D) The equation of a plane in 3D is typically of the form $Ax + By + Cz + D = 0$.

Question 6. Using coordinate geometry, one can derive formulas for geometric properties like distance, area, and volume (for simple shapes like boxes or tetrahedrons in 3D).

(A) True

(B) False

(C) The distance formula is derived using Pythagoras theorem in a coordinate system.

(D) The area of a polygon can be calculated using the coordinates of its vertices (e.g., shoelace formula).

Question 7. Which of the following problems can be solved efficiently using coordinate geometry?

(A) Finding the intersection point of two lines.

(B) Finding the equation of a circle passing through three given points.

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

(D) Proving that the angle in a semicircle is a right angle (by placing the center at the origin and using distance/slope).

Question 8. The concept of locus and its equation is a fundamental application of coordinate geometry, allowing us to describe curves and surfaces algebraically based on geometric conditions.

(A) True

(B) False

(C) The equation of a locus is an algebraic expression that is satisfied by the coordinates of every point on the locus and by no other points.

(D) Finding the equation of a locus often involves setting up an equation based on distance or other geometric relationships.

Question 9. In solid geometry (3D), coordinate methods are used to analyze the properties of points, lines, planes, and solid shapes like spheres, cubes, cones, etc.

(A) True

(B) False

(C) It provides a systematic way to represent 3D objects and their relationships algebraically.

(D) Equations of surfaces (like spheres, cylinders, cones) are derived using 3D coordinate geometry.

Question 10. Coordinate geometry simplifies proofs of many geometric theorems that might be complicated using only synthetic geometry (using diagrams and postulates).

(A) True

(B) False

(C) For example, proving properties of medians, altitudes, or angle bisectors in a triangle.

(D) Placing vertices at convenient coordinates (like origin or on axes) can simplify calculations.