Matching Items MCQs for Sub-Topics of Topic 6: Coordinate Geometry
Introduction to the Cartesian Coordinate System (Two Dimensions)
Question 1. Match the following terms related to the Cartesian coordinate system with their descriptions:
(i) Abscissa
(ii) Ordinate
(iii) Origin
(iv) X-axis
(a) The vertical coordinate of a point.
(b) The point $(0, 0)$.
(c) The horizontal axis.
(d) The horizontal coordinate of a point.
(A) (i)-(a), (ii)-(d), (iii)-(b), (iv)-(c)
(B) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(C) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(D) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
Answer:
Question 2. Match the quadrants with the signs of the coordinates $(x, y)$:
(i) First Quadrant
(ii) Second Quadrant
(iii) Third Quadrant
(iv) Fourth Quadrant
(a) $(-, +)$
(b) $(+, +)$
(c) $(+, -)$
(d) $(-, -)$
(A) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Question 3. Match the points with their location in the Cartesian plane:
(i) $(5, 0)$
(ii) $(0, -3)$
(iii) $(-2, 1)$
(iv) $(-4, -4)$
(a) Third Quadrant
(b) Negative Y-axis
(c) Positive X-axis
(d) Second Quadrant
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
Answer:
Question 4. Match the statements with the correct coordinate component property:
(i) A point $(x, 0)$ has...
(ii) A point $(0, y)$ has...
(iii) A point $(x, y)$ where $xy > 0$ is in...
(iv) A point $(x, y)$ where $xy < 0$ is in...
(a) The second or fourth quadrant.
(b) Abscissa zero.
(c) Ordinate zero.
(d) The first or third quadrant.
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 5. Match the points based on their symmetry with respect to the origin:
(i) $(2, 3)$ is symmetric to $(-2, -3)$ with respect to...
(ii) $(a, b)$ is symmetric to $(a, -b)$ with respect to...
(iii) $(a, b)$ is symmetric to $(-a, b)$ with respect to...
(iv) $(a, b)$ is symmetric to $(b, a)$ with respect to...
(a) The Y-axis.
(b) The line $y=x$.
(c) The X-axis.
(d) The origin.
(A) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(C) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Plotting Points in the Cartesian Plane
Question 1. Match the points with the correct sequence of movements from the origin:
(i) Plotting $(4, 1)$
(ii) Plotting $(-2, 5)$
(iii) Plotting $(3, -3)$
(iv) Plotting $(-1, -2)$
(a) 2 units left, 5 units up.
(b) 3 units right, 3 units down.
(c) 1 unit left, 2 units down.
(d) 4 units right, 1 unit up.
(A) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(B) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
(C) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(D) (i)-(a), (ii)-(d), (iii)-(b), (iv)-(c)
Answer:
Question 2. Match the description of a point's location with its coordinates:
(i) 5 units right of Y-axis, 2 units up from X-axis
(ii) On the negative Y-axis, 4 units from origin
(iii) In the third quadrant, $|x|=3, |y|=1$
(iv) On the X-axis, 6 units to the left of origin
(a) $(-3, -1)$
(b) $(5, 2)$
(c) $(-6, 0)$
(d) $(0, -4)$
(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(B) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 3. Match the properties of a point with its possible location:
(i) Abscissa is positive, Ordinate is negative
(ii) Abscissa is negative, Ordinate is positive
(iii) Abscissa and Ordinate are both negative
(iv) Point is equidistant from axes
(a) Second Quadrant
(b) Third Quadrant
(c) On the line $y = -x$ (excluding origin)
(d) Fourth Quadrant
(A) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(B) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
(C) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(D) (i)-(a), (ii)-(d), (iii)-(b), (iv)-(c)
Answer:
Question 4. Match the statement with the correct coordinate value:
(i) For point $(-5, 8)$, the abscissa is
(ii) For point $(10, -2)$, the ordinate is
(iii) For a point on the Y-axis, the abscissa is
(iv) For a point on the X-axis, the ordinate is
(a) $0$
(b) $-2$
(c) $-5$
(d) $0$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 5. Match the points based on their symmetry across a coordinate axis:
(i) $(a, b)$ symmetric to $(a, -b)$ across...
(ii) $(a, b)$ symmetric to $(-a, b)$ across...
(iii) $(2, -5)$ symmetric to $(2, 5)$ across...
(iv) $(-1, 4)$ symmetric to $(1, 4)$ across...
(a) Y-axis
(b) X-axis
(c) Y-axis
(d) X-axis
(A) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Distance Formula in Two Dimensions
Question 1. Match the pair of points with the distance between them:
(i) $(0, 0)$ and $(3, 4)$
(ii) $(1, 2)$ and $(1, 7)$
(iii) $(2, 3)$ and $(5, 3)$
(iv) $(-1, -1)$ and $(1, 1)$
(a) $5$ units
(b) $3$ units
(c) $5$ units
(d) $2\sqrt{2}$ units
(A) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(B) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(C) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Question 2. Match the geometric property with the condition involving distances:
(i) Points A, B, C are collinear
(ii) Triangle ABC is a right triangle at B
(iii) Triangle ABC is isosceles
(iv) Quadrilateral ABCD is a rhombus
(a) Two sides have equal length (e.g., AB = BC).
(b) $AB^2 + BC^2 = AC^2$ (Pythagoras Theorem).
(c) AB + BC = AC (or other permutations).
(d) All four sides are equal length (AB=BC=CD=DA).
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 3. Match the point with its distance from the origin:
(i) $(-5, 0)$
(ii) $(0, 7)$
(iii) $(-\sqrt{2}, \sqrt{2})$
(iv) $(1, 1)$
(a) $\sqrt{2}$
(b) $5$
(c) $2$
(d) $7$
(A) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(B) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(C) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 4. Match the point with its distance from the specified axis:
(i) $(4, -2)$ distance from X-axis
(ii) $(-5, 3)$ distance from Y-axis
(iii) $(0, 6)$ distance from X-axis
(iv) $(-7, 0)$ distance from Y-axis
(a) $3$
(b) $6$
(c) $2$
(d) $7$
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(C) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(D) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
Answer:
Question 5. Match the pair of points with the description of the line segment connecting them:
(i) $(2, 5)$ and $(2, -1)$
(ii) $(-3, 4)$ and $(5, 4)$
(iii) $(0, 0)$ and $(a, a)$
(iv) $(a, b)$ and $(-a, -b)$
(a) Passes through the origin.
(b) Parallel to the X-axis.
(c) Lies on the line $y=x$.
(d) Parallel to the Y-axis.
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Section Formula in Two Dimensions
Question 1. Match the formula type with its purpose:
(i) Midpoint Formula
(ii) Section Formula (Internal)
(iii) Section Formula (External)
(iv) Centroid Formula
(a) Find point dividing segment outside endpoints.
(b) Find center of gravity of a triangle.
(c) Find point halfway between two points.
(d) Find point dividing segment between endpoints.
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the pair of points with their midpoint:
(i) $(1, 5)$ and $(3, 7)$
(ii) $(-2, 4)$ and $(6, 0)$
(iii) $(0, 0)$ and $(a, b)$
(iv) $(-1, -1)$ and $(1, 1)$
(a) $(2, 6)$
(b) $(0, 0)$
(c) $(2, 2)$
(d) $(a/2, b/2)$
(A) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(B) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(C) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(D) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
Answer:
Question 3. Match the division ratio with the resulting point property (for segment AB, P divides in ratio $m:n$):
(i) $m:n = 1:1$ (internal)
(ii) $m:n = 2:1$ (internal)
(iii) $m:n = 1:2$ (external)
(iv) $m:n = 2:1$ (external)
(a) P is on the extension of AB beyond A.
(b) P is the midpoint of AB.
(c) P is on the extension of AB beyond B.
(d) P divides AB closer to B.
(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Question 4. Match the set of vertices with the coordinates of the centroid:
(i) $(0, 0), (3, 0), (0, 6)$
(ii) $(1, 2), (3, 5), (2, 1)$
(iii) $(-1, 0), (1, 0), (0, 3)$
(iv) $(a, b), (c, d), (e, f)$
(a) $(\frac{a+c+e}{3}, \frac{b+d+f}{3})$
(b) $(0, 1)$
(c) $(2, 8/3)$
(d) $(2, 2)$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the application of section formula with the problem type:
(i) Finding the midpoint of a line segment.
(ii) Determining the ratio in which a coordinate axis divides a line segment.
(iii) Finding the point that divides a segment into three equal parts.
(iv) Locating the intersection of medians in a triangle.
(a) Trisecting a segment.
(b) Finding the centroid.
(c) Using a specific ratio $1:1$ in the section formula.
(d) Setting one coordinate of the division point to zero and solving for the ratio.
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Area of a Triangle and Collinearity in 2D
Question 1. Match the vertices with the area of the triangle:
(i) $(0, 0), (1, 0), (0, 1)$
(ii) $(0, 0), (a, 0), (0, b)$
(iii) $(1, 2), (3, 4), (5, 6)$
(iv) $(1, 1), (1, 5), (4, 1)$
(a) $0$ sq units
(b) $\frac{1}{2}$ sq unit
(c) $\frac{1}{2}|ab|$ sq units
(d) $6$ sq units
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Question 2. Match the condition for collinearity with the method used:
(i) Area of triangle is zero
(ii) Slope of AB = Slope of BC
(iii) AB + BC = AC (sum of distances)
(iv) Points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$
(a) Using Distance Formula
(b) Using Slope Formula
(c) Condition using coordinates
(d) Using Area Formula
(A) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(B) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(C) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(D) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
Answer:
Question 3. Match the set of points with whether they are collinear or not:
(i) $(1, 1), (2, 2), (3, 3)$
(ii) $(1, 0), (0, 1), (1, 1)$
(iii) $(1, 2), (3, 4), (5, 6)$
(iv) $(-2, -3), (1, 0), (4, 3)$
(a) Not Collinear
(b) Collinear
(c) Collinear
(d) Collinear
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(C) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Question 4. Match the properties of the area calculation:
(i) The term $x_1(y_2-y_3)$
(ii) The factor $1/2$ in the formula
(iii) The absolute value sign in the formula
(iv) Area = 0
(a) Ensures area is non-negative.
(b) Indicates collinearity.
(c) Related to the determinant calculation.
(d) Geometric scaling factor.
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the set of points with the type of triangle they form (based on side lengths):
(i) $(0, 0), (3, 0), (0, 4)$
(ii) $(1, 1), (2, 3), (0, 5)$
(iii) $(0, 0), (1, 0), (\frac{1}{2}, \frac{\sqrt{3}}{2})$
(iv) $(1, 2), (5, 2), (3, 6)$
(a) Isosceles triangle
(b) Right-angled triangle
(c) Scalene triangle
(d) Equilateral triangle
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Triangle Centers in Two Dimensions
Question 1. Match the triangle center with the lines whose intersection defines it:
(i) Centroid
(ii) Incenter
(iii) Circumcenter
(iv) Orthocenter
(a) Perpendicular bisectors of sides.
(b) Medians.
(c) Angle bisectors.
(d) Altitudes.
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Question 2. Match the triangle center with the circle it is the center of:
(i) Incenter
(ii) Circumcenter
(iii) Centroid
(iv) Orthocenter
(a) Incircle (tangent to sides).
(b) Circumcircle (passes through vertices).
(c) Pedal circle (passes through feet of altitudes).
(d) Medial circle (passes through midpoints of sides).
(A) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
Answer:
Question 3. Match the type of triangle with the location of its circumcenter:
(i) Acute triangle
(ii) Right-angled triangle
(iii) Obtuse triangle
(iv) Equilateral triangle
(a) Outside the triangle.
(b) Inside the triangle.
(c) At the midpoint of the hypotenuse.
(d) Coincides with all other centers (Centroid, Incenter, Orthocenter).
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 4. Match the property with the triangle center:
(i) Equidistant from the vertices.
(ii) Equidistant from the sides.
(iii) Divides medians in $2:1$ ratio.
(iv) Intersection of lines perpendicular to sides from opposite vertices.
(a) Orthocenter
(b) Incenter
(c) Centroid
(d) Circumcenter
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the type of triangle with the location of its orthocenter:
(i) Acute triangle
(ii) Right-angled triangle
(iii) Obtuse triangle
(iv) Isosceles triangle (non-right)
(a) Outside the triangle.
(b) Inside the triangle.
(c) At the vertex with the right angle.
(d) Lies on the axis of symmetry (median/altitude from apex).
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Locus and its Equation
Question 1. Match the geometric condition with the type of locus:
(i) Equidistant from two fixed points.
(ii) Equidistant from a fixed point and a fixed line.
(iii) Sum of distances from two fixed points is constant.
(iv) Difference of distances from two fixed points is constant.
(a) Parabola.
(b) Hyperbola.
(c) Perpendicular bisector (straight line).
(d) Ellipse.
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 2. Match the geometric description of a locus with its equation type (assuming standard position):
(i) Locus of points at distance $r$ from origin.
(ii) Locus of points whose abscissa is constant $a$.
(iii) Locus of points whose ordinate is constant $b$.
(iv) Locus of points equidistant from X and Y axes.
(a) Pair of lines $y = \pm x$.
(b) Horizontal line $y=b$.
(c) Circle $x^2 + y^2 = r^2$.
(d) Vertical line $x=a$.
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 3. Match the given conditions for a point P$(x, y)$ with the type of locus:
(i) $PA = PB$ where A, B are fixed points.
(ii) $PF = PD$ where F is fixed point, D is on fixed line.
(iii) $PA^2 + PB^2 = \text{constant}$.
(iv) $x^2 + y^2 = k^2$ ($k$ constant).
(a) Circle (or point or imaginary circle).
(b) Parabola.
(c) Perpendicular bisector.
(d) Circle (centered at origin).
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 4. Match the description of a point's movement with the resulting locus:
(i) A point moving on the edge of a spinning wheel.
(ii) A point on a string kept taut while tracing a path around two pins.
(iii) The vertex of a cone rolling on a plane.
(iv) A point on a ladder sliding against a wall and floor.
(a) Cycloid (or related curve).
(b) Ellipse.
(c) Circle (if the point is the center of the base).
(d) Ellipse (if the point divides the ladder in a fixed ratio).
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 5. Match the property of a locus with its general nature:
(i) Described by a linear equation in $x$ and $y$.
(ii) Described by a quadratic equation in $x$ and $y$ (without $xy$ term, and coefficients of $x^2$ and $y^2$ are equal).
(iii) Described by an equation resulting from $PF = e \cdot PD$ with $e=1$.
(iv) The set of points satisfying the condition.
(a) Parabola.
(b) The locus itself.
(c) Straight line.
(d) Circle.
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Transformation of Coordinates: Shifting of Origin
Question 1. Match the original point with its new coordinates after shifting the origin to $(1, 2)$:
(i) $(3, 5)$
(ii) $(1, 2)$
(iii) $(-1, 0)$
(iv) $(1, 1)$
(a) $(-2, -2)$
(b) $(2, 3)$
(c) $(0, 0)$
(d) $(0, -1)$
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Question 2. Match the equation transformation with the shift in origin required:
(i) $x^2 + y^2 - 6x + 8y = 0$ to $X^2 + Y^2 = r^2$
(ii) $y = mx + c$ to $Y = mX$ (if the line passes through the new origin)
(iii) $y^2 - 4y - 4x + 8 = 0$ to $Y^2 = 4aX$
(iv) $x^2 + y^2 = r^2$ to $(X-h)^2 + (Y-k)^2 = r^2$
(a) $(3, -4)$
(b) $(h, k)$
(c) $(3, m(3)+c)$ (any point on the original line)
(d) $(1, 2)$
(A) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(B) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(C) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 3. Match the original equation with the transformed equation after shifting the origin to $(h, k)$:
(i) $x = 5$
(ii) $y = -3$
(iii) $2x + 3y = 6$
(iv) $x^2 + y^2 = 1$
(a) $(X+h)^2 + (Y+k)^2 = 1$
(b) $Y+k = -3$
(c) $X+h = 5$
(d) $2(X+h) + 3(Y+k) = 6$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 4. Match the properties with the transformation of shifting the origin:
(i) Distance between two points
(ii) Equation of a curve
(iii) Shape of a curve
(iv) Area of a triangle
(a) Changes.
(b) Remains invariant.
(c) Changes (algebraic expression).
(d) Remains invariant (geometric property).
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(b)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 5. Match the new coordinate $(X, Y)$ with the shift $(h, k)$ given the original coordinate $(x, y) = (5, 7)$:
(i) Shift to $(1, 1)$
(ii) Shift to $(5, 7)$
(iii) Shift to $(0, 0)$
(iv) Shift to $(2, 3)$
(a) $(3, 4)$
(b) $(5, 7)$
(c) $(0, 0)$
(d) $(4, 6)$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Straight Lines: Slope and Angle Between Lines
Question 1. Match the line equation with its slope:
(i) $y = 3x - 5$
(ii) $2x + y = 7$
(iii) $x - 4y = 1$
(iv) $y = 6$
(a) $1/4$
(b) $0$
(c) $3$
(d) $-2$
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 2. Match the slope value with the angle the line makes with the positive X-axis:
(i) $m = 1$
(ii) $m = 0$
(iii) $m = \text{undefined}$
(iv) $m = -1$
(a) $135^\circ$
(b) $90^\circ$
(c) $45^\circ$
(d) $0^\circ$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(C) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 3. Match the relationship between slopes of two non-vertical lines with their geometric relationship:
(i) $m_1 = m_2$
(ii) $m_1 m_2 = -1$
(iii) $\tan\theta = |\frac{m_1-m_2}{1+m_1m_2}|$ where $\theta$ is angle between lines
(iv) Slopes calculated from three collinear points A, B, C are equal
(a) Condition for perpendicular lines.
(b) Formula for angle between lines.
(c) Condition for parallel lines.
(d) Condition for collinear points.
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 4. Match the line with the slope of a line perpendicular to it:
(i) $y = 2x + 1$
(ii) $y = -x + 5$
(iii) $y = 4$
(iv) $x = -2$
(a) $0$
(b) $1$
(c) $-1/2$
(d) Undefined
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 5. Match the pair of lines with the angle between them:
(i) $y = x$ and $y = -x$
(ii) $y = 2x + 1$ and $y = 2x - 5$
(iii) $y = x$ and $x = 0$
(iv) $y = \sqrt{3}x$ and $y = -\frac{1}{\sqrt{3}}x$
(a) $0^\circ$
(b) $90^\circ$
(c) $45^\circ$
(d) $90^\circ$
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(C) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Straight Lines: Various Forms of Equations
Question 1. Match the equation form with its characteristic information:
(i) Slope-intercept form ($y = mx + c$)
(ii) Point-slope form ($y - y_1 = m(x - x_1)$)
(iii) Intercept form ($\frac{x}{a} + \frac{y}{b} = 1$)
(iv) Normal form ($x \cos\alpha + y \sin\alpha = p$)
(a) Slope and a point on the line.
(b) Perpendicular distance from origin and angle of normal.
(c) X-intercept and Y-intercept.
(d) Slope and Y-intercept.
(A) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(B) (i)-(a), (ii)-(d), (iii)-(b), (iv)-(c)
(C) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
(D) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
Answer:
Question 2. Match the line description with its equation:
(i) Parallel to X-axis, 3 units above origin.
(ii) Parallel to Y-axis, 2 units left of origin.
(iii) Passing through origin with slope -1.
(iv) Y-axis itself.
(a) $x = 0$
(b) $y = 3$
(c) $x = -2$
(d) $y = -x$
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Question 3. Match the information with the most suitable form of equation to use:
(i) Given two points on the line.
(ii) Given slope and Y-intercept.
(iii) Given a point and slope.
(iv) Given X and Y intercepts.
(a) Slope-intercept form.
(b) Two-point form.
(c) Intercept form.
(d) Point-slope form.
(A) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Question 4. Match the equation of the line with its specific type/property:
(i) $y = 5$
(ii) $x = -2$
(iii) $y = x$
(iv) $y = 0$
(a) Line passing through origin with slope 1.
(b) Horizontal line.
(c) Vertical line.
(d) X-axis.
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Question 5. Match the line equation with its equivalent form:
(i) $y = 2x - 4$
(ii) $y - 1 = 3(x - 2)$
(iii) $\frac{x}{5} + \frac{y}{-3} = 1$
(iv) $x \cos 45^\circ + y \sin 45^\circ = \sqrt{2}$
(a) $x + y = 2$
(b) $y = 3x - 5$
(c) $3x - 5y = 15$
(d) $2x - y = 4$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
Answer:
Straight Lines: General Equation and Related Concepts
Question 1. Match the general equation $Ax + By + C = 0$ with the type of line based on coefficients:
(i) $A = 0, B \neq 0$
(ii) $A \neq 0, B = 0$
(iii) $C = 0$
(iv) $A, B, C$ are non-zero
(a) Vertical line.
(b) Line not passing through the origin.
(c) Horizontal line.
(d) Line passing through the origin.
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 2. Match the equation $3x + 4y - 12 = 0$ with its different forms/properties:
(i) Slope
(ii) Y-intercept
(iii) X-intercept
(iv) Intercept form
(a) $4$
(b) $-3/4$
(c) $\frac{x}{4} + \frac{y}{3} = 1$
(d) $3$
(A) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 3. Match the system of linear equations with the number of points of intersection:
(i) $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
(ii) $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
(iii) $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
(iv) Lines are perpendicular
(a) Infinitely many (coincident lines).
(b) No intersection (parallel and distinct lines).
(c) Exactly one intersection (intersecting lines).
(d) Exactly one intersection.
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 4. Match the equation $Ax + By + C = 0$ with the formula for a property derived from it:
(i) Slope (if $B \neq 0$)
(ii) Y-intercept (if $B \neq 0$)
(iii) X-intercept (if $A \neq 0$)
(iv) Perpendicular distance from origin
(a) $-C/A$
(b) $\frac{|C|}{\sqrt{A^2+B^2}}$
(c) $-A/B$
(d) $-C/B$
(A) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(B) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the system of linear equations with its solution (point of intersection):
(i) $x + y = 5, x - y = 1$
(ii) $2x + y = 4, x - y = 2$
(iii) $x = 3, y = 2$
(iv) $y = x, y = -x + 0$
(a) $(0, 0)$
(b) $(2, 0)$
(c) $(3, 2)$
(d) $(3, 2)$
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Straight Lines: Distance and Family of Lines
Question 1. Match the point and line with the distance between them:
(i) $(0, 0)$ and $3x - 4y + 5 = 0$
(ii) $(1, 2)$ and $x + y - 1 = 0$
(iii) $(2, -1)$ and $x = 5$
(iv) $(-3, 4)$ and $y = -2$
(a) $5/\sqrt{2}$
(b) $3$
(c) $1$
(d) $6$
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 2. Match the pair of parallel lines with the distance between them:
(i) $x + y = 1, x + y = 3$
(ii) $2x - y = 5, 4x - 2y = 10$
(iii) $y = 2x + 1, y = 2x + 6$
(iv) $3x + 4y = 10, 3x + 4y = -5$
(a) $\sqrt{2}$
(b) $0$ (coincident lines)
(c) $\sqrt{5}$
(d) $3$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 3. Match the family of lines $L_1 + \lambda L_2 = 0$ with the point it passes through:
(i) $(x + y - 1) + \lambda(x - y - 1) = 0$
(ii) $x + \lambda y = 0$
(iii) $(2x + 3y - 5) + \lambda y = 0$
(iv) $(x - 1) + \lambda (y - 2) = 0$
(a) Origin $(0, 0)$
(b) $(1, 0)$
(c) $(5/2, 0)$
(d) $(1, 2)$
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 4. Match the concept with its representation/formula:
(i) Distance between parallel lines $Ax+By+C_1=0$ and $Ax+By+C_2=0$
(ii) Family of lines through $(x_0, y_0)$
(iii) Perpendicular distance from $(x_0, y_0)$ to $Ax+By+C=0$
(iv) Equation of any line through intersection of $L_1=0, L_2=0$
(a) $|Ax_0+By_0+C|/\sqrt{A^2+B^2}$
(b) $L_1 + \lambda L_2 = 0$
(c) $y-y_0 = m(x-x_0)$
(d) $|C_1-C_2|/\sqrt{A^2+B^2}$
(A) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the equation with the number of lines it represents:
(i) $y = 2x + \lambda$
(ii) $L_1 + \lambda L_2 = 0$ (for varying $\lambda$)
(iii) $Ax + By + C = 0$ (A, B, C fixed, A or B non-zero)
(iv) $(x-1)(x-2) = 0$
(a) A single line.
(b) A family of parallel lines.
(c) A family of lines passing through a fixed point (if L1, L2 intersect).
(d) Two parallel lines.
(A) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Introduction to Three-Dimensional Geometry
Question 1. Match the plane with its equation in 3D Cartesian coordinates:
(i) XY-plane
(ii) YZ-plane
(iii) XZ-plane
(iv) Plane parallel to XY-plane at $z=5$
(a) $y = 0$
(b) $x = 0$
(c) $z = 0$
(d) $z = 5$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 2. Match the point $(x, y, z)$ with its distance from the specified coordinate element:
(i) Distance from XY-plane
(ii) Distance from YZ-plane
(iii) Distance from X-axis
(iv) Distance from origin
(a) $\sqrt{x^2+y^2+z^2}$
(b) $|x|$
(c) $\sqrt{y^2+z^2}$
(d) $|z|$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the properties of points in 3D space:
(i) Points on the X-axis
(ii) Points on the YZ-plane
(iii) Points equidistant from XY and XZ planes
(iv) Origin in 3D
(a) $(0, 0, 0)$
(b) Points $(x, 0, 0)$
(c) Points $(0, y, z)$
(d) Points $(x, y, z)$ with $|y|=|z|$
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Question 4. Match the point $(a, b, c)$ with its projection onto the specified coordinate plane:
(i) Projection onto XY-plane
(ii) Projection onto YZ-plane
(iii) Projection onto XZ-plane
(iv) Projection onto X-axis
(a) $(a, 0, c)$
(b) $(0, b, c)$
(c) $(a, b, 0)$
(d) $(a, 0, 0)$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 5. Match the type of geometric object in 3D with the number of variables needed to describe its points (ignoring boundaries like ends of segments):
(i) A point
(ii) A line
(iii) A plane
(iv) A region of space (volume)
(a) Three parameters (e.g., $x, y, z$).
(b) One parameter (e.g., $t$).
(c) Zero parameters (fixed coordinates).
(d) Two parameters (e.g., $u, v$).
(A) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Distance Formula in Three Dimensions
Question 1. Match the pair of points with the distance between them in 3D:
(i) $(0, 0, 0)$ and $(1, 2, 2)$
(ii) $(1, 0, 0)$ and $(4, 0, 4)$
(iii) $(1, 1, 1)$ and $(2, 2, 2)$
(iv) $(-1, -2, -3)$ and $(1, 2, 3)$
(a) $3$ units
(b) $5$ units
(c) $\sqrt{3}$ units
(d) $2\sqrt{14}$ units
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 2. Match the point with its distance from the specified axis in 3D:
(i) $(3, 4, 5)$ distance from X-axis
(ii) $(-1, 2, -3)$ distance from Y-axis
(iii) $(6, -8, 0)$ distance from Z-axis
(iv) $(1, 1, 1)$ distance from X-axis
(a) $\sqrt{10}$
(b) $\sqrt{41}$
(c) $10$
(d) $\sqrt{2}$
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 3. Match the geometric property in 3D with the condition using distance formula:
(i) Points A, B, C are collinear
(ii) Triangle ABC is equilateral
(iii) Point P is equidistant from points A and B
(iv) Vertices A, B, C, D form a rhombus
(a) $PA = PB$
(b) AB + BC = AC (or permutations)
(c) All sides AB, BC, CD, DA are equal length.
(d) AB = BC = AC.
(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(B) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
Answer:
Question 4. Match the specific point on a coordinate axis with its distance from $(1, 2, 3)$:
(i) $(1, 0, 0)$
(ii) $(0, 2, 0)$
(iii) $(0, 0, 3)$
(iv) $(1, 2, 0)$ (projection on XY-plane)
(a) $\sqrt{13}$
(b) $\sqrt{10}$
(c) $\sqrt{5}$
(d) $3$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
Answer:
Question 5. Match the components used in the 3D distance formula $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ with their meaning:
(i) $x_2-x_1$
(ii) $(y_2-y_1)^2$
(iii) Sum of three terms under square root
(iv) The square root of the sum
(a) Squared difference in y-coordinates.
(b) Difference in x-coordinates.
(c) Square of the distance.
(d) The distance itself.
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Section Formula in Three Dimensions
Question 1. Match the line segment endpoints with their midpoint in 3D:
(i) $(1, 2, 3)$ and $(3, 4, 5)$
(ii) $(-1, -1, -1)$ and $(1, 1, 1)$
(iii) $(0, 0, 0)$ and $(a, b, c)$
(iv) $(2, 0, 0)$ and $(0, 4, 0)$
(a) $(0, 0, 0)$
(b) $(2, 3, 4)$
(c) $(1, 2, 0)$
(d) $(a/2, b/2, c/2)$
(A) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Question 2. Match the description of division with the type of section formula:
(i) Point P is between A and B.
(ii) Point Q is on the extension of AB.
(iii) Ratio of division is $1:1$.
(iv) Ratio of division is $m:n$ where $m, n$ have opposite signs.
(a) External Section Formula.
(b) Midpoint Formula.
(c) Internal Section Formula.
(d) External Section Formula (alternative representation).
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 3. Match the set of vertices with the type of average used for the centroid:
(i) Triangle vertices $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$
(ii) Tetrahedron vertices $(x_1, \dots), \dots, (x_4, \dots)$
(iii) Line segment endpoints $(x_1, y_1, z_1), (x_2, y_2, z_2)$
(iv) Three collinear points
(a) Average of 4 points (for centroid).
(b) Average of 3 points (for centroid).
(c) Average of 2 points (for midpoint).
(d) Centroid is undefined as it's a line segment.
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 4. Match the ratio of internal division $m:n$ for segment AB with the relative position of the division point P:
(i) $m > n$
(ii) $m = n$
(iii) $m < n$
(iv) $m = 0$ or $n = 0$ (and the other is non-zero)
(a) P is the midpoint.
(b) P coincides with one of the endpoints (A or B).
(c) P is closer to B.
(d) P is closer to A.
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 5. Match the point with the ratio it divides the segment A$(1, 2, 3)$ and B$(7, 8, 9)$ internally:
(i) $(3, 4, 5)$
(ii) $(4, 5, 6)$
(iii) $(5, 6, 7)$
(iv) $(1, 2, 3)$
(a) $2:1$
(b) $1:2$
(c) $0:1$
(d) $1:1$
(A) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Introduction to Conic Sections
Question 1. Match the conic section type with its eccentricity ($e$):
(i) Circle
(ii) Ellipse
(iii) Parabola
(iv) Hyperbola
(a) $e > 1$
(b) $e = 1$
(c) $0 \le e < 1$
(d) $e = 0$
(A) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(D) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
Answer:
Question 2. Match the geometric definition with the term:
(i) A fixed point used in the definition of a conic.
(ii) A fixed line used in the definition of a conic.
(iii) The ratio of distances PF/PD.
(iv) Intersection of the conic and its axis.
(a) Directrix.
(b) Vertex.
(c) Focus.
(d) Eccentricity.
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 3. Match the cutting plane's angle ($\beta$) relative to the cone's axis with the conic section formed (where $\alpha$ is semi-vertical angle):
(i) $\beta = 90^\circ$
(ii) $\alpha < \beta < 90^\circ$
(iii) $\beta = \alpha$
(iv) $0 \le \beta < \alpha$
(a) Hyperbola.
(b) Parabola.
(c) Circle.
(d) Ellipse.
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 4. Match the condition for the general second-degree equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ to represent a non-degenerate conic (assuming axes are aligned, so $B=0$):
(i) $A=C$
(ii) $A=0$ or $C=0$ (but not both)
(iii) $A \neq C$, $A, C$ have same sign
(iv) $A, C$ have opposite signs
(a) Parabola.
(b) Hyperbola.
(c) Circle.
(d) Ellipse.
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 5. Match the degenerate conic section with the geometric setup:
(i) Plane passes through vertex, parallel to a generator.
(ii) Plane passes through vertex, perpendicular to axis.
(iii) Plane passes through vertex, contains the axis.
(iv) Plane passes through vertex, makes angle $\beta > \alpha$ with axis.
(a) Pair of intersecting lines.
(b) A single point.
(c) A single line.
(d) A single point.
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
Answer:
Circle in Coordinate Geometry
Question 1. Match the equation of the circle with its center:
(i) $x^2 + y^2 = 9$
(ii) $(x-1)^2 + (y+2)^2 = 4$
(iii) $x^2 + y^2 - 4x + 6y - 3 = 0$
(iv) $2x^2 + 2y^2 + 8x - 12y = 0$
(a) $(-2, 3)$
(b) $(0, 0)$
(c) $(1, -2)$
(d) $(2, -3)$
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(D) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
Answer:
Question 2. Match the equation of the circle with its radius:
(i) $x^2 + y^2 = 16$
(ii) $(x+3)^2 + (y-1)^2 = 25$
(iii) $x^2 + y^2 - 2x - 2y + 1 = 0$
(iv) $x^2 + y^2 + 6x + 8y = 0$
(a) $1$
(b) $5$
(c) $4$
(d) $5$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 3. Match the condition for a second degree equation $x^2 + y^2 + 2gx + 2fy + c = 0$ with the type of circle:
(i) $g^2 + f^2 - c > 0$
(ii) $g^2 + f^2 - c = 0$
(iii) $g^2 + f^2 - c < 0$
(iv) $c = 0$
(a) Imaginary circle.
(b) Point circle.
(c) Real circle.
(d) Circle passing through the origin.
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 4. Match the relative position of a line and a circle with the condition on the distance ($d$) from the center to the line and the radius ($r$):
(i) Line intersects the circle at two distinct points.
(ii) Line is tangent to the circle.
(iii) Line does not intersect the circle.
(iv) Line passes through the center.
(a) $d > r$
(b) $d = 0$
(c) $d = r$
(d) $d < r$
(A) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(D) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
Answer:
Question 5. Match the relative position of two circles ($C_1, C_2$) with the condition on the distance ($d$) between their centers and their radii ($r_1, r_2$):
(i) Circles touch externally.
(ii) Circles touch internally.
(iii) Circles intersect at two distinct points.
(iv) One circle lies inside the other without touching.
(a) $d < |r_1 - r_2|$
(b) $d = r_1 + r_2$
(c) $|r_1 - r_2| < d < r_1 + r_2$
(d) $d = |r_1 - r_2|$
(A) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(B) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(C) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Parabola in Coordinate Geometry
Question 1. Match the standard equation of the parabola (with $a>0$) with its property:
(i) $y^2 = 4ax$
(ii) $y^2 = -4ax$
(iii) $x^2 = 4ay$
(iv) $x^2 = -4ay$
(a) Opens downwards.
(b) Opens upwards.
(c) Opens to the right.
(d) Opens to the left.
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(C) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
(D) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
Answer:
Question 2. Match the standard parabola equation (with $a>0$) with the coordinates of its focus:
(i) $y^2 = 4ax$
(ii) $y^2 = -4ax$
(iii) $x^2 = 4ay$
(iv) $x^2 = -4ay$
(a) $(0, -a)$
(b) $(a, 0)$
(c) $(0, a)$
(d) $(-a, 0)$
(A) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
(B) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(C) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(D) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the standard parabola equation (with $a>0$) with the equation of its directrix:
(i) $y^2 = 4ax$
(ii) $y^2 = -4ax$
(iii) $x^2 = 4ay$
(iv) $x^2 = -4ay$
(a) $x = a$
(b) $y = a$
(c) $x = -a$
(d) $y = -a$
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 4. Match the standard parabola equation (with $a>0$) with the length of its latus rectum:
(i) $y^2 = 4ax$
(ii) $y^2 = -4ax$
(iii) $x^2 = 4ay$
(iv) $x^2 = -4ay$
(a) $4a$
(b) $4a$
(c) $4a$
(d) $4a$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(a), (iii)-(a), (iv)-(a)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer:
Question 5. Match the parabola equation with its vertex:
(i) $y^2 = 8x$
(ii) $x^2 = -12y$
(iii) $(y-1)^2 = 4(x+2)$
(iv) $(x+3)^2 = -16(y-4)$
(a) $(0, 0)$
(b) $(-3, 4)$
(c) $(-2, 1)$
(d) $(0, 0)$
(A) (i)-(a), (ii)-(d), (iii)-(c), (iv)-(b)
(B) (i)-(a), (ii)-(d), (iii)-(b), (iv)-(c)
(C) (i)-(d), (ii)-(a), (iii)-(c), (iv)-(b)
(D) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)
Answer:
Ellipse in Coordinate Geometry
Question 1. Match the standard equation of the ellipse (centered at origin) with the orientation of its major axis:
(i) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$
(ii) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$
(iii) $x^2 + y^2 = r^2$
(iv) $\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1$
(a) Major axis along Y-axis.
(b) No distinct major/minor axis (circle).
(c) Major axis along X-axis.
(d) No distinct major/minor axis (circle).
(A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 2. Match the standard ellipse equation (centered at origin) with the length of its major axis:
(i) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$
(ii) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$
(iii) $\frac{x^2}{25} + \frac{y^2}{9} = 1$
(iv) $\frac{x^2}{4} + \frac{y^2}{36} = 1$
(a) $2b$
(b) $12$
(c) $2a$
(d) $10$
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 3. Match the standard ellipse equation (centered at origin) with its eccentricity formula:
(i) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$
(ii) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$
(iii) $b^2 = a^2(1-e^2)$
(iv) $a^2 = b^2(1-e^2)$
(a) $e = \sqrt{1 - b^2/a^2}$
(b) $e = \sqrt{1 - a^2/b^2}$
(c) $e = \sqrt{1 - b^2/a^2}$
(d) $e = \sqrt{1 - a^2/b^2}$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 4. Match the standard ellipse equation (centered at origin) with the length of its latus rectum:
(i) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$
(ii) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $b > a$
(iii) $\frac{x^2}{25} + \frac{y^2}{16} = 1$
(iv) $\frac{x^2}{9} + \frac{y^2}{36} = 1$
(a) $2b^2/a$
(b) $2a^2/b$
(c) $32/5$
(d) $3$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 5. Match the standard ellipse equation (centered at origin, $a>b$) with its directrices equations:
(i) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
(ii) $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$
(iii) Ellipse with major axis along X-axis, eccentricity $e$
(iv) Ellipse with major axis along Y-axis, eccentricity $e$
(a) $y = \pm a/e$
(b) $x = \pm a/e$
(c) $x = \pm a/e$
(d) $y = \pm a/e$
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Hyperbola in Coordinate Geometry
Question 1. Match the standard equation of the hyperbola (centered at origin) with the orientation of its transverse axis:
(i) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
(ii) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
(iii) $x^2 - y^2 = a^2$
(iv) $y^2 - x^2 = b^2$
(a) Transverse axis along Y-axis.
(b) Transverse axis along X-axis.
(c) Transverse axis along X-axis.
(d) Transverse axis along Y-axis.
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 2. Match the standard hyperbola equation (centered at origin) with the equations of its asymptotes:
(i) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
(ii) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
(iii) $x^2 - y^2 = k$
(iv) $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$
(a) $y = \pm (b/a)x$
(b) $y = \pm (a/b)x$
(c) $y = \pm x$
(d) $y-k = \pm (b/a)(x-h)$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(C) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(D) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
Answer:
Question 3. Match the standard hyperbola equation (centered at origin) with its eccentricity formula:
(i) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
(ii) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
(iii) $c^2 = a^2 + b^2$
(iv) $c^2 = a^2 + b^2$
(a) $e = \sqrt{1 + b^2/a^2}$
(b) $e = \sqrt{1 + b^2/a^2}$
(c) $e = \sqrt{1 + a^2/b^2}$
(d) $e = \sqrt{1 + a^2/b^2}$
(A) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(B) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
Answer:
Question 4. Match the standard hyperbola equation (centered at origin) with the length of its latus rectum:
(i) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
(ii) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
(iii) $\frac{x^2}{16} - \frac{y^2}{9} = 1$
(iv) $\frac{y^2}{25} - \frac{x^2}{144} = 1$
(a) $2b^2/a$
(b) $2a^2/b$
(c) $4.5$
(d) $57.6$
(A) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
Answer:
Question 5. Match the standard hyperbola equation (centered at origin) with its directrices equations:
(i) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
(ii) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
(iii) Hyperbola with transverse axis along X-axis, eccentricity $e$
(iv) Hyperbola with transverse axis along Y-axis, eccentricity $e$
(a) $y = \pm a/e$
(b) $x = \pm a/e$
(c) $x = \pm a/e$
(d) $y = \pm a/e$
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Parametric Equations of Conics (Consolidated)
Question 1. Match the parametric equation set with the corresponding Cartesian equation (centered at origin):
(i) $x = a\cos t, y = a\sin t$
(ii) $x = at^2, y = 2at$
(iii) $x = a\cos \theta, y = b\sin \theta$
(iv) $x = a\sec \theta, y = b\tan \theta$
(a) $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
(b) $x^2 + y^2 = a^2$
(c) $y^2 = 4ax$
(d) $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
(A) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(B) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(C) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
(D) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
Answer:
Question 2. Match the conic section type with a standard parametric representation:
(i) Circle (radius a)
(ii) Parabola ($y^2=4ax$)
(iii) Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$)
(iv) Hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$)
(a) $x = a\sec\theta, y = b\tan\theta$
(b) $x = at^2, y = 2at$
(c) $x = a\cos\theta, y = b\sin\theta$
(d) $x = a\cos t, y = a\sin t$
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Question 3. Match the parametric form with a point it can represent (for $a=1, b=1$):
(i) $x = \cos t, y = \sin t$ (for $t=0$)
(ii) $x = t^2, y = 2t$ (for $t=1$)
(iii) $x = \cos \theta, y = \sin \theta$ (for $\theta=\pi/2$)
(iv) $x = \sec \theta, y = \tan \theta$ (for $\theta=0$)
(a) $(0, 1)$
(b) $(1, 2)$
(c) $(1, 0)$
(d) $(1, 0)$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(b), (ii)-(c), (iii)-(d), (iv)-(a)
(C) (i)-(c), (ii)-(b), (iii)-(d), (iv)-(a)
(D) (i)-(b), (ii)-(c), (iii)-(a), (iv)-(d)
Answer:
Question 4. Match the shifted parametric equation with its center:
(i) $x = 1 + \cos t, y = 2 + \sin t$
(ii) $x = -1 + t^2, y = 3 + 2t$
(iii) $x = 4 + 2\cos \theta, y = 5 + 3\sin \theta$
(iv) $x = -2 + \sec \theta, y = 0 + \tan \theta$
(a) $(4, 5)$
(b) $(-1, 3)$
(c) $(1, 2)$
(d) $(-2, 0)$
(A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
Answer:
Question 5. Match the Cartesian equation with a possible set of parametric equations (centered at origin):
(i) $y^2 = 16x$
(ii) $x^2 + y^2 = 4$
(iii) $\frac{x^2}{9} + \frac{y^2}{4} = 1$
(iv) $\frac{y^2}{1} - \frac{x^2}{1} = 1$
(a) $x = 2\cos t, y = 2\sin t$
(b) $x = \tan \theta, y = \sec \theta$
(c) $x = 4t^2, y = 8t$
(d) $x = 3\cos \theta, y = 2\sin \theta$
(A) (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)
(B) (i)-(a), (ii)-(c), (iii)-(b), (iv)-(d)
(C) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Applications of Coordinate Geometry
Question 1. Match the geometric problem with the coordinate geometry concept/formula used to solve it:
(i) Prove diagonals of a rectangle are equal.
(ii) Find the point dividing a segment in a given ratio.
(iii) Check if three points lie on a line.
(iv) Find the center of the circumcircle of a triangle.
(a) Section Formula.
(b) Distance Formula.
(c) Area of Triangle Formula or Slope Formula.
(d) Finding the intersection of perpendicular bisectors (using equations of lines).
(A) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(B) (i)-(a), (ii)-(b), (iii)-(d), (iv)-(c)
(C) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(D) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
Answer:
Question 2. Match the real-world scenario with the relevant coordinate geometry concept:
(i) Determining a location on a map.
(ii) Calculating the shortest route between two cities given their coordinates.
(iii) Designing the path of a satellite.
(iv) Finding the balancing point of a triangular plate.
(a) Distance Formula.
(b) Cartesian Coordinates.
(c) Centroid.
(d) Equations of Conic Sections (Orbits).
(A) (i)-(b), (ii)-(a), (iii)-(d), (iv)-(c)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(b), (ii)-(a), (iii)-(c), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 3. Match the type of triangle with the coordinate geometry check (assuming vertices are given):
(i) Isosceles triangle
(ii) Equilateral triangle
(iii) Right-angled triangle
(iv) Scalene triangle
(a) All three sides have different lengths.
(b) Sum of squares of two sides equals square of third side.
(c) At least two sides have equal length.
(d) All three sides have equal length.
(A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(B) (i)-(a), (ii)-(b), (iii)-(c), (iv)-(d)
(C) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(d)
(D) (i)-(a), (ii)-(c), (iii)-(d), (iv)-(b)
Answer:
Question 4. Match the coordinate geometry technique with the geometric property it helps prove or find:
(i) Comparing slopes of adjacent sides in a quadrilateral.
(ii) Finding the midpoint of diagonals and checking if they are the same.
(iii) Calculating the lengths of all sides and diagonals of a quadrilateral.
(iv) Using equations of lines to find their intersection.
(a) Intersection point of lines.
(b) Proving a quadrilateral is a parallelogram.
(c) Determining if a quadrilateral is a square, rhombus, or rectangle (by checking side and diagonal lengths).
(d) Checking for perpendicularity of sides/diagonals.
(A) (i)-(d), (ii)-(b), (iii)-(c), (iv)-(a)
(B) (i)-(b), (ii)-(d), (iii)-(a), (iv)-(c)
(C) (i)-(d), (ii)-(b), (iii)-(a), (iv)-(c)
(D) (i)-(b), (ii)-(d), (iii)-(c), (iv)-(a)
Answer:
Question 5. Match the application area with how coordinate geometry is used:
(i) Robotics
(ii) Cartography (Map-making)
(iii) Architectural Design
(iv) Game Development
(a) Representing and manipulating 3D objects and spaces.
(b) Programming movement and interactions of objects in a virtual environment.
(c) Representing positions and distances on Earth's surface.
(d) Calculating positions and movements of robotic arms and sensors.
(A) (i)-(d), (ii)-(c), (iii)-(a), (iv)-(b)
(B) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a)
(C) (i)-(d), (ii)-(c), (iii)-(b), (iv)-(a)
(D) (i)-(c), (ii)-(d), (iii)-(a), (iv)-(b)
Answer: