Regular hexagon ABCDEF is symmetrical about the x-axis and the y-axis in a rectangular coordinate plane, as shown above. If the coordinates of point D are (2,0), what is the area, in square units, of rectangle BCEF?
Given:
Hexagon ABCDEF is symmetrical about the x-axis and the y-axis.
To find: ar(Rectangle BCEF
Approach:
2. ar(Rectangle OGCH)= CG*CH
. Now, OG + GD = OD
4. Also, in right triangle CGD, using trigonometric ratios, we can express CG in terms of hypotenuse a.
Working Out:
Working towards the final answer
In the rectangular coordinate system, lines L_{1} and L_{2} intersect at point C. Are the lines L_{1} and L_{2} perpendicular to each other?
(1) The product of the slopes of lines L_{1} and L_{2} is at a distance of 1 unit from 0 on the number line.
(2) Lines x = 0 and y = 0 are the perpendicular bisectors of lines L_{1} and L_{2 }respectively.
Steps 1 & 2: Understand Question and Draw Inferences
To Find:
Step 3: Analyze Statement 1 independently
(1) Statement 1 states that: "The product of the slopes of lines L_{1} and L_{2} is at a distance of 1 unit from 0 on the number line."
Step 4: Analyze Statement 2 independently
(2) Statement 2 states that: "Lines x = 0 and y = 0 are the perpendicular bisectors of lines L_{1} and L_{2 }respectively."
Step 5: Analyze Both Statements Together (if needed)
As we have got a unique answer from step-4, this step is not required.
Rectangle ABCD is drawn in the xy-plane such that side AB is parallel to the x-axis and the side AD is parallel to the y-axis. What are the coordinates of point D?
(1) The coordinates of point A are (2, 3)
(2) The coordinates of point C are (4, 8)
Steps 1 & 2: Understand Question and Draw Inferences
Given: Rectangle ABCD in the xy-plane
To find: Coordinates of point D
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The coordinates of point A are (2, 3)’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The coordinates of point C are (4, 8)’
Step 5: Analyze Both Statements Together (if needed)
Thus, by combining the 2 statements, we do get a unique answer
Answer: Option C
In a rectangular coordinate plane, AB is the diameter of a circle and point C lies on the circle. If the coordinates of points A and B are (-1,0) and (5,0), and the area of triangle ABC is 6√2 square units, which of the following can be the coordinates of point C?
Given:
In ΔABC
To find: Possible coordinates of point C
Approach and Working:
In the xy-plane, a circle C is drawn with center at (1, 2) and radius equal to 5. Is line l a tangent to the circle C?
(1) Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20.
(2) The x-intercept of line l is 10.
Steps 1 & 2: Understand Question and Draw Inferences
To Find: Is line l a tangent to the circle?
Step 3: Analyze Statement 1 independently
1.Statement 1 states that: "Point A with coordinates (a, b) lies on line l such that a(a-2) +b(b-4) ≤ 20".
Step 4: Analyze Statement 2 independently
2. Statement 2 states that "The x-intercept of line l is 10".
Step 5: Analyze Both Statements Together (if needed)
Thus, the correct answer is Option E .
In the xy–coordinate plane, a line segment is drawn to join the points A (2, 1) and C(4, 3). If point B lies on line segment AC, is AB = BC?
(1) The x-coordinate of point B is 3
(2) The y-coordinate of point B is 2
Steps 1 & 2: Understand Question and Draw Inferences
The given information can be represented visually as under:
We need to find if AB = BC, that is, if B is the mid-point of AC?
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The x-coordinate of point B is 3’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The y-coordinate of point B is 2’
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Steps 3 and 4, this step is not required
Hence, the correct answer is Option D.
Of all the points that satisfy the equation x^{2}+ y^{2} =25 approximately what percentage of the points also satisfy the
inequality xy ≥ 0?
Given
x^{2}+ y^{2} =25
To Find: Percentage of points lying on the circle that satisfies xy ≥ 0?
Approach and Working
In the xy-plane, what is the area of the region bounded by y +2x ≥ 3, y –x ≥ -6 and the line, that is perpendicular to x = 0 and passes
through the origin?
Given
To Find: Area bounded by the region y +2x ≥ 3, y –x ≥ -6 and x-axis
Approach:
1. For finding the area bounded by the region, we need to first draw the line segments y +2x = 3 and y – x = 6
2. Once we draw these line segment, we need to find the side of each line segment where the region specified in the question statement lies
3. Once we have the region, we will find the area of the region using the standard geometry formulas
Working out:
1. The line segment y +2x = 3 will intersect the y-axis at (3,0) and x-axis at
2. Similarly, the line segment y-x ≥ 6 will intersect the y-axis at (-6,0) and x-axis at (6,0).
3. So, the vertex points of the region are ,(6,0) and (3, -3)
4. We know that area of a triangle = ½ * base * height
5. Thus area of the triangle =
Hence the correct answer is Option B.
The line intersects the y-axis at the point P and the line intersects the y-axis at the
point R. If these two lines intersect at point Q, what is the measure of ∠PQR?
Given
To Find: ∠PQR = ?
Approach
Working out
Drawing ΔPQR in the coordinate plane
Finding ∠PQR
∠PQR = ∠PQO + ∠OQR
In the xy- plane, lines l and k intersect at point A whose x and y coordinates are positive. If the lines l and k are not parallel to either of the axes, is the product of the slopes of line l and k greater than zero?
(1) The product of the x-intercepts of the lines l and k is negative.
(2) The product of the y-intercepts of the lines l and k is positive.
Step 1 & 2: Understand Question and Draw Inference
To Find: Is m * m > 0?
Step 3 : Analyze Statement 1 independent
(1) The product of the x-intercepts of the lines l and k is negative.
Insufficient to answer.
Step 4 : Analyze Statement 2 independent
(2) The product of the y-intercepts of the lines l and k is positive.
Insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
Combining both the statements, we can reject case-III, where slopes of both the lines is negative. Now, let’s evaluate the other cases one by one:
Case-II: Lines have opposite slopes
Case Accepted: the product of Slope in such a case will be negetive
Thus, we have a unique answer. Sufficient to answer Answer: C
In the rectangular coordinate system are the points A and B equidistant from point C (1, 0)?
(1) Points A and B lie on circle R, which passes through origin and has a radius equal to 1.
(2) The bisector of the line segment AB passes through point C
step 1 & 2: Understand Question and Draw Inference
To Find: If points A and B are equidistant from C (1, 0)?
Step 3 : Analyze Statement 1 independent
(1) Points A and B lie on circle R, which passes through origin and has a radius equal to 1.
Insufficient to answer
Step 4 : Analyze Statement 2 independent
(2) The bisector of the line segment AB passes through point C
Step 5: Analyze Both Statements Together (if needed)
Both the statements combined does not give us any extra information to answer
the question.
Insufficient to answer
Answer: E
In the xy-plane, a trapezium ABCD has one of its parallel sides AB on the x-axis with vertex A at the origin. The x-coordinate of point B is 6 and the length of the smaller parallel side CD is 2 less than the length of the longer parallel side. If the side AD lies on the line with the equation y = x and the area of the trapezium is 5 square units, what is the coordinate of point C?
Given
Trapezium ABCD with AB on x-axis
To Find: Coordinates of point C?
Approach:
Working out
In the xy-plane, two circles C and R are drawn such that Circle C has its center at the origin and radius equal to 5 and Circle R has its center at (2, 3) and radius equal to 5. Which of the following is the equation of the line that passes through the intersection points of circles C and R?
units (i.e. the radius) from the centre of the circle(2, 3)
To Find: The equation of the line that passes through the intersection point of Circle C and R?
Approach:
Working out:
Thus, the line segment that passes through the intersection points of Circle C
and R should have itsequation as
Answer: D
The figure above shows a circle whose diameter AB lies on the x-axis as shown. Triangle ACB is a right-angled triangle whose side AC makes an angle of 30? with side AB. If the coordinates of points A and B are (0,0) and (4,0) respectively, what is the y-coordinate of point C?
Given
To Find : y – coordinate of point C
Approach:
So, referring to the above figure, we can write: y – coordinate of point C = CP
So, in order to answer the question, we need to know the value of CP
Working out:
Finding BC and AC
In right triangle ACB,
1. Finding the area of triangle ACB
Area of right triangle ACB = ½ AC * BC = ½ * 2√3 * 2 =
2√3 square units
2. Finding CP
½ * CP * AB = 2√3 square units
½ *CP * 4 = 2√3
CP = √3 units
3. Getting to the answer
Therefore, the y-coordinate of point C =CP = √3 units
Looking at the answer choices, we see that the correct answer is OPTION D
In the xy-plane given above, if the parallelogram ABCD has all its sides equal, is ABCD a square?
(1) The lines connecting AC and BD have the product of their slopes equal to -1
(2) Points A and D have the same x-coordinates
Step 1 & 2: Understand Question and Draw Inference
Given: Parallelogram ABCD has all its sides equal
To find: If ABCD is a square?
So, we need to find if one of the angles of ABCD is a right angle.
Step 3 : Analyze Statement 1 independent
(1) Statement 1 states that "The lines connecting AC and BD have the product of their slopes equal to -1"
Step 4 : Analyze Statement 2 independent
Step 5: Analyze Both Statements Together (if needed)
Thus, the correct answer is Option E .
What is the area of the triangle formed between the lines y – x =2, 3x + 4y = 29 and y = 2?
Given
3 lines:
To Find : The area of the triangle formed between these 3 lines
Approach:
Working out:
Finding the vertices of the triangle
Finding the area of the triangle
Looking at the answer choices, we see that the correct answer is Option D
In the xy-plane, Region R is bounded by the line segments with equations, 2x + |4y| = 20 and x=0, whereas Region P is bounded by the line segments with equations |4x| + 2y = 20 and y = 0. If the function A(B) is defined as the area of Region B, what is the value of |A(P) – 2A(R)|?
Given
To Find: |A(P) – 2A(R)|?
Approach
Working out
1. Finding A(R)
a. Assuming y-axis to be the base, we have EF = 5 – (-5) = 10
b. Height = x-coordinate of point D = 10
c. Area of region R = ½ * 10 * 10 = 50
d. A(R) = 50……. (1)
2. Finding A(P)
a. Assuming x-axis to be the base, we have BD = 5 – (-5) = 10
b. Height = y-coordinate of point A = 10
c. Area of region P = ½ * 10 * 10 = 50
d. A(P) = 50…….(2)
3. Using (1) and (2), we have
a. |A(P) – 2A(R)| = | 50 – 2*50| = |-50| = 50
Answer: C
In the xy-plane, the circle C centered at the origin O is intersected by a line l at two points A and B. A line from O is drawn to AB intersecting AB at point D, such that the product of the slopes of OD and AB is -1. If the line l does not pass through origin and the coordinates of point D are (1, -1), what is the radius of the circle?
(1) The x intercept of line l is 2.
(2) The product of the x coordinates of points A and B as well as the product of the y coordinates of points A and B is zero.
Step 1 & 2: Understand Question and Draw Inference
Step 3 : Analyze Statement 1 independent
(1) The x intercept of line l is 2.
We know from the equation of the line, that x- intercept of the line is 2. Hence, there is no added information provided.
Insufficient to answer
Step 4 : Analyze Statement 2 independent
(2) The product of the x coordinates of points A and B as well as the product of the y coordinates of points A and B is zero.
As we know the coordinate of point A, we can calculate the radius of the circle C.
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step-4, this step is not required.
Answer: B
In the xy-plane shown, is the slope of line l non-negative?
1. The line passes through quadrants II and III.
2. For each pair of coordinates (x,y) lying on line l, the product of x and y is not always non-negative.
Step 1 & 2: Understand Question and Draw Inference
To Find:
Step 3 : Analyze Statement 1 independent
(1) Statement 1 states that "The line passes through quadrants II and III".
Hence, Statement 1 is insufficient to answer
Step 4 : Analyze Statement 2 independent
(2) Statement 2 states that "For each pair of coordinates (x, y) lying on the line l, the product of x and y is not always non-negative."
Hence, Statement 2 is insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
1. Line l passes through quadrants-II and III
2. Line l passes through quadrant II, or IV or both
Following cases are possible:
As we do not have a unique answer, hence by combining both the statements also it is insufficient to answer the question.
Thus, the correct answer is Option E .
In a rectangular coordinate plane, points A(3,4), B(6,-5), C(-4,-3) and D(-2,2) are joined to form a quadrilateral. What is the area, in square units, of quadrilateral ABCD?
Given
To Find: Area of quadrilateral ABCD
Approach
2. So, Area of Quadrilateral ABCD = (Area of Rectangle QBPS) – (ar ΔAPB + ar ΔBQC + ar ΔCRD + ar of square DRST + ar ΔATD)
Looking at the answer choices, we see that the correct answer is Option D
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