A Question is given followed by two Statements I and II. Consider the Question and the Statements. Which one of the following is correct in respect of the above Question and the Statements?
Question:
In a quadrilateral ABCD, AB = 6 units, BC = 18 units, CD = 6 units, DA = 9 units. What is the length of diagonal BD?
Statement-I: The length of BD is an integer greater than 13.
Statement-II: The length of BD is an even integer.
Quadrilateral Diagonal Length Analysis
The question asks for the length of the diagonal BD in a quadrilateral ABCD, given the side lengths AB = 6, BC = 18, CD = 6, and DA = 9.
Geometric Constraints: Triangle Inequality Theorem
To find the possible length of the diagonal BD, we can consider the two triangles formed by the diagonal: triangle ABD and triangle BCD.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Applying this to our triangles:
Triangle ABD Analysis:
The sides are AB = 6, DA = 9, and BD.
- \(BD + AB > DA \Rightarrow BD + 6 > 9 \Rightarrow BD > 3\)
- \(BD + DA > AB \Rightarrow BD + 9 > 6 \Rightarrow BD > -3\) (This is always true since length is positive)
- \(AB + DA > BD \Rightarrow 6 + 9 > BD \Rightarrow 15 > BD\)
Combining these inequalities for triangle ABD, we get: \(3 < BD < 15\).
Triangle BCD Analysis:
The sides are BC = 18, CD = 6, and BD.
- \(BD + BC > CD \Rightarrow BD + 18 > 6 \Rightarrow BD > -12\) (Always true)
- \(BD + CD > BC \Rightarrow BD + 6 > 18 \Rightarrow BD > 12\)
- \(BC + CD > BD \Rightarrow 18 + 6 > BD \Rightarrow 24 > BD\)
Combining these inequalities for triangle BCD, we get: \(12 < BD < 24\).
Combining Constraints for Diagonal BD
To satisfy the conditions for both triangles, the length of BD must be within the intersection of the two ranges:
Intersection of \( (3 < BD < 15) \) and \( (12 < BD < 24) \) is \( 12 < BD < 15 \).
The possible lengths for BD must be strictly between 12 and 15.
Evaluating Statement I
Statement I says: "The length of BD is an integer greater than 13."
From our combined constraints, \(12 < BD < 15\). The integers in this range are 13 and 14.
If BD must be greater than 13, the only possibility from the combined range is BD = 14.
Conclusion for Statement I: This statement alone is sufficient to uniquely determine the length of BD as 14.
Evaluating Statement II
Statement II says: "The length of BD is an even integer."
From our combined constraints, \(12 < BD < 15\). The integers in this range are 13 and 14.
If BD must be an even integer, the only possibility from the combined range is BD = 14.
Conclusion for Statement II: This statement alone is sufficient to uniquely determine the length of BD as 14.
Final Conclusion
Since both Statement I and Statement II independently provide enough information to uniquely determine the length of the diagonal BD (which is 14 units), the question can be answered by using either statement alone.
Answer Justification
The question requires finding the length of diagonal BD. Based on the Triangle Inequality Theorem applied to triangles ABD and BCD, we derived the range \(12 < BD < 15\).
- Statement I restricts BD to integers greater than 13. Within the range \(12 < BD < 15\), the only integer greater than 13 is 14. Thus, Statement I is sufficient.
- Statement II restricts BD to be an even integer. Within the range \(12 < BD < 15\), the only even integer is 14. Thus, Statement II is sufficient.
Therefore, the question can be answered using either Statement I alone or Statement II alone.
Similar Questions
- Area of a rectangle with length x and breadth y is P and area of a parallelogram (which is strictly not a rectangle) with adjacent sides of length x and y is Q.
Question : Is \(P > Q\)?
Statement-I: \(x : y = 2 : 1\)
Statement-II: The angle between the two adjacent sides of the parallelogram is \(60^\circ\). - A circle touches all the four sides AB, BC, CD, DA of a quadrilateral ABCD.
Question : What is the perimeter of the quadrilateral?
Statement-I: \(AB + DC = 10\) cm
Statement-II: \(AD + BC = 10\) cm - P, Q, R, S are the mid-points of sides AB, BC, CD, DA respectively of a quadrilateral ABCD.
Question : What is the difference in the area of the quadrilateral ABCD and the area of the quadrilateral PQRS?
Statement-I: Area of the quadrilateral ABCD is 100 square unit.
Statement-II: Area of the quadrilateral PQRS is 50 square unit. - A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question: The largest of five different integers is 8 and least is 2. What is the average of these integers ?
Statement-I: The sum of all the 5 integers is a multiple of 5
Statement-II: The number of odd integers is odd
Which one of the following is correct in respect of the above Question and the Statements? A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
What is the remainder when \(x^{2n}-y^{2n} + 1\) is divided by \(x^n + y^n\), where \(n\) is a natural number?
Statement-I :
\(n\) is odd.
Statement-II :
\(n\) is even.
Which one of the following is correct in respect of the above Question and the Statements?A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
The product of a natural number N and the number M written by the same digits of N in the reverse order is 252. What is the number N?
Statement-I: N+ M = 33
Statement-II: N > MWhich one of the following is correct in respect of the above Question and the Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
In a triangle ABC, \(\angle A = \angle B-\angle C\). Is angle A acute?
Statement-I: ABC is not an obtuse-angled triangle.
Statement-II: Angle C is acute.Which one of the following is correct in respect of the above Question and the Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
ABCD is a parallelogram with \(\angle ABC=60^\circ\). If the area of the parallelogram is \(7\sqrt{3}\) square units, then what is the perimeter of the parallelogram?
Statement-I : The lengths of the sides AB and DA are prime numbers.
Statement-II: The lengths of the sides are natural numbers each greater than 1 unit.Which one of the following is correct in respect of the above Question and the Statements?
A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
AB and CD are chords of a circle intersecting at P. If \(AP \times PB\) = 48 square units, then what is \(CP \times PD\) equal to?
Statement-I : AP = 8 units
Statement-II: CP = 10 units
Which one of the following is correct in respect of the above Question and the Statements?A Question is given followed by two Statements I and II. Consider the Question and the Statements.
Question:
ABC is an isosceles triangle with AB = AC = 10 units. If the area of the triangle is 48 square units, then what is the length of the base BC?
Statement-I : The length of BC is an even integer.
Statement-II: The height of the triangle is greater than the length of half of the base.Which one of the following is correct in respect of the above Question and the Statements?
Important Questions from Data Sufficiency
Consider two Statements and a Question:
Statement 1: Priya is 4 ranks below Seema and is 31 st from the bottom.
Statement 2: Ena is 2 ranks above Seema and is 37" from the bottom.
Question: What is Seema’s rank from the top in the class of 40 students ?
Which one of the following is correct in respect of the Statements and the Question ?
Consider two Statements and a Question:
Statement 1: Each of A and D is heavier than each of B, E and F, but none of them is the-heaviest.
Statement 2: A is heavier than D, but is lighter than C.
Question: Who is the heaviest among A,B,C, D and E?
Which one of the following is correct in respect of the Statements and the Question ?
Consider two Statements and a Question :
Statement-1 : The last day of the month is a Wednesday.
Statement-2 : The third Saturday of the month was the seventeenth day.
Question : What day is the fourteenth of the given month ?
Which one of the following is correct in respect of the Statements and the Question ?
Consider the Question and two Statements given below :
Question : Is xan integer?
Statement-1 : x / 3 is not an integer.
Statement-2 : 3 xis an integer.
Which one of the following is correct in respect of the Question and the Statements?
Consider the Question and two Statements given below:
Question : What is the age of Manisha?
Statement-1: Manisha is 24 years younger than her mother.
Statement-2 : 5 years later, the ages of Manisha and her mother will be in the ratio 3 : 5.
Which one of the following is correct in respect of the Question and the Statements?
