Each item contains a Question followed by two Statements. Answer each item using the following instructions :
Choose option
(a) If the Question can be answered by one of the Statements alone, but not by the other.
(b) If the Question can be answered by either Statement alone.
(c) If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
(d) If the Question cannot be answered even by using both Statements together.
Question : Is the triangle equilateral?
Statement-I: \(a^2 + b^2 + c^2 = (ab + bc + ca)\)
Statement-II: \(3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca\)
Understanding the Question
We are given a triangle with sides \(a, b, c\). The core question is to determine if this triangle is equilateral, meaning if its sides are equal (\(a=b=c\)). We must assess whether two provided statements are individually sufficient to answer this question.
Evaluating Statement-I
Statement-I provides the algebraic condition: \(a^2 + b^2 + c^2 = ab + bc + ca\) To determine if this statement is sufficient, we can rearrange the equation. First, multiply the entire equation by 2:
\(2(a^2 + b^2 + c^2) = 2(ab + bc + ca)\) \(2a^2 + 2b^2 + 2c^2 = 2ab + 2bc + 2ca\)Next, move all terms to the left side of the equation:
\(2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0\)Now, we can group the terms to form perfect squares. Rearrange and combine terms:
\((a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ca + a^2) = 0\)This expression simplifies to the sum of squares:
\((a-b)^2 + (b-c)^2 + (c-a)^2 = 0\)For a triangle, the side lengths \(a, b, c\) must be positive real numbers. The square of a real number is always non-negative. Therefore, the sum of the squares \((a-b)^2\), \((b-c)^2\), and \((c-a)^2\) can only be zero if each individual term is zero:
- \((a-b)^2 = 0\), which implies \(a = b\).
- \((b-c)^2 = 0\), which implies \(b = c\).
- \((c-a)^2 = 0\), which implies \(c = a\).
From these conditions, we conclude that \(a = b = c\). This means the triangle must be equilateral.
Conclusion: Statement-I alone is sufficient to answer the question.
Evaluating Statement-II
Statement-II provides the condition: \(3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca\) Let's rearrange this equation by moving all terms to one side:
\(3a^2 + 3b^2 + 4c^2 - 2ab - 4bc - 4ca = 0\)This statement provides a relationship between the sides \(a, b, c\). However, fulfilling this specific condition does not necessarily mean that the triangle must be equilateral (\(a=b=c\)). It is possible that this equation could hold true for triangles that are not equilateral. Because we cannot definitively conclude \(a=b=c\) using only this statement, it is not sufficient on its own.
Conclusion: Statement-II alone is not sufficient to answer the question.
Final Decision
Based on our analysis:
- Statement-I is sufficient to determine if the triangle is equilateral.
- Statement-II is not sufficient to determine if the triangle is equilateral.
Therefore, the question can be answered using Statement-I alone, but not by Statement-II alone. This corresponds to option (a).
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. - Question : What is the ratio of the lengths of diagonals of a rhombus?
Statement-I: One diagonal of the rhombus is equal to its side.
Statement-II: The longer diagonal of the rhombus is equal to \(\sqrt{3}\) times its side. - The chord of a circle of radius R touches at a point on the circumference of a concentric circle of radius r. The length of the chord is 24 units.
Question : What are the values of r and R?
Statement-I: r is an integer.
Statement-II: R is an integer. - Let a, b, c and d be positive integers.
Question : Which one of a, b, c, d is closest to the product abcd?
Statement-I: \(a > b > c\)
Statement-II: c is not the smallest. - Let \(mn = k\), where m and n are prime numbers and k is an even number.
Question : What is the value of \(mn - n + 1\)?
Statement-I: \(m > n\)
Statement-II: One of the numbers is 2. - Question : If p is a positive integer, then what is the remainder when $p^n$ is divided by $p + 1$?
Statement-I: n is even.
Statement-II: p is even. Question : Is xy positive?
Statement-I: \(x = \sqrt[3]{-0.19683}\)
Statement-II: $y = \(\sqrt[3]{x} \)A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option.
Question : What is the integral value of \(k\) for which the expression \(4x^2-kx+1\)is positive?
Statement-I : \(k<-2\)
Statement-II: \(k>-4\)
Which one of the following is correctin respect of the above Question andthe Statements?
Important Questions from Data Sufficiency
- In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Harsh, Ketan, Mayank, Rohit, and Sidharth, each have different height. Who is the shortest?
(I) Mayank is taller than Harsh but shorter than Ketan.
(II) Rohit is taller than Harsh but shorter than Mayank. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Q. Kunal, Mohit, Rohan, Sandeep, and Tarun - each have different age. Who is the oldest?
(I) Mohit is older than Kunal but younger than Sandeep.
(II) Tarun is younger than Sandeep but older than Mohit. - In this question, a question is followed by two statements numbered (I) and (II). You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and decide the appropriate answer.
Question:
Among 5 trees, L, M, N, T and V each having a different height, which tree is the shortest?
Statements:
(I) N is shorter than only three trees.
(II) N is taller than M but shorter than T. Below is given a pair of questions, which are required to be answered, and a triad of statements. Which group of statements is sufficient to answer both the questions?
Question 1: Who is sitting opposite to Q?
Question 2: Who is sitting to the right of Q?
Statement 1 : P and U are sitting on left and right sides of T, respectively
Statement 2: P, Q, R, S, T and U are sitting around a round table with six chairs
Statement 3: Q is NOT sitting on the immediate left side of R
- 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 \(60^\circ\).
