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?
The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
To determine the integral value of \( k \) for which the expression \( 4x^2 - kx + 1 \) is always positive, let's analyze the given statements:
The expression \( 4x^2 - kx + 1 \) represents a quadratic equation in the form of \( ax^2 + bx + c \), where \( a = 4 \), \( b = -k \), and \( c = 1 \). For the quadratic expression to be positive for all \( x \), the determinant \( \Delta \) must be less than zero. The determinant of a quadratic equation \( ax^2 + bx + c \) is given by:
\(\Delta = b^2 - 4ac\)
Substitute the values of \( a \), \( b \), and \( c \) in this equation:
\((-k)^2 - 4 \times 4 \times 1 \lt 0\)
\(k^2 - 16 \lt 0\)
This inequality implies:
\(-4 \lt k \lt 4\)
Now, let's consider the given statements:
- Statement I: \( k \lt -2 \)
- Statement II: \( k \gt -4 \)
From the range obtained from the determinant inequality, the expression \( 4x^2 - kx + 1 \) is positive when \( k \) lies between \(-4\) and \(4\).
- Statement I: \( k \lt -2 \)
- This statement doesn't ensure the positivity of the expression as it may lead \( k \) to be outside the valid range of \(-4\) to \(4\).
- Therefore, this alone is not specific enough to determine when the expression is positive.
- Statement II: \( k \gt -4 \)
- This statement suggests that \( k \) starts at \(-4\) heading towards \(4\), which is consistent with our derived range criterion for the expression to be positive for all \( x \).
- Thus, the expression is positive for any integer \( k \) within this range based on this condition alone.
Therefore, based on the analysis, the correct option is:
The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.
Similar Questions
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Question : Is \(P > Q\)?
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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 \(60^\circ\).
