Let \(α_1,α_2,….,α_7\) be the roots of the equation \(𝑥^7+3𝑥^5−13𝑥^3−15𝑥=0\) and \(|α_1|≥|α_2|≥⋯≥ |α_7|\). Then \(α_1α_2−α_3α_4+α_5α_6\) is equal to ____.
Correct Answer: 3
Solution and Explanation
Top Questions on Polynomials
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Let \( M \) be a \( 7 \times 7 \) matrix with entries in \( \mathbb{R} \) and having the characteristic polynomial \[ c_M(x) = (x - 1)^\alpha (x - 2)^\beta (x - 3)^2, \] where \( \alpha>\beta \). Let \( {rank}(M - I_7) = {rank}(M - 2I_7) = {rank}(M - 3I_7) = 5 \), where \( I_7 \) is the \( 7 \times 7 \) identity matrix.
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In the following \(p\text{–}V\) diagram, the equation of state along the curved path is given by \[ (V-2)^2 = 4ap, \] where \(a\) is a constant. The total work done in the closed path is:
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Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \) is ___________.
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Concepts Used:
Binomial Distribution
A common probability distribution that models the probability of obtaining one of two outcomes under a given number of parameters is called the binomial distribution. It summarizes the number of trials when each trial has the same probability of attaining one specific outcome. The value of a binomial is acquired by multiplying the number of independent trials by the successes.
Criteria of Binomial Distribution:
Binomial distribution models the probability of happening an event when specific criteria are met. In order to use the binomial probability formula, the binomial distribution involves the following rules that must be present in the process:
- Fixed trials
- Independent trials
- Fixed probability of success
- Two mutually exclusive outcomes