Higher Degree Polynomials — CAT Previous-Year Questions
7 previous-year questions on Higher Degree Polynomials from CAT, with full solutions. Practise free — check answers as you go; sign in to save your progress.
Higher Degree Polynomials · CAT PYQs
The equation x3 + (2r + 1)x2 + (4r - 1)x + 2 = 0 has -2 as one of the roots. If the other roots are real, then the minimum possible non-negative integer value of r is?
Let r and c be real numbers. If r and -r are roots of 5x3 + cx2 - 10x + 9 = 0, then c equals
The smallest integer n such that n3 – 11n2 + 32n – 28 > 0 is
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The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.
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If the roots of the equation x3− ax2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?
Answer the following question based on the information given below.
Consider three circular parks of equal size with centres at A1, A2 and A3 respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A1A2A3, B1B2B3 and C1C2C3, as shown. Three sprinters A, B, and C begin running from points A1, B1 and C1 respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.
The number of roots common between the two equations x3 + 3x2 + 4x + 5 = 0 and x3 + 2x2 + 7x + 3 = 0 is:
m is the smallest positive integer such that for any integer n > m, the quantity n3 – 7n2 + 11n – 5 is positive. What is the value of m?
Answer the following question based on the information given below.
Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.
The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.
If the equation x3 – ax2 + bx – a = 0 has three real roots, then it must be the case that,