Algebra — practice questions, page 7
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QuestionIf α and β are the roots of x² − 5x + 2 = 0, then α⁴ + β⁴ equals:QuestionIf α and β are the roots of 3x² − 7x + 3 = 0, then α² − αβ + β² equals:QuestionIf α and β are the roots of x² − 8x + 3 = 0, the equation with integer coefficients whose roots…QuestionThe roots of 3x² − 10x + k = 0 are in the ratio 2 : 3. Then k equals:QuestionIf α and β are the roots of x² − 4x + 7 = 0, then (α − 1)(β − 1) equals:QuestionIf α is a root of 5x² − 3x − 7 = 0, then the value of 5α² − 3α + 4 is:QuestionLet α and β be the roots of x² − 9x + 14 = 0. The quadratic equation whose two roots are the…QuestionIf α and β are the roots of x² − 3x − 4 = 0, the equation whose roots are α³ and β³ is:QuestionIf α and β are the roots of 2x² − 11x + 5 = 0, the equation whose roots are −α and −β is:QuestionThe equation kx² − 10x + (4k − 9) = 0 has roots that are reciprocals of each other. Then k equals:QuestionFor how many values of k are both roots of x² − 6x + k = 0 positive integers?QuestionThe roots of x² + (2k + 1)x + (k − 3) = 0 are real and opposite in sign exactly when:QuestionIn the equation x² − 20x + k = 0 one root is the square of the other. The positive value of k is:QuestionConstants A, B and C satisfy x² + 3x + 5 = A(x − 1)² + B(x − 1) + C for every real x. Then A + B +…QuestionThe polynomial x⁴ + 2x³ − 3x² + ax + b leaves the remainder 5 on division by (x − 1) and the…QuestionThe polynomial x⁴ + ax² + b is exactly divisible by x² − 4x + 3. Then b equals:QuestionIf (3 + √5)/(3 − √5) = a + b√5 with a and b rational, then b equals:QuestionA positive number x satisfies x = 3 + 1/(3 + 1/(3 + …)), the pattern continuing without end. Then…QuestionThe value of (√5 + √3)/(√5 − √3) + (√5 − √3)/(√5 + √3) is:QuestionThe sum 1/(√1 + √2) + 1/(√2 + √3) + 1/(√3 + √4) + … + 1/(√99 + √100) equals:QuestionThe sum 1/(1 × 2) + 1/(2 × 3) + 1/(3 × 4) + … + 1/(20 × 21) equals:QuestionFor a real number x with x + 1/x = 4, the value of x⁵ + 1/x⁵ is:QuestionA real number x satisfies x − 1/x = 6. Then x² + 1/x² equals:QuestionIf x + y = 4 and 1/x + 1/y = 4/3, then x² + y² equals:QuestionIf a, b and c are non-zero numbers with a + b + c = 0, then a²/(bc) + b²/(ca) + c²/(ab) equals:
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