PROBLEMS BASED ON CUBIC EQUATIONS

(1)  If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.             Solution

(2)  Construct a cubic equation with roots

(i) 1, 2 and 3      (ii) 1,1, and −2         (iii) 2, 1/2 and 1.

Solution

(3)  If α , β and γ are the roots of the cubic equation x³ + 2x² + 3x + 4 = 0 , form a cubic equation whose roots are

(i) 2α , 2β , 2γ

(ii)  1/α, 1/β, 1/γ

(iii) −α, -β, -γ         Solution

(4)  Solve the equation 3x³ −16x² + 23x − 6 = 0 if the product of two roots is 1.          Solution

(5)  Find the sum of squares of roots of the equation 2x⁴ −8x³ + 6x² − 3 = 0.          Solution

(6)  Solve the equation x³ − 9x² + 14x + 24 = 0 if it is given that two of its roots are in the ratio 3: 2.   Solution

(7)  If α, β  and γ are the roots of the polynomial equation ax³ + bx² + cx + d = 0 , find the Value of ∑ α/βγ in terms of the coefficients          Solution

(8)  If α, β, γ and δ are the roots of the polynomial equation 2x⁴+ 5x³ − 7x² + 8 = 0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + δ and αβγδ.     Solution

(9)  If p and q are the roots of the equation lx² + nx + n = 0, show that √(p/q) + √(q/p) + √(n/l)  =  0     Solution

(10)  If the equations x² + px + q = 0 and x² + p'x + q' = 0 have a common root, show that it must be equal to (pq'-p'q)/q q' or (q - q') / (p' - p)     Solution

(11)  Formalate into a mathematical problem to find a number such that when its cube root is added to it, the result is 6.     Solution

(12)  A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was cut away     Solution

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