Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 12 TN Board chapter 3 - Theory of Equations [Latest edition]

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

Construct a cubic equation with roots 1, 2 and 3

Construct a cubic equation with roots 1, 1, and – 2

Construct a cubic equation with roots `2, 1/2, and 1`

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

If α, β and γ are the roots of the cubic equation x³ + 2x² + 3x + 4 = 0, form a cubic equation whose roots are `1/alpha, 1/beta, 1/γ`

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

Solve the equation 3x³ – 16x² + 23x – 6 = 0 if the product of two roots is 1

Find the sum of squares of roots of the equation `2x^4 - 8x^3 + 6x^2 - 3` = 0

Solve the equation x³ – 9x² + 14x + 24 = 0 if it is given that two of its roots are in the ratio 3 : 2

If α, β, and γ are the roots of the polynomial equation ax³ + bx² + cx + d = 0, find the value of `sum  alpha/(betaγ)` in terms of the coefficients

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

If p and q are the roots of the equation lx² + nx + n = 0, show that `sqrt("p"/"q") + sqrt("q"/"p") + sqrt("n"/l)` = 0

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")`

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 left standing

If k is real, discuss the nature of the roots of the polynomial equation 2x² + kx + k = 0, in terms of k

Find a polynomial equation of minimum degree with rational coefficients, having `2 + sqrt(3)"i"` as a root

Find a polynomial equation of minimum degree with rational coefficients, having 2i + 3 as a root

Find a polynomial equation of minimum degree with rational coefficients, having `sqrt(5) - sqrt(3)` as a root

Prove that a straight line and parabola cannot intersect at more than two points

Solve the cubic equation: 2x³ – x² – 18x + 9 = 0 if sum of two of its roots vanishes

Solve the equation 9x³ – 36x² + 44x – 16 = 0 if the roots form an arithmetic progression

Solve the equation 3x³ – 26x² + 52x – 24 = 0 if its roots form a geometric progression

Determine k and solve the equation 2x³ – 6x² + 3x + k = 0 if one of its roots is twice the sum of the other two roots

Find all zeros of the polynomial x⁶ – 3x⁵ – 5x⁴ + 22x³ – 39x² – 39x + 135, if it is known that 1 + 2i and `sqrt(3)` are two of its zeros

Solve the cubic equations:

2x³ – 9x² + 10x = 3

Solve the cubic equations:

8x³ – 2x² – 7x + 3 = 0

Solve the equation:

x⁴ – 14x² + 45 = 0

Solve: (x – 5)(x – 7) (x + 6)(x + 4) = 504

Solve: (x – 4)(x – 2)(x- 7)(x + 1) = 16

Solve: (2x – 1)(x + 3)(x – 2)(2x + 3) + 20 = 0

Solve the following equations
sin² x – 5 sin x + 4 = 0

Solve the following equations
12x³ + 8x = 29x² – 4 = 0

Examine for the rational roots of

2x³ – x² – 1 = 0

Examine for the rational roots of

x⁸ – 3x + 1 = 0

Solve: `8x^(3/(2"n")) - 8x^((-3)/(2"n"))` = 63

Solve: `2sqrt(x/"a") + 3sqrt("a"/x) = "b"/"a" + (6"a")/"b"`

Solve the equation
6x⁴ – 35x³ + 62x² – 35x + 6 = 0

Solve the equation
x⁴ + 3x³ – 3x – 1 = 0

Find all real numbers satisfying 4^x – 3(2^x+2) + 2⁵ = 0

Solve the equation 6x⁴ – 5x³ – 38x² – 5x + 6 = 0 if it is known that `1/3` is a solution

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x⁹ – 4x⁸ + 4x⁷ – 3x⁶ + 2x⁵ + x³ + 7x² + 7x + 2 = 0

Discuss the maximum possible number of positive and negative roots of the polynomial equations x² – 5x + 6 and x² – 5x + 16. Also, draw a rough sketch of the graphs

Show that the equation x⁹ – 5x⁵ + 4x⁴ + 2x² + 1 = 0 has atleast 6 imaginary solutions

Determine the number of positive and negative roots of the equation x⁹ – 5x⁸ – 14x⁷ = 0

Find the exact number of real zeros and imaginary of the polynomial x⁹ + 9x⁷ + 7x⁵ + 5x³ + 3x

Choose the correct alternative:
A zero of x³ + 64 is

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If f and g are polynomials of degrees m and n respectively, and if h(x) = (f o g)(x), then the degree of h is

Choose the correct alternative:
A polynomial equation in x of degree n always has

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If α, β and γ are the zeros of x³ + px² + qx + r, then `sum 1/alpha` is

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According to the rational root theorem, which number is not possible rational root of 4x⁷ + 2x⁷ – 10x³ – 5?

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The polynomial x³ – kx² + 9x has three real roots if and only if, k satisfies

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The number of real numbers in [0, 2π] satisfying sin⁴x – 2 sin²x + 1 is

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If x³ + 12x² + 10ax + 1999 definitely has a positive zero, if and only if 

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The polynomial x³ + 2x + 3 has

Choose the correct alternative:
The number of positive roots of the polynomials `sum_("j" = 0)^"n"  ""^"n""C"_"r" (- 1)^"r" x^"r"` is