HSC Science इयत्ता १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

If to ω ≠ 1 is a cube root of unity, then show that `("a" + "b"omega + "c"omega^2)/("b" + "c"omega + "a"omega^2) + ("a" + "b"omega + "c"omega^2)/("c" + "a"omega + "a"omega^2)` = – 1

Show that `(sqrt(3)/2 + i/2)^5 + (sqrt(3)/2 - i/2)^5 = - sqrt(3)`

Find the value of `[(1 + sin  pi/10 + "i" cos  pi/10)/(1 + sin  pi/10 - "i" cos  pi/10)]^10`

If 2 cos α = `x + 1/x` and 2 cos β = `y + 1/y`, show that `x/y + y/x = 2cos(alpha − beta)`

If 2 cos α = `x + 1/x` and 2 cos β = `y + 1/y`, show that `xy - 1/xy = 2"i" sin(alpha + beta)` 

If 2 cos α = `x + 1/x` and 2 cos β = `y + 1/y`, show that `x^"m" y^"n" + 1/(x^"m" y^"n")` = 2 cos(mα – nβ)

If 2cos α = `x + 1/x` and 2 cos β = `y + 1/x`, show that `x^"m"/y^"n" - y^"n"/x^"m"` = 2i sin(mα – nβ)

Solve the equation z³ + 27 = 0

If ω ≠ 1 is a cube root of unity, show that the roots of the equation (z – 1)³ + 8 = 0 are – 1, 1 – 2ω, 1 – 2ω² 

Find the value of `sum_("k" = 1)^8 (cos  (2"k"pi)/9 + "i" sin  (2"kpi)/9)`

If ω ≠ 1 is a cube root of unity, show that (1 – ω + ω²)⁶ + (1 + ω – ω²)⁶ = 128

If ω ≠ 1 is a cube root of unity, show that (1 + ω)(1 + ω²)(1 + ω⁴)(1 + ω⁸)….. (1 + ω²ⁿ) = 1

If z = 2 – 2i, find the rotation of z by θ radians in the counterclockwise direction about the origin when θ = `pi/3`

If z = 2 – 2i, find the rotation of z by θ radians in the counterclockwise direction about the origin when θ = `(2pi)/3`

If z = 2 – 2i, find the rotation of z by θ radians in the counterclockwise direction about the origin when θ = `(3pi)/3`

Choose the correct alternative:

If ω ≠ 1 is a cubic root of unity and (1 + ω)⁷ = A + Bω, then (A, B) equals

Choose the correct alternative:

The product of all four values of `(cos  pi/3 + "i" sin  pi/3)^(3/4)` is

Choose the correct alternative:

If ω ≠ 1 is a cubic root of unity and `|(1, 1, 1),(1, - omega^2 - 1, omega^2),(1, omega^2, omega^7)|` = 3k, then k is equal to

Choose the correct alternative:

If ω = `cis  (2pi)/3`, then the number of distinct roots of `|(z + 1, omega, omega^2),(omega, z + omega^2, 1),(omega^2, 1, z + omega)|` = 0

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