HSC Arts कक्षा १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions

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`

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

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

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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

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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

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

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

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If cot^–12 and cot^–13 are two angles of a triangle, then the third angle is

Find a parametric form of vector equation of a plane which is at a distance of 7 units from t the origin having 3, – 4, 5 as direction ratios of a normal to it

Find the direction cosines of the normal to the plane 12x + 3y – 4z = 65. Also find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin

Find the vector and Cartesian equation of the plane passing through the point with position vector `2hat"i" + 6hat"j" + 3hat"k"` and normal to the vector `hat"i" + 3hat"j" + 5hat"k"`

A plane passes through the point (− 1, 1, 2) and the normal to the plane of magnitude `3sqrt(3)` makes equal acute angles with the coordinate axes. Find the equation of the plane

Find the intercepts cut off by the plane `vec"r"*(6hat"i" + 45hat"j" - 3hat"k")` = 12 on the coordinate axes

If a plane meets the co-ordinate axes at A, B, C such that the centroid of the triangle ABC is the point (u, v, w), find the equation of the plane

Find the non-parametric form of vector equation and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to thestraight lines `(x - 1)/2 = (y + 1)/3 = (x - 3)/1` and `(x + 3)/2 = (y - 3)/(-5) = (z + 1)/(-3)`