HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

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

Choose the correct alternative:

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)`

Find the parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9

Find the parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, – 2, 3) and parallel to the straight line passing through the points (2, 1, – 3) and (– 1, 5, – 8)

Find the non-parametric form of vector equation and cartesian equation of the plane passing through the point (1, − 2, 4) and perpendicular to the plane x + 2y − 3z = 11 and parallel to the line `(x + 7)/3 = (y + 3)/(-1) = z/1`

Find the parametric form of vector equation, and Cartesian equations of the plane containing the line `vec"r" = (hat"i" - hat"j" + 3hat"k") + "t"(2hat"i" - hat"j" + 4hat"k")` and perpendicular to plane `vec"r"*(hat"i" + 2hat"j" + hat"k")` = 8

Find the parametric vector, non-parametric vector and Cartesian form of the equation of the plane passing through the point (3, 6, – 2), (– 1, – 2, 6) and (6, 4, – 2)

Find the non-parametric form of vector equation and Cartesian equations of the plane `vec"r" = (6hat"i" - hat"j" + hat"k") + "s"(-hat"i" + 2hat"j" + hat"k") + "t"(-5hat"i" - 4hat"j" - 5hat"k")`

Show that the straight lines `vec"r" = (5hat"i" + 7hat"j" - 3hat"k") + "s"(4hat"i" + 4hat"j" - 5hat"k")` and `vec"r"(8hat"i" + 4hat"j" + 5hat"k") + "t"(7hat"i" + hat"j" + 3hat"k")` are coplanar. Find the vector equation of the plane in which they lie