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

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

18x² + 12y² – 144x + 48y + 120 = 0

Identify the type of conic and find centre, foci, vertices, and directrices of the following:

9x² – y² – 36x – 6y + 18 = 0

Choose the correct alternative:

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

Choose the correct alternative:

If P(x, y) be any point on 16x² + 25y² = 400 with foci F(3, 0) then PF₁ + PF₂ is

Choose the correct alternative:

If x + y = k is a normal to the parabola y² = 12x, then the value of k is 14

If `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" * (vec"b" xx vec"c")`

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors `- 6hat"i" + 14hat"j" + 10hat"k", 14hat"i" - 10hat"j" - 6hat"k"` and `2hat"i" + 4hat"j" - 2hat"k"`

The volume of the parallelepiped whose coterminus edges are `7hat"i" + lambdahat"j" - 3hat"k", hat"i" + 2hat"j" - hat"k", -3hat"i" + 7hat"j" + 5hat"k"` is 90 cubic units. Find the value of λ 

If `vec"a", vec"b", vec"c"` are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of `(vec"a" + vec"b") * (vec"b" xx vec"c") + (vec"b" + vec"c")* (vec"c" xx vec"a") + (vec"c" + vec"a") * (vec"a" xx vec"b")`

Find the altitude of a parallelepiped determined by the vectors `vec"a" = - 2hat"i" + 5hat"j" + 3hat"k", vec"b" = hat"i" + 3hat"j" - 2hat"k"` and `vec"c" = - vec"i" + vec"j" + 4vec"k"` if the base is taken as the parallelogram determined by `vec"b"` and `vec"c"`

Determine whether the three vectors `2hat"i" + 3hat"j" + hat"k", hat"i" - 2hat"j" + 2hat"k"` and `3hat"i" + hat"j" + 3hat"k"` are coplanar

Ler `vec"a" = hat"i" + hat"j" + hat"k", vec"b" = hat"i"` and `vec"c" = "c"_1hat"i" + "c"_2hat"j" + "c"_3hat"k"`. If c₁ = 1 and c₂ = 2. find c₃ such that `vec"a", vec"b"` and `vec"c"` are coplanar

If `vec"a" = hat"i" - hat"k", vec"b" = xhat"i" + hat"j" + (1 - x)hat"k", vec"c" = yhat"i" + xhat"j" + (1 + x - y)hat"k"`, show that  `[(vec"a", vec"b", vec"c")]` depends on neither x nor y

If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b

Let `vec"a",  vec"b",  vec"c"` be three non-zero vectors such that `vec"c"` is a unit vector perpendicular to both `vec"a"` and `vec"b"`. If the angle between `vec"a"` and `vec"b"` is `pi/6`, show that `[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`

Write the Maclaurin series expansion of the following functions:

e^x

Write the Maclaurin series expansion of the following functions:

sin x

Write the Maclaurin series expansion of the following functions:

cos x

Write the Maclaurin series expansion of the following functions:

log(1 – x); – 1 ≤ x ≤ 1

Write the Maclaurin series expansion of the following functions:

tan^–1 (x); – 1 ≤ x ≤ 1