Mathematics and Statistics Model set 2 by shaalaa.com 2024-2025 HSC Science (General) 12th Standard Board Exam Question Paper Solution

The direction ratios of the line which is perpendicular to the two lines `(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1) and (x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)` are ______.

4, 5, 7

4, –5, 7

4, –5, –7

–4, 5, 8

Choose correct alternatives :

The vector equation of line 2x – 1 = 3y + 2 = z – 2 is ______.

`barr = (1/2hati - 2/3hatj + 2hatk) + lambda(3hati + 2hatj + 6hatk)`

`barr = hati - hatj + (2hati + hatj + hatk)`

`barr = (1/2hati - hatj) + lambda(hati - 2hatj + 6hatk)`

`barr = (hati + hatj) + lambda(hati - 2hatj + 6hatk)`

If `int_2^e [1/logx - 1/(logx)^2].dx = a + b/log2`, then ______.

a = e, b = –2

a = e, b = 2

a = –e, b = 2

a = –e, b = –2

Select the correct option from the given alternatives:

The general solution of sec x = `sqrt(2)` is ______.

`2npi +- pi/4, n∈Z`

`2npi +- pi/2, n∈Z`

`npi +- pi/2, n∈Z`

`2npi +- pi/3, n∈Z`

Choose the correct option from the given alternatives : 

If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.

1

0

–1

2

Choose correct alternatives :

The shortest distance between the lines `vecr = (hati + 2hatj + hatk) + lambda(hati - hatj + hatk) and vecr = (2hati - hatj - hatk) + μ(2hati + hatj + 2hatk)` is ______.

`(1)/sqrt(3)`

`(1)/sqrt(2)`

`(3)/sqrt(2)`

`sqrt(3)/(2)`

Select the correct option from the given alternatives:

The principal solutions of equation cot θ = `sqrt3` are ______.

`pi/6, (7pi)/6`

`pi/6, (5pi)/6`

`pi/6, (8pi)/6`

`(7pi)/6, pi/3`

The area of triangle formed by the lines x² + 4xy + y² = 0 and x - y - 4 = 0 is ______.

`4/sqrt3` sq units

`8/sqrt3` sq units

`16/sqrt3` sq units

`15/sqrt3` sq units

Evaluate: `int_0^(pi/2) x sin x.dx`

Determine the order and degree of the following differential equation:

`(dy)/(dx) = (2sin x + 3)/(dy/dx)`

Determine the order and degree of the following differential equation:

`[1 + (dy/dx)^2]^(3/2) = 8(d^2y)/dx^2`

Check whether the following matrix is invertible or not:

`[(cos theta, sin theta),(-sin theta, cos theta)]`

Differentiate the following w.r.t.x:

tan[cos(sinx)]

Find the separate equations of the lines represented by the equation 3x² – 10xy – 8y² = 0.

Find the vector equation of the plane passing through the point having position vector `hati + hatj + hatk` and perpendicular to the vector `4hati + 5hatj + 6hatk`.

Show that the line `(x - 2)/(1) = (y - 4)/(2) = (z + 4)/(-2)` passes through the origin.

Solve the following :

Find the area of the region bounded by the curve y = 4x², Y-axis and the lines y = 1, y = 4.

State if the following is not the probability mass function of a random variable. Give reasons for your answer

Write converse, inverse and contrapositive of the following statement. "If voltage increases then current decreases".

Find the value of k if lines represented by kx² + 4xy – 4y² = 0 are perpendicular to each other.

If the vectors `2hati - qhatj + 3hatk` and `4hati - 5hatj + 6hatk` are collinear, find q.

Given X ~ B(n, p) if p = 0.6 and E(X) = 6, find n and Var(X).

Solve graphically: 3x + 2y ≥ 0

Find the inverse of the following matrix.

`[(2, -3),(-1, 2)]`

Show that the following points are collinear:

P = (4, 5, 2), Q = (3, 2, 4), R = (5, 8, 0).

Evaluate the following integrals:

`int(2)/(sqrt(x) - sqrt(x + 3)).dx`

For the differential equation, find the particular solution (x – y²x) dx – (y + x²y) dy = 0 when x = 2, y = 0

Differentiate the following w.r.t. x: `x^(tan^(-1)x`

Find the principal solutions of the following equation:

sin 2θ = `-1/2`

Evaluate: `int_0^π sin^3x (1 + 2cosx)(1 + cosx)^2.dx`

In a box of floppy discs, it is known that 95% will work. A sample of three of the discs is selected at random. Find the probability that none of the floppy disc work.

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`

Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`

Find the Cartesian co-ordinates of the point whose polar co-ordinates are:

`(3/4, (3pi)/4)`

Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die.

Solve the following: 

Find the maximum and minimum values of the function f(x) = cos²x + sinx.

If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.

Solve the following differential equation:

`(cos^2y)/x dy + (cos^2x)/y dx` = 0

In ΔABC, if cot A, cot B, cot C are in A.P. then show that a², b², c² are also in A.P.

Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`

Using the rules in logic, write the negation of the following:

(p ∨ q) ∧ (q ∨ ∼r)

Find the co-ordinates of the foot of the perpendicular drawn from the point `2hati - hatj + 5hatk` to the line `barr = (11hati - 2hatj - 8hatk) + λ(10hati - 4hatj - 11hatk).` Also find the length of the perpendicular.

If `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"c" = hat"j" - hat"k"`. find a vector `vec"b"` satisfying `vec"a" xx vec"b" = vec"c"` and `vec"a"·vec"b"` = 3.