Mathematics and Statistics Model set 2 shaalaa.com 2021-2022 HSC Science (General) 12th Standard Board Exam Question Paper Solution

For the random variable X, if V(X) = 4, E(X) = 3, then E(x²) is ______

9

13

12

7

The area of triangle ΔABC whose vertices are A(1, 1), B(2, 1) and C(3, 3) is ______ sq.units

1

2

3

4

`(hat"i" + hat"j" - hat"k")*(hat"i" - hat"j" + hat"k")` = ______.

`hat"i" - hat"j" - hat"k"`

1

−1

`−hat"j" + hat"k"`

The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______

`pi/3, (11pi)/6`

`pi/6, (11pi)/6`

`pi/4, (11pi)/4`

`pi/6, (11pi)/3`

Negation of p → (p ˅ ∼ q) is ______

∼ p → (∼ p ˅ q)

p ˄ (∼ p ˄ q)

∼ p ˅ (∼ p ˅ ∼ q)

∼ p → (∼ p → q)

The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______.

2, 1, 6

2, 1, – 6

2, – 1, 6

– 2, 1, 6

If the tangent at (1, 1) on y² = x(2 − x)² meets the curve again at P, then P is

(4, 4)

(−1, 2)

(3, 6)

`(9/4, 3/8)`

`int_0^(pi/2) log(tanx)  "d"x` =

`pi/8(log2)`

0

`- pi/8 (log2)`

`pi/2 (log2)`

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

If `bar("a") = 4hat"i" + 3hat"k"` and `bar("b") = -2hat"i" + hat"j" + 5hat"k"`, then find `2bar("a") + 5bar("b")`

State the truth value of (p ˅ ∼p)

`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________

e^x log x + c

e^x (log x)² + c

e^2x log x + c

e^2x (log x)² + c

Find the area of the region bounded by the following curves, X-axis and the given lines: x = 2y, y = 0, y = 4

In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B

Find the perpendicular distance of origin from the plane 6x − 2y + 3z - 7 = 0

Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x)  "d"x`

Write the following compound statements symbolically.

Triangle is equilateral or isosceles

If f(x) = x² − 2x − 3 then find f(A) when A = `[(1, 2),(2, 1)]`

`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`

If y = log [cos(x⁵)] then find `("d"y)/("d"x)`

Find the differential equation of family of all ellipse whose major axis is twice the minor axis

Evaluate : `int_1^9(x + 1)/sqrt(x)*dx`

If a vector has direction angles 45° and 60°, find the third direction angle.

Find the direction ratios of the line perpendicular to the lines

`(x - 7)/2 = (y + 7)/(-3) = (z - 6)/1` and `(x + 5)/1 = (y + 3)/2 = (z - 6)/(-2)`

Solve the following problem :

Let the p. m. f. of the r. v. X be

`"P"(x) = {((3 - x)/(10)", ","for"  x = -1", "0", "1", "2.),(0,"otherwise".):}`
Calculate E(X) and Var(X).

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

Find the condition that the line 4x + 5y = 0 coincides with one of the lines given by ax² + 2hxy + by² = 0 

Use quantifiers to convert the following open sentences defined on N, into a true statement.

n² ≥ 1

A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

`int sqrt((9 + x)/(9 - x))  "d"x`

Solve the differential equation `("d"y)/("d"x) + y` = e^−x 

If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a², b², c² are in A.P.

Find the measure of the acute angle between the line represented by `3"x"^2 - 4sqrt3"xy" + 3"y"^2 = 0` 

Solve the differential equation (x² – yx²)dy + (y² + xy²)dx = 0

If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`

In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.

Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4)

`int (3x + 4)/sqrt(2x^2 + 2x + 1)  "d"x`

If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin²x

Solve the following:

A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.

Find m, if the lines `(1 - x)/3 =(7y - 14)/(2"m") = (z - 3)/2` and `(7 - 7x)/(3"m") = (y - 5)/1 = (6 - z)/5` are at right angles

Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations.

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn.

Maximize z = 7x + 11y subject to 3x + 5y ≤ 26, 5x + 3y ≤ 30, x ≥ 0, y ≥ 0