Mathematics and Statistics Shaalaa.com Model Set 3 2020-2021 HSC Science (General) 12th Standard Board Exam Question Paper Solution

Which of the following is not a statement?

Smoking is injuries to health

2 + 2 = 4

2 is the only even prime number.

Come here

The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______

x = 1, y = 2, z = 3

x = 2, y = 1, z = 3

x = −1, y = 2, z = 3

x = y = z = 3

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`

Given that X ~ B(n = 10, p), E(X) = 8, then value of p = ______

0.4

0.8

0.6

0.7

If a d.r.v. X has the following probability distribution:

then k = ______

`1/7`

`1/8`

`1/9`

`1/10`

Select and write the correct alternative from the given option for the question

The order and degree of `(("d"y)/("d"x))^3 - ("d"^3y)/("d"x^3) + y"e"^x` = 0 are respectively

3, 1

1, 3

3, 3

1, 1

`int  ("e"^x(x - 1))/(x^2)  "d"x` = ______ 

`x"e"^(-x) + c`

`("e"^x)/(x^2) + c`

`(x - 1/x)"e"^x + c`

`("e"^x)/x + c`

If α, β, γ are direction angles of a line and α = 60°, β = 45°, then γ = ______.

30° or 90°

45° or 60°

90° or 130°

60° or 120°

State the truth Value of x² = 25

Find the area bounded by the curve y² = 36x, the line x = 2 in first quadrant 

If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)` 

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. Find the velocity when 𝑡 = 2 sec

If statements p, q are true and r, s are false, determine the truth values of the following.

(p ∧ ~r) ∧ (~q ∨ s)

If A = `[(2, 4),(1, 3)]` and B = `[(1, 1),(0, 1)]` then find (A⁻¹ B⁻¹)

Find A^–1 using adjoint method, where A = `[(cos theta, sin theta),(-sin theta, cos theta)]`

If tan⁻¹x + tan⁻¹y + tan⁻¹z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1

With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`

Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`

Verify y = log x + c is the solution of differential equation `x ("d"^2y)/("d"x^2) + ("d"y)/("d"x)` = 0

Find the derivative of the inverse of function y = 2x³ – 6x and calculate its value at x = −2

Find the values of x, for which the function f(x) = x³ + 12x² + 36𝑥 + 6 is monotonically decreasing

A car is moving in such a way that the distance it covers, is given by the equation s = 4t² + 3t, where s is in meters and t is in seconds. What would be the velocity and the acceleration of the car at time t = 20 seconds?

`int (cos2x)/(sin^2x cos^2x)  "d"x`

Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `-5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3:2
(i) internally
(ii) externally

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

n² ≥ 1

Use quantifiers to convert the given open sentence defined on N into a true statement

3x – 4 < 9

Use quantifiers to convert the given open sentence defined on N into a true statement

Y + 4 > 6

Find the area of the region bounded by the curves x² = 8y, y = 2, y = 4 and the Y-axis, lying in the first quadrant

The probability that certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive

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 combined equation of the following pair of line passing through (−1, 2), one is parallel to x + 3y − 1 = 0 and other is perpendicular to 2x − 3y − 1 = 0

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

If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`

If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`

`int sec^2x sqrt(tan^2x + tanx - 7)  "d"x`

`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`

Find the Cartesian equation of the line passing through (−1, −1, 2) and parallel to the line 2x − 2 = 3y + 1 = 6z – 2

Find acute angle between the lines `(x - 1)/1 = (y - 2)/(-1) = (z - 3)/2` and `(x - 1)/2 = (y - 1)/1 = (z - 3)/1`

In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0

Maximize z = −x + 2y subjected to constraints x + y ≥ 5, x ≥ 3, x + 2y ≥ 6, y ≥ 0 is this LPP solvable? Justify your answer.

Prove that: 

`{:(int_(-a)^a f(x) dx  = 2 int_0^a f(x) dx",", "If"  f(x)  "is an even function"),(                                       = 0",", "if"  f(x)  "is an odd function"):}`

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? `("Given" sqrt(3/2) = 1.2247)`

A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?

Find the Cartesian and vector equation of the plane which makes intercepts 1, 1, 1 on the coordinate axes

If four points `"A"(bar"a"), "B"(bar"b"), "C"(bar"c") and "D"(bar"d")` are coplanar, then show that `[(bar"a", bar"b", bar"c")] + [(bar"b", bar"c", bar"d")] + [(bar"c", bar"a", bar"d")] = [(bar"a", bar"b", bar"c")]`.

If Q is the foot of the perpendicular from P(2, 4, 3) on the line joining the point A(1, 2, 4) and B(3, 4, 5), find coordinates of Q