Mathematics and Statistics Model set 3 shaalaa.com 2021-2022 HSC Science (General) 12th Standard Board Exam Question Paper Solution

In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.

`1 : sqrt(3) : 2`

`2 : sqrt(3) : 1`

`sqrt(3) : 1 : 2`

`sqrt(3) : 2 : 1`

If the sum of two unit vectors is itself a unit vector, then the magnitude of their difference is ______.

`sqrt(2)`

`sqrt(3)`

1

2

If the lines represented by kx² − 3xy + 6y² = 0 are perpendicular to each other, then

k = 6

k = − 6

k = 3

k = − 3

If the foot of the perpendicular drawn from the origin to the plane is (4, −2, -5), then the equation of the plane is ______ 

4x + y + 5z = 14

4x − 2y − 5z = 45

x − 2y − 5z = 10

4x + y + 6z = 11

If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______

35

12

`7/5`

105

The slope of the normal to the curve y = x² + 2e^x + 2 at (0, 4) is ______.

2

−2

`1/2`

`-1/2`

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

The solution of `("d"y)/("d"x)` = 1 is

x + y = c

xy = c

x² + y² = c

y – x = c

A random variable X has the following probability distribution

Then the variance of this distribution is

0.6

0.7

0.77

0.66

Find a unit vector in the opposite direction of `bar("u")`. Where `bar("u") = 8hat"i" + 3hat"j" - hat"k"`

Evaluate cot(tan⁻¹(2x) + cot⁻¹(2x))

If A = `[(-2, 4),(-1, 2)]` then find A² 

Find the vector equation of the line passing through the point having position vector `4hat i - hat j + 2hat"k"` and parallel to the vector `-2hat i - hat j + hat k`.

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are 1, 3, 2 and –1, 1, 2

Reduce the equation `bar"r"*(3hat"i" + 4hat"j" + 12hat"k")` = 8 to normal form

Solve the differential equation sec²y tan x dy + sec²x tan y dx = 0

`int cos^7 x  "d"x`

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

Without using truth table prove that:

~ (p ∨ q) ∨ (~ p ∧ q) ≡ ~ p

Find the matrix X such that AX = I where A = `[(6, 17),(1, 3)]`

Evaluate: `int_1^3 (cos(logx))/x  "d"x`

If the normal to the plane has direction ratios 2, −1, 2 and it’s perpendicular distance from origin is 6, find its equation

`int(log(logx))/x  "d"x`

A r.v. X ~ B(n, p). If the values of mean and variance of X are 18 and 12 respectively, then find total number of positive values of X.

Differentiate sin² (sin⁻¹(x²)) w.r. to x

Show that the points A(2, –1, 0) B(–3, 0, 4), C(–1, –1, 4) and D(0, – 5, 2) are non coplanar

Prove that cot⁻¹(7) + 2 cot⁻¹(3) = `pi/4`

The probability that a person who undergoes a kidney operation will be recovered is 0.5. Find the probability that out of 6 patients who undergo similar operation half of them recover.

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

The surface area of a spherical balloon is increasing at the rate of 2cm²/sec. At what rate the volume of the balloon is increasing when radius of the balloon is 6 cm?

Write the converse, inverse, and contrapositive of the following statement.

"If it snows, then they do not drive the car"

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

Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x

Find the Cartesian equation of the plane passing through the points A(1, 1, 2), B(0, 2, 3) C(4, 5, 6)

Solve the following by inversion method 2x + y = 5, 3x + 5y = −3

Examine whether the following statement pattern is a tautology, a contradiction or a contingency.

(p ∧ ~ q) → (~ p ∧ ~ q)

`int 1/(2 +  cosx - sinx)  "d"x`

D and E divides sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE.

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

If θ is the acute angle between the lines given by ax² + 2hxy + by² = 0 then prove that tan θ = `|(2sqrt("h"^2) - "ab")/("a" + "b")|`. Hence find acute angle between the lines 2x² + 7xy + 3y² = 0 

Minimize z = 2x + 4y is subjected to 2x + y ≥ 3, x + 2y ≥ 6, x ≥ 0, y ≥ 0 show that the minimum value of z occurs at more than two points

A man of height 180 cm is moving away from a lamp post at the rate of 1.2 meters per second. If the height of the lamp post is 4.5 meters, find the rate at which
(i) his shadow is lengthening
(ii) the tip of the shadow is moving

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after `5/2` hours  `("Given"  sqrt(2) = 1.414)`

Evaluate: `int_0^pi 1/(3 + 2sinx + cosx)  "d"x`

Find the area of the region lying between the parabolas 4y² = 9x and 3x² = 16y