Mathematics and Statistics Shaalaa.com Model Set 1 2020-2021 HSC Science (General) 12th Standard Board Exam Question Paper Solution

Which of the following statement is true

3 + 7 = 4 or 3 – 7 = 4

If Pune is in Maharashtra, then Hyderabad is in Kerala

It is false that 12 is not divisible by 3

The square of any odd integer is even

If A = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]`, then A¹⁰ = ______

`[(cos10  alpha, -sin10  alpha),(sin10  alpha, cos10  alpha)]`

`[(cos10  alpha, sin10  alpha),(-sin10  alpha, cos10  alpha)]`

`[(cos10  alpha, sin10  alpha),(-sin10  alpha, -cos10  alpha)]`

`[(cos10  alpha, -sin10  alpha),(-sin10  alpha, -cos10  alpha)]`

Bernoulli distribution is a particular case of binomial distribution if n = ______

4

10

2

1

If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for"  x = 1","  2","  3","),(0",", "otherwise"):}` then E(X) = ______

`343/297`

`294/251`

`297/294`

`294/297`

The separate equations of the lines represented by `3x^2 - 2sqrt(3)xy - 3y^2` = 0 are ______ 

`x + sqrt(3)y` = 0 and `sqrt(3)x + y` = 0

`x - sqrt(3)y` = 0 and `sqrt(3)x - y` = 0

`x - sqrt(3)y` = 0 and `sqrt(3)x + y` = 0

`x + sqrt(3)y` = 0 and `sqrt(3)x - y` = 0

Let I₁ = `int_"e"^("e"^2)  1/logx  "d"x` and I₂ = `int_1^2 ("e"^x)/x  "d"x` then 

I₁ = `1/3 "I"_2`

I₁ + I₂ = 0 

I₁ = 2I₂ 

I₁ = I₂ 

If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______

f(x) − log x + c

f(x) + log x + c

log x − f(x) + c

`1/5x^5` f(x) + c

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

State the truth value of `sqrt(3)` is not an irrational number

Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`

Solve each of the following inequations graphically using XY-plane:

4x - 18 ≥ 0

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

Write the converse and contrapositive of the following statements.

“If a function is differentiable then it is continuous”

Find the principal solutions of tan x = `-sqrt(3)`

Find the combined equation of the following pair of lines passing through point (2, 3) and parallel to the coordinate axes.

Find k, if the sum of the slopes of the lines represented by x² + kxy − 3y² = 0 is twice their product. 

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

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

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

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

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

Find the Cartesian equation of the line passing through A(1, 2, 3) and B(2, 3, 4)

If `bara, barb` and `barc` are position vectors of the points A, B, C respectively and `5bara - 3barb - 2barc = bar0`, then find the ratio in which the point C divides the line segement BA.

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

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

In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled

Maximize z = 10x + 25y subject to x + y ≤ 5, 0 ≤ x ≤ 3, 0 ≤ y ≤ 3

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

A random variable X has the following probability distribution:

Determine:

1. k
2. P(X < 3)
3. P( X > 4)

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

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

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 "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`

If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

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

If three numbers are added, their sum is 2. If 2 times the second number is subtracted from the sum of first and third numbers, we get 8. If three times the first number is added to the sum of second and third numbers, we get 4. Find the numbers using matrices.

Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`

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

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

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)`

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

Show that the lines `(x + 1)/(-10) = (y + 3)/(-1) = (z - 4)/(1)` and `(x + 10)/(-1) = (y + 1)/(-3) = (z - 1)/4` intersect each other.also find the coordinates of the point of intersection

Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r = (mbarb + nbara)/(m + n) `