Mathematics and Statistics Shaalaa.com Model Set 2 2020-2021 HSC Science (General) 12th Standard Board Exam Question Paper Solution

A biconditional statement is the conjunction of two ______ statements.

Negative

Compound

Connective

Conditional

If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______

`(3/(4sqrt(2)), -3/(4sqrt(2)))`

`(3/(4sqrt(2)), 3/(4sqrt(2)))`

`(-3/(4sqrt(2)), 3/(4sqrt(2)))`

`(-3/(4sqrt(2)), -3/(4sqrt(2)))`

The feasible region is the set of point which satisfy.

The object functions

All the given constraints

Some of the given constraints

Only one constraint

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

Differential equation of the function c + 4yx = 0 is

`xy + ("d"y)/("d"x)` = 0

`x  ("d"y)/("d"x) + y` = 0

`("d"y)/("d"x) - 4xy` = 0

`x ("d"y)/("d"x) + 1` = 0

If x = cos⁻¹(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

t

– t

`(-1)/"t"`

`1/"t"`

A ladder 5 m in length is resting against vertical wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 1.5 m /sec. The length of the higher point of the when foot of the ladder is 4 m away from the wall decreases at the rate of ______

1

2

2.5

3

Select the correct option from the given alternatives:

If l, m, n are direction cosines of a line then `"l"hat
"i" + "m"hat"j" + "n"hat"k"` is ______ 

null vector

the unit vector along the line

any vector along the line

a vector perpendicular to the line

Select the correct option from the given alternatives:

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC respectively of a ΔABC. The length of the median through A is

`sqrt(34)/2`

`sqrt(48)/2`

`sqrt(18)`

`sqrt34`

State the truth Value of x² = 25

Evaluate: `int_0^1 "e"^x/sqrt("e"^x - 1)  "d"x`

An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?

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”

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

Find the graphical solution for the system of linear inequation 2x + y ≤ 2, x − y ≤ 1

Find the area enclosed between the X-axis and the curve y = sin x for values of x between 0 to 2π

Find the area of the ellipse `x^2/36 + y^2/64` = 1, using integration

Let the p.m.f. of r.v. X be P(x) = `""^4"C"_x (5/9)^x (4/9)^(4 - x)`, x = 0, 1, 2, 3, 4. Find E(X) and Var(X)

Find the value of h, if the measure of the angle between the lines 3x² + 2hxy + 2y² = 0 is 45°. 

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

Water is being poured at the rate of 36 m³/sec in to a cylindrical vessel of base radius 3 meters. Find the rate at which water level is rising

`int ["cosec"(logx)][1 - cot(logx)]  "d"x`

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.

If a line has the direction ratios 4, −12, 18, then find its direction cosines

Write the following statements in symbolic form

Milk is white if and only if the sky is not blue

Write the following statements in symbolic form

If Kutab – Minar is in Delhi then Taj - Mahal is in Agra

Write the following statements in symbolic form

Even though it is not cloudy, it is still raining

Three chairs and two tables cost ₹ 1850. Five chairs and three tables cost ₹2850. Find the cost of four chairs and one table by using matrices

Transform `[(1, 2, 4),(3, -1, 5),(2, 4, 6)]` into an upper triangular matrix by using suitable row transformations

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.

In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`

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 none will recover

Prove that: `int_0^"a" "f"(x)  "d"x = int_0^"a" "f"("a" - x)  "d"x`. Hence find `int_0^(pi/2) sin^2x  "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 differential equation by eliminating arbitrary constants from the relation x² + y² = 2ax

If log₅ `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`

Find the vector equation of the line passing through the point having position vector `-hat"i"- hat"j" + 2hat"k"` and parallel to the line `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter

Find the equation of the plane passing through the point (7, 8, 6) and parallel to the plane `bar"r"*(6hat"i" + 8hat"j" + 7hat"k")` = 0

The following is the c.d.f. of r.v. X:

Find p.m.f. of X.
i. P(–1 ≤ X ≤ 2)
ii. P(X ≤ 3 / X > 0).

Show that the combined equation of pair of lines passing through the origin is a homogeneous equation of degree 2 in x and y. Hence find the combined equation of the lines 2x + 3y = 0 and x − 2y = 0

If y = cos(m cos^–1x), then show that `(1 - x^2) ("d"^2y)/("d"x^2) - x("d"y)/("d"x) + "m"^2y` = 0

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

`int  ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1])  "d"x`

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

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.

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) `