Maths

The value of the integral $\frac{48}{π^{4}} \int_{0}^{π} (\frac{3 π x^{2}}{2} - x^{3}) \frac{s i n x}{1 + c o s^{2} x} d x$ is equal to……….

0 Follower 4 Views

Contributor-Level 9

$l = \frac{48}{π^{4}} \int_{0}^{π} [{(\frac{π}{2} - x)}^{3} - \frac{3 π^{2}}{4} (\frac{π}{2} - x) + \frac{π^{3}}{4}] \frac{s i n x d x}{1 + c o s^{2} x}$

Using $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x$

we get $l = \frac{48}{π^{4}} \int_{0}^{π} [- {(\frac{π}{2} - x)}^{3} + \frac{3 π^{2}}{4} (\frac{π}{2} - x) + \frac{π^{3}}{4}] \frac{s i n x d x}{1 + c o s^{2} x}$

Adding these two equations, we get

$\Rightarrow l = \frac{12}{π} {[- t a n^{- 1} (c o s x)]}_{0}^{π} = \frac{12}{π} . \frac{π}{2} = 6$

0 0

New answer posted

11 months ago

Let A = ${n \in N : H . C . F . (n, 45) = 1}$ and Let B = ${2 k : k \in {1, 2, . . . . . ., 100}} .$ Then the sum of all the elements of $A \cup B i s . . . . . . . . . .$

0 Follower 2 Views

Contributor-Level 9

Sum of all elements of $A \cap B = 2$ [Sum of natural number upto 100 which are neither divisible by 3 nor by 5]

$= 2 [\frac{100 * 101}{2} - 3 (\frac{33 * 34}{2}) - 5 (\frac{20 * 21}{2}) + 15 (\frac{6 * 7}{2})]$

= 10100 – 3366 – 2100 + 630

= 5264

0 0

New answer posted

11 months ago

If $s i n^{2} (1 0^{0}) s i n (2 0^{0}) s i n (4 0^{0}) s i n (5 0^{0}) s i n (7 0^{0}) = α - \frac{1}{16} s i n (1 0^{0}),$ then 16 + a^-1 is equal to……….

0 Follower 2 Views

Contributor-Level 9

$(s i n 10 ° . s i n 50 ° . s i n 70 °) . (s i n 10 ° . s i n 20 ° . s i n 40 °)$

$= (\frac{1}{4} s i n 30 °) . [\frac{1}{2} s i n 10 ° (c o s 20 ° - c o s 60 °)]$

$= \frac{1}{32} [s i n 30 ° - s i n 10 ° - s i n 10 °]$

$\frac{1}{64} - \frac{1}{16} s i n 10 °$

Clearly $α = \frac{1}{64}$

Hence 16 + a^-1 = 80

0 0

New answer posted

11 months ago

Let the solution curve y = y(x) of the differential equation (4 + x²)dy – 2x(x² + 3y + 4)dx = 0 pass through the origin. They y(2) is equal to………

0 Follower 3 Views

Contributor-Level 9

$(4 + x^{2}) d y - 2 x (x^{2} + 3 y + 4) d x = 0$

$\Rightarrow \frac{d y}{d x} = (\frac{6 x}{x^{2} + 4}) y + 2 x$

$e^{- 3 l n (x^{2} + 4)} = \frac{1}{{(x^{2} + 4)}^{3}}$

so $\frac{y}{{(x^{2} + 4)}^{3}} = \int \frac{2 x}{{(x^{2} + 4)}^{3}} d x + c$

$\Rightarrow y = - \frac{1}{2} (x^{2} + 4) + c {(x^{2} + 4)}^{3}$

When x = 0, y = 0 gives $c = \frac{1}{32}$

So, for x = 2, y = 12

0 0

New answer posted

11 months ago

Let f(x) = $m a x {| x + 1 |, | x + 2 |, . . . . . ., | x + 5 |} . T h e n \int_{- 6}^{0} f (x) d x$ is equal to………

0 Follower 6 Views

Contributor-Level 9

$∴ \int_{- 6}^{0} f (x) d x = 2 * \frac{1}{2} (2 + 5) * 3 = 21$

0 0

New answer posted

11 months ago

LE the common tangents to the curves $4 (x^{2} + y^{2}) = 9$ and y² = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac{l}{e^{2}}$ is equal to…….

0 Follower 16 Views

Contributor-Level 9

Let y = mx + c is the common tangent

$s o c = \frac{1}{m} = \pm \frac{3}{2} \sqrt[]{1 + m^{2}} \Rightarrow m^{2} = \frac{1}{3}$

so equation of common tangents will be

$y = \pm \frac{1}{\sqrt[]{3}} x \pm \sqrt[]{3}$

which intersects at Q (-3, 0)

Major axis and minor axis of ellipse are 12 and 6. So eccentricity

$e^{2} = 1 - \frac{1}{4} = \frac{3}{4}$

...more

Let y = mx + c is the common tangent

$s o c = \frac{1}{m} = \pm \frac{3}{2} \sqrt[]{1 + m^{2}} \Rightarrow m^{2} = \frac{1}{3}$

so equation of common tangents will be

$y = \pm \frac{1}{\sqrt[]{3}} x \pm \sqrt[]{3}$

which intersects at Q (-3, 0)

Major axis and minor axis of ellipse are 12 and 6. So eccentricity

$e^{2} = 1 - \frac{1}{4} = \frac{3}{4}$

and length of latus rectum

$= \frac{2 b^{2}}{a} = 3$

Hence $\frac{l}{e^{2}} = \frac{3}{3 / 4} = 4$

less

0 0

New answer posted

11 months ago

There are ten boys B₁, B₂,…., B₁₀ and five girls G₁, G₂,…., G₅ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B₁ and B₂ together should not be the members of a group, is………….

0 Follower 6 Views

Contributor-Level 9

Boys (10) Girls (5)

(3) (3)

B₁ & B₂ should not be selected together

Total number of ways

= (56 + 56) * 10 = 1120

0 0

New answer posted

11 months ago

The sum of the cubes of all the roots of the equation $x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 0$ is…………

0 Follower 16 Views

Contributor-Level 9

$x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 1 = 0$

$\Rightarrow x = \pm 1$

and let a, b are roots of x² – 3x – 1 = 0

$∴ α + β = 3 α β = - 1$

$∴ 1^{3} + {(- 1)}^{3} + α^{3} + β^{3}$

= 27 + 3 (3) = 36

0 0

New answer posted

11 months ago

Let $Δ, \nabla \in {\land, \lor}$ be such that $p \nabla q \Rightarrow ((p Δ q) \nabla r)$ is a tautology. Then $(p \nabla q) Δ r$ is logically equivalent to:

0 Follower 5 Views

Maths Inverse Trigonometric Functions

Contributor-Level 9

Case – I $Δ \equiv \nabla \equiv \land$

$(p \land q) \to ((p \land q) \land r)$

it can be false if r is false,

so not a tautology

Case – II If $Δ \equiv \nabla \equiv \lor$

$(p \lor q) ? \to ((p \lor q) \lor r) \equiv$ tautology

then $(p \lor q) \lor r \equiv (p Δ r) \lor q$

Case – III If $Δ \equiv v, \nabla \equiv \land$ &nbs

...more

Case – I $Δ \equiv \nabla \equiv \land$

$(p \land q) \to ((p \land q) \land r)$

it can be false if r is false,

so not a tautology

Case – II If $Δ \equiv \nabla \equiv \lor$

$(p \lor q) ? \to ((p \lor q) \lor r) \equiv$ tautology

then $(p \lor q) \lor r \equiv (p Δ r) \lor q$

Case – III If $Δ \equiv v, \nabla \equiv \land$

then $(p \land q) \to {(p \lor q) \land r}$

Not a tautology

Case – IV If $Δ \equiv \land, \nabla \equiv \lor$

$(p \land q) \to {(p \land q) \lor r}$

Not a tautology

less

0 0

New answer posted

11 months ago

Let f(x) = $2 c o s^{- 1} x + 4 c o t^{- 1} x - 3 x^{2} - 2 x + 10, x \in [- 1, 1] . I f [a, b]$ is the range of the function f, then 4a – b is equal to:

0 Follower 4 Views