Maths

The number of solutions of the equation $c o s (x + \frac{π}{3}) c o s (\frac{π}{3} - x) = \frac{1}{4} c o s^{2} 2 x, x \in [- 3 π, 3 π] i s :$

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$c o s (x + \frac{π}{3}) c o s (\frac{π}{3} - x) = \frac{1}{4} c o s^{2} 2 x$

$x = - 3 π, - 2 π, - π, 0, π, 2 π, 3 π$

$∴$ total number of solution = 7.

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a year ago

A random variable X has the following probability distribution

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$?$ x is a random variable.

$\begin{array}{l} ∴ k + 2 k + 4 k + 6 k + 8 k = 1 \\ ∴ k = \frac{1}{21} \end{array}$

$∴ P ((1 < x < 4) | x \leq 2) = \frac{4 k}{7 k} = \frac{4}{7}$

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a year ago

The number of distinct real roots of the equation x⁷ – 7x – 2 = 0 is

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$f (x) = x^{7} - 7 x - 2$

$f' (x) = 7 x^{6} - 7 \Rightarrow f' (x) = 0, x = \pm 1$

$∴ f (1) = < 0 a n d f (- 1) > 0$

Hence number of real roots of f (x) = 0 are 3.

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New answer posted

a year ago

Let the area of the triangle with vertices A $(1, α), B (α, 0) C (0, α)$ be 4 sq. units. If the points $(α, - α), (- α, α) a n d (α^{2}, β)$ are collinear, then $β$ is equal to

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$Δ = 4 \Rightarrow \frac{1}{2} | \begin{matrix} 1 & α & 1 \\ α & 0 & 1 \\ 0 & α & 1 \end{matrix} | = 4$

$\Rightarrow α = \pm 8$

$(α, - α), (- α, α) a n d (α^{2}, β)$ are collinear.

$∴ | \begin{matrix} α & - α & 1 \\ - α & α & 1 \\ α^{2} & β & 1 \end{matrix} | = 0 \Rightarrow β = - 64$

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New answer posted

a year ago

Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1, a > 2,$ having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $6 \sqrt[]{3}$ . Then the eccentricity of the ellipse is

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$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1$

$∴ Δ = \frac{1}{2} * a (1 + c o s θ) . 4 s i n θ$

$∴ Δ_{m a x}^{Δ} = 6 \sqrt[]{3} \Rightarrow a = 4$

$∴ e = \frac{\sqrt[]{3}}{2}$

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New answer posted

a year ago

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with

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Equation of tangent at P (x, y) is Y = $\frac{d y}{d x} (X - x)$

$A / q, 2 x - y \frac{d x}{d y} = 0 \Rightarrow 2 \frac{d y}{y} = \frac{d x}{x}$

$\begin{array}{l} \Rightarrow 2 l n y = l n x + l n c \\ y^{2} = x c \end{array}$

It passes through (3, 3), c = 3

$∴ y^{2} = 3 x ∴$ Length of latus rectum = 3

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New answer posted

a year ago

$\underset{x \to \infty}{l i m} (\frac{n^{2}}{(n^{2} + 1) (n + 1)} + \frac{n^{2}}{(n^{2} + 4) (n + 2)} + \frac{n^{2}}{(n^{2} + 9) (n + 3)} + . . . + \frac{n^{2}}{(n^{2} + n^{2}) (n + n)}) i s e q u a l t o$

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S = $\underset{n \to \infty}{L t} \sum_{r = 1}^{n} \frac{n^{2}}{(n^{2} + r^{2}) (n + r)}$

$\frac{π}{8} + \frac{1}{4} l n 2$

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New answer posted

a year ago

The value of the integral $\int_{- π / 2}^{π / 2} \frac{d x}{(1 + e^{x}) (s i n^{6} x + c o s^{6} x)} i s e q u a l t o$

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$l = \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{d x}{(1 + e^{x}) (s i n^{6} x + c o s^{6} x)}$ ……. (i)

$= 2 \int_{0}^{\infty} \frac{d t}{4 + t^{2}} = 2 {(t a n^{- 1} (\frac{t}{2}))}_{0}^{\infty} = π$

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New answer posted

a year ago

Let f(x) ${\begin{matrix} \frac{s i n (x - [x])}{x - [x]} & , & x \in (- 2, - 1) \\ m a x {2 x, 3 [| x |]} & , & | x | < 1 \\ 1 & , & o t h e r w i s e \end{matrix}$

where [t] denotes greatest integer $\leq t .$ If m is the number of points where f is not continuous and n is the number of points where f is not differentiable, then the ordered pair (m, n) is

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$f (x) = {\begin{cases} \frac{s i n (x + 2)}{x + 2}, x \in (- 2, - 1) \\ 0, x \in (- 1, 0] \\ 2 x, x \in (0, 1) \\ 1, o t h e r w i s e \end{cases}$

$L H D = \underset{h \to 0}{L t} \frac{f (0 - h) - f (0)}{- h} = 0$

$R H D = \underset{h \to 0}{L t} \frac{f (0 + h) - f (0)}{h} = 2$

Hence f (x) is not differentiable at x = 1, 0, 1

$∴ m = 2, n = 3$

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New answer posted

a year ago

Let x,y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is

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