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Class 12th Maths

If y = tan^-1 (sec x³ tanx³), $\frac{π}{2} < x^{3} < \frac{3 π}{2}, t h e n$

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Payal Gupta

Contributor-Level 10

$L e t x^{3} = θ \Rightarrow \frac{θ}{2} \in (\frac{π}{4}, \frac{3 π}{4})$

$∴ y = t a n^{- 1} (s e c θ - t a n θ)$

$t a n^{- 1} (\frac{1 - s i n θ}{c o s θ})$

$\Rightarrow \frac{d y}{d x} = \frac{- 3 x^{2}}{2} \Rightarrow \frac{d^{2} y}{d x^{2}} = - 3 x$

$∴ x^{2} \frac{d^{2} y}{d x^{2}} - 6 y + \frac{3 π}{2} = 0 \Rightarrow x^{2} y^{11} - 6 y + \frac{3 π}{2} = 0$

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New answer posted

6 months ago

Class 12th Maths

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $| (\hat{a} + \hat{b}) + 2 (\hat{a} * \hat{b}) |$ = 2. If $θ \in (0, π)$ is the angle between $\hat{a} a n d \hat{b}$ , then among the statements:

$(S 1) : 2 | \hat{a} * \hat{b} | = | \hat{a} - \hat{b} |$

(S1) : The projection of $\hat{a} o n (\hat{a} + \hat{b}) i s \frac{1}{2}$

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Payal Gupta

Contributor-Level 10

$| \vec{a} + \vec{b} + 2 (\vec{a} * \vec{b}) | = 2, θ \in (0, π)$

squaring on both sides, we get = $\frac{2 π}{3}$

where θ is angle between $\hat{a} a n d \hat{b} .$

$2 | \hat{a} * \hat{b} | = \sqrt[]{3} = | \hat{a} - \hat{b} |, S_{1}$ is correct.

and projection of $\hat{a} o n \hat{a} + \hat{b} = | \frac{\hat{a} . (\hat{a} + \hat{b})}{| \hat{a} + \hat{b} |} | = \frac{1}{2}$

So (S₂) is correct.

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New answer posted

6 months ago

Class 12th Maths

Let the points on the plane P be equidistance from the points (–4, 2, 1) and (2, –2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is

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Payal Gupta

Contributor-Level 10

Let P (x, y, z) be any point on plane P₁ then

${(x + 4)}^{2} + {(y - 2)}^{2} + {(z - 1)}^{2} = {(x - 2)}^{2} + {(y + 2)}^{2} + {(z - 3)}^{2}$

$∴ c o s θ = \frac{| 6 - 2 + 3 |}{14} \Rightarrow θ = \frac{π}{3}$

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New answer posted

6 months ago

Class 12th Maths

If the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{λ}$ and $\frac{x - 2}{1} = \frac{y - 4}{4} = \frac{z - 5}{5} i s \frac{1}{\sqrt[]{3}},$ then the sum of all possible values of $λ$ is

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Payal Gupta

Contributor-Level 10

S. D. = $\frac{1}{\sqrt[]{3}}$

$b_{1} * b_{2} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & λ \\ 1 & 4 & 5 \end{matrix} | = \hat{i} (15 - 4 λ) + \hat{j} (λ - 10) + \hat{k} (5)$

$\frac{| (- \hat{i} - 2 \hat{j} - 2 \hat{k}) . {(15 - 4 λ) \hat{i} + (λ - 10) \hat{j} + 5 \hat{k}} |}{\sqrt[]{{(15 - 4 λ)}^{2} + {(λ - 10)}^{2} + 25}} = \frac{1}{\sqrt[]{3}}$

$\Rightarrow λ = 16$

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New answer posted

6 months ago

Class 12th Maths

A random variable X has the following probability distribution

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Payal Gupta

Contributor-Level 10

$?$ 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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New answer posted

6 months ago

Class 12th Maths

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

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Payal Gupta

Contributor-Level 10

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

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

$∴ f (1) = < 0 a n d f (- 1) > 0$

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

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New answer posted

6 months ago

Class 12th Maths

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

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Payal Gupta

Contributor-Level 10

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

6 months ago

Class 12th Maths

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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Payal Gupta

Contributor-Level 10

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

6 months ago

Class 12th Maths

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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Payal Gupta

Contributor-Level 10

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

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

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

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

$∴ m = 2, n = 3$

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New answer posted

6 months ago

Class 12th Maths Matrices

Let the system of linear equations

x + y + $α z = 2$

3x + y + z = 4

x + 2z = 1

have a unique solution $(x^{*}, y^{*}, z^{*}) .$ If $(α, x^{*}), (y^{*}, α) a n d (x^{*}, - y^{*})$ are collinear points, then the sum of absolute values of all possible values of $α$ is

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Payal Gupta

Contributor-Level 10

x + y + az = 2………… (i)

3x + y + z = 4 ………… (ii)

x + 2z = 1 ……………. (iii)

x = 1, y = 1, z = 0

(for unique solution $a \neq - 3$

$(α, - 1), (1, α) a n d (1, - 1)$ are collinear.

$\frac{α - 1}{1 - α} = \frac{- 1 - α}{1 - 1} \Rightarrow 1 - α^{2} = 0 \Rightarrow α = \pm 1$

Sum of absolute value = 2

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