Maths

Consider the following statements:

A : Rishi is a judge.

B : Rishi is honest.

C : Rishi is not arrogant.

The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is

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$?$ given statement is

$(A \land C) \to B$ then its negation is ${(A \land C) \to B}$

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4 months ago

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

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

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

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

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

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

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4 months ago

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

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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4 months ago

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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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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4 months ago

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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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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4 months ago

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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4 months 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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4 months 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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4 months 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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4 months 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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