Maths

Let Q be the foot of the perpendicular from the point P(7, -2, 13) on the plane containing the lines $\frac{x + 1}{6} = \frac{y - 1}{7} = \frac{z - 3}{8} a n d \frac{x - 1}{3} = \frac{y - 2}{5} = \frac{z - 3}{7} .$ Then (PQ)², is equal to________.

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Contributor-Level 10

${\vec{n}}_{1} = 6 \hat{i} + 7 \hat{j} + 8 \hat{k} {\vec{n}}_{2} = 3 \hat{i} + 5 \hat{j} + 7 \hat{k}$

${\vec{n}}_{1} * {\vec{n}}_{2} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & 7 & 8 \\ 3 & 5 & 7 \end{matrix} | = 9 \hat{i} - 18 \hat{j} + 9 k$

$∴$ the normal to required plane is $\hat{i} - 2 \hat{j} + \hat{k}$

Equation of plane 1(x + 1) – 2(y – 1) + (z – 3) = 0

x – 2y + z = 0

P (7, -2, 13)

$∴ P Q = | \frac{7 + 4 + 13}{\sqrt[]{1 + 4 + 1}} | = \frac{24}{\sqrt[]{6}}$

${(P Q)}^{2} = \frac{24 * 24}{6} = 96$

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3 months ago

Let the mean and variance of four numbers 3, 7, x and y(x > y) be 5 and 10 respectively. Then the mean of four numbers 3 + 2x, 7 + 2y, x + y and x – y is ________.

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$\frac{3 + 7 + x + y}{4} = 5$

$\Rightarrow x + y = 10 . . . . . . . . . (i)$

$\frac{1}{4} (9 + 49 + x^{2} + y^{2}) - 25 = 10$

x² + y² = 82 .(ii)

$∴ \bar{x} = \frac{(3 + 2 x) + (7 + 2 y) + (x + y) + (x - y)}{4}$

$= \frac{10 + 4 x + 2 y}{4}$

Solving (i) & (ii), x = 9, y = 1

$∴ \bar{x} = \frac{48}{4} = 12$

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3 months ago

The least positive integer n such that $\frac{{(2 i)}^{n}}{{(1 - i)}^{n - 2}}, i = \sqrt[]{- 1},$ is a positive integer, is_______.

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Contributor-Level 10

$\frac{{(2 i)}^{n}}{{(1 - i)}^{n - 2}}$

$= \frac{{(2 i)}^{n} {(1 + i)}^{n - 2}}{{(1 + 1)}^{n - 2}}$

$= - 4 {(- 1 + i)}^{n - 2}$

$n o w {(- 1 + i)}^{2} = - 2 i$

$∴ - 4 {(- 1 + i)}^{n - 2} \in (+) v e$ integer for n – 2 = 4

->n = 6

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3 months ago

If the projection of the vector $\hat{i} + 2 \hat{j} + \hat{k}$ on the sum of the two vectors $2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $- λ \hat{i} + 2 \hat{j} + 3 \hat{k}$ is 1, then $λ$ is equal to_________.

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Contributor-Level 10

$\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}$

$\vec{b} = 2 \hat{i} + 4 \hat{j} - 5 \hat{k} \vec{c} = - λ \hat{i} + 2 \hat{j} + 3 \hat{k}$

$\vec{b} + \vec{c} = (2 - λ) \hat{i} + 6 \hat{j} - 2 \hat{k}$

${(12 - λ)}^{2} = {(2 - λ)}^{2} + 40$

$λ = 5$

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3 months ago

The sum of all 3-digit numbers less then or equal to 500, that are formed without using the digit “1” and they all are multiple of 11, is________.

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Contributor-Level 10

The 1^st such digit is 11 * 19 = 209

Sum = [209 + 220 + 231 + .+ 495] - [231 + 319 + 341 + 418 + 451] = 7744

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3 months ago

Let A be a 3 * 3 real matrix. If det(2 adj(2 Adj(adj (2A))) = 2⁴¹, then the value of det (A²) equals________.

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Contributor-Level 10

$| 2 A | = 2^{3} | A |$

replace A by adj 2A

$| 2 a d j 2 A | = 2^{3} | a d j 2 A |$

$= 2^{3} {| 2 A |}^{2} = 2^{3} {(2^{3} | A |)}^{2}$

$= 2^{9} {| A |}^{2}$

Again replace A by (adj A)

|2adj 2 (adj (adj 2A)| = 2⁹ |adj 2A|⁴

= 2⁹ (|2A|²)⁴

->|A²| = 4

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Let $λ \neq 0$ be in R. If a and b are the roots of the equation x² – x + $2 λ = 0,$ and a and $γ$ are the roots of the equation 3x² – 10x + $27 λ = 0,$ then $\frac{β γ}{λ}$ is equal to_________.

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Contributor-Level 10

a + b = 1 $α + γ = \frac{10}{3}$

$α β = 2 λ$ $α γ = 9 λ$

$\frac{β}{γ} = \frac{2}{9},$ $β - γ = 1 - \frac{10}{3} = - \frac{7}{3}$

$\Rightarrow β = \frac{2}{3}$ $γ = 3$

$α = \frac{1}{3}, λ = \frac{1}{9}$

$∴ \frac{β γ}{λ} = 18$

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3 months ago

Let a and b respectively be the points of local maximum and local minimum of the function F(x) = 2x³ – 3x² – 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4 A is equal to________.

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Contributor-Level 10

$f (x) = 2 x^{3} - 3 x^{2} - 12 x$

$f' (x) = 6 x^{2} - 6 x - 12$

= 6 (x – 2) (x + 1)

a = -1, b = 2

A = \int_{- 1}^{0} (2 x^{3} - 3 x^{2} - 12 x) d x - \int_{0}^{2} (2 x^{3} - 3 x^{2} - 12 x) d x = \frac{57}{2}

->4A = 114

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3 months ago

If $A_{k} = \sum_{i = 0}^{9} (\begin{matrix} 9 & i \end{matrix}) [\begin{matrix} 12 & 12 - k + i \end{matrix}] + \sum_{i = 0}^{8} (\begin{matrix} 8 & i \end{matrix}) [\begin{matrix} 13 & 13 - k + i \end{matrix}]$ and A₄ – A₃ = 190 p, then p is equal to__________.

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Contributor-Level 10

= coefficient of x⁴ in (1 + x)²¹ + coefficient of x⁴ in (1 + x)²¹

$=^{21} C_{4} +^{21} C_{4} = 2 .^{21} C_{4}$

$A_{3}$ = coefficient of x³ in (1 + x)²¹ + coefficient of x³ in (1 + x)²¹

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3 months ago

Let be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $\vec{r} \cdot (\hat{i} + \hat{j} + 4 \hat{k}) = 16 a n d \vec{r} \cdot (- \hat{i} + \hat{j} + \hat{k}) = 6 .$

Then which of the following points does NOT lie on P?

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