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a year ago

Maths Class 11th

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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alok kumar singh

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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New answer posted

a year ago

Maths Class 11th

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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alok kumar singh

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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New answer posted

a year ago

Maths Class 12th

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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alok kumar singh

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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New answer posted

a year ago

Maths Class 11th

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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alok kumar singh

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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New answer posted

a year ago

Maths Class 12th

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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alok kumar singh

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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New answer posted

a year ago

Maths Class 11th

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__________.

0 Follower 4 Views

alok kumar singh

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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New answer posted

a year ago

Maths Class 12th

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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alok kumar singh

Contributor-Level 10

Equation of required plane

$(x + y + 4 z - 16) + λ (- x + y + z - 6) = 0$ it passes (1, 2, 3)

$\Rightarrow - 1 + λ (- 2) = 0$

$\Rightarrow λ = - \frac{1}{2}$

$∴$ Equation of plane

$(1 - λ) x + (1 + λ) y + (4 + λ) z - 16 - 6 λ = 0$

$\frac{3}{2} x + \frac{1}{2} y + \frac{7}{2} z - 13 = 0$

$\Rightarrow 3 x + y + 7 z = 26$

(4, 2, 2) not satisfying the plane

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New answer posted

a year ago

Maths Class 11th

Let a circle C in complex plane pass through the points z₁ = 3 + 4i, z₂ = 4 + 3i and z₃ = 5i. If $z (\neq z_{1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then arg(z) is equal to:

0 Follower 4 Views

Vishal Baghel

Contributor-Level 10

$\frac{2}{- 4} = \frac{- 1}{2}$

y – 4 = 2 (x – 3)

y = 2x – 2

x² + (2x – 2)² = 25

$5 x^{2} - 8 x - 21 = 0$

$z (\frac{- 7}{5}, \frac{- 24}{5})$

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New answer posted

a year ago

Maths Class 11th

Let x = 2t, $y = \frac{t^{2}}{3}$ be a conic. Let S be the focus and B the point on the axis of the conic such that $S A ⊥ B A,$ where A is any point on the conic. If k is the ordinate of the centroid of the $Δ S A B,$ then $\underset{t \to 1}{l i m}$ k is equal to:

0 Follower 4 Views

Vishal Baghel

Contributor-Level 10

x = 2t $A (2 t, \frac{t^{2}}{3})$ $\frac{^{}}{}$

$y = \frac{t^{2}}{3}$ S (0, 3)

${\frac{}{}}^{}$ $3 y = {(\frac{x}{2})}^{2}$ $B (0, λ)$

$3 k = \frac{t^{2}}{3} + 3 + λ = \frac{2 t^{2}}{3} + 3 - \frac{12 t^{2}}{9 - t^{2}}$

$\underset{t \to 1}{l i m} 3 k = \frac{2}{3} + 3 - \frac{12}{8} = 3 - \frac{5}{6} = \frac{13}{6}$

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New answer posted

a year ago

Maths Differential Equations

If the solution curve y = y(x) of the differential equation y²dx + (x² – xy + y²)dy = 0, which passes through the point (1, 1) and intersects the line y = $\sqrt[]{3} x$ at the point $(α, \sqrt[]{3} α),$ then value of $l o g_{e} (\sqrt[]{3} α)$ is equal to:

0 Follower 2 Views

Vishal Baghel

Contributor-Level 10

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

$\frac{d x}{d y} + \frac{x^{2} - x y + y^{2}}{y^{2}} = 0 \Rightarrow \frac{d x}{d y} + {(\frac{x}{y})}^{2} - (\frac{x}{y}) + 1 = 0$

Put x = vy $\Rightarrow v + y \frac{d v}{d y} + v^{2} - v + 1 = 0$

$\Rightarrow \frac{y d v}{d y} + v^{2} + 1 = 0$

$\Rightarrow \frac{π}{6} + l n | y | = \frac{π}{4}$

$l n | y | = \frac{π}{12}$

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