Maths

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ration 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (a, β)). Then the value of 7a + 3β) is equal to……….

0 Follower 3 Views

Contributor-Level 9

0 0

New answer posted

2 months ago

If $\vec{a} = 2 \hat{i} + \hat{j} + 3 \hat{k}, \vec{b} = 3 \hat{i} + 3 \hat{j} + \hat{k} a n d \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ are coplanar vectors and $\vec{a} . \vec{c} = 5, \vec{b} ⊥ \vec{c}, t h e n 122 (c_{1} + c_{2} + c_{3})$ is equal to……….

0 Follower 6 Views

Contributor-Level 9

$\vec{a} = 2 \hat{i} + \hat{j} + 3 \hat{k}$

$\begin{array}{l} \vec{b} = 3 \hat{i} + 3 \hat{j} + \hat{k} \\ \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k} \end{array}$

$C o p l n a n a r \Rightarrow | \begin{matrix} 2 & 1 & 3 \\ 3 & 3 & 1 \\ c_{1} & c_{2} & c_{3} \end{matrix} | = 0$

$\begin{array}{l} \Rightarrow - 8 c_{1} + 7 c_{2} + 12 c_{3} = 0 . . . . . . . . (i) \\ \vec{a} . \vec{c} = 5 \Rightarrow 2 c_{1} + c_{2} + 3 c_{3} = 5 . . . . . . . . (i i) \\ \vec{b} . \vec{c} = 0 \Rightarrow 3 c_{1} + 3 c_{2} + c_{3} = 0 . . . . . . . . (i i i) \end{array}$

Solving (i), (ii), (iii)

$C_{1} = \frac{10}{122}, c_{2} = \frac{- 85}{122}, c_{3} = \frac{225}{122}$

$∴ 122 (c_{1} + c_{2} + c_{3}) = 150$

0 0

New question posted

2 months ago

Maths Ncert Solutions Maths class 12th

The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecking it was found that, in the observations, 20 was misread as 5. Then, the correct variance is equal to…………….

0 Follower 2 Views

New answer posted

2 months ago

The number of real solutions of the equation e^4x + 4e^3x – 58e^2x + 4e^x + 1 = 0, is……………

0 Follower 3 Views

Contributor-Level 9

$e^{4 x} + 4 e^{3 x} - 58 e^{2 x} + 4 e^{x} + 1 = 0$

$(e^{2 x} + \frac{1}{e^{2 x}}) + 4 (e^{x} + \frac{1}{e^{x}}) - 58 = 0$

$\Rightarrow {(e^{x} + \frac{1}{e^{x}} + 2)}^{2} = 64$

$e^{x} = \frac{6 \pm \sqrt[]{32}}{2} = 3 \pm 2 \sqrt[]{2}$

0 0

New answer posted

2 months ago

Let R₁ and R₂ be relations on the set {1,2,…., 50} such that

R₁ = ${(p, p^{n}) : p i s a p r i m e a n d n \geq 0 i s a n i n t ? e g e r} a n d$

R₂ = ${(p, p^{n}) : p i s a p r i m e a n d n = 0 o r 1} .$

Then, the number of elements in R₁ – R₂ is…………

0 Follower 5 Views

Contributor-Level 9

A = {1, 2, 3, ….50}

R₁ = (2, 1), (2, 2), (2, 4)…. (2, 32)

(3, 1) (3, 3) (3, 9) (3, 27)

(13, 1) (13, 13) etc.

R₂ = { (2, 1), (2, 2), (3, 1), (3, 3), (5, 1), (5, 5), (7, 1), (7, 7), (1, 1), (11, 11)…}

$∴ R_{1} - R_{2}$ contains only 9 elements.

0 0

New answer posted

2 months ago

Let p, q, r be three logical statements. Consider the compound statements

$S_{1} : ((\sim p) \lor q) \lor ((\sim p) \lor r) a n d$

$S_{2} : p \to (q \lor r)$

Then, which of the following is NOT true?

0 Follower 1 View

Contributor-Level 9

kindly consider the following Image

0 0

New answer posted

2 months ago

Let AB and PQ be two vertical poles, 160m apart from each other. Let C be the middle points of B and Q, which are feet of these two poles. Let $\frac{π}{8}$ and q be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan²q is equal to

0 Follower 4 Views

Contributor-Level 9

0 0

New question posted

2 months ago

The probability, that in a randomly selected 3-digit number at least two digits are odd, is

0 Follower 2 Views

New answer posted

2 months ago

Let the plane $P : \vec{r} . \vec{a} = d$ contains the line of intersection of two planes $\vec{r} . (\hat{i} + 3 \hat{j} - \hat{k}) = 6$ and $\vec{r} . (6 \hat{i} + 5 \hat{j} - \hat{k}) = 7 .$ If the plane P passes through the point $(2, 3, \frac{1}{2})$ , then the value of $\frac{{| 13 \vec{a} |}^{2}}{d^{2}}$ is equal to

0 Follower 2 Views

Contributor-Level 9

$P : (x + 3 y - z - 6) = λ (- 6 x + 5 y - z - 7) = 0$

passes $(2, 3, \frac{1}{2})$

$\Rightarrow (2 + 9 - \frac{1}{2} - 6) = λ (- 12 + 15 - \frac{1}{2} - 7) = 0$

$\Rightarrow λ = 1$

$\frac{{| 13 \vec{a} |}^{2}}{d^{2}} = \frac{{(13)}^{2} (93)}{{(13)}^{2}} = 93$

0 0

New answer posted

2 months ago

The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

0 Follower 3 Views