Class 12th

If the shortest distance between the lines ${\vec{r}}_{1} = α \hat{i} + 2 \hat{j} + 2 \hat{k} + λ (\hat{i} - 2 \hat{j} + 2 \hat{k}), λ \in R,$ a > 0 and ${\vec{r}}_{2} = - 4 \hat{i} - \hat{k} + μ (3 \hat{i} - 2 \hat{j} - 2 \hat{k}), μ \in R$ is 9, the a is equal to………

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Contributor-Level 9

$d i s t . = \frac{{\vec{A}}_{1} A_{2} . ({\vec{b}}_{1} * {\vec{b}}_{2})}{| {\vec{b}}_{1} * {\vec{b}}_{2} |}$

$= \frac{(- 4 - α, - 2, - 3) . (2, 2, 1)}{3}$

$| - 5 - \frac{2 α}{3} | = 9 \Rightarrow - 5 - \frac{2 α}{3} = \pm 9$

$\Rightarrow \frac{2 α}{3} = 4, - 14,$

but a > 0

a = 6

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New answer posted

a year ago

Direction for questions: Answer the questions on the basis of the information given below.

In a school when a survey is conducted on liking of subjects out of three subjects physics, chemistry mathematics then outcome is as follow.

Total number of students who like physics, chemistry and mathematics is 30, 40, 50 respectively. 10 students shown interest in all three subjects, Number of students who like physics and mathematics both is equal to number of students who like chemistry and mathematics both and is equal to 3/4 times number of students who like physics and chemistry both 15 students like only chemistry.

When class teacher saw

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Contributor-Level 10

From the given condition g = 10,

Number of students who like both Physics and Maths = 10 + f

Number of students who like both chemistry and Maths = 10 + e

From given condition 10 + f = 10 + e or e = f

Similarly from 3^rd condition ¾ (10 + e) = 10 + d

Or 30 + 3e = 40 + 4d

d = 3e - 10/4

Since 15 students like only chemistry hence b = 15

Number of students who like only chemistry is 40

Hence e + g + d + b = 40

or e + 10 + (3e – 10)/4 + 15 = 40

or (7e – 10)/4 = 15 or 7e = 70 or e = 10

Hence a = 5, c = 20, f = e = 10, d = 5

Hence the distribution is as follows

As per the observation of class teacher actual numbers of students who lik

...more

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New answer posted

a year ago

If the value of $\underset{x \to 0}{l i m} {(2 - c o s x \sqrt[]{c o s 2 x})}^{(\frac{x + 2}{x^{2}})}$ is equal to e^a, then a is equal to………

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Contributor-Level 9

$\underset{x \to 0}{l i m} {(2 - c o s x . \sqrt[]{c o s 2 x})}^{\frac{x + 2}{x^{2}}}$ $(1^{\infty})$

= $\underset{x \to 0}{l i m} e^{2 (\frac{s i n x \sqrt[]{c o s 2 x} - c o s x . \frac{1 (- 2 s i n 2 x)}{2 \sqrt[]{c o s 2 x}}}{2 x})}$

$= \underset{x \to 0}{l i m} e^{2 (\frac{1}{2} + 1)} = e^{3} = e^{a}$

a = 3

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New answer posted

a year ago

Let P be a plane passing through the points (1,0,1) (1,-2,1) and (0,1,-2). Let a vector $\vec{a} = α \hat{i} + β \hat{j} + γ \hat{k}$ be such $\vec{a}$ that is the parallel to the plane P, perpendicular to $(\hat{i} + 2 \hat{j} + 3 \hat{k}) a n d \vec{a} . (\hat{i} + \hat{j} + 2 \hat{k}) = 2,$ then ${(α - β + γ)}^{2}$ equals………

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Contributor-Level 9

For the plane P,

$\vec{n} = \frac{1}{2} (- 2 \hat{j}) * (- \hat{i} + \hat{j} - 3 \hat{k}) = \hat{j} * \hat{i} + 3 \hat{j} * \hat{k} = - \hat{k} + 3 \hat{i}$

$\vec{a} . \vec{n} = 0 \Rightarrow 3 α - γ = 0 . . . . . . . (i)$

$\vec{a} . (1, 2, 3) = 0 \Rightarrow α + 2 β + 3 γ = 0 . . . . . . . . . . (i i)$

$\vec{a} . (1, 1, 2) = 2 \Rightarrow α + β + 2 γ = 2 . . . . . . . . . . . (i i i)$

(i), (ii) & (iii) Þ a = 1, β = -5, $γ =$ 3

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New answer posted

a year ago

Eight girls – ATHENA, BRISEIS, CALANDRA, DEMETER, ELLIE, FEO, GAIA, and HELEN are sitting in a row facing North. We know the following information about them.

I. The number of girls to the left of ATHENA is equal to the number of girls to the right of GAIA.

II. CALANDRA is sitting three places away to the left of ELLIE.

III. The number of girls sitting between FEO and DEMETER is two.

IV. If BRISEIS and HELEN interchange their positions, then HELEN is fifth to the right of BRISEIS.

V. CALANDRA is sitting at the extreme left end of the row.

VI. GAIA is sitting exactly between BRISEIS and DEMETER.

Who is sitting extreme right

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Contributor-Level 10

According to the information provided in the statement following arrangement will be obtained:

C	H	A	E	D	G	B	F
1	2	3	4	5	6	7	8

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New answer posted

a year ago

The number of rational terms in the binomial expansion of ${(4^{\frac{1}{4}} + 5^{\frac{1}{6}})}^{120}$ is…….

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Contributor-Level 9

$T_{r + 1} =^{120} C_{r} 4^{\frac{120 - r}{4}} . 5^{r / 6}$

For to be rational

r should be multiple of 6

Number of such terms = 21

0 0

New question posted

a year ago

Eight girls – ATHENA, BRISEIS, CALANDRA, DEMETER, ELLIE, FEO, GAIA, and HELEN are sitting in a row facing North. We know the following information about them.

I. The number of girls to the left of ATHENA is equal to the number of girls to the right of GAIA.

II. CALANDRA is sitting three places away to the left of ELLIE.

III. The number of girls sitting between FEO and DEMETER is two.

IV. If BRISEIS and HELEN interchange their positions, then HELEN is fifth to the right of BRISEIS.

V. CALANDRA is sitting at the extreme left end of the row.

VI. GAIA is sitting exactly between BRISEIS and DEMETER.

Who is sitting extreme r

0 Follower 1 View

New answer posted

a year ago

Eight girls – ATHENA, BRISEIS, CALANDRA, DEMETER, ELLIE, FEO, GAIA, and HELEN are sitting in a row facing North. We know the following information about them.

I. The number of girls to the left of ATHENA is equal to the number of girls to the right of GAIA.

II. CALANDRA is sitting three places away to the left of ELLIE.

III. The number of girls sitting between FEO and DEMETER is two.

IV. If BRISEIS and HELEN interchange their positions, then HELEN is fifth to the right of BRISEIS.

V. CALANDRA is sitting at the extreme left end of the row.

VI. GAIA is sitting exactly between BRISEIS and DEMETER.

How many girls are sitting b

0 Follower 1 View

Contributor-Level 10

According to the information provided in the statement following arrangement will be obtained:

C	H	A	E	D	G	B	F
1	2	3	4	5	6	7	8

0 0

New answer posted

a year ago

Eight girls – ATHENA, BRISEIS, CALANDRA, DEMETER, ELLIE, FEO, GAIA, and HELEN are sitting in a row facing North. We know the following information about them.

I. The number of girls to the left of ATHENA is equal to the number of girls to the right of GAIA.

II. CALANDRA is sitting three places away to the left of ELLIE.

III. The number of girls sitting between FEO and DEMETER is two.

IV. If BRISEIS and HELEN interchange their positions, then HELEN is fifth to the right of BRISEIS.

V. CALANDRA is sitting at the extreme left end of the row.

VI. GAIA is sitting exactly between BRISEIS and DEMETER.

Who is the third from the ri

0 Follower 1 View

Contributor-Level 10

According to the information provided in the statement following arrangement will be obtained:

C	H	A	E	D	G	B	F
1	2	3	4	5	6	7	8

0 0

New answer posted

a year ago

Directions for questions: Read the information carefully and answer the questions given below.

Eight girls – ATHENA, BRISEIS, CALANDRA, DEMETER, ELLIE, FEO, GAIA, and HELEN are sitting in a row facing North. We know the following information about them.

I. The number of girls to the left of ATHENA is equal to the number of girls to the right of GAIA.

II. CALANDRA is sitting three places away to the left of ELLIE.

III. The number of girls sitting between FEO and DEMETER is two.

IV. If BRISEIS and HELEN interchange their positions, then HELEN is fifth to the right of BRISEIS.

V. CALANDRA is sitting at the extreme left end of

0 Follower 1 View