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47. Prove that the line through the point (x₁, y₁) and parallel to the line Ax + By + C = 0 is A (x –x₁) + B (y – y₁) = 0.

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Contributor-Level 10

47.

The slope of line 4x + by + c = 0 is

m = -A/B

As the required line is parallel to the line Ax + by + c = 0

They have the same slope ie, m = -A/B

So, equation of line with slope m and passing through (x₁, y₁)

is given by point-slope from as,

⇒ - A (x-x₁) B (y -y₁)

⇒ A (x-x₁) + B (y-y₁)= 0.

Hence proved.

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11 months ago

Does Macquarie Uni have a train station?

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Gunjan Thapa

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Yes, Macquarie Uni has its own train station. It is the only university in Australia with its own metro station. This station is conveniently situated on the corner of the Herring Road and University Avenue, near Macquarie Centre shopping complex. The university's train station is on the T1 Northern Line between Hornsby and the City via Macquarie Park.

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11 months ago

Relations and Functions Maths

2. Check the injectivity and surjectivity of the following functions:

(i) f: N → N given by f(x) = x²

(ii) f: Z → Z given by f(x) = x²

(iii) f: R → R given by f(x) = x²

(iv) f: N → N given by f(x) = x³

(v) f: Z → Z given by f(x) = x³

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Vishal Baghel

Contributor-Level 10

(i) $f : N \to N$ given by $f (x) = x_{}^{2}$

For, $x_{1}, x_{2} \in N$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{2} = x_{2}^{2}$

$\Rightarrow x_{1}^{} = x_{2}^{} \notin N$

So, $f$ is one-one/ injective

For $x \in N$ , i.e., $x = 1, 2, 3 . . . .$

Range of $f (x) = {1_{}^{2}, 2_{}^{2}, 3_{}^{2} . . .} = {1, 4, 9 . . .} \neq N$

i.e., co-domain of $N$

So, $f$ is not onto/ subjective

(ii) $f : Z \to Z$ given by $f (x) = x_{}^{2}$

For, $x_{1}, x_{2} \in Z$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{2} = x_{2}^{2}$

$\Rightarrow x_{1}^{} = \pm x_{2}^{} \notin Z$

i.e., $x_{1} = x_{2}$ and $x_{1} = - x_{2}$

So, $f$ is not one-one/ injective

For $x \in Z$ , $x = 0, \pm 1, \pm 2, \pm 3 . . . .$

Range of $f (x) = {0_{}^{2}, {(\pm 1)}_{}^{2}, {(\pm 2)}_{}^{2}, {(\pm 3)}_{}^{2} . . .}$

${0, 1, 4, 9 . . . .} \neq$ co-domain $Z$

So, $f$ is not onto/ subjective

(iii) $f : R \to R$ given by $f (x) = x_{}^{2}$

For, $x_{1}, x_{2} \in R$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{2} = x_{2}^{2}$

$\Rightarrow x_{1}^{} = \pm x_{2}^{}$

So, $f$ is not injective

For $x \in R$

Range of $f (x) = {x_{}^{2}, x \in R}$ gives a set of all positive real numbers

Hence, range of $f (x \neq)$ co-domain of $R$

So, $f$ is not subjective

(iv) $f : N \to N$ given by&n

...more

(i) $f : N \to N$ given by $f (x) = x_{}^{2}$

For, $x_{1}, x_{2} \in N$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{2} = x_{2}^{2}$

$\Rightarrow x_{1}^{} = x_{2}^{} \notin N$

So, $f$ is one-one/ injective

For $x \in N$ , i.e., $x = 1, 2, 3 . . . .$

Range of $f (x) = {1_{}^{2}, 2_{}^{2}, 3_{}^{2} . . .} = {1, 4, 9 . . .} \neq N$

i.e., co-domain of $N$

So, $f$ is not onto/ subjective

(ii) $f : Z \to Z$ given by $f (x) = x_{}^{2}$

For, $x_{1}, x_{2} \in Z$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{2} = x_{2}^{2}$

$\Rightarrow x_{1}^{} = \pm x_{2}^{} \notin Z$

i.e., $x_{1} = x_{2}$ and $x_{1} = - x_{2}$

So, $f$ is not one-one/ injective

For $x \in Z$ , $x = 0, \pm 1, \pm 2, \pm 3 . . . .$

Range of $f (x) = {0_{}^{2}, {(\pm 1)}_{}^{2}, {(\pm 2)}_{}^{2}, {(\pm 3)}_{}^{2} . . .}$

${0, 1, 4, 9 . . . .} \neq$ co-domain $Z$

So, $f$ is not onto/ subjective

(iii) $f : R \to R$ given by $f (x) = x_{}^{2}$

For, $x_{1}, x_{2} \in R$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{2} = x_{2}^{2}$

$\Rightarrow x_{1}^{} = \pm x_{2}^{}$

So, $f$ is not injective

For $x \in R$

Range of $f (x) = {x_{}^{2}, x \in R}$ gives a set of all positive real numbers

Hence, range of $f (x \neq)$ co-domain of $R$

So, $f$ is not subjective

(iv) $f : N \to N$ given by $f (x) = x_{}^{3}$

For, $x_{1}, x_{2} \in N$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{3} = x_{2}^{3}$

$\Rightarrow x_{1}^{} = x_{2}^{} \in N$

So, $f$ is injective

For $x \in N,$

Range of $f (x) = {1_{}^{3}, 2_{}^{3}, 3_{}^{3} . . . .}$

${1_{}^{}, 8, 2 7_{}^{} . . . .} \neq$ co-domain of $N$

So, $f$ is not subjective

(v) $f : R \to R$ given by $f (x) = x_{}^{3}$

For, $x_{1}, x_{2} \in N$ , $f (x_{1}) = f (x_{2})$

$\Rightarrow x_{1}^{3} = x_{2}^{3}$

$\Rightarrow x_{1}^{} = x_{2}^{} \in N$

So, $f$ is injective

For $x \in R,$

Range of $f (x) = {1_{}^{3}, 2_{}^{3}, 3_{}^{3} . . . .}$

${1_{}^{}, 8, 2 7_{}^{} . . . .} \neq$ co-domain of R

So, $f$ is not subjective

less

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11 months ago

Macquarie University New South Wales

How much does it cost to live in Sydney while studying at Macquarie University?

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Bhumika Kaur

Contributor-Level 10

Macquarie University premium accommodation offers 340+ rooms with the option of studio, twin, four, four or six-bedroom apartments. Additionally, students can also opt for private accommodation that is offered exclusively to the university's students. An estimated living cost for Macquarie University students is given below:

Particulars	Estimated Weekly Cost
Studio room	AUD 500 – 700 (Around INR 27k – 38k)
1 Bedroom apartment	AUD 700 – 900 (Around INR 38k – 49k)
2 Bedroom apartment	AUD 450 – 600 (Around INR 24k – 32k)
Student shared house	AUD 370 – 550 (Around INR 20k – 30k)
Gas, electricity	AUD 30 – 50 (Around INR 1,600 – 2,700)

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11 months ago

46. The line through the points (h, 3) and (4, 1) intersects the line 7x − 9y −19 = 0 at right angle. Find the value of h.

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Business Management Studies MBA/PGDM

Contributor-Level 10

46. Slope of line 1 (passing) through (h. 3) and (4, 1) is

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I got 50 marks in CUET PG in MBA. Can I get admission in central university?

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Where is Macquarie University located?

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Nishtha Singh

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Macquarie University's main campus Wallumattagal is located 15 kms from Sydney's City Centre. The size of the MQ Uni is 126 hectares. The main campus offers everything to students such as cafes, bar and restaurants, a pharmacy, counselling services, etc. MQ Uni is one of the easiest universities to get to by public transport.

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Atul Arora

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Yes, a few colleges offer BA Fashion Design in distance mode in India:

BA Fashion Design Distance Colleges	Tuition Fee
U P Rajarshi Tandon Open University	INR 32,000
Rachna Sansad School of Fashion and Textile Design	NA

Note: The information is sourced from the official/counselling body of the institute/exam.

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Tasbiya Khan

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Yes, there are many affordable top ranked Arts colleges in Mumbai. Some of them are mentioned below along with their tuition fees:

Low-cost Colleges	Tuition Fee
St. Xavier's College, Mumbai	INR 23,361
Mithibai College of Arts	INR 37,634
KJ Somaiya College of Arts and Commerce	INR 2,400
KC College	INR 19,640
BK Birla College of Arts, Science and Commerce	INR 14,365

Disclaimer: This information is sourced from the official website and may vary.

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Chandra Chaudhary

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To apply for Master of Finance course of Macquarie Uni, students can apply online by filling out the application form. For the one-year course, the applicants are required to have AQF Level 7 Bachelor's qualification in a related field or recognised equivalent or AQF Level 8 Bachelor's Honors or Graduate Diploma qualification in a related field or recognised equivalent. With this, two years of relevant approved work experience is required.

Documents required to apply to this course are:

Passport
Evidence of English language proficiency
Previous academic transcripts

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45. Find angles between the lines √3x + y = 1and x + √3y = 1.

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Contributor-Level 10

Kindly go through the solution

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What is the duration of the Macquarie University Master of Business Analytics course?

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Ishita Singh

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The duration of the Macquarie University Master of Business Analytics course is 1.5 years – 2 years. The estimated annual fee for this course is AUD 46,200 (Around INR 25.36 lakh). This course has a strong focus on the real-world application of Business Analytics tools and techniques. Students are going to develop their knowledge of Databases, Computer Programming, Applied Statistics, Machine Learning techniques applied to Big Data.

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What after MAH BCA CET seat allotment?

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Jasleen Shukla

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After MAH BCA CET seat allotment is announced, selected candidates must confirm their admission before the final date by document verification along with fee payment.

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11 months ago

44. Find equation of the line perpendicular to the line x – 7y + 5 = 0 and havingx intercept 3.

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Contributor-Level 10

44.

The slope of line x- 7y + 5 = 0 is

So, equation of line with slope m₂ and having x-intercept 3 is

y = m (x-d), d = x- intercept

⇒y = - 7 (x- 3)

⇒y = - 7x + 21

⇒ 7x + y- 21 = 0.

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11 months ago

43. Find equation of the line parallel to the line 3x − 4y + 2 = 0 and passing throughthe point (–2, 3).

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Contributor-Level 10

43.

The slope of the line 3x - 4y + 2 = 0 is,

$m =- \frac{}{} \frac{3}{4}$
$= - \frac{3}{- 4} = \frac{3}{4}$

So, slope of line parallel to 3x - 4y + 2 = 0 is also m= $\frac{3}{4}$ .

Hence, this parallel line passes through ( -2, 3) we can write,

$m = \frac{y - y_{0}}{x - x_{0}}$

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

3 (x + 2) = 4 (y - 3)

3x + 6 = 4y - 12

3x - 4y + 6 + 12 = 0.

3x - 4y + 18 = 0. Which is the required equation of line.

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11 months ago

42. Find the distance between parallel lines

(i) 15x + 8y – 34 = 0 and 15x + 8y + 31 = 0 (ii) l (x + y) + p = 0 and l (x + y) – r = 0.

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