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The given vertices of the triangle are A(2,0),B(4,5)and C(6,3)

So, equation of line AB is $y-0=\frac{5-0}{4-2}\left(x-2\right)$

$=y=\frac{5}{2}\left(x-2\right)$

Similarly equation of BC is

$\begin{array}{l}y-5=\frac{3-5}{6-4}\left(x-4\right)\\ =y=5+\left(-x+4\right)\\ y=9-x\end{array}$

And equation of AC is $y-0=\frac{3-0}{6-2}\left(x-2\right)$

= $y=\frac{3}{4}\left(x-2\right)$

$\therefore$ Area of $?ABC$

= $area(?ABE)+area(BCDE)-area(?ACD)$

$\begin{array}{l}={\displaystyle \underset{2}{\overset{4}{\int }}{y}_{AB}dx+}{\displaystyle \underset{4}{\overset{6}{\int }}{y}_{BC}dx-{\displaystyle \underset{2}{\overset{6}{\int }}{y}_{BC}dx}}\\ ={\displaystyle \underset{2}{\overset{4}{\int }}\frac{5}{2}\left(x-2\right)dx+{\displaystyle \underset{4}{\overset{6}{\int }}\left(9-x\right)dx-{\displaystyle \underset{2}{\overset{6}{\int }}\frac{3}{4}}}}\left(x-2\right)dx\\ =\frac{5}{2}{\left[\frac{x^2}{2}-2x\right]}_2^4+{\left[9x-\frac{x^2}{2}\right]}_4^6-\frac{3}{4}{\left[\frac{x^2}{2}-2x\right]}_2^6\end{array}$

$\begin{array}{l}=\frac{5}{2}\left[\left(\frac{4^2}{2}-2.4\right)-\left(\frac{2^2}{2}-2.2\right)\right]+\left[\left(9.6-\frac{6^2}{2}\right)-\left(9.4-\frac{4^2}{2}\right)\right]-\frac{3}{4}\left[\left(\frac{6^2}{2}-2.6\right)-\left(\frac{2^2}{2}-2.2\right)\right]\\ =\frac{5}{2}\left[\left(8-8\right)-\left(2-4\right)\right]+\left[54-18-36+8\right]-\frac{3}{4}\left[18-12-2+4\right]\\ =5+8-6=7uni{t}^2\end{array}$

Yes, candidates who have studied Science subjects from PCM and PCB can get admissions to the BPharm course at GTU Ahmedabad. The other requirements to enrol in the course are as follows:

Karmaveer Bhaurao Patil College first merit list 2025 has been released for admission to the BA, BSc, BCom, BVoc and BMS courses at the undergraduate level. During the first cutoff list, the percentages varied between 35 and 70.5 for the students belonging to the General AI quota. Of all the courses, the highest cutoff was for B.Sc. in Biotechnology, followed by B.Sc. in Computer Science and B.Sc. in Medical Laboratory Technology programme. Candidates can refer to the table below to view the Rayat Shikshan Sanstha's Karmaveer Bhaurao Patil College Mumbai cutoff 2025 for all the courses: 

Yes, there are many government Textile Design colleges in India. Some of them are mentioned below along with their tuition fees:

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

Given that equation of

curve $y=x^2$

line $y=\left|x\right|=\left\{x,ifx\ge 0\&-x,ifx\angle 0\right\}$

Since the line passes through A&B in Ist and IInd quadrants

the equation must satisfy

$\Rightarrow y=x^2$

$\Rightarrow x=x^2$ for Ist quadrant and

$\Rightarrow -x=x^2$ for IInd t quadrant

So, $\Rightarrow x^2-x=0$ and $x^2+x=0$

$\Rightarrow x\left(x-1\right)=0$ and $x\left(x+1\right)=0$

$\Rightarrow x=0,1$ $x=0,-1$

$x=0,y=0$

$x=1,y=1$ i.e, A has coordinate (1,1)

$x=-1,y=1$ i.e, B has coordinate (1,1)

Now, area of AODA = area (AOM)-area (ADOM)

$={\displaystyle \underset{0}{\overset{1}{\int }}{y}_{line}dx-{\displaystyle \underset{0}{\overset{1}{\int }}{y}_{curve}dx}}$

$={\displaystyle \underset{0}{\overset{1}{\int }}xdx-{\displaystyle \underset{0}{\overset{1}{\int }}{x}^2}dx={\left[\frac{x^2}{2}\right]}_0^1-}{\left[\frac{a^3}{3}\right]}_0^1$

$=\frac{1}{2}-\frac{1}{3}=\frac{3-2}{6}=\frac{1}{6}unit{s}^2$

$\therefore$ The required area of the region bounded by curve $y=x^2$ and line $y=\left|x\right|$ is $\frac{1}{6}+\frac{1}{6}=\frac{2}{6}=$$\frac{\mathrm{}}{}$$\frac{1}{3}unit{s}^2$

33.  (A) Mn₂O₇

2KMnO₄ +  2H₂SO₄ (conc.) → Mn₂O₇ +    2KHSO₄ + H₂O

Manganese heptoxide is an acid anhydride of permanganic acid ( HMnO₄). It is a dangerous oxidiser, volatile and highly reactive liquid.

Thompson Rivers University provides different housing programs. One of them is Homestay Program. Students will have to pay USD 1200 per month along with a non refundable placement fee of USD 150 and USD 600 for security deposit to avail this program. On campus accomodations cost may range between USD 2520 - 10,300 along with a non refundable placement fee of USD 100.

51.  (i) Both assertion and reason are true and reason is the correct explanation of assertion.

Explanation: In the Mond process Nickel is reacted with carbon monoxide reversibly to give Nickel carbonyl,  Ni (CO)₄. Nickel carbonyl is a volatile compound. It decomposes to nickel and carbon monoxide at 460 K.

The courses offered by SDIET in the admission process 2025-26 are as follows: 

BTech

BBA 

BCA
All the courses offered by SDIET are courses of the UG level. Candidates wishing to apply to these must make sure to follow the admission process that has been set by the institute. Candidates are shortlisted for the BTech course based on their performance in the JEE Mains. 

According to an unofficial source, the University of Alberta acceptance rate is 51%. The percentage shows that the university takes a severe approach to candidate selection and is relatively strict during admission. Official 2024–2025 figures show that over 42,000 candidates were admitted, with over 37,000 being undergraduates and 12.9% being foreign nationals. Admission to the University of Arizona is determined by an applicant's overall academic record and performance in three key areas: grades, English language skills, and course completion.

Saint Louis University (SLU) ranks #108 among all colleges & universities in the USA, also securing a global university rank #461 in the process.  Shiksha, has also ranked the institute nationally, based on the quality of education it imparts with respect to some popular courses.

Given below are the Saint Louis University course rankings as given by Shiksha:

Given equation of the curve is $\left|x\right|+\left|y\right|=1$ , which can be break down into each quadrant .

For Ist quadrant,  $\left|x\right|=x,\left|y\right|=y$

i.e.,  $x+y=1$ - (1)

Similarly for IInd, IIIRd nad IVth quadrant

$-x+y=1$ - (2)

$-x-y=1$ - (3)

$x-y=1$ - (4)

We draw the above focus lines on a graph and find the area enclosed which is a square.

$\therefore$ Required area $\left(?ABCD\right)=4\times area\left(?AOB\right)$ .

$\begin{array}{l}=4\times {\displaystyle \underset{0}{\overset{1}{\int }}y}dx\\ =4{\displaystyle \underset{0}{\overset{1}{\int }}\left(-x\right)}dx\\ =4{\left[x-\frac{x^2}{2}\right]}_0^1\\ =4\left[1-\frac{1}{2}\right]\\ =4\times \frac{1}{2}\\ =2uni{t}^2\end{array}$

You will need the details mentioned on your Karnataka 2nd PUC admit card to check your result. Its okay if you have the photocopies of your hall ticket and lost the original one. 

However, it is always advised to keep the documents like Karnataka 2nd PUC admit card secured till the announcement of results.

Thompson Rivers University tuition fee for undergraduate programs ranges between CAD 25K - 1L. The tuition fees is mentioned below:

The University of Alberta does not ask for GRE scores from international graduate applicants. However, the university recommends that international students submit GRE scores, especially if their degree is not from a well-known university or their GPA score is low. Along with this, students must fulfill the basic eligibility by holding a bachelor's degree from a recognised university and submit some mandatory documents such as transcripts, a letter of recommendation, English language requirements, and more.

Yes, joining the best Textile Design colleges in India can be worthwhile for you. For that, you need to check its ROI. Listed below are some colleges along with their tuition fees and average placement package:

Disclaimer: The above statistics are based on the latest available data.

Yes, for students officially pursuing a degree offered by the institute, UCEOU offers hostel facilities. Students who wish to live on campus premises can opt to reside in hostel facilities. According to the college's official website, the institute houses four boys' hostels and one newly constructed girls' hostel.

Moreover, the official website also states that each male and female hostel is equipped with various necessary amenities. Some of these amenities are electricity and water supply, clean and spacious rooms, and a mess.

The given equation of the parabola is $x^2=y$ ---------(1)

and that the line is $y=x+2$ --------------(2)

Solving (1) and (2) for x and y

$\begin{array}{l}{x}^2=x+2\\ \to x^2-x-2=0\\ =x^2+x-2x-2=0\\ =x\left(x+1\right)-2\left(x+1\right)=0\\ =\left(x+1\right)\left(x-2\right)=0\\ =x=-1\&x=2\end{array}$

When $x=-1,y={\left(-1\right)}^2=1$

And $x=2,y=2^2=4$

$\therefore$ The point of intersection of the parabola and the lines $A(-1,1)\&B(2,4)$

Hence the required area enclosed region is $ea(AOBA)=area(DABCD)-area(DAOBCD)$

$\begin{array}{l}={\displaystyle \underset{-1}{\overset{2}{\int }}{y}_{line}dx-}{\displaystyle \underset{-1}{\overset{2}{\int }}{y}_{curve}dx}\\ ={\displaystyle \underset{-1}{\overset{2}{\int }}\left(x+2\right)dx}-{\displaystyle \underset{-1}{\overset{2}{\int }}{x}^{2}dx}\\ ={\left[\frac{x^2}{2}2x\right]}_{-1}^2-{\left[\frac{x^3}{3}\right]}_{-1}^2\end{array}$

$\begin{array}{l}=\left[\left(\frac{2^2}{2}2.2\right)-\left(\frac{{\left(-1\right)}^2}{2}+2\left(-1\right)\right)\right]-\left[\frac{2^3}{3}-\frac{{\left(-1\right)}^3}{3}\right]\\ =\left[2+4-\frac{1}{2}+2\right]-\left[\frac{8+1}{3}\right]=\frac{15}{2}-3=\frac{9}{2}uni{t}^2\\ \end{array}$