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$Inx\in \left(-2,2\right)$  

 |x|- 1| is not differentiable at x = -1, 0, 1

|cospx| is not differentiable at x =  $\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}$ -  

$\int {\mathrm{c}\mathrm{o}\mathrm{t}}^{-1}?\left(\frac{1}{1+x}\right)dx$

$=\int {\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)dx$

$=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\int \frac{x}{1+(1+x{)}^2}dx;=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\int \frac{x}{x^2+2x+2}dx$

$=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\int \frac{\frac{1}{2}(2x+2)-1}{x^2+2x+2}dx=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\frac{1}{2}\mathrm{l}\mathrm{o}\mathrm{g}?\left(x^2+2x+2\right)+\int \frac{dx}{(x+1{)}^2+1}$

$=x{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(1+x)-\frac{1}{2}\mathrm{l}\mathrm{o}\mathrm{g}?\left(x^2+2x+2\right)+{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}?(x+1)+c$

$A^2{\left|A\right|}^2-{\left|B\right|}^{3.2}B$

$=25A^2-\left|B\right|.B$

$\begin{array}{l}=25A^2+\left|B\right|A^2\\ =0\end{array}$

Put x³/² = t
√xdx = (2/3)dt
⇒ (2/3) ∫ dt/√ (1-t²) = (2/3)sin? ¹t + c
= (2/3)sin? ¹x³/² + c
⇒ g (x) = sin? ¹x
g (0) = 0

Yes, but within the timeline mentioned by the GATE authorities. IIT Guwahati has started the registration process for GATE 2026. The authorities provide two timelines to submit the application form: with late fees and without late fees. So, incase, you have missed out the date to submit, you can still submit the form during the extended period. Candidates can check the GATE 2026 datesheet below -

• GATE application form last date without late fees - September 28
• GATE application form last date with late fees - October 9

? P? = 6!/2! = 360

tanx + tan (x+π/3) + tan (x-π/3) = 3
tanx + (tanx+√3)/ (1-√3tanx) + (tanx-√3)/ (1+√3tanx) = 3
tanx + (8tanx)/ (1-3tan²x) = 3
(tanx (1-3tan²x) + 8tanx)/ (1-3tan²x) = 3
(3 (3tanx – tan³x)/ (1-3tan²x) = 3
⇒ 3tan3x = 3
tan3x = 1

A: runner succeeded exactly 3 times out of 5.
B: runner succeeds on the first trial.
P (B/A) = P (B ∩ A) / P (A) = (p (? C? p² (1−p)²) / (? C? p³ (1-p)²) = 3/5 = 0.6

$\sim \left(\sim q\wedge p\right)=\left(\sim \left(\sim q\right)\right)\vee \left(\sim p\right)$  

$=\sim p\vee q$            

$=p\to q$            

$\therefore contrapositiveis\sim q\to \sim p$            

 Hence, if $\sim$ q then $\sim$ p

Candidates can submit the UPESEAT application form in online mode only. To submit the UPESEAT 2026 appliaction form, aapplicants must register themselves on the official website, upes.ac.in.

|x|³/? = y
⇒ y² – 26y – 27 = 0 ⇒ y = −1 or 27
⇒ y = 27 ⇒ |x|³/? = 3³
|x| = 3? ⇒ x = ±3?
Product of roots = (3? ) (−3? ) = −3¹?

tan α = 1/x, tan β = 2/x, tan γ = 3/x
Now α + β + γ = 180°
⇒ tan α + tan β + tan γ = tan α tan β tan γ
1/x + 2/x + 3/x = (123)/ (xxx)
⇒ x² = 1

Yes, LBSIM offers summer internship opportunities to its PGDM students annually. The key highlights of the batch 2025 are tabulated below:

It is important to check GATE dates 2026 so that students do not miss out on any dates. The exam dates have been announced and can be checked at gate2026.itg.ac.in.

Case-I : $a-2>0$

$F(-2)>0$

$4(a-2)+4a+a>0$

$9\mathrm{a}-8>0$

$a>\frac{8}{9}$

$F\left(1\right)>0$

$a-2-2a+a>0$

  $-2>0$  (not possible)

Case-II : $a-2<0;a<2$

$F(-2)<0\mathrm{}\Rightarrow \mathrm{}a<\frac{8}{9}$

$F\left(1\right)<0\mathrm{}\Rightarrow \mathrm{}a\in R$

$-2<\frac{-b}{2a}<1\mathrm{}a<\frac{4}{3}\mathrm{}a\in \left[0,\frac{8}{9}\right)\cup \left\{2\right\}$

$D\ge 0\mathrm{}a\ge 0$

You can work at various places after gaining hands-on training/ learning of the ERP system. Some of the companies that actively use ERP software are firms working in the manufacturing, retail, banking, healthcare, logistics, construction, and IT sectors. These industries offer Jobs, including ERP End-User, ERP Analyst, and ERP Administrator. Industries like Education, Healthcare, Finance, and Government projects offer School ERP Admin or Hospital ERP Specialist.

The equation of the line through P (x, y) making an angle with the x-axis which is supplementary to the angle made by the tangent at P (x, y) is

$Y-y=-\frac{dy}{dx}\left(X-x\right)$                    …. (1)

At the point where it meets the x-axis

Y = 0, X = x    $+\frac{y}{\frac{dy}{dx}}\Rightarrow OA=x+\frac{y}{\frac{dy}{dx}}$    …. (2)

The line through P (x, y) and perpendicular to (1) is

$Y-y=\frac{dx}{dy}\left(X-x\right)$           

At the point where it meets the y-axis

$X=0,Y=y=\frac{x}{\frac{dy}{dx}}\Rightarrow OB=y-\frac{x}{\frac{dy}{dx}}$           .

$\boxed{x^2+2xy-y^2=2}$            

RHL&LHL lim (x→0) (sin (2x²/a) + cos (3x/b)^ (ab/x²)
= e^ (lim (x→0) (sin (2x²/a) + cos (3x/b) - 1) (ab/x²) = e^ (4b²-9a)/2b)
f (0) = e³
For continuity at x = 0
Limit = f (0)
(4b² - 9a)/2b = 3 ⇒ 4b² – 6b – 9a = 0∀b ∈ R
⇒ D ≥ 0 ⇒ a ≥ -1/4
a? = -1/4
⇒ |1/a? | = 4.

x² + y² – 6x + 8y + 24 = 0 is circle having centre (3, −4) & r = √ (9+16-24) = 1
√x² + y² min. is min. distance from origin = 4
∴ minimum value of log? (x² + y²) = log? 16 = 4

A = {1,2,3,4,5} B = {0,1,2,3,4}
No. of elements common in A&B = 4.
∴ No. of elements common in A × B and B × A = 4 × 4 = 16