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Since, lim (x→0) f (x)/x exist ⇒ f (0) = 0
Now, f' (x) = lim (h→0) (f (x+h)-f (x)/h = lim (h→0) (f (h)+xh²+x²h)/h (take y = h)
= lim (h→0) f (h)/h + lim (h→0) (xh) + x²
⇒ f' (x) = 1 + 0 + x²
⇒ f' (3) = 10

D = |1 -2 3; 2 1; 1 -7 a| = 0 ⇒ a = 8
also, D? = |9 -2 3; b 1; 24 -7 8| = 0 ⇒ b = 3
hence, a-b = 8-3=5

Given (2x² + 3x + 4)¹? = Σ (r=0 to 20) a? x?
replace x by 2/x in above identity :-
2¹? (2x²+3x+4)¹? / x²? = Σ (r=0 to 20) a?2? /x?
⇒ 2¹? Σ (r=0 to 20) a? x? = Σ (r=0 to 20) a?2? x²? (from (i)
now, comparing coefficient of x? from both sides
(take r = 7 in L.H.S. and r = 13 in R.H.S.)
2¹? a? = a?2¹³ ⇒ a? /a? = 2³ = 8

x dy/dx - y = x² (xcosx + sinx), x > 0
dy/dx - y/x = x (xcosx+sinx) ⇒ dy/dx + Py = Q

So, I.F. = e^ (∫-1/x dx) = 1/|x| = 1/x (x > 0)
Thus, y/x = ∫ 1/x (x (xcosx+sinx)dx
⇒ y/x = xsinx + C
? y (π) = π ⇒ C = 1
So, y = x²sinx + x ⇒ (y)? /? = π²/4 + π/2
Also, dy/dx = x²cosx + 2xsinx + 1
⇒ d²y/dx² = -x²sinx + 4xcosx + 2sinx
⇒ [d²y/dx²] at π/2 = -π²/4 + 2

 max {n (A), n (B)} ≤ n (A U B) ≤ n (U)
⇒ 76 ≤ 76 + 63 - x ≤ 100
⇒ -63 ≤ -x ≤ -39
⇒ 63 ≥ x ≥ 39

 f (x) = ∫ (from 1 to 3) (√x dx)/ (1+x)² = ∫ (from 1 to √3) (t⋅2tdt)/ (1+t²)² (put √x = t)
= [ (-t/ (1+t²)] (from 1 to √3) + [tan? ¹t] (from 1 to √3) [Applying by parts]
= (-√3/4 + 1/2) + (π/3 - π/4)
= (-√3+2)/4 + π/12

In Rutherford's model, electrostatic force provides the centripetal force:

$\frac{1}{4\mathrm{\pi E}{ }_{}}$_∈  

Kinetic Energy (K):

$K=\frac{1}{2}m{v}^2=\frac{e^2}{8\pi E_{0}r}$

Potential Energy (U):

$U=-\frac{1}{4\pi E_0}\cdot \frac{e^2}{r}$

Total Energy (E):

$E=K+U=-\frac{e^2}{8\pi E_{0}r}$

Note that, E is the Epsilon symbol. 

  [x]² + 2 [x+2] - 7 = 0
⇒ [x]² + 2 [x] + 4 - 7 = 0
⇒ [x] = 1, -3
⇒ x ∈ [1,2) U [-3, -2)

IIIT Vadodara B.Tech cutoff 2025 has been released in the form of closing and opening ranks for admissions to various B.Tech specialisations. This year, the B.Tech cutoff 2025 ranged from 36583 to 64458 for the General AI category candidates. B.Tech in Computer Science and Engineering, which is the most competitive course offered by IIIT Vadodara, stands at 36583, as the last round closing rank. Please keep in mind that different cutoffs will be reflected for different rounds and categories. For more details, refer to the table below:

In the gold foil scattering experiment by Rutherford and his team, almost all alpha particles passed straight through the foil. There were minor deflections mostly, and only a small amount of them showed deflections at really large angles. Only a few of them rebounded. This was the main observation that led Rutherford to conclude that the positive charge and mass of the atom concentrate at a very tiny section of the atom. That’s the nucleus. The other observation was that the entire atom must be space where the alpha particles could go undeflected. 

Σ (r=30 to 50) 50-rC? =? C? +? C? +? C? + . + ³? C?
=? C? +? C? +? C? + . + (³? C? + ³? C? ) - ³? C?
=? C? +? C? +? C? + . + (³¹C? + ³¹C? ) - ³? C?
=? C? +? C? - ³? C?
=? ¹C? - ³? C?

A² = (cos2θ isin2θ cos2θ)
Similarly, A? = (cos5θ isin5θ cos5θ) = (a b; c d)
(1) a²+b² = cos²5θ - sin²5θ = cos10θ = cos75°
(2) a²-d² = cos²5θ - cos²5θ = 0
(3) a²-b² = cos²5θ + sin²5θ = 1
(4) a²-c² = cos²5θ + sin²5θ = 1

∫ (x/ (xsinx+cosx)²dx = ∫ (xcosx⋅xcosx)/ (xsinx+cosx)² dx
= x/cosx (-1/ (xsinx+cosx) + ∫ (cosx-xsinx)/cosx² (1/ (xsinx+cosx) dx
= -xsecx/ (xsinx+cosx) + ∫sec²xdx
= -xsecx/ (xsinx+cosx) + tanx + C

Since (3,3) lies on x²/a² - y²/b² = 1
9/a² - 9/b² = 1
Now, normal at (3,3) is y-3 = -a²/b² (x-3),

which passes through (9,0) ⇒ b² = 2a²

So, e² = 1 + b²/a² = 3
Also, a² = 9/2
(From (i) and (ii)
Thus, (a², e²) = (9/2, 3)

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The GATE cutoff 2025 for IIIT Vadodara has been released for various categories for admissions to different M.Tech specialisations. The cutoff score ranged from 415 to 443 for students belonging to the General category, All India quota. This is the cutoff for the last round; other rounds will have different cutoffs accordingly. This year, the most difficult M.Tech specialisation proved to be M.Tech. in Computer Science and Engineering (Artificial Intelligence), with a cutoff score ranging from 443 to 539. Therefore, students aiming for a seat should strive for a score above 400 to improve their chances of admission.