Class 12th

sinx = t, cosxdx = dt
I = ∫dt/(t³(1+t?)^(2/3)) = ∫dt/(t?(1+1/t?)^(2/3))
Put 1+1/t? = r³ ⇒ -6/t? dt = 3r²dr
I = (-1/2)∫dr = -1/2 r + c = (-1/2)(1+1/sin?x)^(1/3) + c
f(x) = (-1/2)cosec²x(1+sin?x)^(1/3) and λ=3
⇒ f(π/3) = -2

Note D = |3, 4, 5; 1, 2, 3; 4, 4| = 0 (R? → R? - 2R? + 3R? )
Now let P? = 4x+4y+4z-δ=0. If the system has solutions it will have infinite solution, so P? = αP? + βP?
Hence 3α+β=4 and 4α+2β=4 ⇒ α=2 and β=-2
So for infinite solution 2µ-2=δ ⇒ for 2µ ≠ δ+2 System inconsistent

I = ∫ (cosθ / (2sin²θ + 7sinθ + 3) dθ
sinθ = t ⇒ cosθdθ = dt
= (1/2) ∫ (1 / (t² + (7/2)t + 3/2) dt
= (1/2) ∫ (1 / (t+7/4)² - (5/4)²) dt
= (1/5) ln | (2t+1)/ (t+3)| + c
= (1/5) ln | (2sinθ+1)/ (sinθ+3)| + C
So A = 1/5
B (θ) = 5 (2sinθ+1)/ (sinθ+3)

y = |x| (x-1)
= { 0,   0 ≤ x < 1
{ x-1, 1 ≤ x < 2
Area = ∫? ² 2√x dx - ∫? ² (1) (1)
= [ (4x³/²)/3]? ² - 1/2 = (8√2)/3 - 1/2

e? y'x? + 4x³e? + 2y' / (2√ (y+1) = 0 at (1,0)
y' + 4 + y' = 0 ⇒ y' = -2
equation of tangent at (1,0) is 2x + y - 2 = 0
So option (C) is correct.

dy/dx + 2tanx · y = 2sinx
I.F. = e^ (∫2tanxdx) = sec²x
Solution is y·sec²x = ∫2sinx·sec²xdx + C
ysec²x = 2secx + C
0 = 2·2 + c ⇒ c = -4
ysec²x = 2secx - 4
y (π/4) = √2 - 2

Line are coplanar
so | [α, 5-α, 1], [2, -1, 1], | = 0
−5α + (α – 5)3 + 7 = 0
-2α = 8 ⇒ α = −4
⇒ L? : (x+2)/-4 = (y+1)/9 = (z+1)/1
Now by cross checking option (A) is correct.

So D = 0 → |, [1,3, k²], | = 0 ⇒ k² = 9
x + y + 3z = 0
x + 3y + 9z = 0
3x + y + 3z = 0
(1)- (3)
x = 0 ⇒ y + 3z = 0
y/z = -3
So x + (y/z) = -3

w = I + j - 3k
p = 3i - j + k
q = -3i + 2j + 4k
p * q = |i, j, k; 3, -1, 1; -3, 2, 4| = -6i - 15j + 3k
S.D. = |AD. (p * q)|/|p * q| = | (36+225+9)|/√ (36+225+9) = 3√30

50.00
K? Cr? O? + FeC? O? → Fe³? + CO? + Cr³? + K? + H? O
n_f=6    n_f=3
Now apply equivalent concept
0.02 * 6 * V = (0.288 / (144/3) * 10³
Normality
V = (0.288 * 10³)/ (48 * 6 * 0.02) = 50.00 mL