Let V be the 3-dimensional vector space of all polynomials of order less than or equal to 2 with real coefficients.
(a) Show that the function B: V ×V →R given by B(f,g) = f(−1)g(−1) + f(0)g(0) + f(1)g(1) is an inner product and write out its Gram matrix with respect to the basis (1,t,t²).

DO NOT COPY YOUR SOLUTION FROM OTHER SOLUTIONS

Structural Induction on WFF For a formula α ∈ WFF we let `(α) denote the number of symbols in α that are left brackets ‘(’, let v(α) the number of variable symbols, and c(α) the number of symbols that are the corner symbol ‘¬’. For example in ((p1 → p2) ∧ ((¬p1) → p2)) we have `(α) = 4, v(α) = 4 and c(α) = 1. Prove by induction that he following property holds for all well formed formulas: • `(α) = v(α) + c(α) − 1

During the corrosion of iron, iron is oxidized (Fe → Fe2+ + 2e −) and molecular oxygen is reduced (O2 + 2H2O + 4e −→ 4HO−). The E ◦ for the reduction of Fe2+ to Fe is −0.44 V, the E ◦ for the reduction of O2 is 0.40 V, and the E ◦ for the reduction of Au3+ to Au (Au3+ + 3e −→ Au) is 1.50 V. What would E ◦ cell be for the oxidation of gold by O2? Based on your answer, would you expect gold to corrode in moist air like iron does? Explain your reasoning.

Two capacitors C1 = 2 µF and C2 = 6 µF are connected in parallel across a 11 V battery. They are carefully disconnected so that they are not discharged and are reconnected to each other with positive plate to negative plate and negative plate to positive plate (with no battery).

(a) Find the potential difference across each capacitor after they are connected.

V (2 µF capacitor) .

V (6 µF capacitor)

(b) Find the initial and final energy stored in the capacitors.

mJ (initial)

mJ (final)

A 10.00 mL solution of 0.0500 M AgNO3 was titrated with 0.0250 M NaBr in the cell: S.C.E.(saturated calomel electrode) titration solution Ag (s). (Ksp AgBr(s) = 5.0e-13)

Find the cell voltage (v) when the volume of titrant added is 5.00 ml.

Find the equivalence volume (ml) of titrant added.

Calculate the cell voltage (v) when the volume of titrant added is 24.58 ml

Calculate the cell voltage (v) at the equivalence point.

.

To calibrate a gas chromatograph for blood alcohol assays, the technician must prepare a standard 0.08%(w/v) alcohol solution. Describe how to prepare 100mL of 0.08%(w/w) from a “140 proof” ethanol-water commercial standard. Hint: 140 proof = 70%(w/v). Comment on the total volume of the 0.08%(w/v) calibration standard: Is 100 mL a sufficient volume to allow accurate preparation of the calibration standard… or would 1,000 mL be better? Defend your answer.

a) Find an equation of the plane tangent to the graph of f at the given point P. Write your answer in the form ax + by +cz= d, where a, b, c, and d are integers with no common factor, and a is greater or equal to 0. f(x,y)= 2x³y + 4x-y, P(1,3,7)

b) Use a multivariable chain rule to find a formula for the given derivative or partial derivative. w= f(x,y), x=g(u,v), y=h(u,v);  ∂w/ ∂v

A. Calculate the mass percent (m/m) of a solution prepared by dissolving 51.22 g of NaCl in 151.9 g of H2O.

B. Vinegar is a solution of acetic acid in water. If a 265 mL bottle of distilled vinegar contains 19.2 mL of acetic acid, what is the volume percent (v/v) of the solution?

C. Calculate the mass/volume percent (m/v) of 20.5 g NaCl in 75.0 mL of solution.

D. Calculate the molarity (M) of 155.2 g of H2SO4 in 1.330 L of solution.

A voltaic cell consists of a Zn/Zn²⁺ anode and a Ni/Ni²⁺ cathode at 25 ^∘C. The initial concentrations of Ni²⁺ and Zn²⁺ are 1.7 M and 0.12 M , respectively. The volume of half-cells is the same.

What is the concentration of Ni²⁺ when the cell potential falls to 0.456 V ?   

Enter your answer to 4 decimal places and in units of mM.

Zn²⁺ (aq) + 2 e-  ⟶Zn(s) E° = -0.76 V

Ni²⁺ (aq) + 2 e- ⟶   Ni(s) E° = -0.23 V

A particle of charge q and mass m is accelerated from rest through a potential difference V, after which it encounters a uniform magnetic field B perpendicular to its velocity v. If the particle moves in a plane perpendicular to B,

(a) Find an expression for the radius of its circular path in terms of q, V and B.

(b) If the particle is an electron, what must the potential difference be so it describes a circular path of radius 1.5 m in a magnetic field of 0.05 T?