Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Differentiate sin(x+a)/cosx.
Name the inner functions, differentiate each factor, and then simplify.
Substitution into the original expression verifies the stated domain and final value.
A vertical parabolic arch is \(10\,\mathrm{m}\) high and \(5\,\mathrm{m}\) wide at ground level. Find its width on a horizontal line \(2\,\mathrm{m}\) from the vertex.
Translate every condition into an equation and retain exact values throughout the conic calculation.
The last line satisfies the original conditions and supplies every requested part.
Reflect (3,8) in the mirror line x+3y=7.
Translate every condition into an equation and retain exact values throughout the straight-line calculation.
The last line satisfies the original conditions and supplies every requested part.
A job was planned for 150 workers over a fixed number of days. Four workers left before day 2, four more before day 3, and so on; completion took eight extra days. Find the actual completion time.
Translate the conditions into a sequence equation, then retain every admissible real solution.
The final value satisfies the original recurrence, sum, product or financial condition.
The ionization constant of chloroacetic acid is 1.35 × 10–3. What will be the pH of 0.1M acid and its 0.1M sodium salt solution?
For chloroacetic acid use [H⁺]≈√(KaC). Its sodium salt hydrolyses with Kb=Kw/Ka and [OH⁻]≈√(KbC).
For each pair, identify the orbital whose electron experiences the larger effective nuclear charge: (i) 2s or 3s(ii) 4d or 4f(iii) 3d or 3p.
Greater penetration places an electron closer to the nucleus and reduces shielding. For a given shell, penetration follows s>p>d>f.
Differentiate x⁴(5sinx−3cosx).
Name the inner functions, differentiate each factor, and then simplify.
Substitution into the original expression verifies the stated domain and final value.
A \(100\,\mathrm{m}\) suspension-bridge cable is parabolic. Its centre wire is \(6\,\mathrm{m}\) and each end wire is \(30\,\mathrm{m}\). Find the wire length \(18\,\mathrm{m}\) from the centre.
Translate every condition into an equation and retain exact values throughout the conic calculation.
The last line satisfies the original conditions and supplies every requested part.
The lines of slopes 3 and \(\frac{1}{2}\) are equally inclined to y=mx+4. Determine m.
Translate every condition into an equation and retain exact values throughout the straight-line calculation.
The last line satisfies the original conditions and supplies every requested part.
Ionic product of water at 310 K is 2.7 × 10–14. Determine the pH of neutral water at this temperature?
Neutral water has [H⁺]=[OH⁻]=√Kw. Its neutral pH changes with temperature.
pH=6.78.