Choose a course and topic, attempt each question and open its complete worked solution when you are ready.
Evaluate lim(x→0)(ax+b)/(cx+1).
Simplify the expression before substitution and preserve one-sided information.
Substitution into the original expression verifies the stated domain and final value.
Find the point on the z-axis equidistant from A(1,2,3) and B(3,4,1).
Let P=(0,0,z), set PA²=PB² and solve.
Evaluate 99⁵ using the binomial theorem.
Write 99=100−1 and retain alternating signs.
Compute 8!/(6!2!).
Cancel the common 6! factor.
Solve x+\(\frac{x}{2}\)+\(\frac{x}{3}\)<11.
Use the least common denominator 6.
Given sin x = \(\frac{3}{5}\) and x lies in quadrant II, determine the other five trigonometric functions.
Cosine is negative in quadrant II. Apply the Pythagorean identity.
Suppose \(n(A)=3,n(B)=2\), and \((x,1),(y,2),(z,1)\in A\times B\), where \(x,y,z\) are distinct. Find \(A,B\).
The three distinct first coordinates exhaust \(A\), so \(A=\{x,y,z\}\). The second coordinates 1 and 2 are distinct and exhaust the two-element set \(B\), so \(B=\{1,2\}\).
\(A=\{x,y,z\},\ B=\{1,2\}\).
Classify as finite or infinite:
Infinite: (i), (iii), (v). Finite: (ii), (iv).
Which sequence represents a valid food chain?
A food chain must begin with a producer and continue through organisms that feed on the preceding member. Grass is a producer, a goat feeds on grass, and a human may obtain food from the goat. The other sequences either list only producers, only herbivores, or organisms without the required feeding link.
Grass \(\rightarrow\) goat \(\rightarrow\) human — option (B).
How does force on rod AB change if (i) current increases(ii) a stronger magnet is used(iii) rod length in the field increases?
For a straight conductor, \(F=BIl\sin\theta\). With orientation fixed, increasing I increases force, a stronger magnet increases B and force, and increasing effective length l also increases force. Therefore displacement increases in all three cases.
The force and displacement increase in all three cases.