Choose the correct statements:
Solution to Question 281
Complete workingUse ΔU=−W for a conservative force, and apply momentum conservation to an isolated system.
Establish the four triangle inequalities for |a+b| and |a-b|, including both upper and lower bounds, and state equality cases.
Solution to Question 282
Complete workingPrinciple. Apply the ordinary triangle inequality to the vector triangles formed by a, b and -b.
Find the Cartesian components of angular momentum for r=(x,y,z) and p=(pₓ,pᵧ,p_z). Hence state the result for motion confined to the xy-plane.
Solution to Question 283
Complete workingExpand the determinant for L=r×p.
Values 1,2,3,4 have frequencies 2,3,4,1. Find their mean.
Solution to Question 284
Complete workingUse the weighted mean Σfx/Σf.
Prove that P(−2,3,5), Q(1,2,3), and R(7,0,−1) are collinear.
Solution to Question 285
Complete workingUse rectangular-coordinate definitions and keep corresponding coordinates aligned.
The final relation satisfies the original coordinate or distance condition, completing the verification.
Determine the centre and radius of (x+5)²+(y−3)²=36.
Solution to Question 286
Complete workingTranslate 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.
Evaluate 96³ using the binomial theorem.
Solution to Question 287
Complete workingWrite 96 as 100−4.
Solve 3x−7>5x−1 for real x.
Solution to Question 288
Complete workingCollect like terms, then divide by a negative coefficient and reverse the sign.
Two circles contain arcs of equal length that subtend 60° and 75° at their centres. Find the ratio of their radii.
Solution to Question 289
Complete workingEqual arc lengths give r₁θ₁=r₂θ₂.
If \(A\times B=\{(a,x),(a,y),(b,x),(b,y)\}\), find \(A\) and \(B\).
Solution to Question 290
Complete workingThe set of first coordinates is \(A\), and the set of second coordinates is \(B\). Therefore \(A=\{a,b\}\) and \(B=\{x,y\}\).
\(A=\{a,b\},\ B=\{x,y\}\).