Daily Practice Questions

Explanation: The greatest possible value for y occurs when y = x + 2, since that would maximize its value within the inequality. Set x + 2 equal to − x + 4, since the value of y would be at its maximum in both inequalities, and then solve for x :

Explanation: The figure shows a square with a perimeter of 20 units. Since all four sides of a square have the same length, each side of this square has a length of 20 units ÷ 4 = 5 units; this is Quantity A. The area of a square is equal to the square of one of its sides, so the area of the square is (5 units)^2 = 25 units^2. Since each side of a square is the same length, the four triangles created by the two diagonals in the figure have the same area, which means that Quantity B, the area of the shaded region, is one-fourth the area of the square, or 25 units^2 ÷ 4. This is clearly greater than Quantity A, which is 20 units ÷ 4, so there is no need to do any further calculation. (B) is the correct answer.

Explanation: The Mean Value Theorem applies to any function that is continuous on $[0, \pi]$ and differentiable on $(0, \pi)$; since $f(x) = 2\sin x + \sin 4x$ is a sum of sine functions, it satisfies these conditions everywhere. Therefore, the theorem guarantees exactly one value $c$ for every specific average rate of change $\frac{f(\pi)-f(0)}{\pi-0}$. However, the question asks for the number of points where the function *satisfies* the theorem, which implies finding how many distinct values of $c$ exist in $(0, \pi)$ such that $f'(c) = \frac{f(\pi)-f(0)}{\pi}$. Calculating the average rate of change yields 0, so we solve $f'(c) = 2\cos c + 4\cos 4c = 0$. By analyzing the graph or using trigonometric identities, there are exactly 4 distinct solutions for $c$ in the interval $(0, \pi)$, making (D) the correct answer.

Explanation: “Derive” means “obtain,” making this a logical option to express how aerobic organisms gather their energy. It is not “discover” or “discern,” since these are advanced thought processes that are not integral to obtaining energy. It is not “inhale” because while oxygen may be inhaled, the energy is gathered from the process of cellular respiration.

Explanation: 21. (E) It is wise to have an equals method that is compatible with the compareTo method; namely, n1.equals(n2) and n1.compareTo(n2)==0 have the same value if n1 and n2 are Name objects. However, nothing in the Java language mandates that if a class has a compareTo method, it must also have an equals method. Choice A is true. You know this because the Name class has no mutator methods. Thus, Name objects can never be changed. Choice B is true: If a Name is initialized with null references, each of the methods will throw a NullPointerException . Choice C is true: If n1.equals(n2) is true, then n1.compareTo(n2) == 0 is true, because both are conditions for equality of n1 and n2 and should therefore be consistent. Choice D is true: If the parameter is null, the compareTo method will throw a NullPointerException .

Explanation: The determinant of this matrix is 3(6) – [(–5) n ] = 18 + 5 n . The question states that the determinant is 3, so 18 + 5 n = 3. Subtract 18 from both sides to get 5 n = –15; n = –3, which is (B).