Part 1 – Multiple Choice Direction: From the Option Letter A To D, Circle the Letter of the Options That Best Answer the Question.

Here are the solutions to the problems: 1. If f(x) = x^2 - 4x + 7, the minimum value of f(x) is: Step 1: Identify the function as a parabola opening upwards, meaning its minimum value occurs at the vertex. The x-coordinate of the vertex for a quadratic function ax^2 + bx + c is given by x = -(b)/(2a). For f(x) = x^2 - 4x + 7, we have a=1 and b=-4. x = -(-4)/(2(1)) = (4)/(2) = 2 Step 2: Substitute the x-coordinate of the vertex into the function to find the minimum value. f(2) = (2)^2 - 4(2) + 7 f(2) = 4 - 8 + 7 f(2) = 3 The minimum value is 3. The correct option is C. C. 3 2. Solve: _2(x-1) + _2(x+1) = 3 Step 1: Use the logarithm property _b M + _b N = _b (MN) to combine the terms on the left side. _2((x-1)(x+1)) = 3 _2(x^2 - 1) = 3 Step 2: Convert the logarithmic equation to an exponential equation using the definition _b M = N b^N = M. 2^3 = x^2 - 1 8 = x^2 - 1 Step 3: Solve for x. x^2 = 8 + 1 x^2 = 9 x = ±sqrt(9) x = ± 3 Step 4: Check for valid solutions. The arguments of logarithms must be positive. For x=3: x-1 = 3-1 = 2 > 0 x+1 = 3+1 = 4 > 0 So, x=3 is a valid solution. For x=-3: x-1 = -3-1 = -4 < 0 So, x=-3 is not a valid solution. The only valid solution is x=3. The correct option is B. B. 3 3. The derivative of x^x is: Step 1: Let y = x^x. To differentiate this, use logarithmic differentiation. Take the natural logarithm of both sides. y = (x^x) y = x x Step 2: Differentiate both sides with respect to x. Use the chain rule on the left side and the product rule on the right side. (1)/(y) (dy)/(dx) = (1)( x) + (x)((1)/(x)) (1)/(y) (dy)/(dx) = x + 1 Step 3: Solve for (dy)/(dx) by multiplying both sides by y. (dy)/(dx) = y( x + 1) Substitute y = x^x back into the equation. (dy)/(dx) = x^x( x + 1) The correct option is B. B. x^x( x + 1) 4. A geometric sequence has first term 3 and ratio 2. Sum of first 6 terms: Step 1: Identify the given values for the geometric sequence. First term, a = 3. Common ratio, r = 2. Number of terms, n = 6. Step 2: Use the formula for the sum of the first n terms of a geometric sequence: S_n = (a(r^n - 1))/(r-1). S_6 = (3(2^6 - 1))/(2-1) Step 3: Calculate the sum. S_6 = (3(64 - 1))/(1) S_6 = 3(63) S_6 = 189 The correct option is A. A. 189 5. If = 3/5 and is acute, = Step 1: Since is an acute angle, it is in the first quadrant, where both sine and cosine are positive. Use the Pythagorean identity ^2 + ^2 = 1. ((3)/(5))^2 + ^2 = 1 Step 2: Solve for ^2 . (9)/(25) + ^2 = 1 ^2 = 1 - (9)/(25) ^2 = (25)/(25) - (9)/(25) ^2 = (16)/(25) Step 3: Take the square root to find . Since is acute, must be positive. = sqrt((16)/(25)) = (4)/(5) The correct option is A. A. (4)/(5) 6. The solution of |2x-5|=7 is: Step 1: For an absolute value equation |A|=B, there are two possibilities: A=B or A=-B. So, we have two equations: 1) 2x - 5 = 7 2) 2x - 5 = -7 Step 2: Solve the first equation. 2x - 5 = 7 2x = 7 + 5 2x = 12 x = (12)/(2) x = 6 Step 3: Solve the second equation. 2x - 5 = -7 2x = -7 + 5 2x = -2 x = (-2)/(2) x = -1 The solutions are x=6 and x=-1. The correct option is A. A. 6, -1 7. The determinant of [[1,2],[3,4]] is: Step 1: For a 2 × 2 matrix a & b \\ c & d , the determinant is given by the formula ad - bc. For the given matrix 1 & 2 \\ 3 & 4 , we have a=1, b=2, c=3, and d=4. Step 2: Calculate the determinant. det = (1)(4) - (2)(3) det = 4 - 6 det = -2 The correct option is A. A. -2 8. _x 0 ( x)/(x) = Step 1: This is a fundamental limit in calculus. The limit _x 0 ( x)/(x) is a standard result. Step 2: Recall the standard limit. _x 0 ( x)/(x) = 1 The correct option is B. B. 1 9. If x^2-5x+6=0, then x equals: Step 1: This is a quadratic equation. We can solve it by factoring. Look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. (x-2)(x-3) = 0 Step 2: Set each factor equal to zero and solve for x. x - 2 = 0 x = 2 x - 3 = 0 x = 3 The solutions are x=2 or x=3. The correct option is A. A. 2 or 3 Drop the next question!