How does Test Citadel mirror the real Digital SAT Bluebook exam?

Test Citadel features an authentic Bluebook-calibrated countdown timer with visibility toggle, question navigation grid, choice eliminator cross-out tool, embedded Desmos graphing calculator, and official College Board formula sheets.

Can I try Test Citadel questions for free?

Yes. Every visitor can complete 4 free diagnostic problems (2 Math with Desmos and 2 Reading & Writing) with full Socratic explanations before upgrading to Citadel Pro.

What is included in the Citadel Pro pass?

Citadel Pro grants unlimited access to 2,600+ adaptive questions, multi-tier Socratic hint trees, Desmos graphing shortcuts, distractor trap breakdowns, and simulated full-length SAT and ACT exams.

SAT Math Difficulty: Hard (800) • Calibrated to 2026 Testing Standard

Digital SAT Math Practice Question #1191

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

The concentration of a certain drug in a patient's bloodstream, in milligrams per liter, $ t $ hours after administration, can be modeled by the polynomial function $ C(t) = 2t^3 - 5t^2 + at + b $. Understanding the drug's behavior at specific time points is crucial for dosage adjustments and ensuring patient safety. When the drug concentration is measured 1 hour after administration, it is found to be 0 mg/L. Furthermore, when the polynomial $ C(t) $ is divided by $ (t-2) $, the remainder is $ -10 $. What is the value of $ C(3) $?

Select Your Answer:

Socratic AI Engine Step-by-Step Derivation

0ms Precomputed

Tactical Insight (Hint 1)

Recall the Remainder Theorem: If a polynomial $ P(x) $ is divided by $ (x-k) $, the remainder is $ P(k) $. How can this be applied to the given conditions?

Elimination Framework (Hint 2)

You have two unknown coefficients, $ a $ and $ b $. Each piece of information provided in the problem statement will allow you to set up an equation involving $ a $ and $ b $. How can you combine these equations to solve for the unknowns?

⚡ 15-Second Desmos Speed Hack

⚡ Hack the Test: While there isn't a 'shortcut' to bypass the core algebra, efficiency is key. Once you've correctly determined $ a = -9 $ and $ b = 12 $, you have the full polynomial $ C(t) = 2t^3 - 5t^2 - 9t + 12 $. To evaluate $ C(3) $ quickly and accurately, use the 'table' or 'function evaluation' feature on a graphing calculator. Input the function and then find the value when $ t=3 $. This minimizes arithmetic errors, especially with multiple terms and exponents.

Masterclass Solution & Distractor Trap Analysis

The problem asks us to find the value of $ C(3) $ for the polynomial function $ C(t) = 2t^3 - 5t^2 + at + b $, given two conditions. We need to use these conditions to determine the unknown coefficients $ a $ and $ b $ first. **Step...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

🔒 Citadel Pro Scholar Vault