Before You Begin

Question #40291

Difficulty: 8/10 | 🟢 Free

The Question

Consider the following system of equations: $x + \sqrt{y} - 2^z = -1$ $2x + 3\sqrt{y} - 4 \cdot 2^z = 1$ $3x + 4\sqrt{y} - (a^2 - 3a + 7) 2^z = a^2 + a - 2$ Find the number of integers $a$ for which the system has exactly one real solution $(x, y, z)$ .

Options

A.
1
B.
2
C.
3
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

The Empty Set Trap|Cramer's Rule Constraints|The Zero-Factor Cancellation Trap|Domain Restrictions

Why This Question Is Tricky

Common Mistakes

This question is built on the 'Zero-Factor Cancellation Trap' combined with domain restrictions.

Students typically compute the variables using Cramer's rule and cancel the common factor (a-1) from the numerator and denominator.

They then solve the inequalities Y >= 0 and Z < 0 on the reduced rational functions, obtaining an interval that includes the integer a=1.

The trap is that at a=1, the original determinant D=0, and the system actually has infinitely many solutions that satisfy the domain constraints, violating the 'exactly one solution' requirement.

Blindly using the reduced formulas hides the fact that a=1 changes the nature of the system from independent to dependent, forcing the student to manually verify the boundary cases.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	The Empty Set Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free