**Core Concept**
Amenorrhea of 6 weeks duration in a woman can be caused by pregnancy, while a lump in the right iliac fossa may indicate appendicitis or an ectopic pregnancy. The investigation of choice should be able to diagnose both conditions.

**Why the Correct Answer is Right**
The correct investigation of choice is an ultrasound of the abdomen, specifically a transabdominal and transvaginal ultrasound. This imaging modality is useful for diagnosing pregnancy and evaluating the adnexal region for signs of ectopic pregnancy or appendicitis. Ultrasound can also assess the ovaries for any evidence of ovarian torsion or other pathology. Additionally, it is non-invasive and does not involve radiation, making it a safe option for pregnant women.

**Why Each Wrong Option is Incorrect**
**Option A:** **CT Scan**. While a CT scan can diagnose appendicitis and ectopic pregnancy, it involves radiation and is not the first-line imaging modality for diagnosing pregnancy.

**Option B:** **MRI**. MRI is not typically used as a first-line imaging modality for diagnosing appendicitis or ectopic pregnancy due to its higher cost and limited availability.

**Option C:** **X-ray**. An X-ray is not useful for diagnosing appendicitis or ectopic pregnancy and is not the preferred imaging modality for evaluating a lump in the right iliac fossa.

**Clinical Pearl / High-Yield Fact**
In a woman of childbearing age presenting with amenorrhea and a lump in the right iliac fossa, always consider the possibility of ectopic pregnancy, and ultrasound is the imaging modality of choice to evaluate for both appendicitis and ectopic pregnancy.

**Correct Answer:** C. Ultrasound.