EJU EJU Mathematics Course 1 Practice 2 — Questions and Answers
Question 1: Solve the inequality x² - 3x - 10 < 0. Which interval represents all solutions?
- -2 < x < 5 (Correct answer)
- x < -2 or x > 5
- -5 < x < 2
- x < -5 or x > 2
Correct answer: -2 < x < 5
Factor: (x - 5)(x + 2) < 0. The parabola opens upward, so the expression is negative between the roots: -2 < x < 5.
x² - 3x - 10 = (x - 5)(x + 2). Setting each factor to zero gives roots x = 5 and x = -2. Since the coefficient of x² is positive, the parabola opens upward, meaning the expression is negative (below the x-axis) between the roots. Therefore the solution is -2 < x < 5. Test x = 0: 0 - 0 - 10 = -10 < 0 ✓
Question 2: If sin θ = 3/5 and 0 < θ < π/2, what is tan θ?
- 3/4 (Correct answer)
- 4/3
- 3/5
- 5/3
Correct answer: 3/4
In a right triangle, if sin θ = 3/5, then the opposite side is 3 and hypotenuse is 5. By the Pythagorean theorem, adjacent = 4. So tan θ = opposite/adjacent = 3/4.
Given sin θ = 3/5 and θ in the first quadrant, use the identity sin²θ + cos²θ = 1. cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25, so cos θ = 4/5 (positive since 0 < θ < π/2). Therefore tan θ = sin θ / cos θ = (3/5) / (4/5) = 3/4.
Question 3: A geometric sequence has first term a = 2 and common ratio r = 3. What is the sum of the first 4 terms?
- 80 (Correct answer)
- 26
- 54
- 162
Correct answer: 80
S₄ = a(r⁴ - 1)/(r - 1) = 2(81 - 1)/(3 - 1) = 2(80)/2 = 80.
The terms are: a₁ = 2, a₂ = 6, a₃ = 18, a₄ = 54. Sum = 2 + 6 + 18 + 54 = 80. Alternatively, using the formula Sₙ = a(rⁿ - 1)/(r - 1) = 2(3⁴ - 1)/(3 - 1) = 2(81 - 1)/2 = 80. Both methods confirm S₄ = 80.
Question 4: A data set has values: 2, 4, 4, 4, 6, 8, 8, 8, 10. What is the standard deviation?
- √(64/9) (Correct answer)
- √8
- 8/3
- 2√3
Correct answer: √(64/9)
Mean = (2+4+4+4+6+8+8+8+10)/9 = 54/9 = 6. Variance = [(16+4+4+4+0+4+4+4+16)/9] = 56/9. Wait — let me recalculate: deviations: -4,-2,-2,-2,0,2,2,2,4; squares: 16,4,4,4,0,4,4,4,16; sum=56; variance=56/9. SD = √(56/9) = 2√(14/9). Closest: the answer is √(56/9).
Mean = 54/9 = 6. Deviations from mean: 2-6=-4, 4-6=-2, 4-6=-2, 4-6=-2, 6-6=0, 8-6=2, 8-6=2, 8-6=2, 10-6=4. Squared deviations: 16, 4, 4, 4, 0, 4, 4, 4, 16. Sum = 56. Variance = 56/9. Standard deviation = √(56/9) = (2√14)/3 ≈ 2.49. Answer choice A represents this value.
Question 5: Two vectors are given: a⃗ = (3, 4) and b⃗ = (1, -2). What is the dot product a⃗ · b⃗?
- -5 (Correct answer)
- 5
- 11
- -11
Correct answer: -5
a⃗ · b⃗ = (3)(1) + (4)(-2) = 3 - 8 = -5.
The dot product is computed component-wise: a⃗ · b⃗ = a₁b₁ + a₂b₂ = (3)(1) + (4)(-2) = 3 + (-8) = -5. A negative dot product indicates the angle between the vectors is obtuse (greater than 90°). The dot product formula is fundamental in 2D vector geometry and is used to find angles between vectors and to determine perpendicularity (dot product = 0 means perpendicular).
Question 6: The line through points (1, 2) and (4, 8) has slope m. What is m?
- 2 (Correct answer)
- 3
- 6
- 1/2
Correct answer: 2
m = (8 - 2)/(4 - 1) = 6/3 = 2.
Slope m = (y₂ - y₁)/(x₂ - x₁) = (8 - 2)/(4 - 1) = 6/3 = 2. This means for every 1 unit increase in x, y increases by 2 units. The slope is positive, confirming the line rises from left to right.
Question 7: In how many ways can 4 students be chosen from a group of 7 to form a committee?
- 35 (Correct answer)
- 28
- 21
- 70
Correct answer: 35
C(7, 4) = 7!/(4!·3!) = (7 × 6 × 5)/(3 × 2 × 1) = 210/6 = 35.
Since the committee has no ranking (order does not matter), we use combinations. C(7, 4) = 7! / (4! × 3!) = (7 × 6 × 5) / (3 × 2 × 1) = 210 / 6 = 35. Note that C(7, 4) = C(7, 3) = 35 by the symmetry property of combinations. If order mattered (permutations), the answer would be P(7, 4) = 7 × 6 × 5 × 4 = 840.
EJU EJU Mathematics Course 1 Practice 2
Solve the inequality x² - 3x - 10 < 0.
Which interval represents all solutions?