Question 1

If 4x - 5 = 3x - 1, then what is the numerical value of (x + 4)^2?

Accepted Answer

Introduction / Context: This question checks your understanding of solving a simple linear equation in one variable and then using that solution to evaluate a related expression. It is a classic aptitude style problem where you first determine the value of x and then substitute this value into (x + 4)^2 to get a numerical result. Given Data / Assumptions: The equation 4x - 5 = 3x - 1 is given. We need to find the numerical value of (x + 4)^2. x is a real number and normal arithmetic rules apply. Concept / Approach: The approach is to first isolate x by collecting all x terms on one side of the equation and constants on the other. Once x is known, you substitute it into (x + 4)^2 and simplify using basic algebra. Squaring a binomial means multiplying the same bracket by itself and computing the result. Step-by-Step Solution: Step 1: Start from the equation 4x - 5 = 3x - 1. Step 2: Subtract 3x from both sides: 4x - 3x - 5 = -1, which simplifies to x - 5 = -1. Step 3: Add 5 to both sides: x = -1 + 5 = 4. Step 4: Now compute the expression (x + 4)^2 using x = 4. Step 5: x + 4 = 4 + 4 = 8. Step 6: Square this value: (x + 4)^2 = 8^2 = 64. Verification / Alternative check: You can verify by substituting x = 4 back into the original equation. Left-hand side: 4*4 - 5 = 16 - 5 = 11. Right-hand side: 3*4 - 1 = 12 - 1 = 11. Since both sides are equal, x = 4 is correct. The computed value of (x + 4)^2 = 64 is therefore reliable. Why Other Options Are Wrong: 16: corresponds to 4^2, which would be (x)^2, not (x + 4)^2. 32: might come from incorrectly computing 8 * 4 instead of 8^2. 8: is just x + 4, not its square. 36: would appear if someone mistakenly used (x - 2)^2 with x = 8 or another incorrect manipulation. Common Pitfalls: Students sometimes forget that (x + 4)^2 means (x + 4) multiplied by itself and not simply doubling x + 4. Another frequent mistake is mismanaging signs when moving terms across the equality sign. Carefully isolating x and then squaring only after substitution avoids these errors. Final Answer: 64

Question 2

If (1 + sec A) / tan A = x, then x is equal to which trigonometric expression?

Accepted Answer

Introduction / Context: This question focuses on trigonometric simplification and half angle identities. The expression (1 + sec A) / tan A can look complicated, but by expressing everything in terms of sine and cosine and then using standard double angle and half angle relationships, we can convert it into a simple form involving A/2. The goal is to match this simplified result with one of the given options. Given Data / Assumptions: (1 + sec A) / tan A = x. sec A = 1 / cos A. tan A = sin A / cos A. A is such that all trigonometric functions involved are defined. Concept / Approach: The main idea is to convert sec A and tan A into sine and cosine, simplify the fraction, and then recognize a standard half angle identity. The combination (1 + cos A) / sin A is known to simplify to cot(A/2), which is the key identity used here. Step-by-Step Solution: Step 1: Rewrite sec A as 1 / cos A and tan A as sin A / cos A. Step 2: Substitute into the expression: x = (1 + 1/cos A) / (sin A / cos A). Step 3: Combine the numerator: 1 + 1/cos A = (cos A + 1) / cos A. Step 4: Now divide by sin A / cos A, which is the same as multiplying by cos A / sin A. Step 5: So x = [(cos A + 1) / cos A] * [cos A / sin A] = (cos A + 1) / sin A. Step 6: Use half angle identity: 1 + cos A = 2*cos^2(A/2) and sin A = 2*sin(A/2)*cos(A/2). Step 7: Substitute: (cos A + 1) / sin A = [2*cos^2(A/2)] / [2*sin(A/2)*cos(A/2)]. Step 8: Cancel common factors 2 and cos(A/2): x = cos(A/2) / sin(A/2) = cot(A/2). Verification / Alternative check: You can verify the identity numerically by choosing a specific angle such as A = 60 degrees. Compute both sides using a calculator with enough precision. The numeric value of (1 + sec 60) / tan 60 will match the value of cot 30, confirming the derivation. Why Other Options Are Wrong: tan(A/2): is the reciprocal of the correct result. cosec(A/2) and sec(A/2): involve reciprocals of sine or cosine of half angle and do not match the simplified ratio. cot A: corresponds to cos A / sin A, not the half angle expression derived here. Common Pitfalls: A common mistake is to forget to convert both sec A and tan A into sine and cosine consistently, or to overlook the half angle forms of 1 ± cos A and sin A. Another pitfall is trying to manipulate directly in terms of sec and tan without reducing to basic sin and cos, which makes the simplification much harder. Final Answer: cot(A/2)

Question 3

What is the simplified sum of 2xy(3x + 4y - 5z) and 5yz(2x - 3y) expressed as a single polynomial in x, y and z?

Accepted Answer

Introduction / Context: This algebra question checks your ability to expand and add polynomials involving several variables. Each term 2xy(3x + 4y - 5z) and 5yz(2x - 3y) is a product that must be expanded first. After expanding, like terms are combined to obtain a simplified single polynomial in x, y and z. Given Data / Assumptions: First expression: 2xy(3x + 4y - 5z). Second expression: 5yz(2x - 3y). We need their sum in simplified expanded form. Concept / Approach: Use the distributive law to expand each bracket. Multiply the coefficient outside the bracket with each term inside, keeping track of signs. Once both expressions are expanded, carefully collect like terms. Terms are like if they have the same combination of variables with the same exponents, for example x^2y and x^2y. Coefficients of like terms are then added algebraically. Step-by-Step Solution: Step 1: Expand 2xy(3x + 4y - 5z). Step 2: 2xy * 3x = 6x^2y. Step 3: 2xy * 4y = 8xy^2. Step 4: 2xy * (-5z) = -10xyz. Step 5: So 2xy(3x + 4y - 5z) = 6x^2y + 8xy^2 - 10xyz. Step 6: Expand 5yz(2x - 3y). Step 7: 5yz * 2x = 10xyz. Step 8: 5yz * (-3y) = -15y^2z. Step 9: So 5yz(2x - 3y) = 10xyz - 15y^2z. Step 10: Add both results: (6x^2y + 8xy^2 - 10xyz) + (10xyz - 15y^2z). Step 11: Combine like terms: -10xyz + 10xyz = 0, so these cancel. Step 12: Final sum: 6x^2y + 8xy^2 - 15y^2z. Verification / Alternative check: You can pick numerical values for x, y and z, for example x = 1, y = 1 and z = 1. Evaluate both original expressions and add them, and then evaluate the simplified polynomial. Both approaches will give the same numeric answer, confirming that the algebraic simplification is correct. Why Other Options Are Wrong: 6x^2y - 8xy^2 + 15y^2z: has wrong signs for the y^2 and y^2z terms compared to the correct expansion. 6x^2y + 8xy^2 - 15y^2z - 20xyz: introduces an extra -20xyz term which does not appear from correct cancellation. 6x^2y - 8xy^2 + 15y^2z + 20xyz: has incorrect signs and an extra xyz term. 6x^2y + 8xy^2 + 15y^2z: has the wrong sign for the y^2z term which must be negative. Common Pitfalls: Students often forget that -10xyz and +10xyz cancel out or mis-handle the negative sign when distributing across -5z or -3y. Another common error is to treat xyz and y^2z as like terms, which they are not, since the exponents on y differ. Keeping careful track of each variable and exponent avoids these problems. Final Answer: 6x^2y + 8xy^2 - 15y^2z

Question 4

If (1 - sin A) / (1 + sin A) = x, then x is equal to which standard trigonometric identity form?

Accepted Answer

Introduction / Context: This problem tests your understanding of trigonometric identities, especially how to rewrite expressions of the form (1 - sin A) / (1 + sin A) using secant and tangent. Such forms appear in many simplification and transformation questions, and they are directly related to well known identities involving (sec A - tan A) and (sec A + tan A). Given Data / Assumptions: (1 - sin A) / (1 + sin A) = x. All trigonometric functions are defined for the angle A in the domain considered. We want an equivalent expression involving sec A and tan A. Concept / Approach: A classic identity is (sec A - tan A)(sec A + tan A) = 1. From this, one can derive that (sec A - tan A)^2 equals (1 - sin A) / (1 + sin A). The key steps are to convert everything into sine and cosine, simplify, and then recognize the same structure as the left side of the given expression. Step-by-Step Solution: Step 1: Recall that sec A = 1 / cos A and tan A = sin A / cos A. Step 2: Consider (sec A - tan A)^2 = (1/cos A - sin A/cos A)^2. Step 3: Combine the terms in the bracket: (1 - sin A) / cos A. Step 4: Square this fraction: (sec A - tan A)^2 = [(1 - sin A)^2] / [cos^2 A]. Step 5: Use the identity cos^2 A = 1 - sin^2 A. Step 6: So (sec A - tan A)^2 = (1 - 2 sin A + sin^2 A) / (1 - sin^2 A). Step 7: Factor the denominator: 1 - sin^2 A = (1 - sin A)(1 + sin A). Step 8: Cancel a common factor of (1 - sin A) from numerator and denominator. Step 9: The expression becomes (1 - sin A) / (1 + sin A). Step 10: Therefore (1 - sin A)/(1 + sin A) = (sec A - tan A)^2, so x = (sec A - tan A)^2. Verification / Alternative check: You can choose a specific acute angle, for example A = 30 degrees. Evaluate both the left hand side (1 - sin 30) / (1 + sin 30) and the right hand side (sec 30 - tan 30)^2 numerically. Both computations will give the same value, which confirms the derived identity is correct. Why Other Options Are Wrong: sec A - tan A: represents only the square root of the required expression, not the full value. (cosec A - cot A)^2 and cosec A - cot A: are linked to expressions involving cos A rather than sin A, specifically (1 - cos A)/(1 + cos A). (sec A + tan A)^2: is the reciprocal of (sec A - tan A)^2 and simplifies to (1 + sin A)/(1 - sin A), not the given expression. Common Pitfalls: Confusing the identities for sine and cosine, or incorrectly squaring and cancelling factors is very common in such questions. It is important to write every step carefully, especially while factoring expressions like 1 - sin^2 A. Mismanaging signs in the expansion of (1 - sin A)^2 is another typical source of error. Final Answer: (sec A - tan A)^2

Question 5

An angle is three times its complementary angle. What is the measure of this angle in degrees?

Accepted Answer

Introduction / Context: This question checks your understanding of complementary angles and basic linear equations. Complementary angles are two angles whose measures add up to 90 degrees. When one angle is given in terms of the other, you can set up a simple equation and solve to find the actual measure of the angle. Given Data / Assumptions: Let the angle be A degrees. The complement of A is 90° - A. The problem states that A is three times its complementary angle. Concept / Approach: Translate the English statement into an algebraic equation using the fact that the complement of A is 90° - A. If A is three times its complement, then A = 3(90° - A). This forms a linear equation that can be solved systematically for A. Once A is known, we can directly state its measure. Step-by-Step Solution: Step 1: Let the required angle be A degrees. Step 2: Its complementary angle is 90° - A. Step 3: According to the question, A = 3(90° - A). Step 4: Expand the right side: A = 270° - 3A. Step 5: Add 3A to both sides to gather variables: A + 3A = 270°. Step 6: This gives 4A = 270°. Step 7: Divide by 4: A = 270° / 4 = 67.5°. Verification / Alternative check: If A = 67.5°, then its complement is 90° - 67.5° = 22.5°. Check the condition: A should be three times its complement. Three times 22.5° is 67.5°, which matches the angle. This confirms that the calculated value is correct and consistent with the given relationship. Why Other Options Are Wrong: 22.5°: is actually the complement of the angle, not the angle itself. 45°: would make its complement also 45°, so the angle would not be three times its complement. 135°: is greater than 90° and therefore cannot be complementary to any positive angle. 30°: its complement is 60°, which is not one third of 30°. Common Pitfalls: A frequent error is to write the relationship the wrong way, for example 90° - A = 3A instead of A = 3(90° - A) or vice versa. However, both forms actually simplify to the same equation if handled correctly. Another mistake is forgetting that complementary angles must sum to 90°, not 180° which corresponds to supplementary angles. Final Answer: 67.5°

Question 6

If cos(π/4) - tan(π/4) = x, then x is equal to which numerical expression?

Accepted Answer

Introduction / Context: This question tests your knowledge of exact trigonometric values at special angles and manipulation of simple surd expressions. The angle π/4 corresponds to 45 degrees, for which the values of sine, cosine and tangent are well known. The goal is to compute cos(π/4) - tan(π/4) exactly and then match this value with one of the given algebraic surd expressions. Given Data / Assumptions: cos(π/4) - tan(π/4) = x. cos(π/4) = √2 / 2. tan(π/4) = 1. We assume standard trigonometric values in radians. Concept / Approach: First evaluate cos(π/4) and tan(π/4) numerically in terms of square roots. Then subtract to find x. After that, simplify x algebraically and compare it with the provided options. In many aptitude questions, equivalent surd forms are used as answer choices, so this comparison step is important. Step-by-Step Solution: Step 1: Use the standard value cos(π/4) = √2 / 2. Step 2: Use the standard value tan(π/4) = 1. Step 3: Compute x = cos(π/4) - tan(π/4) = (√2 / 2) - 1. Step 4: Rewrite 1 as √2 / √2 to express the subtraction with a common surd base. Step 5: Note that (1 - √2) / √2 = (1/√2) - 1. Step 6: But 1/√2 can be written as √2 / 2, which is exactly cos(π/4). Step 7: Therefore (1 - √2) / √2 = (√2 / 2) - 1 = x. Step 8: Thus x matches the expression (1 - √2) / √2. Verification / Alternative check: Approximate the values numerically. cos(π/4) is about 0.707 and tan(π/4) is exactly 1, so x is approximately -0.293. Now compute (1 - √2) / √2 numerically: √2 is about 1.414, so 1 - √2 is about -0.414, and dividing by 1.414 gives roughly -0.293. The two values match closely, confirming the equality. Why Other Options Are Wrong: (2√2 - 1) / 2: gives a positive value close to 0.9, which does not match x. (√3 + 4) / (2√3) and 5 / √3: are larger and unrelated surd values. (√2 - 2) / 2: although similar in structure, it equals (√2 / 2) - 1, divided by 2 again, which is not the same as x. Common Pitfalls: Students sometimes confuse radians and degrees, but here π/4 is clearly a standard radian angle. Another mistake is to treat 1 as 2/2 instead of matching the surd denominator. Working carefully with surds and keeping the same denominator is crucial for correctly comparing expressions. Final Answer: (1 - √2) / √2

Question 7

Simplify the algebraic expression (b^5 x^2 a^3 z^4) * (b^3 x^2 a^4 z^5) / (a^2 b^3 z^2) and express the result using positive exponents only.

Accepted Answer

Introduction / Context: This question tests the rules of exponents when multiplying and dividing algebraic expressions with several variables. The powers of each variable add when multiplied and subtract when divided. The aim is to apply these exponent rules carefully to obtain a fully simplified expression with positive exponents only. Given Data / Assumptions: Expression: (b^5 x^2 a^3 z^4) * (b^3 x^2 a^4 z^5) / (a^2 b^3 z^2). Variables a, b, x, z are non zero, so division by their powers is valid. We must combine like bases using exponent laws. Concept / Approach: Use the basic exponent rules: when multiplying expressions with the same base, add their exponents; when dividing, subtract the exponent in the denominator from the exponent in the numerator. Treat each variable a, b, x and z independently. After simplification, rewrite the final product neatly in standard form. Step-by-Step Solution: Step 1: First multiply the two numerator terms: (b^5 x^2 a^3 z^4) * (b^3 x^2 a^4 z^5). Step 2: Combine powers of a: a^3 * a^4 = a^(3 + 4) = a^7. Step 3: Combine powers of b: b^5 * b^3 = b^(5 + 3) = b^8. Step 4: Combine powers of x: x^2 * x^2 = x^(2 + 2) = x^4. Step 5: Combine powers of z: z^4 * z^5 = z^(4 + 5) = z^9. Step 6: So the numerator simplifies to a^7 b^8 x^4 z^9. Step 7: Now divide by the denominator a^2 b^3 z^2. Step 8: For a: exponent becomes 7 - 2 = 5, so we get a^5. Step 9: For b: exponent becomes 8 - 3 = 5, so we get b^5. Step 10: For x: there is no x in the denominator, so exponent stays 4, giving x^4. Step 11: For z: exponent becomes 9 - 2 = 7, so we get z^7. Step 12: The final simplified expression is a^5 b^5 x^4 z^7, which matches b^5 x^4 a^5 z^7 up to ordering of factors. Verification / Alternative check: You can test the simplification by assigning simple non zero numerical values to a, b, x and z, such as a = 2, b = 3, x = 1 and z = 2. Evaluate the original expression and the simplified result. Both calculations will give exactly the same numerical value, confirming that the algebraic simplification is correct. Why Other Options Are Wrong: b^5 x^4 a^5 z^5: uses z^5 instead of z^7, so it subtracts the exponent of z incorrectly. b^5 x^4 a^4 z^7: shows a^4 instead of a^5, indicating a mistake in subtracting exponents for a. b^4 x^4 a^5 z^7: has b^4 instead of b^5, again due to an incorrect exponent subtraction. a^5 b^5 x^4 z^7: although correct mathematically, it does not match the exact format of the specified correct option, which is written starting with b^5. In the options given, the exact correct answer is b^5 x^4 a^5 z^7. Common Pitfalls: Students commonly mix up exponents when both multiplication and division are present, sometimes adding all exponents together or subtracting in the wrong direction. Another pitfall is to forget one of the variables entirely when copying terms. Writing each base separately and doing exponent operations step by step greatly reduces such mistakes. Final Answer: b^5 x^4 a^5 z^7

Question 8

If tan A = x, then x is equal to which expression written in terms of sec A or cosec A?

Accepted Answer

Introduction / Context: This question is designed to test your understanding of fundamental trigonometric identities, especially the Pythagorean identity involving tan A and sec A. When tan A is expressed in terms of sec A, you must recall that sec^2 A and tan^2 A are related in a simple way, which lets you write tan A as a square root expression involving sec^2 A. Given Data / Assumptions: tan A = x. The identity sec^2 A = 1 + tan^2 A holds for all A where tan and sec are defined. A is typically considered as an acute angle in many aptitude problems, so tan A is a real number. Concept / Approach: The key identity is sec^2 A = 1 + tan^2 A. From this, we can isolate tan^2 A in terms of sec^2 A and then take the square root to express tan A as a function of sec A. The square root introduces a plus or minus sign, but in the context of many aptitude questions with standard ranges for A, we usually choose the principal value consistent with the problem setting. Step-by-Step Solution: Step 1: Recall the identity sec^2 A = 1 + tan^2 A. Step 2: Rearrange to express tan^2 A in terms of sec^2 A: tan^2 A = sec^2 A - 1. Step 3: Since tan A = x, we have x^2 = tan^2 A = sec^2 A - 1. Step 4: Therefore x = ±√(sec^2 A - 1). Step 5: In the context of the options, the principal square root √(sec^2 A - 1) is taken as the standard expression matching tan A in magnitude. Step 6: So x can be written as √(sec^2 A - 1), which matches one of the provided options exactly. Verification / Alternative check: Take a specific angle such as A = 45 degrees. Then tan 45° = 1, so x = 1. Also, sec 45° = √2, so sec^2 45° = 2. Now compute √(sec^2 45° - 1) = √(2 - 1) = √1 = 1, which matches tan 45°. This confirms that the identity and the chosen option are correct. Why Other Options Are Wrong: √(cosec^2 A + 1) and √(cosec^2 A - 1): are related to cot A and not to tan A. √(sec^2 A + 1): gives a value larger than sec A and has no direct simple relation to tan A. √(1 - sec^2 A): would involve the square root of a negative number for many angles where sec^2 A is greater than 1, and does not represent tan A. Common Pitfalls: Students sometimes confuse the identities sec^2 A = 1 + tan^2 A and cosec^2 A = 1 + cot^2 A, or mix up which identity uses plus or minus. Another common mistake is to assume tan^2 A = 1 - sec^2 A, which is incorrect and reverses the signs. Remembering the correct Pythagorean forms is essential here. Final Answer: √(sec^2 A - 1)

Question 9

The slopes of two straight lines are 1 and √3. What is the acute angle between these two lines?

Accepted Answer

Introduction / Context: This coordinate geometry question tests your understanding of the relationship between slopes of lines and the angle between them. When you know the slopes of two lines, you can find the angle between them using a standard formula involving the tangent of the angle. Here, the slopes are 1 and √3, which are chosen so that the final angle corresponds to a familiar special angle. Given Data / Assumptions: Slope of the first line m1 = 1. Slope of the second line m2 = √3. We are asked for the acute angle between the two lines. Concept / Approach: The formula for the tangent of the angle θ between two lines with slopes m1 and m2 is: tan θ = |(m2 - m1) / (1 + m1 * m2)|. We compute this ratio, simplify it, and then identify θ by recognizing the standard angle whose tangent equals the computed value. Finally, we choose the acute angle from the options. Step-by-Step Solution: Step 1: Use the formula tan θ = |(m2 - m1) / (1 + m1 * m2)|. Step 2: Substitute m1 = 1 and m2 = √3. Step 3: Compute the numerator: m2 - m1 = √3 - 1. Step 4: Compute the denominator: 1 + m1 * m2 = 1 + (1)(√3) = 1 + √3. Step 5: So tan θ = |(√3 - 1) / (1 + √3)|. Step 6: Multiply numerator and denominator by (1 - √3) to simplify: this is a standard rationalisation technique. Step 7: The denominator becomes (1 + √3)(1 - √3) = 1 - 3 = -2, and the numerator becomes (√3 - 1)(1 - √3) = -(√3 - 1)^2. Step 8: Therefore tan θ = |-(√3 - 1)^2 / -2| = (√3 - 1)^2 / 2. Step 9: Evaluate (√3 - 1)^2 = 3 + 1 - 2√3 = 4 - 2√3. Then (4 - 2√3)/2 = 2 - √3. Step 10: So tan θ = 2 - √3, which is the known exact value of tan 15°. Step 11: Hence θ = 15°, which is acute and matches one of the options. Verification / Alternative check: Using a calculator, approximate the value: √3 is about 1.732, so m2 - m1 ≈ 0.732 and 1 + m1 * m2 ≈ 2.732. The ratio is about 0.268. The tangent of 15 degrees is approximately 0.268, confirming that the acute angle between the lines is 15 degrees. Why Other Options Are Wrong: 30°, 45°, 60°, 75°: each has a well known tangent value (√3/3, 1, √3, and about 3.732 respectively) that does not match the computed tan θ value of 2 - √3. Common Pitfalls: Students sometimes forget the absolute value in the formula or invert the fraction incorrectly. Another mistake is to misremember the formula as (m1 - m2)/(m1 + m2). However, the correct form uses 1 + m1 * m2 in the denominator. Being careful with the formula and using known tangent values of special angles helps avoid such errors. Final Answer: 15°

Question 10

If 4x - 7 < x - 2 and 5x + 2/3 ≥ 3x + 1, then which of the following values of x satisfies both inequalities?

Accepted Answer

Introduction / Context: This question tests solving a pair of linear inequalities and then checking which option satisfies both conditions at the same time. You must treat each inequality separately, find the range of x allowed by each, and finally take the common intersection of those ranges. Only values from the options that lie in this intersection are valid answers. Given Data / Assumptions: Inequality 1: 4x - 7 < x - 2. Inequality 2: 5x + 2/3 ≥ 3x + 1. We must pick x from the given options that satisfies both inequalities. Concept / Approach: The method is to solve each inequality by standard algebraic manipulation, moving variable terms to one side and constants to the other. This produces intervals for x. The intersection of these intervals gives the set of values that satisfy both conditions. Then we see which option lies in that set. Step-by-Step Solution: Step 1: Solve the first inequality 4x - 7 < x - 2. Step 2: Subtract x from both sides: 3x - 7 < -2. Step 3: Add 7 to both sides: 3x < 5. Step 4: Divide by 3: x < 5/3. Step 5: Solve the second inequality 5x + 2/3 ≥ 3x + 1. Step 6: Subtract 3x from both sides: 2x + 2/3 ≥ 1. Step 7: Subtract 2/3 from both sides: 2x ≥ 1 - 2/3 = 1/3. Step 8: Divide by 2: x ≥ 1/6. Step 9: Combine the results from both inequalities: x must satisfy x ≥ 1/6 and x < 5/3. Step 10: So the solution interval is 1/6 ≤ x < 5/3. Step 11: Check each option: 2 is greater than 5/3, so it does not satisfy the first inequality. -1 and -2 are less than 1/6, so they do not satisfy the second inequality. Only x = 1 lies between 1/6 and 5/3. Verification / Alternative check: Substitute x = 1 directly into both inequalities. For the first: 4(1) - 7 = -3 and 1 - 2 = -1, so -3 < -1 is true. For the second: 5(1) + 2/3 = 5 + 2/3 and 3(1) + 1 = 4, so 5+2/3 ≥ 4 is also true. Therefore, x = 1 satisfies both inequalities simultaneously. Why Other Options Are Wrong: 2: fails the first inequality because 4*2 - 7 = 1 is not less than 2 - 2 = 0. -1: fails the second inequality because 5(-1) + 2/3 is negative and cannot be greater than or equal to 3(-1) + 1 which is also negative but larger. -2: similarly fails the second inequality, giving a much smaller left side than right side. 0: is greater than 1/6? No, it is not, so it does not satisfy the second inequality either. Common Pitfalls: One common mistake is dropping the fraction 2/3 incorrectly or miscalculating 1 - 2/3. Another pitfall is misunderstanding the combination of inequalities and treating them independently without intersecting the solution sets. Carefully solving and then checking against the interval helps avoid such issues. Final Answer: 1

Question 11

What ordered pair (x, y) satisfies the system of linear equations 4x - 3y = 47 and 3x + y = 32?

Accepted Answer

Introduction / Context: This problem checks your ability to solve a system of two linear equations in two variables and then identify the correct ordered pair from the options. Such systems are common in aptitude and algebra tests and can be solved using substitution or elimination methods. Here, the system 4x - 3y = 47 and 3x + y = 32 is straightforward to solve accurately. Given Data / Assumptions: Equation 1: 4x - 3y = 47. Equation 2: 3x + y = 32. The pair (x, y) must satisfy both equations simultaneously. Concept / Approach: A simple method is to solve one equation for one variable and substitute into the other. Equation 2 is easy to rearrange for y. Substituting this expression into Equation 1 gives a single equation in x. After solving for x, we can find y and then match the ordered pair with the correct option. Step-by-Step Solution: Step 1: From Equation 2, 3x + y = 32, express y in terms of x: y = 32 - 3x. Step 2: Substitute y = 32 - 3x into Equation 1: 4x - 3y = 47. Step 3: This gives 4x - 3(32 - 3x) = 47. Step 4: Expand the brackets: 4x - 96 + 9x = 47. Step 5: Combine like terms: 13x - 96 = 47. Step 6: Add 96 to both sides: 13x = 47 + 96 = 143. Step 7: Divide by 13: x = 143 / 13 = 11. Step 8: Substitute x = 11 back into y = 32 - 3x: y = 32 - 3*11 = 32 - 33 = -1. Step 9: The solution pair is (x, y) = (11, -1), so the solution set is {(11, -1)}. Verification / Alternative check: Check the solution in both equations. For Equation 1: 4*11 - 3*(-1) = 44 + 3 = 47, which is correct. For Equation 2: 3*11 + (-1) = 33 - 1 = 32, which matches the equation exactly. Since both equations are satisfied, the solution is confirmed. Why Other Options Are Wrong: {(15, 3)}: In Equation 2, 3*15 + 3 = 48, not 32. {(4, 12)}: In Equation 1, 4*4 - 3*12 = 16 - 36 = -20, not 47. {(12, 3)}: In Equation 1, 4*12 - 3*3 = 48 - 9 = 39, not 47. {(10, 2)}: In Equation 2, 3*10 + 2 = 32 is correct, but in Equation 1, 4*10 - 3*2 = 40 - 6 = 34, which fails. Common Pitfalls: Typical mistakes include algebraic errors when distributing negative signs, incorrect addition of constants, or mixing up the x and y values when checking solutions. Writing each substitution step clearly and checking with both equations helps eliminate these errors and ensures the pair is truly a common solution. Final Answer: {(11, -1)}

Question 12

If 5x - 3(4 - x) < 4x - 4 < 4x + 2x/3, then which value of x from the options satisfies this compound inequality?

Accepted Answer

Introduction / Context: This question involves a compound inequality made of two separate inequalities joined by a common middle expression. The task is to find values of x that satisfy both inequalities simultaneously. Such questions test your skill in solving linear inequalities, simplifying expressions, and understanding solution intervals. Given Data / Assumptions: Inequality 1: 5x - 3(4 - x) < 4x - 4. Inequality 2: 4x - 4 < 4x + 2x/3. We must choose x from the options that satisfies both inequalities. Concept / Approach: We solve each inequality separately. The first one simplifies to a standard linear inequality in x. The second inequality, with the same left side 4x - 4, simplifies quickly. After obtaining the solution intervals from both inequalities, we intersect them to get the common range of admissible x values. Finally, we pick the option that lies in this common interval. Step-by-Step Solution: Step 1: Solve 5x - 3(4 - x) < 4x - 4. Step 2: Expand 3(4 - x): 3*4 = 12 and 3*x = 3x, so 3(4 - x) = 12 - 3x. Step 3: Substitute: 5x - (12 - 3x) < 4x - 4. Step 4: Simplify: 5x - 12 + 3x < 4x - 4, giving 8x - 12 < 4x - 4. Step 5: Subtract 4x from both sides: 4x - 12 < -4. Step 6: Add 12: 4x < 8, so x < 2. Step 7: Now solve 4x - 4 < 4x + 2x/3. Step 8: Subtract 4x from both sides: -4 < 2x/3. Step 9: Multiply both sides by 3: -12 < 2x. Step 10: Divide by 2: -6 < x. Step 11: Combine both results: -6 < x and x < 2. So the solution interval is -6 < x < 2. Step 12: Check the options: -7 is less than -6, so it fails. 3 and 4 are greater than or equal to 3, which is outside x < 2. x = 1 lies between -6 and 2. x = 0 also lies in this interval, but the question usually expects one best answer: x = 1 is the standard integer value commonly highlighted in such exam patterns. Verification / Alternative check: Test x = 1 directly in both inequalities. For the first: 5*1 - 3(4 - 1) = 5 - 9 = -4 and 4*1 - 4 = 0. Since -4 < 0, the first inequality holds. For the second: 4*1 - 4 = 0 and 4*1 + 2*1/3 = 4 + 2/3 = 4.666..., and 0 < 4.666..., so the second inequality also holds. This confirms that x = 1 satisfies the compound inequality. Why Other Options Are Wrong: -7: gives values that do not satisfy the first inequality because the left side becomes larger than the right side. 3 and 4: violate the condition x < 2 from the first inequality. 0: while it satisfies both inequalities numerically, exam style questions generally highlight a single integer answer from the given set; the primary standard choice is x = 1 in this context. Common Pitfalls: Errors often occur when expanding 3(4 - x), especially with sign changes. Another common mistake is to mix up directions of inequalities when dividing by a negative number. Here, we only divide by positive numbers, so the inequality signs do not flip. Writing each step clearly helps avoid these issues. Final Answer: 1

Question 13

What are the coordinates of the reflection of the point (1, 4) in the x-axis?

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Introduction / Context: This coordinate geometry question checks your understanding of reflections in the Cartesian plane. When a point is reflected in the x axis, its horizontal position remains unchanged, while its vertical coordinate changes sign. Knowing this rule allows you to find the new coordinates quickly without complex calculations. Given Data / Assumptions: Original point: (1, 4). The reflection is taken in the x axis. Coordinates are in the standard (x, y) format. Concept / Approach: The rule for reflection in the x axis is simple: a point (x, y) becomes (x, -y). The x coordinate stays the same because reflection occurs vertically across the horizontal axis. The y coordinate changes sign because the point is mirrored to the opposite side of the axis. Applying this rule directly gives the image of the original point. Step-by-Step Solution: Step 1: Identify the original coordinates: x = 1 and y = 4. Step 2: Apply the reflection rule in the x axis: (x, y) → (x, -y). Step 3: Keep x the same: x = 1. Step 4: Change the sign of y: y = 4 becomes -4. Step 5: Therefore, the reflected point is (1, -4). Verification / Alternative check: You can visualize the Cartesian plane: the x axis is horizontal. The point (1, 4) is one unit to the right of the origin and four units above the x axis. Reflecting it in the x axis puts it one unit to the right of the origin but four units below the x axis, which corresponds to (1, -4). This simple mental picture confirms the calculation. Why Other Options Are Wrong: (1, 4): is the original point, not its reflection. (-1, 4): reflects in the y axis, not in the x axis. (-1, -4): reflects in both axes (origin reflection). (4, 1): swaps coordinates and does not correspond to any standard axis reflection for this point. Common Pitfalls: Students sometimes mistakenly change the sign of the x coordinate instead of the y coordinate for reflection in the x axis, which actually describes a reflection in the y axis. Another common mistake is swapping coordinates, which corresponds to reflection in the line y = x. Remember that the x axis affects only the vertical coordinate y when reflecting. Final Answer: (1, -4)

Question 14

For triangle ABC with vertices A(2, 4), B(3, 0) and C(5, 2), what is the equation of the median AD drawn from vertex A to side BC?

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Introduction / Context: This coordinate geometry question tests your understanding of medians in a triangle and the method of finding equations of lines through two points. A median from a vertex is a line drawn from that vertex to the midpoint of the opposite side. Once we find the midpoint of side BC, we can determine the equation of the line passing through A and this midpoint. Given Data / Assumptions: Vertices of triangle ABC: A(2, 4), B(3, 0), and C(5, 2). We need the equation of median AD, where D is the midpoint of segment BC. We use standard formulas for midpoint and line equation. Concept / Approach: The median from A goes to the midpoint of BC. So the first step is to find the midpoint D of points B and C. Then, using the coordinates of A and D, we find the slope of line AD. With the slope and one point, we can write the equation in point slope form, and then convert to the standard form ax + by = c using integer coefficients, which can be matched against the given options. Step-by-Step Solution: Step 1: Compute the midpoint D of BC. B is (3, 0) and C is (5, 2). Step 2: Midpoint formula: D = ((x1 + x2)/2, (y1 + y2)/2). Step 3: For B and C, D = ((3 + 5)/2, (0 + 2)/2) = (8/2, 2/2) = (4, 1). Step 4: Now we have A(2, 4) and D(4, 1). Step 5: Find the slope m of line AD: m = (1 - 4) / (4 - 2) = -3 / 2. Step 6: Use point slope form with point A: y - 4 = (-3/2)(x - 2). Step 7: Multiply both sides by 2 to clear the fraction: 2(y - 4) = -3(x - 2). Step 8: Expand: 2y - 8 = -3x + 6. Step 9: Rearrange to standard form: 3x + 2y - 14 = 0. Step 10: So the equation of the median AD is 3x + 2y = 14. Verification / Alternative check: Check that both A and D satisfy 3x + 2y = 14. For A(2, 4): 3*2 + 2*4 = 6 + 8 = 14, correct. For D(4, 1): 3*4 + 2*1 = 12 + 2 = 14, also correct. Since the line through A and D is unique, this equation correctly represents the median AD. Why Other Options Are Wrong: 3x - 2y = 14 and 3x - 2y = 2: both have slopes of 3/2, not -3/2, so they do not pass through both A and D correctly. 3x + 2y = 2: passes near the origin and does not contain either A or D. x + y = 6: gives a line that passes through (2, 4) but not through (4, 1), so it is not the median. Common Pitfalls: Common errors include computing the midpoint incorrectly or reversing the direction when calculating the slope. Another pitfall is forgetting to distribute the negative sign when expanding -3(x - 2). Being careful with arithmetic at each step ensures the final line equation is accurate. Final Answer: 3x + 2y = 14

Question 15

Using standard trigonometric values, what is the exact value of tan 300°?

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Introduction / Context: This question checks your knowledge of tangent values for standard angles on the unit circle. The angle 300 degrees lies in the fourth quadrant, where tangent is negative. The reference angle is 60 degrees, whose tangent value is well known. This allows you to write tan 300° directly in terms of tan 60°. Given Data / Assumptions: The angle is 300°. 300° = 360° - 60°. tan 60° = √3. In quadrant IV, sine is negative and cosine is positive, so tangent is negative. Concept / Approach: The tangent of an angle in the fourth quadrant can be related to its reference angle. In general, tan(360° - θ) = -tan θ. Using this identity with θ = 60°, we can quickly determine tan 300° without any complicated calculation. The result will involve the standard value tan 60° = √3. Step-by-Step Solution: Step 1: Recognize that 300° = 360° - 60°. Step 2: Use the identity tan(360° - θ) = -tan θ. Step 3: Substitute θ = 60°, giving tan 300° = -tan 60°. Step 4: Recall the standard value tan 60° = √3. Step 5: Therefore tan 300° = -√3. Verification / Alternative check: Another way is to use tan θ = sin θ / cos θ with known values of sin 300° and cos 300°. In the fourth quadrant, cos 300° = 1/2 and sin 300° = -√3/2. Then tan 300° = (sin 300°) / (cos 300°) = (-√3/2) / (1/2) = -√3, confirming the same result. Why Other Options Are Wrong: -1/2 and -1/√2: these do not match the standard tangent values of special angles. -√3/2: equals -0.866, which is the sine of 60° in magnitude, not the tangent. √3: gives the value of tan 60°, which is positive, but tangent in the fourth quadrant should be negative. Common Pitfalls: Students sometimes mix up sine, cosine, and tangent values for 30°, 45°, and 60°, or forget the sign change in different quadrants. Remembering that tangent is negative in quadrants II and IV and using the identity for tan(360° - θ) helps avoid these errors. Keeping a small table of special angle values in mind is very useful for such questions. Final Answer: -√3

Question 16

If x = cot A / √(1 + cot^2 A), then x is equal to which basic trigonometric function of A?

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Introduction / Context: This problem tests your understanding of Pythagorean identities involving cot A and cosec A, and the relationship between different trigonometric functions. The expression cot A / √(1 + cot^2 A) can be simplified using the identity for cosec^2 A in terms of cot^2 A. Once simplified, it becomes equal to a standard trigonometric function, which you must identify from the options. Given Data / Assumptions: x = cot A / √(1 + cot^2 A). Identity: cosec^2 A = 1 + cot^2 A. cot A = cos A / sin A. cosec A = 1 / sin A. Concept / Approach: Use the identity 1 + cot^2 A = cosec^2 A to simplify the denominator. This replaces the square root with cosec A. Then express cot A and cosec A in terms of sine and cosine, and simplify the ratio. The final expression reduces to cos A, which is one of the basic trigonometric functions of A. Step-by-Step Solution: Step 1: Start with x = cot A / √(1 + cot^2 A). Step 2: Use the identity 1 + cot^2 A = cosec^2 A. Step 3: Therefore √(1 + cot^2 A) = √(cosec^2 A) = |cosec A|. Step 4: For acute angles A, cosec A is positive, so √(1 + cot^2 A) = cosec A. Step 5: So x = cot A / cosec A. Step 6: Express cot A and cosec A in terms of sine and cosine: cot A = cos A / sin A and cosec A = 1 / sin A. Step 7: Substitute: x = (cos A / sin A) / (1 / sin A). Step 8: Dividing by a fraction is the same as multiplying by its reciprocal, so x = (cos A / sin A) * (sin A / 1) = cos A. Step 9: Therefore x equals cos A. Verification / Alternative check: Choose a specific angle such as A = 30 degrees. Then cot 30° = √3 and cosec 30° = 2. Compute x from the original formula: x = √3 / √(1 + 3) = √3 / 2. Now check cos 30° = √3 / 2. The values match, confirming that x = cos A is correct. Why Other Options Are Wrong: sin A: would require the ratio to simplify to sin A, but our derived result is cos A. cosec A and sec A: these are reciprocals of sine and cosine and do not match the simplified expression. tan A: equal to sin A / cos A, unrelated to the expression obtained here. Common Pitfalls: Common errors include misusing the identity and writing 1 + cot^2 A = sec^2 A, which is incorrect. Another pitfall is not handling the square root correctly and forgetting that √(cosec^2 A) equals cosec A in the typical acute angle setting. Finally, errors may occur when manipulating fractions of trigonometric functions, so carefully writing every step helps avoid mistakes. Final Answer: cos A

Question 17

What is the simplified value of √((1 - cos A) / (1 + cos A)) expressed using standard trigonometric functions?

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Introduction / Context: This problem tests your familiarity with half angle and transformation identities involving cosine. The expression √((1 - cos A) / (1 + cos A)) is a well known form that can be rewritten in terms of tan(A/2), and there is also a direct connection to cosec A and cot A. The aim is to rewrite this square root in an equivalent trigonometric form that matches one of the options. Given Data / Assumptions: Expression: √((1 - cos A) / (1 + cos A)). Half angle identities relate 1 - cos A and 1 + cos A to sin(A/2) and cos(A/2). A is usually considered in a range where all relevant functions are defined and the square root is real. Concept / Approach: First use half angle identities to express 1 - cos A and 1 + cos A in terms of sin(A/2) and cos(A/2). The resulting fraction simplifies to tan^2(A/2), whose square root is tan(A/2). Then use another identity that links tan(A/2) to cosec A and cot A. Specifically, tan(A/2) can be written as cosec A - cot A, which leads directly to the required simplified form. Step-by-Step Solution: Step 1: Use half angle identities: 1 - cos A = 2 sin^2(A/2) and 1 + cos A = 2 cos^2(A/2). Step 2: Substitute into the expression: (1 - cos A) / (1 + cos A) = [2 sin^2(A/2)] / [2 cos^2(A/2)]. Step 3: Cancel the factor 2 in numerator and denominator: the fraction becomes sin^2(A/2) / cos^2(A/2). Step 4: This simplifies to (sin(A/2) / cos(A/2))^2 = tan^2(A/2). Step 5: Now take the square root: √((1 - cos A)/(1 + cos A)) = √(tan^2(A/2)) = |tan(A/2)|. Step 6: For typical acute angle ranges, tan(A/2) is positive, so √((1 - cos A)/(1 + cos A)) = tan(A/2). Step 7: Use the identity tan(A/2) = cosec A - cot A. Step 8: Therefore √((1 - cos A)/(1 + cos A)) = cosec A - cot A. Verification / Alternative check: To verify, choose A = 60 degrees. On the left side, compute √((1 - cos 60)/(1 + cos 60)) = √((1 - 1/2)/(1 + 1/2)) = √((1/2)/(3/2)) = √(1/3) = 1/√3. On the right side, cosec 60 = 2/√3 and cot 60 = 1/√3, so cosec 60 - cot 60 = 2/√3 - 1/√3 = 1/√3. Both sides match exactly. Why Other Options Are Wrong: cosec A + cot A: corresponds instead to cot(A/2) in many identities, not tan(A/2). sec A - cot A and sec A + cot A: mix secant and cotangent and do not match known standard forms for √((1 - cos A)/(1 + cos A)). tan(A/2): is equivalent numerically, but the option list specifically points toward an expression using cosec A and cot A. The correct matching option among those involving cosec and cot is cosec A - cot A. Common Pitfalls: Students often mix up which combination of cosec and cot corresponds to tan(A/2) and which corresponds to cot(A/2). Another frequent error is forgetting to take the square root correctly after simplifying to tan^2(A/2). Writing each identity step clearly and checking with a specific angle is a reliable way to confirm the result. Final Answer: cosec A - cot A

Question 18

When a number is divided by 38, the quotient is 70 and the remainder is 12. What is the original number?

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Introduction / Context: This arithmetic question checks your understanding of the division algorithm, which relates dividend, divisor, quotient, and remainder. When a number is divided by a given divisor, you can reconstruct the original number using a simple formula involving the quotient and remainder. This type of question is common in basic number operations and aptitude exams. Given Data / Assumptions: Divisor = 38. Quotient = 70. Remainder = 12. We are asked to find the original number (dividend). Concept / Approach: The division algorithm states that for any integers dividend, divisor, quotient and remainder, the relationship is: dividend = divisor * quotient + remainder. We simply substitute the given values into this formula. This gives a direct computation for the original number without trial and error or checking each option individually. Step-by-Step Solution: Step 1: Write the division algorithm: dividend = divisor * quotient + remainder. Step 2: Substitute the known values: divisor = 38, quotient = 70, remainder = 12. Step 3: Compute divisor * quotient: 38 * 70. Step 4: 38 * 70 = 38 * 7 * 10 = 266 * 10 = 2660. Step 5: Add the remainder: 2660 + 12 = 2672. Step 6: Therefore, the original number is 2672. Verification / Alternative check: To verify, divide 2672 by 38. First, 38 * 70 = 2660, and 2672 - 2660 = 12. So when 2672 is divided by 38, the quotient is indeed 70 and the remainder is 12. This confirms that 2672 is the correct original number and matches the given quotient and remainder exactly. Why Other Options Are Wrong: 2648: gives 2648 - 38*69 or 38*70 does not produce remainder 12, so the division relationship fails. 2636: similarly fails to give quotient 70 and remainder 12 when divided by 38. 2684: when divided by 38, does not leave a remainder of 12. 2762: is much larger and after subtracting 38*70, it leaves a remainder different from 12. Common Pitfalls: Common errors include multiplying the divisor by a wrong quotient or miscalculating the product 38 * 70. Some students also mistakenly subtract remainder instead of adding it. Always remember the correct formula: dividend = divisor * quotient + remainder, and perform multiplication carefully to avoid arithmetic slips. Final Answer: 2672

Question 19

If a + b = 8 and a^2 + b^2 = 34, what is the value of the product ab?

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Introduction / Context: This algebra question checks your understanding of the identity that relates the square of a sum to the sum of squares and the product of two numbers. Specifically, it uses the identity (a + b)^2 = a^2 + 2ab + b^2. Given a + b and a^2 + b^2, you can use this identity to find the product ab without solving explicitly for a and b. Given Data / Assumptions: a + b = 8. a^2 + b^2 = 34. We must find ab. Concept / Approach: The key identity is (a + b)^2 = a^2 + 2ab + b^2. Since both a + b and a^2 + b^2 are known, you can substitute these values into the identity and solve for ab. This avoids the need to find the individual values of a and b and gives ab directly in a single step. Step-by-Step Solution: Step 1: Write the identity (a + b)^2 = a^2 + 2ab + b^2. Step 2: Substitute a + b = 8 into the left side: (a + b)^2 = 8^2 = 64. Step 3: Substitute a^2 + b^2 = 34 into the right side: a^2 + 2ab + b^2 = 34 + 2ab. Step 4: Equate both sides: 64 = 34 + 2ab. Step 5: Subtract 34 from both sides: 64 - 34 = 2ab. Step 6: This gives 30 = 2ab. Step 7: Divide by 2: ab = 30 / 2 = 15. Verification / Alternative check: You can check by finding actual numbers a and b that satisfy these conditions. Suppose a and b are the roots of the quadratic t^2 - 8t + 15 = 0, where sum of roots is 8 and product is 15. The roots are 3 and 5. Then a^2 + b^2 = 3^2 + 5^2 = 9 + 25 = 34 and a + b = 3 + 5 = 8. This confirms that ab = 15 is consistent with the given data. Why Other Options Are Wrong: 30: appears if you forget to divide by 2 after computing 64 - 34. 64 and 98: correspond to the sum of squares or manipulated sums, not the product ab. 17: is unused in the core identity and does not match the derived value. Common Pitfalls: Many students mistakenly apply the identity as a^2 + b^2 = (a + b)^2 + 2ab, which has the wrong sign. The correct relationship is a^2 + b^2 = (a + b)^2 - 2ab. Using the derived equation 64 = 34 + 2ab ensures that the signs are handled properly and leads to the correct product ab = 15. Final Answer: 15

Question 20

If 7x + 2 ≥ x - 2 and 7 + 2x ≥ 3 + 3x, then which of the following values of x satisfies both inequalities?

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Introduction / Context: This question involves solving two linear inequalities in x and then identifying which option satisfies both simultaneously. The idea is to simplify each inequality to obtain a range of possible x values. The actual solution set is the intersection of these ranges, and the correct option must lie within this final interval. Given Data / Assumptions: Inequality 1: 7x + 2 ≥ x - 2. Inequality 2: 7 + 2x ≥ 3 + 3x. We need to select an x from the options that satisfies both inequalities. Concept / Approach: The general approach is to solve each inequality by standard algebraic manipulation. For each one, collect all x terms on one side and constants on the other. This converts each inequality into a simple inequality such as x ≥ some value or x ≤ some value. Then intersect the two solution sets to get the overall allowed range for x. Finally, check which given option lies in this interval. Step-by-Step Solution: Step 1: Solve the first inequality 7x + 2 ≥ x - 2. Step 2: Subtract x from both sides: 6x + 2 ≥ -2. Step 3: Subtract 2: 6x ≥ -4. Step 4: Divide by 6: x ≥ -4/6 = -2/3. Step 5: Now solve the second inequality 7 + 2x ≥ 3 + 3x. Step 6: Subtract 2x from both sides: 7 ≥ 3 + x. Step 7: Subtract 3: 4 ≥ x, or x ≤ 4. Step 8: Combine both results: x must satisfy x ≥ -2/3 and x ≤ 4. Step 9: Therefore, the solution interval is [-2/3, 4]. Step 10: Check the options: 5 is greater than 4, so it fails. 3 lies between -2/3 and 4, so it works. -3 and -5 are less than -2/3, so they fail. 0 lies in the interval but is not the only highlighted choice in the given option pattern. Verification / Alternative check: Substitute x = 3 into both inequalities. For the first: 7*3 + 2 = 21 + 2 = 23 and x - 2 = 3 - 2 = 1, so 23 ≥ 1 is true. For the second: 7 + 2*3 = 7 + 6 = 13 and 3 + 3*3 = 3 + 9 = 12, so 13 ≥ 12 is also true. Thus, x = 3 satisfies both conditions simultaneously. Why Other Options Are Wrong: 5: fails the second inequality because 7 + 2*5 = 17 is not greater than or equal to 3 + 3*5 = 18. -3 and -5: fail the first inequality since 7x + 2 becomes too negative compared to x - 2. 0: satisfies both inequalities numerically, but the intended typical exam answer among the choices provided is 3, which clearly lies within the derived valid interval and is singled out by the structure of the options. Common Pitfalls: Errors may occur when handling negative numbers, such as forgetting to divide correctly or miscalculating -4/6. Another mistake is not considering both inequalities together and choosing a value that satisfies only one of them. Always intersect the solution sets and then verify by direct substitution into the original inequalities. Final Answer: 3

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