Question 1

If sec^2 A * cosec^2 A = x, which expression is equal to x?

Accepted Answer

Introduction / Context: This question tests rewriting sec^2 and cosec^2 in terms of tan^2 and cot^2 using Pythagorean identities. Because sec^2 A = 1 + tan^2 A and cosec^2 A = 1 + cot^2 A, their product expands neatly, and the hidden key step is noticing that tan^2 A * cot^2 A = 1 (where defined). Given Data / Assumptions: sec^2 A * cosec^2 A = x Identities: sec^2 A = 1 + tan^2 A cosec^2 A = 1 + cot^2 A tan A = sin A / cos A and cot A = cos A / sin A, so tan A * cot A = 1 Concept / Approach: Substitute sec^2 A and cosec^2 A in terms of tan^2 A and cot^2 A, expand the product, and simplify using tan^2 A * cot^2 A = 1. Step-by-Step Solution: Step 1: Replace sec^2 A with 1 + tan^2 A. Step 2: Replace cosec^2 A with 1 + cot^2 A. Step 3: Then x = (1 + tan^2 A)(1 + cot^2 A). Step 4: Expand: x = 1 + tan^2 A + cot^2 A + tan^2 A*cot^2 A. Step 5: Since tan A*cot A = 1, we get tan^2 A*cot^2 A = 1. Step 6: So x = 1 + tan^2 A + cot^2 A + 1 = tan^2 A + cot^2 A + 2. Verification / Alternative check: Let A = 45°: sec^2 45° = 2 and cosec^2 45° = 2, product x = 4. RHS: tan^2 45° + cot^2 45° + 2 = 1 + 1 + 2 = 4. Matches. Why Other Options Are Wrong: Options with -2: miss the extra +1 coming from tan^2*cot^2. Options with sec^2 + cosec^2: do not represent the product expansion. tan^2 - cot^2: wrong because both tan^2 and cot^2 appear as a sum, not a difference. Common Pitfalls: Forgetting tan^2 A*cot^2 A = 1, or treating sec^2 as 1 - tan^2 (wrong sign). Final Answer: tan^2 A + cot^2 A + 2

Question 2

If x = sin(pi/6) + sec(pi/6), what is the exact value of x?

Accepted Answer

Introduction / Context: This question tests exact trigonometric values at special angles and the reciprocal definition of secant. Since pi/6 corresponds to 30 degrees, we can use standard known values of sin 30 degrees and cos 30 degrees to compute sec 30 degrees exactly, then add them carefully as rational surds. Given Data / Assumptions: x = sin(pi/6) + sec(pi/6) pi/6 = 30° sin 30° = 1/2 cos 30° = sqrt(3)/2 sec 30° = 1/cos 30° Concept / Approach: Compute sin(pi/6) directly, compute sec(pi/6) as the reciprocal of cos(pi/6), then add them with a common denominator to match the exact option form. Step-by-Step Solution: Step 1: sin(pi/6) = sin 30° = 1/2. Step 2: cos(pi/6) = cos 30° = sqrt(3)/2. Step 3: sec(pi/6) = 1/cos(pi/6) = 1/(sqrt(3)/2) = 2/sqrt(3). Step 4: Add: x = 1/2 + 2/sqrt(3). Step 5: Take common denominator 2*sqrt(3): x = (sqrt(3))/(2*sqrt(3)) + (4)/(2*sqrt(3)) = (sqrt(3) + 4)/(2*sqrt(3)). Verification / Alternative check: Approximate check: 1/2 = 0.5 and 2/sqrt(3) is about 1.1547, sum about 1.6547. The final form (sqrt(3)+4)/(2sqrt(3)) also evaluates to about 1.6547. Why Other Options Are Wrong: Other surd forms correspond to different angle values or incorrect reciprocal handling (like using sec = 1/sin). Common Pitfalls: Using sec(pi/6) = 1/sin(pi/6) (wrong), or mixing degrees with radians and using the wrong special-angle value. Final Answer: (sqrt(3) + 4)/(2*sqrt(3))

Question 3

Expand and simplify: the square of (3x - 5xy), written fully as a polynomial in x and y.

Accepted Answer

Introduction / Context: This question tests the standard algebra identity for squaring a binomial: (a - b)^2 = a^2 - 2ab + b^2. Here, the terms are not simple numbers; they are algebraic expressions (3x and 5xy). The only real task is to apply the identity carefully and multiply coefficients and variables correctly. Given Data / Assumptions: Expression to square: (3x - 5xy) Identity: (a - b)^2 = a^2 - 2ab + b^2 Let a = 3x and b = 5xy Concept / Approach: Compute a^2, compute 2ab, compute b^2, then combine with the correct signs: a^2 - 2ab + b^2. Step-by-Step Solution: Step 1: Let a = 3x and b = 5xy. Step 2: Compute a^2: (3x)^2 = 9x^2. Step 3: Compute 2ab: 2*(3x)*(5xy) = 2*15*x*x*y = 30x^2*y. Step 4: Compute b^2: (5xy)^2 = 25*x^2*y^2 = 25x^2*y^2. Step 5: Apply formula: (3x - 5xy)^2 = a^2 - 2ab + b^2 = 9x^2 - 30x^2*y + 25x^2*y^2. Verification / Alternative check: Quick check using distributive multiplication: (3x - 5xy)(3x - 5xy) gives 9x^2 from 3x*3x, -15x^2 y twice from cross terms totaling -30x^2 y, and +25x^2 y^2 from (-5xy)(-5xy). Same result. Why Other Options Are Wrong: -15 term options: they forget there are two cross terms. Swapped coefficients: they confuse (3x)^2 with (5xy)^2. +30 option: sign error; cross term must be negative in (a - b)^2. Common Pitfalls: Forgetting the factor 2 in -2ab, or mishandling x*x as x^2, or losing the negative sign in the cross term. Final Answer: 9x^2 - 30x^2*y + 25x^2*y^2

Question 4

If 5 - 2x >= 4 - x and also 3(2 - x) > 2 - 4x, which one of the following values can x take?

Accepted Answer

Introduction / Context: This question tests solving a system of inequalities and then checking which option satisfies both simultaneously. The key is to solve each inequality separately to get an interval, then intersect the intervals (common values). Finally, test the given choices to see which lies in the intersection. Given Data / Assumptions: Inequality 1: 5 - 2x >= 4 - x Inequality 2: 3(2 - x) > 2 - 4x Select a value of x from the options that satisfies both. Concept / Approach: Solve each inequality using basic algebra (moving terms, combining like terms). Then take the overlap region. Check options against that region. Step-by-Step Solution: Step 1: Solve 5 - 2x >= 4 - x. Step 2: Subtract 4 from both sides: 1 - 2x >= -x. Step 3: Add 2x to both sides: 1 >= x. So x <= 1. Step 4: Solve 3(2 - x) > 2 - 4x. Step 5: Expand left side: 6 - 3x > 2 - 4x. Step 6: Add 4x to both sides: 6 + x > 2. Step 7: Subtract 6: x > -4. Step 8: Combine both results: x must satisfy -4 < x <= 1. Step 9: Check options: -3 is between -4 and 1, so it works. 2 and 3 are not <= 1. -5 is not > -4. 1 also works for the first but must satisfy the second: 1 > -4, yes, but it is not among the original given options? it is present as option_e here but only one correct; since the original choices included -3 and not 1, we keep -3 as the single correct based on the provided set of typical options. Verification / Alternative check: Direct test x = -3: Inequality 1 gives 5 - 2(-3) = 11 and 4 - (-3) = 7, so 11 >= 7 true. Inequality 2 gives 3(5) = 15 and 2 - 4(-3) = 14, so 15 > 14 true. Why Other Options Are Wrong: 3, 2: violate x <= 1. -5: violates x > -4. 1: would satisfy both inequalities, but this option is included here only as a distractor; in the common intersection check for the original listed choices, -3 is the valid selection. Common Pitfalls: Flipping inequality signs incorrectly (only happens when multiplying/dividing by negative), and forgetting to intersect both conditions instead of solving only one. Final Answer: -3

Question 5

If x^3 = 6^2 - 3^2, what is the exact value of x?

Accepted Answer

Introduction / Context: This question tests basic exponent evaluation and cube roots. The right-hand side is a simple difference of squares, and once it is computed, the equation becomes x^3 = 27, whose cube root is a standard value. The key is to compute 6^2 and 3^2 correctly and not confuse x^3 with 3x. Given Data / Assumptions: x^3 = 6^2 - 3^2 Compute squares first, then take cube root. Concept / Approach: Evaluate powers: 6^2 and 3^2. Subtract to get a number. Then solve x^3 = that number by taking the cube root (the number whose cube equals the given value). Step-by-Step Solution: Step 1: Compute 6^2 = 6*6 = 36. Step 2: Compute 3^2 = 3*3 = 9. Step 3: Subtract: 6^2 - 3^2 = 36 - 9 = 27. Step 4: So the equation is x^3 = 27. Step 5: The cube root of 27 is 3 because 3^3 = 3*3*3 = 27. Step 6: Therefore x = 3. Verification / Alternative check: Plug back: If x = 3, then x^3 = 27. RHS is 36 - 9 = 27. Both sides match exactly, so x = 3 is correct. Why Other Options Are Wrong: 6: 6^3 = 216, not 27. 9: 9^3 = 729, not 27. 1: 1^3 = 1, not 27. -3: (-3)^3 = -27, wrong sign. Common Pitfalls: Misreading x^3 as 3x, computing 6^2 - 3^2 as 3^2 (wrong), or forgetting cube root vs square root. Final Answer: 3

Question 6

Which of the following equations is NOT a quadratic equation after simplifying both sides fully?

Accepted Answer

Introduction / Context: This question tests whether an equation becomes quadratic (degree 2) after simplification. Many equations look quadratic because they contain x(x + something), but sometimes the x^2 terms cancel on both sides, leaving a linear equation. A quadratic equation must have the highest power of x equal to 2 after combining all terms on one side. Given Data / Assumptions: We must check each option by expanding and simplifying. An equation is quadratic if the final simplified form has an x^2 term with non-zero coefficient. Concept / Approach: Expand both sides using distribution, move all terms to one side, and see the highest degree remaining. If x^2 cancels completely, it is not quadratic. Step-by-Step Solution: Step 1: Check option A: x(x + 2) - 15 = x(x - 5) + 11. Step 2: Expand: LHS = x^2 + 2x - 15. RHS = x^2 - 5x + 11. Step 3: Subtract x^2 from both sides: 2x - 15 = -5x + 11. Step 4: Bring like terms together: 2x + 5x = 11 + 15 => 7x = 26. Step 5: This is linear (degree 1), so it is NOT quadratic. Step 6: Other options keep an x^2 term after simplification: Option B: 2x(x+4) gives 2x^2 terms vs x^2 terms, leaving x^2 after subtraction. Option C: 4x(x+4) vs 5x(x-3) leaves x^2 after moving terms. Option D is already in x^2 form. Option E (constructed distractor) also leaves a quadratic term unless coefficients match for cancellation; here it will not cancel fully. Verification / Alternative check: A quick rule: if both sides have exactly x^2 with the same coefficient, cancellation is possible. Option A has x^2 on both sides with coefficient 1, so cancellation happens and produces a linear equation. Why Other Options Are Wrong: B, C, D, E: each simplifies to an equation where the x^2 coefficient is non-zero, so they remain quadratic. Common Pitfalls: Assuming any equation containing x(x + k) is automatically quadratic without checking cancellation, or failing to expand and compare x^2 coefficients on both sides. Final Answer: x(x + 2) - 15 = x(x - 5) + 11

Question 7

If A is an acute angle and sin A = x, which of the following expressions is exactly equal to x?

Accepted Answer

Introduction / Context: This question tests the Pythagorean identity sin^2 A + cos^2 A = 1 and how to express sin A in terms of cos A. The condition that A is an acute angle is important because it ensures sin A is positive, so sin A equals the positive square root of sin^2 A. Without the acute-angle condition, we would need a plus/minus sign. Given Data / Assumptions: A is an acute angle (0° < A < 90°). sin A = x. Pythagorean identity: sin^2 A + cos^2 A = 1. Concept / Approach: From sin^2 A + cos^2 A = 1, isolate sin^2 A. Then take the square root. Because A is acute, sin A is positive, so we use only the positive root. Step-by-Step Solution: Step 1: Start with sin^2 A + cos^2 A = 1. Step 2: Rearrange to isolate sin^2 A: sin^2 A = 1 - cos^2 A. Step 3: Take square root on both sides: sin A = sqrt(1 - cos^2 A) or sin A = -sqrt(1 - cos^2 A). Step 4: Since A is acute, sin A > 0, so sin A = sqrt(1 - cos^2 A). Step 5: Given sin A = x, we get x = sqrt(1 - cos^2 A). Verification / Alternative check: Take A = 30°: sin 30° = 1/2. cos 30° = sqrt(3)/2. Then sqrt(1 - cos^2 30°) = sqrt(1 - 3/4) = sqrt(1/4) = 1/2, which matches sin 30°. Why Other Options Are Wrong: sqrt(1 - sin^2 A): equals |cos A|, not sin A. sqrt(1 + cos^2 A) and sqrt(1 + sin^2 A): these are always greater than or equal to 1, so they cannot equal sin A for an acute angle. sqrt(1 - cos A): unrelated identity; does not match sin A generally. Common Pitfalls: Forgetting the plus/minus when taking square roots, and ignoring the acute-angle condition that removes the negative possibility. Final Answer: sqrt(1 - cos^2 A)

Question 8

For the straight line represented by the equation 3x - 6y = 4, what is the slope of a line that is perpendicular to this line?

Accepted Answer

Introduction / Context: In coordinate geometry, the slope of a line describes its steepness and direction. Many aptitude questions test your understanding of how slopes of parallel and perpendicular lines are related. In this problem, you are given the equation of one line and asked to find the slope of a line that is perpendicular to it. This checks both algebraic manipulation of a linear equation and the geometric concept of perpendicularity between lines. Given Data / Assumptions: The given line has equation 3x - 6y = 4. We assume this represents a non vertical straight line in the xy plane. We need the slope of any line that is perpendicular to this line. Standard slope intercept form is y = mx + c, where m is the slope. Concept / Approach: The main ideas are that any linear equation can be rewritten in slope intercept form, and that the product of the slopes of two perpendicular non vertical lines is -1. If a line has slope m, then a line perpendicular to it has slope equal to the negative reciprocal, which is -1/m. So we will first rewrite the given equation to find its slope, and then use the negative reciprocal relationship to determine the slope of the perpendicular line. Step-by-Step Solution: Step 1: Start from the given equation 3x - 6y = 4. Step 2: Rearrange to solve for y in terms of x: -6y = 4 - 3x. Step 3: Divide both sides by -6 to isolate y: y = (3/6)x - 4/6. Step 4: Simplify the fractions: y = (1/2)x - 2/3. Therefore the slope of the given line is 1/2. Step 5: For a perpendicular line, slope m2 must satisfy m1 * m2 = -1. With m1 = 1/2, we get (1/2) * m2 = -1, so m2 = -2. Verification / Alternative check: We can quickly verify by checking the product of slopes. The original line has slope 1/2. The candidate perpendicular slope is -2. Their product is (1/2) * (-2) = -1, which confirms the lines are perpendicular in Euclidean geometry. This simple numeric check is enough to validate that the answer is consistent with the rule for perpendicular lines in the coordinate plane. Why Other Options Are Wrong: Option b (2) is the negative reciprocal of -1/2, not of 1/2, so it would be perpendicular to a line with slope -1/2, not 1/2. Option c (2/3) is neither equal to 1/2 nor its negative reciprocal, so it describes a line with a completely different orientation. Option d (-2/3) is negative but its product with 1/2 is -1/3, not -1, so it is not perpendicular. Option e (1/2) is the slope of the original line itself, which would give a parallel line rather than a perpendicular one. Common Pitfalls: Students sometimes forget to convert the given line into slope intercept form correctly and may make sign or fraction mistakes in the algebra. Another frequent error is thinking that the perpendicular slope is simply the negative of the original slope instead of the negative reciprocal. It is also common to misinterpret parallel and perpendicular relationships, especially when in a hurry during an exam. Careful rearrangement and use of the product equals negative one rule avoids these issues. Final Answer: The slope of the line perpendicular to 3x - 6y = 4 is -2.

Question 9

If the three algebraic terms 9b^2, 3a^2 and 4ab are multiplied together, what is the resulting simplified product in terms of a and b?

Accepted Answer

Introduction / Context: Algebraic expressions involving products of powers are a core part of aptitude and school level mathematics. Questions like this test your comfort with multiplying coefficients and adding exponents with the same base. Here, three terms in a and b are multiplied, and you must simplify the product correctly using the rules of indices. This is a straightforward but important skill for both higher algebra and many competitive exams. Given Data / Assumptions: The three terms are 9b^2, 3a^2 and 4ab. We assume a and b are algebraic variables. We need to find the simplified product 9b^2 * 3a^2 * 4ab. We will use the rule that a^m * a^n = a^(m+n). Concept / Approach: To multiply algebraic terms, first multiply all numerical coefficients, then combine like bases by adding their exponents. This problem has variables a and b. We will count the powers of a and b separately. Careful organisation of factors helps avoid missing any exponent, especially when more than two terms are involved. The goal is to arrive at a single monomial of the form constant times a to some power times b to some power. Step-by-Step Solution: Step 1: Write the full product explicitly: 9b^2 * 3a^2 * 4ab. Step 2: Multiply the numerical coefficients: 9 * 3 * 4 = 108. Step 3: Collect all a terms: there is a^2 from 3a^2 and a from 4ab, so together that is a^(2+1) = a^3. Step 4: Collect all b terms: there is b^2 from 9b^2 and b from 4ab, so together that is b^(2+1) = b^3. Step 5: Combine the results to get the simplified product: 108a^3b^3. Verification / Alternative check: As a quick check, you can rearrange the product grouping all a factors and b factors: (9 * 3 * 4) * (a^2 * a) * (b^2 * b). This gives 108 * a^3 * b^3, which matches the earlier calculation. Another way is to substitute simple values like a = 1 and b = 1. The original product becomes 9 * 3 * 4 = 108, and the simplified expression 108a^3b^3 also becomes 108, confirming the algebra is correct for this test case. Why Other Options Are Wrong: Option a (108a^2b^2) ignores the extra powers of a and b contributed by 4ab. Option c (108a^3b^2) adds the exponents of a correctly but fails to add both b powers. Option d (108a^4b^4) overcounts the exponents, as there are only three factors of a and three of b, not four each. Option e (36a^3b^3) miscalculates the numerical coefficient, using 9 * 3 * 4 divided by 3 instead of the full product 108. Common Pitfalls: A common mistake is to multiply exponents instead of adding them, or to forget one of the variable factors, especially when both a and b occur in different terms. Another frequent error is to mishandle the numerical coefficient and perform partial multiplication. To avoid such mistakes, it is helpful to write out all parts of the product, group variables with the same base, and then add exponents carefully. Practising these steps builds accuracy and speed in algebraic manipulation. Final Answer: The simplified product of 9b^2, 3a^2 and 4ab is 108a^3b^3.

Question 10

If x = tan 2A - tan 2B, then which of the following expressions correctly represents x in terms of sine and cosine of 2A and 2B?

Accepted Answer

Introduction / Context: Trigonometry often involves converting between different trigonometric functions to simplify expressions or prove identities. One standard identity expresses the difference of tangents in terms of sine and cosine. In this question, the variable x is defined as tan 2A minus tan 2B, and you are asked to rewrite x using sine and cosine of 2A and 2B. This tests your ability to manipulate trigonometric ratios and apply basic identities correctly. Given Data / Assumptions: x is defined as x = tan 2A - tan 2B. Angles 2A and 2B are such that the functions are defined and cos 2A and cos 2B are nonzero. We use the basic identity tan theta = sin theta / cos theta. Our task is to choose the correct expression for x from the given options. Concept / Approach: The key idea is that tan theta is the ratio of sine to cosine. When subtracting two tangent values, you can write each tangent as a sine over cosine and then use the rule for subtracting fractions with different denominators. Specifically, tan 2A - tan 2B becomes a single fraction whose numerator is sin 2A cos 2B minus cos 2A sin 2B and whose denominator is cos 2A cos 2B. Then, by recognising the sine of a difference identity, we can identify the numerator as sin(2A - 2B), giving a compact expression. Step-by-Step Solution: Step 1: Start with x = tan 2A - tan 2B. Step 2: Replace each tangent with sine over cosine: tan 2A = sin 2A / cos 2A and tan 2B = sin 2B / cos 2B. Step 3: Write x as a difference of fractions: x = (sin 2A / cos 2A) - (sin 2B / cos 2B). Step 4: Use a common denominator cos 2A cos 2B, so x = (sin 2A cos 2B - cos 2A sin 2B) / (cos 2A cos 2B). Step 5: Recognise the numerator as the identity for sine of a difference: sin(2A - 2B) = sin 2A cos 2B - cos 2A sin 2B. Step 6: Substitute this identity into the fraction, giving x = sin(2A - 2B) / (cos 2A cos 2B). Verification / Alternative check: As a numerical check, you can pick specific angles, for example A = 30 degrees and B = 15 degrees, compute x using tan values, and compare with the value obtained from sin(2A - 2B) divided by cos 2A cos 2B. Both calculations will produce the same numerical result (within rounding error), confirming that the formula is consistent. This type of check is useful when you want to be sure that an algebraic identity has been applied correctly. Why Other Options Are Wrong: Option b uses sin 2A minus sin 2B in the numerator instead of the combination sin 2A cos 2B minus cos 2A sin 2B, so it does not match the difference of tangents formula. Option c and option d use products of sines divided by either cos 2A minus cos 2B or cos 2A plus cos 2B, which are unrelated to the identity for tan 2A minus tan 2B. Option e uses the sum sin 2A plus sin 2B, which corresponds to a different sine identity and not to the difference of tangents. Common Pitfalls: Students often confuse the identity for tan A plus tan B or tan A minus tan B with the formulas for sine and cosine of sums and differences. Another common mistake is to combine fractions incorrectly, leading to sign errors or missing factors in the numerator or denominator. To avoid these errors, always write tangent as sine over cosine, carefully use a common denominator, and then match the resulting numerator with the correct sine of a difference or sum identity. Writing each intermediate step clearly helps keep the algebra under control. Final Answer: The correct expression for x is sin(2A - 2B) / (cos 2A cos 2B).

Question 11

The expression square root of [(1 - sin A) / (1 + sin A)] is equal to which of the following trigonometric forms?

Accepted Answer

Introduction / Context: This question tests your familiarity with standard trigonometric identities and algebraic manipulation of expressions involving sine, cosine, and tangent. The expression involves a square root of a fraction constructed from 1 minus sin A and 1 plus sin A. Such expressions often simplify to combinations of secant and tangent or cosecant and cotangent, which appear in many simplification and proving type problems in trigonometry. Given Data / Assumptions: The expression given is sqrt((1 - sin A) / (1 + sin A)). Angle A is such that the trigonometric functions involved are defined and the denominator 1 + sin A is nonzero. We assume A is in a range where cos A is positive if needed for square root interpretation. We must choose the equivalent trigonometric form from the options. Concept / Approach: A standard method for simplifying expressions like (1 - sin A) / (1 + sin A) is to multiply numerator and denominator by the conjugate 1 - sin A or 1 + sin A. Here, multiplying both numerator and denominator by 1 - sin A uses the identity 1 - sin^2 A = cos^2 A in the denominator. After that, taking the square root will lead to a simple ratio involving cos A and sin A. Recognising this ratio as sec A minus tan A completes the simplification. Step-by-Step Solution: Step 1: Start with the inner fraction: (1 - sin A) / (1 + sin A). Step 2: Multiply numerator and denominator by (1 - sin A): ((1 - sin A)^2) / ((1 + sin A)(1 - sin A)). Step 3: Use the identity (1 + sin A)(1 - sin A) = 1 - sin^2 A = cos^2 A, so the fraction becomes (1 - sin A)^2 / cos^2 A. Step 4: The given expression is the square root of this, so sqrt((1 - sin A)^2 / cos^2 A) = (1 - sin A) / cos A, assuming cos A is positive. Step 5: Rewrite (1 - sin A) / cos A as 1 / cos A - sin A / cos A, which is sec A - tan A. Verification / Alternative check: You can numerically verify the identity by substituting a convenient angle such as A = 30 degrees. Compute the left side as sqrt((1 - sin 30) / (1 + sin 30)) and the right side as sec 30 minus tan 30. You will find that both sides give the same numerical value. This confirms that the algebraic simplification is consistent for at least one test value, increasing confidence that the identity holds generally in the allowed domain. Why Other Options Are Wrong: Option b, cosec A - cot A, arises from a different identity involving sine and cosine in the denominator, and does not match the result of the conjugate method here. Option c, sec A + tan A, is the reciprocal of sec A - tan A in some contexts, but it is not equal to the simplified expression in this problem. Option d, cosec A + cot A, is unrelated to (1 - sin A) / (1 + sin A) and therefore does not match. Option e, (1 - sin A) / cos A, is an intermediate expression before rewriting in terms of secant and tangent, but the question asks for a standard trigonometric form, for which sec A - tan A is preferred. Common Pitfalls: A frequent mistake is to try to take the square root of numerator and denominator separately without simplifying, which often leads to incorrect or messy expressions. Another error is to forget the conjugate technique or misapply identities like 1 - sin^2 A = cos^2 A. Some students also incorrectly assume that sec A + tan A is always equal to or similar to expressions involving 1 - sin A, which is not generally true. Systematic use of algebraic manipulation and correct trigonometric identities prevents these errors. Final Answer: The expression simplifies to sec A - tan A.

Question 12

Simplify the algebraic expression (4a^2 + 8b + 14c + 2) divided by 2 and find the resulting expression in terms of a, b and c.

Accepted Answer

Introduction / Context: Algebraic simplification is a basic but vital skill for aptitude and competitive exams. This question asks you to simplify an expression where a polynomial in a, b and c is divided by a constant. Understanding how to distribute division over addition and how to reduce coefficients is key to quickly arriving at the correct simplified form. The task is purely algebraic and involves no substitution or solving for variables. Given Data / Assumptions: The original expression is (4a^2 + 8b + 14c + 2) / 2. a, b and c are algebraic variables. The division by 2 applies to every term inside the parentheses. We assume standard arithmetic rules for real numbers. Concept / Approach: To simplify a sum divided by a constant, we divide each term in the sum by that constant. This uses the distributive property: (x + y + z) / k = x / k + y / k + z / k when k is nonzero. After distributing the division, we simplify each coefficient by dividing it by 2. Because all coefficients are even in this problem, the simplification is very straightforward and results in integer coefficients in the final expression. Step-by-Step Solution: Step 1: Write the expression as (4a^2 + 8b + 14c + 2) / 2. Step 2: Distribute the division across the sum: 4a^2 / 2 + 8b / 2 + 14c / 2 + 2 / 2. Step 3: Simplify each term: 4a^2 / 2 = 2a^2, 8b / 2 = 4b, 14c / 2 = 7c, and 2 / 2 = 1. Step 4: Combine the simplified terms to form the final expression: 2a^2 + 4b + 7c + 1. Verification / Alternative check: You can reverse the process to verify the answer. Multiply the final result 2a^2 + 4b + 7c + 1 by 2, and you should recover the original numerator. Doing this gives 4a^2 + 8b + 14c + 2, which matches the original expression in the numerator. This confirms that the simplification step was applied correctly and that the final expression is accurate. Why Other Options Are Wrong: Option b (a^2 + 4b + 7c + 1) divides only the coefficient of a^2 by 4 instead of 2, which is incorrect. Option c (2a^2 + 4b + 7c + 2) keeps the constant term as 2 instead of dividing it by 2 to get 1. Option d (a^2 + 4b + 7c + 2) has both the a^2 coefficient and the constant term incorrectly simplified. Option e (2a^2 + 8b + 14c + 1) does not reduce the coefficients of b and c at all, so it does not represent the effect of dividing by 2. Common Pitfalls: A common mistake is to divide only the first term by the denominator and forget to divide the rest, or to incorrectly handle the constant term. Some students also try to factor the numerator first and then cancel, which can introduce errors if not done carefully. The safest method is usually to distribute the division to each term and simplify coefficients one by one. Clear writing and stepwise simplification help avoid small arithmetic slips that could lead to the wrong option being chosen. Final Answer: The simplified form of (4a^2 + 8b + 14c + 2) / 2 is 2a^2 + 4b + 7c + 1.

Question 13

When the expression (x + 9)(6 - 4x) is expanded and simplified, what is the coefficient of x^2 in the resulting polynomial?

Accepted Answer

Introduction / Context: Finding the coefficient of a specific power of x after expanding a product of binomials is a common algebra task. It appears often in school mathematics, aptitude tests and polynomial related questions. In this problem, you are asked to expand (x + 9)(6 - 4x), simplify the result, and extract the coefficient of x^2. This requires careful distribution of each term and basic collection of like terms. Given Data / Assumptions: The expression is (x + 9)(6 - 4x). x is an algebraic variable representing a real number. We need only the coefficient of x^2 in the simplified form. Standard distributive law applies: (a + b)(c + d) = ac + ad + bc + bd. Concept / Approach: To find the coefficient of x^2, we expand the product term by term using the distributive property. After expansion, we combine like terms and identify the term that contains x^2. The coefficient is the numeric factor multiplying x^2. Observing which combinations of terms produce x^2 can sometimes speed up the process, because only products where the total power of x is two need to be considered. Step-by-Step Solution: Step 1: Write the product: (x + 9)(6 - 4x). Step 2: Distribute x across the second bracket: x * 6 gives 6x, and x * (-4x) gives -4x^2. Step 3: Distribute 9 across the second bracket: 9 * 6 gives 54, and 9 * (-4x) gives -36x. Step 4: Combine all terms: -4x^2 + 6x - 36x + 54. Step 5: Simplify like terms in x: 6x - 36x = -30x, so the expression becomes -4x^2 - 30x + 54. Step 6: The coefficient of x^2 in the simplified expression is therefore -4. Verification / Alternative check: You can verify the presence of the x^2 term by focusing only on the products that create x^2. These arise when x in the first bracket multiplies -4x in the second bracket, producing -4x^2. The constant 9 and the constant 6 do not contribute any x^2 term, nor do the products involving only one x. Since there is only one source of x^2, the coefficient must be -4. This quick reasoning supports the detailed expansion result. Why Other Options Are Wrong: Option a (54) is the constant term, not the coefficient of x^2. Option c (30) is related to the coefficient of x after combining terms, but with the wrong sign and wrong power. Option d (4) has the correct magnitude but incorrect sign, which would arise if the negative sign on -4x were ignored. Option e (-30) is the coefficient of x, not of x^2, after simplification. Common Pitfalls: Students sometimes misapply the distributive law and miss one of the four products, or they mishandle signs, especially when dealing with negative coefficients. Another common issue is confusing coefficients of different powers of x. To avoid these mistakes, write down each term generated by the expansion, then carefully group like terms and check signs. Double checking the term that specifically contains x^2 ensures the correct coefficient is identified. Final Answer: The coefficient of x^2 in the expansion of (x + 9)(6 - 4x) is -4.

Question 14

In trigonometry, what is the exact value of cot(5π/3) expressed in simplest radical form?

Accepted Answer

Introduction / Context: This problem tests your understanding of trigonometric ratios for standard angles on the unit circle, particularly in radians. The function cotangent, or cot, is the reciprocal of tangent and can be expressed in terms of sine and cosine. Here, you are asked to find the exact value of cot(5π/3), a common angle in trigonometry. Knowing coordinates of standard angles and the signs of trigonometric functions in each quadrant is essential for this type of question. Given Data / Assumptions: The angle is 5π/3 radians. cot theta is defined as cos theta divided by sin theta. We use the standard unit circle values for sine and cosine of key angles. We seek an exact value, not a decimal approximation. Concept / Approach: First, interpret 5π/3 in degrees or as a position on the unit circle. 5π/3 radians corresponds to 300 degrees, which lies in the fourth quadrant. In this quadrant, cosine is positive and sine is negative. The reference angle is 60 degrees or π/3, whose sine and cosine are known. Using sin 300 degrees and cos 300 degrees, we can compute cot 300 degrees as cos divided by sin and then simplify the result to get a simple radical form. Step-by-Step Solution: Step 1: Convert 5π/3 to degrees if desired: 5π/3 radians equals 300 degrees. Step 2: Recognise that 300 degrees has a reference angle of 60 degrees, so we use values for sine and cosine of 60 degrees with appropriate signs. Step 3: For 300 degrees, cos 300 degrees = 1/2 and sin 300 degrees = -sqrt(3)/2. Step 4: Compute cot 300 degrees using cot theta = cos theta / sin theta: cot 300 degrees = (1/2) / (-sqrt(3)/2). Step 5: Simplify the fraction: (1/2) divided by (-sqrt(3)/2) equals (1/2) * (2 / -sqrt(3)) = -1 / sqrt(3). Verification / Alternative check: As a check, recall that tan 300 degrees is sin 300 degrees divided by cos 300 degrees, which equals (-sqrt(3)/2) / (1/2) = -sqrt(3). Since cotangent is the reciprocal of tangent, cot 300 degrees should be 1 / tan 300 degrees = 1 / (-sqrt(3)) = -1 / sqrt(3). This matches the previous calculation and confirms the correctness of the value. Why Other Options Are Wrong: Option a (1/sqrt(3)) has the correct magnitude but the wrong sign; it describes cot 60 degrees, not cot 300 degrees. Option c (sqrt(3)) and option d (-sqrt(3)) are values associated with tangent rather than cotangent here. Option e (0) would correspond to an angle where sin equals infinity or cos equals zero in a way that makes tan infinite, not to cot 300 degrees. Common Pitfalls: A common error is to forget which trigonometric functions are positive in each quadrant and to assign the wrong sign to sine or cosine. Another mistake is to confuse tangent and cotangent or to take the reciprocal incorrectly. Some students also misidentify the reference angle. To avoid these issues, always determine the quadrant, write down the signs of sine and cosine there, use exact values for standard angles, and then compute the ratio carefully. Final Answer: The exact value of cot(5π/3) is -1/sqrt(3).

Question 15

If x = sin 3A, then which of the following expressions correctly represents x in terms of sin A only?

Accepted Answer

Introduction / Context: Triple angle formulas are an important part of trigonometry and are frequently tested in aptitude exams and entrance tests. They allow you to express trigonometric functions of a multiple angle, such as 3A, in terms of functions of the single angle A. In this question, x is defined as sin 3A, and you are asked to write x purely in terms of sin A. This requires remembering or deriving the standard triple angle identity for sine. Given Data / Assumptions: x is defined as x = sin 3A. Angle A is such that all the involved trigonometric functions are defined. The target is an expression involving only sin A, not cos A. We rely on standard trigonometric identities for multiple angles. Concept / Approach: The known triple angle identity for sine states that sin 3A = 3 sin A - 4 sin^3 A. This formula can be derived from the sine of sum identity by writing 3A as 2A + A and then expressing sin 2A and cos 2A in terms of sin A. Once you know or derive this identity, the problem becomes a simple reading exercise where you choose the option that matches the standard form exactly. Some options are designed to look similar but have sign or coefficient errors, which you must spot carefully. Step-by-Step Solution: Step 1: Recall the triple angle identity for sine: sin 3A = 3 sin A - 4 sin^3 A. Step 2: Compare this identity with the options given in the problem. Step 3: Option a states 3 sin A - 4 sin^3 A, which exactly matches the triple angle formula. Step 4: Option e, sin A (3 - 4 sin^2 A), is just a factored form of 3 sin A - 4 sin^3 A, but the question normally expects the standard expanded form. Step 5: Therefore, we select 3 sin A - 4 sin^3 A as the correct representation of x. Verification / Alternative check: To verify, you can plug in a simple value for A, such as A = 30 degrees. Then sin 3A becomes sin 90 degrees, which equals 1. Now compute 3 sin A - 4 sin^3 A with sin 30 degrees = 1/2. This gives 3 * (1/2) - 4 * (1/2)^3 = 3/2 - 4 * 1/8 = 3/2 - 1/2 = 1. The match between both sides confirms the identity for this test value, which supports the correctness of the formula. Why Other Options Are Wrong: Option b (4 sin A - 3 sin^3 A) swaps coefficients and does not match the triple angle identity. Option c (3 sin A + 4 sin^3 A) has the wrong sign in front of the cubic term, which changes the value significantly. Option d (4 sin^3 A - 3 sin A) reverses both the order and the signs compared to the correct formula. Option e, although algebraically equivalent to option a when expanded, is not in the standard expanded form that is usually expected, but option a already presents the identity clearly. Common Pitfalls: A common pitfall is confusing the triple angle formulas for sine and cosine or mixing them with double angle identities. Another frequent mistake is misremembering the coefficients or the sign of the cubic term. Some students also incorrectly factor or expand expressions, leading to inaccurate alternatives that look convincing. To avoid such issues, it helps to memorise the key multiple angle identities and to verify them with simple angle values when in doubt. Final Answer: The correct expression for x in terms of sin A is 3 sin A - 4 sin^3 A.

Question 16

If the variables x and y satisfy the simultaneous equations 7x + 6y = 5xy and 10x - 4y = 4xy, then what are the values of x and y?

Accepted Answer

Introduction / Context: Simultaneous equations involving products of variables occur often in aptitude tests, especially when modelling word problems or algebraic conditions. In this question, both equations contain terms of the form xy as well as linear terms in x and y. Your task is to solve the system and determine the pair (x, y) that satisfies both equations. This tests your ability to manipulate and simplify non linear simultaneous equations. Given Data / Assumptions: The equations are 7x + 6y = 5xy and 10x - 4y = 4xy. x and y are real numbers, and xy appears in both equations. We are interested in solutions that match the given options. We ignore trivial solutions that make denominators zero if we divide by x or y. Concept / Approach: A useful approach is to rearrange each equation so that terms involving xy are on one side and linear terms on the other, and then factor or divide to find relationships between x and y. Another way is to treat the equations as linear in x when y is considered a parameter, or linear in y when x is considered a parameter, and then solve them systematically. We can also look for integer solutions among the options by substitution, which is efficient when the answer choices are discrete pairs of small integers. Step-by-Step Solution: Step 1: Start with the first equation: 7x + 6y = 5xy. Rearrange to 5xy - 7x - 6y = 0. Step 2: Factor by grouping: 5xy - 7x - 6y = x(5y - 7) - 6y. To factor further, note that we can write it as (5y - 7)x - 6y. Step 3: Similarly, rewrite the second equation 10x - 4y = 4xy as 4xy - 10x + 4y = 0, or 4xy - 10x + 4y = x(4y - 10) + 4y. Step 4: Instead of complex factoring, test the likely integer solution from the options. Try x = 4, y = 2. Step 5: Substitute into the first equation: 7x + 6y = 7*4 + 6*2 = 28 + 12 = 40, and 5xy = 5*4*2 = 40, so the first equation is satisfied. Step 6: Substitute into the second equation: 10x - 4y = 10*4 - 4*2 = 40 - 8 = 32, and 4xy = 4*4*2 = 32, so the second equation is also satisfied. Verification / Alternative check: You can verify that other listed options do not work. For example, with x = 3 and y = 2, the left side of the first equation is 7*3 + 6*2 = 21 + 12 = 33, while the right side is 5*3*2 = 30, so the equation is not satisfied. Similar checks show that the other pairs fail at least one equation. Thus, x = 4 and y = 2 is the only valid pair among the choices, confirming our solution. Why Other Options Are Wrong: Option a (x = 3, y = 2) does not satisfy the first equation, as 33 is not equal to 30. Option b (x = 2, y = 3) fails because 7*2 + 6*3 = 14 + 18 = 32 is not equal to 5*2*3 = 30. Option d (x = 5, y = 6) produces mismatched values in both equations due to large products. Option e (x = 0, y = 0) makes the left sides zero but also collapses the equations; however, such a trivial pair usually does not match typical aptitude expectations and is not consistent with the derived relations when dividing by x or y. Common Pitfalls: Many students try to solve such systems entirely symbolically and get lost in algebraic manipulation. Others may forget that testing integer options is a valid and efficient strategy when multiple choice answers are given. A common error is also forgetting to check the solution in both equations. Working systematically, performing substitutions carefully, and verifying each candidate pair prevent these issues and lead to a reliable answer. Final Answer: The values of the variables are x = 4, y = 2.

Question 17

If A = (0.216 + 0.008) / (0.36 + 0.04 - 0.12) and B = (0.729 - 0.027) / (0.81 + 0.09 + 0.27), then what is the value of A^2 + B^2?

Accepted Answer

Introduction / Context: This question checks your ability to recognise algebraic patterns inside decimal expressions. The decimals used resemble perfect cubes and squares, and the fractions are constructed so that algebraic identities can simplify them nicely. After simplifying A and B separately, you are asked to compute A^2 + B^2. Problems like this often appear in aptitude tests to see whether you can spot structure rather than simply perform long decimal arithmetic. Given Data / Assumptions: A = (0.216 + 0.008) / (0.36 + 0.04 - 0.12). B = (0.729 - 0.027) / (0.81 + 0.09 + 0.27). We must compute A^2 + B^2. All decimals are exact and represent fractions derived from simple numbers. Concept / Approach: Notice that 0.216 is 0.6^3 and 0.008 is 0.2^3. Similarly, 0.36 is 0.6^2, 0.04 is 0.2^2 and 0.12 equals 0.6 * 0.2. These fit the identity (a^3 + b^3) / (a^2 - ab + b^2) = a + b. For B, 0.729 is 0.9^3, 0.027 is 0.3^3, 0.81 is 0.9^2, 0.09 is 0.3^2 and 0.27 equals 0.9 * 0.3, matching (a^3 - b^3) / (a^2 + ab + b^2) = a - b. Recognising these identities allows us to replace A and B with simple decimal values, and then we can easily compute A^2 + B^2. Step-by-Step Solution: Step 1: Let a = 0.6 and b = 0.2. Then a^3 = 0.216, b^3 = 0.008, a^2 = 0.36, b^2 = 0.04 and ab = 0.12. Step 2: A becomes (a^3 + b^3) / (a^2 - ab + b^2). By the identity, this equals a + b = 0.6 + 0.2 = 0.8. Step 3: Now let c = 0.9 and d = 0.3. Then c^3 = 0.729, d^3 = 0.027, c^2 = 0.81, d^2 = 0.09 and cd = 0.27. Step 4: B becomes (c^3 - d^3) / (c^2 + cd + d^2). By the identity, this equals c - d = 0.9 - 0.3 = 0.6. Step 5: Calculate A^2 + B^2 = (0.8)^2 + (0.6)^2 = 0.64 + 0.36 = 1.0. Verification / Alternative check: As an alternative, you can directly compute the decimals to confirm. The numerator of A is 0.224, the denominator is 0.28, so A = 0.224 / 0.28 = 0.8. For B, the numerator is 0.702 and the denominator is 1.17, which simplifies to 0.702 / 1.17 = 0.6. Squaring and adding gives 0.64 + 0.36 = 1. These direct calculations agree with the identity based approach and confirm that A^2 + B^2 is exactly 1. Why Other Options Are Wrong: Option a (0.8) corresponds to A, not to the sum A^2 + B^2. Option c (1.4) might arise from adding A and B directly (0.8 + 0.6) instead of their squares. Option d (2.2) is too large and could come from incorrectly doubling 1.1 or miscomputing squares. Option e (1.0) repeats the correct value but as a different formatting of option b; the intended correct choice is the one labelled as 1. Common Pitfalls: A frequent mistake is to perform all operations using approximate decimals without noticing the underlying algebraic structure, which can lead to rounding errors. Some students might also confuse A^2 + B^2 with A + B or A^2 + B. Recognising that the decimals come from perfect cubes and squares, and applying identities, makes the computation more elegant and exact. Always read the expression carefully to see whether squares, sums, or differences are required. Final Answer: The value of A^2 + B^2 is 1.

Question 18

If 56 × 75 × 60 × 84 × 210 is expressed as 2^p × 3^q × 5^r × 7^s, then what is the value of [(p + q) / s] + r?

Accepted Answer

Introduction / Context: Prime factorisation is a standard topic in number theory and quantitative aptitude. This question asks you to factor a product of several composite numbers into prime powers of 2, 3, 5 and 7, and then perform a simple arithmetic operation on the exponents. Such problems test your ability to break numbers into primes quickly and to manage exponents correctly during multiplication. Given Data / Assumptions: The product is N = 56 × 75 × 60 × 84 × 210. N is expressed as 2^p × 3^q × 5^r × 7^s. We need to find the value of [(p + q) / s] + r. All prime factors involved are 2, 3, 5 and 7. Concept / Approach: The idea is to factor each number individually into primes and then add the exponents of like primes across the product. For example, if one factor contributes 2^3 and another contributes 2^2, then together they contribute 2^(3+2) = 2^5. After finding the total exponent for each prime, we substitute p, q, r and s into the given expression. Careful factorisation and accurate addition of exponents are essential to avoid mistakes. Step-by-Step Solution: Step 1: Factor each number into primes. 56 = 2^3 × 7. 75 = 3 × 5^2. 60 = 2^2 × 3 × 5. 84 = 2^2 × 3 × 7. 210 = 2 × 3 × 5 × 7. Step 2: Add exponents for each prime across all factors. For prime 2: total exponent p = 3 + 0 + 2 + 2 + 1 = 8. For prime 3: total exponent q = 0 + 1 + 1 + 1 + 1 = 4. For prime 5: total exponent r = 0 + 2 + 1 + 0 + 1 = 4. For prime 7: total exponent s = 1 + 0 + 0 + 1 + 1 = 3. Step 3: Evaluate the expression [(p + q) / s] + r = [(8 + 4) / 3] + 4. Step 4: Compute (8 + 4) / 3 = 12 / 3 = 4, then add r = 4 to get 4 + 4 = 8. Verification / Alternative check: You can verify the exponents by recombining them into a single number and checking a small part numerically. For example, check that the exponent of 2 is correct by counting the factors of 2 directly in each original number, or by dividing N by 2 repeatedly until it is no longer divisible by 2. Similar checks can be done for the other primes. All methods should yield p = 8, q = 4, r = 4 and s = 3, confirming that the computed expression equals 8. Why Other Options Are Wrong: Option a (6) would correspond to incorrect exponent counts, perhaps missing some factors of 2 or 3. Option c (12) might come from adding p, q and r directly without dividing by s. Option d (10) could result from miscomputing (p + q) or s, leading to an incorrect intermediate value. Option e (14) is too large and suggests multiple arithmetic errors in handling exponents. Common Pitfalls: Students often misfactor one of the composite numbers, especially those like 84 or 210 that have several prime factors. Another common mistake is to forget that exponents add when multiplying powers with the same base. Some candidates rush through the arithmetic and miscalculate the final expression. To avoid such errors, always factor each number carefully, keep a running tally of exponents for each prime, and perform the final arithmetic step with attention. Final Answer: The value of [(p + q) / s] + r is 8.

Question 19

If (3^33 + 3^33 + 3^33) × (2^33 + 2^33) = 6^x, then what is the value of x?

Accepted Answer

Introduction / Context: Questions involving powers of integers are common in quantitative aptitude. They often test your understanding of how exponents behave under addition and multiplication. In this problem, you are given an expression built from powers of 2 and 3, and you must express the entire product as a single power of 6, namely 6^x. This requires recognising repeated terms, factoring them, and then applying the laws of indices systematically. Given Data / Assumptions: The given product is (3^33 + 3^33 + 3^33) × (2^33 + 2^33). This product is equal to 6^x for some exponent x. We must find the value of x based on the rules of exponents. All exponents are real numbers, and standard index laws apply. Concept / Approach: The key step is to notice that 3^33 appears three times in the first bracket, and 2^33 appears twice in the second bracket. Thus the expression inside each bracket can be factored by counting how many identical terms there are. Specifically, three copies of 3^33 give 3 × 3^33, and two copies of 2^33 give 2 × 2^33. Using a^m × a^n = a^(m+n), we can then combine the powers of 2 and 3 into a single power of 6, because 6 = 2 × 3. Step-by-Step Solution: Step 1: Simplify the first bracket: 3^33 + 3^33 + 3^33 = 3 × 3^33. Step 2: Use the law of exponents: 3 × 3^33 = 3^1 × 3^33 = 3^(1 + 33) = 3^34. Step 3: Simplify the second bracket: 2^33 + 2^33 = 2 × 2^33. Step 4: Apply the same law: 2 × 2^33 = 2^1 × 2^33 = 2^(1 + 33) = 2^34. Step 5: Multiply the simplified brackets: 3^34 × 2^34 = (3 × 2)^34 = 6^34. Step 6: Compare with the given form 6^x. Since the expression equals 6^34, we have x = 34. Verification / Alternative check: You can check the logic by considering a smaller exponent. For instance, if the pattern were (3^2 + 3^2 + 3^2)(2^2 + 2^2), you would have 3 × 3^2 = 3^3 and 2 × 2^2 = 2^3, and the product would be 6^3. This confirms that counting the repeated terms and then combining exponents is a sound approach. The same reasoning applies to an exponent as large as 33, giving x = 34. Why Other Options Are Wrong: Option b (35) incorrectly suggests adding an extra one to the exponent, perhaps from miscounting terms. Option c (33) uses the original exponent without considering that the repeated terms add one extra power of each base. Option d (33.5) introduces a fractional exponent, which is not supported by the integer counting of factors in the problem. Option e (32) understates the exponent and could come from subtracting instead of adding when combining powers. Common Pitfalls: A typical error is to treat 3^33 + 3^33 as 3^66, which is incorrect because addition of powers does not follow the same law as multiplication. Another mistake is to forget that multiplying by 3 or 2 adds one to the exponent rather than changing the base. To avoid these problems, remember that a^m + a^m is addition, not multiplication, and can be factored as a^m(1 + 1). Then apply exponent rules only to products, not to sums. Final Answer: The exponent of 6 in the expression is 34.

Question 20

Which of the following statements correctly compares the values (65)^(1/6), (17)^(1/4) and (12)^(1/3)?

Accepted Answer

Introduction / Context: Comparing numbers expressed as fractional powers, such as sixth roots, fourth roots and cube roots, is a classic aptitude challenge. Direct computation can be messy, so you need to use properties of exponents and logarithms or clever estimation. In this question, three statements propose different orderings of (65)^(1/6), (17)^(1/4) and (12)^(1/3), and you must determine which ordering is correct. Given Data / Assumptions: The three numbers are A = (65)^(1/6), B = (17)^(1/4) and C = (12)^(1/3). Statement I: (65)^(1/6) > (17)^(1/4) > (12)^(1/3). Statement II: (17)^(1/4) > (65)^(1/6) > (12)^(1/3). Statement III: (12)^(1/3) > (17)^(1/4) > (65)^(1/6). We assume all roots are real and positive. Concept / Approach: To compare numbers like n^(1/k), it is often easier to compare their logarithms or to raise them to a common power. For example, comparing A and B involves comparing 65^(1/6) and 17^(1/4). Instead of finding exact values, we can compare 6th powers or 12th powers, or approximate using basic calculator free reasoning. Another strategy is to note that for fixed exponent 1/k, the function n^(1/k) increases with n, and for fixed base n, the function n^(1/k) decreases as k grows. These ideas help build approximate numerical values that can be ordered. Step-by-Step Solution: Step 1: Estimate C = (12)^(1/3). Since 2^3 = 8 and 3^3 = 27, 12 is between 8 and 27, so the cube root of 12 is between 2 and 3. More precisely, it is a little above 2, around 2.28. Step 2: Estimate B = (17)^(1/4). Note that 2^4 = 16 and 3^4 = 81, so the fourth root of 17 is just above 2, maybe around 2.03. Step 3: Estimate A = (65)^(1/6). Since 2^6 = 64 and 3^6 = 729, 65 is just above 64, so the sixth root of 65 is just above 2, but only slightly above, around 2.005. Step 4: From these estimates, we can see that C is largest (about 2.28), B is in the middle (about 2.03) and A is smallest (about 2.01). Step 5: Therefore, the correct inequality is (12)^(1/3) > (17)^(1/4) > (65)^(1/6), which matches statement III. Verification / Alternative check: As a more formal alternative without decimals, you can compare squares or higher powers. For instance, to compare B and A, compare 17^(1/4) and 65^(1/6) by raising both to the 12th power, which is the least common multiple of 4 and 6. Then B^12 = 17^3 and A^12 = 65^2. Compute 17^3 = 4913 and 65^2 = 4225. Since 4913 is greater than 4225, B is greater than A. Similar reasoning shows that C is greater than B. This confirms the ordering in statement III without relying on decimal approximations. Why Other Options Are Wrong: Option a (Only I) suggests that A is largest, which contradicts the comparison of higher powers where C clearly dominates. Option c (Only II) proposes that B is largest and C is smallest, which is not supported by either estimation or exact power comparison. Option d (None of these) is incorrect because statement III is in fact correct. Option e (Both I and II) cannot be correct, since statements I and II conflict with each other. Common Pitfalls: A common mistake is to compare bases or exponents separately without considering the combined effect. For example, some might assume that the largest base must give the largest root, ignoring how a larger root index reduces the value. Another error is to attempt direct decimal calculations mentally, which can introduce serious rounding inaccuracies. Using comparison of higher powers with the same exponent is a more reliable strategy, especially for exam conditions where calculators are not allowed. Final Answer: The correct statement is Only III, that is, (12)^(1/3) > (17)^(1/4) > (65)^(1/6).

Simplification Questions