Q: In trigonometry, the triple-angle identity rewrites cos(3A) using only cos(A). Which of the following expressions is the correct identity for cos 3A in terms of cos A?

Introduction / Context: This question tests a standard trigonometric identity: the triple-angle formula for cosine. Triple-angle identities are used to simplify expressions, solve trigonometric equations, and convert higher-angle terms like cos(3A) into powers of cos(A). Given Data / Assumptions: We need an identity for cos(3A) written only in terms of cos(A). A is any real angle (identity must hold for all A). Concept / Approach: The known triple-angle identity for cosine is derived from angle addition: cos(3A) = cos(2A + A) and then applying cos(2A) and sin(2A) formulas. The final form becomes a cubic polynomial in cos(A). Step-by-Step Solution: Start with cos(3A) = cos(2A + A) Use cos(u+v) = cos u * cos v - sin u * sin v So cos(3A) = cos(2A)*cos(A) - sin(2A)*sin(A) Replace cos(2A) = 2cos^2(A) - 1 and sin(2A) = 2sin(A)cos(A) Then cos(3A) = (2cos^2(A) - 1)cos(A) - (2sin(A)cos(A))sin(A) = 2cos^3(A) - cos(A) - 2sin^2(A)cos(A) Use sin^2(A) = 1 - cos^2(A) = 2cos^3(A) - cos(A) - 2(1 - cos^2(A))cos(A) = 2cos^3(A) - cos(A) - 2cos(A) + 2cos^3(A) = 4cos^3(A) - 3cos(A) Verification / Alternative check: Check A = 0: LHS cos(0) = 1. RHS 4*1 - 3*1 = 1, so it matches. This quick check supports the identity choice. Why Other Options Are Wrong: 3 cos A - 4 cos^3 A is just the negative of the correct expression. 4 cos^3 A + 3 cos A has the wrong sign on the linear term and fails at A = 0. cos A + 4 cos^3 A and 4 cos A - 3 cos^3 A do not match the known cubic structure and fail simple substitutions. Common Pitfalls: Mixing up the signs, or confusing the cosine triple-angle identity with the sine triple-angle identity (sin 3A). Another common error is writing cos(3A) as 3cos(A) - 4cos^3(A) without noticing it is the negative. Final Answer: cos(3A) = 4 cos^3 A - 3 cos A

Q: In right-triangle trigonometry, cosec X = 1/sin X relates an angle to the ratio of hypotenuse and opposite side. In triangle XYZ, angle Y = 90 degrees. If cosec X = 13/12 and XY = 1 cm, what is the length of side YZ (in cm)?

Introduction / Context: This problem tests how to convert a trigonometric ratio (cosec) into side ratios in a right-angled triangle and then scale those ratios using a given side length. It combines knowledge of cosec X, the definition of sine, and the idea of similar right-triangle side triples. Given Data / Assumptions: Triangle XYZ is right-angled at Y (angle Y = 90 degrees). cosec X = 13/12, so sin X = 12/13. XY = 1 cm. For angle X: opposite side is YZ, adjacent side is XY, hypotenuse is XZ. Concept / Approach: Use cosec X = hypotenuse/opposite. From cosec X = 13/12, we get XZ:YZ = 13:12. A right triangle with sides in the ratio 5:12:13 is a standard Pythagorean triple, meaning the adjacent side to angle X is 5 parts when hypotenuse is 13 parts and opposite is 12 parts. Step-by-Step Solution: cosec X = 13/12 means XZ / YZ = 13/12 So let XZ = 13k and YZ = 12k for some scale factor k Because angle Y is 90 degrees, Pythagoras applies: XY^2 + YZ^2 = XZ^2 XY^2 = XZ^2 - YZ^2 = (13k)^2 - (12k)^2 = 169k^2 - 144k^2 = 25k^2 So XY = 5k Given XY = 1 cm, we get 5k = 1 => k = 0.2 Then YZ = 12k = 12 * 0.2 = 2.4 cm Verification / Alternative check: Compute hypotenuse: XZ = 13k = 2.6 cm. Check: 1^2 + 2.4^2 = 1 + 5.76 = 6.76 and 2.6^2 = 6.76, so the triangle is consistent. Why Other Options Are Wrong: 2.0 and 1.8 are too small; they would not keep the 12:13 sin ratio with XY = 1. 2.6 confuses YZ with the hypotenuse XZ. 1.5 corresponds to a different side ratio and fails the Pythagoras check with hypotenuse 2.6. Common Pitfalls: Mixing up which side is opposite angle X (it is YZ), or treating XY as the opposite side. Another common mistake is assuming cosec is opposite/hypotenuse instead of hypotenuse/opposite. Final Answer: YZ = 2.4 cm

Question 1

For an acute angle θ with 0° < θ < 90°, consider the expression 4 sin^2 θ + 1. For every such θ, this expression is always greater than or equal to which of the following options?

Accepted Answer

Introduction / Context: This question combines trigonometric expressions with inequality reasoning. We are given an expression involving sin^2 θ and asked which of the listed expressions is always less than or equal to it, for all acute angles θ. Such problems encourage you to manipulate expressions algebraically and use the fact that sine values lie between 0 and 1 for 0° < θ < 90°. Given Data / Assumptions: 0° < θ < 90° so θ is acute. The expression of interest is 4 sin^2 θ + 1. We must find which option is always less than or equal to 4 sin^2 θ + 1 for all such θ. Concept / Approach: We test each option by forming an inequality of the form 4 sin^2 θ + 1 ≥ (candidate expression). If we can show that this inequality holds for every θ in (0°, 90°), then that candidate is valid. The cleanest candidate to test is 4 sin θ, because both sides are simple functions of sin θ. For that one, a simple algebraic manipulation will reveal a perfect square, which is always non negative. For other options, you can quickly see counterexamples by picking convenient angles like 0°, 30°, 60°, or 90°. Step-by-Step Solution: Let s = sin θ, where 0 < s < 1 for 0° < θ < 90°. Consider the inequality 4s^2 + 1 ≥ 4s. Rearrange to one side: 4s^2 - 4s + 1 ≥ 0. Factor the quadratic: 4s^2 - 4s + 1 = 4(s^2 - s + 1/4) = 4(s - 1/2)^2. Since a square is always ≥ 0, (s - 1/2)^2 ≥ 0 for all real s. Therefore, 4(s - 1/2)^2 ≥ 0, which implies 4s^2 + 1 ≥ 4s for all 0 < s < 1. So 4 sin^2 θ + 1 is always greater than or equal to 4 sin θ. Now briefly test other options to see that they fail for some angles. For option A (2): at very small θ, sin θ is near 0, so 4 sin^2 θ + 1 is close to 1, which is less than 2. So it is not always ≥ 2. For option C (4 cos θ): at θ = 0°, 4 sin^2 θ + 1 = 1, but 4 cos θ = 4, so the inequality fails. For option D (4 tan θ): near θ = 90°, tan θ becomes very large while 4 sin^2 θ + 1 stays bounded between 1 and 5, so it fails. For option E (sin θ + cos θ): a detailed analysis shows 4 sin^2 θ + 1 is not tightly bounded by this expression in the same guaranteed way, and option B is the clear algebraic match. Verification / Alternative check: We can verify option B numerically at a few angles. At θ = 30°, sin θ = 1/2, so 4 sin^2 θ + 1 = 4 * 1/4 + 1 = 2, and 4 sin θ = 2. They are equal. At θ = 60°, sin θ = sqrt(3) / 2, so 4 sin^2 θ + 1 = 4 * 3/4 + 1 = 4, and 4 sin θ = 2 sqrt(3) ≈ 3.464, which is less than 4. At θ close to 0°, sin θ is very small, so 4 sin^2 θ + 1 is slightly above 1, and 4 sin θ is near 0, so the inequality holds there too. Why Other Options Are Wrong: As seen, option A fails at small angles, C fails at 0°, D fails near 90°, and E does not have a guaranteed inequality of the same simple squared form. Only 4 sin θ is always less than or equal to 4 sin^2 θ + 1 for all acute θ. Common Pitfalls: A common mistake is to guess based on values at special angles like 45° without checking the whole range. Another is to assume the largest looking expression must be the answer. Always convert to a single variable and check if the difference simplifies to a square or obviously non negative form. Final Answer: The expression 4 sin^2 θ + 1 is always greater than or equal to 4 sin θ for 0° < θ < 90°.

Question 2

In a regular hexagon ABCDEF, two vertical towers stand at vertices B and C. From vertex A on the ground, the angle of elevation to the top of the tower at B is 30°, and the angle of elevation to the top of the tower at C is 45°. Assuming all vertice…

Accepted Answer

Introduction / Context: This is a geometry and trigonometry hybrid problem involving a regular hexagon and angles of elevation. By using properties of a regular hexagon and basic tangent definitions, we can relate the horizontal distances from A to B and from A to C with the heights of the towers placed at those vertices. Such questions appear in height and distance sections of aptitude exams, often combined with polygon properties. Given Data / Assumptions: ABCDEF is a regular hexagon. Vertical towers of heights h_B and h_C stand at vertices B and C respectively. From vertex A, angle of elevation to the top at B is 30°. From vertex A, angle of elevation to the top at C is 45°. All vertices of the hexagon lie on the same horizontal plane. We need the ratio h_B : h_C. Concept / Approach: In a regular hexagon of side length s, each side is equal, and all interior angles are 120°. If the hexagon is inscribed in a circle, the central angle between adjacent vertices is 60°. The distances AB and AC can be expressed in terms of the side length s. Then, using tan(angle) = height / horizontal distance, we can write h_B and h_C in terms of s and the given angles. Finally, dividing h_B by h_C eliminates s and gives the required ratio. Step-by-Step Solution: Let the side length of the regular hexagon be s. Then AB = s, because adjacent vertices are one side apart. Vertices of a regular hexagon on a circle are 60° apart at the center. The distance AC corresponds to a chord that spans two sides, that is, 120° at the center. For a circle of radius R, chord length for 60° is AB = 2R sin 30° = R, so s = R. Chord length for 120° is AC = 2R sin 60° = 2R * (sqrt(3) / 2) = R sqrt(3) = s sqrt(3). Now use tangents for the right triangles formed with vertical heights. tan 30° = h_B / AB = h_B / s. So h_B = s * tan 30° = s * (1 / sqrt(3)) = s / sqrt(3). tan 45° = h_C / AC = h_C / (s sqrt(3)). Since tan 45° = 1, we have h_C = s sqrt(3). Therefore, h_B : h_C = (s / sqrt(3)) : (s sqrt(3)). Cancel s from both terms: h_B : h_C = 1 / sqrt(3) : sqrt(3). Multiply numerator and denominator by sqrt(3) to simplify: 1 : 3. Verification / Alternative check: You can assign a convenient value to s to check the ratio numerically. For example, let s = sqrt(3). Then AB = sqrt(3) and AC = 3. From A, h_B = AB * tan 30° = sqrt(3) * (1 / sqrt(3)) = 1. Also, h_C = AC * tan 45° = 3 * 1 = 3. Thus h_B : h_C = 1 : 3, matching the algebraic result. Why Other Options Are Wrong: 1 : sqrt(3) would be the ratio if h_C were s rather than s sqrt(3). 1 : 2 and 1 : 2sqrt(3) do not arise from the geometry and correspond to incorrect distance assumptions. sqrt(3) : 1 is the inverse of the correct ratio and would imply h_B is taller than h_C, which contradicts the angles of elevation (45° for C should correspond to a relatively taller or closer tower than 30° for the same horizontal scale). Common Pitfalls: A frequent error is to assume AC = 2s instead of s sqrt(3). Another mistake is to forget that the angle of elevation is measured from A to the top of each tower, not from the center of the hexagon. Using wrong chord lengths or misapplying tangent formulas will quickly lead to incorrect ratios. Drawing a rough diagram often helps to visualise the relationships. Final Answer: The ratio of the heights of the towers at B and C is 1 : 3.

Question 3

Evaluate the exact value of the product tan 1° × tan 2° × tan 3° × ... × tan 89°, where the angles are in degrees and the product runs through all integer angles from 1° to 89°.

Accepted Answer

Introduction / Context: This is a classic trigonometric product problem that highlights the complementary angle property of the tangent function. Instead of calculating each tangent individually, you should observe how tan θ and tan(90° - θ) relate to each other. This insight allows you to pair terms and simplify a very long product into something extremely simple, which is a common trick in competitive exam problems. Given Data / Assumptions: The product is tan 1° × tan 2° × tan 3° × ... × tan 89°. Angles are in degrees. All angles from 1° to 89° are included exactly once. Concept / Approach: The key identity is tan(90° - θ) = cot θ, and the fact that tan θ * cot θ = 1 for all angles θ where both are defined. In the given product, terms can be paired: tan 1° with tan 89°, tan 2° with tan 88°, and so on. Each such pair multiplies to 1 because tan(90° - θ) is the reciprocal of tan θ. The only term that does not have a distinct complement in the range 1° to 89° is tan 45°, because 45° is its own complement. Since tan 45° = 1, it does not change the product. Step-by-Step Solution: Write the product as P = tan 1° × tan 2° × ... × tan 44° × tan 45° × tan 46° × ... × tan 89°. Notice that 89° is 90° - 1°, 88° is 90° - 2°, and so on up to 46° which is 90° - 44°. Pair terms: (tan 1° × tan 89°), (tan 2° × tan 88°), ..., (tan 44° × tan 46°). Use tan(90° - θ) = cot θ, so tan(90° - θ) = 1 / tan θ. Therefore, tan θ × tan(90° - θ) = tan θ × (1 / tan θ) = 1 for each such pair. There are 44 such pairs from 1° through 44° combined with 89° down to 46°. The remaining middle term is tan 45°. Since tan 45° = 1, it does not change the product. Thus the total product P is 1 × 1 × ... × 1 × tan 45° = 1 × 1 = 1. Verification / Alternative check: As a sanity check, you might approximate a few paired values: tan 10° ≈ 0.176 and tan 80° ≈ 5.671. Their product is close to 1. Similarly, tan 30° ≈ 0.577 and tan 60° ≈ 1.732, whose product is almost exactly 1.33? Actually, tan 30° * tan 60° = (1 / sqrt(3)) * sqrt(3) = 1 exactly. These examples illustrate the general pairing pattern and give confidence that every complementary pair yields a product of 1. Why Other Options Are Wrong: 0 would require at least one term to be zero, but tan θ is zero only when θ is a multiple of 180°, which does not occur between 1° and 89°. Infinity is not valid because tangent remains finite on (0°, 90°). The values 2 and -1 are not supported by the pairing argument, which shows that the product equals 1 exactly. Common Pitfalls: Some students attempt to compute many of the tangents numerically, which is impossible under exam conditions and unnecessary. Others forget the complementary angle property and treat the product as something that must grow very large. Always look for patterns in symmetrical ranges like 1° to 89°, especially involving tan θ and tan(90° - θ). Final Answer: The product tan 1° × tan 2° × ... × tan 89° is exactly 1.

Question 4

Solve the exponential equation 3^(x + 2) + 3^(−x) = 10 and identify which of the following ordered pair of real numbers correctly lists all solutions for x.

Accepted Answer

Introduction / Context: This problem tests your ability to solve equations involving exponents with the same base, here base 3. By introducing a substitution, the equation can be turned into a quadratic in a single variable. This technique is common in algebra and aptitude tests whenever exponential expressions appear both as direct powers and as reciprocals. Given Data / Assumptions: The equation is 3^(x + 2) + 3^(−x) = 10. x is a real number. We must determine all real solutions for x and then choose the option that lists them correctly. Concept / Approach: Because both terms use base 3, we can simplify by making a substitution such as t = 3^x. Then 3^(x + 2) becomes 9t and 3^(−x) becomes 1 / t. The original equation changes into an equation in t that is quadratic in form after we clear denominators. Solving that quadratic gives possible values for t, and from each t value we solve 3^x = t to find x. Finally, we check that the values satisfy the original equation. Step-by-Step Solution: Let t = 3^x. Then t > 0 for all real x. Compute 3^(x + 2) = 3^x * 3^2 = 9t. Compute 3^(−x) = 1 / 3^x = 1 / t. Substitute into the equation: 9t + 1 / t = 10. Multiply both sides by t (t > 0) to clear the denominator: 9t^2 + 1 = 10t. Rearrange to standard quadratic form: 9t^2 - 10t + 1 = 0. Solve using the quadratic formula: t = [10 ± sqrt(10^2 - 4 * 9 * 1)] / (2 * 9). The discriminant is 100 - 36 = 64, so t = [10 ± 8] / 18. Thus t = (10 + 8) / 18 = 18 / 18 = 1, or t = (10 - 8) / 18 = 2 / 18 = 1 / 9. Recall t = 3^x, so we solve 3^x = 1 and 3^x = 1 / 9. 3^x = 1 gives x = 0. 3^x = 1 / 9 = 3^(−2) gives x = -2. Verification / Alternative check: Check x = 0: 3^(0 + 2) + 3^(−0) = 3^2 + 3^0 = 9 + 1 = 10, which works. Check x = -2: 3^(−2 + 2) + 3^(2) = 3^0 + 3^2 = 1 + 9 = 10, which also works. No other t values came from the quadratic, and both t values were positive, so we have found all real solutions. Why Other Options Are Wrong: Pairs like 0 and 2, 1 and -1, or 1 and 2 arise from solving incorrect quadratics or misinterpreting the exponent rules. The pair -2 and 2 includes 2 as a solution, but x = 2 does not satisfy the equation, since 3^(2 + 2) + 3^(−2) = 81 + 1/9, which is far from 10. Only the pair 0 and -2 matches the correct solutions. Common Pitfalls: Mistakes often occur when clearing denominators or when using the quadratic formula. Some students incorrectly set t = 3^(−x) instead of 3^x and then mix up the transformations. Being careful with exponent rules (like 3^(x + 2) = 3^x * 9) and double checking the discriminant avoids these errors. Final Answer: The solutions of the equation are x = 0 and x = -2, so the correct option is 0 and -2.

Question 5

Solve the pair of linear inequalities 3x - 4 > 2 - x/3 and 3x + 5 > 4x - 5, and determine which of the following given values of x satisfies both inequalities simultaneously.

Accepted Answer

Introduction / Context: Many aptitude exams include questions where you must solve two inequalities together and then verify which given option lies in the common solution set. This question involves linear inequalities with a fractional term and requires careful algebra. Once you solve each inequality separately, you combine their solution ranges and then test the candidate values of x. Given Data / Assumptions: Inequality 1: 3x - 4 > 2 - x/3. Inequality 2: 3x + 5 > 4x - 5. We must find which among the listed values {7, 10, -11, 1, 0} satisfies both inequalities. Concept / Approach: The approach is to solve each inequality step by step using standard algebraic manipulations, taking care when multiplying or dividing by positive constants. Since there is no multiplication by negative numbers, the inequality signs will not flip. After obtaining each inequality's solution range, we intersect these ranges to find the common interval for x. Finally, we check which option lies inside that interval. Step-by-Step Solution: First inequality: 3x - 4 > 2 - x/3. Add x/3 to both sides: 3x + x/3 - 4 > 2. Write 3x as 9x/3: (9x/3 + x/3) - 4 > 2, so 10x/3 - 4 > 2. Add 4 to both sides: 10x/3 > 6. Multiply both sides by 3: 10x > 18. Divide by 10: x > 18/10 = 9/5 = 1.8. Second inequality: 3x + 5 > 4x - 5. Subtract 3x from both sides: 5 > x - 5. Add 5 to both sides: 10 > x, which is x < 10. Combine both: x > 1.8 and x < 10. So the common solution set is 1.8 < x < 10. Now test the options: 7 is between 1.8 and 10, so it satisfies both. 10 is not allowed because the second inequality is strict (>), and at x = 10 we have equality 3x + 5 = 4x - 5. -11, 1, and 0 are all less than or equal to 1.8, so they do not satisfy the first inequality. Verification / Alternative check: Substitute x = 7 into both inequalities. For the first: 3 * 7 - 4 = 21 - 4 = 17, and 2 - 7/3 = 2 - 2.333..., which is about -0.333..., so 17 > -0.333... holds. For the second: 3 * 7 + 5 = 21 + 5 = 26 and 4 * 7 - 5 = 28 - 5 = 23, so 26 > 23 holds. This confirms that x = 7 satisfies both inequalities. Why Other Options Are Wrong: At x = 10, the second inequality becomes 35 > 35, which is false because it is not strictly greater. Values -11, 1, and 0 fail the first inequality since they do not satisfy x > 1.8. Therefore none of those values belong to the common solution interval 1.8 < x < 10. Common Pitfalls: A common error is to forget to combine the solution sets and instead treat each inequality separately when checking options. Another frequent mistake is mishandling the fractional term x/3 or making arithmetic slips when simplifying. Always rewrite terms with a common denominator and carefully track each step when solving inequalities with fractions. Final Answer: The only option that satisfies both inequalities is 7.

Question 6

Five bottles cost as much as two bags. The combined cost of 15 bottles and 4 bags is Rs 2000. Based on this information, what is the price of a single bag in rupees?

Accepted Answer

Introduction / Context: This is a standard linear equations problem involving prices of two items. You know the relationship between the cost of bottles and bags, and the total cost of a combination of these items. Using algebra, you can set up equations in terms of the price of one bottle and one bag, and then find the unknown price. Such questions appear frequently in aptitude tests to assess comfort with forming and solving simple equations from word statements. Given Data / Assumptions: Five bottles cost as much as two bags. Total cost of 15 bottles and 4 bags is Rs 2000. Prices of bottles and bags are constant and positive. We must find the price of one bag. Concept / Approach: Let the price of one bottle be B rupees and the price of one bag be A rupees. The first statement gives us one linear relationship between A and B. The second statement gives us a second equation involving the total cost of a specific combination of bottles and bags. From the first relation, we can express B in terms of A, and then substitute into the second equation to solve for A directly. This avoids dealing with two unknowns at the end of the problem. Step-by-Step Solution: Let A = price of one bag in rupees. Let B = price of one bottle in rupees. Given that five bottles cost as much as two bags: 5B = 2A. From this, express B in terms of A: B = (2/5)A. We are also told that 15 bottles and 4 bags together cost Rs 2000. So 15B + 4A = 2000. Substitute B = (2/5)A into this equation. 15 * (2/5)A + 4A = 2000. Compute 15 * (2/5)A = (30/5)A = 6A. So the equation becomes 6A + 4A = 2000. Combine like terms: 10A = 2000. Solve for A: A = 2000 / 10 = 200. Therefore, the price of one bag is Rs 200. Verification / Alternative check: We can check our result by computing the price of one bottle and verifying the total cost. With A = 200, B = (2/5) * 200 = 80. Now compute the cost of 15 bottles and 4 bags: 15 * 80 + 4 * 200 = 1200 + 800 = 2000. This matches the given total cost, so the value A = 200 is correct. Why Other Options Are Wrong: If you assume A = 215, 240, 250, or 300 and then compute the implied bottle price B from 5B = 2A, substituting into 15B + 4A will not give exactly 2000. These alternative values arise only from algebraic errors or misinterpretation of the initial relation between bottles and bags. Common Pitfalls: Common mistakes include reversing the relationship and writing 5B = A instead of 5B = 2A, or forgetting to multiply the bottle price by 15 when forming the total cost equation. Another pitfall is dropping a factor when simplifying 15 * (2/5)A. Writing equations carefully from the word statement and double checking each step helps avoid these issues. Final Answer: The price of a single bag is Rs 200.

Question 7

Solve the compound inequality 4(2x - 4) - 2 > 3x - 1 ≥ 4x - 7 and determine which of the following values x can take so that the entire chained inequality holds true.

Accepted Answer

Introduction / Context: This question involves a chained inequality with three algebraic expressions. The notation 4(2x - 4) - 2 > 3x - 1 ≥ 4x - 7 means that both inequalities must hold at the same time. Your goal is to solve this compound inequality and then see which given value of x lies in the resulting solution set. Such chained inequalities appear in aptitude tests to check comfort with solving and combining multiple linear inequalities. Given Data / Assumptions: First part: 4(2x - 4) - 2 > 3x - 1. Second part: 3x - 1 ≥ 4x - 7. Both must be satisfied simultaneously. We must choose which option value of x satisfies the compound inequality. Concept / Approach: We break the chained inequality into two separate inequalities and solve each one step by step. Each part produces an interval of x values. The final solution set is the intersection (overlap) of these intervals because both conditions must hold at once. After finding the intersection, we check which option lies inside this combined range. Only one option is chosen so that exactly one is correct. Step-by-Step Solution: First inequality: 4(2x - 4) - 2 > 3x - 1. Expand: 4 * 2x = 8x and 4 * (-4) = -16, so 8x - 16 - 2 > 3x - 1. Simplify left side: 8x - 18 > 3x - 1. Subtract 3x from both sides: 5x - 18 > -1. Add 18 to both sides: 5x > 17. Divide by 5 (positive): x > 17/5 = 3.4. Second inequality: 3x - 1 ≥ 4x - 7. Subtract 3x from both sides: -1 ≥ x - 7. Add 7 to both sides: 6 ≥ x, which is x ≤ 6. Now combine: x > 3.4 and x ≤ 6. So the solution set is 3.4 < x ≤ 6. Among the options {1, 3, 4, 7, 10}, only x = 4 lies strictly above 3.4 and at or below 6. Verification / Alternative check: Check x = 4 in both inequalities. First part: 4(2 * 4 - 4) - 2 = 4(8 - 4) - 2 = 4 * 4 - 2 = 16 - 2 = 14. Right side is 3x - 1 = 12 - 1 = 11. Since 14 > 11, the first inequality holds. Second part: 3x - 1 = 11 and 4x - 7 = 16 - 7 = 9, so 11 ≥ 9, which also holds. Thus x = 4 satisfies the chain. Why Other Options Are Wrong: x = 1 and x = 3 are not greater than 3.4, so they fail the first inequality. x = 7 and x = 10 are greater than 6, so they fail the second inequality. Hence none of these values belong to the solution set 3.4 < x ≤ 6. Common Pitfalls: Some learners try to manipulate the entire chain in one go and lose track of signs, or they forget that the inequalities describe an interval that must be intersected, not merely united. Another mistake is to ignore the strictness of the inequality and accidentally include boundary points that should be excluded. Always solve each piece separately and clearly write the final interval before testing options. Final Answer: The value of x that satisfies the full compound inequality is 4.

Question 8

If a and b are real numbers such that a + b = 11 and a^2 + b^2 = 61, then using standard algebraic identities, what is the value of the product ab?

Accepted Answer

Introduction / Context: This question checks your understanding of the identity relating the square of a sum to the sum of squares and the product of two numbers. Instead of solving for a and b individually, you can use the relation (a + b)^2 = a^2 + 2ab + b^2 and the given values of a + b and a^2 + b^2 to find ab quickly. This is a common pattern in algebra questions on aptitude exams. Given Data / Assumptions: a + b = 11. a^2 + b^2 = 61. a and b are real numbers. We are asked to find ab. Concept / Approach: Use the identity (a + b)^2 = a^2 + 2ab + b^2. We already know a + b and a^2 + b^2, so substituting these into the identity gives an equation in terms of ab. From this, we can isolate 2ab, find its value, and then divide by 2 to get ab. There is no need to find explicit values of a and b, though that is also possible in principle. Step-by-Step Solution: Start from the identity (a + b)^2 = a^2 + 2ab + b^2. We are given a + b = 11, so (a + b)^2 = 11^2 = 121. We are also given a^2 + b^2 = 61. Substitute into the identity: 121 = 61 + 2ab. Rearrange to solve for 2ab: 2ab = 121 - 61. Compute the difference: 121 - 61 = 60. So 2ab = 60. Divide both sides by 2: ab = 60 / 2 = 30. Verification / Alternative check: We can verify by constructing a and b as roots of a quadratic equation. If a and b are roots of t^2 - (a + b)t + ab = 0, then the quadratic is t^2 - 11t + 30 = 0. Factor this: t^2 - 11t + 30 = (t - 5)(t - 6). So a and b could be 5 and 6. Then a + b = 11 and a^2 + b^2 = 25 + 36 = 61, both matching the given values. The product ab = 5 * 6 = 30, confirming our result. Why Other Options Are Wrong: Values like 12, 24, 18, or 96 would not satisfy the relationship 2ab = 60. If any of those were used as ab, then a^2 + b^2 computed from (a + b)^2 - 2ab would not equal 61. For example, if ab were 24, then a^2 + b^2 would be 121 - 48 = 73, which contradicts the given 61. Common Pitfalls: A common mistake is to misapply the identity as a^2 + b^2 = (a + b)^2 + 2ab instead of (a + b)^2 - 2ab. Another error is in arithmetic when squaring 11 or subtracting 61 from 121. Carefully writing the identity and substituting the given values avoids these issues. Final Answer: The value of the product ab is 30.

Question 9

Evaluate the exact value of tan(-240°) using trigonometric periodicity and symmetry properties.

Accepted Answer

Introduction / Context: This question tests your understanding of the periodicity and symmetry of the tangent function for angles measured in degrees. Instead of computing tan(-240°) directly, you can reduce the angle to a simpler equivalent angle using known properties such as tan(θ + 180°) = tan θ and tan(-θ) = -tan θ. Recognising these patterns is crucial in trigonometry and aptitude tests. Given Data / Assumptions: The angle is -240°. We are asked to find tan(-240°). All trigonometric functions are considered in degrees. Concept / Approach: The tangent function has period 180°, which means tan(θ + 180°) = tan θ for all θ where tangent is defined. Also, tangent is an odd function, so tan(-θ) = -tan θ. We can use either or both of these properties to simplify -240° to one of the standard angles like 60° or -60°, for which exact tangent values are known. In this case, working with 180° shifts and symmetry around 0° is convenient. Step-by-Step Solution: Start with tan(-240°). Use periodicity: add 180° to the angle, since tan(θ + 180°) = tan θ. -240° + 180° = -60°. Therefore, tan(-240°) = tan(-60°). Now use the odd function property: tan(-θ) = -tan θ. So tan(-60°) = -tan 60°. We know tan 60° = sqrt(3). Hence, tan(-60°) = -sqrt(3). Therefore, tan(-240°) = -sqrt(3). Verification / Alternative check: An alternative path is to add 360° to -240° to get a coterminal angle in the standard 0° to 360° range. -240° + 360° = 120°. Now compute tan 120°. Since 120° = 180° - 60°, the reference angle is 60°, and tangent in the second quadrant (between 90° and 180°) is negative. So tan 120° = -tan 60° = -sqrt(3). This is the same value we obtained earlier, confirming the result. Why Other Options Are Wrong: 1 / sqrt(3) and -1 / sqrt(3) correspond to tangent values at 30° and -30°, not 60° or -60°. sqrt(3) is the tangent of 60°, which would be correct for tan 60° or tan 240°, but not tan(-240°). The value 0 is tangent at angles like 0°, 180°, and 360°, which are not equivalent to -240° in terms of tangent's behaviour. Common Pitfalls: One common mistake is to treat tangent as having a 360° period instead of 180°, which can lead to unnecessary steps. Another error is mishandling signs when dealing with negative angles or angles in different quadrants. Always reduce angles step by step and use the known exact values for 30°, 45°, and 60° to guide you. Final Answer: The value of tan(-240°) is -sqrt(3).

Question 10

Which of the following trigonometric expressions is exactly equal to tan(A / 2) for an angle A, assuming all trigonometric functions involved are defined?

Accepted Answer

Introduction / Context: This question tests your knowledge of standard half angle formulas in trigonometry. The function tan(A / 2) can be expressed in several equivalent ways in terms of sine and cosine of A. However, in a multiple choice setting, we usually look for one canonical form. Recognising the correct half angle expression is important in many simplification and equation solving problems in trigonometry. Given Data / Assumptions: We are asked to identify an expression that equals tan(A / 2). Angle A is such that all trigonometric functions used are defined. Options involve combinations of sin A and cos A. Concept / Approach: The standard half angle identities for tangent are: tan(A / 2) = (1 - cos A) / sin A, and tan(A / 2) = sin A / (1 + cos A). In this question, we select the exact expression that matches tan(A / 2) without simultaneously matching several options. Option A clearly matches the first canonical form directly. Other options can be analysed by simplifying them in terms of sin A and cos A to see whether they reduce to tan(A / 2) or not. Step-by-Step Solution: Recall the half angle identity: tan(A / 2) = (1 - cos A) / sin A. Look at option A: (1 - cos A) / sin A. Option A is exactly the same as the identity, so it equals tan(A / 2). Check option B: sin A / (1 - cos A). This is the reciprocal of option A only if cos A is replaced by -cos A; it does not match the known half angle formula. Check option C: (1 + cos A) / sin A. This is closer to the reciprocal of sin A / (1 + cos A), not tan(A / 2). Check option D: 1 / (sec A + cot A). Writing in sine and cosine gives 1 / (1 / cos A + cos A / sin A), which does not simplify to tan(A / 2). Check option E: cos A / (1 + sin A). This also does not match any standard half angle form for tangent. Therefore, only option A matches the standard half angle identity for tangent. Verification / Alternative check: We can verify using a specific angle, for example A = 60°. Then A / 2 = 30°, so tan(A / 2) = tan 30° = 1 / sqrt(3). Compute option A: (1 - cos 60°) / sin 60° = (1 - 1/2) / (sqrt(3) / 2) = (1/2) / (sqrt(3) / 2) = 1 / sqrt(3). This matches tan 30°. Evaluating other options at A = 60° gives different values, confirming that only option A equals tan(A / 2). Why Other Options Are Wrong: Option B gives the reciprocal of option A, which corresponds to cot(A / 2) rather than tan(A / 2). Option C is not one of the canonical half angle formulas and evaluates to a different value at test angles. Option D simplifies to an expression that involves both sine and cosine in a more complex manner and does not match tan(A / 2). Option E also fails to match the half angle identity when tested with specific angles. Common Pitfalls: A common mistake is to misremember which way the sine and cosine go in the half angle formulas, leading to confusion between tan(A / 2) and cot(A / 2). Another error is to assume that any ratio of linear combinations of sine and cosine might represent tan(A / 2) without checking the actual identity. Always refer back to the standard forms and, if unsure, test with a simple angle like 60° to distinguish between options. Final Answer: The expression that equals tan(A / 2) is (1 - cos A) / sin A.

Question 11

If x^2 + 1/x^2 = 2 for a nonzero real number x, then using algebraic identities, what is the value of the expression x - 1/x?

Accepted Answer

Introduction / Context: This question uses a standard technique where a symmetric expression like x^2 + 1/x^2 is related to another symmetric expression like x - 1/x. By recalling and applying the correct identity, you can quickly go from one expression to the other without solving for x explicitly. These transformations are common in algebra and aptitude questions to test recognition of patterns and manipulation of exponents and reciprocals. Given Data / Assumptions: x^2 + 1/x^2 = 2. x is a nonzero real number. We must find the value of x - 1/x. Concept / Approach: We use the identity (x - 1/x)^2 = x^2 + 1/x^2 - 2. This connects the given quantity x^2 + 1/x^2 with the desired quantity x - 1/x. After substituting the given value 2 into this identity, we will find the square of x - 1/x. Then we can deduce the actual value of x - 1/x. Importantly, we must consider which values are possible given the algebraic equation that x satisfies. Step-by-Step Solution: Start with the identity: (x - 1/x)^2 = x^2 + 1/x^2 - 2. We are given x^2 + 1/x^2 = 2. Substitute this into the identity: (x - 1/x)^2 = 2 - 2 = 0. So (x - 1/x)^2 = 0. The only real number whose square is 0 is 0 itself. Therefore, x - 1/x = 0. This simplifies to x = 1/x, which implies x^2 = 1. Hence x = 1 or x = -1, but in either case x - 1/x = 0. Verification / Alternative check: We can verify by solving for x directly from x^2 + 1/x^2 = 2. Multiply both sides by x^2 (x ≠ 0): x^4 + 1 = 2x^2. Rearrange to x^4 - 2x^2 + 1 = 0, which factors as (x^2 - 1)^2 = 0. So x^2 = 1, giving x = 1 or x = -1. Compute x - 1/x for both values. For x = 1, x - 1/x = 1 - 1 = 0. For x = -1, x - 1/x = -1 - (-1) = -1 + 1 = 0. In both cases, the expression equals 0, confirming our answer. Why Other Options Are Wrong: Values like 2, -2, 1, or -1 would give nonzero squares when substituted into (x - 1/x)^2. For example, if x - 1/x were 2, then x^2 + 1/x^2 would be 2 + 2 = 4, not 2. Therefore these other options contradict the given condition x^2 + 1/x^2 = 2. Common Pitfalls: Some students mistakenly use the formula for (x + 1/x)^2 instead of (x - 1/x)^2, which leads to x^2 + 1/x^2 + 2 and an incorrect value. Another mistake is to assume x must be a single numeric value like 1 and not consider -1, but here both possible x values give the same result for x - 1/x. Focus on the identity with the correct sign to avoid confusion. Final Answer: The value of the expression x - 1/x is 0.

Question 12

The straight line passing through the points (−2, 5) and (6, b) is perpendicular to the line given by 20x + 5y = 3. Using the concept of slopes of perpendicular lines, what is the value of b?

Accepted Answer

Introduction / Context: This coordinate geometry question tests your understanding of slopes of straight lines and the condition for perpendicularity. Two non vertical lines are perpendicular if the product of their slopes is -1. By finding the slope of the given line from its equation and relating it to the slope of the line through the two points, you can determine the unknown coordinate b. Given Data / Assumptions: A line L1 passes through points (−2, 5) and (6, b). A line L2 is given by the equation 20x + 5y = 3. L1 is perpendicular to L2. We need to find the value of b. Concept / Approach: First, find the slope of the line 20x + 5y = 3 by rewriting it in slope intercept form y = mx + c. Then use the fact that if two lines are perpendicular, their slopes m1 and m2 satisfy m1 * m2 = -1. Next, compute the slope of the line passing through (−2, 5) and (6, b) in terms of b. Equating the product of slopes to -1 gives an equation in b that we can solve easily. Step-by-Step Solution: Rewrite the line 20x + 5y = 3 in the form y = mx + c. Solve for y: 5y = -20x + 3. Divide by 5: y = -4x + 3/5. So the slope of this line (L2) is m2 = -4. Let the slope of line L1, passing through (−2, 5) and (6, b), be m1. Slope m1 = (b - 5) / (6 - (−2)) = (b - 5) / 8. Since L1 is perpendicular to L2, m1 * m2 = -1. So (b - 5) / 8 * (−4) = -1. Simplify: (b - 5) * (−4) / 8 = -1. Note that −4 / 8 = −1/2, so (b - 5) * (−1/2) = -1. Multiply both sides by -2: b - 5 = 2. Therefore, b = 5 + 2 = 7. Verification / Alternative check: Compute slopes explicitly with b = 7. The slope of L1 is (7 - 5) / (6 - (−2)) = 2 / 8 = 1/4. The slope of L2 is -4. Their product is (1/4) * (−4) = -1, which confirms that L1 and L2 are perpendicular when b = 7. Why Other Options Are Wrong: If b were -7, 4, -4, or 0, the slope (b - 5)/8 would not be 1/4, and the product with -4 would not equal -1. For example, if b = 4, the slope would be (4 - 5)/8 = -1/8, and the product with -4 would be 1/2, not -1. Hence those values do not satisfy the perpendicularity condition. Common Pitfalls: Typical errors include miscomputing the slope from the standard form equation, forgetting that the product of slopes for perpendicular lines is -1 (not 1), or mixing up the coordinates in the slope formula. Carefully rewriting the given line in y = mx + c form and plugging points correctly into the slope formula avoids these issues. Final Answer: The value of b that makes the line through (−2, 5) and (6, b) perpendicular to 20x + 5y = 3 is 7.

Question 13

If X = tan(A + B), then using the addition formula for tangent, which of the following expressions correctly represents X in terms of tan A and tan B?

Accepted Answer

Introduction / Context: This question directly tests your knowledge of the tangent addition formula in trigonometry. Knowing how to express tan(A + B) in terms of tan A and tan B is essential for simplifying expressions and solving trigonometric equations. Recognising the correct formula from among several similar looking options is a common test of attention to detail in exams. Given Data / Assumptions: X is defined as tan(A + B). We are asked to express X in terms of tan A and tan B. A and B are angles for which all functions involved are defined. Concept / Approach: The standard tangent addition identity is: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). We simply recall this identity and match it with one of the given options. Other options involve sign changes or extra terms that correspond to different, incorrect formulas. Being able to distinguish the correct formula from minor variations is crucial. Step-by-Step Solution: Recall the exact identity: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). Since X = tan(A + B), we have X = (tan A + tan B) / (1 - tan A tan B). Now compare this with the options: Option A: (tan A - tan B) / (1 + tan A tan B) has wrong signs in both numerator and denominator. Option B: (tan A + tan B) / (1 - tan A tan B) matches the identity exactly. Option C: (tan A + tan B) / (1 + tan A tan B) has a plus in the denominator, which is incorrect for tan(A + B). Option D: (tan A - tan B) / (1 - tan A tan B) corresponds to tan(A - B), not tan(A + B). Option E adds +1 at the end and does not match any standard formula. Therefore, option B is the correct expression for X. Verification / Alternative check: As a quick check, pick A = B = 45°. Then A + B = 90°, so tan(A + B) is undefined (infinite). In our formula, tan 45° = 1, so the denominator 1 - tan A tan B becomes 1 - 1 * 1 = 0, while the numerator is 1 + 1 = 2, giving division by zero, which matches the undefined nature of tan 90°. The other formulas would not all show this behaviour consistently. Why Other Options Are Wrong: Option A corresponds to a different combination of angles. Option C would be the form for something like tan(A + B) if the identity had a plus in the denominator, which it does not. Option D is actually the formula for tan(A - B). Option E modifies the correct formula by adding 1, which changes the functional behaviour entirely and does not represent a standard trigonometric identity. Common Pitfalls: Students often mix up tan(A + B) and tan(A - B) or misremember the sign in the denominator. Another frequent mistake is to flip the numerator or denominator when trying to recall the formula from memory. Practising these identities regularly and checking them with simple angles helps prevent such errors. Final Answer: The correct expression for X is (tan A + tan B) / (1 - tan A tan B).

Question 14

Solve the linear equation 5x - 40 = 3x and then find the numerical value of the expression 2x - 11.

Accepted Answer

Introduction / Context: This is a straightforward linear equation problem followed by an evaluation of a simple expression in x. Such questions test your basic algebraic manipulation skills: isolating x and then substituting the value into another expression. Although simple, they appear frequently in the arithmetic and algebra sections of aptitude tests and must be done quickly and accurately. Given Data / Assumptions: The equation is 5x - 40 = 3x. x is a real number. We are asked to find the value of 2x - 11 after solving for x. Concept / Approach: We solve the equation 5x - 40 = 3x by bringing all x terms to one side and constants to the other. Once we obtain the value of x, we directly substitute it into 2x - 11. There is no need for any advanced technique; the main focus is on careful and correct algebraic manipulation. Step-by-Step Solution: Start with the equation: 5x - 40 = 3x. Subtract 3x from both sides to collect x terms: 5x - 3x - 40 = 0. This simplifies to 2x - 40 = 0. Add 40 to both sides: 2x = 40. Divide by 2: x = 40 / 2 = 20. Now evaluate the expression 2x - 11. Substitute x = 20: 2x - 11 = 2 * 20 - 11. Compute 2 * 20 = 40. So 2x - 11 = 40 - 11 = 29. Verification / Alternative check: We can verify x = 20 satisfies the original equation. Substitute into 5x - 40: 5 * 20 - 40 = 100 - 40 = 60. Substitute into 3x: 3 * 20 = 60. The two sides are equal, confirming that x = 20 is correct. With x confirmed, the calculation 2x - 11 = 40 - 11 = 29 is reliable. Why Other Options Are Wrong: If you misplaced a sign or made an arithmetic mistake, you might get 19, 9, or even 39. For example, forgetting to add 40 correctly or miscomputing 2 * 20 could lead to these errors. The value 0 would arise only if 2x - 11 were mistakenly set equal to the equation itself instead of being evaluated after solving. None of these incorrect values satisfy both the equation and the expression simultaneously. Common Pitfalls: A frequent error is to subtract 5x from both sides and write 3x - 5x - 40 = 0 incorrectly, leading to -2x - 40 = 0 and then mistakes with signs. Another pitfall is to forget that you must substitute the solved value of x into 2x - 11, not into the original equation again. Careful stepwise operations prevent these issues. Final Answer: The numerical value of 2x - 11 is 29.

Question 15

What is the area of a sector of a circle whose central angle is 90° and whose radius is 14 cm, using the standard formula for the area of a sector in square centimetres?

Accepted Answer

Introduction / Context: This geometry question involves the area of a sector of a circle, a common topic in school mathematics and aptitude exams. Instead of the full area of a circle, we only want the area corresponding to a central angle of 90°, which represents a quarter of the full circle. The problem tests your ability to use the sector area formula correctly and substitute the given radius and angle. Given Data / Assumptions: Radius r of the circle is 14 cm. Central angle θ of the sector is 90°. We assume π is used in the usual way, often approximated as 22/7 in aptitude problems. We are asked to find the area of the sector in square centimetres. Concept / Approach: The area of a full circle is πr^2. A sector with central angle θ degrees represents θ/360 of the full circle. Therefore, the area of a sector is (θ / 360) * πr^2. With θ = 90°, the sector is exactly one quarter of the full circle, so its area is (1/4) * πr^2. Substituting r = 14 cm and π = 22/7 (a typical exam approximation) gives a numerical value. Step-by-Step Solution: Use the formula for sector area: Area = (θ / 360) * π * r^2. Here θ = 90° and r = 14 cm. So Area = (90 / 360) * π * 14^2. Simplify 90 / 360 = 1/4. Therefore, Area = (1/4) * π * 14^2. Compute 14^2 = 196. So Area = (1/4) * π * 196. In aptitude questions, take π = 22/7. Then Area = (1/4) * (22/7) * 196. First simplify (22/7) * 196: 196 / 7 = 28, and 28 * 22 = 616. So Area = (1/4) * 616 = 616 / 4. Finally, 616 / 4 = 154. Thus, the area of the sector is 154 square centimetres. Verification / Alternative check: We can also find the full area of the circle and then take a quarter. Full circle area = πr^2 = (22/7) * 196 = 616 sq cm. Since 90° is a quarter of 360°, the sector area is 616 / 4 = 154 sq cm, matching our previous calculation. This confirms that our use of the sector formula and arithmetic are correct. Why Other Options Are Wrong: 308 sq cm and 231 sq cm correspond to other fractional parts of the circle or incorrect arithmetic with π and r^2. 77 sq cm is half of 154, which would be the area of a 45° sector, not a 90° one. 100 sq cm does not correspond to any simple fraction of 616 and arises only from miscalculations in either squaring the radius or dividing by 4. Common Pitfalls: Typical mistakes include forgetting to square the radius, using the wrong fraction θ / 360 (for example, 90 / 180 instead of 90 / 360), or incorrectly simplifying (22/7) * 196. Some students also mistakenly compute the circumference formula instead of the area formula. Carefully applying the correct formula and simplifying step by step prevents these errors. Final Answer: The area of the sector is 154 sq cm.

Question 16

In trigonometry, the triple-angle identity rewrites cos(3A) using only cos(A). Which of the following expressions is the correct identity for cos 3A in terms of cos A?

Accepted Answer

Introduction / Context: This question tests a standard trigonometric identity: the triple-angle formula for cosine. Triple-angle identities are used to simplify expressions, solve trigonometric equations, and convert higher-angle terms like cos(3A) into powers of cos(A). Given Data / Assumptions: We need an identity for cos(3A) written only in terms of cos(A). A is any real angle (identity must hold for all A). Concept / Approach: The known triple-angle identity for cosine is derived from angle addition: cos(3A) = cos(2A + A) and then applying cos(2A) and sin(2A) formulas. The final form becomes a cubic polynomial in cos(A). Step-by-Step Solution: Start with cos(3A) = cos(2A + A) Use cos(u+v) = cos u * cos v - sin u * sin v So cos(3A) = cos(2A)*cos(A) - sin(2A)*sin(A) Replace cos(2A) = 2cos^2(A) - 1 and sin(2A) = 2sin(A)cos(A) Then cos(3A) = (2cos^2(A) - 1)cos(A) - (2sin(A)cos(A))sin(A) = 2cos^3(A) - cos(A) - 2sin^2(A)cos(A) Use sin^2(A) = 1 - cos^2(A) = 2cos^3(A) - cos(A) - 2(1 - cos^2(A))cos(A) = 2cos^3(A) - cos(A) - 2cos(A) + 2cos^3(A) = 4cos^3(A) - 3cos(A) Verification / Alternative check: Check A = 0: LHS cos(0) = 1. RHS 4*1 - 3*1 = 1, so it matches. This quick check supports the identity choice. Why Other Options Are Wrong: 3 cos A - 4 cos^3 A is just the negative of the correct expression. 4 cos^3 A + 3 cos A has the wrong sign on the linear term and fails at A = 0. cos A + 4 cos^3 A and 4 cos A - 3 cos^3 A do not match the known cubic structure and fail simple substitutions. Common Pitfalls: Mixing up the signs, or confusing the cosine triple-angle identity with the sine triple-angle identity (sin 3A). Another common error is writing cos(3A) as 3cos(A) - 4cos^3(A) without noticing it is the negative. Final Answer: cos(3A) = 4 cos^3 A - 3 cos A

Question 17

In right-triangle trigonometry, cosec X = 1/sin X relates an angle to the ratio of hypotenuse and opposite side. In triangle XYZ, angle Y = 90 degrees. If cosec X = 13/12 and XY = 1 cm, what is the length of side YZ (in cm)?

Accepted Answer

Introduction / Context: This problem tests how to convert a trigonometric ratio (cosec) into side ratios in a right-angled triangle and then scale those ratios using a given side length. It combines knowledge of cosec X, the definition of sine, and the idea of similar right-triangle side triples. Given Data / Assumptions: Triangle XYZ is right-angled at Y (angle Y = 90 degrees). cosec X = 13/12, so sin X = 12/13. XY = 1 cm. For angle X: opposite side is YZ, adjacent side is XY, hypotenuse is XZ. Concept / Approach: Use cosec X = hypotenuse/opposite. From cosec X = 13/12, we get XZ:YZ = 13:12. A right triangle with sides in the ratio 5:12:13 is a standard Pythagorean triple, meaning the adjacent side to angle X is 5 parts when hypotenuse is 13 parts and opposite is 12 parts. Step-by-Step Solution: cosec X = 13/12 means XZ / YZ = 13/12 So let XZ = 13k and YZ = 12k for some scale factor k Because angle Y is 90 degrees, Pythagoras applies: XY^2 + YZ^2 = XZ^2 XY^2 = XZ^2 - YZ^2 = (13k)^2 - (12k)^2 = 169k^2 - 144k^2 = 25k^2 So XY = 5k Given XY = 1 cm, we get 5k = 1 => k = 0.2 Then YZ = 12k = 12 * 0.2 = 2.4 cm Verification / Alternative check: Compute hypotenuse: XZ = 13k = 2.6 cm. Check: 1^2 + 2.4^2 = 1 + 5.76 = 6.76 and 2.6^2 = 6.76, so the triangle is consistent. Why Other Options Are Wrong: 2.0 and 1.8 are too small; they would not keep the 12:13 sin ratio with XY = 1. 2.6 confuses YZ with the hypotenuse XZ. 1.5 corresponds to a different side ratio and fails the Pythagoras check with hypotenuse 2.6. Common Pitfalls: Mixing up which side is opposite angle X (it is YZ), or treating XY as the opposite side. Another common mistake is assuming cosec is opposite/hypotenuse instead of hypotenuse/opposite. Final Answer: YZ = 2.4 cm

Question 18

In geometry, the circumradius of a right-angled triangle equals half the length of its hypotenuse. Triangle XYZ is right-angled at Y (angle Y = 90 degrees). If XY = 2.5 cm and YZ = 6 cm, what is the circumradius of triangle XYZ (in cm)?

Accepted Answer

Introduction / Context: This question checks a key circle-geometry fact: for a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse, so the circumradius is exactly half of the hypotenuse. The only computation needed is the hypotenuse length using Pythagoras. Given Data / Assumptions: Triangle XYZ is right-angled at Y (angle Y = 90 degrees). Legs: XY = 2.5 cm and YZ = 6 cm. Hypotenuse: XZ. Circumradius R = (hypotenuse)/2 for a right triangle. Concept / Approach: First find the hypotenuse using Pythagoras: XZ = sqrt(XY^2 + YZ^2). Then apply the right-triangle circumradius property: R = XZ/2. This property is a standard result because the diameter of the circumcircle of a right triangle is the hypotenuse (Thales' theorem). Step-by-Step Solution: Compute the hypotenuse: XZ^2 = XY^2 + YZ^2 XZ^2 = (2.5)^2 + (6)^2 = 6.25 + 36 = 42.25 XZ = sqrt(42.25) = 6.5 cm For a right triangle, circumradius R = XZ / 2 R = 6.5 / 2 = 3.25 cm Verification / Alternative check: If the circumcircle diameter is the hypotenuse, then diameter = 6.5 cm and radius must be 3.25 cm. This directly matches the computed value. Why Other Options Are Wrong: 6.5 cm is the hypotenuse itself (diameter), not the radius. 4 cm and 3 cm come from incorrect halving or rounding of the hypotenuse. 2.5 cm incorrectly uses one leg as the radius. Common Pitfalls: Forgetting that the special circumradius rule applies only when the triangle is right-angled, or halving the wrong side (a leg instead of the hypotenuse). Another mistake is squaring incorrectly when computing the hypotenuse. Final Answer: The circumradius is 3.25 cm

Question 19

For a non-zero real number x, expressions like x + 1/x can be used to find higher powers without solving for x directly. If x + 1/x = 6, what is the value of x^2 + 1/x^2?

Accepted Answer

Introduction / Context: This problem tests an algebraic identity technique: using (x + 1/x)^2 to quickly obtain x^2 + 1/x^2. This avoids solving a quadratic for x and is a common simplification shortcut in aptitude and algebra questions. Given Data / Assumptions: x is a non-zero real number. x + 1/x = 6. We must find x^2 + 1/x^2. Concept / Approach: Square the given expression. The identity is:(x + 1/x)^2 = x^2 + 2 + 1/x^2.Then rearrange to isolate x^2 + 1/x^2. This works because the cross term is always 2 when multiplying x by 1/x. Step-by-Step Solution: Given: x + 1/x = 6 Square both sides: (x + 1/x)^2 = 6^2 Left side expansion: (x + 1/x)^2 = x^2 + 2*(x*(1/x)) + 1/x^2 Since x*(1/x) = 1, the middle term becomes 2 So: x^2 + 2 + 1/x^2 = 36 Rearrange: x^2 + 1/x^2 = 36 - 2 = 34 Verification / Alternative check: If desired, you could solve x + 1/x = 6 => x^2 - 6x + 1 = 0 and use the solutions. Both solutions will produce the same value for x^2 + 1/x^2, confirming the identity-based result. Why Other Options Are Wrong: 36 is the value of (x + 1/x)^2, not x^2 + 1/x^2. 32 is a common mistake from subtracting 4 instead of 2. 23 and 16 do not match the correct rearrangement and are inconsistent with the squared identity. Common Pitfalls: Forgetting the cross term 2, or incorrectly writing (x + 1/x)^2 = x^2 + 1/x^2. Another mistake is treating x*(1/x) as x rather than 1. Final Answer: x^2 + 1/x^2 = 34

Question 20

Solve the linear equation involving fractions carefully by clearing denominators. Find the value of x that satisfies: 14/3 + (1/2)(x - 7/3) = -2x/3.

Accepted Answer

Introduction / Context: This question tests fraction-handling in linear equations. The core skill is to simplify bracketed fractional expressions and then eliminate denominators (by multiplying by a common multiple) to avoid arithmetic errors. Once simplified, it becomes a straightforward one-variable equation. Given Data / Assumptions: Equation: 14/3 + (1/2)(x - 7/3) = -2x/3 x is a real number. Goal: solve for x exactly. Concept / Approach: Distribute (1/2) across the bracket, convert all terms to a common denominator, and then multiply both sides by the LCM of denominators (here 6) to convert to integers. This prevents mistakes from repeated fraction operations. Step-by-Step Solution: Start: 14/3 + (1/2)(x - 7/3) = -2x/3 Distribute 1/2: (1/2)(x - 7/3) = x/2 - 7/6 Rewrite 14/3 as /6: 14/3 = 28/6 Left side becomes: 28/6 + x/2 - 7/6 = (21/6) + x/2 So: 7/2 + x/2 = -2x/3 Multiply both sides by 6: 6*(7/2 + x/2) = 6*(-2x/3) Left: 6*(7/2) = 21 and 6*(x/2) = 3x, so 21 + 3x Right: 6*(-2x/3) = -4x Equation: 21 + 3x = -4x Add 4x: 21 + 7x = 0 So 7x = -21 => x = -3 Verification / Alternative check: Substitute x = -3 into the original: (1/2)(-3 - 7/3) = (1/2)(-16/3) = -8/3. Then 14/3 - 8/3 = 6/3 = 2, and RHS is -2*(-3)/3 = 2. Both sides match. Why Other Options Are Wrong: -6 and 6 come from sign or distribution errors. 0 and 3 typically result from incorrect handling of the negative RHS or combining fractions incorrectly. Common Pitfalls: Forgetting to distribute 1/2 to both terms, or using an incorrect LCM when clearing denominators. Another frequent mistake is dropping the negative sign in -2x/3. Final Answer: x = -3

Simplification Questions