Question 1

If x = a cos θ cos φ, y = a cos θ sin φ and z = a sin θ, then what is the simplified value of x² + y² + z² in terms of a?

Accepted Answer

Introduction / Context: This question comes from coordinate geometry and trigonometry, where variables are expressed in terms of angles and a common parameter a. The expression x² + y² + z² often represents the square of a distance in three dimensional space. The task is to simplify this sum using trigonometric identities. Given Data / Assumptions: x = a cos θ cos φ. y = a cos θ sin φ. z = a sin θ. We want to find x² + y² + z² in terms of a only. Concept / Approach: Important ideas: Square each of x, y and z and add the results. Use the identity cos²φ + sin²φ = 1. Use the identity sin²θ + cos²θ = 1. Factor out common constants and terms like a² to simplify. Step-by-Step Solution: Step 1: Compute x²: x² = (a cos θ cos φ)² = a² cos²θ cos²φ. Step 2: Compute y²: y² = (a cos θ sin φ)² = a² cos²θ sin²φ. Step 3: Compute z²: z² = (a sin θ)² = a² sin²θ. Step 4: Add them: x² + y² + z² = a² cos²θ cos²φ + a² cos²θ sin²φ + a² sin²θ. Step 5: Factor out a²: x² + y² + z² = a²[cos²θ cos²φ + cos²θ sin²φ + sin²θ]. Step 6: Inside the bracket, factor cos²θ: cos²θ(cos²φ + sin²φ) + sin²θ. Step 7: Use cos²φ + sin²φ = 1, giving cos²θ × 1 + sin²θ = cos²θ + sin²θ. Step 8: Apply sin²θ + cos²θ = 1 to get x² + y² + z² = a² × 1 = a². Verification / Alternative check: Step 1: Choose specific angles, for example θ = 0 and φ = 0, and a = 1. Step 2: Then x = 1 × cos 0 × cos 0 = 1, y = 1 × cos 0 × sin 0 = 0, z = 1 × sin 0 = 0. Step 3: Compute x² + y² + z² = 1² + 0² + 0² = 1, which equals a² when a = 1. Step 4: Trying other values of θ and φ numerically will also give x² + y² + z² = a², reinforcing the algebraic result. Why Other Options Are Wrong: Option 2a² or 4a² would require the sum inside the bracket to be 2 or 4 instead of 1, which contradicts the identities used. Option 0 would imply x² + y² + z² is always zero, which is impossible unless a is zero. Option a has incorrect units, since squaring variables produces a quantity proportional to a², not a. Common Pitfalls: Forgetting to factor out a² early often leads to more complicated algebra than necessary. Some learners misapply cos²φ + sin²φ = 1 and try to combine it incorrectly with θ instead of φ. It is also common to overlook that cos²θ + sin²θ = 1 regardless of θ, as long as θ is a real angle. Final Answer: The simplified value of x² + y² + z² is a².

Question 2

A 1.6 m tall observer stands 45 m away from the foot of a vertical tower. If the angle of elevation from the observer's eye level to the top of the tower is 30°, what is the height of the tower in metres?

Accepted Answer

Introduction / Context: This is a standard height and distance problem in trigonometry. It involves using the tangent of an angle of elevation to relate the vertical height difference and the horizontal distance. The observer's eye level must be included to obtain the total height of the tower, a detail that is often overlooked by students. Given Data / Assumptions: Height of observer = 1.6 m. Horizontal distance between observer and tower = 45 m. Angle of elevation from eye level to top of tower = 30°. The tower is vertical and the ground is level. We use tan 30° = 1/√3 and approximate √3 ≈ 1.732 when needed. Concept / Approach: Key ideas: In a right triangle, tan θ = opposite side / adjacent side. The opposite side here is the vertical height difference between the top of the tower and the observer's eye level. The adjacent side is the horizontal distance from the observer to the base of the tower, which is 45 m. Add the observer's own height to this vertical difference to get the full height of the tower. Step-by-Step Solution: Step 1: Let H be the total height of the tower in metres. Step 2: The vertical height difference between the top of the tower and the observer's eye level is H − 1.6. Step 3: Using tan 30° = opposite / adjacent = (H − 1.6) / 45. Step 4: Since tan 30° = 1/√3, we have (H − 1.6) / 45 = 1 / √3. Step 5: Rearrange to get H − 1.6 = 45 / √3. Step 6: Simplify 45 / √3 by rationalising if desired: 45 / √3 = (45√3) / 3 = 15√3. Step 7: Approximate √3 ≈ 1.732, so H − 1.6 ≈ 15 × 1.732 = 25.98. Step 8: Therefore H ≈ 25.98 + 1.6 = 27.58 metres. Verification / Alternative check: Step 1: Take H = 27.58 m and recompute the angle of elevation using tan θ = (H − 1.6) / 45. Step 2: Then H − 1.6 ≈ 25.98, so tan θ ≈ 25.98 / 45 ≈ 0.5773. Step 3: tan 30° is exactly 1/√3 ≈ 0.5773, so the computed value is consistent with a 30° angle. Why Other Options Are Wrong: Option 25.98 corresponds only to the vertical rise above the observer's eye and ignores the 1.6 m eye level height. Option 26.58 is too small and does not match the precise calculation using tan 30°. Option 27.98 is slightly larger than the correct sum and would give a tangent less than 1/√3. Option 29.58 is much larger and clearly inconsistent with the trigonometric relationship. Common Pitfalls: A very common mistake is forgetting to add the observer's height to the vertical difference, leading to the answer 25.98 instead of the correct total height. Some students use sin or cos instead of tan, which does not fit this right triangle configuration. Rounding errors or using an incorrect approximate value for √3 can slightly shift the final decimal, so careful approximation is important. Final Answer: The height of the tower is approximately 27.58 metres.

Question 3

A solid spherical lead ball of radius 6 cm is melted and recast into smaller solid spherical lead balls, each of radius 3 mm. How many such small balls can be formed in total?

Accepted Answer

Introduction / Context: This problem is a classic example of volume conservation in solid geometry. A large sphere is melted and recast into several smaller spheres of a different radius. The total volume of material remains the same, so the number of smaller spheres is found by dividing the volume of the large sphere by the volume of one small sphere. Given Data / Assumptions: Radius of the large spherical lead ball R = 6 cm. Radius of each small spherical ball r = 3 mm. 1 cm = 10 mm, so 3 mm = 0.3 cm. Material is conserved: total volume of large sphere = sum of volumes of all small spheres. Volume of a sphere with radius r is (4/3)πr³. Concept / Approach: Key steps: Convert all measurements to the same unit, here centimetres. Write the volume of the large sphere using radius 6 cm. Write the volume of a small sphere using radius 0.3 cm. Divide the larger volume by the smaller volume to get the number of small spheres. Cancel common factors and avoid using π explicitly where it cancels. Step-by-Step Solution: Step 1: Volume of the large sphere V₁ = (4/3)πR³ = (4/3)π(6)³. Step 2: Compute 6³ = 216, so V₁ = (4/3)π × 216 = 288π cubic centimetres. Step 3: Convert the small radius to centimetres: r = 3 mm = 0.3 cm. Step 4: Volume of a small sphere V₂ = (4/3)πr³ = (4/3)π(0.3)³. Step 5: Compute 0.3³ = 0.027, so V₂ = (4/3)π × 0.027 = 0.036π cubic centimetres. Step 6: Let n be the number of small spheres. Then n × V₂ = V₁, so n = V₁ / V₂ = 288π / 0.036π. Step 7: Cancel π: n = 288 / 0.036. Step 8: Compute the division: 0.036 = 36/1000, so 288 / 0.036 = 288 × (1000/36) = (288/36) × 1000 = 8 × 1000 = 8000. Verification / Alternative check: Step 1: Note that the ratio of radii is R / r = 6 / 0.3 = 20. Step 2: The number of small spheres is the cube of the ratio of radii: (R / r)³ = 20³ = 8000. Step 3: This quick method confirms the detailed volume calculation. Why Other Options Are Wrong: Option 4000 corresponds to half the correct value and would imply the radius ratio cubed is 10³ instead of 20³. Option 4250 or 8005 are arbitrary and not linked to the exact volume ratio. Option 6000 also does not arise from any correct combination of the radius ratio or volume formulas. Common Pitfalls: Using 3 mm directly without converting to centimetres leads to incorrect radius ratio and wrong volume values. Some students mistakenly take the ratio of areas or linear dimensions instead of the ratio of volumes, forgetting that volume scales with the cube of the radius. Another error is to approximate π differently in numerator and denominator instead of recognising that it cancels out completely. Final Answer: The total number of small spherical balls that can be formed is 8000.

Question 4

If x² + 5x + 6 = 0, what is the value of the rational expression 2x / (x² − 7x + 6)?

Accepted Answer

Introduction / Context: This algebra problem requires you to use the fact that x satisfies a given quadratic equation to simplify a rational expression involving x. Instead of solving for x and substituting both roots separately, you can use factorisation and algebraic manipulation to show that the expression has the same constant value for both roots. Given Data / Assumptions: x² + 5x + 6 = 0. We need to compute 2x / (x² − 7x + 6). x is a root of the quadratic equation, so it satisfies the relation x² = −5x − 6. Concept / Approach: Key ideas: Factorise the quadratic x² + 5x + 6 to find its roots or to express x² in terms of x. Simplify the denominator x² − 7x + 6 by factorisation. Substitute the values of x corresponding to the roots and verify that the expression gives the same result for both, confirming that it is constant. Step-by-Step Solution: Step 1: Factorise the quadratic: x² + 5x + 6 = (x + 2)(x + 3) = 0. Step 2: The roots are x = −2 and x = −3. Step 3: Factorise the denominator: x² − 7x + 6 = (x − 1)(x − 6). Step 4: Evaluate the expression for x = −2: numerator 2x = 2(−2) = −4. Step 5: Denominator for x = −2: (−2)² − 7(−2) + 6 = 4 + 14 + 6 = 24. Step 6: So for x = −2, the expression equals −4 / 24 = −1/6. Step 7: Evaluate the expression for x = −3: numerator 2x = 2(−3) = −6. Step 8: Denominator for x = −3: 9 − 7(−3) + 6 = 9 + 21 + 6 = 36, so the expression is −6 / 36 = −1/6 again. Step 9: Since both roots give the same value, the expression equals −1/6 whenever x satisfies the quadratic. Verification / Alternative check: Step 1: Use the relation from the quadratic: x² = −5x − 6. Step 2: Substitute in the denominator: x² − 7x + 6 = (−5x − 6) − 7x + 6 = −12x. Step 3: So the expression becomes 2x / (−12x) = −1/6 for any x that also makes the original quadratic zero and x ≠ 0. Step 4: Neither root is zero, so this simplification is valid, confirming the value −1/6. Why Other Options Are Wrong: Options 1/6 and 1/3 have the wrong sign and do not match the computed value. Option −1/3 would appear if the denominator were −6x instead of −12x. Option 0 would require the numerator to be zero, which is not true for either root of the quadratic. Common Pitfalls: Some learners incorrectly cancel x from numerator and denominator before using the quadratic relation, which is only safe if x is known to be nonzero and the simplification is valid. Another common error is to substitute only one root and mistakenly assume the expression is equal only for that root, without checking the second root for consistency. Misfactorising either the quadratic or the denominator can lead to incorrect numeric results. Final Answer: The value of 2x / (x² − 7x + 6) is −1/6 for any x satisfying x² + 5x + 6 = 0.

Question 5

If a + b = 5 and ab = 3 for two real numbers a and b, what is the exact value of a² + b²?

Accepted Answer

Introduction / Context: This algebra problem focuses on symmetric expressions in two variables, namely sums and products of a and b. Given a + b and ab, we can find a² + b² without solving for a and b individually by using the formula for the square of a binomial. This kind of manipulation is common in quadratic equations and identities. Given Data / Assumptions: a + b = 5. ab = 3. We are asked to find a² + b². Concept / Approach: Key concepts: Use the identity (a + b)² = a² + 2ab + b². Rearrange this identity to express a² + b² in terms of a + b and ab. Substitute the given numerical values to compute the result directly. Step-by-Step Solution: Step 1: Recall the identity (a + b)² = a² + 2ab + b². Step 2: Rearrange to express a² + b²: a² + b² = (a + b)² − 2ab. Step 3: Substitute a + b = 5: (a + b)² = 5² = 25. Step 4: Substitute ab = 3: 2ab = 2 × 3 = 6. Step 5: Compute a² + b² = 25 − 6 = 19. Verification / Alternative check: Step 1: One can construct the quadratic equation whose roots are a and b: t² − (a + b)t + ab = 0, so t² − 5t + 3 = 0. Step 2: Solve this quadratic to find explicit values of a and b using the quadratic formula. Step 3: After obtaining numeric values for a and b, compute a² + b² directly, which will simplify numerically to 19, confirming the identity based approach. Why Other Options Are Wrong: Option 17 would result from incorrectly computing 25 − 8 instead of 25 − 6. Option 18 may appear if someone uses 2ab = 7 by mistake or misreads the problem data. Option 20 could result from adding instead of subtracting, that is, taking 25 + (−5) incorrectly. Option 16 does not correspond to any correct manipulation of the given relationships. Common Pitfalls: Students sometimes misremember the identity and write (a + b)² = a² + b² instead of including the 2ab term. Another error is to confuse ab with a²b² or to square ab accidentally while manipulating. Some learners try to solve for a and b directly, which is unnecessary work when symmetric identities give the answer quickly. Final Answer: The value of a² + b² is 19.

Question 6

From the centre of a circle, a perpendicular of length 8 cm is drawn to a chord of length 12 cm. What is the diameter of the circle in centimetres?

Accepted Answer

Introduction / Context: This geometry question uses properties of chords in a circle. When a radius or line from the centre is drawn perpendicular to a chord, it bisects the chord. This fact, combined with Pythagoras theorem, allows us to determine the radius and hence the diameter of the circle from the chord length and the perpendicular distance. Given Data / Assumptions: Length of the chord AB is 12 cm. Perpendicular from the centre O to the chord has length OM = 8 cm. M is the midpoint of chord AB because OM is perpendicular from the centre. We need the diameter, which is 2 times the radius. Concept / Approach: Key concepts: A perpendicular from the centre of a circle to a chord bisects the chord. In right triangle OMA, OA is the radius, OM is one leg and AM is half of the chord. Use Pythagoras theorem: OA² = OM² + AM². Once the radius is found, multiply by 2 to get the diameter. Step-by-Step Solution: Step 1: Let AB be the chord and O be the centre of the circle. Step 2: Since OM is perpendicular to AB, point M is the midpoint of AB and AM = MB = 12 / 2 = 6 cm. Step 3: Consider the right triangle OMA with OM = 8 cm and AM = 6 cm. Step 4: Let the radius R = OA. Apply Pythagoras theorem: R² = OM² + AM². Step 5: Compute R² = 8² + 6² = 64 + 36 = 100. Step 6: Therefore R = √100 = 10 cm. Step 7: The diameter D = 2R = 2 × 10 = 20 cm. Verification / Alternative check: Step 1: If the diameter is 20 cm, the radius is 10 cm. Recompute the perpendicular distance using OM² = R² − AM². Step 2: OM² = 10² − 6² = 100 − 36 = 64, so OM = 8 cm, which matches the given perpendicular length. Step 3: This confirms that the radius and hence the diameter are correctly calculated. Why Other Options Are Wrong: Option 10 cm is the radius, not the diameter, and misses the final step. Options 12 cm, 16 cm and 18 cm do not satisfy Pythagoras theorem with legs 8 cm and 6 cm when taken as diameters. For example, a diameter of 16 cm would give a radius of 8 cm, where R² = 64, which is less than OM² + AM² = 100. Common Pitfalls: Some learners forget that the perpendicular from the centre bisects the chord and mistakenly use the full chord length instead of half. Another frequent mistake is to stop after computing the radius and not converting it to the diameter as required. Incorrect application of Pythagoras theorem, such as using R = OM − AM, leads to wrong results. Final Answer: The diameter of the circle is 20 cm.

Question 7

If x = a + 1/a and y = a − 1/a for a ≠ 0, what is the value of the expression x⁴ + y⁴ − 2x²y²?

Accepted Answer

Introduction / Context: This algebra problem involves symmetric expressions and powers of two derived variables x and y. The given definitions of x and y in terms of a and 1/a suggest using identities for squares and differences to simplify the expression x⁴ + y⁴ − 2x²y² without expanding everything fully. Given Data / Assumptions: x = a + 1/a. y = a − 1/a. a is nonzero. We want to find x⁴ + y⁴ − 2x²y². Concept / Approach: Main ideas: Observe that x⁴ + y⁴ − 2x²y² can be written as (x² − y²)². Compute x² and y² separately using binomial expansion. Find x² − y², then square it to obtain the final value. This avoids expanding the fourth powers directly. Step-by-Step Solution: Step 1: Note that x⁴ + y⁴ − 2x²y² = (x² − y²)², which is a standard identity for the square of a difference. Step 2: Compute x² = (a + 1/a)² = a² + 2 + 1/a². Step 3: Compute y² = (a − 1/a)² = a² − 2 + 1/a². Step 4: Find x² − y² = (a² + 2 + 1/a²) − (a² − 2 + 1/a²). Step 5: Simplify: x² − y² = a² + 2 + 1/a² − a² + 2 − 1/a² = 4. Step 6: Now compute (x² − y²)² = 4² = 16. Step 7: Therefore x⁴ + y⁴ − 2x²y² = 16. Verification / Alternative check: Step 1: Choose a specific nonzero value for a, for example a = 2. Step 2: Then x = 2 + 1/2 = 2.5 and y = 2 − 1/2 = 1.5. Step 3: Compute x² ≈ 6.25 and y² ≈ 2.25. Step 4: Compute x⁴ + y⁴ − 2x²y² numerically and you will find that the result is very close to 16, confirming the algebraic solution. Why Other Options Are Wrong: Option 4 is only the value of x² − y², not its square. Options 8 and 64 arise from incorrect squaring or misinterpreting intermediate results. Option 0 would require x² = y², which clearly is not the case because x and y are different when a is nonzero. Common Pitfalls: Learners sometimes try to expand x⁴ and y⁴ directly, which is lengthy and increases the chance of arithmetic mistakes. Another mistake is to misapply the identity for (x² − y²)² or to forget that x⁴ + y⁴ − 2x²y² has that special form. Students may also incorrectly expand (a ± 1/a)², missing the middle term or miscomputing 2(a)(1/a). Final Answer: The value of x⁴ + y⁴ − 2x²y² is 16.

Question 8

If 2^(x + 4) − 2^(x + 2) = 3, what is the value of x?

Accepted Answer

Introduction / Context: This is an exponential equation involving powers of 2. The exponents differ by a constant, which suggests factoring out a common power of 2. Solving such equations efficiently is a key skill in algebra and aptitude tests. Given Data / Assumptions: Equation: 2^(x + 4) − 2^(x + 2) = 3. Base of the exponents is 2, a positive number. We need to solve for the real value of x. Concept / Approach: Important ideas: Factor out the smaller power of 2 from both terms to simplify the equation. Use exponent rules: 2^(x + 4) = 2^(x + 2) · 2². Transform the equation into a simple linear equation in terms of 2^(x + 2). Recognise that 2^k = 1 only when k = 0, which helps solve for x. Step-by-Step Solution: Step 1: Rewrite 2^(x + 4) as 2^(x + 2) × 2² = 4 · 2^(x + 2). Step 2: Substitute into the equation: 4 · 2^(x + 2) − 2^(x + 2) = 3. Step 3: Factor out 2^(x + 2): 2^(x + 2)(4 − 1) = 3. Step 4: Simplify the bracket: 4 − 1 = 3, so 2^(x + 2) × 3 = 3. Step 5: Divide both sides by 3: 2^(x + 2) = 1. Step 6: The equation 2^k = 1 holds only when k = 0 because base 2 is not 1. Step 7: Therefore x + 2 = 0 and x = −2. Verification / Alternative check: Step 1: Substitute x = −2 into the original equation. Step 2: Compute 2^(x + 4) = 2^2 = 4 and 2^(x + 2) = 2^0 = 1. Step 3: Evaluate 4 − 1 = 3, which matches the right side of the equation, confirming x = −2 is correct. Why Other Options Are Wrong: Option 0 would give 2^4 − 2^2 = 16 − 4 = 12, not 3. Option 2 yields 2^6 − 2^4 = 64 − 16 = 48, which is far from 3. Option −1 gives 2^3 − 2^1 = 8 − 2 = 6, not 3. Option 1 gives 2^5 − 2^3 = 32 − 8 = 24, again incorrect. Common Pitfalls: A common mistake is to treat 2^(x + 4) − 2^(x + 2) as 2^(2x + 6) or combine exponents incorrectly. Some learners try to take logarithms prematurely rather than factoring, which makes the problem unnecessarily complicated. Another error is to forget that 2^k = 1 only for k = 0, not for other values. Final Answer: The solution of the equation is x = −2.

Question 9

If x + 1/x = 5 for a nonzero real number x, what is the value of the expression 6x / (x² + x + 1)?

Accepted Answer

Introduction / Context: This algebra question uses a relation between x and 1/x to simplify another rational expression in x. Instead of solving a quadratic equation for x, we can manipulate the given identity to rewrite the denominator, leading to a quick simplification. These techniques are very useful in aptitude exams to save time. Given Data / Assumptions: x + 1/x = 5. x ≠ 0, since 1/x exists. We need to compute 6x / (x² + x + 1). Concept / Approach: Main ideas: From x + 1/x = 5, derive a relationship involving x² and 1. Try to express the denominator x² + x + 1 in terms of x and the known relation. Once the denominator is expressed as a multiple of x, cancel with the numerator 6x to simplify. Step-by-Step Solution: Step 1: Start with the given equation: x + 1/x = 5. Step 2: Multiply both sides by x to clear the fraction: x² + 1 = 5x. Step 3: Rearrange to get x² − 5x + 1 = 0, which is a quadratic that x satisfies. Step 4: We want to simplify x² + x + 1. From x² + 1 = 5x, add x to both sides to get x² + x + 1 = 5x + x = 6x. Step 5: Substitute this into the expression: 6x / (x² + x + 1) = 6x / (6x). Step 6: Since x ≠ 0, 6x cancels, giving 6x / 6x = 1. Verification / Alternative check: Step 1: Solve the quadratic x² − 5x + 1 = 0 using the quadratic formula if desired. Step 2: For each root obtained, compute x + 1/x and confirm it equals 5. Step 3: Substitute those numeric values of x back into 6x / (x² + x + 1) and verify that the result is 1 in both cases, confirming that the expression is constant for the given condition. Why Other Options Are Wrong: Options 3 and 2 would result only if the denominator were 2x or 3x rather than 6x. Option 0 would require the numerator to be zero, which is not true since x ≠ 0. Option 5 is simply the value of x + 1/x and has no direct connection to the simplified expression after substitution. Common Pitfalls: Some students incorrectly treat x² + x + 1 as x(x + 1) + 1 and then attempt unrelated factoring rather than using the relation x² + 1 = 5x. Another mistake is to solve for x immediately, which is not necessary and consumes more time. Careless algebraic manipulation when adding x to both sides can also lead to errors such as writing x² + x + 1 = 5x − x. Final Answer: The value of 6x / (x² + x + 1) is 1.

Question 10

A man's expenditure for 8 months is equal to his income for 6 months. Over one full year he saves a total of ₹6000. What is his average monthly income in rupees?

Accepted Answer

Introduction / Context: This question involves a typical income and expenditure scenario based on simple arithmetic and ratio reasoning. The man spends in eight months what he earns in six months, and we are told his annual savings. By relating income and expenditure per month, we can determine his monthly income. Given Data / Assumptions: Let monthly income be I rupees. Let monthly expenditure be E rupees. Total expenditure for 8 months equals total income for 6 months: 8E = 6I. Total savings in 12 months = ₹6000. Saving in each month is I − E. Concept / Approach: Key ideas: Use the relation 8E = 6I to express E in terms of I. Compute monthly saving S = I − E using this relationship. Total annual saving is 12S, which is given as 6000, so solve for I. This leads to a straightforward linear equation in I. Step-by-Step Solution: Step 1: From 8E = 6I, divide both sides by 2 to get 4E = 3I. Step 2: Express E in terms of I: E = (3/4)I. Step 3: Monthly saving S = I − E = I − (3/4)I = (1/4)I. Step 4: Total saving for 12 months is 12S = 12 × (1/4)I = 3I. Step 5: We are told that 3I = 6000. Step 6: Solve for I: I = 6000 / 3 = 2000. Step 7: Therefore the average monthly income is ₹2000. Verification / Alternative check: Step 1: With income I = 2000, expenditure E = (3/4) × 2000 = 1500 per month. Step 2: Income in 6 months = 6 × 2000 = 12000. Expenditure in 8 months = 8 × 1500 = 12000, which matches the condition. Step 3: Monthly saving S = 2000 − 1500 = 500. Annual saving over 12 months is 12 × 500 = 6000, which matches the given figure. Why Other Options Are Wrong: Option Rs. 2400 would give E = 1800, so annual saving becomes 12 × 600 = 7200, not 6000. Option Rs. 1800 would lead to a lower saving and would not satisfy both 8E = 6I and the given annual saving. Option Rs. 2150 or Rs. 3000 similarly do not satisfy both the ratio condition and the exact annual saving amount. Common Pitfalls: Students sometimes misinterpret the statement and set 8I = 6E instead of 8E = 6I, reversing the relationship. Another mistake is to forget that savings accumulate over 12 months, not just 6 or 8 months. Careless manipulation of fractions, especially when computing E = (3/4)I, can lead to slight numerical errors. Final Answer: The man's average monthly income is Rs. 2000.

Question 11

If a = √2 + 1 and b = √2 − 1 in basic surd manipulation, what is the exact simplified value of the expression 1/(a + 1) + 1/(b + 1)?

Accepted Answer

Introduction / Context: This question checks understanding of surds, rationalisation, and algebraic simplification using conjugate based ideas. Such questions are common in aptitude exams to test manipulation of radicals without a calculator. Given Data / Assumptions: a = √2 + 1 b = √2 − 1 Required expression is 1/(a + 1) + 1/(b + 1) All quantities are real numbers Concept / Approach: We simplify each denominator first, then add the two fractions. The key idea is that √2 + 1 and √2 − 1 are conjugate like expressions. Adding 1 to each makes simple surds that can be handled directly. We then compute a common denominator and simplify carefully. No special formulas beyond basic fraction addition are required. Step-by-Step Solution: a + 1 = (√2 + 1) + 1 = √2 + 2 b + 1 = (√2 − 1) + 1 = √2 So 1/(a + 1) = 1/(√2 + 2) And 1/(b + 1) = 1/√2 Required sum S = 1/(√2 + 2) + 1/√2 Take common denominator √2(√2 + 2) S = [√2 + (√2 + 2)] / [√2(√2 + 2)] S = [√2 + √2 + 2] / [√2(√2 + 2)] = [2√2 + 2] / [√2(√2 + 2)] Factor numerator: 2(√2 + 1) Denominator: √2(√2 + 2) = √2 * √2 + √2 * 2 = 2 + 2√2 = 2(1 + √2) So S = 2(√2 + 1) / [2(1 + √2)] = 1 Verification / Alternative check: We can approximate numerically. √2 ≈ 1.414. Then a ≈ 2.414 and b ≈ 0.414. So a + 1 ≈ 3.414 and b + 1 ≈ 1.414. Then 1/(a + 1) ≈ 0.293 and 1/(b + 1) ≈ 0.707. Their sum is approximately 1.0 which matches the exact result. Why Other Options Are Wrong: 2 is too large and would require each term to be roughly 1 which is not true. 0 would require the two fractions to cancel exactly which they do not. −1 contradicts the clearly positive nature of both fractions. 3 is far too large compared to the approximate numerical sum which is near 1. Common Pitfalls: Learners sometimes try to rationalise each surd separately, creating unnecessary steps. Another common mistake is using an incorrect common denominator while adding fractions with surd denominators. Some students also forget that both denominators are positive and incorrectly consider negative values. Final Answer: Therefore, the exact simplified value of 1/(a + 1) + 1/(b + 1) is 1.

Question 12

In triangle ABC, points D on AB and E on AC satisfy AD = 8 cm, DB = 12 cm, AE = 6 cm and EC = 9 cm. If DE is drawn parallel to side BC, then BC is equal to which multiple of segment DE?

Accepted Answer

Introduction / Context: This question tests the concept of similar triangles formed by a line drawn parallel to one side of a triangle. Using the basic proportionality theorem, we can relate side lengths of the smaller and larger triangles to find how many times BC is compared with DE. Given Data / Assumptions: Triangle ABC with D on AB and E on AC AD = 8 cm, DB = 12 cm so AB = 20 cm AE = 6 cm, EC = 9 cm so AC = 15 cm DE ∥ BC Need BC expressed as a multiple of DE Concept / Approach: If a line segment is drawn parallel to one side of a triangle, joining the other two sides, the two resulting triangles are similar. That means the ratios of corresponding sides are equal. Here, triangle ADE is similar to triangle ABC. The scale factor between them equals AD/AB and also AE/AC. We use this factor to relate BC and DE. Step-by-Step Solution: AB = AD + DB = 8 + 12 = 20 cm AC = AE + EC = 6 + 9 = 15 cm Since DE ∥ BC, triangle ADE ~ triangle ABC So AD/AB = AE/AC = DE/BC Compute ratio AD/AB = 8/20 = 2/5 Hence DE/BC = 2/5 So BC/DE = 5/2 Therefore BC = (5/2) DE Verification / Alternative check: We can check with a scaled numeric example. Suppose DE were 2 units. Then BC would be (5/2)*2 = 5 units. The ratio of corresponding sides BC:DE is 5:2 which equals AB:AD = 20:8 = 5:2, confirming consistency with similarity properties. Why Other Options Are Wrong: 2/5 DE and 2/3 DE correspond to DE larger than BC, which conflicts with the full side being longer. 3/2 DE implies BC/DE = 3/2 which does not match AD/AB = 2/5. 4/3 DE also gives a ratio inconsistent with the verified similarity ratio of 5/2. Common Pitfalls: Some learners confuse DE/BC with BC/DE and invert ratios incorrectly. Others add lengths incorrectly or forget to check that both AD/AB and AE/AC give the same proportion, which is required if DE is parallel to BC. Mixing up which triangle is larger can also cause reversed ratios. Final Answer: Thus, side BC is equal to (5/2) DE.

Question 13

A, B and C can complete a piece of work alone in 72 days, 24 days and 36 days respectively. If all three start the job together and continue working at their constant rates, in how many days will they finish the entire work?

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Introduction / Context: This question uses the standard time and work concept where total work is considered as one complete job and individual rates are added to get a combined rate. It checks the ability to convert days per job into jobs per day and manipulate fractions correctly. Given Data / Assumptions: A alone completes the work in 72 days B alone completes the work in 24 days C alone completes the work in 36 days All three work together from the beginning Their work rates are constant and additive Concept / Approach: If a person finishes one job in T days, the work rate is 1/T jobs per day. When multiple people work together, their rates add. We first find the combined rate of A, B and C, then take the reciprocal to get the total time taken to finish one full job. Step-by-Step Solution: Rate of A = 1/72 job per day Rate of B = 1/24 job per day Rate of C = 1/36 job per day Combined rate R = 1/72 + 1/24 + 1/36 Take LCM of 72, 24 and 36 which is 72 1/72 + 1/24 + 1/36 = 1/72 + 3/72 + 2/72 = 6/72 So R = 6/72 = 1/12 jobs per day Time taken T = 1 / R = 1 / (1/12) = 12 days Verification / Alternative check: We can check by imagining the amount of work done in 12 days. At 1/12 job per day, in 12 days the team completes (1/12)*12 = 1 full job. This confirms that 12 days is consistent with the combined rate and completes exactly one job with no extra or missing work. Why Other Options Are Wrong: 9 days would imply a rate of 1/9 which is greater than 1/12 and inconsistent with the calculated sum. 15 and 18 days correspond to combined rates that are too slow. 10 days also produces too large a rate. Only 12 days matches the exact sum of individual rates. Common Pitfalls: Typical mistakes include adding the times directly instead of adding rates, or computing the LCM incorrectly. Some students may invert the final rate incorrectly or stop at a fractional answer without simplifying. It is important to remember that time and rate are reciprocals of each other in these problems. Final Answer: Therefore, all three working together will finish the work in 12 days.

Question 14

Solve the linear fractional equation [7(5x/3 − 3/2)]/2 + 3/2 = 1/4 and determine the exact value of the variable x.

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Introduction / Context: This is a linear equation with fractions inside brackets. Such questions examine handling of nested fractions, distribution, and simplification of rational expressions. The main skill is to clear denominators systematically without making arithmetic errors. Given Data / Assumptions: Equation: [7(5x/3 − 3/2)]/2 + 3/2 = 1/4 x is a real number We seek the exact simplified value of x Concept / Approach: We first simplify the inner bracket, then multiply by 7 and divide by 2. After that we combine constant terms and isolate x. A good method is to multiply through by the least common multiple of denominators so that the equation becomes an integer linear equation. Step-by-Step Solution: Start with [7(5x/3 − 3/2)]/2 + 3/2 = 1/4 Let us simplify the bracket: 5x/3 − 3/2 LCM of 3 and 2 is 6 5x/3 − 3/2 = (10x/6) − (9/6) = (10x − 9)/6 Multiply by 7: 7 * (10x − 9)/6 = (70x − 63)/6 Divide by 2: [(70x − 63)/6] / 2 = (70x − 63)/12 Equation becomes (70x − 63)/12 + 3/2 = 1/4 Rewrite 3/2 with denominator 12: 3/2 = 18/12 So (70x − 63)/12 + 18/12 = 1/4 Combine numerators: (70x − 63 + 18)/12 = (70x − 45)/12 So (70x − 45)/12 = 1/4 Cross multiply: 4(70x − 45) = 12 280x − 180 = 12 280x = 192 x = 192/280 = 24/35 Verification / Alternative check: Substitute x = 24/35 back into the original expression. The inner term 5x/3 becomes 5*(24/35)/3 and evaluates correctly, and the entire left side simplifies to 1/4. A quick check with approximate decimals also confirms equality of both sides, validating the solution. Why Other Options Are Wrong: 35/24 and −35/24 are reciprocals or sign changes that do not satisfy the equation when substituted. −24/35 changes the sign of x and reverses the direction of the linear term, giving a different left side. Zero clearly fails because the constant terms alone do not satisfy the equality. Common Pitfalls: Sometimes learners forget to divide by 2 after multiplying by 7, or they mishandle the LCM of 3 and 2. Losing track of signs when combining −63 and +18 is also common. Always work carefully with numerators and denominators and only cross multiply once the fractions are correctly arranged. Final Answer: Hence, the solution of the equation is x = 24/35.

Question 15

For real numbers a and b, if a³ + b³ = 19 and the product ab = −6, then using identities for sums of cubes, what is the value of the sum a + b?

Accepted Answer

Introduction / Context: This question tests the algebraic identity that relates the cube of a sum to the sum of cubes and the product term. It is a standard pattern in polynomial manipulation and factorisation used to determine sums of numbers from their cubes and product. Given Data / Assumptions: a³ + b³ = 19 ab = −6 a and b are real numbers We must find S = a + b Concept / Approach: We use the identity (a + b)³ = a³ + b³ + 3ab(a + b). This links the sum of cubes, the product ab and the sum a + b. By substituting the given values, we form an equation in S and solve the cubic, searching for a simple real root that matches the answer options. Step-by-Step Solution: Let S = a + b Use identity: (a + b)³ = a³ + b³ + 3ab(a + b) So S³ = 19 + 3(−6)S S³ = 19 − 18S Rearrange to standard form: S³ + 18S − 19 = 0 Now test simple integer values from the options For S = 1: 1³ + 18(1) − 19 = 1 + 18 − 19 = 0 Therefore S = 1 is a root So a + b = 1 Verification / Alternative check: Once we know S = 1 and ab = −6, we can set up the quadratic t² − S t + ab = 0, that is t² − t − 6 = 0. This factors to (t − 3)(t + 2) = 0 with roots 3 and −2. Then a³ + b³ = 27 − 8 = 19 which matches the given condition, confirming that S = 1 is consistent. Why Other Options Are Wrong: For S = 5, S³ + 18S − 19 = 125 + 90 − 19 ≠ 0. For S = 7 or S = −5 or S = −7, substituting into S³ + 18S − 19 also gives non zero results. Hence none of these candidate sums satisfy the derived cubic equation. Common Pitfalls: Some learners mistakenly use the identity for a³ − b³ instead of a³ + b³, leading to a wrong sign pattern. Others forget the factor 3ab(a + b) in the formula. It is also common to try to solve the cubic purely algebraically when checking integer options is faster in exam settings. Final Answer: Thus, the required sum is a + b = 1.

Question 16

In an arithmetic progression, the third term is −13 and the sixth term is −4. Using properties of arithmetic progressions, what is the sum of the first 12 terms of this sequence?

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Introduction / Context: This problem tests core arithmetic progression concepts, including finding the first term, common difference and then using the formula for the sum of a finite number of terms. It requires manipulation of two term values to extract progression parameters. Given Data / Assumptions: Third term T₃ = −13 Sixth term T₆ = −4 We consider a standard arithmetic progression with first term a and common difference d We need the sum of the first 12 terms S₁₂ Concept / Approach: In an arithmetic progression, the nth term is Tₙ = a + (n − 1)d. We use the given T₃ and T₆ to form two equations in a and d. Once a and d are known, we use Sₙ = n/2 [2a + (n − 1)d] to compute the sum of the first 12 terms. Step-by-Step Solution: T₃ = a + 2d = −13 T₆ = a + 5d = −4 Subtract the first from the second: (a + 5d) − (a + 2d) = −4 − (−13) 3d = 9, so d = 3 Substitute d into a + 2d = −13 a + 2*3 = −13 gives a + 6 = −13 a = −19 Now use sum formula Sₙ = n/2 [2a + (n − 1)d] For n = 12: S₁₂ = 12/2 [2(−19) + 11*3] S₁₂ = 6[−38 + 33] = 6(−5) = −30 Verification / Alternative check: We can list a few terms: a = −19, d = 3. So terms are −19, −16, −13, −10, −7, −4, and so on. The 3rd term is −13 and the 6th term is −4, matching the data. Pairing terms T₁ with T₁₂, T₂ with T₁₁, and so forth also yields a consistent sum of −30 when calculated carefully. Why Other Options Are Wrong: 67 and 45 are positive and ignore the heavily negative early terms. −48 has too large a magnitude and does not match the formula based calculation. 30 has the wrong sign because the progression remains largely negative over the first 12 terms. Common Pitfalls: Errors often arise when forming equations for T₃ and T₆, particularly with the (n − 1)d term. Another pitfall is mixing up signs while substituting a and d into the sum formula. Students sometimes use n in place of (n − 1) inside the bracket which changes the final answer significantly. Final Answer: Therefore, the sum of the first 12 terms of the arithmetic progression is −30.

Question 17

A point P(5, 2) is reflected in the vertical line x = −3 in the coordinate plane. After this reflection, what are the coordinates of the image of point P?

Accepted Answer

Introduction / Context: This problem checks understanding of reflections of points in vertical lines in coordinate geometry. Reflecting across a vertical line changes the x coordinate but preserves the y coordinate, and the new x is symmetric with respect to the given line. Given Data / Assumptions: Original point P has coordinates (5, 2) Reflection line is x = −3 We assume the usual Cartesian coordinate system We must find coordinates of the reflected point Concept / Approach: For reflection across a vertical line x = k, the y coordinate remains unchanged while the x coordinate is mirrored across k. If the original x is x₁, then the distance from x₁ to k equals the distance from the reflected x coordinate to k, but on the opposite side. Algebraically, reflected x is x′ = 2k − x₁. Step-by-Step Solution: Line of reflection is x = −3 Original point P has x coordinate 5 and y coordinate 2 Distance from P to the line along x axis is 5 − (−3) = 8 units The reflected point must lie 8 units on the other side of x = −3 So new x coordinate x′ = −3 − 8 = −11 The y coordinate does not change, so y′ = 2 Reflected point is (−11, 2) Verification / Alternative check: We can use the formula x′ = 2k − x₁ directly. Here k = −3 and x₁ = 5. So x′ = 2(−3) − 5 = −6 − 5 = −11. The y coordinate remains 2. This matches the coordinate pair (−11, 2) obtained from the distance based reasoning, confirming the result. Why Other Options Are Wrong: (−11, −2) incorrectly changes the y coordinate. (11, −2) and (11, 2) lie on the opposite side of the origin and are not symmetric about x = −3 relative to point (5, 2). The option (5, 2) is just the original point and represents no reflection at all. Common Pitfalls: Students sometimes reflect across the y axis x = 0 instead of the given line x = −3. Others mistakenly negate both coordinates or only change the sign of x without considering the exact distance from the line of reflection. Remember that reflection preserves perpendicular distance to the mirror line. Final Answer: The image of P after reflection in x = −3 is (−11, 2).

Question 18

Find the coordinates of the centroid of a triangle whose vertices in the coordinate plane are A(1, −5), B(4, 0) and C(−2, 2), using the standard centroid formula.

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Introduction / Context: This question focuses on the centroid of a triangle in coordinate geometry. The centroid is the intersection point of the medians and represents the average of the vertex coordinates. It is a basic but important concept in analytic geometry. Given Data / Assumptions: Vertex A = (1, −5) Vertex B = (4, 0) Vertex C = (−2, 2) We use the centroid formula for a triangle Concept / Approach: For a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃), the centroid G has coordinates G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3). We simply sum the x coordinates, sum the y coordinates and divide each sum by three. Step-by-Step Solution: x coordinates: x₁ = 1, x₂ = 4, x₃ = −2 Sum of x coordinates = 1 + 4 + (−2) = 3 Centroid x coordinate = 3/3 = 1 y coordinates: y₁ = −5, y₂ = 0, y₃ = 2 Sum of y coordinates = −5 + 0 + 2 = −3 Centroid y coordinate = −3/3 = −1 So centroid G = (1, −1) Verification / Alternative check: We can verify by noting that the centroid must lie inside the triangle and should have coordinates between the extreme x and y values of the vertices. Here x ranges from −2 to 4 and y ranges from −5 to 2. The point (1, −1) fits comfortably within those ranges, confirming the plausibility of the result. Why Other Options Are Wrong: (−1, 1) simply flips the signs and does not use the centroid formula. (2, −2) and (−2, 2) are not equal to the averages of the given coordinates. (0, 0) is frequently guessed but does not reflect the asymmetric distribution of the vertices and fails the average calculations. Common Pitfalls: Errors usually happen when adding or dividing the coordinates, especially with negative values. Some learners incorrectly divide by 2 instead of 3, confusing the centroid with the midpoint between two points. Always remember that a triangle has three vertices so the coordinate averages must be divided by three. Final Answer: Therefore, the centroid of triangle ABC is (1, −1).

Question 19

The slope of line segment AB is −4/3. The coordinates of A and B are A(x, −5) and B(−5, 3) respectively. What is the value of x?

Accepted Answer

Introduction / Context: This question tests your understanding of the slope formula in coordinate geometry and your ability to use it to find an unknown coordinate. The slope gives the steepness of the line, and knowing two points with an unknown coordinate allows you to set up an equation and solve it for that coordinate. Given Data / Assumptions: • Line segment AB has slope m = −4/3. • Point A has coordinates (x, −5). • Point B has coordinates (−5, 3). • We must find the value of x that makes the slope correct. Concept / Approach: The slope m of a line passing through points (x₁, y₁) and (x₂, y₂) is given by m = (y₂ − y₁) / (x₂ − x₁). By substituting the coordinates of A and B into this formula and equating the result to the given slope −4/3, we obtain a simple linear equation in x that can be solved easily. Step-by-Step Solution: Step 1: Label A(x₁, y₁) = (x, −5) and B(x₂, y₂) = (−5, 3). Step 2: Use the slope formula m = (y₂ − y₁) / (x₂ − x₁). Step 3: Compute the numerator: y₂ − y₁ = 3 − (−5) = 3 + 5 = 8. Step 4: Compute the denominator: x₂ − x₁ = −5 − x. Step 5: So the slope from A to B is 8 / (−5 − x) = −4/3. Step 6: Set up the equation 8 / (−5 − x) = −4/3. Step 7: Cross multiply: 8 * 3 = −4 * (−5 − x). Step 8: This gives 24 = −4(−5 − x) = 20 + 4x. Step 9: Solve for x: 24 − 20 = 4x, so 4 = 4x and x = 1. Verification / Alternative check: Substitute x = 1 back into the coordinates of A, giving A(1, −5). Now the slope between A(1, −5) and B(−5, 3) is (3 − (−5)) / (−5 − 1) = 8 / (−6) = −4/3, which matches the given slope. Therefore, x = 1 is confirmed as correct. Why Other Options Are Wrong: Option −1: With A(−1, −5), the slope becomes 8 / (−5 − (−1)) = 8 / (−4) = −2, not −4/3. Option 2: With A(2, −5), the slope is 8 / (−7), which is not equal to −4/3. Option −2: With A(−2, −5), the slope is 8 / (−3), which is again not equal to −4/3. Common Pitfalls: A frequent mistake is reversing the order of points incorrectly, which changes the signs in both numerator and denominator, though here that would still give the same slope. A more serious error is mismanaging the negative signs during cross multiplication, leading to wrong values of x. Writing each step clearly and checking the final slope with the found value of x helps avoid these errors. Final Answer: The correct value of x is 1.

Question 20

Using standard trigonometric ratios, evaluate the exact value of the expression √2 · sec 45° − tan 30° and choose the equivalent simplified form.

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Introduction / Context: This problem checks familiarity with exact trigonometric ratios for standard angles 30° and 45°. It combines these values in a simple algebraic expression involving secant and tangent and asks for an equivalent simplified form in terms of square roots. Given Data / Assumptions: Expression: √2 · sec 45° − tan 30° Angles are in degrees We use exact values of trigonometric functions at 30° and 45° Concept / Approach: We recall that sec 45° is the reciprocal of cos 45°, and tan 30° is sin 30° divided by cos 30°. Substituting these standard values, we simplify the resulting expression using rational operations with surds and then match it to one of the given options. Step-by-Step Solution: cos 45° = √2 / 2, so sec 45° = 1 / (√2 / 2) = 2/√2 = √2 Thus √2 · sec 45° = √2 · √2 = 2 sin 30° = 1/2, cos 30° = √3 / 2 Hence tan 30° = (1/2) / (√3 / 2) = 1/√3 Given expression = 2 − 1/√3 Write 2 as 2√3/√3 2 − 1/√3 = (2√3/√3) − 1/√3 = (2√3 − 1)/√3 Verification / Alternative check: Approximate √3 ≈ 1.732. Then (2√3 − 1)/√3 ≈ (3.464 − 1)/1.732 ≈ 2.464/1.732 ≈ 1.4226. Evaluate the original expression numerically: √2 ≈ 1.414, sec 45° ≈ 1.414 so the product is 2, and tan 30° ≈ 0.577. Then 2 − 0.577 ≈ 1.423 which matches the approximate value, confirming the simplification. Why Other Options Are Wrong: (2√3 − 1)/3 has a different denominator and evaluates to a smaller value. (√3 − 1)/√3 and (√3 − 1)/3 correspond to entirely different expressions. (2√3 + 1)/3 would be larger than 1.5 and does not match the calculated value near 1.42. Common Pitfalls: Learners sometimes misremember standard values, such as swapping tan 30° and tan 60°, or incorrectly simplifying 2/√2. Another issue is rationalising denominators in the wrong step, which is not necessary here beyond matching the option forms. Always compute primary trig values accurately first. Final Answer: Thus, √2 · sec 45° − tan 30° simplifies exactly to (2√3 − 1)/√3.

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