Question 1

If 3 sinθ + 4 cosθ = 5 and 0 degrees < θ < 90 degrees, then what is the value of sinθ?

Accepted Answer

Introduction / Context: This trigonometry question tests your skill in manipulating linear combinations of sinθ and cosθ. Recognizing patterns involving Pythagorean triples can greatly simplify the problem. This is a very common style in entrance and aptitude tests. Given Data / Assumptions: 3 sinθ + 4 cosθ = 5. The range of θ is 0 degrees < θ < 90 degrees, so sinθ and cosθ are positive. We must find sinθ. Concept / Approach: Notice the coefficients 3 and 4 and the right side 5. These resemble the 3, 4, 5 Pythagorean triple, where 3^2 + 4^2 = 5^2. This suggests that we can interpret sinθ and cosθ as legs of a right triangle and use this triple structure. One neat way is to suppose sinθ = 3/5 and cosθ = 4/5, then check whether the equation holds. Step-by-Step Solution: Step 1: Recall that for any angle θ, sin^2θ + cos^2θ = 1.Step 2: Observe that 3, 4, and 5 satisfy 3^2 + 4^2 = 9 + 16 = 25 = 5^2.Step 3: Consider the possibility sinθ = 3/5 and cosθ = 4/5. These values satisfy sin^2θ + cos^2θ = (3/5)^2 + (4/5)^2 = 9/25 + 16/25 = 25/25 = 1.Step 4: Substitute into the original equation: 3 sinθ + 4 cosθ = 3 * (3/5) + 4 * (4/5).Step 5: Compute: 3 * (3/5) = 9/5 and 4 * (4/5) = 16/5.Step 6: Add them: 9/5 + 16/5 = 25/5 = 5, which matches the given right side exactly.Step 7: Since 0 degrees < θ < 90 degrees, both sinθ and cosθ must be positive, and sinθ = 3/5 is consistent with this. Verification / Alternative check: Alternatively, you can write 3 sinθ + 4 cosθ in the form R sin(θ + φ), using R = square root of (3^2 + 4^2) = 5 and an angle φ such that cosφ = 3/5 and sinφ = 4/5. Then 3 sinθ + 4 cosθ = 5 sin(θ + φ) = 5, giving sin(θ + φ) = 1, so θ + φ = 90 degrees and sinθ evaluates to 3/5 again. Why Other Options Are Wrong: The values 1/5 and 2/5 are too small to satisfy the equation with any matching cosθ value. The value 4/5 would make cosθ smaller than 3/5 to preserve sin^2θ + cos^2θ = 1, which breaks the given linear relation. The value 1 would force cosθ = 0, which does not satisfy 3 sinθ + 4 cosθ = 5. Common Pitfalls: Learners may try to square both sides and create a more complicated equation, or they may forget that both sinθ and cosθ must be positive in the first quadrant. Recognizing the 3, 4, 5 pattern quickly leads to an elegant solution. Final Answer: The value of sinθ is 3/5.

Question 2

In an algebraic simplification problem, if 2x + 9/x = 9 for a real non zero number x, then what is the minimum possible value of the expression x^2 + 1/x^2?

Accepted Answer

Introduction / Context: This aptitude question tests algebraic manipulation and the use of quadratic equations to find the minimum value of an expression in x and 1/x. Problems of this type are very common in bank exams, SSC exams, and other competitive tests because they combine equation solving with optimization in a single short statement. Given Data / Assumptions: 2x + 9/x = 9 for a real non zero number x. The required expression is x^2 + 1/x^2. We assume x is real so that the minimum value is interpreted over the real numbers. Concept / Approach: The key idea is to clear the denominator to obtain a quadratic equation in x, solve for its real roots, and then evaluate x^2 + 1/x^2 for each root. Since there are only two possible real values of x, the minimum value of x^2 + 1/x^2 will be the smaller of the two corresponding values. This approach avoids unnecessary calculus and relies only on standard quadratic and algebraic identities. Step-by-Step Solution: Start from 2x + 9/x = 9.Multiply both sides by x (x is non zero): 2x^2 + 9 = 9x.Rearrange to get a quadratic: 2x^2 - 9x + 9 = 0.Compute the discriminant: D = (-9)^2 - 4 * 2 * 9 = 81 - 72 = 9.Roots are x = [9 ± 3] / (2 * 2) = (9 ± 3) / 4.So x = 12/4 = 3 or x = 6/4 = 3/2.For x = 3, x^2 + 1/x^2 = 9 + 1/9 = 82/9.For x = 3/2, x^2 + 1/x^2 = (9/4) + (4/9) = (81 + 16) / 36 = 97/36.Between 82/9 and 97/36, clearly 97/36 is smaller. Verification / Alternative check: We can also use the identity x^2 + 1/x^2 = (x - 1/x)^2 + 2, which shows the expression is always at least 2. Our computed value 97/36 is greater than 2 and therefore reasonable. Substituting x = 3 and x = 3/2 back into the original equation 2x + 9/x = 9 confirms that both satisfy the given condition, so no other real x values need to be considered. Why Other Options Are Wrong: Values like 95/36, 87/25, 623/27, and 25/4 do not arise from any real solution of the quadratic 2x^2 - 9x + 9 = 0. They either correspond to incorrect algebraic manipulation or to evaluating x^2 + 1/x^2 for values of x that do not satisfy the original equation. Among all valid candidates, 97/36 is strictly the smallest value. Common Pitfalls: Many learners forget to multiply the equation by x correctly and make sign or coefficient errors in the quadratic. Another frequent mistake is to assume that the first value found for x gives the minimum without checking the second root. Some students also try to square the whole equation unnecessarily, which creates extra work and potential extraneous solutions. Carefully forming the quadratic and evaluating both resulting values of x^2 + 1/x^2 avoids these errors. Final Answer: The minimum possible value of x^2 + 1/x^2, given that 2x + 9/x = 9, is 97/36.

Question 3

If (5x − y) / (5x + y) = 3/7 for real numbers x and y with non zero denominator, then what is the value of the expression (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy)?

Accepted Answer

Introduction / Context: This question checks algebraic simplification skills and the ability to recognize expressions as perfect squares. By rewriting the numerator and denominator in a suitable form, the value of the big fraction can often be obtained without explicitly solving for x and y, although solving is also possible in this case. Given Data / Assumptions: (5x − y) / (5x + y) = 3/7. x and y are real numbers and 5x + y is not zero. The required value is (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy). Concept / Approach: The numerator and denominator can be recognized as squares of linear expressions in x and y. In particular, 4x^2 + y^2 − 4xy equals (2x − y)^2, and 9x^2 + 16y^2 + 24xy equals (3x + 4y)^2. If we can relate 2x − y and 3x + 4y to the given ratio involving 5x − y and 5x + y, we may evaluate the expression. An even simpler approach here is to use the given ratio to find y in terms of x and then substitute directly. Step-by-Step Solution: Start with (5x − y) / (5x + y) = 3/7.Cross multiply: 7(5x − y) = 3(5x + y).This gives 35x − 7y = 15x + 3y.Rearrange: 35x − 15x = 3y + 7y, so 20x = 10y.Hence y = 2x.Now compute the numerator: 4x^2 + y^2 − 4xy = 4x^2 + (2x)^2 − 4x(2x) = 4x^2 + 4x^2 − 8x^2 = 0.Therefore the numerator is zero for every non zero x, while the denominator is positive.Thus (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy) = 0 / (positive) = 0. Verification / Alternative check: We can verify that the denominator is not zero for y = 2x. Substitute y = 2x: 9x^2 + 16(4x^2) + 24x(2x) = 9x^2 + 64x^2 + 48x^2 = 121x^2, which is non zero when x is non zero. Hence the fraction is well defined. Also, as noted earlier, 4x^2 + y^2 − 4xy = (2x − y)^2, which becomes (2x − 2x)^2 = 0, confirming the result in a more conceptual way. Why Other Options Are Wrong: The options 3/7, 18/49, 1/6, and 1/3 might tempt students who try to connect the result directly to the given ratio, but none of these values is consistent with the exact algebraic substitution y = 2x. Since the numerator reduces to zero for all valid values of x and y that satisfy the given ratio, the only correct value is 0. Common Pitfalls: One common mistake is to assume (4x^2 + y^2 − 4xy) is equal to (2x + y)^2 or another incorrect square, which leads to algebraic errors. Another pitfall is trying to manipulate the ratio (5x − y) / (5x + y) into the desired expression without first solving for y. Being comfortable with recognizing perfect squares and systematically solving the ratio equation helps avoid these issues. Final Answer: The value of the expression (4x^2 + y^2 − 4xy) / (9x^2 + 16y^2 + 24xy) is 0.

Question 4

If (x + y)^2 = xy + 1 and x^3 − y^3 = 1 for real numbers x and y, then what is the value of the difference x − y?

Accepted Answer

Introduction / Context: This algebra question combines symmetric expressions and identities involving x + y, xy, and x^3 − y^3. It is designed to test the ability to use algebraic identities cleverly instead of expanding everything directly. Such problems appear frequently in aptitude exams and algebra sections of school level mathematics. Given Data / Assumptions: (x + y)^2 = xy + 1. x^3 − y^3 = 1. x and y are real numbers. We need to find x − y. Concept / Approach: The identity for the difference of cubes, x^3 − y^3 = (x − y)(x^2 + xy + y^2), is central here. The expression x^2 + xy + y^2 can be written in terms of x + y and xy. The equation (x + y)^2 = xy + 1 gives a direct relationship between these symmetric parts. The trick is to express x^2 + xy + y^2 in a form that can be evaluated using the given relation without solving for x and y individually. Step-by-Step Solution: Start with the identity: x^3 − y^3 = (x − y)(x^2 + xy + y^2).We are given x^3 − y^3 = 1, so 1 = (x − y)(x^2 + xy + y^2).Now use (x + y)^2 = x^2 + 2xy + y^2 = xy + 1.From this, x^2 + 2xy + y^2 = xy + 1.Rearrange: x^2 + y^2 = (xy + 1) − 2xy = 1 − xy.Now compute x^2 + xy + y^2 = (x^2 + y^2) + xy = (1 − xy) + xy = 1.Thus x^2 + xy + y^2 equals 1.Substitute back into 1 = (x − y)(x^2 + xy + y^2): 1 = (x − y) * 1.Therefore x − y = 1. Verification / Alternative check: We can check the result by constructing a concrete pair (x, y) that satisfies both given conditions. If x − y = 1 and x^2 + xy + y^2 = 1, then x and y will satisfy x^3 − y^3 = 1 by the cube identity. Solving the system to find actual values is possible but not required, because the algebraic manipulation already shows that the only value consistent with x^3 − y^3 = 1 and x^2 + xy + y^2 = 1 is x − y = 1. Why Other Options Are Wrong: Values such as 0, −1, 2, or 3 for x − y would change x^3 − y^3 to some number other than 1 when multiplied by x^2 + xy + y^2, which we have shown equals 1. Therefore they are inconsistent with the given condition x^3 − y^3 = 1 and must be rejected. Common Pitfalls: Students sometimes expand everything and attempt to solve for x and y directly, which leads to unnecessary complexity. Others misapply the square identity and incorrectly compute x^2 + xy + y^2. Keeping track of the symmetric forms and using identities systematically is the fastest and safest route. Remember that x^2 + xy + y^2 = (x + y)^2 − xy, and that can be easily evaluated from the given relation. Final Answer: The required value of x − y is 1.

Question 5

If sin θ + sin 5θ = sin 3θ for an acute angle θ with 0° < θ < 60°, then what is the value of θ in degrees?

Accepted Answer

Introduction / Context: This trigonometry question tests the use of sum to product formulas and basic equation solving with sine functions. It also illustrates how adding a condition on the range of the angle ensures a unique answer, which is important for multiple choice questions in competitive exams. Given Data / Assumptions: sin θ + sin 5θ = sin 3θ. 0° < θ < 60°, so θ is an acute angle less than 60 degrees. We work in degrees and standard trigonometric definitions. Concept / Approach: We use the identity sin C + sin D = 2 sin((C + D) / 2) cos((C − D) / 2). Applying this to sin θ + sin 5θ allows us to rewrite the left side in terms of sin 3θ and cos 2θ. Then we can equate both sides and solve for θ. The additional condition on θ eliminates extra solutions that might otherwise satisfy the equation but fall outside the specified range. Step-by-Step Solution: Apply the sum to product formula: sin θ + sin 5θ = 2 sin((θ + 5θ) / 2) cos((θ − 5θ) / 2).This simplifies to 2 sin(3θ) cos(−2θ).Since cos is an even function, cos(−2θ) = cos 2θ, so the left side becomes 2 sin 3θ cos 2θ.The given equation becomes 2 sin 3θ cos 2θ = sin 3θ.Rearrange: sin 3θ (2 cos 2θ − 1) = 0.So either sin 3θ = 0 or 2 cos 2θ − 1 = 0.From 2 cos 2θ − 1 = 0, we get cos 2θ = 1/2, so 2θ = 60° or 300°.For 0° < θ < 60°, only 2θ = 60° is valid, giving θ = 30°. Verification / Alternative check: Check θ = 30° in the original equation: sin 30° + sin 150° = 1/2 + 1/2 = 1. On the right side, sin 3θ = sin 90° = 1. Both sides are equal, so θ = 30° is indeed a solution. The other possibility sin 3θ = 0 would give 3θ = 180°, 360° and so on, which leads to θ = 60° or larger values. Because of the condition 0° < θ < 60°, these are excluded, ensuring a unique answer. Why Other Options Are Wrong: Angles 15°, 45°, 60°, and 75° do not satisfy sin θ + sin 5θ = sin 3θ within the specified interval. Substituting any of these values either breaks the equality or falls outside the allowed range. For example, θ = 45° gives sin 45° + sin 225° which does not equal sin 135°. Thus these options are incorrect. Common Pitfalls: Some students cancel sin 3θ from both sides without considering that it might be zero, which can hide valid solutions. Others forget to apply the range condition and report θ = 60° as a solution. Always use sum to product identities carefully, solve all resulting cases, and then apply the interval restriction to select the correct answer. Final Answer: The angle θ that satisfies the given condition in the range 0° < θ < 60° is 30°.

Question 6

If A/3 = B/2 = C/5 for three real numbers A, B and C, then what is the ratio (C + A)^2 : (A + B)^2 : (B + C)^2?

Accepted Answer

Introduction / Context: This is a ratio based algebra question that checks understanding of proportional relationships and simple substitution. By expressing A, B, and C in terms of a common parameter, we can easily compute the required squared ratio. Such questions are standard in quantitative aptitude sections of many examinations. Given Data / Assumptions: A/3 = B/2 = C/5. A, B, and C are real numbers, and denominators are non zero. The required ratio is (C + A)^2 : (A + B)^2 : (B + C)^2. Concept / Approach: Whenever three fractions are equal, we introduce a common parameter, say k. We then write A = 3k, B = 2k, C = 5k. The expressions involving A, B, and C can then be simplified in terms of k alone. Since any common factor in the numerator and denominator of a ratio cancels out, k will drop out and we will obtain a simple numerical ratio. Step-by-Step Solution: Let A/3 = B/2 = C/5 = k.Then A = 3k, B = 2k, C = 5k.Compute C + A = 5k + 3k = 8k, so (C + A)^2 = (8k)^2 = 64k^2.Compute A + B = 3k + 2k = 5k, so (A + B)^2 = (5k)^2 = 25k^2.Compute B + C = 2k + 5k = 7k, so (B + C)^2 = (7k)^2 = 49k^2.Therefore the ratio (C + A)^2 : (A + B)^2 : (B + C)^2 = 64k^2 : 25k^2 : 49k^2.Divide every term by k^2: 64 : 25 : 49. Verification / Alternative check: We can choose a convenient value of k, for example k = 1. Then A = 3, B = 2, C = 5. Substitute: (C + A)^2 = (5 + 3)^2 = 8^2 = 64, (A + B)^2 = (3 + 2)^2 = 5^2 = 25, and (B + C)^2 = (2 + 5)^2 = 7^2 = 49. This gives the ratio 64 : 25 : 49, which confirms our algebraic derivation. Why Other Options Are Wrong: The other options are permutations of similar numbers but do not match the actual order that comes from the expressions involving A, B, and C. For example, 25 : 4 : 9 ignores the contribution from the coefficients 3, 2, and 5 in the correct combinations. Only 64 : 25 : 49 matches (8k)^2 : (5k)^2 : (7k)^2. Common Pitfalls: A common mistake is to misassign the parameter, such as writing A = k/3 instead of A = 3k, which reverses the relationship. Another error is to forget to square the sums, leading to the ratio 8 : 5 : 7 instead of 64 : 25 : 49. Carefully following the substitution and respecting the squares avoids these issues. Final Answer: The required ratio (C + A)^2 : (A + B)^2 : (B + C)^2 is 64 : 25 : 49.

Question 7

If a + b = 2c for distinct real numbers a, b and c, then what is the value of the expression a / (a − c) + c / (b − c)?

Accepted Answer

Introduction / Context: This question tests algebraic manipulation and substitution using a linear relation between three variables. It shows how expressions can simplify dramatically when the given condition is used correctly. Such problems are typical in algebraic simplification sections of quantitative aptitude tests. Given Data / Assumptions: a + b = 2c. a, b, and c are distinct real numbers, so denominators are non zero in the given expression. The required value is a / (a − c) + c / (b − c). Concept / Approach: The key idea is to express one variable in terms of the others using the relation a + b = 2c. It is convenient to write b in terms of a and c. Once b is written as 2c − a, we substitute this into the expression and simplify. Because the structure of the expression is symmetric in a and c in a certain way, terms nicely cancel to give a constant result independent of the particular values of a, b, and c. Step-by-Step Solution: From a + b = 2c, express b as b = 2c − a.Now consider the expression a / (a − c) + c / (b − c).Substitute b = 2c − a into the second fraction: b − c = (2c − a) − c = c − a.Thus the expression becomes a / (a − c) + c / (c − a).Note that c / (c − a) = c / (−(a − c)) = −c / (a − c).So the expression simplifies to a / (a − c) − c / (a − c).Combine over a common denominator (a − c): (a − c) / (a − c) = 1.Therefore a / (a − c) + c / (b − c) = 1. Verification / Alternative check: We can verify this with specific numbers that satisfy a + b = 2c. For example, take c = 3 and a = 1, then b = 2c − a = 6 − 1 = 5. Compute the expression: a / (a − c) = 1 / (1 − 3) = 1 / (−2) = −1/2, and c / (b − c) = 3 / (5 − 3) = 3/2. Their sum is −1/2 + 3/2 = 1, confirming the algebraic simplification. Why Other Options Are Wrong: The values −1, 0, 1/2, and 2 would require the combined numerator after simplification to be a different multiple of (a − c). However, as shown, the expression simplifies exactly to (a − c) / (a − c), which is 1 for all allowed values of a and c. Therefore only the option 1 is consistent with the identity for all valid triples (a, b, c) satisfying a + b = 2c. Common Pitfalls: Learners sometimes substitute incorrectly for b or mishandle the sign when changing c / (c − a) to an equivalent form. Forgetting that c − a is the negative of a − c can easily lead to incorrect cancellation. Always check denominators carefully and keep track of signs when factoring out a minus sign. Final Answer: The value of a / (a − c) + c / (b − c) is the constant 1.

Question 8

If sin θ + cos θ = 1 for an angle θ (in degrees), then what is the value of the product sin θ · cos θ?

Accepted Answer

Introduction / Context: This trigonometric simplification question tests the use of the Pythagorean identity sin^2 θ + cos^2 θ = 1, together with basic algebra. It is a classic type of problem used to check understanding of relationships between sine and cosine and the effect of squaring an equation involving them. Given Data / Assumptions: sin θ + cos θ = 1. θ is an angle measured in degrees or radians, with standard trigonometric definitions. We need to find sin θ · cos θ. Concept / Approach: The idea is to square both sides of the equation sin θ + cos θ = 1. When we square the left side, we obtain sin^2 θ + cos^2 θ plus twice the product sin θ cos θ. Using the identity sin^2 θ + cos^2 θ = 1 allows us to solve for sin θ cos θ directly. This method is very efficient and requires no explicit angle values. Step-by-Step Solution: Starting from sin θ + cos θ = 1, square both sides.Left side: (sin θ + cos θ)^2 = sin^2 θ + 2 sin θ cos θ + cos^2 θ.Right side: 1^2 = 1.Use the identity sin^2 θ + cos^2 θ = 1, so the left side becomes 1 + 2 sin θ cos θ.Set this equal to 1: 1 + 2 sin θ cos θ = 1.Subtract 1 from both sides: 2 sin θ cos θ = 0.Therefore sin θ cos θ = 0. Verification / Alternative check: If sin θ cos θ = 0, then at least one of sin θ or cos θ must be zero. Consider θ = 0°, then sin θ = 0 and cos θ = 1, and sin θ + cos θ = 0 + 1 = 1, which satisfies the given condition. Similarly, θ = 90° gives sin θ = 1 and cos θ = 0, and their sum is again 1. In both cases, the product sin θ cos θ equals 0, confirming our algebraic result. Why Other Options Are Wrong: If sin θ cos θ were 1, 1/2, −1/2, or −1, then the squared equation would become 1 + 2 sin θ cos θ = 1, which would not hold. For example, if sin θ cos θ = 1/2, the left side becomes 1 + 1 = 2, which is not equal to 1. Hence all other options are inconsistent with the relation sin θ + cos θ = 1. Common Pitfalls: Students sometimes attempt to guess an angle without using algebra, or they mistakenly square each term separately without considering the cross term 2 sin θ cos θ. Remember that (a + b)^2 expands to a^2 + 2ab + b^2. Also, forgetting the fundamental identity sin^2 θ + cos^2 θ = 1 can prevent reaching the correct conclusion quickly. Final Answer: The product sin θ · cos θ is 0.

Question 9

If a^2/(b + c) = b^2/(c + a) = c^2/(a + b) = 1 for real numbers a, b and c, then what is the value of the sum 2/(1 + a) + 2/(1 + b) + 2/(1 + c)?

Accepted Answer

Introduction / Context: This problem combines symmetry and simple algebraic manipulation. The equal fractions involving a^2, b^2, and c^2 suggest that a, b, and c share a common pattern. Once we find specific values consistent with the given condition, we can evaluate the required sum. Problems like this appear frequently in algebra and aptitude tests to encourage recognition of symmetry and substitution strategies. Given Data / Assumptions: a^2/(b + c) = b^2/(c + a) = c^2/(a + b) = 1. a, b, and c are real numbers, and none of the denominators b + c, c + a, a + b is zero. The required sum is 2/(1 + a) + 2/(1 + b) + 2/(1 + c). Concept / Approach: From a^2/(b + c) = 1 we immediately obtain a^2 = b + c. Similarly, b^2 = c + a and c^2 = a + b. This is a symmetric system of three equations. Often such a system has a simple solution where all variables are equal. Testing a = b = c is an efficient first step. If a = b = c, the equations reduce to a single equation, which can be solved easily. Once a, b, and c are known, evaluating the required sum is straightforward. Step-by-Step Solution: From a^2/(b + c) = 1 we have a^2 = b + c.From b^2/(c + a) = 1 we have b^2 = c + a.From c^2/(a + b) = 1 we have c^2 = a + b.Assume a = b = c due to symmetry and test this possibility.If a = b = c, then a^2 = b + c becomes a^2 = a + a = 2a.So a^2 − 2a = 0 gives a(a − 2) = 0, hence a = 0 or a = 2.If a = 0, then b and c are also 0, but this makes denominators like a + b equal to 0, which is not allowed.Therefore a = b = c = 2 is the valid solution.Now evaluate the required sum: 2/(1 + a) + 2/(1 + b) + 2/(1 + c) = 3 * 2/(1 + 2) = 3 * (2/3) = 2. Verification / Alternative check: Substitute a = b = c = 2 back into the original equalities: a^2/(b + c) = 4/(2 + 2) = 4/4 = 1, and similarly for b and c. This confirms that a = b = c = 2 satisfies all the given conditions. The function 2/(1 + a) + 2/(1 + b) + 2/(1 + c) is fully symmetric in a, b, and c, so evaluating it at the symmetric solution is sufficient for this problem. Why Other Options Are Wrong: Values 0, 1, 3, and 6 would correspond to different specific values of a, b, and c. However, under the constraint a^2 = b + c, b^2 = c + a, and c^2 = a + b, the only non trivial symmetric solution that keeps denominators non zero is a = b = c = 2, producing a sum equal to 2 and no other value in the options. Common Pitfalls: Some students attempt to solve the three equations simultaneously in a completely general way, which is algebraically heavy and unnecessary. Others may forget to check the case a = b = c or ignore the requirement that denominators must not be zero. Recognizing symmetry and testing simple equal solutions is an important problem solving strategy for such questions. Final Answer: The value of 2/(1 + a) + 2/(1 + b) + 2/(1 + c) is 2.

Question 10

In triangle ABC, the median AD is drawn from vertex A to side BC, and it is given that the length AD equals one half of side BC. What is the measure of angle BAC (in degrees)?

Accepted Answer

Introduction / Context: This geometry question explores a special property of medians in a triangle. It connects the length of a median to the type of triangle, specifically to right angled triangles. Recognizing this property allows us to find the measure of an angle without heavy calculation. Such questions often appear in geometry sections of school exams and competitive tests. Given Data / Assumptions: Triangle ABC is given. AD is the median from vertex A to side BC, so D is the midpoint of BC. The length of the median AD satisfies AD = (1/2) * BC. We need to find angle BAC in degrees. Concept / Approach: A well known theorem states that in a right angled triangle, the median drawn from the right angle to the hypotenuse is equal to half the length of the hypotenuse. The converse is also true: if the median from a vertex to the opposite side has length equal to half the opposite side, then the triangle is right angled at that vertex, and the opposite side is the hypotenuse. Here BC plays the role of the hypotenuse, and AD is the median from A, so we expect angle BAC to be a right angle. Step-by-Step Solution: AD is a median, so D is the midpoint of BC and BD = DC = BC/2.Given AD = BC/2.Thus AD = BD = DC, which means triangle BDC is isosceles with BD = DC and point A lies such that AD is also equal to these segments.Consider triangle ABC. If angle BAC is a right angle, then BC is the hypotenuse.In a right triangle, the midpoint of the hypotenuse is equidistant from all three vertices.So if ABC is right angled at A, the midpoint D of BC will satisfy AD = BD = CD = BC/2.This matches the given condition AD = BC/2.By the converse of this result, the fact that the median from A to BC is half of BC implies that triangle ABC is right angled at A.Therefore angle BAC = 90°. Verification / Alternative check: We can also verify using coordinates and distance formulas. Place B and C so that BC is horizontal and let D be the origin. If BD = DC = r, then B and C are at (−r, 0) and (r, 0). Let A be at (0, h). Then AD = h, BD = DC = r, and BC = 2r. Condition AD = BC/2 implies h = r. The distances AB and AC are both √(r^2 + r^2) = r√2, so triangle ABC is isosceles with equal legs AB and AC and vertex A at (0, r). The angle at A is 90° because the vectors AB and AC are perpendicular. This confirms the theoretical argument. Why Other Options Are Wrong: Angles 30°, 45°, 60°, and 75° correspond to non right angled triangles where the median from A to BC is not exactly half of BC. They would not satisfy the geometric property that the midpoint of the hypotenuse is equidistant from all three vertices. Therefore none of these angles fit the given length condition for AD. Common Pitfalls: Students sometimes try to apply the medians length formula or Apollonius theorem directly without remembering the simpler special property of right triangles. Others confuse the roles of sides and medians or assume that the equality AD = BC/2 holds in many types of triangles, which is not true. Recognizing this as a right triangle configuration simplifies the problem enormously. Final Answer: The measure of angle BAC in triangle ABC is 90°.

Question 11

If (sin θ + cos θ) / (sin θ − cos θ) = 3 for an angle θ where sin θ ≠ cos θ, then what is the value of the expression sin^4 θ − cos^4 θ?

Accepted Answer

Introduction / Context: This trigonometry question involves manipulating an equation with a ratio of sums and differences of sine and cosine. The goal is to evaluate a higher power expression sin^4 θ − cos^4 θ. Such questions test comfort with identities, algebraic manipulation, and recognizing patterns like difference of squares. Given Data / Assumptions: (sin θ + cos θ) / (sin θ − cos θ) = 3. sin θ ≠ cos θ, so the denominator is non zero. We need to find sin^4 θ − cos^4 θ. Concept / Approach: The expression sin^4 θ − cos^4 θ can be written as (sin^2 θ − cos^2 θ)(sin^2 θ + cos^2 θ). Since sin^2 θ + cos^2 θ = 1, the expression simplifies to sin^2 θ − cos^2 θ. Also, sin^2 θ − cos^2 θ equals (sin θ + cos θ)(sin θ − cos θ). Therefore, once we know the product (sin θ + cos θ)(sin θ − cos θ), we directly obtain sin^4 θ − cos^4 θ. Given their ratio, we can express the squares of the sum and difference in terms of sin θ cos θ and solve for that product. Step-by-Step Solution: Let S = sin θ + cos θ and D = sin θ − cos θ.Given S / D = 3, so S = 3D.Compute S^2: S^2 = (sin θ + cos θ)^2 = sin^2 θ + 2 sin θ cos θ + cos^2 θ = 1 + 2k, where k = sin θ cos θ.Compute D^2: D^2 = (sin θ − cos θ)^2 = sin^2 θ − 2 sin θ cos θ + cos^2 θ = 1 − 2k.From S = 3D we get S^2 = 9D^2.So 1 + 2k = 9(1 − 2k) = 9 − 18k.Rearrange: 1 + 2k = 9 − 18k gives 20k = 8, so k = 8 / 20 = 2 / 5.Now D^2 = 1 − 2k = 1 − 4/5 = 1/5.Then S * D = (sin θ + cos θ)(sin θ − cos θ) = sin^2 θ − cos^2 θ = D^2 * (S / D) = D * S = (±√(1/5)) * (±3√(1/5)) = 3/5.Thus sin^2 θ − cos^2 θ = 3/5.Finally, sin^4 θ − cos^4 θ = (sin^2 θ − cos^2 θ)(1) = 3/5. Verification / Alternative check: Once we know sin^2 θ − cos^2 θ = 3/5 and sin^2 θ + cos^2 θ = 1, we can solve for sin^2 θ and cos^2 θ explicitly and check the original ratio. Solving the system gives sin^2 θ = 4/5 and cos^2 θ = 1/5 (or vice versa, depending on signs). Suitable choices of signs for sin θ and cos θ can then be used to verify that (sin θ + cos θ) / (sin θ − cos θ) equals 3. This confirms that sin^4 θ − cos^4 θ = 3/5 is consistent with the given condition. Why Other Options Are Wrong: The values 4/3, 3/4, 5/3, and 1/3 do not match the product (sin θ + cos θ)(sin θ − cos θ) computed from the ratio S / D = 3. They arise from incorrect manipulation of the squares or from forgetting that sin^4 θ − cos^4 θ reduces to sin^2 θ − cos^2 θ. Only 3/5 respects both the identity and the given ratio. Common Pitfalls: Some students attempt to square the ratio directly without tracking the relationship between S^2 and D^2. Others incorrectly treat sin^4 θ − cos^4 θ as (sin θ − cos θ)^4, which is not true. Remember to use the difference of squares factorization, and be careful when introducing the variable k for sin θ cos θ. Systematic algebra prevents such mistakes. Final Answer: The value of sin^4 θ − cos^4 θ is 3/5.

Question 12

If x^(1/4) + x^(−1/4) = 2 for a positive real number x, then what is the value of x^81 + 1/x^81?

Accepted Answer

Introduction / Context: This problem tests exponent manipulation and the use of a simple but important inequality: for a positive real t, the expression t + 1/t has a minimum of 2 when t = 1. The equation x^(1/4) + x^(−1/4) = 2 is of this form and forces x^(1/4) to be equal to 1. Once x is found, evaluating very high powers like x^81 + 1/x^81 becomes trivial. Given Data / Assumptions: x^(1/4) + x^(−1/4) = 2. x is a positive real number (so its real fourth root is defined and positive). The required value is x^81 + 1/x^81. Concept / Approach: Let t = x^(1/4). Then x^(−1/4) is 1/t. The equation becomes t + 1/t = 2. For positive t, this equation has a unique solution t = 1, since t + 1/t is minimized at 2 and increases for t not equal to 1. Once t is known, x = t^4 can be computed and then used to evaluate x^81 and 1/x^81. Step-by-Step Solution: Let t = x^(1/4). Then x^(−1/4) = 1/t.The equation becomes t + 1/t = 2.Multiply both sides by t (t is positive and non zero): t^2 + 1 = 2t.Rearrange: t^2 − 2t + 1 = 0.This factors as (t − 1)^2 = 0.Hence t = 1.Recall that t = x^(1/4), so x^(1/4) = 1 implies x = 1^4 = 1.Now compute x^81 + 1/x^81: since x = 1, we have x^81 = 1 and 1/x^81 = 1.Therefore x^81 + 1/x^81 = 1 + 1 = 2. Verification / Alternative check: The function f(t) = t + 1/t for t > 0 has derivative f'(t) > 0, so it has a global minimum at t = 1, where it equals 2. This means that the equation t + 1/t = 2 has only the solution t = 1 in positive reals. Since we explicitly assume x is positive, t is positive and therefore t = 1 is the only possibility. This confirms that x must be 1 and that any other value for x^81 + 1/x^81 would contradict the original equation. Why Other Options Are Wrong: The values −2, 0, 1, and 4 would arise only if x were complex or negative in a way that yields multiple fourth roots, or if the algebraic reasoning were incorrect. Under the condition that x is positive, the only acceptable solution is x = 1, which forces x^81 + 1/x^81 to equal 2. Therefore all other options are ruled out. Common Pitfalls: One common mistake is to treat x^(1/4) as x/4 or to misinterpret the exponents. Another is to attempt to directly raise the equation to high powers without first solving for x. This leads to very large exponents and unnecessary complexity. Introducing t = x^(1/4) and using the simple quadratic equation is the clean and reliable method. Final Answer: The value of x^81 + 1/x^81 is 2.

Question 13

If a(a + b + c) = 45, b(a + b + c) = 75 and c(a + b + c) = 105 for real numbers a, b and c, then what is the value of the sum a^2 + b^2 + c^2?

Accepted Answer

Introduction / Context: This algebra problem tests how to use a common factor (a + b + c) to express several related products and then deduce the individual values of a, b, and c. Once those values are known, computing a^2 + b^2 + c^2 is straightforward. Questions of this type encourage thinking in terms of shared structure and factoring. Given Data / Assumptions: a(a + b + c) = 45. b(a + b + c) = 75. c(a + b + c) = 105. a, b, and c are real numbers. We need a^2 + b^2 + c^2. Concept / Approach: We notice that each equation has the common factor S = a + b + c. Writing aS = 45, bS = 75, and cS = 105 allows us to express a, b, and c in terms of S. Then we can use the fact that a + b + c equals S itself to form a single equation in S and solve it. Once S is known, we find a, b, and c individually and compute their squares and sum. Step-by-Step Solution: Let S = a + b + c.From a(a + b + c) = 45 we get aS = 45, so a = 45 / S.From b(a + b + c) = 75 we get bS = 75, so b = 75 / S.From c(a + b + c) = 105 we get cS = 105, so c = 105 / S.Now sum a + b + c and set equal to S: (45 / S) + (75 / S) + (105 / S) = S.This gives (45 + 75 + 105) / S = S.Compute 45 + 75 + 105 = 225, so 225 / S = S.Therefore S^2 = 225, so S = 15 or S = −15.From the original equations, a, b, and c are 45 / S, 75 / S, and 105 / S. Taking S = 15 gives a = 3, b = 5, c = 7, which are simple positive values.If S = −15, then a, b, and c are negative but their squares will be the same as in the positive case, so a^2 + b^2 + c^2 will be unchanged.Compute a^2 + b^2 + c^2 = 3^2 + 5^2 + 7^2 = 9 + 25 + 49 = 83. Verification / Alternative check: Substitute a = 3, b = 5, c = 7, and S = 15 into the original equations. We get aS = 3 * 15 = 45, bS = 5 * 15 = 75, and cS = 7 * 15 = 105, which matches the given conditions. This confirms that our values are consistent and that a^2 + b^2 + c^2 really is 83. Why Other Options Are Wrong: Values such as 75, 217, 225, and 133 would correspond to different combinations of a, b, and c. However, those combinations would not satisfy all three proportional equations simultaneously. Only the combination that leads to 3, 5, and 7 (or their negatives) fits the given products and therefore only 83 is valid for the sum of squares. Common Pitfalls: Some students attempt to eliminate variables pairwise without introducing the sum S, which quickly becomes messy. Others forget that a + b + c equals S and treat the three equations as independent. Introducing S explicitly and solving for it keeps the algebra organized and simple. Final Answer: The value of a^2 + b^2 + c^2 is 83.

Question 14

If x^2 + 1/x^2 = 1 for a non zero real number x, then what is the value of the expression x^48 + x^42 + x^36 + x^30 + x^24 + x^18 + x^12 + x^6 + 1?

Accepted Answer

Introduction / Context: This algebraic powers question tests familiarity with cyclic patterns of powers when a base satisfies a special relation such as x^2 + 1/x^2 = 1. Instead of computing each term separately, we exploit the structure of the equation to deduce repeating behavior for even powers of x and then sum them efficiently. Such problems often appear in higher level algebra and aptitude tests to check deeper understanding of exponents and identities. Given Data / Assumptions: x is a non zero real number (or possibly complex), and x^2 + 1/x^2 = 1. We need to evaluate S = x^48 + x^42 + x^36 + x^30 + x^24 + x^18 + x^12 + x^6 + 1. Concept / Approach: Let t = x^2. Then t + 1/t = 1. This implies t^2 − t + 1 = 0, a quadratic equation whose roots are complex numbers on the unit circle. Powers of t will repeat with a certain period because t is a root of unity up to sign. We want powers of x that are multiples of 6, 12, 18, and so on, which correspond to integer powers of t. Observing the pattern in t^n for different n allows us to compute each term in the sum S as either 1 or −1 and then add them. Step-by-Step Solution: Let t = x^2. Then x is non zero, so t is non zero.Given x^2 + 1/x^2 = 1 becomes t + 1/t = 1.Multiply by t: t^2 + 1 = t, so t^2 − t + 1 = 0.From this quadratic, t is a complex number of modulus 1, and it satisfies t^2 = t − 1.One can show that t^6 = 1, so powers of t repeat every 6 steps.Now rewrite each term in S in terms of t: x^48 = (x^2)^24 = t^24, x^42 = (x^2)^21 = t^21, and so on.Compute exponents modulo 6: 24 mod 6 = 0, 21 mod 6 = 3, 18 mod 6 = 0, 15 mod 6 = 3, 12 mod 6 = 0, 9 mod 6 = 3, 6 mod 6 = 0, 3 mod 6 = 3, and 0 mod 6 = 0 for the constant 1 term, considered as t^0.For t satisfying t^6 = 1 and t^2 − t + 1 = 0, we have t^0 = 1 and t^3 = −1.Thus terms with exponent multiple of 6 contribute 1, and terms with exponent congruent to 3 mod 6 contribute −1.In S, x^48, x^36, x^24, x^12, and 1 correspond to t^0 and give 1 each.Terms x^42, x^30, x^18, and x^6 correspond to t^3 and give −1 each.So S = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 = 1. Verification / Alternative check: We can check with a specific complex solution of t^2 − t + 1 = 0, such as t = (1 + i√3)/2, which has modulus 1 and satisfies t^6 = 1. Evaluating t^0 and t^3 for this t confirms that t^0 = 1 and t^3 = −1. Substituting into the pattern for S yields exactly the same alternating sum of ones and minus ones equal to 1, confirming the reasoning. Why Other Options Are Wrong: The values −9, 0, 9, and 3 would correspond to different counts of 1 and −1 terms in the sum or to an incorrect assumption that all terms are the same. Our careful modular exponent analysis shows exactly five terms equal to 1 and four terms equal to −1, giving 1. Any other total would contradict the algebraic structure implied by t^2 − t + 1 = 0. Common Pitfalls: Students may try to compute large powers directly or forget to switch to t = x^2. Another frequent error is to assume that because x^2 + 1/x^2 = 1, x itself must be 1, which is not true here since x can be complex. The correct path is to exploit the quadratic relation for t and observe the periodic behavior of t^n. Final Answer: The value of x^48 + x^42 + x^36 + x^30 + x^24 + x^18 + x^12 + x^6 + 1 is 1.

Question 15

What is the simplified value of the trigonometric expression sin^2(90° − θ) − [sin(90° − θ) · sin θ / tan θ]?

Accepted Answer

Introduction / Context: This trigonometric simplification problem checks familiarity with complementary angle identities and the relationship between sine, cosine, and tangent. The expression may look complicated at first, but it reduces quickly if we systematically apply basic identities for 90° − θ and for tan θ. Given Data / Assumptions: Expression: sin^2(90° − θ) − [sin(90° − θ) · sin θ / tan θ]. θ is such that tan θ is defined and non zero (so sin θ and cos θ are not zero simultaneously and cos θ is non zero). Concept / Approach: We use the complementary angle identities: sin(90° − θ) = cos θ and cos(90° − θ) = sin θ. We also use tan θ = sin θ / cos θ. By replacing sin(90° − θ) with cos θ, the expression becomes easier to handle. Then we simplify the fraction involving sin θ divided by tan θ and look for cancellation. Step-by-Step Solution: First, replace sin(90° − θ) with cos θ.Then sin^2(90° − θ) becomes cos^2 θ.The expression becomes cos^2 θ − [cos θ · sin θ / tan θ].Recall that tan θ = sin θ / cos θ.Thus sin θ / tan θ = sin θ / (sin θ / cos θ) = cos θ, provided sin θ is not zero.So the bracketed term becomes cos θ · cos θ = cos^2 θ.Therefore the whole expression simplifies to cos^2 θ − cos^2 θ.Finally, cos^2 θ − cos^2 θ = 0. Verification / Alternative check: Choose a convenient angle, for example θ = 30°. Then sin(90° − 30°) = sin 60° = √3/2 and cos 30° = √3/2. Compute the original expression numerically: sin^2 60° = (√3/2)^2 = 3/4. The second term is sin 60° · sin 30° / tan 30°. Now sin 30° = 1/2, tan 30° = 1/√3. So sin 60° · sin 30° / tan 30° = (√3/2) * (1/2) / (1/√3) = (√3/4) * √3 = 3/4. Hence the expression is 3/4 − 3/4 = 0, confirming the algebraic simplification. Why Other Options Are Wrong: Values such as 1, cosec θ, cos θ, or sin θ would remain dependent on θ, while our simplification shows the expression collapses to a constant 0 for all admissible θ. Therefore any option that depends on θ cannot be correct, leaving only 0 as the consistent result. Common Pitfalls: A frequent error is to forget that sin(90° − θ) equals cos θ and instead confuse it with sin θ or tan θ. Another mistake is mishandling the division by tan θ, such as treating sin θ / tan θ as sin θ * tan θ rather than sin θ / (sin θ / cos θ). Keeping the identities clear and writing intermediate steps avoids these issues. Final Answer: The simplified value of the expression is 0.

Question 16

What is the simplified value of the trigonometric expression cosec 2A + cot 2A in terms of the angle A?

Accepted Answer

Introduction / Context: This question tests knowledge of a standard trigonometric identity that connects the sum cosec x + cot x with half angles. Recognizing this identity allows us to simplify expressions involving double angles quickly. Such identities are extremely useful in solving trigonometric equations and proving other results. Given Data / Assumptions: Expression: cosec 2A + cot 2A. A is an angle for which the required trigonometric functions are defined and finite. We must express the result in terms of A using basic identities. Concept / Approach: There is a well known identity: cosec x + cot x = cot(x / 2). This can be derived from half angle formulas or from geometric considerations in a right triangle. If we set x = 2A, then cosec 2A + cot 2A becomes cot A directly. Using this identity is the fastest route, but we can also verify it using sine and cosine definitions of the functions. Step-by-Step Solution: Recall that cosec x = 1 / sin x and cot x = cos x / sin x.Therefore cosec x + cot x = 1 / sin x + cos x / sin x = (1 + cos x) / sin x.Now use the half angle formulas: 1 + cos x = 2 cos^2(x / 2) and sin x = 2 sin(x / 2) cos(x / 2).Substitute into (1 + cos x) / sin x to get [2 cos^2(x / 2)] / [2 sin(x / 2) cos(x / 2)].Simplify by cancelling 2 and cos(x / 2): we obtain cos(x / 2) / sin(x / 2).But cos(x / 2) / sin(x / 2) is exactly cot(x / 2).Hence cosec x + cot x = cot(x / 2).Set x = 2A. Then cosec 2A + cot 2A = cot(2A / 2) = cot A. Verification / Alternative check: Take a specific angle, say A = 30°. Then 2A = 60°. Compute the left side: cosec 60° = 1 / (√3/2) = 2/√3, cot 60° = 1/√3, so their sum is 2/√3 + 1/√3 = 3/√3 = √3. On the right side, cot A = cot 30° = √3. Both sides match, giving confidence in the identity and the result. Why Other Options Are Wrong: The expressions sec A, sec(A/2), cot^2 A, and cosec A do not follow from the identity for cosec x + cot x. In particular, none of them can be simplified to (1 + cos 2A) / sin 2A, whereas cot A can. Therefore only cot A is consistent with the algebraic derivation. Common Pitfalls: Students sometimes confuse the identity cosec x + cot x = cot(x / 2) with similar looking identities for tan or sec, or they incorrectly try to express everything in terms of tan A without using half angles. Remembering and practicing standard half angle identities helps avoid such confusion. Final Answer: The simplified value of cosec 2A + cot 2A is cot A.

Question 17

In aptitude (algebra and powers), consider the quadratic equation x^2 - 2x + 1 = 0. First solve this equation for its real root x, and then, using this value, compute the exact value of the expression x^4 + 1/x^4.

Accepted Answer

Introduction / Context: This aptitude question combines a very basic quadratic equation with manipulation of powers and reciprocals. The key idea is that once we know the exact value of x from the quadratic, the expression x^4 + 1/x^4 becomes a simple substitution problem. Such questions are common in school algebra and competitive exams because they test both equation solving and comfort with indices in a single step. Given Data / Assumptions: - We are given the quadratic equation x^2 - 2x + 1 = 0. - We are asked to find x^4 + 1/x^4 once x is known. - x must be non zero, otherwise 1/x^4 would not be defined. The quadratic itself will confirm this. Concept / Approach: First, solve the quadratic equation using factoring. Then, once the root is known, substitute this root into the expression x^4 + 1/x^4. Because the quadratic has a repeated root, we only get one possible real value of x, which makes the resulting expression unique. The arithmetic in powers is very simple once x is found. Step-by-Step Solution: 1) Rewrite the quadratic: x^2 - 2x + 1 = 0. 2) Factor it as (x - 1)(x - 1) = 0, which means x - 1 = 0. 3) Therefore x = 1 is the only real root, and it is non zero, so 1/x^4 is defined. 4) Compute x^4 = 1^4 = 1. 5) Compute 1/x^4 = 1/1^4 = 1. 6) Add the two values: x^4 + 1/x^4 = 1 + 1 = 2. Verification / Alternative check: We can quickly verify the factorisation by expanding (x - 1)^2. This gives x^2 - 2x + 1, which matches the original quadratic exactly. Also, substituting x = 1 directly into the original equation gives 1 - 2 + 1 = 0, confirming that x = 1 is correct. Once that is confirmed, the values 1^4 and 1/1^4 are clearly equal to 1, so the sum 2 is secure. Why Other Options Are Wrong: Option A (1) would be correct if the expression were x^4 or 1/x^4 alone, but not their sum. Option C (0) would require the two terms to cancel, which would need x^4 = -1/x^4, impossible for real x. Option D (3) and Option E (4) are simple distractors obtained by guessing or by incorrectly adding extra terms, but none matches the exact computed sum of 2. Common Pitfalls: A common mistake is to mis factor the quadratic or to assume two different roots instead of recognising it as a perfect square. Another error is to forget that both x^4 and 1/x^4 must be evaluated at the same root. Some learners also rush and write x^4 + 1/x^4 = (x + 1/x)^4, which is algebraically incorrect. Careful stepwise substitution avoids all of these issues. Final Answer: The quadratic equation has root x = 1, and substituting this value into the expression gives x^4 + 1/x^4 = 2, so the correct answer is 2.

Question 18

In geometry (equilateral triangles and medians), the length of a median of an equilateral triangle is given as 12√3 cm. Using this information, determine the exact area of the equilateral triangle in square centimetres.

Accepted Answer

Introduction / Context: This problem tests properties of an equilateral triangle, specifically the relationship between its side length, height, and median, and how that relationship leads to the area. In an equilateral triangle, the median, altitude, and angle bisector from any vertex coincide as a single segment. Competitive exams often include such questions to check whether a learner remembers standard formulas and can apply them correctly to compute area from partial information. Given Data / Assumptions: - The triangle is equilateral, so all three sides are equal in length. - The length of a median (which is also the height) is 12√3 cm. - We need to find the area of the triangle in square centimetres. - Standard formulas for equilateral triangles and right triangles apply. Concept / Approach: In an equilateral triangle of side a, the altitude or median from any vertex to the opposite side has length (√3/2) * a. We can invert this formula to find the side length when the median is known. Once the side a is found, the area of an equilateral triangle is given by (√3/4) * a^2. The question is essentially a two step substitution problem using these standard geometric results. Step-by-Step Solution: 1) Let the side of the equilateral triangle be a cm. Then the median (also the height) is h = (√3/2) * a. 2) We are told that h = 12√3, so (√3/2) * a = 12√3. 3) Divide both sides by √3 to get a/2 = 12, which gives a = 24 cm. 4) Use the area formula for an equilateral triangle: Area = (√3/4) * a^2. 5) Substitute a = 24: Area = (√3/4) * 24^2 = (√3/4) * 576. 6) Simplify: 576/4 = 144, so Area = 144√3 square centimetres. Verification / Alternative check: We can confirm the side length by forming a right triangle made from half of the equilateral triangle. Half the base is a/2 = 12, and the height is 12√3. By the Pythagoras relation, a^2 = (a/2)^2 + h^2 = 12^2 + (12√3)^2 = 144 + 432 = 576, giving a = 24, which matches our earlier result. Substituting back again produces 144√3, confirming the calculation. Why Other Options Are Wrong: Option A (288 sq. cm) and Option B (144 sq. cm) ignore the √3 factor required in the area formula of an equilateral triangle. Option C (288√3 sq. cm) doubles the true area, which might arise from mis using the altitude formula. Option E (96√3 sq. cm) results from an incorrect intermediate side length. Only Option D correctly matches the exact area based on the given median length. Common Pitfalls: Many students confuse the role of the median with a general median in scalene triangles and forget that in an equilateral triangle it is also the altitude. Others mistakenly use 1/2 * base * height but insert the median value without first verifying the correct base, especially when they do not remember the altitude side relationship. Another common error is missing or misplacing the √3 factor. Keeping a clear list of standard formulas for equilateral triangles helps avoid these mistakes. Final Answer: Using the relation between the median and side length in an equilateral triangle and then applying the area formula, the area of the triangle is 144√3 sq. cm.

Question 19

In circle geometry, O is the centre of the circumcircle of triangle ABC, and the point A and the chord BC lie on opposite sides of O. If the central angle ∠BOC is 150 degrees, find the measure of the inscribed angle ∠BAC in degrees.

Accepted Answer

Introduction / Context: This question checks understanding of the relationship between central angles and inscribed angles in a circle. When a triangle is inscribed in a circle, each angle at the circumference corresponds to a central angle subtended by the same chord. Mastery of this relationship is very important in coordinate geometry, Euclidean geometry, and many entrance exam problems involving circles and cyclic quadrilaterals. Given Data / Assumptions: - O is the centre of the circumcircle of triangle ABC. - Chord BC and vertex A are on the circle, and A and BC lie on opposite sides of O, ensuring a proper triangle configuration. - The central angle ∠BOC is given as 150 degrees. - We must find the inscribed angle ∠BAC, which also subtends chord BC. Concept / Approach: The key theorem is that an inscribed angle in a circle is half the measure of the central angle subtending the same chord. Here, both the central angle ∠BOC and the inscribed angle ∠BAC stand on chord BC. Therefore, ∠BAC should be exactly half of ∠BOC, as long as they are on the same arc of the circle. This is a direct application question with no additional construction needed. Step-by-Step Solution: 1) Identify that chord BC subtends the central angle ∠BOC at the centre O. 2) Recognise that the same chord BC subtends the inscribed angle ∠BAC at the point A on the circumference. 3) Apply the theorem: inscribed angle = half of the central angle standing on the same chord. 4) Compute ∠BAC = (1/2) * ∠BOC = (1/2) * 150 degrees. 5) Therefore, ∠BAC = 75 degrees. Verification / Alternative check: We can reason in reverse: if an inscribed angle were 75 degrees, then the central angle standing on the same chord must be 150 degrees. This is consistent with the given data. Additionally, this result does not depend on the precise shape of triangle ABC, only on the circle property, so any configuration with chord BC and vertex A on the circle satisfying the condition will always give the same angle value. Why Other Options Are Wrong: Option B (70 degrees), Option D (65 degrees), and Option E (80 degrees) are simply numbers near the correct result, often used to trap students who attempt rough estimates without applying the exact half relation. Option C (60 degrees) might come from confusing the given angle with an interior triangle angle of a regular hexagon or other special figure, but it does not match half of 150 degrees. Only Option A matches one half of the central angle exactly. Common Pitfalls: A frequent error is to confuse which angle is central and which is inscribed, leading to doubling instead of halving, or to using the wrong chord. Some learners also mistakenly think that the inscribed angle is always half of any angle in the diagram rather than specifically the central angle on the same chord. Drawing a quick sketch and marking the chord BC and vertex A clearly can prevent such misunderstandings. Final Answer: Using the circle theorem that an inscribed angle is half the corresponding central angle, we get ∠BAC = 150° / 2 = 75°.

Question 20

In trigonometry, the real angle X (with 0° ≤ X ≤ 90°) satisfies the equation sin X − cos X = 1. Using this information, determine the exact value of the sum sin X + cos X.

Accepted Answer

Introduction / Context: This question tests basic trigonometric identities and reasoning about angles in the first quadrant. Instead of demanding memorisation of special angle values directly, it uses a simple equation relating sin X and cos X to guide you to a specific angle. Once that angle is known, another simple trigonometric expression has to be evaluated. Such questions check both conceptual understanding and careful algebraic manipulation. Given Data / Assumptions: - X is an angle measured in degrees with 0° ≤ X ≤ 90° (first quadrant). - The equation sin X − cos X = 1 holds exactly. - We must find sin X + cos X for the same angle X. - Standard trigonometric values for special angles are valid, and sin X and cos X are non negative in this interval. Concept / Approach: The strategy is to use the Pythagoras identity sin^2 X + cos^2 X = 1 along with the given relation sin X − cos X = 1. Squaring the given equation helps eliminate the square root behaviour and relate sin X cos X to the identity. Once possible values for sin X and cos X are found, we can quickly compute their sum. Because we are in the first quadrant, we can discard extraneous possibilities based on sign. Step-by-Step Solution: 1) Start with the equation sin X − cos X = 1. 2) Square both sides: (sin X − cos X)^2 = 1^2. 3) Expand the left side: sin^2 X + cos^2 X − 2 sin X cos X = 1. 4) Use sin^2 X + cos^2 X = 1 to simplify, giving 1 − 2 sin X cos X = 1. 5) Subtract 1 from both sides: −2 sin X cos X = 0, so sin X cos X = 0. 6) Thus either sin X = 0 or cos X = 0. In 0° to 90°, sin X = 0 only at X = 0°, which gives 0 − 1 = −1, not 1. So X must be 90°. 7) At X = 90°, sin 90° = 1 and cos 90° = 0, so sin X + cos X = 1 + 0 = 1. Verification / Alternative check: We can directly test the candidate angle X = 90° in the original equation. Substituting gives sin 90° − cos 90° = 1 − 0 = 1, which matches the condition exactly. No other angle in the first quadrant yields sin X − cos X = 1, so this solution is unique. Therefore, the corresponding sum sin X + cos X must be 1, confirming our result. Why Other Options Are Wrong: Option B (0) would correspond to sin X = −cos X, which cannot hold in the first quadrant where both are non negative. Option C (1/2) and Option D (2) are typical guesses without solving the equation. Option E (√2) comes from confusing this with the known value of sin 45° + cos 45°, but X is not 45° here. Only Option A matches the exact computed sum from the valid angle X = 90°. Common Pitfalls: Learners may try to guess the angle by memory rather than using algebra, which can be risky. Another frequent error is to forget that squaring an equation can introduce extraneous solutions, so the result must always be checked in the original equation. Some students also wrongly assume that sin X and cos X are always equal in the first quadrant, which is only true at 45°. Careful use of identities and verification avoids these mistakes. Final Answer: The only first quadrant angle that satisfies sin X − cos X = 1 is X = 90°, and for that angle sin X + cos X = 1.

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