Question 1

What is the exact simplified value of the trigonometric expression cot 45° + (1/3) cosec 60°, expressed in standard surd form?

Accepted Answer

Introduction / Context: This question checks basic trigonometric values for standard angles and the ability to combine them into a single simplified expression. Values at 45 degrees and 60 degrees are frequently tested because they produce neat surds involving sqrt(2) and sqrt(3). The goal is to simplify cot 45° and cosec 60° and then combine them carefully into one fraction in exact form. Given Data / Assumptions: The expression is cot 45° + (1/3) cosec 60°. We work in degrees. We use standard exact values for trigonometric functions at 45° and 60°. We are required to express the answer in exact surd form without decimal approximation. Concept / Approach: Recall the standard values: tan 45° = 1, so cot 45° = 1 / tan 45° = 1. For 60 degrees, sin 60° = sqrt(3) / 2, so cosec 60° = 1 / sin 60° = 2 / sqrt(3). Substitute these values into the expression and then simplify systematically. To present the result cleanly, rationalise the denominator where necessary and combine the constant 1 with the surd term over a common denominator. Step-by-Step Solution: Compute cot 45°. Since tan 45° = 1, cot 45° = 1.Compute cosec 60°. Since sin 60° = sqrt(3) / 2, cosec 60° = 1 / sin 60° = 2 / sqrt(3).Now evaluate (1/3) cosec 60°: (1/3) * (2 / sqrt(3)) = 2 / (3 sqrt(3)).So the expression becomes 1 + 2 / (3 sqrt(3)).Rationalise the small fraction: 2 / (3 sqrt(3)) = (2 sqrt(3)) / (9).Thus the total is 1 + (2 sqrt(3) / 9) = (9 / 9) + (2 sqrt(3) / 9) = (9 + 2 sqrt(3)) / 9. Verification / Alternative check: We can check numerically using approximate values. Take sqrt(3) approximately equal to 1.732. Then sin 60° is about 0.866, cosec 60° is about 1.155, and (1/3) cosec 60° is about 0.385. Adding 1 gives approximately 1.385. Now evaluate (9 + 2 sqrt(3)) / 9 numerically: 2 sqrt(3) is about 3.464, so 9 + 3.464 ≈ 12.464. Divide by 9 to get about 1.3849, which matches our previous estimate, confirming the correctness. Why Other Options Are Wrong: √3 and √3 + 2 are much larger and do not match the approximate value near 1.385. (2√2 + 3)/√6 does not simplify to the same numerical value when computed. (3 + √3)/3 gives roughly 1.244, which is different from 1.385. Only (9 + 2√3)/9 agrees with both exact simplification and numerical checking. Common Pitfalls: Using wrong standard values for sine or cosine at 60 degrees or for tangent at 45 degrees. Forgetting to multiply cosec 60° by 1/3 carefully, which alters the final answer. Not rationalising the denominator correctly and therefore failing to match any of the given options. Final Answer: (9 + 2√3)/9

Question 2

In right-angled triangle DEF, where angle E = 90° and angle F = 45°, what is the exact value of the product sin F × tan F?

Accepted Answer

Introduction / Context: This question is a straightforward test of standard trigonometric values for a special angle, 45 degrees. Right-angled triangles with one acute angle equal to 45 degrees are isosceles, and the trigonometric ratios for 45 degrees are well known. The task is to compute sin 45 degrees and tan 45 degrees and then multiply them together correctly. Given Data / Assumptions: Triangle DEF is right-angled at E, so angle E = 90°. Angle F = 45°, so the remaining angle D is also 45°. We are asked to find sin F × tan F where F = 45°. Standard exact values for trigonometric functions at 45° are used. Concept / Approach: For an angle of 45 degrees, consider the standard right-angled isosceles triangle with legs of equal length, say 1, and hypotenuse sqrt(2). From this, sin 45° is opposite over hypotenuse, which is 1 / sqrt(2). Tan 45° is opposite over adjacent, which is 1. After recalling these values, we simply multiply sin 45° by tan 45° and simplify the result. Because tan 45° equals 1, the product is just sin 45° itself. Step-by-Step Solution: In a 45°–45°–90° triangle, let each leg be 1 and the hypotenuse be sqrt(2).Compute sin 45°. Sin 45° = opposite / hypotenuse = 1 / sqrt(2).Compute tan 45°. Tan 45° = opposite / adjacent = 1 / 1 = 1.Now find the product sin F × tan F: sin 45° × tan 45° = (1 / sqrt(2)) * 1.This equals 1 / sqrt(2). No further simplification is needed unless rationalisation is required, but the option given uses 1/√2. Verification / Alternative check: Use approximate decimal values to check. Sin 45° is about 0.707, tan 45° is exactly 1, so their product is about 0.707. The value 1/√2 is approximately 1 / 1.414 ≈ 0.707, which matches. None of the other options produce this value, confirming the correctness of 1/√2 as the exact form. Why Other Options Are Wrong: √2 is about 1.414, which is the reciprocal of the correct value. 1/√3 and 2/√3 involve sqrt(3) and are not related to the 45 degree ratios in this case. 1 would be the result of tan 45° alone, not the product with sin 45°. Only 1/√2 matches sin 45° × tan 45°. Common Pitfalls: Confusing sin 45° with cos 45°. For 45 degrees they are equal but sometimes students misremember the exact form. Thinking tan 45° is something other than 1 and thus changing the product incorrectly. Misinterpreting the right angle location and trying to use triangle side lengths instead of known trigonometric values. Final Answer: 1/√2

Question 3

If cot θ = 21/20 for an acute angle θ in a right-angled triangle, what is the exact value of sec θ expressed as a rational fraction?

Accepted Answer

Introduction / Context: This problem tests the relationship among the six basic trigonometric ratios and how to move from one ratio to another using right triangle geometry and the Pythagorean theorem. Given cot θ, you are asked to find sec θ. This involves interpreting cot θ as a ratio of sides in a right triangle and then computing cos θ and its reciprocal, sec θ. Given Data / Assumptions: cot θ = 21/20. θ is an acute angle, so all basic trigonometric ratios are positive. We consider a right triangle where the angle θ is one of the acute angles. We must determine sec θ as a simplified fraction. Concept / Approach: By definition, cot θ = adjacent / opposite in a right triangle. So if cot θ = 21/20, we can take the side adjacent to θ as 21 units and the side opposite θ as 20 units. Then we use the Pythagorean theorem to find the hypotenuse. Once we know all three sides, cos θ = adjacent / hypotenuse, and sec θ is its reciprocal, hypotenuse / adjacent. The numbers 20, 21, and 29 form a well known Pythagorean triple because 20^2 + 21^2 = 29^2. Step-by-Step Solution: Interpret cot θ = 21/20 as adjacent side = 21 and opposite side = 20.Use Pythagoras theorem to find the hypotenuse h: h^2 = 21^2 + 20^2.Compute 21^2 = 441 and 20^2 = 400, so h^2 = 441 + 400 = 841.Thus h = sqrt(841) = 29.Now cos θ = adjacent / hypotenuse = 21 / 29.Secant is the reciprocal of cosine: sec θ = 1 / cos θ = 29 / 21. Verification / Alternative check: We can verify by computing cot θ and sec θ directly from the sides. Cot θ = adjacent / opposite = 21 / 20, which matches the given ratio. Sec θ = hypotenuse / adjacent = 29 / 21. Also check the identity 1 + tan^2 θ = sec^2 θ. Here tan θ = opposite / adjacent = 20 / 21, so tan^2 θ = 400 / 441. Then sec^2 θ should be 1 + 400 / 441 = 841 / 441, and sec θ is sqrt(841 / 441) = 29 / 21, consistent with our result. Why Other Options Are Wrong: 21/29 is cos θ, not sec θ. 29/20 and 20/29 correspond to using the wrong pair of sides to form the ratio. 20/21 equals tan θ, not sec θ. Only 29/21 correctly represents the reciprocal of cos θ. Common Pitfalls: Confusing cot θ with tan θ and swapping the opposite and adjacent sides. Forgetting that sec θ is the reciprocal of cos θ and instead dividing in the wrong order. Making arithmetic mistakes while computing the hypotenuse, especially when squaring and adding 20 and 21. Final Answer: 29/21

Question 4

Using properties of powers and number theory, find a common factor of the numbers 37^57 + 43^57 and 37^37 + 43^37.

Accepted Answer

Introduction / Context: This question is about recognizing algebraic patterns in expressions involving large powers. Instead of trying to compute gigantic numbers like 37^57 or 43^57, we use known divisibility properties for expressions of the form a^n + b^n where n is an odd integer. This is a standard idea in number theory used in many aptitude and Olympiad style questions. Given Data / Assumptions: Two numbers are given: N1 = 37^57 + 43^57 and N2 = 37^37 + 43^37. Both exponents 57 and 37 are odd numbers. We want a common factor shared by N1 and N2. We are given options involving simple combinations of 37 and 43. Concept / Approach: For any integers a and b, if n is odd, then a^n + b^n is divisible by a + b. This can be seen from the factorization of a^n + b^n when n is odd, which includes the factor (a + b). In our problem, both 37^57 + 43^57 and 37^37 + 43^37 have odd exponents, so each is divisible by 37 + 43. Therefore, the sum 37 + 43 is a common factor of both numbers. Step-by-Step Solution: Let a = 37 and b = 43.Observe that 57 and 37 are odd integers.For any odd exponent n, the expression a^n + b^n has a factor (a + b).Thus 37^57 + 43^57 is divisible by 37 + 43.Similarly, 37^37 + 43^37 is also divisible by 37 + 43.Therefore, 37 + 43 is a common factor of both numbers.Compute 37 + 43 = 80 for verification, though the option is given symbolically. Verification / Alternative check: We can check the idea on a smaller example. Let a = 2 and b = 3. Then 2^3 + 3^3 = 8 + 27 = 35, which is divisible by 2 + 3 = 5. Similarly, 2^5 + 3^5 = 32 + 243 = 275, which is also divisible by 5. This confirms the pattern for odd powers. Returning to our problem, the same algebraic principle applies to 37 and 43 with exponents 57 and 37, confirming that 37 + 43 is indeed a common factor. Why Other Options Are Wrong: (43 − 37) equals 6. While 6 may or may not divide these sums, it is not guaranteed by the standard identity for a^n + b^n. (37 + 37) equals 74, which does not have a special role in the factorization of a^n + b^n for a = 37, b = 43. The number 10 is arbitrary in this context and not linked to the algebraic structure. 6(43 − 37) equals 36, which again has no guaranteed relation to the given expressions. Common Pitfalls: Attempting to compute or estimate huge powers directly, which is not practical. Confusing the identities for a^n − b^n and a^n + b^n and using the wrong factorization. Overlooking the condition that n must be odd for a^n + b^n to have a factor (a + b). Final Answer: (43 + 37)

Question 5

Using the identity for the difference of squares, calculate the value of the numerical expression (61681^2 − 31681^2) ÷ 30000.

Accepted Answer

Introduction / Context: This question demonstrates how algebraic identities can simplify apparently huge numerical calculations. Directly squaring 61681 and 31681 would be very tedious. However, the expression has the special form a^2 − b^2, which can be rewritten as (a − b)(a + b). Recognizing and using this identity simplifies the calculation dramatically. Given Data / Assumptions: The expression to evaluate is (61681^2 − 31681^2) ÷ 30000. We assume normal integer arithmetic. The numbers 61681 and 31681 are just large integers with no further hidden structure. The task is to compute the exact integer result. Concept / Approach: Use the identity a^2 − b^2 = (a − b)(a + b). Here a = 61681 and b = 31681. After factorization, we see that the numerator becomes (a − b)(a + b), and in this specific problem a − b equals 30000, which matches the divisor in the denominator. This allows immediate cancellation and leaves only the sum a + b to compute, which is manageable even by mental arithmetic. Step-by-Step Solution: Let a = 61681 and b = 31681.Compute a − b: 61681 − 31681 = 30000.Compute a + b: 61681 + 31681 = 93362.Using the identity a^2 − b^2 = (a − b)(a + b), the numerator becomes 30000 * 93362.The full expression is (30000 * 93362) ÷ 30000.Cancel the factor 30000 in numerator and denominator: the result is simply 93362. Verification / Alternative check: An approximate check can be done by noticing that 61681 is roughly 62000 and 31681 is roughly 32000. The difference of squares of those rough values, divided by 30000, would be close to the sum of the two numbers, which is about 94000. Our exact answer 93362 is near that rough estimate, indicating that the calculation is plausible. More precise checks would involve computing the exact product, but the algebraic identity is exact and guarantees the correctness once the basic subtraction and addition are correct. Why Other Options Are Wrong: Values like 94362, 95362, and 96362 are close but do not equal the exact sum 61681 + 31681. 93352 differs by 10 from the correct answer and would result from a minor arithmetic mistake when adding 61681 and 31681. Only 93362 corresponds exactly to a + b after correct computation. Common Pitfalls: Failing to recognize the difference of squares pattern and trying to square the large numbers directly. Subtracting or adding the large numbers incorrectly, especially when carrying digits. Forgetting to cancel the common factor 30000 in numerator and denominator after factorization. Final Answer: 93362

Question 6

For which value of the parameter g do the linear equations 6x + 12y + 9 = 0 and 2x + g y + 3 = 0 represent the same straight line, that is, have infinitely many common solutions?

Accepted Answer

Introduction / Context: This question checks understanding of when two linear equations in x and y represent the same straight line. If they represent the same line, then every point on one line lies on the other, meaning the equations are proportional. This is a standard topic in coordinate geometry and linear equations in two variables. Given Data / Assumptions: The first line is given by 6x + 12y + 9 = 0. The second line is 2x + g y + 3 = 0, where g is a real parameter. We must find g such that these two equations describe exactly the same geometric line. All coefficients are real numbers. Concept / Approach: Two linear equations a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 represent the same straight line if and only if there exists a non zero constant k such that a1 = k a2, b1 = k b2, and c1 = k c2. In other words, the ratios a1/a2, b1/b2, and c1/c2 are all equal. We apply this idea to the given pair of equations and solve for g so that the three ratios match. Step-by-Step Solution: Write the equations in coefficient form: For the first line, a1 = 6, b1 = 12, c1 = 9.For the second line, a2 = 2, b2 = g, c2 = 3.If the lines are the same, then 6/2 = 12/g = 9/3.Compute 6/2 = 3 and 9/3 = 3, so the common ratio must be 3.Set 12/g equal to 3: 12/g = 3.Solve for g: multiply both sides by g to get 12 = 3g, so g = 12/3 = 4. Verification / Alternative check: With g = 4, the second equation becomes 2x + 4y + 3 = 0. Multiply this entire equation by 3 to see if it matches the first: 3 * (2x + 4y + 3) = 6x + 12y + 9 = 0. This is exactly the first equation. Therefore, for g = 4, each equation is a constant multiple of the other, and they represent the same line. Any other value of g would make the ratios of corresponding coefficients unequal and hence describe a different line or a line that intersects at a single point. Why Other Options Are Wrong: For g = 3, 6/2 = 3 but 12/3 = 4, so the ratios are not equal. For g = 6, 12/6 = 2, which does not match 6/2 = 3. For g = 9, 12/9 = 4/3, again different from 3. For g = 0, the second equation becomes 2x + 3 = 0, which is a vertical shift and not a multiple of the first equation. Common Pitfalls: Equating only two of the ratios and forgetting to check the third constant term, which must also match the same ratio. Mixing up the order of coefficients when forming ratios a1/a2, b1/b2, and c1/c2. Assuming that having equal slopes alone is enough; equal slopes give parallel lines, but equal ratios including constants give exactly the same line. Final Answer: 4

Question 7

In right-angled triangle ABC, right angled at B, if ∠A = 45° and hence ∠C = 45°, what is the exact simplified value of the expression (cosec C + 1/√3)?

Accepted Answer

Introduction / Context: This problem uses trigonometric values for a 45 degree angle and asks you to combine cosecant with a constant fraction involving sqrt(3). The triangle is right angled at B with the other two angles both equal to 45 degrees, making it a special isosceles right triangle. The purpose is to apply standard trigonometric values and then perform a careful algebraic simplification. Given Data / Assumptions: Triangle ABC is right angled at B, so angle B = 90°. Angle A = 45°, so angle C is also 45°. We must compute the expression cosec C + 1/√3. Standard exact values for trigonometric ratios at 45° are used. Concept / Approach: For an angle of 45 degrees in a right triangle, sin 45° is 1 / sqrt(2), so cosec 45° is its reciprocal, sqrt(2). Once we know cosec C, which is cosec 45°, we simply add 1/√3. To match the answer options, it is useful to combine these two terms into a single fraction and simplify the surds. This involves using a common denominator and possibly rationalising if needed. Step-by-Step Solution: Because angle C = 45°, cosec C = cosec 45°.Recall sin 45° = 1 / sqrt(2), so cosec 45° = 1 / sin 45° = sqrt(2).The expression becomes sqrt(2) + 1/√3.Write both terms over a common denominator of √3: sqrt(2) = (sqrt(2) * √3) / √3 = sqrt(6) / √3.Thus sqrt(2) + 1/√3 = (sqrt(6) / √3) + (1 / √3) = (sqrt(6) + 1) / √3.Therefore, the simplified expression is (√6+1)/√3. Verification / Alternative check: Use approximate decimal values. Take sqrt(2) approximately equal to 1.414 and sqrt(3) approximately equal to 1.732. Then 1/√3 is about 0.577. The original expression sqrt(2) + 1/√3 is about 1.414 + 0.577 = 1.991. Now evaluate (√6+1)/√3. Since sqrt(6) is about 2.449, the numerator is about 3.449. Dividing by 1.732 gives approximately 1.991, which matches, confirming the algebraic simplification. Why Other Options Are Wrong: (3√2+1)/3 and (2+√3)/2 evaluate to different decimal values and do not equal about 1.991. 5/√3 equals roughly 5/1.732 ≈ 2.886, which is much larger. The value 2 is only an approximation and does not match the exact surd expression. Only (√6+1)/√3 matches both the exact and approximate value of the given expression. Common Pitfalls: Using the wrong value for sin 45° or cosec 45°. Failing to find a common denominator when adding sqrt(2) and 1/√3. Making mistakes when multiplying surds, for example confusing sqrt(2) * sqrt(3) with sqrt(5) instead of sqrt(6). Final Answer: (√6+1)/√3

Question 8

In right-angled triangle UVW, with angle V = 90°, if cosec U = 13/12 and side UV = 2.5 cm, what is the length (in cm) of side VW opposite angle U?

Accepted Answer

Introduction / Context: This question combines trigonometric ratios with right triangle geometry and a real length. You are given cosec U and one side length, and you must find another side of the right triangle. It tests understanding of how cosec relates to the ratio of hypotenuse to opposite side, and also how to recognise standard Pythagorean triples. Given Data / Assumptions: Triangle UVW is right-angled at V, so angle V = 90°. cosec U = 13/12. Side UV has length 2.5 cm. We must find the length of side VW, which is opposite angle U. Concept / Approach: By definition, cosec U = hypotenuse / opposite. In triangle UVW, the side opposite angle U is VW, and the hypotenuse is UW. If cosec U = 13/12, then we can take UW = 13k and VW = 12k for some positive scale factor k. The remaining side UV is adjacent to angle U and must satisfy the Pythagorean triple relation, giving UV = 5k in a 5, 12, 13 triangle. Since UV is given as 2.5 cm, we can find k and then compute VW = 12k. Step-by-Step Solution: From cosec U = 13/12, set hypotenuse UW = 13k and opposite side VW = 12k.In a 5, 12, 13 right triangle, the third side is 5k. Here that side corresponds to UV.Thus UV = 5k. We are given UV = 2.5 cm.So 5k = 2.5, which gives k = 2.5 / 5 = 0.5.Now compute VW = 12k = 12 * 0.5 = 6 cm.Therefore, the length of side VW is 6 cm. Verification / Alternative check: Check consistency with cosec U. With k = 0.5, UW = 13k = 6.5 cm and VW = 6 cm. Then sin U = opposite / hypotenuse = VW / UW = 6 / 6.5 = 12 / 13. Hence cosec U = 1 / sin U = 13 / 12, which matches the given value. Also, confirm the Pythagorean theorem: UV^2 + VW^2 = 2.5^2 + 6^2 = 6.25 + 36 = 42.25, and UW^2 = 6.5^2 = 42.25. This verifies that the sides form a valid right triangle. Why Other Options Are Wrong: 6.5 is the hypotenuse, not the opposite side VW. 4, 5, and 5.6 do not maintain the correct cosec U value or the 5, 12, 13 ratio when combined with UV = 2.5. Only 6 cm produces both the correct triangle shape and the given cosec U. Common Pitfalls: Misidentifying which side is opposite angle U and which is the hypotenuse in the right triangle. Forgetting that once you introduce k in the triple 5, 12, 13, all sides must be scaled by the same factor. Using the wrong trigonometric ratio (for example, treating cosec as if it were sec or something else). Final Answer: 6

Question 9

For which real value of x do the linear expressions 7x + 13 and 13x − 7 become equal to each other?

Accepted Answer

Introduction / Context: This is a basic algebra question about solving a simple linear equation. The equality of two linear expressions is used to find the value of x. Such questions test your ability to rearrange and simplify equations to isolate the variable and are a foundation for more advanced algebra topics. Given Data / Assumptions: The two expressions are 7x + 13 and 13x − 7. They are equal at some real value of x. We must solve 7x + 13 = 13x − 7 for x. Standard properties of equality and basic arithmetic operations apply. Concept / Approach: To solve an equation like 7x + 13 = 13x − 7, we bring all terms involving x to one side and constant terms to the other side. This gives a simple form kx = constant, from which x is easily obtained by dividing. Throughout the process, whatever operation is done to one side of the equation must be done to the other side to preserve equality. Step-by-Step Solution: Start with the equation 7x + 13 = 13x − 7.Subtract 7x from both sides to collect x terms on the right: 13 = 6x − 7.Add 7 to both sides to move constants to the left: 13 + 7 = 6x.Compute 13 + 7 = 20, so 20 = 6x.Divide both sides by 6: x = 20 / 6 = 10 / 3 after simplifying the fraction by dividing numerator and denominator by 2. Verification / Alternative check: Substitute x = 10/3 into both expressions. For 7x + 13: 7 * (10/3) + 13 = 70/3 + 13 = 70/3 + 39/3 = 109/3. For 13x − 7: 13 * (10/3) − 7 = 130/3 − 7 = 130/3 − 21/3 = 109/3. Since both expressions give the same value 109/3 when x = 10/3, the solution is correct. Why Other Options Are Wrong: For x = −10/3, 7x + 13 and 13x − 7 produce different values. For x = 3/10 or −3/10, the values of the expressions are not equal. For x = 0, 7x + 13 equals 13 and 13x − 7 equals −7, which clearly are not equal. Only x = 10/3 satisfies the equation. Common Pitfalls: Moving terms across the equals sign without changing their sign correctly. Making arithmetic errors while adding or subtracting constants like 13 and 7. Failing to simplify the final fraction 20/6 to 10/3, though even 20/6 would still represent the same numeric value. Final Answer: 10/3

Question 10

When the algebraic expression 256a^2 b^2 c^2 is divided by 64a^2, what is the resulting simplified expression in terms of a, b, and c?

Accepted Answer

Introduction / Context: This question tests basic manipulation of algebraic expressions involving powers and coefficients. It is essentially a division of one monomial by another. The goal is to simplify by dividing coefficients and subtracting exponents for matching variables, a common skill required in algebra and polynomial operations. Given Data / Assumptions: The numerator is 256a^2 b^2 c^2. The denominator is 64a^2. All variables a, b, and c are assumed non zero for division to make sense. We are asked to write the simplified result in standard algebraic form. Concept / Approach: When dividing monomials, divide the numerical coefficients and subtract the exponents of like variables. For example, a^m / a^n = a^(m − n) when a is not zero. Here the coefficient 256 is divided by 64, and a^2 in numerator is divided by a^2 in denominator. The remaining variables b^2 and c^2 stay as they are since there are no matching powers of b or c in the denominator. Step-by-Step Solution: Write the expression as (256a^2 b^2 c^2) / (64a^2).First divide the coefficients: 256 / 64 = 4.Next divide the powers of a: a^2 / a^2 = a^(2 − 2) = a^0 = 1.The variables b^2 and c^2 remain unchanged because there are no b or c factors in the denominator.Thus the result is 4 * b^2 * c^2.So the simplified expression is 4b^2c^2. Verification / Alternative check: To verify, choose simple non zero values for a, b, and c, such as a = 1, b = 2, c = 3. Compute the original fraction: numerator = 256 * 1^2 * 2^2 * 3^2 = 256 * 4 * 9 = 9216. Denominator = 64 * 1^2 = 64. Then 9216 / 64 = 144. Now compute 4b^2c^2 directly: 4 * 2^2 * 3^2 = 4 * 4 * 9 = 144. The results match, confirming the simplification is correct. Why Other Options Are Wrong: 2bc^2 and 2b^2c result from dividing the coefficient incorrectly by a factor of 2 instead of 4. 4bc drops one power of both b and c, ignoring the original exponents b^2 and c^2. b^2c^2 misses the coefficient 4 entirely. Only 4b^2c^2 correctly accounts for both coefficients and exponents. Common Pitfalls: Forgetting that a^2 / a^2 equals 1 and mistakenly leaving a factor of a in the final result. Dividing 256 by 64 incorrectly, especially if doing mental arithmetic under time pressure. Misreading the original expression and changing exponents of b or c when no division affects them. Final Answer: 4b^2c^2

Question 11

Rehman walks at 8 km/h and Roma cycles at 13 km/h towards each other. If they meet after 36 minutes, what distance in kilometres initially separated them?

Accepted Answer

Introduction / Context: This is a classic relative speed problem from time, speed, and distance. Two people move towards each other from opposite directions, and you are given their speeds and the time until they meet. The concept of relative speed allows you to treat the situation as if one person is stationary and the other is moving at the sum of their speeds, which simplifies the calculation of the initial distance between them. Given Data / Assumptions: Rehman walks at 8 km/h. Roma cycles at 13 km/h. They are moving directly towards each other. They meet after 36 minutes. We must find the initial distance between them in kilometres. Concept / Approach: When two objects move towards each other along a straight line, their relative speed is the sum of their individual speeds. Distance covered before they meet is then relative speed multiplied by time. Since the time is given in minutes, it must be converted to hours to match the units of km/h. Once we find the distance using Distance = Speed * Time, we simply express it to one decimal place as needed. Step-by-Step Solution: Compute the relative speed: 8 km/h + 13 km/h = 21 km/h.Convert time from minutes to hours: 36 minutes = 36/60 hours = 0.6 hours.Use the formula Distance = Speed * Time with relative speed.Distance = 21 km/h * 0.6 h = 12.6 km.Therefore, the initial distance between Rehman and Roma was 12.6 kilometres. Verification / Alternative check: We can split the total distance into individual portions covered by each person. In 0.6 hours, Rehman walking at 8 km/h covers 8 * 0.6 = 4.8 km. Roma cycling at 13 km/h covers 13 * 0.6 = 7.8 km. Adding these gives 4.8 + 7.8 = 12.6 km, which matches the distance computed via relative speed. This double check confirms the calculation is correct. Why Other Options Are Wrong: 18.9 km and 15.8 km are larger than the correct distance and would require either more time or higher speeds. 9.5 km and 21.0 km do not match the product of relative speed and time and fail the individual distance split check. Only 12.6 km satisfies both the relative speed formula and the individual distance calculations. Common Pitfalls: Forgetting to convert minutes into hours, which changes the distance by a factor of 60. Incorrectly subtracting the speeds instead of adding them when the persons move towards each other. Making arithmetic mistakes in multiplying 21 by 0.6 or adding the individual distances. Final Answer: 12.6 km

Question 12

If the simultaneous linear equations 3x + 2y = 7 and 4x − y = 24 hold for real numbers x and y, what is the value of (x − y)?

Accepted Answer

Introduction / Context: This question involves solving a system of two linear equations in two variables and then computing a simple expression involving x and y. Such problems are central in algebra and appear frequently in aptitude exams. The aim is to find x and y and then evaluate x − y. You can use substitution or elimination; elimination works quickly here. Given Data / Assumptions: The first equation is 3x + 2y = 7. The second equation is 4x − y = 24. x and y are real numbers. We must find x − y once the system is solved. Concept / Approach: To solve two linear equations, we can express one variable in terms of the other and substitute, or we can eliminate one variable by adding or subtracting equations after scaling them. Here, solving the second equation for y and substituting into the first equation gives a simple single variable equation. Once x is found, y is obtained easily and then x − y can be computed. Step-by-Step Solution: Start with the second equation: 4x − y = 24.Solve for y: y = 4x − 24.Substitute this expression for y into the first equation 3x + 2y = 7.This gives 3x + 2(4x − 24) = 7.Expand: 3x + 8x − 48 = 7, so 11x − 48 = 7.Add 48 to both sides: 11x = 55, so x = 55 / 11 = 5.Now find y from y = 4x − 24: y = 4 * 5 − 24 = 20 − 24 = −4.Compute x − y: 5 − (−4) = 5 + 4 = 9. Verification / Alternative check: Substitute x = 5 and y = −4 back into both original equations. For 3x + 2y: 3 * 5 + 2 * (−4) = 15 − 8 = 7, which is correct. For 4x − y: 4 * 5 − (−4) = 20 + 4 = 24, which also matches. Thus (x, y) = (5, −4) satisfies both equations. The value x − y = 9 is therefore reliable. Why Other Options Are Wrong: Values 1 and −1 would require different values of x and y that do not satisfy both equations simultaneously. −9 is the negative of the correct result and might arise from reversing the expression as y − x instead of x − y. 0 would correspond to x = y, which is not true here. Only 9 is consistent with the correct solution pair (5, −4). Common Pitfalls: Mistakes in solving for y from the second equation, for example getting y = 24 − 4x instead of 4x − 24. Errors in distributing 2 when substituting 2(4x − 24). Forgetting that x − y is x minus y, not y minus x, so the sign matters. Final Answer: 9

Question 13

What is the exact value of the trigonometric expression cosec(11π/6), where the angle is measured in radians?

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Introduction / Context: This question requires evaluation of a trigonometric function at a standard angle expressed in radians. The angle 11π/6 corresponds to 330 degrees. To find cosec(11π/6), we first find sin(11π/6) and then take its reciprocal. Understanding the unit circle and the relationship between radians and degrees is very helpful here. Given Data / Assumptions: The angle is 11π/6 radians. We must compute the exact value of cosec(11π/6). Cosecant is defined as 1 / sin θ. We use standard exact values of sine for special angles on the unit circle. Concept / Approach: First convert 11π/6 to degrees to recognise the reference angle. Since π radians = 180 degrees, 11π/6 = 11 * 180 / 6 = 330 degrees. The sine of 330 degrees is negative because the angle lies in the fourth quadrant, where sine is negative. Specifically, 330 degrees has a reference angle of 30 degrees, and sin 330 degrees = −sin 30 degrees. Once sin(11π/6) is known, cosec(11π/6) is simply its reciprocal. Step-by-Step Solution: Convert 11π/6 to degrees: 11π/6 * (180/π) = 11 * 30 = 330 degrees.In the unit circle, sin 330 degrees = −sin 30 degrees because 330 degrees is in the fourth quadrant.Recall sin 30 degrees = 1/2, so sin 330 degrees = −1/2.Therefore, sin(11π/6) = −1/2.Cosecant is the reciprocal of sine: cosec(11π/6) = 1 / sin(11π/6) = 1 / (−1/2) = −2. Verification / Alternative check: Another way is to consider the unit circle coordinates. The angle 11π/6 has coordinates (cos 11π/6, sin 11π/6) = (cos 330°, sin 330°) = (√3/2, −1/2). These are standard values. Since sin 11π/6 = −1/2, its reciprocal is −2, confirming the previous calculation. The sign is negative in the fourth quadrant, where sine is negative but cosine is positive. Why Other Options Are Wrong: 2 would be cosec of 30 degrees, not 330 degrees where sine is negative. −2/√3 and ±√2 involve sqrt(3) or sqrt(2) and correspond to different standard angles such as 60 degrees or 45 degrees. Only −2 matches the reciprocal of sin(11π/6) = −1/2. Common Pitfalls: Forgetting that 11π/6 lies in the fourth quadrant and therefore sine is negative. Mixing up the sine values of 30 degrees and 60 degrees. Incorrectly converting from radians to degrees, which changes the angle and hence the trigonometric values. Final Answer: -2

Question 14

Which of the following expressions is identically equal to 1 / (tan A + cot A) for an angle A where all trigonometric functions are defined?

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Introduction / Context: This question tests algebraic manipulation of trigonometric expressions and the ability to recognise useful identities. We are asked to find an expression that equals 1 / (tan A + cot A). The options involve combinations of cosec, sin, sec, and cos. The key is to rewrite everything in terms of sin A and cos A and simplify step by step. Given Data / Assumptions: We must find an expression equal to 1 / (tan A + cot A). Angle A is such that all functions tan A, cot A, sin A, cos A, sec A, and cosec A are defined and non zero where used. Options involve products or square roots of expressions built from cosecA, sinA, secA, and cosA. We look for an identity that holds for all such angles A. Concept / Approach: First simplify tan A + cot A in terms of sine and cosine. We have tan A = sin A / cos A and cot A = cos A / sin A. Thus tan A + cot A = (sin^2 A + cos^2 A) / (sin A cos A) = 1 / (sin A cos A). Therefore 1 / (tan A + cot A) = sin A cos A. Next, we simplify each option in terms of sin A and cos A and see which one is equal to sin A cos A. We expect the correct option to give exactly sin A cos A after simplification. Step-by-Step Solution: Simplify the target: tan A + cot A = sin A / cos A + cos A / sin A = (sin^2 A + cos^2 A) / (sin A cos A) = 1 / (sin A cos A).Thus 1 / (tan A + cot A) = sin A cos A.Now examine option d: (cosecA - sinA)(secA - cosA).Rewrite in sin and cos: cosecA = 1 / sin A and secA = 1 / cos A.So (cosecA - sinA) = (1 / sin A - sin A) = (1 - sin^2 A) / sin A = cos^2 A / sin A.Similarly (secA - cosA) = (1 / cos A - cos A) = (1 - cos^2 A) / cos A = sin^2 A / cos A.Multiply them: (cos^2 A / sin A) * (sin^2 A / cos A) = (cos^2 A sin^2 A) / (sin A cos A) = sin A cos A.Therefore option d equals sin A cos A, which matches 1 / (tan A + cot A). Verification / Alternative check: We can test using a specific angle, for example A = 45 degrees. Then tan 45° = 1 and cot 45° = 1, so tan A + cot A = 2, and 1 / (tan A + cot A) = 1/2. Compute option d at A = 45 degrees. Sin 45° = cos 45° = sqrt(2)/2, cosec 45° = sec 45° = sqrt(2). Then cosecA - sinA = sqrt(2) - sqrt(2)/2 = sqrt(2)/2, and secA - cosA is the same, sqrt(2)/2. Their product is (sqrt(2)/2)^2 = 1/2, matching 1 / (tan 45° + cot 45°). This confirms the identity works at least for one test angle and together with algebraic simplification shows it is generally valid. Why Other Options Are Wrong: Option a gives a different product; it simplifies to 1 / (sin A cos A), not sin A cos A. Options b and c involve square roots and thus represent the square root of some product, not the product itself. Option e changes the sign before cosA and does not simplify to sin A cos A. Only option d simplifies exactly to sin A cos A, which is 1 / (tan A + cot A). Common Pitfalls: Not rewriting cosec and sec in terms of sin and cos and instead trying to manipulate them directly. Making algebraic mistakes when simplifying the products or forgetting to use sin^2 A + cos^2 A = 1. Confusing 1 / (sin A cos A) with sin A cos A, which are reciprocals. Final Answer: (cosecA - sinA)(secA - cosA)

Question 15

Using √3 ≈ 1.732, calculate the value of the expression √3 / (√3 + 3) correct to three decimal places.

Accepted Answer

Introduction / Context: This question tests simple numerical approximation skills and basic fraction manipulation. You are asked to evaluate a surd expression numerically using an approximate value for sqrt(3) and then round the result to three decimal places. This is a common type of question in quantitative aptitude tests, especially when calculators are not allowed. Given Data / Assumptions: Use the approximation √3 ≈ 1.732. The expression is √3 / (√3 + 3). We need the value correct to three decimal places. Standard rules of decimal division and rounding apply. Concept / Approach: First substitute the approximate value 1.732 for √3 in both numerator and denominator. Then compute the numerator and denominator as decimals and perform the division. Finally, round the result to three decimal places. Because both numerator and denominator contain the same approximation for √3, the final value is stable and does not require any rationalisation of the denominator. Step-by-Step Solution: Substitute √3 ≈ 1.732 into the expression.The numerator becomes 1.732.The denominator becomes 1.732 + 3 = 4.732.Now compute the fraction 1.732 / 4.732.Performing the division (for example by long division or an approximate method) gives a value close to 0.366.Rounded to three decimal places, the result is 0.366. Verification / Alternative check: You can check this division by multiplication. Take 0.366 and multiply by 4.732. Compute 4.732 * 0.3 = 1.4196. Then 4.732 * 0.06 = 0.28392, and 4.732 * 0.006 = 0.028392. Adding these parts: 1.4196 + 0.28392 + 0.028392 ≈ 1.7319, which is extremely close to the original numerator 1.732, confirming that 0.366 is a very good approximation of the quotient. Why Other Options Are Wrong: 0.566, 0.356, 0.346, and 0.376 yield products with 4.732 that are noticeably different from 1.732. For example, 4.732 * 0.356 is approximately 1.685, which is too small. 0.566 would give a product above 2.6, which is far too large. Only 0.366 reproduces the numerator accurately when multiplied back by the denominator. Common Pitfalls: Forgetting to add 3 to √3 in the denominator and instead computing 1.732 / 3. Incorrect rounding, such as cutting off at two decimal places instead of three. Basic calculation errors during long division of 1.732 by 4.732. Final Answer: 0.366

Question 16

Determine the value of the exponent a that satisfies the equation 9^a + 40^a = 41^a, where all bases are positive integers.

Accepted Answer

Introduction / Context: This question connects exponents with the geometry of Pythagorean triples. The numbers 9, 40, and 41 suggest a relation similar to a right triangle with sides forming a triple. The task is to find an exponent a such that 9^a + 40^a equals 41^a. Instead of trying random large exponents blindly, we can look for patterns and test small integer exponents. Given Data / Assumptions: The equation is 9^a + 40^a = 41^a. All bases 9, 40, and 41 are positive integers. a is a real exponent, but we look for integer values among the options. Options provided are 1, 2, 3, 4, and 5. Concept / Approach: Recognise that 9, 40, and 41 resemble a Pythagorean triple because 9^2 + 40^2 = 41^2. This suggests that a = 2 might satisfy the equation exactly. We test a = 2 first. If it works, there is no need to try higher values. In general, for a larger exponent, the largest base dominates, making equality very unlikely unless there is a special relationship like the one for squares. Step-by-Step Solution: Test a = 2. Compute 9^2 = 81.Compute 40^2 = 1600.Add them: 81 + 1600 = 1681.Compute 41^2: 41^2 = 1681 as well.Thus 9^2 + 40^2 = 41^2, so a = 2 satisfies the equation exactly.There is no need to test other exponents because this provides a perfect equality. Verification / Alternative check: To check that other exponents do not work, note that for a = 1 we have 9^1 + 40^1 = 49 while 41^1 = 41, so the left side is greater. For a = 3, 9^3 + 40^3 = 729 + 64000 = 64729, while 41^3 = 68921, and the right side is greater. As a increases, the term with the largest base 41 will dominate more strongly, so 41^a will grow faster than 9^a + 40^a. Therefore, equality occurs only at a = 2 among positive integer exponents. Why Other Options Are Wrong: a = 1 gives 49 on the left and 41 on the right, not equal. a = 3, 4, and 5 produce values where 41^a is larger than the sum of the other two terms. None of these satisfy the required equality. Only a = 2 produces an exact match between left and right sides. Common Pitfalls: Not recognising the Pythagorean triple pattern and trying random exponents without a plan. Making calculation errors while squaring or cubing the numbers. Assuming that if one exponent works, others might also work, without checking the growth behaviour of exponential functions. Final Answer: 2

Question 17

In right-angled triangle ABC, right angled at B, if ∠C = 60° (so ∠A = 30°), what is the simplified exact value of the expression (cot A − 1/2)?

Accepted Answer

Introduction / Context: This trigonometry question uses a standard right triangle with angles 30 degrees, 60 degrees, and 90 degrees. You are asked to evaluate an expression involving cot A when angle C is 60 degrees. Once the relationship between angles is clear, the problem reduces to using well known trigonometric ratios for 30 degrees and then subtracting a fraction. Given Data / Assumptions: Triangle ABC is right angled at B. Angle C = 60°, so angle A = 30°. The expression to evaluate is cot A − 1/2. We use standard exact values of trigonometric ratios for 30 degrees. Concept / Approach: For angle A = 30 degrees in a 30°–60°–90° triangle, the sides can be taken in the ratio 1 : √3 : 2, where 1 is opposite 30 degrees, √3 is opposite 60 degrees, and 2 is the hypotenuse. Cotangent is defined as adjacent / opposite. For a 30 degree angle, cot 30° equals √3. Once cot A = √3 is known, we simply compute √3 − 1/2 as the required expression. Step-by-Step Solution: Since triangle ABC is right angled at B and angle C = 60°, the remaining acute angle A must be 30°.In a 30°–60°–90° triangle, let the side opposite 30° be 1, the side opposite 60° be √3, and the hypotenuse be 2.For angle A = 30°, the opposite side is 1 and the adjacent side is √3.Cot A = adjacent / opposite = √3 / 1 = √3.The expression to evaluate is cot A − 1/2 = √3 − 1/2.No further algebraic simplification is required; the exact value is √3 − 1/2. Verification / Alternative check: We can check numerically. Take √3 approximately equal to 1.732. Then cot 30° ≈ 1.732. Compute 1.732 − 0.5 = 1.232. Now evaluate each option numerically. Option a, √3 − 1/2, gives about 1.232 as expected. Option b gives about 2.232, option c gives about 0.077, option d gives about −1.232, and option e gives about 0.577. None of these except option a match the expected numeric value, confirming that √3 − 1/2 is correct. Why Other Options Are Wrong: √3 + 1/2 is larger than cot 30° and does not match the expression cot A − 1/2. 1/√3 − 1/2 uses tan 30° rather than cot 30° and leads to a much smaller positive value. 1/2 − √3 is simply the negative of the correct answer. 1/√3 is tan 30°, not cot 30°, so it does not match the expression being evaluated. Common Pitfalls: Confusing cot 30° with tan 30° and using 1/√3 instead of √3. Mixing up which angle is 30 degrees and which is 60 degrees in the triangle. Forgetting to subtract 1/2 after finding cot A and instead stopping at √3. Final Answer: √3 - 1/2

Question 18

If the linear equation 5x + 6(3 − 2x) = 4 holds for a real number x, what is the value of x that satisfies this equation?

Accepted Answer

Introduction / Context: This is a straightforward linear equation involving a bracket. The task is to expand the bracket, combine like terms, and solve for the variable x. Such questions test basic algebra skills, especially the correct application of the distributive property and manipulation of equations. Given Data / Assumptions: The equation is 5x + 6(3 − 2x) = 4. x is a real number. We must find the value of x that makes this equality true. Standard algebraic operations such as expansion and simplification are allowed. Concept / Approach: The main steps are: expand the bracket 6(3 − 2x), then collect all terms involving x on one side of the equation and constants on the other side. This will give a simple equation of the form kx = constant. Solving this equation yields the value of x. Care must be taken with the signs when distributing the 6 across the bracket. Step-by-Step Solution: Start with 5x + 6(3 − 2x) = 4.Expand the bracket: 6 * 3 = 18 and 6 * (−2x) = −12x.So the equation becomes 5x + 18 − 12x = 4.Combine like terms for x: 5x − 12x = −7x, giving −7x + 18 = 4.Subtract 18 from both sides: −7x = 4 − 18 = −14.Divide both sides by −7: x = (−14) / (−7) = 2. Verification / Alternative check: Substitute x = 2 back into the original equation. Compute the bracket first: 3 − 2x = 3 − 4 = −1. Then 6(3 − 2x) = 6(−1) = −6. Now compute 5x + 6(3 − 2x) = 5 * 2 + (−6) = 10 − 6 = 4, which matches the right-hand side. Therefore x = 2 is indeed the correct solution. Why Other Options Are Wrong: For x = 1, the left side becomes 5 * 1 + 6(3 − 2) = 5 + 6 = 11, not 4. For x = 3, LHS = 15 + 6(3 − 6) = 15 − 18 = −3, not 4. For x = 4, LHS = 20 + 6(3 − 8) = 20 − 30 = −10. For x = 0, LHS = 0 + 18 = 18, again not equal to 4. Common Pitfalls: Incorrectly expanding 6(3 − 2x), sometimes writing 18 + 12x instead of 18 − 12x. Not combining the x terms correctly when simplifying 5x − 12x. Dropping the negative sign when dividing by −7, which would give x = −2 instead of 2. Final Answer: 2

Question 19

If a and b are real numbers such that a + b = 1 and ab = -6, find the exact value of a^3 + b^3. Use standard algebraic identities (do not approximate), and keep the given values a + b = 1 and ab = -6 unchanged.

Accepted Answer

Introduction / Context: This question tests the identity that relates the sum of cubes (a^3 + b^3) to the sum (a + b) and product (ab). Instead of solving for a and b individually, we can use an algebraic shortcut that works for all real numbers where the given conditions hold. This is a common simplification pattern in aptitude: convert a complicated-looking expression into something you can compute directly from provided symmetric values. Given Data / Assumptions: • a + b = 1 • ab = -6 • a and b are real numbers • Required: a^3 + b^3 Concept / Approach: Use the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b). This identity is derived from expanding (a + b)^3 and regrouping terms. It is especially useful when a + b and ab are given directly, because it avoids solving a quadratic for a and b. Step-by-Step Solution: 1) Start with the identity: a^3 + b^3 = (a + b)^3 - 3ab(a + b) 2) Substitute a + b = 1: (a + b)^3 = 1^3 = 1 3) Substitute ab = -6 and a + b = 1 into the second term: -3ab(a + b) = -3 * (-6) * 1 = 18 4) Add the results: a^3 + b^3 = 1 + 18 = 19 Verification / Alternative check: If you form the quadratic with roots a and b: t^2 - (a + b)t + ab = 0 gives t^2 - t - 6 = 0, which factors to (t - 3)(t + 2) = 0. So (a, b) could be (3, -2) in some order. Then a^3 + b^3 = 27 + (-8) = 19, confirming the identity-based result. Why Other Options Are Wrong: • 17, 15, 13, 21: these do not match the identity result using the fixed values a + b = 1 and ab = -6. Common Pitfalls: • Using a^3 + b^3 = (a + b)^3 + 3ab(a + b) with the wrong sign. • Trying to guess a and b without using the symmetric identity. Final Answer: 19

Question 20

In coordinate geometry, the reflection of a point in the origin means both its x-coordinate and y-coordinate change sign. What are the coordinates of the reflection of the point (3, -5) in the origin of the Cartesian plane?

Accepted Answer

Introduction / Context: This question tests a basic coordinate-geometry transformation: reflection in the origin. Reflections are common in aptitude and analytic geometry, and the origin reflection is the simplest because it flips the point across both axes at the same time. The key idea is that the origin (0, 0) is the midpoint of a point and its image under this reflection. Given Data / Assumptions: • Original point P = (3, -5) • Reflection is taken in the origin (0, 0) • Required: coordinates of the reflected point Concept / Approach: Reflection in the origin maps any point (x, y) to (-x, -y). This happens because reflecting across the y-axis changes x to -x, and reflecting across the x-axis changes y to -y. Doing both together is exactly reflection through the origin. Step-by-Step Solution: 1) Start with the rule for origin reflection: (x, y) → (-x, -y) 2) Identify x and y from the given point: x = 3 and y = -5 3) Change the sign of x: 3 becomes -3 4) Change the sign of y: -5 becomes 5 5) Therefore, the reflected point is: (-3, 5) Verification / Alternative check: Check using midpoint logic: The midpoint of (3, -5) and (-3, 5) is ((3 + -3)/2, (-5 + 5)/2) = (0, 0). Since the origin is the midpoint, this confirms it is a reflection through the origin. Why Other Options Are Wrong: • (3, 5): only y sign changes (reflection in x-axis, not origin). • (-3, -5): only x sign changes (reflection in y-axis, not origin). • (5, -3) and (-5, 3): these swap coordinates, which is a different transformation. Common Pitfalls: • Changing only one sign instead of both. • Swapping x and y (confusing reflection with rotation or interchange). Final Answer: (-3, 5)

Simplification Questions