Question 1

Surd decomposition and parameter sum Given that √28 − 6√3 is attempted to be written in the form √3·a + b with a, b rational numbers, determine the value of (a + b).

Accepted Answer

Introduction / Context: This question explores expressing a radical combination in a restricted “linear in √3” form. We test whether √28 − 6√3 can be written as √3·a + b where a and b are rational numbers, and then interpret what (a + b) would be. The core idea is to compare like radicals and rational parts consistently. Given Data / Assumptions: Expression: √28 − 6√3. Target form: √3·a + b with a, b ∈ rationals. We treat √7 and √3 as linearly independent over the rationals (they cannot be converted into each other with rational coefficients). Concept / Approach: Break √28 as 2√7. Then √28 − 6√3 = 2√7 − 6√3. If the target is √3·a + b, then any √7 term must vanish or be representable using √3 and rationals, which is not possible since √7 is irrational and independent from √3 over Q. Step-by-Step Solution: Rewrite √28 = 2√7, so the expression is 2√7 − 6√3.Target form is a√3 + b with a, b rational.There is no way to express 2√7 as q√3 + r for rational q, r, because √7 and √3 are distinct quadratic irrationals.Therefore, the proposed representation does not exist with rational a and b. Verification / Alternative check: Assume, for contradiction, 2√7 = a√3 + b. Squaring gives 28 = 3a^2 + b^2 + 2ab√3. The left side has no √3 term; hence 2ab must be 0. If ab = 0, then either a = 0 or b = 0, which still cannot yield 28 from 3a^2 + b^2 without contradicting the absence of √3 and the presence of √7 upon taking square roots. Hence impossible. Why Other Options Are Wrong: Numerical choices like −2, 2, 1, −1 assume a valid (a, b) exists; but no rational pair exists, so those sums are meaningless for this constraint. Common Pitfalls: Confusing √28 with 2√7 and then trying to blend √7 into a combination of √3 and rationals; these radicals cannot be interchanged with rational coefficients. Final Answer: Not possible (no such rational a, b)

Question 2

Cube–reciprocal identity with a = (√2 − 1)^(1/3) Evaluate (a − a^−1)^3 + 3(a − a^−1).

Accepted Answer

Introduction / Context: This problem tests recognition of a compact algebraic identity that relates (a − 1/a)^3 + 3(a − 1/a) to a^3 − 1/a^3. Using the given a = (√2 − 1)^(1/3), we can evaluate the expression exactly without messy decimals. Given Data / Assumptions: a = (√2 − 1)^(1/3). Target: (a − a^−1)^3 + 3(a − a^−1). All algebra performed with exact radicals. Concept / Approach: Identity: (t)^3 + 3t = a^3 − 1/a^3 where t = a − 1/a. Expand (a − 1/a)^3 to verify this, or use the well-known result to avoid expansion. Then compute a^3 and a^−3 directly from the definition of a. Step-by-Step Solution: Let t = a − 1/a. Then t^3 + 3t = a^3 − 1/a^3.Given a = (√2 − 1)^(1/3) ⇒ a^3 = √2 − 1.Compute a^−3 = 1/(√2 − 1) = (√2 + 1)/[(√2 − 1)(√2 + 1)] = √2 + 1.Hence a^3 − a^−3 = (√2 − 1) − (√2 + 1) = −2. Verification / Alternative check: Expand (a − 1/a)^3 = a^3 − 3a + 3/a − 1/a^3. Adding 3(a − 1/a) gives a^3 − 1/a^3 indeed, confirming the identity. Why Other Options Are Wrong: 2, 2√2, and √2 result from sign or reciprocal mistakes; 0 would require a^3 = 1/a^3, which is false here. Common Pitfalls: Forgetting to rationalize 1/(√2 − 1), or misapplying the identity by using plus instead of minus in the reciprocal term. Final Answer: -2

Question 3

Surd ratios and symmetric powers Let x = (√3 + 1)/(√3 − 1) and y = (√3 − 1)/(√3 + 1). Evaluate x^2/y + y^2/x.

Accepted Answer

Introduction / Context: Expressions featuring x and its reciprocal often reduce via symmetry. Here, y is the reciprocal of x, enabling a simplification from mixed fractions to a sum of cubes of x and 1/x. Given Data / Assumptions: x = (√3 + 1)/(√3 − 1), y = (√3 − 1)/(√3 + 1). We need S = x^2/y + y^2/x. Concept / Approach: Notice y = 1/x. Then S = x^2 * x + (1/x^2) * (1/x) = x^3 + 1/x^3. Next compute x in simplest form, then use the identity x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x). Step-by-Step Solution: Simplify x: (√3 + 1)/(√3 − 1) = ((√3 + 1)^2)/(3 − 1) = (3 + 1 + 2√3)/2 = 2 + √3.Thus 1/x = 2 − √3 (since (2 + √3)(2 − √3) = 1).Then x + 1/x = (2 + √3) + (2 − √3) = 4.Compute x^3 + 1/x^3 = (x + 1/x)^3 − 3(x + 1/x) = 4^3 − 3*4 = 64 − 12 = 52. Verification / Alternative check: Directly cube x and 1/x numerically using √3 ≈ 1.732; sums confirm the exact value 52. Why Other Options Are Wrong: 76 and 64 come from squaring 4 or forgetting the subtraction term; 4 corresponds to x + 1/x, not the cube sum; 28 is a rough, incorrect estimate. Common Pitfalls: Not noticing y = 1/x; failing to rationalize x first, which hides the elegant cancellation. Final Answer: 52

Question 4

Use x + a/x = b to simplify a rational expression If x + a/x = b, evaluate (x^2 + b x + a) / (b x^2 − x^3).

Accepted Answer

Introduction / Context: This problem relies on manipulating the relation x + a/x = b. Converting it to a quadratic in x lets us replace x^2 terms and reduce a complicated rational expression to a simple ratio. Given Data / Assumptions: x + a/x = b, with x ≠ 0 and a ≠ 0. Expression: E = (x^2 + b x + a) / (b x^2 − x^3). Concept / Approach: From x + a/x = b, multiply by x to get x^2 + a = b x ⇒ x^2 − b x + a = 0. Use this to replace x^2 and also to simplify b − x in terms of a/x, which collapses the denominator neatly. Step-by-Step Solution: From x^2 − b x + a = 0, we have x^2 = b x − a.Numerator: x^2 + b x + a = (b x − a) + b x + a = 2 b x.Also from x + a/x = b ⇒ b − x = a/x.Denominator: b x^2 − x^3 = x^2 (b − x) = x^2 * (a/x) = a x.Therefore, E = (2 b x) / (a x) = 2b/a. Verification / Alternative check: Pick sample values a = 2, x satisfying x + 2/x = b; compute both sides numerically to confirm E = 2b/a regardless of the particular root x. Why Other Options Are Wrong: a + b, b/a, and ab ignore the key transformations; a/(2b) flips the correct ratio. Common Pitfalls: Losing track of signs when substituting x^2 = b x − a; not noticing that b − x = a/x. Final Answer: 2b/a

Question 5

Triangles with a line parallel to base – Using similarity: In ∆ABC, D and E lie on AB and AC respectively such that AD = 8 cm, DB = 12 cm (so AB = 20 cm), AE = 6 cm, EC = 9 cm (so AC = 15 cm). Show that DE ∥ BC and find BC in terms of DE.

Accepted Answer

Introduction / Context: When a segment joins points on two sides of a triangle in the same ratio, it is parallel to the third side (Converse of Basic Proportionality Theorem), leading to similar triangles and a fixed scale factor. Given Data / Assumptions: AD/DB = 8/12 = 2/3. AE/EC = 6/9 = 2/3. Therefore DE ∥ BC. Concept / Approach: Similarity of ∆ADE and ∆ABC gives scale factor AD/AB = 8/20 = 2/5. Corresponding sides satisfy DE/BC = 2/5 ⇒ BC = (5/2)DE. Step-by-Step Solution: AD/DB = AE/EC ⇒ DE ∥ BC.AD/AB = 8/20 = 2/5 ⇒ DE/BC = 2/5 ⇒ BC = (5/2)DE. Final Answer: 5/2 DE

Question 6

A tailor has 37.5 metres of cloth. He can cut 8 equal pieces from each metre of cloth. How many pieces can he make from exactly half of the cloth he has?

Accepted Answer

Introduction / Context: This is a straightforward application of fractions and unitary method. We are given the total length of cloth and the number of pieces obtainable from one metre. The problem then asks for the number of pieces that can be cut from half of the total cloth. Such questions are standard in aptitude tests involving proportional reasoning. Given Data / Assumptions: Total length of cloth = 37.5 metres. From 1 metre of cloth, the tailor can cut 8 equal pieces. We are interested in half of the total cloth, that is 37.5 / 2 metres. All pieces are of equal size. Concept / Approach: First, determine the length of cloth in half the total amount. Next, use the rate of 8 pieces per metre to find the number of pieces from that half length. This is essentially multiplying the number of metres in half by the number of pieces per metre. Step-by-Step Solution: Step 1: Compute half of 37.5 metres: 37.5 / 2 = 18.75 metres. Step 2: From 1 metre, the tailor can make 8 pieces. Step 3: From 18.75 metres, total pieces = 18.75 × 8. Step 4: Calculate 18.75 × 8 = (18 × 8) + (0.75 × 8) = 144 + 6 = 150. Step 5: Therefore, the tailor can make 150 pieces from half of the cloth. Verification / Alternative check: We can find the total pieces from the full 37.5 metres and then divide by 2. From 37.5 metres, total pieces = 37.5 × 8 = 300. Half of 300 is 150, which matches the earlier result. Thus the answer is consistent. Why Other Options Are Wrong: 300: This is the number of pieces from the full 37.5 metres, not from half of it. 175 and 200: These are higher than the correct count for 18.75 metres and do not match any logical calculation based on 8 pieces per metre. 250: This corresponds to 31.25 metres of cloth, which is more than half of 37.5 metres, so it is not correct. Common Pitfalls: Some learners mistakenly multiply the total 37.5 metres by 8 and then forget to halve the number of pieces. Others miscalculate half of 37.5. Always handle the fraction step carefully and then apply the pieces-per-metre rate. Final Answer: The tailor can make 150 pieces from half of the cloth he has.

Question 7

In triangle ABC, points D and E lie on sides AB and AC respectively, and segment DE is drawn parallel to side BC. The segments on side AB satisfy the ratio AD : DB = 2 : 5. The area of the smaller triangle ADE is 8 cm². Using properties of similar t…

Accepted Answer

Introduction: This geometry question tests the concept of similar triangles formed by a line drawn parallel to the base of a triangle, and how their areas are related to the ratios of corresponding sides. Once similarity is understood, the area of the quadrilateral can be found by subtracting the smaller triangle area from the larger triangle area. Given Data / Assumptions: In triangle ABC, D lies on AB and E lies on AC. DE is parallel to BC. AD : DB = 2 : 5, so AB is divided internally in this ratio. Area of triangle ADE = 8 cm². We must find the area of quadrilateral BDEC. Concept / Approach: When a line segment is drawn parallel to one side of a triangle and cutting the other two sides, the smaller triangle formed at the vertex is similar to the original triangle. The ratio of similarity is the ratio of corresponding sides, and the ratio of their areas is the square of the ratio of their corresponding sides. After obtaining the area of the whole triangle ABC, the quadrilateral area is simply the difference between the total area and the area of the smaller triangle. Step-by-Step Solution: Let AB = AD + DB. Given AD : DB = 2 : 5, so AD : AB = 2 : (2 + 5) = 2 : 7. Thus, the ratio of similarity (ADE to ABC) is AD / AB = 2 / 7. Area scale factor = (2 / 7)² = 4 / 49. Let area of triangle ABC = K. Then area(ADE) = (4 / 49) * K. Given area(ADE) = 8, so 8 = (4 / 49) * K. Solve for K: K = 8 * (49 / 4) = 2 * 49 = 98 cm². Quadrilateral BDEC is the remaining part: area(BDEC) = area(ABC) − area(ADE) = 98 − 8 = 90 cm². Verification / Alternative check: As a ratio, area(ADE) : area(ABC) = 4 : 49. That means area(BDEC) : area(ADE) = (49 − 4) : 4 = 45 : 4. If area(ADE) is 8, then area(BDEC) = (45 / 4) * 8 = 45 * 2 = 90 cm², which confirms our previous result. Why Other Options Are Wrong: Values 86, 94, and 98 cm² do not satisfy the required area ratio. In particular, 98 cm² is the total area of triangle ABC, not the quadrilateral. Any value other than 90 would break the fixed similarity ratio implied by AD : AB. Common Pitfalls: Learners sometimes assume that area ratios are directly proportional to side ratios instead of the square of side ratios. Another error is to forget that AB is the sum AD + DB when forming the 2 : 7 ratio. Being precise with ratios and squaring for area is essential. Final Answer: The area of quadrilateral BDEC is 90 cm².

Question 8

Consider the straight line that passes through the points (8, 2) and (3, 1). What is the slope of the line that is perpendicular to this line?

Accepted Answer

Introduction: This coordinate geometry question checks your understanding of slopes of lines and the relationship between the slopes of perpendicular lines. Once you find the slope of the given line, you can use the negative reciprocal to obtain the slope of any line perpendicular to it. Given Data / Assumptions: The original line passes through points (8, 2) and (3, 1). We need the slope of a line that is perpendicular to this line. All work is in the standard Cartesian coordinate system. Concept / Approach: The slope m of a line passing through (x₁, y₁) and (x₂, y₂) is: m = (y₂ − y₁) / (x₂ − x₁). If two lines are perpendicular and neither is vertical, then the product of their slopes is −1. Thus, if one line has slope m, the perpendicular line has slope m⊥ = −1 / m. This negative reciprocal relationship is the key to the solution. Step-by-Step Solution: First find the slope of the given line through (8, 2) and (3, 1). m = (1 − 2) / (3 − 8) = (−1) / (−5) = 1/5. So the slope of the given line is 1/5. For a line perpendicular to this, we need a slope m⊥ such that m * m⊥ = −1. So (1/5) * m⊥ = −1. Therefore m⊥ = −1 / (1/5) = −5. Verification / Alternative check: You can verify by checking the product: (1/5) * (−5) = −1, which confirms perpendicularity. This is the standard condition for two non vertical lines to be perpendicular in the plane. Why Other Options Are Wrong: The slope 5 has product 5 * (1/5) = 1, indicating parallelism in a different sense, not perpendicularity. The slopes 1/5 and −1/5 are either the original slope or a line that is not perpendicular because their product with 1/5 is not −1. Only −5 satisfies the negative reciprocal condition. Common Pitfalls: A common error is to change only the sign or only invert the slope instead of doing both. Another mistake is miscalculating the original slope by mixing up x and y differences. Always compute the initial slope carefully, then take its negative reciprocal for a perpendicular line. Final Answer: The slope of the line perpendicular to the given line is −5.

Question 9

In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If he attempts all 60 questions and secures 130 marks, the no of questions he attempts correctly is :

Accepted Answer

Let the number of correct answers be X. Number of incorrect answers = (60 – X). 4x – (60 – x) = 130 => 5x = 190 => x = 38

Question 10

If 2 tables and 3 chairs cost Rs, 3500 and 3 tables and 2 chairs cost Rs. 4000, then how much does a table cost ?

Accepted Answer

Let the cost of a table and that of a chair be Rs. x and Rs, y respectively. Then, 2x + 3y = 3500 ...(i) and 3x + 2y = 4000 .....(ii) solving (i) and (ii) we get x = 1000, y = 500

Question 11

A fires 5 shots to B's 3 but A kills only once in 3 shots while B kills once in 2 shots. When B has missed 27 times, A has killed :

Accepted Answer

Let the no of shots be x. Then, Shots fired by A =(5/8)x Shots fired by B = (3/8)x Killing shots by A = 1/3 of (5/8)x = (5/24)x Shots missed by B = 1/2 of (3/8)x = (3/16)x (3/16)x = 27 ⇒ x =144 Birds killed by A = (5/24)x = (5/24)*144 = 30

Question 12

If a * b = 2a - 3b + ab, then 3 * 5 + 5 * 3 is equal to :

Accepted Answer

3 * 5 + 5 * 3 = (2 * 3 - 3 * 5 + 3 * 5) + ( 2 * 5 - 3 * 3 + 5 * 3) = (6 + 10 - 9 + 15) = 22.

Question 13

If a + b + c = 13, a2+b2+c2=69 then find the ab + bc + ca :

Accepted Answer

a + b + c 2 = a 2 + b 2 + c 2 + 2 a b + b c + c a 2 a b + b c + c a = a + b + c 2 - a 2 + b 2 + c 2 = 169 - 69 = 100 = a b + b c + c a = 50

Question 14

The difference of 18 and its reciprocal is equal to :

Accepted Answer

Required difference = 19/16 - 16/19 = 105/304..

Question 15

To fill a tank, 25 buckets of water is required. How many buckets of water will be required to fill the same tank if the capacity of bucket is reduced to two-fifth of its present ?

Accepted Answer

62.5

Question 16

A tank is 2/5 full. If 16 litres of water is added to the tank, it becomes 6/7 full. The capacity of the tank is:

Accepted Answer

Let the capacity of the tank be x litres. Then, (6x/7 - 2x/5) = 16 <=> 30x - 14x = 16*35 <=> x = 35

Question 17

Simplify the following equation. 1943 + 12 + 13 + 11 + 14

Accepted Answer

After solving we get, 62/931. None of the options satisfy.

Question 18

In a regular week, there are 5 working days and for each day, the working hours are 8. A man gets Rs. 2.40 per hour for regular work and Rs. 3.20 per hours for overtime. If he earns Rs. 432 in 4 weeks, then how many hours does he work for ?

Accepted Answer

Suppose the man works overtime for x hours. Now, working hours in 4 weeks = (5 x 8 x 4) = 160. Therefore, 160 * 2.40 + x * 3.20 = 432 => 3.20x = 432 - 384 = 48 => x = 15. Hence, total hours of work = (160 + 15) = 175.

Question 19

How many 18 ' s are there in 3712 ?

Accepted Answer

37 1 2 = 75 2 Required number = 75 2 1 8 = 75 x 4 = 300

Question 20

The total of 324 of 20 paise and 25 paise make a sum of Rs. 71. The no of 20 paise coins is :

Accepted Answer

Let the number of 20 paise coins be x. Then the no of 25 paise coins = (324 - x). 0.20 * (x) + 0.25 (324 - x) = 71 => x = 200..

Simplification Questions