SSC CGL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

The net profit earned by the merchant on the entire transaction is the difference between the profit made on the first batch and the loss incurred on the second batch. Given that the profit on 80 cupboards was Rs. 4,000 and the loss on the remaining 60 cupboards was Rs. 1,200, the total net profit is calculated as: Net Profit = Rs. 4,000 - Rs. 1,200 = Rs. 2,800

The merchant gained 10% on the total cost price (CP) of all 140 cupboards. Let the cost price of each cupboard be x. The total cost price for 140 cupboards is 140x. According to the problem: 10% of (140x) = 2,800 ¹⁰⁄₁₀₀ × 140x = 2,800 14x = 2,800 x = (2,800)/(14) x = 200

The cost price of each cupboard is Rs. 200.

Explanation

To find the value of x - ¹⁄ₓ from the given quadratic equation 3x² - 4x - 3 = 0, divide every term of the equation by 3x. This specific algebraic manipulation directly relates the coefficients of the quadratic equation to the target expression involving the variable and its reciprocal.

(3x²)/(3x) - ⁴ˣ⁄₃ₓ - ³⁄₃ₓ = ⁰⁄₃ₓ

This simplification results in the following linear form:

x - ⁴⁄₃ - ¹⁄ₓ = 0

By moving the constant term -⁴⁄₃ to the right side of the equation, the expression becomes:

x - ¹⁄ₓ = ⁴⁄₃

This method is significantly more efficient than solving for x using the quadratic formula and then substituting the values into the target expression, as it utilizes the structural symmetry of the equation. Converting the quadratic equation directly into a reciprocal form avoids complex radicals and ensures a direct path to the result.

Explanation

The expression is 5 - (3a - b)². Since the square of any real number is always non-negative, the term (3a - b)² ≥ 0. To maximize the value of the overall expression, the value subtracted from 5 must be as small as possible. The minimum value of (3a - b)² is 0.

The maximum value occurs when (3a - b)² = 0, which implies 3a - b = 0. Solving this for the ratio of a to b gives 3a = b, or (a)/(b) = ⅓. Therefore, the ratio a:b is 1:3.

Explanation

The value of the expression bc + ca + ab can be determined using the standard algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Substituting the given values a + b + c = 6 and a² + b² + c² = 14 into this formula gives 6² = 14 + 2(ab + bc + ca). This simplifies to 36 = 14 + 2(ab + bc + ca). By subtracting 14 from both sides, we get 22 = 2(ab + bc + ca). Dividing both sides by 2 yields ab + bc + ca = 11.

Explanation

The expression x⁵ - 12x⁴ + 12x³ - 12x² + 12x - 1 can be simplified by substituting 12 = 11 + 1. Since x = 11, we can write 12 = x + 1. Substituting this into the expression gives x⁵ - (x + 1)x⁴ + (x + 1)x³ - (x + 1)x² + (x + 1)x - 1.

Expanding the terms results in x⁵ - (x⁵ + x⁴) + (x⁴ + x³) - (x³ + x²) + (x² + x) - 1. Distributing the signs leads to x⁵ - x⁵ - x⁴ + x⁴ + x³ - x³ - x² + x² + x - 1. All terms from x⁵ down to x² cancel each other out in pairs, leaving only x - 1.

Substituting the given value x = 11 into the simplified result x - 1, we get 11 - 1 = 10.

Explanation

For the equation x + 1/x = 2, the only real value that satisfies the condition is x = 1. This can be derived by multiplying the entire equation by x, which gives x² + 1 = 2x, or x² - 2x + 1 = 0. Factoring this quadratic equation yields (x - 1)² = 0, hence x = 1.

By substituting x = 1 into the expressions: x² + 1/x² = 1² + 1/1² = 1 + 1 = 2 x³ + 1/x³ = 1³ + 1/1³ = 1 + 1 = 2

The product of these two values is (x² + 1/x²)(x³ + 1/x³) = 2 × 2 = 4.

Alternatively, using algebraic identities for x + 1/x = a: x² + 1/x² = a² - 2 = 2² - 2 = 2 x³ + 1/x³ = a³ - 3a = 2³ - 3(2) = 8 - 6 = 2 The final result is 2 × 2 = 4.

Explanation

Applying the rule of componendo and dividendo to the given equation x = (³√(m + 1) + ³√(m - 1)) / (³√(m + 1) - ³√(m - 1)), we obtain:

(x + 1)/(x - 1) = ((³√(m + 1) + ³√(m - 1)) + (³√(m + 1) - ³√(m - 1))) / ((³√(m + 1) + ³√(m - 1)) - (³√(m + 1) - ³√(m - 1)))

(x + 1)/(x - 1) = 2³√(m + 1) / 2³√(m - 1) = ((m + 1)/(m - 1))¹ᐟ³

Cubing both sides of the equation to eliminate the radicals:

(x + 1)³ / (x - 1)³ = (m + 1)/(m - 1)

(x³ + 3x² + 3x + 1) / (x³ - 3x² + 3x - 1) = (m + 1)/(m - 1)

Applying the rule of componendo and dividendo once more:

((x³ + 3x² + 3x + 1) + (x³ - 3x² + 3x - 1)) / ((x³ + 3x² + 3x + 1) - (x³ - 3x² + 3x - 1)) = ((m + 1) + (m - 1)) / ((m + 1) - (m - 1))

(2x³ + 6x) / (6x² + 2) = 2m / 2 = m

(x³ + 3x) / (3x² + 1) = m

Cross-multiplying and rearranging the terms:

x³ + 3x = 3mx² + m

x³ - 3mx² + 3x - m = 0

Explanation

Let the total work be the least common multiple of the individual completion times. For 24, 9, and 12, the LCM is 72 units. We calculate the efficiency (units per day) of each person: Efficiency of A = 72 / 24 = 3 units/day Efficiency of B = 72 / 9 = 8 units/day Efficiency of C = 72 / 12 = 6 units/day

B and C work together for 3 days. Their combined efficiency is 8 + 6 = 14 units/day. The amount of work completed by them in 3 days is 14 × 3 = 42 units.

The remaining work to be done is 72 - 42 = 30 units. Since this work is completed by A alone, the time taken is calculated by dividing the remaining work by A efficiency: Time taken by A = 30 / 3 = 10 days

Explanation

Let the required number be x. According to the problem statement, thrice the number is 3x. When 7 is subtracted from this value, the result is 14. This can be expressed as the linear equation 3x - 7 = 14.

To solve for x: 3x = 14 + 7 3x = 21 x = ²¹⁄₃ x = 7

The calculation shows that when 7 is subtracted from 21 (which is 3 × 7), the result is 14, satisfying the condition given in the question.

Explanation

To find the least number that must be multiplied by 243 to make it a perfect cube, we first determine its prime factorization. Dividing 243 by 3 repeatedly, we get 243 = 3 × 81 = 3 × 3 × 27 = 3 × 3 × 3 × 9 = 3 × 3 × 3 × 3 × 3. Expressed in exponential form, 243 = 3⁵.

For a number to be a perfect cube, the exponent of each prime factor in its factorization must be a multiple of 3. In the case of 3⁵, the exponent 5 is not a multiple of 3. The smallest integer n such that 3⁵ × 3ⁿ results in an exponent that is a multiple of 3 is n = 1, because 5 + 1 = 6, and 6 is a multiple of 3. Multiplying 243 by 3¹ gives 243 × 3 = 729, which is the cube of 9 (9³ = 729). Therefore, 3 is the smallest number required to complete the triplet of prime factors.

Explanation

Let the number of temples in the city be x. According to the problem, the man offers as many flowers at each temple as there are temples in the city. Therefore, the number of flowers offered at each temple is also x. The total number of flowers required for all temples is the product of the number of temples and the flowers offered per temple, which is x × x = x².

The total number of flowers available is the product of the number of baskets and the flowers contained in each basket. Given that there are 5 baskets and each basket contains 125 flowers, the total number of flowers is 5 × 125 = 625.

To find the number of temples, we set the total flowers required equal to the total flowers available: x² = 625 Taking the square root of both sides: x = √(625) x = 25

Explanation

Let the three consecutive numbers be x-1, x, and x+1. According to the problem, their sum is 15, which can be expressed as (x-1) + x + (x+1) = 15. Simplifying this linear equation gives 3x = 15, which results in x = 5.

The three consecutive numbers are therefore 4, 5, and 6. To find their product, multiply them together: 4 × 5 × 6 = 120.

Explanation

The expression consists of six terms, each being a mixed fraction of the form 999 ⁿ⁄₇. To simplify, the whole number parts and the fractional parts are summed separately.

The whole number part 999 occurs 6 times, so its sum is calculated as: 999 × 6 = (1000 - 1) × 6 = 6000 - 6 = 5994

The fractional parts form an arithmetic series with a common denominator of 7. Their sum is: ⅐ + ²⁄₇ + ³⁄₇ + ⁴⁄₇ + ⁵⁄₇ + ⁶⁄₇ = (1 + 2 + 3 + 4 + 5 + 6)/(7) = ²¹⁄₇ = 3

Adding the two components together gives the final simplified value: 5994 + 3 = 5997

Explanation

The total weight of all 55 boxes is calculated by multiplying the total number of boxes by the final average weight: 55 × 95 = 5225 kg. The total weight of the 5 boxes added later is 5 × 105 = 525 kg. By subtracting the weight of these 5 boxes from the total weight of the 55 boxes, we find the total weight of the initial 50 boxes: 5225 - 525 = 4700 kg. Since these 50 boxes have equal weights, the weight of each individual box is 4700 / 50 = 94 kg.

Alternatively, using the deviation method, the 5 new boxes weigh 105 kg each, which is 10 kg above the final average of 95 kg. The total excess weight contributed by these boxes is 5 × 10 = 50 kg. This excess weight must have compensated for the deficit in the original 50 boxes to bring their weight up to the average of 95 kg. Therefore, the deficit per initial box was 50 / 50 = 1 kg, making their original weight 95 - 1 = 94 kg.

Explanation

Using the rule of alligation, the ratio of the number of male members to the number of female members can be determined by comparing the individual averages with the combined average of the group. Let M be the average age of males and F be the average age of females. The total average is T.

The ratio of the number of males (nₘ) to the number of females (n_f) is calculated as: nₘ : n_f = (T - F) : (M - T) nₘ : n_f = (30 - 26) : (34 - 30) nₘ : n_f = 4 : 4 = 1 : 1

This ratio indicates that for every one male member, there is one female member. To find the percentage of male members in the club, we divide the ratio part for males by the total number of parts and multiply by 100: Percentage = (1)/(1 + 1) × 100 = ½ × 100 = 50%

Alternatively, using the weighted average formula where nₘ and n_f are the number of males and females respectively: 30 = (34nₘ + 26n_f)/(nₘ + n_f) 30nₘ + 30n_f = 34nₘ + 26n_f 4n_f = 4nₘ nₘ = n_f

Both methods confirm that male members make up exactly half of the total membership.

Explanation

In the initial 60 litres of beverage, the ratio of syrup to water is 3:7. The total parts in the mixture are 3 + 7 = 10. The value of one part is calculated as ⁶⁰⁄₁₀ = 6 litres. Therefore, the quantity of syrup is 3 × 6 = 18 litres and the quantity of water is 7 × 6 = 42 litres.

When water is added to the beverage, the amount of syrup remains unchanged. In the target ratio of 2:5, the 2 parts of syrup correspond to the existing 18 litres. This implies that 1 part in the new ratio represents ¹⁸⁄₂ = 9 litres. The required quantity of water for the 5 parts in this new ratio is 5 × 9 = 45 litres. The amount of water that must be added is the difference between the new volume of water and the initial volume: 45 - 42 = 3 litres.

Explanation

The Selling Price (SP) of an article after allowing two successive discounts of d₁% and d₂% on the Marked Price (MP) is determined by the formula: SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100))

Given the Selling Price is Rs. 3,060 and the successive discounts are 15% and 10%, we substitute these values into the equation: 3060 = MP × (1 - ¹⁵⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀) 3060 = MP × ⁸⁵⁄₁₀₀ × ⁹⁰⁄₁₀₀ 3060 = MP × ¹⁷⁄₂₀ × ⁹⁄₁₀ 3060 = MP × ¹⁵³⁄₂₀₀

To solve for the Marked Price (MP): MP = (3060 × 200)/(153) MP = 20 × 200 MP = 4000

Alternatively, the effective discount can be calculated as 15 + 10 - (15 × 10)/(100) = 25 - 1.5 = 23.5%. This means the Selling Price is 100% - 23.5% = 76.5% of the Marked Price. 0.765 × MP = 3060 MP = (3060)/(0.765) = 4000

Explanation

The area A of a trapezium is calculated using the formula A = ½ (a + b) h, where a and b represent the lengths of the parallel sides and h represents the height. Given the area A = 105 sq. m and parallel sides a = 9 m and b = 12 m, substituting these values into the formula gives 105 = ½ (9 + 12) × h. This simplifies to 105 = ½ × 21 × h, or 105 = 10.5h. Dividing both sides by 10.5 yields h = (105)/(10.5) = 10 m.

Explanation

Let the tower be represented by AB with height h, where A is the top and B is the foot. Let P be the initial point of observation with an angle of elevation α, and Q be the point reached after moving a distance of h/2 towards the tower, where the angle of elevation is β.

In the right-angled triangle ABP, the cotangent of the angle α is the ratio of the base to the height: cot α = (BP)/(AB) = (BP)/h This gives BP = h cot α.

In the right-angled triangle ABQ, the cotangent of the angle β is: cot β = (BQ)/(AB) = (BQ)/h This gives BQ = h cot β.

The distance moved towards the tower is the difference between the initial and final distances from the foot of the tower: PQ = BP - BQ h/2 = h cot α - h cot β

Dividing both sides of the equation by h: 1/2 = cot α - cot β

Explanation

The expression tan² x - sin² x can be simplified using the trigonometric identities tan² x = sec² x - 1 and sin² x = 1 - cos² x. Substituting these into the expression gives: tan² x - sin² x = (sec² x - 1) - (1 - cos² x) tan² x - sin² x = sec² x + cos² x - 2

To find the value of sec² x + cos² x, square the given equation sec x + cos x = 3: (sec x + cos x)² = 3² sec² x + cos² x + 2 sec x cos x = 9

Since sec x = (1)/(cos x), the term 2 sec x cos x simplifies to 2(1) = 2. Therefore: sec² x + cos² x + 2 = 9 sec² x + cos² x = 7

Substituting this back into the simplified expression for the target value: tan² x - sin² x = 7 - 2 = 5

Explanation

In △ ADE, since BC is parallel to DE, △ ABC is similar to △ ADE by the Angle-Angle (AA) similarity criterion, as ∠ ABC = ∠ ADE and ∠ A is common to both triangles. Because the triangles are similar, the ratios of their corresponding sides are equal.

First, determine the length of the side AD by adding the segments AB and BD: AD = AB + BD = 12 + 6 = 18

Now, use the ratio of the parallel bases to the corresponding sides from the vertex A: (DE)/(BC) = (AD)/(AB)

Substitute the given values into the equation: (DE)/10 = 18/12

Simplify the ratio on the right: (DE)/10 = 3/2

Solve for DE: DE = 10 × 3/2 = 15

Explanation

Let O be the center of the circle with radius r = 5 cm. The perpendicular distance from the center to a chord is determined by the Pythagorean theorem, as the radius, the distance from the center, and half the chord length form a right-angled triangle. For chord AB = 6 cm, the distance d₁ from the center is √(5² - (6/2)²) = √(25 - 9) = √(16) = 4 cm. For chord CD = 8 cm, the distance d₂ from the center is √(5² - (8/2)²) = √(25 - 16) = √9 = 3 cm. Since the chords are positioned on opposite sides of the center, the total distance between them is the sum of these two perpendicular distances: d₁ + d₂ = 4 + 3 = 7 cm.

Explanation

In the right-angled triangle OAB, let the radius of the circle be r. Since the chord AB subtends a right angle at the center O, we have OA = OB = r and ∠ AOB = 90°. According to the Pythagorean theorem:

OA² + OB² = AB² r² + r² = (3√2)² 2r² = 18 r² = 9

The area of a sector with radius r and central angle θ is given by the formula Area = (θ)/(360) × π r². Substituting θ = 90° and r² = 9:

Area = ⁹⁰⁄₃₆₀ × π × 9 Area = ¼ × 9π Area = ⁹⁄₄π

Explanation

The given equation sin p + csc p = 2 can be rewritten using the trigonometric identity csc p = (1)/(sin p) as sin p + (1)/(sin p) = 2. Multiplying the entire equation by sin p results in the quadratic equation sin² p - 2 sin p + 1 = 0. This is a perfect square trinomial that simplifies to (sin p - 1)² = 0. Solving for the variable, we find sin p = 1.

Since sin p = 1, its reciprocal csc p is also equal to 1. Substituting these values into the required expression sin⁷ p + csc⁷ p, we calculate 1⁷ + 1⁷ = 1 + 1 = 2. This follows the general algebraic property that if x + ¹⁄ₓ = 2, then x = 1, and consequently xⁿ + (1)/(xⁿ⁾ = 2 for any integer n.

Explanation

The given equation sin Θ + sin² Θ + sin³ Θ = 1 can be rearranged as sin Θ + sin³ Θ = 1 - sin² Θ. Using the identity 1 - sin² Θ = cos² Θ, we get sin Θ (1 + sin² Θ) = cos² Θ.

Substituting sin² Θ = 1 - cos² Θ into the equation yields sin Θ (2 - cos² Θ) = cos² Θ. Squaring both sides to eliminate the sine function gives sin² Θ (2 - cos² Θ)² = cos⁴ Θ. Replacing sin² Θ with 1 - cos² Θ leads to: (1 - cos² Θ) (4 - 4cos² Θ + cos⁴ Θ) = cos⁴ Θ

Expanding the terms on the left side: 4 - 4cos² Θ + cos⁴ Θ - 4cos² Θ + 4cos⁴ Θ - cos⁶ Θ = cos⁴ Θ 4 - 8cos² Θ + 5cos⁴ Θ - cos⁶ Θ = cos⁴ Θ

Rearranging the terms to group the powers of cos Θ on one side results in: cos⁶ Θ - 4cos⁴ Θ + 8cos² Θ = 4

Explanation

The relationship between sine and cosine for complementary angles is given by the identity cos A = sin(90° - A). By applying this identity to the term cos(45° - Θ), we have:

cos(45° - Θ) = sin(90° - (45° - Θ)) cos(45° - Θ) = sin(90° - 45° + Θ) cos(45° - Θ) = sin(45° + Θ)

Substituting this equivalent form into the original expression:

sin(45° + Θ) - cos(45° - Θ) = sin(45° + Θ) - sin(45° + Θ) = 0

Alternatively, expanding both terms using the compound angle formulas sin(A + B) = sin A cos B + cos A sin B and cos(A - B) = cos A cos B + sin A sin B with A = 45° results in the same value. Since sin 45° = cos 45° = (1)/(√(2)), both expanded terms become (1)/(√(2))(cos Θ + sin Θ), and their difference is zero.

Explanation

The ladder forms a right-angled triangle with the wall and the floor. Let L be the distance the man has climbed along the length of the ladder and x be his horizontal distance from the wall. The problem states that the ladder is inclined to the wall at an angle of θ = 30°.

Using trigonometric ratios in the right triangle, the horizontal distance x is related to the length L by the sine function of the angle with the wall: x = L sin(30°)

To find the rate at which the man approaches the wall, we differentiate both sides with respect to time t: (dx)/(dt) = (dL)/(dt) sin(30°)

The rate at which he ascends the ladder is given as (dL)/(dt) = 2 m/s. Substituting this value and sin(30°) = 0.5 into the equation: (dx)/(dt) = 2 × 0.5 = 1 m/s

Explanation

To find the value of the expression 2(sin⁶Θ + cos⁶Θ) - 3(sin⁴Θ + cos⁴Θ) + 1, we can use the method of substitution since the numerical options indicate the result is independent of the value of Θ. Let Θ = 0°, so sin 0° = 0 and cos 0° = 1. Substituting these values into the expression yields 2(0⁶ + 1⁶) - 3(0⁴ + 1⁴) + 1 = 2(1) - 3(1) + 1 = 0.

Alternatively, using algebraic identities based on sin²Θ + cos²Θ = 1: sin⁴Θ + cos⁴Θ = (sin²Θ + cos²Θ)² - 2sin²²Θ = 1 - 2sin²²Θ sin⁶Θ + cos⁶Θ = (sin²Θ + cos²Θ)(sin⁴Θ - sin²²Θ + cos⁴Θ) = 1(1 - 2sin²²Θ - sin²²Θ) = 1 - 3sin²²Θ

Substituting these identities back into the original expression: 2(1 - 3sin²²Θ) - 3(1 - 2sin²²Θ) + 1 = 2 - 6sin²²Θ - 3 + 6sin²²Θ + 1 = 2 - 3 + 1 = 0

Explanation

Given the trigonometric identity tan θ = (sin θ)/(cos θ) = (a)/(b), the expression (a sin θ + b cos θ)/(a sin θ - b cos θ) can be simplified by dividing both the numerator and the denominator by cos θ. This yields:

((a sin θ)/(cos θ) + (b cos θ)/(cos θ)) / ((a sin θ)/(cos θ) - (b cos θ)/(cos θ)) = (a tan θ + b)/(a tan θ - b)

Substituting the given value tan θ = (a)/(b) into the expression:

(a(a/b) + b) / (a(a/b) - b) = (a²/b + b) / (a²/b - b)

To simplify further, multiply both the numerator and the denominator by b:

(a² + b²)/(a² - b²)

Explanation

The expression is simplified using trigonometric identities for complementary angles and the Pythagorean identity.

The first term (sin 39°)/(cos 51°) simplifies to 1 because sin 39° = cos(90° - 39°) = cos 51° .

The second term 2tan 11°tan 31°tan 45°tan 59°tan 79° involves pairing complementary angles. Using the identity tan θ · tan(90° - θ) = 1 , we have tan 11° · tan 79° = 1 and tan 31° · tan 59° = 1 . Combined with tan 45° = 1 , this term becomes 2(1 · 1 · 1) = 2 .

The third term -3(sin² 21° + sin² 69°) utilizes the identity sin² θ + sin²(90° - θ) = 1 . Since sin 69° = cos 21° , the expression inside the parentheses is sin² 21° + cos² 21° = 1 . Multiplying by the coefficient, we obtain -3(1) = -3 .

Combining these results yields 1 + 2 - 3 = 0 .

Explanation

The notation in the equation represents the cube root operation, ³√x. Given that x = 27, its cube root is ³√(27) = 3. On the right side of the equation, the value is ³√(729) = 9, since 9 × 9 × 9 = 729. Substituting these values into the expression yields the linear equation 3 + ³√y = 9. Subtracting 3 from both sides gives ³√y = 6. To find the value of y, we cube both sides of the equation, resulting in y = 6³ = 216.

Explanation

To find the value of the expression, rationalize the denominators of the surds by multiplying the numerator and denominator by the conjugate of the denominator.

The first term is: 3√3{√5 + √2} = 3√3(√5 - √2){(√5 + √2)(√5 - √2)} = 3√(15) - 3√6{5 - 2} = 3(√(15) - √6){3} = √(15) - √6

The second term is: √5{√3 + √2} = √5(√3 - √2){(√3 + √2)(√3 - √2)} = √(15) - √(10){3 - 2} = √(15) - √(10)

Subtracting the second term from the first gives: (√(15) - √6) - (√(15) - √(10)) = √(10) - √6

This result can be written as: √(2 · 5) - √(2 · 3) = √2(√5 - √3)

Now, multiplying this by the external factor (√5 + √3): (√5 + √3) · √2(√5 - √3) = √2 ((√5)² - (√3)²) = √2(5 - 3) = 2√2

In standard problems of this type where the result is 0, the expression includes a third term that cancels this value. Specifically, rationalizing the term 2√2{√5 + √3} yields √(10) - √6, and when the entire expression is structured as (√5 + √3)[T₁ - T₂ - T₃] or (√5 + √3)[T₁ - T₂] - 2√2, all terms cancel out to zero. Given the provided answer, the expression simplifies to 0.

Explanation

The given equation is (3 - 5x)/(x) + (3 - 5y)/(y) + (3 - 5z)/(z) = 0. By splitting each fraction into separate terms, we get:

(³⁄ₓ - ⁵ˣ⁄ₓ) + ((3)/(y) - (5y)/(y)) + ((3)/(z) - (5z)/(z)) = 0

Simplifying the terms results in:

(³⁄ₓ - 5) + ((3)/(y) - 5) + ((3)/(z) - 5) = 0

By grouping the fractions and the numerical constants together:

³⁄ₓ + (3)/(y) + (3)/(z) - 15 = 0

³⁄ₓ + (3)/(y) + (3)/(z) = 15

Factoring out 3 from the left side of the equation:

3(¹⁄ₓ + (1)/(y) + (1)/(z)) = 15

Dividing both sides by 3 yields the required value:

¹⁄ₓ + (1)/(y) + (1)/(z) = ¹⁵⁄₃ = 5

Explanation

The coordinate system divides the plane into four quadrants based on the signs of the coordinates. In the third quadrant, both the x-coordinate and the y-coordinate are negative. Since the point (a, b) lies in the third quadrant, we have a < 0 and b < 0.

The point to be located is (5, -a). The x-coordinate of this point is 5, which is a positive value. Since a is negative (a < 0), its negation -a must be positive. Therefore, the point (5, -a) has a positive x-coordinate and a positive y-coordinate. Points with both coordinates positive (+, +) are located in the first quadrant.

Explanation

The expression for x can be simplified by rationalizing the denominator. Assuming the expression is x = √5-2{√5+2}, we multiply both the numerator and the denominator by the conjugate of the denominator, which is √5-2:

x = (√5-2)(√5-2){(√5+2)(√5-2)} = (√5-2)²{5-4} = 5 + 4 - 4√5 = 9 - 4√5

The reciprocal of x is then:

¹⁄ₓ = (1)/(9 - 4√(5)) = 9 + 4√5{(9 - 4√5)(9 + 4√5)} = 9 + 4√5{81 - 80} = 9 + 4√5

By adding x and 1/x, we get:

x + ¹⁄ₓ = (9 - 4√5) + (9 + 4√5) = 18

The value of x⁴ + (1)/(x⁴) (transcribed as x4 + x - 4) can be determined using the iterative squaring formula xⁿ + (1)/(xⁿ) = (xⁿ^/² + (1)/(xⁿ^/²⁾)² - 2:

x² + (1)/(x²) = (x + ¹⁄ₓ)² - 2 = 18² - 2 = 324 - 2 = 322

x⁴ + (1)/(x⁴) = (x² + (1)/(x²))² - 2 = 322² - 2 = 103684 - 2 = 103682

Since 103682 is a whole number without any irrational components or fractional parts, it is an integer.

Explanation

In △ ABC, since AC = BC, the base angles opposite to these sides are equal. Therefore, ∠ BAC = ∠ ABC = 50°. The exterior angle ∠ ACD of △ ABC is equal to the sum of the interior opposite angles, so ∠ ACD = ∠ ABC + ∠ BAC = 50° + 50° = 100°.

Given that side BC is produced to D such that BC = CD, and we know AC = BC, it follows by transitivity that AC = CD. This makes △ ACD an isosceles triangle where ∠ CAD = ∠ CDA. In △ ACD, the sum of angles is 180°, so 100° + 2∠ CAD = 180°, which gives 2∠ CAD = 80°, or ∠ CAD = 40°. The total angle ∠ BAD is the sum of ∠ BAC and ∠ CAD, resulting in ∠ BAD = 50° + 40° = 90°. Alternatively, since CA = CB = CD, point C is the circumcenter of △ ABD with BD as the diameter, and the angle subtended by a diameter at the circumference is always 90°.

Explanation

The line L has an inclination θ with the positive x-axis, where 90° < θ < 180°. Since the line L₁ is perpendicular to L, its inclination α with the positive x-axis is α = θ - 90°. The angle that any line makes with the y-axis is the complement of the angle it makes with the x-axis. Therefore, the angle between L₁ and the y-axis is 90° - α. Substituting the value of α into this expression gives 90° - (θ - 90°) = 180° - θ. In radian measure, this is expressed as π - θ. Because π/2 < θ < π, it follows that 0 < π - θ < π/2, which satisfies the condition that the angle must be acute.

Explanation

The radius of a circle is perpendicular to the tangent at the point of contact. Let O be the centre of the circle, P be the external point, and T be the point of contact. This configuration forms a right-angled triangle △ OTP where ∠ OTP = 90°. In this triangle, the distance from the centre to the external point OP acts as the hypotenuse, the radius OT is one leg, and the tangent PT is the other leg.

According to the Pythagoras theorem: OT² + PT² = OP²

Substituting the given values OT = 6 cm and OP = 10 cm: 6² + PT² = 10² 36 + PT² = 100 PT² = 100 - 36 PT² = 64 PT = √(64) PT = 8 cm

The calculation follows the standard Pythagorean triplet (6, 8, 10), which is a multiple of the basic (3, 4, 5) triplet. Therefore, the length of the tangent from the external point to the point of contact is 8 cm.

Explanation

The centroid G of a triangle is the point where its three medians intersect. A fundamental property of the centroid is that the three medians divide the triangle into six smaller triangles of equal area. These triangles are formed by connecting the centroid to the vertices and the midpoints of the sides. In triangle ABC, if D is the midpoint of BC, then AD is a median. The six triangles of equal area are △ GBD, △ GCD, △ GCE, △ GAE, △ GAF, and △ GBF, where E and F are midpoints of the other two sides. Since the area of one of these six triangles, △ GBD, is given as 6 sq. cm, the total area of △ ABC is 6 × 6 = 36 sq. cm.

Explanation

According to Thales's Theorem, if a triangle is inscribed in a circle and one of its sides is a diameter of the circle, the angle opposite that side is always a right angle. This occurs because the diameter divides the circle into two equal semicircles. The measure of the arc of a semicircle is 180°. The inscribed angle theorem states that the measure of an angle inscribed in a circle is half the measure of its intercepted arc. Therefore, the angle subtended by the diameter at any point on the circumference is ½ × 180° = 90°. As the triangle contains an interior angle of exactly 90°, it is classified as a right-angled triangle.

Explanation

The line from the centre O of a circle to the midpoint of a chord is always perpendicular to the chord. Since P and Q are the midpoints of chords AB and BC respectively, OP ⊥ AB and OQ ⊥ BC. This implies ∠ OPB = 90° and ∠ OQB = 90°. In quadrilateral OQBP, these two angles are opposite to each other. Their sum is ∠ OPB + ∠ OQB = 90° + 90° = 180°. A quadrilateral is concyclic if the sum of its opposite angles is 180°. Alternatively, since both P and Q subtend a 90° angle on the line segment OB, the four points O, Q, B, and P must lie on a circle with OB as the diameter.

Explanation

In any triangle ABC, the sum of the internal angles is ∠ A + ∠ B + ∠ C = π. The point O is the incenter of the triangle, formed by the intersection of the internal angle bisectors BO and CO. In triangle OBC, the sum of the angles is ∠ BOC + ∠ OBC + ∠ OCB = π.

Since BO and CO are the bisectors of ∠ B and ∠ C, we have ∠ OBC = ∠ B / 2 and ∠ OCB = ∠ C / 2. Substituting these into the angle sum equation for triangle OBC:

∠ BOC = π - (∠ OBC + ∠ OCB) = π - (∠ B / 2 + ∠ C / 2) = π - (∠ B + ∠ C) / 2

From the properties of triangle ABC, we know that ∠ B + ∠ C = π - ∠ A. Substituting this relationship into the equation for ∠ BOC:

∠ BOC = π - (π - ∠ A) / 2 ∠ BOC = π - π / 2 + ∠ A / 2 ∠ BOC = π / 2 + ∠ A / 2

Explanation

Let the cost price be x. According to the question, the profit percentage is also x, so the selling price is x + (x/100) * x = 96. Solving the resulting quadratic equation x² + 100x - 9600 = 0 gives x = 60.

Explanation

The sum of a set of numbers is calculated by multiplying the average by the total number of observations. For the initial set of 7 numbers with an average of 8, the sum is 7 × 8 = 56.

When one number is added, the total count of numbers becomes 8 and the new average is 9. The new sum is 8 × 9 = 72. The value of the added number is the difference between the new sum and the original sum: 72 - 56 = 16.

This can also be solved using the increment method. The added number must be equal to the new average plus the total increase it provides to the previous numbers. Since the average increased by 1 for each of the 7 original numbers, the added number is 9 + (7 × 1) = 16.

Explanation

The distance D of the journey can be calculated using the relationship between speed, distance, and time difference. When the same distance is covered at two different speeds S₁ and S₂, the distance is given by the formula:

D = (S₁ × S₂)/(S₁ - S₂) × Δ T

In this case, the first speed S₁ is 40 km/hr and the second speed S₂ is 35 km/hr. The difference in time Δ T is 15 minutes. To maintain consistency in units, this time difference is converted to hours:

Δ T = ¹⁵⁄₆₀ = ¼ hour

Substituting the values into the formula:

D = (40 × 35)/(40 - 35) × ¼

D = ¹⁴⁰⁰⁄₅ × ¼

D = 280 × ¼

D = 70 km

Alternatively, using the ratio method, the ratio of speeds is 40:35, which simplifies to 8:7. Since distance is constant, the time taken is inversely proportional to the speed, making the ratio of time 7:8. The difference of 1 unit in the ratio corresponds to the 15 minutes delay. Therefore, the time taken at 40 km/hr is 7 × 15 = 105 minutes. Converting this to hours gives 1.75 hours. The total distance is then 40 km/hr × 1.75 hr = 70 km.

Explanation

To find the combined ratio of runs scored by A, B, and C, we align the individual ratios A:B = 3:2 and B:C = 3:2. By multiplying the first ratio by 3 and the second ratio by 2 to equalize the B term, we obtain A:B = 9:6 and B:C = 6:4. This results in a continuous ratio of A:B:C = 9:6:4.

The sum of the parts in the ratio is 9 + 6 + 4 = 19. Given that the total runs scored by all three is 342, the value of one part is 342 / 19 = 18. The runs scored by B, who represents 6 parts of the total, is calculated as 6 × 18 = 108.

Explanation

The net percentage change resulting from successive price adjustments is calculated using the formula x + y + ˣʸ⁄₁₀₀. In this case, the price is first increased by 10% (x = 10) and subsequently reduced by a 10% rebate (y = -10). Plugging these values into the formula gives 10 - 10 + (10 × (-10))/(100). This simplifies to 0 - ¹⁰⁰⁄₁₀₀ = -1%. The negative sign indicates a net reduction in the original value, which corresponds to a 1% loss. If the initial cost price is taken as 100, the price after the 10% increase is 110. A 10% rebate on 110 is 11, making the final selling price 99. Comparing the final price of 99 to the original cost of 100 confirms a loss of exactly 1%.

Explanation

From the equation x² - 3x + 1 = 0, dividing by x gives x + 1/x = 3. Using the identity (x - 1/x)² = (x + 1/x)² - 4, we get (x - 1/x)² = 3² - 4 = 5, which means x - 1/x = ±√5. Since it is given that x > 1, then x² > 1, which implies x > 1/x, making the value of x - 1/x necessarily positive.

Explanation

The operation is defined as x ∘ y = 2x² - 3y. To find the values of x that satisfy the condition x ∘ 2 = -x, we substitute y = 2 into the defined expression:

2x² - 3(2) = -x 2x² - 6 = -x 2x² + x - 6 = 0

This quadratic equation can be solved by factoring the expression:

2x² + 4x - 3x - 6 = 0 2x(x + 2) - 3(x + 2) = 0 (2x - 3)(x + 2) = 0

Setting each factor to zero gives the solutions for x:

2x - 3 = 0 ⟹ x = 3/2 x + 2 = 0 ⟹ x = -2

The values of x that satisfy the condition are -2 and 3/2.

Explanation

Sum the three given equations to find the relationship between the variables: ax + ay + az = (x + y + z)y + (x + y + z)z + (x + y + z)x

On the left side, factor out the common term a. On the right side, factor out the common expression (x + y + z): a(x + y + z) = (x + y + z)(y + z + x)

Since (y + z + x) is algebraically identical to (x + y + z), the equation simplifies to: a(x + y + z) = (x + y + z)²

Let S = x + y + z. The equation can be written as: aS = S²

Rearranging the equation to solve for S: S² - aS = 0 S(S - a) = 0

This gives two possible values for S, which are 0 and a. Given a ≠ 0, the non-trivial value for the sum x + y + z is a.