SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The perimeter of a sector of a circle is the sum of its arc length and the two radii that bound it. The formula for the arc length l is given by l = (θ)/(360) × 2π r, where θ is the angle subtended at the centre and r is the radius.

Substituting the given values r = 24 cm, θ = 60°, and π = 3.14: l = ⁶⁰⁄₃₆₀ × 2 × 3.14 × 24 l = ⅙ × 48 × 3.14 l = 8 × 3.14 = 25.12 cm

The total perimeter P is calculated as: P = l + 2r P = 25.12 + 2(24) P = 25.12 + 48 = 73.12 cm

Explanation

Given the equation 3 sin A + 4 cos A = 5, we observe that the coefficients 3, 4, and 5 form a Pythagorean triplet since 3² + 4² = 5². For any equation of the form a sin θ + b cos θ = √(a² + b²), we can compare it to the fundamental identity sin² θ + cos² θ = 1 by dividing the equation by the constant term.

Dividing the given equation by 5, we get: ⅗ sin A + ⅘ cos A = 1

Comparing this to sin A · sin A + cos A · cos A = 1, we identify that sin A = ⅗ and cos A = ⅘. The value of tan A is defined as the ratio of sine to cosine: tan A = (sin A)/(cos A) = (3/5)/(4/5) = ¾

Explanation

Let the upstream speed be U and the downstream speed be D. Based on the relationship Time = Distance / Speed, we can set up two linear equations from the given conditions:

24/U + 40/D = 17 ... (i) 12/U + 32/D = 10 ... (ii)

To solve for 1/D, multiply equation (ii) by 2 to equate the 1/U coefficients: 24/U + 64/D = 20 ... (iii)

Subtracting equation (i) from equation (iii) eliminates the U term: (24/U + 64/D) - (24/U + 40/D) = 20 - 17 24/D = 3 D = 8 km/h

Substituting D = 8 back into equation (ii): 12/U + 32/8 = 10 12/U + 4 = 10 12/U = 6 U = 2 km/h

The speed of the stream is calculated as half the difference between the downstream and upstream speeds: Speed of stream = (D - U) / 2 = (8 - 2) / 2 = 6 / 2 = 3 km/h

Explanation

Since △ ABC ≅ △ QPR, the corresponding parts of congruent triangles (CPCT) must be equal. This means the vertices correspond in the order given: A ↔ Q, B ↔ P, and C ↔ R. Consequently, the side lengths correspond as AB = QP, BC = PR, and AC = QR, while the angles correspond as ∠ A = ∠ Q, ∠ B = ∠ P, and ∠ C = ∠ R.

Given AB = 7 cm, we find that PQ = AB = 7 cm. Similarly, given ∠ C = 75°, it follows that ∠ R = ∠ C = 75°. Therefore, both PQ = 7 cm and ∠ R = 75° are true based on the congruence relation.

Explanation

To find the time when A and B meet for the first time on a circular track while running in opposite directions, we use the concept of relative speed. First, convert the speeds from km/h to m/s by multiplying with the conversion factor ⁵⁄₁₈.

Speed of A = 36 × ⁵⁄₁₈ = 10 m/s. Speed of B = 54 × ⁵⁄₁₈ = 15 m/s.

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. Relative speed = 10 + 15 = 25 m/s.

The time taken for them to meet for the first time anywhere on the track is given by the formula: Time = (Distance)/(Relative Speed)

Substituting the total track length and the relative speed: Time = ⁴⁸⁰⁰⁄₂₅ = 192 seconds.

Explanation

The weight of the new student is calculated using the standard replacement formula: Weight of new person = Weight of replaced person + (Total number of items × Change in average). Since the average weight of 10 students increases by 0.5 kg, the total weight of the group increases by 10 × 0.5 = 5 kg. This increase occurs because the weight of the new student exceeds the weight of the replaced student by exactly this amount. Substituting the given values, the weight of the new student is 51 + 5 = 56 kg.

Explanation

Let the total capacity of the tank be 24 units, which is the least common multiple (LCM) of the individual filling times 6 and 8. Based on this, the filling rate of Pipe A is 24 / 6 = 4 units per hour, and the filling rate of Pipe B is 24 / 8 = 3 units per hour.

When both pipes are opened together, their combined rate of filling is 4 + 3 = 7 units per hour. In the first 3 hours, the amount of the tank filled by both pipes is 7 × 3 = 21 units.

The remaining capacity of the tank is 24 - 21 = 3 units. Since Pipe A is closed after 3 hours, Pipe B must fill the remaining 3 units alone. The time taken by Pipe B to fill this remainder is 3 / 3 = 1 hour. Consequently, the total time elapsed from the start until the tank is completely filled is 3 hours (with both pipes) plus 1 hour (with Pipe B only), resulting in 4 hours.

Explanation

The gain is given as ⅕ of the selling price (SP). To find the gain percentage, we first determine the cost price (CP) using the relation CP = SP - Gain.

Substituting the given gain value: CP = SP - ⅕ SP = ⅘ SP

The gain percentage is calculated based on the cost price using the formula: Gain % = (Gain)/(CP) × 100

Substituting the expressions for gain and cost price in terms of SP: Gain % = (1)/(5) SP{⅘ SP} × 100 Gain % = ¼ × 100 = 25%

While the selling price is provided as ₹71,785, it is not required for the calculation because the percentage is derived from the constant ratio between the gain and the cost price.

Explanation

The trigonometric identity for complementary angles states that tan(90° - θ) = cot θ. In the given expression (tan 55°)/(cot 35°), we observe that the sum of the angles is 55° + 35° = 90°, meaning they are complementary. By rewriting the numerator using this relationship, we get tan 55° = tan(90° - 35°) = cot 35°. Substituting this back into the original expression yields (cot 35°)/(cot 35°), which simplifies to 1.

Explanation

The mean proportional of two numbers p and q is given by the formula √(p × q). In this problem, the two numbers are 4a² and x, and their mean proportional is ¹⁄₅ₐ. Substituting these values into the formula:

¹⁄₅ₐ = √(4a² · x)

To find x, square both sides of the equation:

(¹⁄₅ₐ)² = (√(4a² · x))²

(1)/(25a²) = 4a² · x

Isolate x by dividing both sides by 4a²:

x = (1)/(25a² · 4a²)

x = (1)/(100a⁴)

Explanation

The final selling price SP after two successive discounts d₁ and d₂ on a marked price MP is determined by the relationship SP = MP × (1 - d₁{100}) × (1 - d₂{100}).

Given MP = 150, SP = 105, and d₁ = 12.5%, we substitute these values into the formula. Since 12.5% is equivalent to the fraction ⅛, the value remaining after the first discount is 150 × (1 - ⅛) = 150 × ⅞ = 131.25.

Now, we calculate the second discount d₂ using the intermediate price: 105 = 131.25 × (1 - d₂{100}) 1 - d₂{100} = (105)/(131.25) To simplify this calculation, we can express 131.25 as ⁵²⁵⁄₄: 1 - d₂{100} = 105 × ⁴⁄₅₂₅ 1 - d₂{100} = ⁴²⁰⁄₅₂₅ = ⅘ 1 - d₂{100} = 0.8 d₂{100} = 1 - 0.8 = 0.2 d₂ = 20%

Explanation

Let the present ages of A and B be 3x and 4x years respectively. Five years ago, their ages were 3x - 5 and 4x - 5. According to the given condition, the ratio of their ages five years back was 2 : 3, which can be expressed as the equation (3x - 5)/(4x - 5) = ⅔.

By cross-multiplying, we get 3(3x - 5) = 2(4x - 5), which simplifies to 9x - 15 = 8x - 10. Rearranging the terms to solve for x, we find 9x - 8x = 15 - 10, which gives x = 5. The present age of A is represented by 3x, so A's present age is 3 × 5 = 15 years.

Alternatively, using the ratio method, the difference between the ratio parts for both A and B is 3 - 2 = 1 unit and 4 - 3 = 1 unit. This 1 unit difference corresponds to the 5-year time gap. Since 1 unit equals 5 years, A's present age of 3 units equals 3 × 5 = 15 years.

Explanation

The area of an equilateral triangle is determined by the formula Area = √3{4} a², where a represents the length of the side. Given the side length a = 18 cm, the calculation is performed as follows:

Area = √3{4} × (18)²

Squaring the side length gives:

Area = √3{4} × 324

Dividing 324 by 4 yields 81, which results in a final area of 81√3 cm².

Explanation

To find the remainder when 19²⁰⁰ is divided by 20, we use the properties of modular arithmetic. We observe that 19 is equivalent to -1 when taken modulo 20, because 19 = 20 - 1. This can be written as:

19 ≡ -1 20

According to the power property of congruences, if a ≡ b n, then aᵏ ≡ bᵏ n for any positive integer k. Applying this property to our expression with k = 200:

19²⁰⁰ ≡ (-1)²⁰⁰ 20

Since 200 is an even integer, any negative number raised to this power results in a positive value. Specifically, (-1)²⁰⁰ = 1. Therefore:

19²⁰⁰ ≡ 1 20

Alternatively, this can be solved using the binomial expansion of (20 - 1)²⁰⁰. Every term in the expansion contains a factor of 20 except for the very last term, which is 200{200} (-1)²⁰⁰. As all other terms are divisible by 20, the remainder is determined solely by this last term:

1 × (-1)²⁰⁰ = 1

Explanation

Using the rule of alligation to determine the ratio of the two investments based on their individual simple interest rates and the overall effective rate:

Rate of the first part (R₁) = 4% Rate of the second part (R₂) = 6% Average effective rate (Rₘ) = 4.7%

The ratio of the principal invested at 4% (P₁) to the principal invested at 6% (P₂) is calculated as: (P₁)/(P₂) = (R₂ - Rₘ)/(Rₘ - R₁) = (6 - 4.7)/(4.7 - 4) = (1.3)/(0.7) = ¹³⁄₇

The total principal of ₹2,00,000 is divided according to this ratio. The total number of parts is 13 + 7 = 20. Value of one part = (2,00,000)/(20) = 10,000

The investment in the first bank (at 4%) = 13 × 10,000 = 1,30,000 The investment in the second bank (at 6%) = 7 × 10,000 = 70,000

Thus, the amounts invested are ₹1,30,000 and ₹70,000.

Explanation

The expression is simplified by first substituting the identity cot A = (cos A)/(sin A), which gives: (sin A)/(1 + cos A) + (cos A)/(sin A) Combining these fractions using the common denominator sin A(1 + cos A) results in: (sin² A + cos A(1 + cos A))/(sin A(1 + cos A)) Expanding the terms in the numerator leads to: (sin² A + cos A + cos² A)/(sin A(1 + cos A)) According to the Pythagorean identity, sin² A + cos² A = 1. Substituting this into the numerator simplifies the expression to: (1 + cos A)/(sin A(1 + cos A)) Canceling the common term (1 + cos A) in the numerator and denominator leaves (1)/(sin A). By trigonometric definition, the reciprocal of the sine function is the cosecant function, or csc A.

Explanation

The curved surface area of a right circular cone is calculated using the formula CSA = π r l, where r represents the radius of the base and l represents the slant height of the cone.

Given the radius r = 9 cm and the slant height l = 49 cm, we substitute these values into the formula using π ≈ ²²⁄₇:

CSA = ²²⁄₇ × 9 × 49

By simplifying the expression, the 7 in the denominator divides 49 to give 7:

CSA = 22 × 9 × 7

CSA = 22 × 63

CSA = 1386 cm²

Explanation

In a circle, if two arcs are congruent, they subtend equal angles at the center of the circle. According to the property of circles, equal arcs of a circle subtend equal chords. Let the center of the circle be O and the radius be r. If arc PAQ is congruent to arc RBS, the central angles ∠ POQ and ∠ ROS are equal. Using the law of cosines or the property of isosceles triangles formed by the radii, the length of chord PQ is given by 2r sin(∠ POQ / 2) and the length of chord RS is given by 2r sin(∠ ROS / 2). Since the central angles are equal, it follows that PQ = RS. Therefore, the ratio of the lengths of the chords PQ and RS is 1 : 1.

Explanation

In an election with two candidates where all votes are valid, the sum of the percentages of votes received by both candidates is 100%. If the winning candidate received 88% of the total votes, the losing candidate received 100% - 88% = 12% of the total votes.

The majority by which a candidate is elected is defined as the difference between the votes of the winner and the loser. In terms of percentages, the majority is:

Majority percentage = 88% - 12% = 76%

Given that the majority is 684 votes, we can establish the relationship where 76% of the total votes x equals 684:

⁷⁶⁄₁₀₀ × x = 684

To find the total number of votes x, rearrange the formula:

x = (684 × 100)/(76)

Dividing 684 by 76 yields 9. Therefore:

x = 9 × 100 = 900

Explanation

To identify the largest fraction, we evaluate the numerical magnitude of each term. Converting 79/90 into decimal form results in approximately 0.878, which is the highest value in the given set. This can be confirmed by comparing the proximity of each fraction to 1. The distance of 79/90 from the whole is 1 - 79/90 = 11/90, which is approximately 0.122. This gap is significantly smaller than the gaps for the other fractions, which are at least 0.25 or greater. Since 79/90 has the smallest deficit from 1, it is the largest value provided. Comparing the numerators to denominators also shows that 79/90 represents nearly 88 percent of a whole, exceeding the proportions of the other ratios.

Explanation

In an isosceles triangle where PQ = PR, the altitude PT drawn from the vertex P to the base QR bisects the base. This implies QT = TR and creates a right-angled triangle △ PTQ. Applying the Pythagorean theorem to △ PTQ:

PQ² = PT² + QT² 17² = 15² + QT² 289 = 225 + QT² QT² = 64 QT = 8 cm

This calculation corresponds to the standard Pythagorean triplet (8, 15, 17). Since PT bisects QR, the total length of the base is QR = 2 × QT = 2 × 8 = 16 cm.

Explanation

The equation x + ¹⁄ₓ = 2 can be rearranged by multiplying both sides by x to obtain x² + 1 = 2x, which is x² - 2x + 1 = 0. This is the expansion of the perfect square (x - 1)² = 0, which yields the unique solution x = 1. Substituting x = 1 into the expression (x²⁷²³ + (1)/(x²³⁵⁶⁾), we get 1²⁷²³ + (1)/(1²³⁵⁶⁾. Since 1 raised to any finite power is 1, the value becomes 1 + ¹⁄₁ = 1 + 1 = 2.

Explanation

The expression follows the standard algebraic identity for the difference of two cubes: (a³ - b³)/(a² + ab + b²) = a - b

In the given expression, although the numerator is written with squares, the structure of the denominator, 7.3² + 7.3 × 4.7 + 4.7², is the characteristic quadratic factor (a² + ab + b²) of the cubic identity. This indicates that the intended expression to be simplified is based on a = 7.3 and b = 4.7.

Applying the result of the identity: a - b = 7.3 - 4.7 7.3 - 4.7 = 2.6

Explanation

When two circles of equal radius r pass through each other's centers, the distance between their centers is also r. Let the centers be C₁ and C₂, and the intersection points be A and B. The segment C₁C₂ has length r. The distances AC₁ and AC₂ are both radii, so AC₁ = AC₂ = r. This makes triangle AC₁C₂ an equilateral triangle with side length r. The common chord AB is perpendicular to the line joining the centers and is bisected by it. The length of the common chord is twice the altitude of the equilateral triangle AC₁C₂. The altitude h of an equilateral triangle with side r is given by the formula h = √3{2}r. Therefore, the length of the common chord AB = 2h = 2 × √3{2}r = √3r. Substituting the given radius r = 13 cm, the length of the common chord is 13√3 cm.

Explanation

For a number to be divisible by 36, it must be simultaneously divisible by 4 and 9, as 36 = 4 × 9 and these factors are coprime.

The divisibility rule for 4 states that the last two digits of the number must be divisible by 4. Here, the last two digits are y6. Possible values for the digit y that make y6 divisible by 4 are 1, 3, 5, 7, and 9 (16, 36, 56, 76, 96). The question requires the largest possible value of y, which is y = 9.

The divisibility rule for 9 states that the sum of the digits of the number must be divisible by 9. The sum of the digits of 5x79856y6 is 5 + x + 7 + 9 + 8 + 5 + 6 + y + 6 = 46 + x + y. Substituting y = 9, the sum becomes 46 + x + 9 = 55 + x. For 55 + x to be divisible by 9, the nearest multiple of 9 greater than 55 is 63. Thus, 55 + x = 63, which gives x = 8.

We need to calculate the negative value of √(2x + y). Substituting x = 8 and y = 9 into the expression: √(2(8) + 9) = √(16 + 9) = √(25) = 5

The negative value of this result is -5.

Explanation

Let A and B represent the number of marbles Anna and Ben have, respectively. According to the first condition, if Anna gives one marble to Ben, they have equal amounts: A - 1 = B + 1. This equation simplifies to A = B + 2.

According to the second condition, if Ben gives one marble to Anna, she has three times as many marbles as Ben: A + 1 = 3(B - 1). Substituting the first relationship into the second gives: (B + 2) + 1 = 3B - 3 B + 3 = 3B - 3 2B = 6 B = 3

Using A = B + 2, we find that A = 3 + 2 = 5. Therefore, the total number of marbles they both have is A + B = 5 + 3 = 8.

Explanation

To find the joint ratio A:B:C from the given ratios A:B = 6:8 and B:C = 7:12, we equate the common term B in both expressions. The ratio A:B can be simplified to 3:4. To make the value of B identical in both ratios, we find the least common multiple of 4 (from A:B) and 7 (from B:C), which is 28. Multiplying the terms of A:B by 7 gives 21:28, and multiplying the terms of B:C by 4 gives 28:48.

Alternatively, using the standard compounding method: A = 6 × 7 = 42 B = 8 × 7 = 56 C = 8 × 12 = 96

The resulting ratio is 42:56:96. Dividing each term by the common factor 2 yields the simplified ratio 21:28:48.

Explanation

The speed of the man in still water is u = 6 km/hr. The man rows 6 km in 45 minutes with the current, which is his downstream speed v_d. Converting time to hours, 45 minutes is 0.75 hours, so v_d = 6 / 0.75 = 8 km/hr.

The speed of the current v can be found using the relation v_d = u + v. Substituting the known values, 8 = 6 + v, which gives v = 2 km/hr. The speed of the man against the current, or upstream speed vᵤ, is given by vᵤ = u - v = 6 - 2 = 4 km/hr.

To find the total time taken for a 16 km round trip, we calculate the time taken for each leg. The time taken to travel downstream is t_d = 16 / v_d = 16 / 8 = 2 hours. The time taken to return upstream is tᵤ = 16 / vᵤ = 16 / 4 = 4 hours. The total time taken is t_d + tᵤ = 2 + 4 = 6 hours.

Explanation

Using the algebraic identity for the sum of cubes, we have (3a + 2b)³ = (3a)³ + (2b)³ + 3(3a)(2b)(3a + 2b). Substituting the given values 3a + 2b = 27 and 27a³ + 8b³ = 1458 into this formula:

27³ = 1458 + 3(6ab)(27)

Calculating the cube of 27, we get 27³ = 19683. Substituting this back into the equation:

19683 = 1458 + 18ab(27)

Subtracting 1458 from both sides:

19683 - 1458 = 18ab(27) 18225 = 486ab

To find the value of 2ab, we can rearrange the equation as:

2ab = (18225 × 2)/(486) 2ab = ¹⁸²²⁵⁄₂₄₃

Dividing 18225 by 243, we find:

2ab = 75

Explanation

In parallelogram MNOP, opposite sides are equal and parallel, so MN = OP and MN ∥ OP. The problem states that MN is produced to Q such that MN = NQ. Therefore, NQ = OP and NQ ∥ OP. Consider the triangles △ RNQ and △ ROP. Because NQ ∥ OP, the alternate interior angles ∠ RNQ = ∠ ROP and ∠ RQN = ∠ RPO are equal. Since we also have NQ = OP, the triangles △ RNQ and △ ROP are congruent by the Angle-Side-Angle (ASA) criterion. In congruent triangles, corresponding parts are equal, which means RN = RO. Since RN and RO are equal segments of the line ON, the point R is the midpoint of ON, dividing it in the ratio 1 : 1.

Explanation

The numerator 4x² + 9y² - 12xy is a perfect square trinomial that follows the algebraic identity (a - b)² = a² + b² - 2ab. By identifying the terms, we see that a² = 4x² (which means a = 2x) and b² = 9y² (which means b = 3y). The middle term 2ab = 2(2x)(3y) = 12xy matches the given expression. Therefore, the numerator can be factored as (2x - 3y)².

In algebra, the square of a binomial (A - B)² is identical to (B - A)² because (-(B - A))² = (B - A)². This allows the numerator to be expressed as (3y - 2x)². Substituting this into the original fraction gives:

(3y - 2x)²{3y - 2x}

By applying the law of exponents aᵐ{aⁿ} = aᵐ⁻ⁿ, we subtract the power of the denominator from the power of the numerator. This simplifies the expression to (3y - 2x)²⁻¹, which equals 3y - 2x.

Explanation

The product-to-sum trigonometric identity for sin A sin B is derived from the cosine compound angle formulas. The expansions for the cosine of the difference and the sum of two angles are: cos(A - B) = cos A cos B + sin A sin B cos(A + B) = cos A cos B - sin A sin B By subtracting the expression for cos(A + B) from the expression for cos(A - B), the cos A cos B terms cancel out: cos(A - B) - cos(A + B) = (cos A cos B + sin A sin B) - (cos A cos B - sin A sin B) cos(A - B) - cos(A + B) = 2 sin A sin B Dividing both sides of the equation by 2 isolates the product of the sine functions: sin A sin B = ½ {cos(A - B) - cos(A + B)}

Explanation

The net percentage change for successive increases or decreases is calculated using the formula x + y + ˣʸ⁄₁₀₀.

First, calculate the net change for the first two days (18% increase and 15% decrease): Net₁_,₂ = 18 - 15 + (18 × (-15))/(100) = 3 - ²⁷⁰⁄₁₀₀ = 3 - 2.7 = 0.3% increase.

Next, combine this result with the third day change (3% decrease): Netₜₒₜₐₗ = 0.3 - 3 + (0.3 × (-3))/(100) = -2.7 - (0.9)/(100) = -2.7 - 0.009 = -2.709%.

A negative result indicates a decrease. Rounding 2.709% to one decimal place yields a 2.7% decrease.

Explanation

A chord is defined as a line segment whose endpoints lie on the circumference of a circle. When this segment is extended infinitely in both directions, it becomes a straight line that passes through the circle at two distinct points. In geometry, a line that intersects a circle at exactly two points is called a secant. Therefore, the infinite extension of a chord results in a secant line.

Explanation

To find the final payment at the end of the second year, we first calculate the interest accrued during the first year and the remaining principal after the partial repayment.

Initial principal P₁ = 192208. Interest for the first year I₁ = P₁ × R × T = 192208 × 0.15 × 1 = 28831.2.

At the end of the first year, Aman pays 20196 towards the principal. The remaining principal for the second year P₂ = 192208 - 20196 = 172012.

Interest for the second year is calculated on the remaining principal P₂ because simple interest is typically calculated on the outstanding balance in such repayment scenarios. Interest for the second year I₂ = 172012 × 0.15 × 1 = 25801.8.

The total amount to be paid at the end of the second year to clear all dues includes the remaining principal P₂, the interest from the first year I₁, and the interest from the second year I₂.

Total payment = P₂ + I₁ + I₂ = 172012 + 28831.2 + 25801.8 = 172012 + 54633 = 226645.

Explanation

The seller's cost price for a standard batch of 20 cookies (representing one kg) is ₹76. From this, the cost price of a single cookie is calculated as 76 / 20 = 3.8. Although the customer expects one kg (20 cookies), the seller cheats and only delivers 19 cookies. The actual cost incurred by the seller for the quantity delivered is the cost of 19 cookies, which is 19 × 3.8 = 72.2. The seller receives a selling price of ₹95 for this transaction. The gain is the difference between the selling price and the cost of the cookies actually given: Gain = 95 - 72.2 = 22.8.

Explanation

The area A of a sector of a circle is calculated using the formula A = ½ r l, where r is the radius and l is the length of the arc. This formula is derived from the ratio of the sector area to the total circle area being equal to the ratio of the arc length to the total circumference.

By substituting the given radius r = 24 cm and the arc length l = 7 cm into the formula:

A = ½ × 24 × 7

A = 12 × 7 = 84

This calculation results in an area of 84 cm².

Explanation

The selling price (SP) of an article after a discount is related to the marked price (MP) by the formula SP = MP × (1 - ᵈ⁄₁₀₀), where d is the discount percentage.

Given the selling price SP = 1792 and the discount d = 20%, the relationship is:

1792 = MP × (1 - ²⁰⁄₁₀₀) 1792 = MP × 0.80

To find the marked price, divide the selling price by the remaining percentage:

MP = (1792)/(0.8) MP = ¹⁷⁹²⁰⁄₈ MP = 2240

Alternatively, if an article is sold at a 20% discount, then 80% of the marked price equals the selling price. Therefore, 80% of MP = 1792. Calculating 100% gives ¹⁷⁹²⁄₈₀ × 100 = 22.4 × 100 = 2240.

Explanation

Let the total capacity of the tank be the least common multiple of 60 and 45, which is 180 units. The filling rate of pipe Q is 180 / 60 = 3 units per hour, and the filling rate of pipe R is 180 / 45 = 4 units per hour. In the first 6 hours, pipes Q and R together fill 6 × (3 + 4) = 42 units.

After 6 hours, pipe W is opened to empty the tank. Let its emptying rate be w units per hour. For the next 24 hours, all three pipes are active, contributing a net volume of 24 × (3 + 4 - w) = 24(7 - w) units. The total volume in the tank after these 24 hours is the half-level mark, which is 180 / 2 = 90 units.

The equation representing the total volume is 42 + 24(7 - w) = 90. Subtracting 42 from both sides yields 24(7 - w) = 48. Dividing by 24 gives 7 - w = 2, which results in w = 5 units per hour. To find the time pipe W alone takes to empty the entire 180-unit tank, we divide the total capacity by its emptying rate: 180 / 5 = 36 hours.

Explanation

To simplify the given expression, we find a common denominator by combining the two square root terms:

(√(secθ + 1))² + (√(secθ - 1))²{√((secθ - 1)(secθ + 1))}

Expanding the squares in the numerator and applying the algebraic identity (a-b)(a+b) = a² - b² in the denominator, we get:

(secθ + 1 + secθ - 1)/(√(sec²θ - 1)) = (2secθ)/(√(sec²θ - 1))

Using the fundamental trigonometric identity sec²θ - 1 = tan²θ, the expression simplifies to:

(2secθ)/(tanθ)

By substituting secθ = (1)/(cosθ) and tanθ = (sinθ)/(cosθ), the calculation proceeds as follows:

(2)/(cosθ) · (cosθ)/(sinθ) = (2)/(sinθ) = 2cscθ

Explanation

The trigonometric identity sin² θ + cos² θ = 1 and the complementary angle relations sin(90° - θ) = cos θ and cos(90° - θ) = sin θ are applied here.

In the numerator, since 26° + 64° = 90°, we have sin 64° = cos 26°. Thus, sin² 64° = cos² 26°. Substituting this, the numerator becomes 9sin² 26° + 9cos² 26° + 36 = 9(sin² 26° + cos² 26°) + 36 = 9(1) + 36 = 45.

In the denominator, since 32° + 58° = 90°, we have cos 58° = sin 32°. Thus, cos² 58° = sin² 32°. Substituting this, the denominator becomes 32 - 7cos² 32° - 7sin² 32° = 32 - 7(cos² 32° + sin² 32°) = 32 - 7(1) = 25.

The complete expression is √(⁴⁵⁄₂₅) = √(⁹⁄₅) = (3)/(√(5)). Multiplying the numerator and denominator by √5 to rationalize gives 3√5{5}.

Explanation

The average of the first n natural numbers is calculated using the formula (n + 1)/(2). This formula is derived from the sum of the first n natural numbers, which is (n(n + 1))/(2). To find the average, the sum is divided by the total number of terms n, resulting in (n + 1)/(2).

In this problem, n = 125. Substituting this value into the formula:

Average = (125 + 1)/(2)

Average = ¹²⁶⁄₂

Average = 63

This can also be understood as the middle term of an arithmetic progression. For a sequence of consecutive natural numbers, the average is simply the mean of the first term (1) and the last term (125), which yields (1 + 125)/(2) = 63.

Explanation

The volume of a cone is calculated using the formula V = ⅓π r²h, where r is the radius and h is the vertical height. To find the height, we use the relationship between slant height l, radius r, and height h given by the Pythagorean theorem: l² = r² + h².

Substituting the given values r = 3 and l = 3√2 into the equation: (3√2)² = 3² + h² 18 = 9 + h² h² = 9 h = 3 units.

Now, substituting r = 3 and h = 3 into the volume formula: V = ⅓ × π × 3² × 3 V = ⅓ × π × 9 × 3 V = 9π cubic units.

Explanation

In the quadrilateral WXYZ, the diagonals ZX and WY intersect at point O. To find the length of ZX, we consider the triangles △ XQO and △ ZPO. Given that XQ ⊥ WY and ZP ⊥ WY, it follows that ∠ XQO = ∠ ZPO = 90°. Additionally, ∠ XOQ and ∠ ZOP are vertically opposite angles, which makes them equal. The problem states that XQ = ZP, providing the necessary conditions for the Angle-Angle-Side (AAS) congruence criterion. Since △ XQO ≅ △ ZPO, their corresponding parts must be equal, meaning OX = OZ. Given that OX = 12 cm, the length of OZ is also 12 cm. The diagonal ZX is composed of the segments OX and OZ, so its total length is ZX = OX + OZ = 12 + 12 = 24 cm.

Explanation

The area of a parallelogram is determined by the formula Area = base × height, where the height is the perpendicular distance between the base and its opposite parallel side. This geometric property exists because any parallelogram can be rearranged into a rectangle with the same base and height by shifting a triangular section from one end to the other, thus maintaining the same surface area.

For a parallelogram with a base of 26 cm and a height of 18 cm, the calculation is: Area = 26 cm × 18 cm

To solve this efficiently using mental math, the calculation can be broken down as follows: 26 × 18 = 26 × (20 - 2) 26 × 20 = 520 26 × 2 = 52 520 - 52 = 468

The resulting product yields a total area of 468 cm², which represents the two-dimensional space enclosed by the four sides of the figure.

Explanation

According to the Power of a Point theorem, for a point R outside a circle with center O and radius r, the power of the point is given by d² - r², where d is the distance from the point to the center. This value is equal to the product of the distances from point R to the two intersection points of any secant passing through R.

Given that the secant intersects the circle at points A and B, the power of point R is RA · RB. We are given RA = 7 cm and AB = 5 cm. Since A and B are on the same secant line starting from R, the distance RB = RA + AB = 7 + 5 = 12 cm.

Substituting the known values into the theorem: RA · RB = d² - r² 7 · 12 = 18² - r² 84 = 324 - r² r² = 324 - 84 r² = 240

Taking the square root to find the radius: r = √(240) = √(16 · 15) = 4√(15) cm

Explanation

The given expression is sin x + csc x = 2. Since csc x = (1)/(sin x), let sin x = t. The equation is transformed into t + ¹⁄ₜ = 2. Multiplying through by t yields t² + 1 = 2t, which can be rearranged as t² - 2t + 1 = 0. This is the expansion of the identity (t - 1)² = 0, which implies t = 1.

Consequently, sin x = 1. Since csc x is the reciprocal of sin x, csc x = ¹⁄₁ = 1. To find the value of sin⁷ x + csc⁷ x, we substitute these values: 1⁷ + 1⁷ = 1 + 1 = 2. This follows the algebraic principle where if x + ¹⁄ₓ = 2, then xⁿ + (1)/(xⁿ⁾ = 2 for any positive integer n, as the only real value satisfying the initial condition is x = 1.

Explanation

To find the percentage scored by student C, we first calculate the total marks obtained by summing the scores in all six subjects: English, Hindi, Maths, Physics, Chemistry, and Biology. Based on the table, the marks for student C are 80, 80, 70, 60, 70, and 60 respectively.

Total marks obtained by C = 80 + 80 + 70 + 60 + 70 + 60 = 420

As each subject is out of 100, the total maximum marks for the six subjects is 6 × 100 = 600. The percentage score is calculated using the following formula:

Percentage = (Total marks obtained / Maximum marks) × 100

Substituting the values:

Percentage = (420 / 600) × 100 = 420 / 6 = 70%

Explanation

To find the percentage share of sales for the months of June and July relative to the total sales, we first calculate the total sales across all months mentioned in the table and then determine the combined sales for the target months.

Total Sales = 25,000 (May) + 24,000 (June) + 18,000 (July) + 35,000 (November) + 26,000 (December) + 22,000 (January) = 150,000

Combined Sales (June and July) = 24,000 + 18,000 = 42,000

The percentage share is calculated using the formula: Percentage Share = ( (Sales in June and July)/(Total Sales) ) × 100

Substituting the values: Percentage Share = ( (42,000)/(150,000) ) × 100 = ⁴²⁄₁₅₀ × 100 = ¹⁴⁄₅₀ × 100 = 14 × 2 = 28%

Explanation

Let H, W, and S represent the present ages of the husband, wife, and son, respectively. According to the problem:

1. H + W + S = 90
2. H - W = (S)/(3)
3. S + 10 = ½(H + 10)

Rearranging equation (3) to solve for H: 2S + 20 = H + 10 H = 2S + 10

Substituting this expression for H into equation (2) allows us to express W in terms of S: (2S + 10) - W = (S)/(3) W = 2S + 10 - (S)/(3) = (5S + 30)/(3)

Now, substitute the expressions for H and W into equation (1): (2S + 10) + (5S + 30)/(3) + S = 90 3S + 10 + (5S + 30)/(3) = 90 3(3S + 10) + (5S + 30) = 270 9S + 30 + 5S + 30 = 270 14S + 60 = 270 14S = 210 S = 15

With the son's age S = 15, we find the husband's age: H = 2(15) + 10 = 40

The present age of the mother (wife) is then: W = H - (S)/(3) = 40 - ¹⁵⁄₃ = 40 - 5 = 35