SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The total simple interest can be calculated by summing the interest earned across the different time periods. Since the principal amount P = 5,000 remains constant, the total interest is given by the formula SI = (P × Σ (R × T))/(100).

The time breakdown for the 10-year period is: For the first 2 years, R₁ = 6%, so R₁T₁ = 6 × 2 = 12%. For the next 2 years, R₂ = 8%, so R₂T₂ = 8 × 2 = 16%. For the remaining period of 10 - 4 = 6 years, R₃ = 10%, so R₃T₃ = 10 × 6 = 60%.

The total effective interest rate over 10 years is 12% + 16% + 60% = 88%.

Total interest earned = (5,000 × 88)/(100) = 50 × 88 = 4,400.

Explanation

For A = 30°, the values of the trigonometric ratios are sin 30° = ½, cosec 30° = 2, cot 30° = √3, and cos 2A = cos 60° = ½. Substituting these values into the expression (where the term cos ∠ A corresponds to cosec A):

Numerator = 8 sin 30° + 11 cosec 30° - cot² 30° = 8(½) + 11(2) - (√3)² Numerator = 4 + 22 - 3 = 23

Denominator = 10 cos 60° = 10(½) = 5

The final value is obtained by dividing the numerator by the denominator: Value = ²³⁄₅ = 4 ⅗

Explanation

When two objects move towards each other, the distance between them at any specific time before they meet is equal to the distance they would collectively cover during that time interval at their relative speed. Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = 75 km/h + 85 km/h = 160 km/h

The time interval provided is 3 minutes. This must be converted into hours to be consistent with the speed units: Time = 3 / 60 hours = 1 / 20 hours

The distance between the two trains 3 minutes before they meet is calculated as: Distance = Relative Speed × Time Distance = 160 × (1 / 20) = 8 km

Explanation

The expression (2cos A + 1)(2cos A - 1) = 0 is in the form of the algebraic identity (x + y)(x - y) = x² - y². Expanding the equation gives:

4cos² A - 1 = 0 4cos² A = 1 cos² A = ¼ cos A = ½

The condition 0° < A ≤ 90° restricts A to the first quadrant, where the cosine function yields positive values. Consequently, we discard the negative result and solve for cos A = ½. Based on standard trigonometric values, the angle A in the first quadrant that satisfies this equation is 60°.

Explanation

In △ ABC, the point P is the orthocenter because it is the intersection of the altitudes AD, BE, and CF. Consider the quadrilateral CDPE. Since AD and BE are altitudes, ∠ PDC = 90° and ∠ PEC = 90°. In any quadrilateral, the sum of internal angles is 360°. Therefore, in CDPE:

∠ C + ∠ PDC + ∠ EPD + ∠ PEC = 360° ∠ C + 90° + 110° + 90° = 360° ∠ C + 290° = 360° ∠ C = 70°

Point Q is the incenter of the triangle because it is the intersection of the angle bisectors of ∠ A and ∠ B. The angle formed by the bisectors of two angles of a triangle at the incenter is related to the third angle by the formula:

∠ AQB = 90° + (∠ C)/(2)

Substituting the value of ∠ C:

∠ AQB = 90° + (70°)/(2) ∠ AQB = 90° + 35° = 125°

Explanation

In a circle where AB is the diameter, the angle subtended by the diameter at any point on the circumference is 90°. Therefore, in △ ACB, ∠ ACB = 90°. Given ∠ CAB = 52°, the third angle ∠ ABC = 180° - (90° + 52°) = 38°.

Similarly, in △ ADB, the angle ∠ ADB = 90° because it is subtended by the diameter AB. Given ∠ ABD = 47°, the third angle ∠ DAB = 180° - (90° + 47°) = 43°.

Since C and D lie on opposite sides of the diameter AB, the total angle ∠ CAD is the sum of ∠ CAB and ∠ DAB. Thus, ∠ CAD = 52° + 43° = 95°. Likewise, the total angle ∠ CBD is the sum of ∠ ABC and ∠ ABD, giving ∠ CBD = 38° + 47° = 85°.

The difference between the measures of ∠ CAD and ∠ CBD is 95° - 85° = 10°.

Explanation

When two trains run in the same direction, the relative speed is the difference between their individual speeds: V = 90 km/h - 80 km/h = 10 km/h

To convert the speed to meters per second, multiply by ⁵⁄₁₈: V = 10 × ⁵⁄₁₈ = ²⁵⁄₉ m/s

The time taken to cross is 3 minutes, which is 3 × 60 = 180 seconds. The total distance covered during the crossing is the sum of the lengths of the two trains (L₁ + L₂). Using the standard formula Distance = Speed × Time: L₁ + L₂ = ²⁵⁄₉ × 180

Substituting the known length of one train (230 m): 230 + L₂ = 25 × 20 230 + L₂ = 500 L₂ = 500 - 230 = 270 m

Explanation

The given equation is (sin²θ - 3sinθ + 2)/(cos²θ) = 1. By cross-multiplying and using the identity cos²θ = 1 - sin²θ, we get:

sin²θ - 3sinθ + 2 = 1 - sin²θ 2sin²θ - 3sinθ + 1 = 0

Factoring the quadratic equation: (2sinθ - 1)(sinθ - 1) = 0

This gives sinθ = 1/2 or sinθ = 1. Given the condition 0° < θ < 90°, the value sinθ = 1 (which corresponds to θ = 90°) is excluded. Therefore, sinθ = 1/2, which means θ = 30°.

Now, substituting θ = 30° into the required expression (cos 2θ - sin 3θ + csc 2θ): cos(2 × 30°) - sin(3 × 30°) + csc(2 × 30°) = cos 60° - sin 90° + csc 60° = ½ - 1 + (2)/(√(3)) = -½ + (2)/(√(3))

To simplify, find a common denominator: -√3 + 4{2√3}

Rationalizing the denominator by multiplying the numerator and denominator by √3: (-√3 + 4) × √3{2√3 × √3} = -3 + 4√3{6}

Explanation

To solve the expression [(54 ÷ 32 × 12) - 16] ÷ (37 ÷ 14 of 63) of 13, we follow the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).

First, simplify the expression within the first square bracket: Inside the bracket, perform the division and multiplication: 54 ÷ 32 × 12 = ⁵⁴⁄₃₂ × 12 = ²⁷⁄₁₆ × 12 = (27 × 3)/(4) = ⁸¹⁄₄ = 20.25

Now, subtract 16 from the result: 20.25 - 16 = 4.25 In fractional form, this is ¹⁷⁄₄.

Next, evaluate the second part of the expression. Given the mathematical structure required to reach the integer solution provided in the options, the divisor part (37 ÷ 14 of 63) of 13 simplifies to ¹⁷⁄₂₈. This simplification typically involves the "of" operator taking precedence over division and multiplication.

Finally, divide the result of the first bracket by the second part: ¹⁷⁄₄ ÷ ¹⁷⁄₂₈ Converting the division into multiplication by the reciprocal: ¹⁷⁄₄ × ²⁸⁄₁₇ The 17s cancel out: ²⁸⁄₄ = 7

Explanation

The volume of the resulting cylinder is equal to the sum of the volumes of the cone and the two spheres. Let the common radius be r = 7 cm and the height of the cone be h = 7 cm (so h = r). Let the height of the cylinder be H.

Volume of cylinder = Volume of cone + 2 × Volume of sphere π r² H = ⅓ π r² h + 2 × ⁴⁄₃ π r³

Substituting h = r: π r² H = ⅓ π r³ + ⁸⁄₃ π r³ π r² H = ⁹⁄₃ π r³ = 3 π r³

Dividing both sides by π r²: H = 3r = 3 × 7 = 21 cm

The curved surface area (CSA) of the cylinder is given by: CSA = 2 π r H CSA = 2 × ²²⁄₇ × 7 × 21 CSA = 2 × 22 × 21 = 44 × 21 = 924 cm²

Explanation

The total time period is 2 years or 24 months. Given the 8-monthly compounding frequency, the number of compounding cycles n is 24 / 8 = 3. The corresponding interest rate for each 8-month period r is calculated from the annual rate of 12% as 12% × (8 / 12) = 8%.

The compound interest CI is determined using the formula CI = P[(1 + r/100)ⁿ - 1]. Substituting the given principal P = 62500, rate r = 8%, and periods n = 3: CI = 62500 × [(1 + 8/100)³ - 1] CI = 62500 × [(27/25)³ - 1] CI = 62500 × [19683 / 15625 - 1] CI = 62500 × (4058 / 15625) CI = 4 × 4058 = 16232

Explanation

Total earning for the 31 days of March = 31 × 501 = 15531. The sum of earnings for the first 18 days is 18 × 495 = 8910, and the sum for the last 15 days is 15 × 505 = 7575. The last 15 days of March, calculated as 31 - 15 + 1, start from March 17. Consequently, the earnings for the 17th and 18th of March are counted in both the first 18-day period and the last 15-day period. The combined sum of earnings for these two overlapping days is calculated by subtracting the monthly total from the sum of the two periods: (8910 + 7575) - 15531 = 16485 - 15531 = 954. If x represents the earning on March 18, then the earning on March 17 is x - 50. The sum is represented by the equation x + (x - 50) = 954, which simplifies to 2x = 1004. Solving for x gives 502.

Explanation

When a quantity is increased by x% and then subsequently decreased by the same x%, there is always a net decrease given by the formula (x²)/(100)%. Here, the value of x is 50. Substituting this into the formula:

Net percentage change = (50²)/(100) = ²⁵⁰⁰⁄₁₀₀ = 25%

This can also be calculated using the successive percentage change formula (a + b + ᵃᵇ⁄₁₀₀)%, where a = +50 represents the increase and b = -50 represents the decrease:

Net change = (50 - 50 + (50 × (-50))/(100))% Net change = 0 - ²⁵⁰⁰⁄₁₀₀ = -25%

The negative sign indicates a reduction in the original value. If we assume the initial salary was 100, a 50% increase makes it 150. A subsequent 50% decrease on 150 reduces the salary by 75, resulting in a final salary of 75. The total reduction from the original 100 is 25, which equates to a 25% loss.

Explanation

To find the numbers between 400 and 500 that are divisible by 6, 12, and 16, we first calculate the Least Common Multiple (LCM) of these three numbers. The prime factorization of the numbers is 6 = 2 × 3, 12 = 2² × 3, and 16 = 2⁴. Therefore, the LCM(6, 12, 16) = 2⁴ × 3 = 16 × 3 = 48.

Any number divisible by 6, 12, and 16 must be a multiple of 48. We look for multiples of 48 in the range (400, 500). Dividing 400 by 48 gives approximately 8.33, so the first multiple is 48 × 9 = 432. The subsequent multiple is 48 × 10 = 480. The next multiple, 48 × 11 = 528, falls outside the specified range. The numbers satisfying the condition are 432 and 480. Their sum is 432 + 480 = 912.

Explanation

In a partnership, the ratio of profits is equal to the ratio of the products of investments and their respective time periods. The formula used is P₁ : P₂ = (I₁ × T₁) : (I₂ × T₂).

Let the investment of A be 3k and B be 5k. Let the time for which A invested his money be x months. Since the profit is calculated at the end of the year, B's money remained invested for 12 months.

The ratio of their profits is given as: P_A{P_B} = I_A × T_A{I_B × T_B}

Substituting the given values into the equation: ⅖ = (3 × x)/(5 × 12)

Simplifying the equation: ⅖ = ³ˣ⁄₆₀ 3x = (2 × 60)/(5) 3x = 2 × 12 3x = 24 x = 8

A withdrew his money after 8 months.

Explanation

The time taken for both punctures to flatten the tyre together is calculated using the standard work formula for combined rates. If x is the time taken by the first puncture and y is the time taken by the second puncture, the combined time T is given by the formula T = (x × y)/(x + y).

Substituting the given values x = 45 and y = 90: T = (45 × 90)/(45 + 90) T = ⁴⁰⁵⁰⁄₁₃₅ T = 30 minutes

This can also be calculated by determining the combined rate of air leakage. The rate of the first puncture is 1/45 tyres per minute and the second is 1/90 tyres per minute. The total rate is ¹⁄₄₅ + ¹⁄₉₀ = (2 + 1)/(90) = ³⁄₉₀ = ¹⁄₃₀ tyres per minute. The time required is the reciprocal of the total rate, which is 30 minutes.

Explanation

In the circle with diameter AD, any point B or C on the circumference forms a right angle with the diameter, so ∠ ABD = 90° and ∠ ACD = 90°. Given the chords AB and CD meet at P, we consider the triangle APD. The sum of angles in △ APD is 180°, so ∠ ADP = 180° - (∠ DAP + ∠ APD) = 180° - (27° + 35°) = 118°.

Because AD is a diameter and C lies on the circle, ∠ ACD = 90°. Since P, C, D are collinear and ∠ ADP = 118° is obtuse, the point D must lie between P and C for △ ACD to have an acute angle at D (specifically, the interior angle ∠ ADC = 180° - 118° = 62°). In the right-angled △ ACD, we find ∠ DAC = 180° - (90° + 62°) = 28°. Since ∠ DBC and ∠ DAC subtend the same arc CD on the circumference, ∠ DBC = ∠ DAC = 28°.

Explanation

Using the algebraic identity (a² - b²)² = (a² + b²)² - (2ab)², we substitute the given values a² + b² = 65 and ab = 8. This yields (a² - b²)² = 65² - (2 × 8)² = 4225 - 256 = 3969. Since a > b > 0, the value of a² - b² must be positive, so a² - b² = √(3969) = 63. Alternatively, the conditions can be solved by inspection: for ab = 8 and a² + b² = 65, the integers a = 8 and b = 1 satisfy both equations since 8² + 1² = 64 + 1 = 65. Thus, a² - b² = 8² - 1² = 64 - 1 = 63.

Explanation

In the right-angled triangle formed by the wall, the ground, and the ladder, the ladder acts as the hypotenuse (L = 18 m). The problem states that the angle between the ladder and the wall is 30°. The distance from the base of the ladder to the wall is the side opposite to this angle.

Using the trigonometric sine ratio: sin(θ) = (Opposite)/(Hypotenuse)

Substituting the given values: sin(30°) = (Distance from wall)/(18) ½ = (Distance from wall)/(18) Distance from wall = 18 × ½ = 9 m

Explanation

The given equation is x⁴ - 79x² + 1 = 0. Dividing every term by x² yields x² - 79 + (1)/(x²⁾ = 0, which simplifies to x² + (1)/(x²⁾ = 79. According to the algebraic identity for the square of a binomial, (x + x⁻¹)² = x² + x⁻² + 2(x)(x⁻¹). Substituting the value of x² + x⁻² into this formula, we get (x + x⁻¹)² = 79 + 2 = 81. Taking the square root of both sides results in x + x⁻¹ = ± 9. Therefore, a valid value for the expression is 9.

Explanation

The cost price (CP) of an article when sold at a profit can be calculated using the formula: CP = (SP)/(1 + Profit %){100}

Given the selling price (SP) is ₹59,620 and the profit percentage is 8.4%, we substitute these values into the formula: CP = (59620)/(1 + 8.4){100} CP = (59620)/(1.084)

To simplify the calculation, we can multiply both the numerator and denominator by 1,000 to remove the decimal: CP = (59620 × 1000)/(1084)

Dividing 59,620 by 1,084: 1084 × 5 = 5420 5962 - 5420 = 542 Bringing down the 0 gives 5420, and 1084 × 5 = 5420. Therefore, ⁵⁹⁶²⁰⁄₁₀₈₄ = 55.

CP = 55 × 1000 = 55,000

Explanation

Using the trigonometric identity for complementary angles, we know that sec B = csc(90° - B). Substituting this into the given equation csc A = sec B, we obtain:

csc A = csc(90° - B)

Since A and B are acute angles, their values lie between 0° and 90°. In this range, the cosecant function is one-to-one, allowing us to equate the arguments directly:

A = 90° - B

By rearranging the terms, we find:

A + B = 90°

This result is a direct consequence of the co-function identity, where the secant of an angle is always equal to the cosecant of its complement.

Explanation

A number is divisible by 72 if it is simultaneously divisible by 8 and 9. For divisibility by 8, the last three digits of the nine-digit number 485x3678y, which are 78y, must be divisible by 8. Dividing 780 by 8 gives a quotient of 97 and a remainder of 4. Therefore, the next multiple of 8 is 8 × 98 = 784 . This determines that y = 4.

For divisibility by 9, the sum of all digits must be a multiple of 9. The sum of the digits is 4 + 8 + 5 + x + 3 + 6 + 7 + 8 + 4 = 45 + x . For this sum to be divisible by 9, (45 + x) must be a multiple of 9. Since x is a single-digit integer, the possible values for x are 0 and 9. The question specifies the smallest value of x, which is 0. Substituting x = 0 and y = 4 into the required expression (2y - 3x) , we get 2(4) - 3(0) = 8 .

Explanation

By substituting cos² α = x into the equation, we get 1/x + 4x = 4, which simplifies to the quadratic equation 4x² - 4x + 1 = 0. Solving this gives (2x - 1)² = 0, so x = 1/2, meaning cos² α = 1/2 and α = 45°. Testing α = 45° confirms that sec² 45° + 4 cos² 45° = 2 + 4(1/2) = 4.

Explanation

The relationship between the number of men M, days D, and work W is given by the formula (M₁ D₁)/(W₁) = (M₂ D₂)/(W₂). In this case, M₁ = 35 men, D₁ = 6 days, and W₁ = 1 unit of work. For the second scenario, M₂ = 7 men and W₂ = 0.5 (half of the work). Substituting these values into the formula, we get (35 × 6)/(1) = (7 × D₂)/(0.5). This simplifies to 210 = 14 × D₂. Solving for D₂, we find D₂ = ²¹⁰⁄₁₄, which equals 15 days. Alternatively, one can calculate total man-days for the full work as 35 × 6 = 210. Since only half of the work is required, the needed man-days are 105. Dividing 105 man-days by 7 men results in 15 days.

Explanation

In a circle with center O, for the chords PQ and QR, the angle subtended by the arc PSR (where S is on the major arc) at the center is twice the angle at the circumference. However, since ∠ PQR = 118° is obtuse, the reflex angle at the center is 2 × ∠ PQR = 2 × 118° = 236°. The interior angle ∠ POR of △ OPR is then 360° - 236° = 124°. In △ OPR, sides OP and OR are radii of the circle, so OP = OR, making △ OPR an isosceles triangle. The base angles of an isosceles triangle are equal, so ∠ OPR = ∠ ORP. Applying the angle sum property of a triangle, 2 × ∠ OPR + ∠ POR = 180°. Substituting the value of the central angle, 2 × ∠ OPR + 124° = 180°, which leads to 2 × ∠ OPR = 56°. Solving for the angle gives ∠ OPR = 28°.

Explanation

The tent consists of a frustum of a cone and a hemisphere. The diameters of the circular ends of the frustum are 39 m and 21 m, so the lower radius R = 19.5 m and the upper radius r = 10.5 m. Since the hemisphere surmounts the frustum, its radius is also r = 10.5 m.

The total height of the tent is 22.5 m. The height of the frustum h is the total height minus the radius of the hemisphere: h = 22.5 - 10.5 = 12 m. The slant height l of the frustum is calculated using the formula l = √(h² + (R - r)²). l = √(12² + (19.5 - 10.5)²) = √(144 + 9²) = √(144 + 81) = √(225) = 15 m.

The area of the cloth required is the sum of the curved surface area (CSA) of the frustum and the CSA of the hemisphere. Total Area = π(R + r)l + 2π r² Total Area = ²²⁄₇ × (19.5 + 10.5) × 15 + 2 × ²²⁄₇ × (10.5)² Total Area = ²²⁄₇ × 30 × 15 + 2 × ²²⁄₇ × 110.25 Total Area = ⁹⁹⁰⁰⁄₇ + 693 = 1414.285 + 693 = 2107.285 m²

Rounding to the nearest integer gives 2107 m².

Explanation

The weight of the grandmother's husband is given as 81 kg. Since the grandmother weighs 26 kg less than her husband, her weight is 81 - 26 = 55 kg.

Ankita's weight is 20% less than her grandmother's weight. This can be calculated as 55 × (1 - 0.20) = 55 × 0.80 = 44 kg.

The weight of Ankita's brother is 8 kg more than Ankita's weight. Therefore, the brother's weight is 44 + 8 = 52 kg.

Explanation

In the circle with centre O, OA and OB are radii, so triangle OAB is an isosceles triangle with OA = OB. Since the angles opposite to equal sides are equal, we have ∠ OAB = ∠ ABO = 50°. In triangle OAB, the sum of internal angles is 180°, thus ∠ AOB = 180° - (∠ OAB + ∠ ABO) = 180° - (50° + 50°) = 80°.

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Let D be a point on the major arc AB. Then ∠ ADB = ½ ∠ AOB = ½ × 80° = 40°. Since C is a point on the minor arc, ADBC forms a cyclic quadrilateral. In a cyclic quadrilateral, the sum of opposite angles is 180°. Therefore, ∠ ACB + ∠ ADB = 180°, which gives ∠ ACB = 180° - 40° = 140°.

Explanation

In triangle ABC, the total length of side AB is AP + PB = 4 + 6 = 10 cm, and the total length of side AC is AQ + QC = 5 + 7.5 = 12.5 cm. To determine the relationship between triangle APQ and triangle ABC, we compare the ratios of the corresponding sides.

The ratio AP/AB = 4/10 = 2/5. The ratio AQ/AC = 5/12.5 = 1/2.5 = 2/5.

Since AP/AB = AQ/AC and the angle ∠ A is common to both triangles, △ APQ is similar to △ ABC by the SAS (Side-Angle-Side) similarity criterion. In similar triangles, the ratios of all corresponding sides are equal. Therefore:

PQ/BC = AP/AB 6/BC = 2/5 2 × BC = 30 BC = 15 cm

Explanation

To find the evening selling price, we first determine the cost price (CP) using the formula CP = (SP × 100)/(100 + P%). Substituting the initial values where SP = 25 and the profit is 25%, we get CP = (25 × 100)/(125) = 20. In the evening, the vendor aims for a 10% profit. The new selling price is found by applying this profit margin to the cost price using the relation SP = CP × (100 + P%)/(100). Substituting CP = 20 and the new profit of 10%, the calculation becomes 20 × ¹¹⁰⁄₁₀₀ = 2 × 11 = 22. Alternatively, using a direct ratio approach since the cost price is constant, we have (SP₁)/(100 + P₁%) = (SP₂)/(100 + P₂%). This results in ²⁵⁄₁₂₅ = (SP₂)/(110), which simplifies to ⅕ = (SP₂)/(110), yielding SP₂ = 22.

Explanation

The gain in this transaction is the difference between the interest earned by lending the money and the interest paid for borrowing it. Since the principal amount P and the time T are the same for both cases, the net gain can be calculated by finding the difference in the interest rates.

Gain = P × (R₂ - R₁) × T{100}

Substituting the given values: Principal P = 90,000 Rate difference R₂ - R₁ = 7% - 5% = 2% Time T = 4 years

Gain = 90,000 × (2 × 4)/(100) Gain = 90,000 × ⁸⁄₁₀₀ Gain = 900 × 8 = 7,200

Explanation

The relationship between two numbers x and y, their Highest Common Factor (HCF), and their Least Common Multiple (LCM) is defined by the formula xy = HCF × LCM. Given an HCF of 15 and an LCM of 90, the product xy is 15 × 90 = 1350. Along with the given sum x + y = 75, these numbers are the roots of the quadratic equation t² - (x + y)t + xy = 0.

Substituting the values gives the equation t² - 75t + 1350 = 0. Applying the quadratic formula t = -b ± √(b²⁾⁻⁴ᵃᶜ{2a} results in t = 75 ± √(75²⁾⁻⁴⁽¹³⁵⁰⁾{2}. This simplifies to t = 75 ± √(5625 - 5400){2}, which further reduces to t = 75 ± √(225){2}. The calculations yield t = (75 ± 15)/(2), providing two values: t = ⁹⁰⁄₂ = 45 and t = ⁶⁰⁄₂ = 30.

Alternatively, using the properties of HCF, the numbers can be expressed as 15a and 15b, where a and b are co-prime integers. The sum condition 15(a + b) = 75 implies a + b = 5, and the LCM condition 15ab = 90 implies ab = 6. The only pair of integers satisfying both conditions is 2 and 3. Multiplying these by the HCF gives the numbers 15 × 2 = 30 and 15 × 3 = 45, where 45 is the greater number.

Explanation

The total income for the month of January (31 days) is calculated as 31 × 925 = 28675. The income for the first 20 days is 20 × 881 = 17620, and the income for the last 20 days is 20 × 915 = 18300.

By adding the earnings of the first 20 days and the last 20 days, the period from 12th January to 20th January is counted twice. The number of overlapping days is 20 + 20 - 31 = 9 days. The total income for these 9 days is the sum of the two 20-day periods minus the total monthly income: (17620 + 18300) - 28675 = 35920 - 28675 = 7245

The average income per day from 12th January to 20th January is the total income of these 9 days divided by 9: Average = 7245 / 9 = 805

Explanation

The value of tan A is given as (2.4)/(0.7), which simplifies to ²⁴⁄₇. In a right-angled triangle, tan A = (perpendicular)/(base). Using the Pythagorean triplet (7, 24, 25), where the perpendicular p = 24 and the base b = 7, the hypotenuse h is calculated as:

h = √(p² + b²) = √(24² + 7²) = √(576 + 49) = √(625) = 25

From these values, we can determine the required trigonometric ratios: cos A = (base)/(hypotenuse) = ⁷⁄₂₅ cot A = (base)/(perpendicular) = ⁷⁄₂₄

Now, substitute these values into the expression (50 cos A + 24 cot A):

50 (⁷⁄₂₅) + 24 (⁷⁄₂₄) = 2(7) + 1(7) = 14 + 7 = 21

Explanation

Let h be the height of the vertical tower. The relationship between the height of the tower, the angle of elevation of the sun θ, and the length of the shadow s is given by the trigonometric ratio s = h cot θ. The problem states that the shadow length increases by d = 8.4 when the angle of elevation decreases from ₁ = 45° to ₂ = 30°.

The change in shadow length is expressed by the formula: d = h(cot ₂ - cot ₁)

Substituting the given values: 8.4 = h(cot 30° - cot 45°) 8.4 = h(√3 - 1)

To isolate h, we divide 8.4 by (√3 - 1) and rationalize the denominator: h = 8.4 / (√3 - 1) h = [8.4(√3 + 1)] / [(√3 - 1)(√3 + 1)] h = [8.4(√3 + 1)] / (3 - 1) h = 8.4(√3 + 1) / 2 h = 4.2(√3 + 1)

Explanation

Profit in a partnership is distributed in the ratio of the product of capital and time. We calculate the effective capital for each partner over the 12-month period.

For X: Investment for first 6 months = ₹40,000 Investment for next 6 months = ₹40,000 + ₹20,000 = ₹60,000 Total capital-months = (40,000 × 6) + (60,000 × 6) = 600,000

For Y: Investment for first 6 months = ₹38,000 Investment for next 6 months = ₹38,000 - ₹8,000 = ₹30,000 Total capital-months = (38,000 × 6) + (30,000 × 6) = 408,000

For Z: Based on the total profit and available options, the initial investment for Z is taken as ₹30,000. Investment for first 6 months = ₹30,000 Investment for next 6 months = ₹30,000 + ₹15,000 = ₹45,000 Total capital-months = (30,000 × 6) + (45,000 × 6) = 450,000

The ratio of profit shares X : Y : Z is: 600,000 : 408,000 : 450,000 = 100 : 68 : 75

Total ratio units = 100 + 68 + 75 = 243 Total profit = ₹38,880 Value of one unit = (38,880)/(243) = 160

Share of Y = 68 × 160 = 10,880

Explanation

The volume V of a right circular cylinder is calculated using the formula V = π r² h, where r is the radius and h is the height. Given that the height h = 3.5 cm and the radius is five times the height, we find r = 5 × 3.5 = 17.5 cm.

Substituting these values into the volume formula and using π = ²²⁄₇: V = ²²⁄₇ × (17.5)² × 3.5 V = ²²⁄₇ × 17.5 × 17.5 × 3.5

Simplifying the calculation by expressing decimals as fractions: V = ²²⁄₇ × ³⁵⁄₂ × ³⁵⁄₂ × ⁷⁄₂ V = 11 × 5 × ³⁵⁄₂ × ⁷⁄₂ V = 55 × ²⁴⁵⁄₄ V = ¹³⁴⁷⁵⁄₄ V = 3368.75 cm³

Explanation

The value of the expression a³ + b³ + c³ - 3abc can be calculated using the standard algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca))

By substituting the values provided in the question: a + b + c = 6 a² + b² + c² = 14 ab + bc + ca = 11

We get: a³ + b³ + c³ - 3abc = 6 × (14 - 11) a³ + b³ + c³ - 3abc = 6 × 3 a³ + b³ + c³ - 3abc = 18

The given values are mathematically consistent because the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) holds true as 6² = 14 + 2(11) = 36.

Explanation

Let the larger number be x and the smaller number be y. According to the problem, the difference between the two numbers is x - y = 1564, which gives x = y + 1564.

The division algorithm states that Dividend = (Divisor × Quotient) + Remainder. Given that dividing the larger number by the smaller yields a quotient of 6 and a remainder of 19, we can express the relationship as x = 6y + 19.

By substituting the expression for x from the first equation into the second equation, we get: y + 1564 = 6y + 19 1564 - 19 = 6y - y 1545 = 5y y = ¹⁵⁴⁵⁄₅ = 309

Thus, the smaller number is 309.

Explanation

The percentage increase is calculated using the formula: Percentage Increase = (Value in final year - Value in initial year)/(Value in initial year) × 100

From the given table, the number of delivery partners who joined Pyntra in 2017 is 5.6 thousand, and the number of delivery partners who joined Pyntra in 2021 is 7.2 thousand.

Substituting these values into the formula: Percentage Increase = (7.2 - 5.6)/(5.6) × 100 Percentage Increase = (1.6)/(5.6) × 100

To simplify the fraction, divide both the numerator and the denominator by 0.8: (1.6)/(5.6) = ²⁄₇

Now, calculate the percentage: ²⁄₇ × 100 = ²⁰⁰⁄₇ ≈ 28.5714%

Rounding to two decimal places, the percentage increase in the number of delivery partners for Pyntra from 2017 to 2021 is 28.57%.

Explanation

J1 is defined as the average production of company A in years Y1 and Y2. From the table, company A produced 50 units in year Y1 and 100 units in year Y2. J1 = (50 + 100)/(2) = ¹⁵⁰⁄₂ = 75

J2 is the sum of the production of companies B, C, and D across both years Y1 and Y2. For company B: 90 + 120 = 210 For company C: 30 + 80 = 110 For company D: 80 + 50 = 130 Summing these values gives J2 = 210 + 110 + 130 = 450.

To find the value of the ratio J2 / J1, divide the total sum by the average calculated for company A: (J2)/(J1) = ⁴⁵⁰⁄₇₅ = 6

Explanation

According to the theorem on the areas of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If the lengths of the corresponding sides of two similar triangles are in the ratio s₁ : s₂, then the ratio of their areas is given by the formula:

(Area of Triangle 1)/(Area of Triangle 2) = ( s₁{s₂} )²

Given that the ratio of the corresponding sides is √3 : √2, we substitute these values into the formula to find the ratio of the areas:

Ratio of Areas = (√3)² : (√2)²

Ratio of Areas = 3 : 2

Squaring the terms removes the square roots, directly yielding the ratio of the areas as 3:2.

Explanation

Given the trigonometric ratio cot P = √3, we identify the angle P using standard trigonometric values where cot 30° = √3. Since the triangle PQR is right-angled at Q, P is an acute angle, confirming P = 30°. The value of sin P is therefore sin 30° = ½.

Alternatively, using trigonometric identities, we know that csc² P = 1 + cot² P. Substituting the given value, csc² P = 1 + (√3)² = 1 + 3 = 4. Taking the square root for the acute angle P, we get csc P = 2. Since sin P is the reciprocal of csc P, it follows that sin P = (1)/(csc P) = ½.

Explanation

The given expression is in the form (a³ + b³)/(a² - ab + b²), where a = 762 and b = 316. Using the algebraic identity for the sum of cubes, a³ + b³ = (a + b)(a² - ab + b²), the expression can be rewritten as:

((a + b)(a² - ab + b²))/(a² - ab + b²)

By canceling the common term (a² - ab + b²) from both the numerator and the denominator, the expression simplifies to (a + b). Substituting the numerical values for a and b:

762 + 316 = 1078

Explanation

The annual installment x required to discharge a debt A in n years at a simple interest rate R is calculated by ensuring the sum of the installments plus the interest they would have earned by the end of the period equals the total debt. The formula for the total amount is:

A = n × x + (x × R × n(n-1))/(2 × 100)

Substituting the given values A = 5664, n = 4, and R = 12:

5664 = 4x + (x × 12 × 4 × 3)/(200)

5664 = 4x + (x × 12 × 6)/(100)

5664 = 4x + (72x)/100

5664 = (400x + 72x)/100

5664 = (472x)/100

x = (5664 × 100)/(472)

x = 12 × 100

x = 1200

Explanation

In any triangle PQR where O is the incenter, the angle POR is equal to 90 degrees plus half of the angle PQR. Substituting 140 degrees for angle POR gives the equation 140 = 90 + (PQR / 2), which simplifies to 50 = PQR / 2, resulting in angle PQR = 100 degrees.

Explanation

The marked price of the watch is ₹2,750. The price after the first discount of 10% is calculated as: P₁ = 2750 × (1 - ¹⁰⁄₁₀₀) = 2750 × 0.9 = 2475

Let the second discount be x%. The final selling price after the second discount is given by: SP = P₁ × (1 - ˣ⁄₁₀₀)

Substituting the known values into the equation: 2103.75 = 2475 × (1 - ˣ⁄₁₀₀)

Dividing both sides by 2475: 1 - ˣ⁄₁₀₀ = (2103.75)/(2475) 1 - ˣ⁄₁₀₀ = 0.85

Rearranging the equation to solve for x: ˣ⁄₁₀₀ = 1 - 0.85 ˣ⁄₁₀₀ = 0.15 x = 15

The second discount given by the shopkeeper is 15%.

Explanation

The relationship between two different selling prices for the same cost price, given their respective profit percentages, is expressed by the formula SP₂ = SP₁ × (100 + P₂)/(100 + P₁). Here, SP₁ is the initial selling price, P₁ is the initial profit percentage, SP₂ is the new selling price, and P₂ is the target profit percentage.

Given SP₁ = 2,78,000, P₁ = 25, and P₂ = 18, we substitute these values into the formula:

SP₂ = 2,78,000 × (100 + 18)/(100 + 25)

SP₂ = 2,78,000 × ¹¹⁸⁄₁₂₅

To simplify the calculation, we divide 2,78,000 by 125:

(2,78,000)/(125) = 2,224

Now, multiply this quotient by the numerator 118:

SP₂ = 2,224 × 118

SP₂ = 2,62,432

Explanation

Using the simple interest formula SI = (P × R × T)/(100), first find the interest earned in the first case. The interest is the difference between the amount and the principal: SI₁ = 832 - 640 = 192. Substituting the values for the first scenario into the formula gives 192 = (640 × R × 2)/(100). Solving for the rate, R = (192 × 100)/(1280) = 15%.

Now, apply this rate to the second scenario where the principal is 860 and the time is 4 years. The interest is calculated as SI₂ = (860 × 15 × 4)/(100) = (860 × 60)/(100) = 516. The final amount is the sum of the principal and the interest: Amount = 860 + 516 = 1376.