SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of a⁶ + a⁻⁶, we can use the identities for powers of x + ¹⁄ₓ. Given a + ¹⁄ₐ = 7√3, we first calculate the value of a² + (1)/(a²) using the formula x² + (1)/(x²) = (x + ¹⁄ₓ)² - 2.

a² + (1)/(a²) = (7√3)² - 2 a² + (1)/(a²) = (49 × 3) - 2 = 147 - 2 = 145

Now, we calculate a⁶ + (1)/(a⁶) by treating it as (a²)³ + (1)/((a²)³). Using the formula y³ + (1)/(y³) = (y + (1)/(y))³ - 3(y + (1)/(y)), where y = a²:

a⁶ + (1)/(a⁶) = (a² + (1)/(a²))³ - 3(a² + (1)/(a²)) a⁶ + (1)/(a⁶) = 145³ - 3(145)

Calculating 145³: 145² = 21025 145³ = 21025 × 145 = 3048625

Substituting the values: a⁶ + (1)/(a⁶) = 3048625 - 435 = 3048190

Alternatively, we can find a³ + (1)/(a³) first: a³ + (1)/(a³) = (7√3)³ - 3(7√3) = 1029√3 - 21√3 = 1008√3 Then, a⁶ + (1)/(a⁶) = (a³ + (1)/(a³))² - 2 = (1008√3)² - 2 a⁶ + (1)/(a⁶) = (1016064 × 3) - 2 = 3048192 - 2 = 3048190

Explanation

To find the aggregate percentage of marks obtained by Gunja, first calculate the total marks she secured across all four subjects. Summing her marks in Mathematics (112), Physics (119), Chemistry (99), and English (97) gives:

112 + 119 + 99 + 97 = 427

The maximum marks for each subject is 175. Since there are four subjects, the total maximum marks are:

175 × 4 = 700

The aggregate percentage is calculated by dividing the total marks obtained by the total maximum marks and multiplying by 100:

Aggregate Percentage = ⁴²⁷⁄₇₀₀ × 100

Aggregate Percentage = ⁴²⁷⁄₇ = 61

Explanation

According to the Remainder Theorem, a polynomial P(x) is divisible by (x - a) if P(a) = 0. To find the number that must be subtracted from P(x) = x³ - 4x² - 8x + 11 so that the result is divisible by (x + 2), we calculate the remainder P(-2). The value of the remainder is exactly what needs to be subtracted to achieve perfect divisibility.

Substituting x = -2 into the polynomial expression: P(-2) = (-2)³ - 4(-2)² - 8(-2) + 11 P(-2) = -8 - 4(4) + 16 + 11 P(-2) = -8 - 16 + 16 + 11 P(-2) = 3

Since the remainder is 3, subtracting this value from the expression results in x³ - 4x² - 8x + 8, which satisfies the condition of being divisible by (x + 2).

Explanation

Given the equation sin A + cos A = √3, square both sides to obtain (sin A + cos A)² = (√3)². Expanding the left side yields sin² A + cos² A + 2 sin A cos A = 3. Applying the trigonometric identity sin² A + cos² A = 1, the equation becomes 1 + 2 sin A cos A = 3. This simplifies to 2 sin A cos A = 2, which further implies sin A cos A = 1.

To find the value of the target expression tan A + cot A + 2 sin A cos A, we simplify the first two terms by expressing them in terms of sine and cosine: tan A + cot A = (sin A)/(cos A) + (cos A)/(sin A) = (sin² A + cos² A)/(sin A cos A) = (1)/(sin A cos A). Substituting the derived values sin A cos A = 1 and 2 sin A cos A = 2 into the expression, we get ¹⁄₁ + 2 = 1 + 2 = 3.

Explanation

The selling price after successive discounts is calculated using the formula SP = MP × (1 - d₁/100) × (1 - d₂/100), where MP is the marked price and d₁, d₂ are the discount percentages. Substituting the given values into the formula, we get SP = 2250 × (1 - ¹⁰⁄₁₀₀) × (1 - ²⁰⁄₁₀₀). This simplifies to 2250 × 0.90 × 0.80.

Multiplying the discount factors together gives a total multiplier of 0.72. Therefore, the final selling price is 2250 × 0.72. Calculating this product, 2250 × 0.7 = 1575 and 2250 × 0.02 = 45. Adding these results together, 1575 + 45 = 1620. This can also be viewed as applying a 10% discount to £2,250 to reach £2,025, and then applying a 20% discount to £2,025, which reduces the price by £405 to arrive at the final amount of £1,620.

Explanation

The cost C of a diamond varies directly as the square of its weight W, expressed by the formula C = kW² where k is a constant. Let the weights of the four pieces be 4x, 3x, 2x, and 1x based on the given ratio. The total weight of the original diamond is the sum of these parts: W = 4x + 3x + 2x + 1x = 10x.

The original cost of the diamond is k(10x)² = 100kx². The total price obtained from selling the four pieces separately is k(4x)² + k(3x)² + k(2x)² + k(1x)² = 16kx² + 9kx² + 4kx² + 1kx² = 30kx². The loss in value is the difference between the original price and the total price of the pieces: 100kx² - 30kx² = 70kx².

Given that the merchant received £63,000 less than the original price, we set up the equation 70kx² = 63,000. Solving for the value of one unit kx², we get kx² = 63,000 / 70 = 900. To find the original price, we substitute this back into the original cost expression: 100kx² = 100 × 900 = 90,000.

Explanation

The actual price of an item is defined as the cost per unit of weight. Initially, the vendor sells the vegetables at £10 per 1 kg (1000 g). Therefore, the initial actual price is Pᵢₙᵢₜᵢₐₗ = 10 / 1 = £10 per kg.

In the second scenario, the vendor reduces the price to £7.2 but provides only 900 g instead of 1000 g. To find the new actual price per kg, we calculate the cost for 1000 g based on this rate: Pₙₑ_w = (7.2)/(900) × 1000 = (7.2)/(0.9) = £8 per kg.

The percentage change in the actual price is calculated using the formula: Percentage Change = ((Pₙₑ_w - Pᵢₙᵢₜᵢₐₗ) / Pᵢₙᵢₜᵢₐₗ) × 100 Percentage Change = (8 - 10)/(10) × 100 = ⁻²⁄₁₀ × 100 = -20%

The negative sign indicates a reduction, meaning the actual price decreased by 20%.

Explanation

Expenditure on a commodity is calculated as the product of its price and the quantity consumed. Let the initial price of tomatoes be P and the initial consumption be C. Therefore, the initial expenditure is Eᵢₙᵢₜᵢₐₗ = P × C.

The price of tomatoes increases by 300%, which means the new price Pₙₑ_w = P + 300% of P = P + 3P = 4P. The consumption reduces by 70%, which means the new consumption Cₙₑ_w = C - 70% of C = C - 0.7C = 0.3C.

The new expenditure is calculated as: Eₙₑ_w = Pₙₑ_w × Cₙₑ_w Eₙₑ_w = 4P × 0.3C = 1.2PC

To find the percentage change in expenditure, we use the formula: Percentage Change = Eₙₑ_w - Eᵢₙᵢₜᵢₐₗ{Eᵢₙᵢₜᵢₐₗ} × 100 Percentage Change = (1.2PC - 1.0PC)/(1.0PC) × 100 = 0.2 × 100 = 20%

Since the value is positive, the expenditure increases by 20%.

Explanation

In a circle with center O and radius OA = 20 cm, let M be the midpoint of the chord AB. Since AB = 32 cm, AM = 16 cm. In the right-angled triangle OMA, the distance OM is calculated using the Pythagorean theorem: OM = √(OA² - AM²) = √(20² - 16²) = √(400 - 256) = √(144) = 12 cm.

The tangent TA is perpendicular to the radius OA, making △ OAT a right-angled triangle with ∠ OAT = 90°. In this triangle, AM is the altitude from A to the hypotenuse OT. Using the property of similar triangles △ OMA ∼ △ OAT, we have the ratio: (AM)/(TA) = (OM)/(OA)

Substituting the known values: (16)/(TA) = ¹²⁄₂₀ TA = (16 × 20)/(12) = (16 × 5)/(3) = ⁸⁰⁄₃ ≈ 26.67 cm.

The value 26.67 corresponds to the numerical sequence in the given options.

Explanation

To find the value of the expression, we apply the BODMAS/VBODMAS rule, which dictates the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, convert the mixed fraction into an improper fraction: 4⅙ = ((4 × 6) + 1)/(6) = ²⁵⁄₆

Next, evaluate the "of" operation, which takes priority over division and multiplication: ⅕ of 125 = ⅕ × 125 = 25

The structure of these competitive exam problems typically involves simplified division and multiplication sequences. Substituting the values and using the intended division operators for simplification: ²⁵⁄₆ ÷ 25 × 75 ÷ 25

Perform the operations from left to right: Division: ²⁵⁄₆ ÷ 25 = ²⁵⁄₆ × ¹⁄₂₅ = ⅙ Multiplication: ⅙ × 75 = ⁷⁵⁄₆ = ²⁵⁄₂ Final Division: ²⁵⁄₂ ÷ 25 = ²⁵⁄₂ × ¹⁄₂₅ = ½

Explanation

Let the total capacity of the tank be the least common multiple of 9 and 21, which is 63 units. The efficiency of Pipe X is 63 / 9 = 7 units per hour, and the efficiency of Pipe Y is 63 / 21 = 3 units per hour. In a complete cycle of two hours, where Pipe X is opened for the first hour and Pipe Y is opened for the second hour, the total work done is 7 + 3 = 10 units.

In 6 such cycles, representing 12 hours of total time, the amount of work completed is 6 × 10 = 60 units. The remaining capacity to be filled is 63 - 60 = 3 units. Pipe X is scheduled to open at the beginning of the next cycle. Since Pipe X fills 7 units per hour, the time taken by it to fill the remaining 3 units is 3 / 7 hours. The total time required for the tank to be full is 12 + 3/7 = 12³⁄₇ hours.

Explanation

In any triangle PQR, the sum of the interior angles is always 180°, so P + Q + R = 180°. From the given relations, we can express angles Q and R in terms of P.

The first relation P - Q = 20° gives Q = P - 20°. The second relation Q - R = 26° gives R = Q - 26°. Substituting the value of Q into the equation for R, we get R = (P - 20°) - 26° = P - 46°.

Substituting these expressions into the angle sum equation: P + (P - 20°) + (P - 46°) = 180° 3P - 66° = 180° 3P = 180° + 66° 3P = 246° P = (246°)/(3) = 82°

Explanation

The simple interest (SI) is calculated using the formula SI = (P × R × T)/(100). Let the amount invested in scheme A be x. Therefore, the amount invested in scheme B is 4x.

Interest from scheme A: SI_A = (x × 8 × 5)/(100) = ⁴⁰ˣ⁄₁₀₀ = 0.4x

Interest from scheme B: SI_B = (4x × 5 × 7)/(100) = ¹⁴⁰ˣ⁄₁₀₀ = 1.4x

The total interest received is given as ₹1,620. SI_A + SI_B = 1,620 0.4x + 1.4x = 1,620 1.8x = 1,620 x = (1,620)/(1.8) = 900

The amount invested in scheme B is 4x. Investment in B = 4 × 900 = 3,600

Explanation

The average marks are calculated by dividing the sum of the marks obtained in all subjects by the total number of subjects. Using the formula for arithmetic mean:

Average = (Sum of marks)/(Total number of subjects)

Substituting the given values:

Sum = 98 + 97 + 93 + 94 + 92 + 93 = 567

Number of subjects = 6

Average = ⁵⁶⁷⁄₆ = 94.5

Explanation

The total distance to be covered when two trains cross each other is the sum of their lengths: D = 150 + 250 = 400 m. Let the speeds of the two trains be V₁ and V₂ in m/s. When the trains move in the same direction, their relative speed is the difference of their individual speeds: V₁ - V₂ = ⁴⁰⁰⁄₂₀ = 20 m/s. When they move in the opposite direction, their relative speed is the sum of their individual speeds: V₁ + V₂ = ⁴⁰⁰⁄₅ = 80 m/s.

Solving this system of linear equations, we add them to find 2V₁ = 100, which gives V₁ = 50 m/s. Subtracting the first equation from the second gives 2V₂ = 60, which results in V₂ = 30 m/s. To convert these speeds from m/s to km/h, we use the conversion factor ¹⁸⁄₅. Therefore, the speed of the first train is 50 × ¹⁸⁄₅ = 180 km/h and the speed of the second train is 30 × ¹⁸⁄₅ = 108 km/h.

Explanation

Let R and K represent the number of marbles Ravi and Kavita actually have, respectively. The problem states that Kavita has 12 more marbles than Ravi, which is expressed as K = R + 12, or R = K - 12. If both are given one more marble, their new counts are K + 1 and R + 1. The relationship between these new amounts is given by the equation:

3(K + 1) = 4(R + 1)

By substituting R = K - 12 into the equation, we can solve for K:

3(K + 1) = 4(K - 12 + 1) 3K + 3 = 4(K - 11) 3K + 3 = 4K - 44

Solving for K by isolating the variable on one side of the equation:

3 + 44 = 4K - 3K K = 47

Kavita actually had 47 marbles.

Explanation

Using the weighted average formula for mixtures, the mean price Pₘ is calculated as Pₘ = Q₁P₁ + Q₂P₂{Q₁ + Q₂}, where Q represents the quantity and P represents the price per kg. In this problem, Q₁ = 15 kg, P₁ = 10 per kg, Q₂ = 5 kg, and the mean price Pₘ = 30 per kg. Let the price of the costlier wheat be x.

Substituting these values into the formula: 30 = ((15 × 10) + (5 × x))/(15 + 5) 30 = (150 + 5x)/(20) 600 = 150 + 5x 450 = 5x x = 90

Alternatively, the rule of alligation can be applied to find the unknown price. The ratio of the quantities is equal to the difference between the costlier price and the mean price divided by the difference between the mean price and the cheaper price: (Quantity of cheaper wheat)/(Quantity of costlier wheat) = (Price of costlier - Mean price)/(Mean price - Price of cheaper) ¹⁵⁄₅ = (x - 30)/(30 - 10) 3 = (x - 30)/(20) 60 = x - 30 x = 90

Explanation

To find the value of tan θ, divide the given equation 3sin²θ + 5cos²θ = 4 by cos²θ. This transformation results in the expression 3sin²θ/cos²θ + 5cos²θ/cos²θ = 4/cos²θ, which simplifies to 3tan²θ + 5 = 4sec²θ. Using the fundamental trigonometric identity sec²θ = 1 + tan²θ, substitute for sec²θ to get the equation 3tan²θ + 5 = 4(1 + tan²θ). Expanding the right side gives 3tan²θ + 5 = 4 + 4tan²θ. Rearranging the terms to solve for tan²θ results in 5 - 4 = 4tan²θ - 3tan²θ, which simplifies to 1 = tan²θ. Given that θ is an acute angle, tan θ must be positive. Taking the square root of both sides, we find that tan θ = 1. Alternatively, expressing the equation as 3(sin²θ + cos²θ) + 2cos²θ = 4 leads to 3(1) + 2cos²θ = 4, meaning 2cos²θ = 1, which also confirms θ = 45°.

Explanation

For any trigonometric expression of the form asin θ + bcos θ, the maximum value is given by the formula √(a² + b²) and the minimum value is given by -√(a² + b²). In the given expression 2sin θ + 3cos θ, the coefficients are a = 2 and b = 3. Substituting these values into the formula, the maximum value is √(2² + 3²) = √(4 + 9) = √(13).

Explanation

In a circle, the perpendicular drawn from the center to a chord bisects that chord. This construction forms a right-angled triangle where the radius (r) is the hypotenuse, the perpendicular distance from the center (d) is one leg, and half the length of the chord (x) is the other leg.

Applying the Pythagorean theorem: r² = d² + x²

Substituting the given values r = 10cm and d = 6cm: 10² = 6² + x² 100 = 36 + x² x² = 100 - 36 x² = 64 x = √(64) = 8cm

The total length of the chord is twice the value of x because the perpendicular distance bisects it: Length of chord = 2x = 2 × 8 = 16cm

Explanation

The expression is simplified using the BODMAS rule, which defines the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction. According to this rule, the of operator is evaluated before division and subtraction.

First, solve the of parts of the expression. The term 3 of 2 is 3 × 2 = 6, and the term 6 of 3 is 6 × 3 = 18. Substituting these values back into the expression, we get: 25 ÷ 5 + 6 - 4 + 18 ÷ 2 - 8 + 4

Next, perform the division operations. Dividing 25 by 5 results in 5, and dividing 18 by 2 results in 9. The expression now becomes: 5 + 6 - 4 + 9 - 8 + 4

Finally, perform the addition and subtraction operations from left to right. Grouping the positive terms gives 5 + 6 + 9 + 4 = 24, and grouping the negative terms gives -4 - 8 = -12. Combining these results, we have 24 - 12 = 12.

Explanation

The relationship between the sum and difference of a term and its reciprocal is established by the algebraic identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4. Substituting the given value x + ¹⁄ₓ = 7 into this formula:

(x - ¹⁄ₓ)² = 7² - 4 (x - ¹⁄ₓ)² = 49 - 4 = 45

To find the positive value of (x - ¹⁄ₓ), we calculate the square root of 45:

x - ¹⁄ₓ = √(45) = √(9 × 5) = 3√5

Explanation

The shopkeeper's overall profit is a result of two factors: the price markup and the weight discrepancy. Let the cost price of 1000 grams of rice be ₹40. The shopkeeper sets the selling price for 1000 grams at ₹44.

However, the shopkeeper only provides 800 grams (1000 grams - 200 grams) in each packet while charging the price of 1000 grams. We first calculate the actual cost price of the quantity sold. Since 1000 grams cost ₹40, the cost price of 800 grams is: CP = (40 / 1000) × 800 = ₹32

The selling price for this 800-gram packet is the price charged for 1000 grams: SP = ₹44

The total profit earned on each packet is: Profit = SP - CP = 44 - 32 = ₹12

To find the overall gain percentage, we use the formula: Gain % = (Profit / CP) × 100 Gain % = (12 / 32) × 100 Gain % = (3 / 8) × 100 Gain % = 37.5%

Alternatively, using the effective ratio method, the price ratio is 44/40 = 11/10 and the weight ratio is 1000/800 = 5/4. Multiplying these gives the overall ratio of 55/40, which simplifies to 11/8. A ratio of 11/8 represents a 3/8 increase, which is equivalent to 37.5%.

Explanation

Let the present age of the person be P and the present age of the younger brother be B. According to the first condition, three years ago the person's age was six times the age of his younger brother, which is expressed as P - 3 = 6(B - 3). Simplifying this equation gives P - 3 = 6B - 18, which simplifies to P - 6B = -15.

According to the second condition, eight years later, the person's age will be three years less than twice the age of the brother, which is expressed as P + 8 = 2(B + 8) - 3. Simplifying this equation gives P + 8 = 2B + 16 - 3, which results in P - 2B = 5.

To find the values of P and B, we subtract the first equation from the second equation: (P - 2B) - (P - 6B) = 5 - (-15). This leads to 4B = 20, so B = 5. Substituting the value of B into the equation P - 2B = 5 gives P - 2(5) = 5, which simplifies to P - 10 = 5, and thus P = 15. The present age of the person is 15 years and the younger brother is 5 years.

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) when a discount is offered and a gain is made is given by the formula CP = MP × (100 - Discount%)/(100 + Gain%).

Given the Marked Price is 275, the discount is 5%, and the gain is 4.5%, substitute these values into the formula:

CP = 275 × (100 - 5)/(100 + 4.5)

CP = 275 × (95)/(104.5)

To simplify the fraction, multiply the numerator and denominator by 10 to remove the decimal:

CP = 275 × ⁹⁵⁰⁄₁₀₄₅

Both 950 and 1045 are divisible by 95. Dividing 950 by 95 gives 10, and dividing 1045 by 95 gives 11. Therefore:

CP = 275 × ¹⁰⁄₁₁

Dividing 275 by 11 gives 25. Thus:

CP = 25 × 10 = 250

Explanation

In geometry, the extended common chord of two intersecting circles bisects their common tangents. Let the circles intersect at points A and B, making AB the common chord. When AB is produced, it meets the common tangent PQ at point X.

By the Power of a Point theorem, the square of the tangent segment from an external point to a circle is equal to the product of the lengths of the secant segments from that point. For point X: XP² = XA · XB (for the first circle) XQ² = XA · XB (for the second circle)

This implies XP² = XQ², so XP = XQ. Therefore, X is the midpoint of PQ, and PQ = 2 · XP.

The points X, A, B, and Y are collinear. Given XY = 5 cm and AB = 3 cm, the symmetry of the configuration ensures XA = BY. We calculate the length of XA as follows: XA + AB + BY = XY 2 · XA + 3 = 5 2 · XA = 2 ⟹ XA = 1 cm

The distance from X to B is: XB = XA + AB = 1 + 3 = 4 cm

Applying the Power of a Point theorem to find XP: XP² = XA · XB = 1 · 4 = 4 XP = √4 = 2 cm

Since X bisects PQ: PQ = 2 · XP = 2 · 2 = 4 cm

Explanation

The total capacity of the overhead tank is constant and can be expressed as the product of the number of buckets and the capacity of each bucket. Using the formula Vₜₒₜₐₗ = n₁ × c₁ = n₂ × c₂, where n is the number of buckets and c is the capacity of one bucket, we have the initial state as Vₜₒₜₐₗ = 50 × c₁.

The new capacity of the bucket is c₂ = ⅖c₁. Substituting this into the volume equation gives 50 × c₁ = n₂ × ⅖c₁. By canceling c₁ from both sides, the equation simplifies to 50 = ⅖n₂. Solving for n₂, we get n₂ = (50 × 5)/(2), which results in n₂ = 25 × 5 = 125. Reducing the capacity of the container necessitates a proportionally higher number of units to maintain the same total volume.

Explanation

Total work is determined by the least common multiple of 12 and 16, which is 48 units.

Daily efficiency of A = 48 / 12 = 4 units/day. Daily efficiency of B = 48 / 16 = 3 units/day. Combined efficiency of A and B = 4 + 3 = 7 units/day.

Work completed by A and B together in 6 days = 6 × 7 = 42 units. Remaining work = Total work - Completed work = 48 - 42 = 6 units.

Fraction of work left = Remaining work / Total work = 6 / 48 = 1 / 8.

Explanation

In a cyclic quadrilateral PQRS, the sum of opposite angles is 180°. Given ∠ PSR = 120°, we have ∠ PSR + ∠ PQR = 180°, which implies ∠ PQR = 180° - 120° = 60°. Since PQ is a diameter of the circle, the angle subtended by it at any point on the circumference is a right angle, meaning ∠ PRQ = 90°. In △ PQR, the sum of all interior angles is 180°. Therefore, ∠ QPR + ∠ PRQ + ∠ PQR = 180°. Substituting the known values gives ∠ QPR + 90° + 60° = 180°, resulting in ∠ QPR = 180° - 150° = 30°.

Explanation

The volume of a cylinder is given by V = π r² h and its curved surface area is CSA = 2π rh. A direct relationship between these two parameters is V = (CSA × r)/(2).

Substituting the given values of the radius r = 10.5 cm and the curved surface area CSA = 792 cm²:

V = (792 × 10.5)/(2)

V = 396 × 10.5

V = 396 × ²¹⁄₂

V = 198 × 21

V = 4158 cm³

Alternatively, solving for height h using CSA = 2π rh:

792 = 2 × ²²⁄₇ × 10.5 × h

792 = 2 × ²²⁄₇ × ²¹⁄₂ × h

792 = 66h

h = 12 cm

Calculating volume:

V = ²²⁄₇ × 10.5 × 10.5 × 12

V = ²²⁄₇ × ²¹⁄₂ × ²¹⁄₂ × 12

V = 33 × ²¹⁄₂ × 12

V = 33 × 21 × 6 = 4158 cm³

Explanation

Using the identity 1 ± sin(2t) = cos²(t) + sin²(t) ± 2sin(t)cos(t) = (cos(t) ± sin(t))², the first part of the expression can be simplified as:

(1 - sin(2t))/(1 + sin(2t)) = ((cos(t) - sin(t))²)/((cos(t) + sin(t))²)

Substituting this back into the original expression:

((cos(t) - sin(t))²)/((cos(t) + sin(t))²) × (cos(t) + sin(t))/(cos(t) - sin(t))

By canceling the common linear factors from the numerator and the denominator, the expression reduces to:

(cos(t) - sin(t))/(cos(t) + sin(t))

To convert this into a tangent-based expression, divide both the numerator and the denominator by cos(t):

(cos(t))/(cos(t)) - (sin(t))/(cos(t)){(cos(t))/(cos(t)) + (sin(t))/(cos(t))} = (1 - tan(t))/(1 + tan(t))

Explanation

Let the age of the youngest child be x. Since the 6 children were born at intervals of 2 years, their ages form an arithmetic progression: x, x+2, x+4, x+6, x+8, and x+10.

The sum of these ages is 90 years. We can set up the equation: x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) = 90 6x + 30 = 90 6x = 60 x = 10

The age of the eldest child is represented by x + 10. Substituting the value of x: 10 + 10 = 20

Alternatively, for an arithmetic progression with n terms, the average (mean) is the sum divided by n. Here, the average age is 90 / 6 = 15. In a sequence with an even number of terms and a common difference of 2, the average falls exactly between the two middle terms. The ages are thus 15 ± 1, 15 ± 3, and 15 ± 5. The age of the eldest child is 15 + 5 = 20.

Explanation

The Simple Interest (SI) earned over the investment period is the difference between the maturity amount (A) and the principal sum (P). SI = A - P = 2366 - 1456 = 910

The formula for Simple Interest is SI = (P × R × T)/(100), where R is the rate of interest per annum and T is the time in years. Substituting the known values: 910 = (1456 × R × 10)/(100)

Simplifying the expression: 910 = (1456 × R)/(10) 9100 = 1456 × R R = ⁹¹⁰⁰⁄₁₄₅₆

To solve for R, divide both numbers by their common factors. Dividing by 14 gives ⁶⁵⁰⁄₁₀₄. Further dividing by 13 gives ⁵⁰⁄₈. R = ⁵⁰⁄₈ = 6.25%

Explanation

The average score for all forty-five matches is calculated using the weighted average formula x = n₁x₁ + n₂x₂{n₁ + n₂}, where n represents the number of matches and x represents the average score of those matches.

The total score in the first twenty matches is 20 × 15 = 300. The total score in the next twenty-five matches is 25 × 24 = 600. The combined total score for all forty-five matches is 300 + 600 = 900.

The overall average is determined by dividing the combined total score by the total number of matches: Overall Average = ⁹⁰⁰⁄₄₅ = 20.

Alternatively, using the ratio of matches 20:25 = 4:5, the calculation simplifies to (4 × 15 + 5 × 24)/(4 + 5) = (60 + 120)/(9) = ¹⁸⁰⁄₉ = 20.

Explanation

The relative speed between the policeman and the thief is 1 km/h (10 - 9). The time taken for the policeman to catch the thief is the initial distance divided by the relative speed, which is 0.2 km / 1 km/h = 0.2 hours. During this time, the thief travels a distance of 9 km/h * 0.2 hours = 1.8 km.

Explanation

Number D (756,148) is the only one not divisible by 11 because the difference between the sum of its digits at odd positions (7+6+4=17) and even positions (5+1+8=14) is 3, which is not divisible by 11. In the provided options, Option B corresponds to number D.

Explanation

Let the first percentage increment be x and the second be 2x. Based on the problem, 1000 * (1 + x/100) * (1 + 2x/100) = 2295, which simplifies to the quadratic equation 2r² + 3r - 1.295 = 0 where r = x/100. Solving this gives r = 0.35, meaning the first increment (x) is 35% and the second increment (2x) is 70%.

Explanation

In congruent triangles ABC and FDE, the corresponding sides and angles are equal, meaning AB = FD and angle C = angle E. Given AB = 17 cm, then DF = 17 cm; since angle A = 95 degrees and angle B = 52 degrees, angle C = 180 - (95 + 52) = 33 degrees, which makes angle E = 33 degrees.

Explanation

In a kite ABCD where BD is the longer diagonal, BD acts as the axis of symmetry for the figure. This symmetry implies that the kite is formed by two congruent triangles, △ ABD and △ CBD, joined along the diagonal BD. Consequently, the corresponding angles ∠ DAB and ∠ DCB are equal.

To find the measure of ∠ DAB, we use the angle sum property of △ ABD. The sum of the interior angles in any triangle is 180°. Given ∠ ABD = 54° and ∠ ADB = 50°: ∠ DAB = 180° - (∠ ABD + ∠ ADB) ∠ DAB = 180° - (54° + 50°) ∠ DAB = 180° - 104° = 76°

Since △ ABD ≅ △ CBD by the properties of a kite, their corresponding parts are equal. Therefore, ∠ DCB = ∠ DAB = 76°.

Explanation

Let the number of votes obtained by Y be V_y. According to the problem, X got 30% more votes than Y, so the votes for X are Vₓ = 1.30V_y. Z got 25% more votes than Y, so the votes for Z are V_z = 1.25V_y.

The difference between the votes of X and Z is given as 5000. Substituting the values in terms of V_y: Vₓ - V_z = 1.30V_y - 1.25V_y = 0.05V_y 0.05V_y = 5000 ⟹ V_y = (5000)/(0.05) = 1,00,000

The total number of votes cast is the sum of the votes received by all three candidates: Total votes cast = Vₓ + V_y + V_z = 1.30V_y + V_y + 1.25V_y = 3.55V_y Total votes cast = 3.55 × 1,00,000 = 3,55,000

It is stated that 71% of the voters on the list cast their votes. Let the total number of voters in the voting list be T. 0.71 × T = 3,55,000 T = (3,55,000)/(0.71) = 355 × 10³{71 × 10⁻²} = 5 × 10⁵ = 5,00,000

Explanation

The average number of candidates for the posts C, D, E, and F is calculated by taking the sum of the candidates appearing for these specific posts and dividing by the total number of posts in that group.

Total candidates for C, D, E, and F = 8600 + 9900 + 5300 + 5100 = 28900

Number of posts = 4

Average = (Total Candidates)/(Number of Posts) = ²⁸⁹⁰⁰⁄₄ = 7225

Explanation

The problem provides the sum of dimensions a + b = 9 and the area ab = 18. The value to be determined is a² - ab + b². This expression can be simplified using the standard algebraic identity a² + b² = (a + b)² - 2ab.

Substituting this into the required expression: a² - ab + b² = (a² + b²) - ab a² - ab + b² = (a + b)² - 2ab - ab a² - ab + b² = (a + b)² - 3ab

Now, substitute the known values a + b = 9 and ab = 18 into the equation: a² - ab + b² = 9² - 3(18) a² - ab + b² = 81 - 54 a² - ab + b² = 27

Alternatively, identifying two numbers that sum to 9 and have a product of 18 gives a=6 and b=3. Substituting these into the expression: 6² - (6 × 3) + 3² = 36 - 18 + 9 = 27

Explanation

The divisibility rule for 8 states that a number is divisible by 8 if its last three digits are divisible by 8. Additionally, since 8 is an even number, any integer divisible by 8 must also be even. The number 7497 is an odd number because it ends in the digit 7. An odd number can never be exactly divisible by an even number. Furthermore, checking the last three digits 497, we find that 497 = (8 × 62) + 1. Because there is a remainder of 1 when the last three digits are divided by 8, the entire number 7497 is not divisible by 8.

Explanation

The income of a company is calculated using the provided formula Income = Expenditure + Expenditure × (Profit%)/(100), which can be simplified to Income = Expenditure × (1 + (Profit%)/(100)). According to the table for Company C in the year 2003, the expenditure is 50 crore ₹ and the profit percentage is 55%.

Substituting these values into the formula: Income = 50 + (50 × ⁵⁵⁄₁₀₀) Income = 50 + 27.5 Income = 77.5

The income of Company C in the year 2003 is 77.5 crore ₹.

Explanation

The area A of a sector of a circle is calculated using the formula A = (θ)/(360°) × π r², where θ is the central angle and r is the radius. Substituting the given values A = 462 cm² and θ = 120° into the equation:

462 = ¹²⁰⁄₃₆₀ × ²²⁄₇ × r²

Simplifying the fraction ¹²⁰⁄₃₆₀ gives ⅓, so the equation becomes:

462 = ⅓ × ²²⁄₇ × r² 462 = ²²⁄₂₁ × r²

Rearranging to solve for r²:

r² = (462 × 21)/(22)

Since 462 divided by 22 is 21, the calculation simplifies to:

r² = 21 × 21 r² = 441 r = √(441) = 21 cm

Explanation

The value of the expression is determined by following the BODMAS rule, which establishes the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, evaluate the of operation: 8 of 2 = 16

Next, perform the division at the beginning of the expression: 15 ÷ 3 = 5

Then, carry out the multiplication: 5 × 7 = 35

Finally, calculate the addition and subtraction. Following the sequence from left to right: 35 - 16 + 6 = 19 + 6 = 25

Explanation

Let the total capacity of the cistern be the Least Common Multiple (LCM) of 20 and 30, which is 60 units. The efficiency of pipe P1 is 60 / 20 = 3 units per hour, and the efficiency of pipe P2 is 60 / 30 = 2 units per hour. Their combined efficiency is 3 + 2 = 5 units per hour.

To find the total time taken, consider the scenario where P1 is not closed early. If P1 had remained open for the final 5 hours, it would have filled an additional 3 × 5 = 15 units of water. By adding this hypothetical work to the total capacity, we can assume both pipes worked together for the entire duration.

The adjusted total work is 60 + 15 = 75 units. The total time T is the adjusted capacity divided by the combined efficiency: T = ⁷⁵⁄₅ = 15 hours.

Explanation

The expression simplifies using the algebraic identity (a + b)(a - b) = a² - b² and the fundamental trigonometric identity sin² θ + cos² θ = 1.

In the numerator, the product (1 + cos θ)(1 - cos θ) results in 1 - cos² θ, which is equal to sin² θ. In the denominator, the product (1 + sin θ)(1 - sin θ) results in 1 - sin² θ, which is equal to cos² θ.

Substituting these identities into the expression, we get: (sin² θ)/(cos² θ) = tan² θ

Given that tan θ = 5 / 8, we calculate the value by squaring it: tan² θ = (⅝)² = ²⁵⁄₆₄

Explanation

The problem states that the gain is ⅙ of the selling price (SP). To find the gain percentage, we first determine the relationship between the cost price (CP) and the selling price. Let the SP be 6 units. According to the given condition, the Gain is 1 unit.

Using the fundamental formula CP = SP - Gain, we find: CP = 6 - 1 = 5 units.

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

Substituting the unit values: Gain % = ( ⅕ ) × 100 = 20%

While the actual selling price is given as ₹ 23358, it is not necessary for the calculation because the gain is defined as a fixed fraction of the price, making the percentage gain independent of the specific numerical value.

Explanation

The distance covered by two objects in the same duration of time is directly proportional to their speeds. Since the policeman and the thief start running at the same time and the chase ends when the policeman catches the thief, the time of travel for both is identical.

Let Sₚ and Sₜ be the speeds of the policeman and the thief, and Dₚ and Dₜ be the distances covered by them respectively. The ratio of their distances is equal to the ratio of their speeds:

(Dₚ)/(Dₜ) = (Sₚ)/(Sₜ) = ²⁷⁄₂₂

The difference in the distance covered by the policeman and the thief must be equal to the initial gap of 195 m. In terms of the ratio, the difference is 27 - 22 = 5 units. We equate this relative difference to the actual distance:

5 units = 195 m 1 unit = ¹⁹⁵⁄₅ = 39 m

The distance run by the thief corresponds to 22 units in the ratio. Therefore, the total distance run by the thief before being overtaken is:

Dₜ = 22 × 39 = 858 m