SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, the left side of the equation can be expanded as follows: (√3 + 1)² = (√3)² + (1)² + 2(√3)(1) = 3 + 1 + 2√3 = 4 + 2√3

By comparing the resulting expression 4 + 2√3 with the given form x + √3y, we equate the rational parts and the coefficients of the irrational parts. This yields x = 4 and y = 2.

The value of x + y is calculated as 4 + 2 = 6.

Explanation

For any point outside a circle, the lengths of the two tangents drawn from that point to the circle are equal. Point T is located on the common tangent line passing through the point of contact P. From point T, two tangents are drawn to the first circle: the common tangent segment TP and the tangent segment TA. According to the property of tangents from an external point, TA = TP. Similarly, from point T, two tangents are drawn to the second circle: the common tangent segment TP and the tangent segment TB. This implies TB = TP. By equating these two expressions through the common segment TP, we find that TA = TB.

Explanation

The fundamental Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Mathematically, for a triangle with sides XY, YZ, and XZ, this is expressed as XY + XZ > YZ, YZ + XZ > XY, and XY + YZ > XZ. By rearranging these inequalities, we derive the Side Difference Theorem, which posits that the absolute difference between the lengths of any two sides of a triangle is always less than the length of the third side. Starting from the inequality XY < YZ + XZ and subtracting YZ from both sides, we obtain XY - YZ < XZ. This condition ensures that the three segments can actually meet to form a closed polygon with three vertices. If the difference were equal to or greater than the third side, the vertices would be collinear or the sides would not meet, failing to form a triangle.

Explanation

The trigonometric identity sin A = cos B holds true when the angles are complementary, such that A + B = 90°. Applying this property to the given equation sin (3x - 20°) = cos (3y + 20°), we equate the sum of the two angles to 90°:

(3x - 20°) + (3y + 20°) = 90°

Simplifying the left side of the equation, the constant terms -20° and +20° cancel out, leaving:

3x + 3y = 90°

By factoring out 3 from the left side, we obtain:

3(x + y) = 90°

Dividing both sides by 3 gives the final value:

x + y = 30°

Explanation

Let the required number be x. According to the problem statement, 1/2 is added to the number and the resulting sum is multiplied by 3 to give a final value of 21. This relationship is represented by the linear equation:

3(x + ½) = 21

To find the value of x, first divide both sides of the equation by 3:

x + 0.5 = ²¹⁄₃

x + 0.5 = 7

Next, subtract 0.5 from both sides of the equation to isolate x:

x = 7 - 0.5

x = 6.5

The number is 6.5, as adding 0.5 to it results in 7, and multiplying that sum by 3 yields exactly 21.

Explanation

The total temperature for Monday, Tuesday, Wednesday, and Thursday is calculated as 4 × 48 = 192°. The total temperature for Tuesday, Wednesday, Thursday, and Friday is calculated as 4 × 52 = 208°.

By subtracting the first sum from the second, we eliminate the common days (Tuesday, Wednesday, and Thursday) to find the difference between Friday and Monday: (Friday + Tuesday + Wednesday + Thursday) - (Monday + Tuesday + Wednesday + Thursday) = 208 - 192 Friday - Monday = 16

Given that the temperature on Monday was 42°, we substitute this value into the equation: Friday - 42 = 16 Friday = 16 + 42 = 58

The temperature on Friday was 58°.

Explanation

Given the equations m - n = 2 and mn = 15 for positive integers m and n, we can identify the values by looking for two numbers whose product is 15 and whose difference is 2. The factors of 15 are (1, 15) and (3, 5). The pair m = 5 and n = 3 satisfies both conditions because 5 - 3 = 2 and 5 × 3 = 15.

Using these values, we calculate the required components: m² - n² = 5² - 3² = 25 - 9 = 16 m³ - n³ = 5³ - 3³ = 125 - 27 = 98

Multiplying these results together: (m² - n²)(m³ - n³) = 16 × 98 16 × 98 = 16 × (100 - 2) = 1600 - 32 = 1568

Alternatively, using algebraic identities: (m + n)² = (m - n)² + 4mn = 2² + 4(15) = 64 ⟹ m + n = 8 m² - n² = (m - n)(m + n) = 2 × 8 = 16 m³ - n³ = (m - n)(m² + mn + n²) = (m - n)((m - n)² + 3mn) = 2(2² + 3(15)) = 2(4 + 45) = 98 The product is 16 × 98 = 1568.

Explanation

Substituting the given values into the expression, we have P² = 9² = 81 and Q² = (√(17))² = 17. The term inside the parentheses is P² - Q² = 81 - 17 = 64. The expression then simplifies to (64)⁻¹^/³. Since 64 can be expressed as 4³, we apply the exponent rule (aᵐ)ⁿ = aᵐⁿ to get (4³)⁻¹^/³ = 4³^×⁽⁻¹^/³⁾ = 4⁻¹. The value of 4⁻¹ is equal to 1/4.

Explanation

Two angles are considered supplementary if the sum of their measures is exactly 180°. Given the two angles (5y + 62°) and (22° + y), we can establish the following linear equation:

(5y + 62) + (22 + y) = 180

To solve for y, first combine the like terms on the left side of the equation:

6y + 84 = 180

Next, isolate the term containing y by subtracting 84 from both sides:

6y = 180 - 84 6y = 96

Finally, divide both sides by 6 to find the value of y:

y = ⁹⁶⁄₆ y = 16

Explanation

In a triangle ABC with circumcentre I, the angle subtended by a side at the centre is twice the angle subtended by it at any point on the remaining part of the circle. According to this theorem, ∠ BIC = 2 × ∠ BAC. Given that ∠ BAC = 60°, it follows that ∠ BIC = 2 × 60° = 120°. In the triangle BIC, the segments IB and IC are equal as they are both radii of the circumcircle. This makes △ BIC an isosceles triangle. The perpendicular ID drawn from the vertex I to the base BC of an isosceles triangle acts as the angle bisector of the vertex angle ∠ BIC. Consequently, ∠ BID = ½ ∠ BIC = ½ × 120° = 60°.

Explanation

Given the definitions for m and n, we square both sides to obtain m² = (cos² α)/(cos² β) and n² = (cos² α)/(sin² β). Substituting these into the required expression:

(m² + n²) cos² β = ( (cos² α)/(cos² β) + (cos² α)/(sin² β) ) cos² β

Factoring out cos² α from the terms within the parentheses:

cos² α ( (1)/(cos² β) + (1)/(sin² β) ) cos² β

Using a common denominator for the terms inside the parentheses:

cos² α ( (sin² β + cos² β)/(cos² β sin² β) ) cos² β

Applying the fundamental trigonometric identity sin² β + cos² β = 1, the expression becomes:

cos² α ( (1)/(cos² β sin² β) ) cos² β

Canceling the cos² β term from the numerator and the denominator yields:

(cos² α)/(sin² β) = ( (cos α)/(sin β) )²

Since it is given that n = (cos α)/(sin β), the result is n².

Explanation

The weight W of a sphere is given by the product of its volume V and its density or weight per unit volume ρ. The volume of a sphere is proportional to the cube of its radius r, such that V = ⁴⁄₃ π r³. Therefore, the weight of a sphere is expressed as W = ⁴⁄₃ π r³ ρ.

Given the ratio of weights W₁ : W₂ = 8 : 27 and the ratio of weights per 1 cc (densities) ₁ : ₂ = 8 : 1, we can set up the following equation:

(W₁)/(W₂) = (4)/(3) π r₁³ ₁{⁴⁄₃ π r₂³ ₂}

⁸⁄₂₇ = (r₁³)/(r₂³) × ⁸⁄₁

By dividing both sides of the equation by 8, we get:

¹⁄₂₇ = (r₁³)/(r₂³)

Taking the cube root of both sides to find the ratio of the radii:

(r₁)/(r₂) = ³√(¹⁄₂₇) = ⅓

Thus, the ratio of the radii of the two spheres is 1 : 3.

Explanation

Let h be the height of the vertical tower and p be the height of the flagpole mounted on top of it. The point on the ground is located at a distance of 9 m from the foot of the tower. According to the problem, the angle of elevation of the bottom of the flagpole (the top of the tower) is 30°. Using the trigonometric relation in the right-angled triangle formed by the tower and the ground distance:

tan 30° = ʰ⁄₉ (1)/(√(3)) = ʰ⁄₉ h = (9)/(√(3)) = 3√3 m

The angle of elevation of the top of the flagpole from the same point is 60°. In the larger right-angled triangle formed by the combined height of the tower and flagpole:

tan 60° = (h + p)/(9) √3 = (h + p)/(9) h + p = 9√3 m

To find the height of the flagpole p, we subtract the height of the tower from the total height:

p = (h + p) - h p = 9√3 - 3√3 = 6√3 m

Explanation

The time taken for two individuals to complete a piece of work together is determined by the formula (xy)/(x + y), where x and y are the number of days each takes to complete the work alone. Substituting the given values x = 8 and y = 10, the time taken is (8 × 10)/(8 + 10) = ⁸⁰⁄₁₈. Simplifying this fraction by dividing both the numerator and the denominator by 2 yields ⁴⁰⁄₉ days.

Alternatively, using the LCM method, let the total work be 40 units, which is the Least Common Multiple of 8 and 10. The daily work rate of A is 40 / 8 = 5 units per day, and the daily work rate of B is 40 / 10 = 4 units per day. The combined daily rate is 5 + 4 = 9 units per day. The total time taken to complete the 40 units of work together is therefore ⁴⁰⁄₉ days.

Explanation

The area of a rhombus is determined by the formula Area = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the two diagonals. Substituting the given values into the formula, we have Area = ½ × 8 cm × 6 cm. Calculating the product of the diagonals gives 48, and dividing by 2 results in an area of 24 cm². This formula is derived from the fact that the diagonals of a rhombus bisect each other at right angles, dividing the shape into four congruent right-angled triangles whose total area equals half the product of the diagonals.

Explanation

To find the single discount equivalent to two successive discounts of x% and y%, we use the formula (x + y - ˣʸ⁄₁₀₀)%. Applying this formula with the given values x = 10 and y = 20, the calculation becomes 10 + 20 - (10 × 20)/(100). This simplifies to 30 - ²⁰⁰⁄₁₀₀, which results in 30 - 2 = 28%.

Alternatively, using the multiplier method, an initial price can be considered as 100 units. A 10% discount reduces the price to 90 units (100 × 0.90). A subsequent 20% discount is then applied to this reduced price of 90 units, which is a reduction of 90 × 0.20 = 18 units. The final price after both discounts is 90 - 18 = 72 units. The total effective discount is the total reduction from the original price, calculated as 100 - 72 = 28, which corresponds to a single discount of 28%.

Explanation

Let the amounts with A and B be A and B respectively. The total amount is given as A + B = 6300. According to the condition, ⁵⁄₁₉A = ⅖B. To find the ratio of their amounts, we rearrange the equation to solve for A/B:

(A)/(B) = (2/5)/(5/19) = ⅖ × ¹⁹⁄₅ = ³⁸⁄₂₅

This implies the ratio A:B is 38:25. The sum of the ratio parts is 38 + 25 = 63 units. Since 63 units correspond to the total amount of Rs. 6300, the value of one unit is 6300 / 63 = Rs. 100. Therefore, the amount of B is 25 units × 100 = Rs. 2500.

Explanation

To find the gain or loss percentage when the selling price of m articles equals the cost price of n articles, we use the formula:

Gain or Loss Percent = (n - m)/(m) × 100

In this problem, the selling price of 40 articles (m = 40) is equal to the cost price of 50 articles (n = 50). Substituting these values into the formula:

Percentage = (50 - 40)/(40) × 100

Percentage = ¹⁰⁄₄₀ × 100

Percentage = ¼ × 100 = 25%

Since the result is positive, it represents a gain. Alternatively, if we assume the cost price of one article is 1 unit, the cost price of 50 articles is 50 units. This is the same as the selling price of 40 articles. Therefore, the selling price of one article is 50 / 40 = 1.25 units. The profit per article is 1.25 - 1 = 0.25 units, which corresponds to a 25% gain.

Explanation

Let B's income be 100. Since A's income is 25% more than B's, A's income is calculated as 100 + 25 = 125. To find B's income as a percentage of A's income, we use the following relation:

Percentage = (B's income)/(A's income) × 100

Substituting the values into the formula:

Percentage = ¹⁰⁰⁄₁₂₅ × 100

Simplifying the fraction ¹⁰⁰⁄₁₂₅ to ⅘ and multiplying by 100 gives:

⅘ × 100 = 80

Alternatively, if A is x% more than B, then B is ¹⁰⁰⁄₁₀₀₊ₓ × 100 percent of A. By substituting x = 25:

¹⁰⁰⁄₁₂₅ × 100 = 80

Explanation

To find the time taken by a train to pass a stationary object of negligible width like a telegraph post, the total distance to be covered is equal to the length of the train.

First, convert the speed of the train from km/hr to m/s: Speed = 36 × ⁵⁄₁₈ = 10 m/s

The distance to be covered is the length of the train, which is 60 metres. Using the standard formula for time: Time = (Distance)/(Speed) Time = ⁶⁰⁄₁₀ = 6 seconds

Explanation

To find the value of x² + 1/25x², first divide the given equation 5x + 1/x = 10 by 5 to get x + 1/5x = 2. Squaring both sides gives x² + 2(x)(1/5x) + 1/25x² = 4, which simplifies to x² + 1/25x² = 4 - 2/5 = 18/5. The value 18/5 is equivalent to the mixed fraction 3 3/5, represented by option C.

Explanation

To find the ratio r : h, isolate the radical term and square both sides of the equation: 4r - h = √(r² + h²) (4r - h)² = r² + h²

Expanding the left side using the identity (a - b)² = a² - 2ab + b²: 16r² - 8rh + h² = r² + h²

Subtracting h² from both sides simplifies the equation to: 16r² - 8rh = r² 15r² = 8rh

Given that r ≠ 0, we can divide both sides by r: 15r = 8h ʳ⁄ₕ = ⁸⁄₁₅

This yields the ratio r : h = 8 : 15.

Explanation

Given the equation sin A + sin² A = 1, we can rearrange it as sin A = 1 - sin² A. Using the fundamental trigonometric identity sin² A + cos² A = 1, we know that 1 - sin² A = cos² A. Substituting this into our rearranged equation gives sin A = cos² A.

To find the value of cos² A + cos⁴ A, we square both sides of the relation sin A = cos² A to get sin² A = cos⁴ A. Now, substituting these values back into the expression we wish to evaluate: cos² A + cos⁴ A = sin A + sin² A. Since the original equation states that sin A + sin² A = 1, the value of cos² A + cos⁴ A must also be 1.

Explanation

The sum of the interior angles of any quadrilateral is constant and equal to 360°. To find the measure of the fourth angle, the measures of the three given angles are summed and then subtracted from the total. The sum of the given angles is 60° + 90° + 100° = 250°. Using the formula for the sum of angles, the fourth angle is calculated as 360° - 250° = 110°.

Explanation

In the given range 0° < θ < 90°, the value of cos θ always lies strictly between 0 and 1, such that 0 < cos θ < 1. For any real number x where 0 < x < 1, multiplying the inequality x < 1 by the positive value x results in x² < x. Applying this property to the trigonometric function by substituting x = cos θ, it follows that cos² θ < cos θ, or equivalently, cos θ > cos² θ. Since the boundary values of 0° and 90° are excluded, cos θ cannot be equal to 1 or 0, making the inequality strictly greater than rather than greater than or equal to.

Explanation

To find the value of the expression, convert all terms to a common base of 3. In the numerator, 3 × 9ⁿ⁺¹ = 3¹ × (3²)ⁿ⁺¹ = 3²ⁿ⁺³ and 9 × 3²ⁿ⁻¹ = 3² × 3²ⁿ⁻¹ = 3²ⁿ⁺¹. In the denominator, 9 × 3²ⁿ = 3² × 3²ⁿ = 3²ⁿ⁺² and 6 × 9ⁿ⁻¹ = (2 × 3) × (3²)ⁿ⁻¹ = 2 × 3¹ × 3²ⁿ⁻² = 2 × 3²ⁿ⁻¹.

Substituting these back into the expression, we get: 3²ⁿ⁺³ + 3²ⁿ⁺¹{3²ⁿ⁺² - 2 × 3²ⁿ⁻¹}

Factor out the lowest power of 3, which is 3²ⁿ⁻¹, from both the numerator and the denominator: 3²ⁿ⁻¹ (3⁴ + 3²){3²ⁿ⁻¹ (3³ - 2 × 3⁰)}

Canceling 3²ⁿ⁻¹ from both terms, the expression simplifies to: (81 + 9)/(27 - 2) = ⁹⁰⁄₂₅

Dividing both the numerator and denominator by 5 gives ¹⁸⁄₅, which can be written as the mixed fraction 3 ⅗.

Explanation

The total age of the four children is calculated by multiplying their average age by the number of children: 12 × 4 = 48 years. When the father is included, the group consists of five people. The total age of these five individuals is calculated by multiplying their combined average age by the count: 20 × 5 = 100 years. To find the age of the father, subtract the total age of the children from the total age of the entire group: 100 - 48 = 52 years.

Explanation

Using the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b), we can evaluate the given expression by cubing the equation a+b=3. Cubing both sides gives (a+b)³ = 3³, which expands to a³ + b³ + 3ab(a + b) = 27. Substituting the given value a + b = 3 into this expansion results in a³ + b³ + 3ab(3) = 27. This simplifies to a³ + b³ + 9ab = 27. By rearranging the terms to match the required expression, we subtract 27 from both sides to get a³ + b³ + 9ab - 27 = 0. This result can also be verified by substituting specific values that satisfy the condition a+b=3, such as a=3 and b=0, where the expression becomes 3³ + 0³ + 9(3)(0) - 27 = 27 - 27 = 0.

Explanation

The given equation is x + ¹⁄ₓ = 2. Multiplying both sides by x gives x² + 1 = 2x, which can be rearranged into the quadratic form x² - 2x + 1 = 0. This is the expansion of the identity (x - 1)² = 0, which solves to x = 1.

Substituting x = 1 into the target expression: x⁷ + (2)/(x⁶) = (1)⁷ + (2)/((1)⁶) 1 + ²⁄₁ = 1 + 2 = 3

Explanation

In Δ ABC, since BE and CF are medians, E is the midpoint of AC and F is the midpoint of AB. According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of its length. Therefore, in Δ ABC, we have FE ∥ BC and FE = ½ BC.

Similarly, in Δ OBC, P is given as the midpoint of BO and Q is the midpoint of CO. Applying the Midpoint Theorem to Δ OBC, the segment joining these midpoints is PQ ∥ BC and PQ = ½ BC.

By comparing the two relations, we find that FE = ½ BC and PQ = ½ BC, which implies FE = PQ. Given that the length of PQ is 3 cm, the length of FE must also be 3 cm.

Explanation

According to the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Since ST ∥ QR in triangle PQR, the segments on PQ and PR are proportional:

(PS)/(SQ) = (PT)/(TR)

We are given (PS)/(SQ) = ⅗, so (PT)/(TR) = ⅗. To find PT in terms of the whole side PR, we use the ratio of the part to the whole:

(PT)/(PR) = (PT)/(PT + TR) = (3)/(3 + 5) = ⅜

Given that PR = 6 cm, we substitute this value into the equation:

PT = ⅜ × PR = ⅜ × 6 = ¹⁸⁄₈ = 2.25 cm

Explanation

The given equation is 5 sin²θ + 4 cos²θ = ⁹⁄₂. We can decompose the first term to utilize the fundamental identity sin²θ + cos²θ = 1. Rewriting the equation, we get sin²θ + 4 sin²θ + 4 cos²θ = 4.5, which simplifies to sin²θ + 4(sin²θ + cos²θ) = 4.5.

Substituting the identity, the equation becomes sin²θ + 4(1) = 4.5, leading to sin²θ = 0.5 = ½. Since the problem specifies 0 < θ < (π)/(2), we take the positive square root to find sinθ = (1)/(√(2)). This value corresponds to an angle of θ = (π)/(4) or 45°. Evaluating the tangent for this angle, we obtain tanθ = tan(45°) = 1.

Explanation

The formula for the total amount A in compound interest is given by A = P(1 + ʳ⁄₁₀₀)ᵗ, where P is the principal, r is the annual rate of interest, and t is the time in years. Substituting the given values:

11664 = 10000(1 + ʳ⁄₁₀₀)²

Dividing both sides by 10000 gives:

¹¹⁶⁶⁴⁄₁₀₀₀₀ = (1 + ʳ⁄₁₀₀)²

1.1664 = (1 + ʳ⁄₁₀₀)²

To find the value of 1 + ʳ⁄₁₀₀, we take the square root of both sides. Since 108² = 11664, the square root of 1.1664 is 1.08:

1.08 = 1 + ʳ⁄₁₀₀

Subtracting 1 from both sides:

0.08 = ʳ⁄₁₀₀

r = 8

Thus, the annual rate of compound interest is 8%.

Explanation

Let the height of the tower be H = 60 m and the height of the pole be h. Let the distance between the tower and the pole be x.

From the top of the tower, the angle of depression to the bottom of the pole is 60°. In the right-angled triangle formed by the tower height and the ground distance: tan 60° = H/x √3 = 60/x ⟹ x = 60/√3 = 20√3 m

The angle of depression to the top of the pole is 45°. The vertical distance from the top of the tower to the level of the top of the pole is H - h. In the right-angled triangle formed by this height difference and the same horizontal distance x: tan 45° = (H - h)/x 1 = (60 - h)/x ⟹ x = 60 - h

Substituting the value of x from the first calculation: 20√3 = 60 - h h = 60 - 20√3 h = 20(3 - √3) m

Explanation

The total work done is constant and is defined by the product of the number of men and the number of days taken. This relationship is expressed by the formula M₁D₁ = M₂D₂, where M represents the number of men and D represents the number of days.

Given that M₁ = 24 men and D₁ = 17 days, the total work is 24 × 17 man-days. We need to find the number of men M₂ required to complete the same work in D₂ = 51 days.

24 × 17 = M₂ × 51

To find M₂, rearrange the equation: M₂ = (24 × 17)/(51)

Since 51 is a multiple of 17 (17 × 3 = 51), the calculation simplifies to: M₂ = ²⁴⁄₃ M₂ = 8

By increasing the duration of the work from 17 days to 51 days (a threefold increase), the number of men required decreases inversely by the same factor, resulting in 8 men.

Explanation

In a cyclic quadrilateral, the exterior angle formed by extending one side is equal to the interior opposite angle. Given that side AB is extended to X, the exterior angle ∠ XBC is equal to the interior opposite angle ∠ ADC.

Therefore, ∠ ADC = ∠ XBC = 82°.

The angle ∠ ADC is composed of two adjacent angles, ∠ ADB and ∠ BDC, which are formed by the diagonal BD. We can express this relationship as: ∠ ADC = ∠ ADB + ∠ BDC

Substituting the known values into the equation: 82° = 47° + ∠ BDC ∠ BDC = 82° - 47° ∠ BDC = 35°

Explanation

When a price is increased by a certain percentage and then decreased by the same percentage, the net result is always a loss. This can be calculated using the successive percentage change formula: x + y + ˣʸ⁄₁₀₀, where x represents the percentage increase (markup) and y represents the percentage decrease (discount).

Substituting x = 10 and y = -10 into the formula: Net change = 10 - 10 + (10 × (-10))/(100) Net change = 0 - ¹⁰⁰⁄₁₀₀ = -1%

The negative result indicates a loss of 1% relative to the original cost price.

Alternatively, if the cost price is assumed to be 100, a 10% markup makes the marked price 110. Applying a 10% discount on 110 results in a reduction of 11, making the final selling price 110 - 11 = 99. Since the selling price is 1 unit less than the cost price of 100, the seller incurs a 1% loss.

Explanation

For a sum S to be divided into two integers in a ratio x:y, the sum of the ratio parts (x + y) must be a factor of S. In this case, the total sum is 84 and the sum of the ratio terms is 3 + 2 = 5. Dividing the total sum by the sum of the ratio parts yields 84 ÷ 5 = 16.8. Since the result is not an integer, it is impossible for two whole numbers to sum to 84 while maintaining a ratio of 3:2.

Explanation

The new selling price SPₙₑ_w required to earn a specific profit percentage from a known loss-making selling price SPₒₗ_d can be calculated using the ratio: SPₙₑ_w = SPₒₗ_d × (100 + Profit%)/(100 - Loss%) Given SPₒₗ_d = 9, Loss% = 10, and Profit% = 15, the substitution yields: SPₙₑ_w = 9 × (100 + 15)/(100 - 10) SPₙₑ_w = 9 × ¹¹⁵⁄₉₀ SPₙₑ_w = ¹¹⁵⁄₁₀ = 11.50

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. Using the relative speed formula: Relative speed Vᵣₑₗ = Vₜᵣₐᵢₙ + Vₘₐₙ = 60 + 6 = 66 kmph.

To convert the speed from kmph to m/s, we multiply by ⁵⁄₁₈: Vᵣₑₗ = 66 × ⁵⁄₁₈ = (11 × 5)/(3) = ⁵⁵⁄₃ m/s.

The distance to be covered to pass the man is equal to the length of the train, which is 110 metres. The time taken is calculated using the formula: Time = (Distance)/(Speed) Time = (110)/(55){3} Time = (110 × 3)/(55) Time = 2 × 3 = 6 seconds.

Explanation

To find the relationship between x and y, we eliminate the parameter t from the given parametric equations x = at² and y = 2at.

Squaring the equation for y yields: y² = (2at)² = 4a²t²

We can rewrite this expression to isolate the terms matching the equation for x: y² = 4a(at²)

Since x = at², we substitute x into the equation: y² = 4ax

Alternatively, we can solve for t from the second equation, giving t = y / 2a. Substituting this into the first equation results in x = a(y / 2a)², which simplifies to x = ay² / 4a². Multiplying both sides by 4a gives the Cartesian equation y² = 4ax, which represents a parabola.

Explanation

The value of the expression can be determined by substituting the given condition xy + yz + zx = 1 directly into the numerator. Replacing 1 in 1 + y² yields: 1 + y² = (xy + yz + zx) + y²

Rearranging the terms to facilitate factoring: 1 + y² = (y² + xy) + (yz + zx)

Factoring by grouping: 1 + y² = y(y + x) + z(y + x) = (x + y)(y + z)

Substituting this factored form back into the original expression: ((x + y)(y + z))/((x + y)(y + z)) = 1

Alternatively, using a value-substitution shortcut, assume y = 0. The condition xy + yz + zx = 1 then simplifies to zx = 1. By choosing x = 1 and z = 1, the value of the expression is calculated as: (1 + 0²)/((1 + 0)(0 + 1)) = ¹⁄₁ = 1

Explanation

The line segment joining the center of the circle to the external point from which tangents are drawn bisects the angle between those tangents. Given ∠ QPR = 120°, the angle ∠ OPQ is half of the total angle, calculated as (120°)/(2) = 60°. In a circle, the radius OQ is perpendicular to the tangent PQ at the point of contact, which establishes that ∠ OQP = 90°. In the triangle POQ, the sum of all interior angles is 180°. Therefore, the value of ∠ POQ is determined by subtracting the known angles from the total sum: ∠ POQ = 180° - 90° - 60° = 30°.

Explanation

In △ ABC, the sum of all interior angles is 180°, expressed as ∠ BAC + ∠ ABC + ∠ ACB = 180°. Given that ∠ BAC = 90°, the sum of the other two angles is ∠ ABC + ∠ ACB = 90°. Since the problem states AB = AC, the triangle is isosceles. In an isosceles triangle, the angles opposite to the equal sides are equal, meaning ∠ ABC = ∠ ACB. Substituting this into the sum equation gives 2 × ∠ ABC = 90°. Solving for the required angle, we find ∠ ABC = 45°.

Explanation

The value of the trigonometric expression is determined using the properties of angles and periodicity. For the first term, we use the identity tan(360° - θ) = -tan θ. Substituting θ = 45° gives tan 315° = tan(360° - 45°) = -tan 45° = -1.

For the second term, we use the identity cot(-θ) = -cot θ. Thus, cot(-405°) = -cot 405°. Since the cotangent function has a period of 360°, we have cot(360° + θ) = cot θ. This gives -cot(360° + 45°) = -cot 45° = -1.

Multiplying these results together, we get tan 315° × cot(-405°) = (-1) × (-1) = 1.

Explanation

Let the second number be 6x, which is the least common multiple of 2 and 3. Since the second number is twice the first and thrice the third, the first number is 3x and the third number is 2x. The average of the three numbers is given as 33, meaning their sum is 33 × 3 = 99. Summing the individual terms gives 3x + 6x + 2x = 99, which simplifies to 11x = 99. Solving for x yields x = 9. The largest of the three numbers is the second number, which is 6x = 6 × 9 = 54.

Explanation

The sum of the items is calculated using the formula Sum = Average × Number of items. The initial incorrect sum is 30 × 100 = 3000. To find the correct sum, we subtract the values that were wrongly included and add the correct values.

The sum of the wrongly taken items is 32 + 12 = 44, while the sum of the correct items is 23 + 11 = 34. The net difference to be applied to the total sum is 34 - 44 = -10.

The corrected sum is 3000 - 10 = 2990. Therefore, the correct average is 2990 / 100 = 29.9. Alternatively, the change in average can be calculated directly by dividing the net difference by the total number of items: -10 / 100 = -0.1. Subtracting this from the original average gives 30 - 0.1 = 29.9.

Explanation

Let the weight of the empty box be B and the weight of the sand required to fill the box completely be S. Based on the problem statement, we can form two linear equations:

B + S = 8.5 B + 0.5S = 5.5

Subtracting the second equation from the first allows us to find the weight of half the sand: (B + S) - (B + 0.5S) = 8.5 - 5.5 0.5S = 3.0

Multiplying by 2 gives the weight of the sand when the box is full: S = 6.0 kg

To find the weight of the empty box, substitute the weight of the sand S into the first equation: B + 6.0 = 8.5 B = 8.5 - 6.0 = 2.5 kg

Explanation

Given the equations a + (1)/(b) = 1 and b + (1)/(c) = 1, we can express c and ¹⁄ₐ in terms of b. From the first equation, a = 1 - (1)/(b) = (b - 1)/(b), which implies ¹⁄ₐ = (b)/(b - 1). From the second equation, (1)/(c) = 1 - b, which gives c = (1)/(1 - b).

To find the value of c + ¹⁄ₐ, we substitute these expressions: c + ¹⁄ₐ = (1)/(1 - b) + (b)/(b - 1)

By adjusting the denominator of the second term, we get: c + ¹⁄ₐ = (1)/(1 - b) - (b)/(1 - b) c + ¹⁄ₐ = (1 - b)/(1 - b) c + ¹⁄ₐ = 1

Explanation

When two circles touch externally, the distance between their centers is equal to the sum of their radii. Let R and r be the radii of the two circles, where R represents the radius of the bigger circle. According to the problem, R + r = 14.

The sum of the areas of the two circles is given as 130π square centimeters. Using the formula for the area of a circle A = π r², we can establish the equation π R² + π r² = 130π, which simplifies to R² + r² = 130.

To find the values of R and r, we can use the algebraic identity (R + r)² = R² + r² + 2Rr. Substituting the known values: 14² = 130 + 2Rr 196 = 130 + 2Rr 2Rr = 66 Rr = 33

We now have a system where the sum of the radii is 14 and their product is 33. These values are the roots of the quadratic equation x² - 14x + 33 = 0. Factoring the equation: (x - 11)(x - 3) = 0 The roots are x = 11 and x = 3. Since R is the radius of the bigger circle, its value is 11 cm.