SSC CGL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of the expression (1)/(1 - a) + (1)/(1 - b) + (1)/(1 - c), we can utilize the algebraic identity (x)/(1 - x) + 1 = (x + (1 - x))/(1 - x) = (1)/(1 - x).

Given the equation: (a)/(1 - a) + (b)/(1 - b) + (c)/(1 - c) = 1

Add 1 to each of the three terms on the left side of the equation. To maintain equality, we must add 3 to the right side: ((a)/(1 - a) + 1) + ((b)/(1 - b) + 1) + ((c)/(1 - c) + 1) = 1 + 3

Simplifying each term inside the parentheses: (a + 1 - a)/(1 - a) + (b + 1 - b)/(1 - b) + (c + 1 - c)/(1 - c) = 4

This results in: (1)/(1 - a) + (1)/(1 - b) + (1)/(1 - c) = 4

Explanation

For any set of real numbers, the sum of their squares is equal to zero if and only if each individual term is equal to zero. Applying this principle to the given equation (x - 3)² + (y - 5)² + (z - 4)² = 0, we must have (x - 3)² = 0, (y - 5)² = 0, and (z - 4)² = 0. Solving these individual equations yields the values x = 3, y = 5, and z = 4.

To find the value of the expression x² / 9 + y² / 25 + z² / 16, substitute these values into the formula: 3² / 9 + 5² / 25 + 4² / 16 = 9/9 + 25/25 + 16/16 = 1 + 1 + 1 = 3.

Explanation

To find the value of P when x = 6, substitute the given value into the linear equation ⁴ˣ⁄₃ + 2P = 12. This yields the following expression:

⁴⁽⁶⁾⁄₃ + 2P = 12

By simplifying the first term, we calculate ²⁴⁄₃, which results in 8. The equation is now:

8 + 2P = 12

Subtracting 8 from both sides of the equation to isolate the term containing P gives:

2P = 12 - 8

2P = 4

Dividing both sides by 2 provides the final value for P:

P = ⁴⁄₂ = 2

Explanation

The expression 4 + 3√3{7 + 4√3} is simplified by rationalizing the denominator. To do this, multiply both the numerator and the denominator by the conjugate of the denominator, which is 7 - 4√3.

For the denominator, apply the algebraic identity (a + b)(a - b) = a² - b²: (7 + 4√3)(7 - 4√3) = 7² - (4√3)² = 49 - (16 × 3) = 49 - 48 = 1

For the numerator, expand the terms: (4 + 3√3)(7 - 4√3) = 4(7) + 4(-4√3) + 3√3(7) + 3√3(-4√3) = 28 - 16√3 + 21√3 - 12(3) = 28 + 5√3 - 36 = 5√3 - 8

Substituting these results back into the fraction gives: 5√3 - 8{1} = 5√3 - 8

Explanation

To make the expression 16a² - 12a a perfect square, it must be written in the form (x - y)² = x² - 2xy + y². Comparing the terms, we identify x² = 16a², which implies x = 4a. The middle term -2xy corresponds to -12a. Substituting x = 4a into the middle term equation gives -2(4a)y = -12a. Simplifying this results in -8ay = -12a, which gives y = ¹²ᵃ⁄₈ₐ = ³⁄₂. The value that must be added to complete the square is y². Calculating this, we get y² = (³⁄₂)² = ⁹⁄₄. Adding this value results in the perfect square trinomial (4a - ³⁄₂)².

Explanation

The equation 2x + 3y = 12 can be converted into the intercept form ˣ⁄ₐ + (y)/(b) = 1 by dividing every term by 12. This yields ²ˣ⁄₁₂ + ³ʸ⁄₁₂ = ¹²⁄₁₂, which simplifies to ˣ⁄₆ + ʸ⁄₄ = 1. This shows that the line has a positive x-intercept at x = 6 and a positive y-intercept at y = 4.

The line segment connecting these two intercepts (6, 0) and (0, 4) lies in the 1st quadrant, where both x and y are positive. As the line extends beyond the y-axis to the left, the x-coordinates become negative while the y-coordinates remain positive, meaning the line passes through the 2nd quadrant. As the line extends beyond the x-axis to the right, the x-coordinates remain positive while the y-coordinates become negative, meaning the line passes through the 4th quadrant. Substituting negative values for both x and y into the expression 2x + 3y will always result in a negative value, confirming the line can never satisfy the equation for points in the 3rd quadrant.

Explanation

In any triangle ABC, let hₐ, h_b, and h_c be the lengths of the altitudes dropped from vertices A, B, and C to the opposite sides a, b, and c respectively. In the right-angled triangle formed by an altitude, the altitude is a leg and one of the adjacent sides of the original triangle acts as the hypotenuse. Since the hypotenuse is always the longest side in a right-angled triangle, each altitude is shorter than the sides of the triangle meeting at that vertex. Specifically, hₐ < c and hₐ < b, h_b < a and h_b < c, and h_c < a and h_c < b. By summing the relations hₐ < c, h_b < a, and h_c < b, we obtain hₐ + h_b + h_c < a + b + c. This demonstrates that the sum of the three altitudes is strictly less than the perimeter of the triangle.

Explanation

The tangent to a circle is perpendicular to the radius at the point of contact, forming a right-angled triangle where the distance from the external point to the center serves as the hypotenuse. According to the Pythagorean theorem, the length of the tangent L is calculated using the formula L = √(d²⁾⁻ʳ^^^², where d is the distance from the center and r is the radius. Substituting the given values, L = √(5²⁾⁻⁴^^^² = √(25 - 16) = √9 = 3 cm. This result follows the standard (3, 4, 5) Pythagorean triplet.

Explanation

In the cyclic quadrilateral ABCD, the equality of sides AD = DC establishes that △ ADC is an isosceles triangle. Since the base angles of an isosceles triangle are equal, the angle ∠ ACD = ∠ CAD = θ. The sum of the interior angles of a triangle is 180°, so the angle at vertex D is calculated as ∠ ADC = 180° - (∠ CAD + ∠ ACD) = 180° - 2θ. In any cyclic quadrilateral, opposite angles are supplementary, meaning their sum is 180°. Therefore, ∠ ABC + ∠ ADC = 180°. By substituting the expression for ∠ ADC into this relationship, we obtain ∠ ABC + (180° - 2θ) = 180°. Solving this equation for the unknown angle yields ∠ ABC = 2θ.

Explanation

The area of an equilateral triangle can be expressed directly in terms of its height h using the formula Area = (h²)/(√(3)). Given that the height h is 15 cm, the area is calculated as:

Area = (15²)/(√(3)) = (225)/(√(3))

To simplify this expression, multiply the numerator and denominator by √3:

Area = 225 · √3{√3 · √3} = 225√3{3} = 75√3 sq. cm.

Explanation

The radius R of the circle is half the diameter, so R = 10 cm. For any chord of length L, the perpendicular distance d from the center to the chord is given by the formula d = √(R²⁾⁻⁽^L^/²⁾^^^², derived from the Pythagorean theorem in the right-angled triangle formed by the radius, the distance from the center, and half the chord length.

For the first chord of length 12 cm, the distance from the center is d₁ = √(10²⁾⁻⁽¹²^/²⁾^^^² = √(100 - 36) = √(64) = 8 cm. For the second chord of length 16 cm, the distance from the center is d₂ = √(10²⁾⁻⁽¹⁶^/²⁾^^^² = √(100 - 64) = √(36) = 6 cm. Since the chords lie on opposite sides of the center, the total distance between them is the sum of these perpendicular distances: d₁ + d₂ = 8 + 6 = 14 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, it divides those two sides in the same ratio. In Δ ABC, given that DE ∥ AC, the theorem states that (BD)/(DA) = (BE)/(EC).

The length of segment BD is found by subtracting AD from the total length of side AB, giving BD = AB - AD = 10 cm - 4 cm = 6 cm. Substituting the known lengths into the proportionality relation, we have (BE)/(EC) = (BD)/(DA) = ⁶⁄₄. Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, 2, results in ³⁄₂. Therefore, the ratio BE : CE is 3 : 2.

Explanation

In a circle with center O, the line segments OA, OB, and OC are radii, meaning OA = OB = OC. This property forms two isosceles triangles, △ OAB and △ OAC. In △ OAB, the sum of the interior angles is 180°. Since OA = OB, the base angles are equal, so ∠ OAB = ∠ OBA. The calculation is 2∠ OAB + ∠ AOB = 180°, which simplifies to 2∠ OAB + 90° = 180°, resulting in ∠ OAB = 45°.

Similarly, in △ OAC, the equality OA = OC implies that ∠ OAC = ∠ OCA. The angle sum property gives 2∠ OAC + ∠ AOC = 180°, which leads to 2∠ OAC + 110° = 180°, resulting in ∠ OAC = 35°. Considering the case where B and C are on opposite sides of the radius OA, the total angle ∠ BAC is the sum of these two angles: ∠ BAC = ∠ OAB + ∠ OAC = 45° + 35° = 80°.

Explanation

Let h be the height of the balloon and d be the distance between the two consecutive kilometer stones, which is 1 km. When the angles of elevation α = 30° and β = 60° are measured from two points on the same side of the balloon, the height is determined by the formula h = (d)/(cot α - cot β). Substituting the known values into the equation, we have h = (1)/(cot 30° - cot 60°). Using the trigonometric values cot 30° = √3 and cot 60° = 1 / √3, the denominator becomes √3 - 1 / √3, which simplifies to (3 - 1) / √3 = 2 / √3. Therefore, h = (1)/(2 / √(3)) = √3{2} km.

Explanation

The expression is evaluated using the complementary angle identity cos θ = sin(90° - θ) and the reciprocal relationship csc θ = (1)/(sin θ).

In the first part of the expression, 3 cos 80° csc 10°, we substitute cos 80° with sin(90° - 80°) = sin 10°. Since csc 10° = (1)/(sin 10°), the product becomes: 3 sin 10° · (1)/(sin 10°) = 3

In the second part of the expression, 2 cos 59° csc 31°, we substitute cos 59° with sin(90° - 59°) = sin 31°. Since csc 31° = (1)/(sin 31°), the product becomes: 2 sin 31° · (1)/(sin 31°) = 2

Summing the results of both parts provides the final value: 3 + 2 = 5

Explanation

The relationship between radians and degrees is established by the identity π radians = 180°. To convert any angle from radians to degrees, the radian measure is multiplied by the conversion factor 180°/π.

Applying this to the given value:

Angle in degrees = (3π)/(5) × (180°)/(π)

The factor of π cancels out from the numerator and the denominator, simplifying the expression to:

Angle in degrees = (3 × 180°)/(5)

Dividing 180° by 5 yields 36°. Multiplying this result by 3 gives:

3 × 36° = 108°

Explanation

The given equation is tan Θ + cot Θ = 2. Using the reciprocal relationship cot Θ = (1)/(tan Θ), the expression is equivalent to tan Θ + (1)/(tan Θ) = 2. Squaring both sides of this equation according to the algebraic identity (a + b)² = a² + b² + 2ab results in the following steps:

(tan Θ + cot Θ)² = 2²

tan² Θ + cot² Θ + 2 tan Θ cot Θ = 4

Because tan Θ and cot Θ are reciprocals, their product tan Θ · cot Θ equals 1. Substituting this value into the equation yields:

tan² Θ + cot² Θ + 2(1) = 4

tan² Θ + cot² Θ = 4 - 2 = 2

Alternatively, the equation x + ¹⁄ₓ = 2 for x = tan Θ is satisfied only when x = 1. This implies tan Θ = 1 and cot Θ = 1. Substituting these specific values into the sum of the squares gives 1² + 1² = 2. This relationship ensures that for any positive integer power n, the sum tanⁿ Θ + cotⁿ Θ will equal 2 when tan Θ + cot Θ = 2.

Explanation

To eliminate Θ from the given system, square both equations and add them together. The provided equations are xcos Θ - ysin Θ = 2 and xsin Θ + ycos Θ = 4.

Squaring the first equation yields: x² cos² Θ + y² sin² Θ - 2xy sin Θ cos Θ = 4

Squaring the second equation yields: x² sin² Θ + y² cos² Θ + 2xy sin Θ cos Θ = 16

Adding these two equations cancels the cross-product terms -2xy sin Θ cos Θ and +2xy sin Θ cos Θ, resulting in: x² (cos² Θ + sin² Θ) + y² (sin² Θ + cos² Θ) = 4 + 16

Using the fundamental trigonometric identity sin² Θ + cos² Θ = 1, the expression simplifies to: x²(1) + y²(1) = 20 x² + y² = 20

Explanation

The expression is evaluated using the algebraic identities for the sum and difference of cubes, x³ + y³ = (x + y)(x² - xy + y²) and x³ - y³ = (x - y)(x² + xy + y²).

The first component, (sin³ A + cos³ A)/(sin A + cos A), can be expanded as ((sin A + cos A)(sin² A - sin A cos A + cos² A))/(sin A + cos A). After canceling the common factor (sin A + cos A) in the numerator and denominator, the expression remains as sin² A + cos² A - sin A cos A. Using the fundamental trigonometric identity sin² A + cos² A = 1, this simplifies to 1 - sin A cos A.

The second component, (cos³ A - sin³ A)/(cos A - sin A), is similarly expanded as ((cos A - sin A)(cos² A + cos A sin A + sin² A))/(cos A - sin A). Upon canceling the common factor (cos A - sin A), the expression becomes cos² A + sin² A + cos A sin A, which simplifies to 1 + sin A cos A.

Adding the two simplified components together results in (1 - sin A cos A) + (1 + sin A cos A). The terms -sin A cos A and +sin A cos A cancel each other out, leaving a final value of 1 + 1 = 2.

Explanation

Let a = 1/x, b = 1/y, and c = 1/z. The given equation 1 / x² + 1 / y² + 1 / z² = 1 / xy + 1 / yz + 1 / zx transforms into the algebraic identity a² + b² + c² = ab + bc + ca.

By multiplying the entire equation by 2 and rearranging the terms, we get: 2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0

This expression can be grouped into a sum of perfect squares: (a² - 2ab + b²) + (b² - 2bc + c²) + (c² - 2ca + a²) = 0 (a - b)² + (b - c)² + (c - a)² = 0

For the sum of squares of real numbers to equal zero, each individual squared term must independently be zero. This leads to the following set of equations: a - b = 0 ⟹ a = b b - c = 0 ⟹ b = c c - a = 0 ⟹ c = a

Therefore, a = b = c. Substituting the original variables back into the relation gives 1/x = 1/y = 1/z, which simplifies to x = y = z.

Explanation

To find the value of a cos Θ - b sin Θ, let this expression be represented by x. By squaring both the given equation a sin Θ + b cos Θ = c and the expression for x, we can use the trigonometric identity (a sin Θ + b cos Θ)² + (a cos Θ - b sin Θ)² = a² + b². This identity is derived by expanding the squares, which results in a² sin² Θ + b² cos² Θ + 2ab sin Θ cos Θ + a² cos² Θ + b² sin² Θ - 2ab sin Θ cos Θ. The cross-product terms cancel out, and the remaining terms simplify to a²(sin² Θ + cos² Θ) + b²(sin² Θ + cos² Θ), which equals a² + b² since sin² Θ + cos² Θ = 1.

Substituting the given value c into the identity yields c² + x² = a² + b². Solving for x gives x² = a² + b² - c². Taking the square root of both sides, the value of the expression is x = ± √(a² + b² - c²).

Explanation

Given the equations x = a sec Θ cos Φ, y = b sec Θ sin Φ, and z = c tan Θ, we substitute these into the required expression:

(x²)/(a²) + (y²)/(b²) - (z²)/(c²) = ((a sec Θ cos Φ)²)/(a²) + ((b sec Θ sin Φ)²)/(b²) - ((c tan Θ)²)/(c²)

Simplifying the terms by canceling the constants a², b², and c²:

= sec² Θ cos² Φ + sec² Θ sin² Φ - tan² Θ

Factoring out sec² Θ from the first two terms:

= sec² Θ (cos² Φ + sin² Φ) - tan² Θ

Since the trigonometric identity cos² Φ + sin² Φ = 1 holds true, the expression becomes:

= sec² Θ (1) - tan² Θ

= sec² Θ - tan² Θ

Using the fundamental identity sec² Θ - tan² Θ = 1, the final value is 1.

Explanation

Let x be the number of sums Mohan solved correctly. Since he attempted a total of 30 sums, the number of sums he solved incorrectly is 30 - x.

Based on the marking scheme, he receives 3 marks for every correct answer and loses 2 marks for every wrong answer. The total score can be represented by the equation: 3x - 2(30 - x) = 40

Expanding and simplifying the equation: 3x - 60 + 2x = 40 5x - 60 = 40 5x = 100 x = 20

Mohan solved 20 sums correctly, resulting in 60 marks, while the 10 incorrect sums resulted in a deduction of 20 marks, leaving him with a final score of 40.

Explanation

Applying the componendo and dividendo rule to the equation (sec Θ + tan Θ)/(sec Θ - tan Θ) = ⁵⁄₃ allows for direct simplification:

((sec Θ + tan Θ) + (sec Θ - tan Θ))/((sec Θ + tan Θ) - (sec Θ - tan Θ)) = (5 + 3)/(5 - 3)

(2 sec Θ)/(2 tan Θ) = ⁸⁄₂

(sec Θ)/(tan Θ) = 4

By substituting the trigonometric identities sec Θ = (1)/(cos Θ) and tan Θ = (sin Θ)/(cos Θ), the expression becomes:

(1 / cos Θ)/(sin Θ / cos Θ) = 4

(1)/(sin Θ) = 4

sin Θ = ¼

Explanation

The price after the first discount of 10% on the marked price of 720 is: 720 × (1 - ¹⁰⁄₁₀₀) = 720 × 0.9 = 648

The second discount is applied to this intermediate price of 648 to result in the final price of 550.80. The magnitude of the second discount is: 648 - 550.80 = 97.20

The rate of the second discount is calculated using the formula: Second Discount % = ((Discount Amount)/(Price after 1st discount)) × 100 ((97.20)/(648)) × 100 = ⁹⁷²⁰⁄₆₄₈ = 15%

Explanation

Let the total capacity of the cistern be the least common multiple of 12 and 15, which is 60 units. The filling rate of pipe P is 60 / 12 = 5 units per minute, and the filling rate of pipe Q is 60 / 15 = 4 units per minute. When both pipes are opened together, their combined rate is 5 + 4 = 9 units per minute.

In the first 3 minutes, both pipes fill 9 × 3 = 27 units. The remaining volume to be filled in the cistern is 60 - 27 = 33 units. After pipe P is closed, pipe Q continues to fill the cistern alone at its rate of 4 units per minute. Therefore, the additional time required by pipe Q to fill the remaining portion is 33 / 4 minutes, which simplifies to 8 ¼ minutes.

Explanation

The teacher's age is determined by calculating the difference between the total age of the group including the teacher and the total age once the teacher is excluded. Initially, there are 15 individuals (14 girls and 1 teacher) with an average age of 15 years, giving a total sum of 15 × 15 = 225 years. When the teacher is excluded, the number of individuals becomes 14 and the average age decreases by 1 to 14 years, resulting in a new total sum of 14 × 14 = 196 years. The teacher's age is the difference between these two sums: 225 - 196 = 29 years. This can also be solved using the shortcut formula where the excluded value equals the old average plus the product of the new count and the change in average: 15 + (14 × 1) = 29 years.

Explanation

To find the square root of 33 - 4√(35), we express the expression in the form (x - y)² = x² + y² - 2xy.

Comparing the given expression 33 - 4√(35) with the identity, we equate the irrational part to 2xy: 2xy = 4√(35) xy = 2√(35) = √(4 × 35) = √(140)

We need to find two factors of √(140) whose squares sum to 33. Let the factors be 2√7 and √5. Checking the sum of their squares: (2√7)² + (√5)² = (4 × 7) + 5 = 28 + 5 = 33

Since the sum of the squares matches the constant term 33, we can rewrite the original expression as: 33 - 4√(35) = (2√7)² + (√5)² - 2(2√7)(√5) = (2√7 - √5)²

Taking the square root of (2√7 - √5)² yields ± (2√7 - √5).

Explanation

The relationship between the marked price (MP) and the cost price (CP) when a profit is desired after offering a discount is given by the ratio:

(MP)/(CP) = (100 + Profit%)/(100 - Discount%)

Given that the cost of manufacture (CP) is 900, the desired profit is 25%, and the discount to be offered is 10%, we substitute these values into the formula:

(MP)/(900) = (100 + 25)/(100 - 10)

(MP)/(900) = ¹²⁵⁄₉₀

By simplifying the equation to find the marked price:

MP = ¹²⁵⁄₉₀ × 900

MP = 125 × 10

MP = 1250

By setting the marked price at 1250, the trader ensures that even after a 10% reduction (125), the selling price of 1125 yields a 25% profit on the original cost of 900.

Explanation

The sides of the triangular field are 15 m, 20 m, and 25 m. We can check if this is a right-angled triangle by applying the Pythagorean theorem, where the square of the longest side must equal the sum of the squares of the other two sides. Here, 15² + 20² = 225 + 400 = 625, and 25² = 625. Since 15² + 20² = 25², the field is a right-angled triangle with a base of 15 m and a height of 20 m.

The area of a right-angled triangle is calculated using the formula Area = ½ × base × height. Substituting the values, we get Area = ½ × 15 × 20 = 15 × 10 = 150 sq.m.

The cost of sowing seeds is given as 5 rupees per sq.m. Therefore, the total cost is Total Cost = Area × Rate = 150 × 5 = 750 rupees.

Explanation

The given equation is a² + 1 = a, which can be rearranged as a² - a + 1 = 0. This expression is a factor in the algebraic identity for the sum of cubes: (a + 1)(a² - a + 1) = a³ + 1. Since a² - a + 1 = 0, it follows that a³ + 1 = 0, which gives a³ = -1.

The expression to be evaluated is a¹² + a⁶ + 1. We can rewrite the higher powers of a in terms of a³: a¹² = (a³)⁴ = (-1)⁴ = 1 a⁶ = (a³)² = (-1)² = 1

Substituting these values into the expression, we get: 1 + 1 + 1 = 3

By identifying the relationship between the quadratic equation and the cubic identity, we determine that any power of a that is a multiple of 6 will equal 1, while any odd multiple of 3 will equal -1. This cyclical property allows for the direct substitution and calculation of the final value.

Explanation

If a person pays ¹⁄ₙ of what others are paying, their share of the total cost T is given by ¹⁄ₙ₊₁ × T. This is because if X = ¹⁄ₙ(T - X), then nX = T - X, which simplifies to (n+1)X = T or X = (T)/(n+1).

Given the total cost T = 60:

A pays ½ of what others pay, so A = ¹⁄₂₊₁ × 60 = ⁶⁰⁄₃ = 20. B pays ⅓ of what others pay, so B = ¹⁄₃₊₁ × 60 = ⁶⁰⁄₄ = 15. C pays ¼ of what others pay, so C = ¹⁄₄₊₁ × 60 = ⁶⁰⁄₅ = 12.

The total amount paid by A, B, and C is 20 + 15 + 12 = 47.

The amount paid by D is the remaining balance of the total gift price: D = 60 - (A + B + C) = 60 - 47 = 13.

Explanation

The given binary operation is defined by the formula ab = a + b + (a)/(b). To find the value of 124, we substitute the values a = 12 and b = 4 into this expression.

Substituting the values, we get: 12^*4 = 12 + 4 + ¹²⁄₄

Calculating the terms: 12 + 4 = 16 ¹²⁄₄ = 3

Adding these results together: 16 + 3 = 19

Explanation

In any triangle ABC, let BC be the base and AM be the median drawn to this base. Since M is the midpoint of BC, we have BM = MC = ½ BC. The problem states that the length of the median is half the length of the base, so AM = ½ BC. Consequently, AM = BM = MC.

This equality implies that the point M is equidistant from the three vertices A, B, and C. In geometry, the point equidistant from all vertices of a triangle is the circumcenter. When the circumcenter of a triangle lies on one of its sides (specifically, at the midpoint of the hypotenuse), that triangle is a right-angled triangle. This is a direct consequence of Thales's Theorem, which states that an angle inscribed in a semicircle is a right angle. Here, BC serves as the diameter of a circle centered at M, and since A lies on this circle, ∠ BAC must be 90°.

Alternatively, consider the angles in the two isosceles triangles formed by the median. In △ ABM, since AM = BM, let ∠ MAB = ∠ MBA = α. In △ ACM, since AM = MC, let ∠ MAC = ∠ MCA = β. The sum of the interior angles of △ ABC is ∠ ABC + ∠ BCA + ∠ BAC = 180°. Substituting the angle variables gives α + β + (α + β) = 180°, which simplifies to 2(α + β) = 180°. This results in α + β = 90°. Since ∠ BAC = α + β, the triangle contains a right angle at vertex A.

Explanation

For any three line segments to form a triangle, they must satisfy the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be strictly greater than the length of the third side: a + b > c. Given the four segments of 2 cm, 3 cm, 5 cm, and 6 cm, we can test the possible combinations of three segments by checking if the sum of the two shortest segments exceeds the longest segment in that set.

For the combination (2, 3, 5), the sum 2 + 3 = 5, which is not greater than 5. For the combination (2, 3, 6), the sum 2 + 3 = 5, which is less than 6. For the combination (2, 5, 6), the sum 2 + 5 = 7, which is greater than 6, satisfying the condition. For the combination (3, 5, 6), the sum 3 + 5 = 8, which is greater than 6, also satisfying the condition. Consequently, exactly two sets of segments can form a triangle.

Explanation

The given equation x² + y² + z² = 2(x - y - z) - 3 can be rearranged by bringing all terms to the left side of the equation: x² - 2x + y² + 2y + z² + 2z + 3 = 0. By partitioning the constant 3 into 1 + 1 + 1, the expression can be grouped into perfect squares for each variable: (x² - 2x + 1) + (y² + 2y + 1) + (z² + 2z + 1) = 0. This simplifies to the sum of squares: (x - 1)² + (y + 1)² + (z + 1)² = 0.

For real numbers x, y, and z, the sum of squared terms is zero if and only if each individual term is equal to zero. Therefore, we set x - 1 = 0, y + 1 = 0, and z + 1 = 0, which gives the values x = 1, y = -1, and z = -1. Substituting these specific values into the required expression 2x - 3y + 4z yields 2(1) - 3(-1) + 4(-1). Calculating the arithmetic results in 2 + 3 - 4, which equals 1.

Explanation

Let the speed of the swimmer in still water be u and the speed of the current be v.

When the swimmer moves against the current (upstream) for 5 minutes, the distance covered from point A to an intermediate point C is: AC = (u - v) × 5

When the swimmer moves in favor of the current (downstream) for the next 5 minutes from point C to point B, the distance covered is: CB = (u + v) × 5

The distance AB represents the net displacement from the starting point. Since the downstream speed is greater than the upstream speed, point B lies downstream from point A. The relationship between the distances is: AB = CB - AC 100 = 5(u + v) - 5(u - v) 100 = 5u + 5v - 5u + 5v 100 = 10v v = 10 meters/minute

To convert the speed from meters per minute to kilometers per hour: v = (10 × 60)/(1000) = ⁶⁰⁰⁄₁₀₀₀ = 0.6 km/h

Explanation

Let O₁ and O₂ be the centres of the two circles and M be the intersection point of the line segment O₁O₂ and the common chord AB. In geometry, the line joining the centres of two intersecting circles is the perpendicular bisector of their common chord. Thus, AM = MB = (AB)/(2) = ⁸⁄₂ = 4 cm and ∠ O₁MA = 90°.

In the right-angled triangle O₁MA, the side O₁A is the radius of the circle, given as 5 cm. Using the Pythagorean theorem: O₁M = √(O₁₎_A_²₋_A_M_² O₁M = √(5² - 4²) = √(25 - 16) = √9 = 3 cm.

Since both circles have the same radius, the distance from the midpoint M to each centre is the same, meaning O₁M = MO₂ = 3 cm. The total distance between the centres O₁O₂ is the sum of these two segments: O₁O₂ = O₁M + MO₂ = 3 + 3 = 6 cm.

Explanation

Let the side of rhombus ABCD be a. Since E, A, B, F are collinear and EA = AB = BF, we have EA = AB = BF = BC = CD = DA = a.

Let ∠ DAB = θ. In △ EAD, because EA = AD = a, the triangle is isosceles. The exterior angle ∠ DAB = θ is equal to the sum of the interior opposite angles ∠ AED + ∠ ADE. Since these angles are equal, ∠ AED = θ/2.

In △ BCF, the exterior angle ∠ CBF is equal to ∠ DAB (corresponding angles as AD ∥ BC), so ∠ CBF = θ. Since BC = BF = a, △ BCF is also isosceles. The sum of its interior angles is 180°, so 2 ∠ BFC + ∠ CBF = 180°. This gives ∠ BFC = (180° - θ)/2 = 90° - θ/2.

Consider the triangle formed by the intersection of ED and CF with the base EF. Let the intersection point be P. In △ PEF, the sum of angles must be 180°. We have ∠ PEF = ∠ AED = θ/2 and ∠ PFE = ∠ BFC = 90° - θ/2.

The angle at the intersection is ∠ EPF = 180° - (∠ PEF + ∠ PFE) = 180° - (θ/2 + 90° - θ/2) = 180° - 90° = 90°.

Therefore, ED is perpendicular to CF, which is expressed as ED ⊥ CF.

Explanation

In a right-angled triangle, the circumradius R is half the hypotenuse c, expressed as R = ᶜ⁄₂. Given R = 15 cm, the hypotenuse c = 2 × 15 = 30 cm. The inradius r of a right-angled triangle is given by the formula r = (a + b - c)/(2), where a and b are the other two sides. Substituting the given values, 6 = (a + b - 30)/(2), which simplifies to a + b = 42 cm.

Using the Pythagorean theorem, a² + b² = c² = 30² = 900. We can find the sides by solving the system a + b = 42 and a² + b² = 900. Substituting b = 42 - a into the second equation gives a² + (42 - a)² = 900, which simplifies to 2a² - 84a + 1764 = 900, or a² - 42a + 432 = 0. Solving this quadratic equation using the factors 18 × 24 = 432 and 18 + 24 = 42, we find a = 18 and b = 24. The sides of the triangle are therefore 18 cm, 24 cm, and 30 cm.

Explanation

The number of trees planted along one side of a straight road is determined by dividing the total length of the road by the distance between the trees and adding one for the tree at the starting point. Using the formula N = (L)/(d) + 1, where L is the length of the road and d is the distance between trees, we get ¹⁷⁶⁰⁄₂₀ + 1 = 88 + 1 = 89 trees for one side. Since the trees are planted on both sides of the road, the total number of trees is 89 × 2 = 178.

Explanation

Let the sides of the triangle be 4k, 5k, and 6k. The semi-perimeter s is given by s = (4k + 5k + 6k)/(2) = ¹⁵ᵏ⁄₂ = 7.5k. The area of a triangle Δ can be expressed in terms of its inradius r and semi-perimeter s using the formula Δ = r · s. Substituting the given inradius r = 3 cm, the area is Δ = 3 · 7.5k = 22.5k.

The area of a triangle is also calculated as Δ = ½ · base · altitude. For the altitude h corresponding to the largest side (6k) as the base, the area is Δ = ½ · 6k · h = 3kh.

Equating the two expressions for the area gives 3kh = 22.5k. Since k represents a side length ratio factor and cannot be zero, we divide both sides by 3k to solve for h. This yields h = (22.5)/(3) = 7.5 cm.

Explanation

In the given configuration, both circles have radius a and pass through each other's centers O and O'. Therefore, the distance between the centers OO' = a. Since the points A and B are the intersection points of the two circles, they are at a distance a from both centers O and O'. Consequently, OA = O'A = OB = O'B = a.

The quadrilateral AOBO' is composed of two equilateral triangles, △ AOO' and △ BOO', which share the common side OO'. All sides of these triangles (OA, O'A, OO', OB, O'B) are equal to a. The area of an equilateral triangle with side length a is given by the formula √3{4}a². Since the quadrilateral consists of two such triangles, its total area is 2 × √3{4}a² = √3{2}a².

Explanation

The arithmetic mean of a combined group is calculated as the weighted sum of the means of its constituent parts. Let x be the mean score of the remaining 55 percent of the students. Considering the total group as 100 percent, the total sum of scores can be represented by the following equation:

100 × 52 = (20 × 80) + (25 × 31) + (55 × x)

5200 = 1600 + 775 + 55x

5200 = 2375 + 55x

55x = 5200 - 2375

55x = 2825

x = ²⁸²⁵⁄₅₅ ≈ 51.3636

The calculation yields a value of approximately 51.4 percent when rounded to one decimal place.

Explanation

The sum of the angles in a triangle is 180°. Given the ratio ∠ A: ∠ B: ∠ C = 2: 3: 4, we can represent the angles as 2x, 3x, and 4x. Summing them gives 2x + 3x + 4x = 180°, which simplifies to 9x = 180°, resulting in x = 20°. Therefore, ∠ A = 2 × 20° = 40°.

Since the line CD is drawn parallel to AB, the side AC acts as a transversal intersecting these parallel lines. According to the properties of parallel lines, the alternate interior angles are equal. Thus, ∠ ACD = ∠ BAC = ∠ A. Substituting the value of ∠ A, we find that ∠ ACD = 40°.

Explanation

The curved surface area (CSA) of a cylinder is 2π rh and the total surface area (TSA) is 2π r(h+r). Given the ratio CSA : TSA = 1:2, we have:

(2π rh)/(2π r(h+r)) = ½ ʰ⁄ₕ₊ᵣ = ½ 2h = h+r ⇒ h = r

The total surface area is given as 616 cm². Substituting h=r into the TSA formula:

2π r(r+r) = 4π r² = 616 4 × ²²⁄₇ × r² = 616 r² = (616 × 7)/(88) = 7 × 7 = 49 r = 7 cm

Since h = r, the height h is also 7 cm. The volume V of the cylinder is calculated as:

V = π r² h = ²²⁄₇ × 7² × 7 V = 22 × 7 × 7 = 1078 cm³

Explanation

Let the radius of the circular ground be r meters. The distance covered along the boundary is the circumference C = 2π r, and the distance covered along the diameter is D = 2r. Using the formula Time = Distance / Speed, with speed v = 30 m/minute and a time difference of 30 seconds, which is 0.5 minutes, the relationship is expressed as:

(2π r)/(30) - ²ʳ⁄₃₀ = 0.5

Factoring out common terms from the left side of the equation:

²ʳ⁄₃₀ (π - 1) = 0.5

Substituting π = 22/7:

ʳ⁄₁₅ (²²⁄₇ - 1) = 0.5

ʳ⁄₁₅ × ¹⁵⁄₇ = 0.5

ʳ⁄₇ = 0.5

r = 7 × 0.5 = 3.5

The radius of the circular ground is 3.5 meters.

Explanation

Given the condition a + b + c = 0, we can substitute parts of this identity into the expression. The expression is interpreted as the product of two sums: ((a + b)/c + (b + c)/a + (c + a)/b) and (a/(b + c) + b/(c + a) + c/(a + b)).

From a + b + c = 0, we have: a + b = -c ⟹ (a + b)/c = (-c)/c = -1 b + c = -a ⟹ (b + c)/a = -a/a = -1 c + a = -b ⟹ (c + a)/b = (-b)/b = -1

The first bracket evaluates to: (-1) + (-1) + (-1) = -3

Using the same identities for the denominators in the second bracket: a/(b + c) = a/(-a) = -1 b/(c + a) = b/(-b) = -1 c/(a + b) = c/(-c) = -1

The second bracket evaluates to: (-1) + (-1) + (-1) = -3

The value of the total expression is the product of the two results: (-3) × (-3) = 9

Explanation

Let the total sum of money be S. The daily wage of A is S/21 and the daily wage of B is S/28. When both are paid from the same sum, their combined daily wage is the sum of their individual daily wages.

Combined daily wage = (S)/(21) + (S)/(28)

To find the number of days D the money will last for both, we divide the total sum by the combined daily wage:

D = (S)/(S){21 + (S)/(28)} = (1)/(1){21 + ¹⁄₂₈}

Taking the least common multiple of 21 and 28, which is 84, the expression becomes:

D = (1)/(4 + 3){84} = (1)/(7){84} = ⁸⁴⁄₇ = 12 days

Explanation

Let the labelled price of the wrist watch be L = 100. Arvind purchased the watch at a 30% discount, so his purchase price or cost price C is calculated as C = 100 - (0.30 × 100) = 70. He then sold the watch at a profit of 40% based on his cost price. The selling price S is therefore S = 70 + (0.40 × 70) = 70 + 28 = 98. The loss amount with respect to the original labelled price is L - S = 100 - 98 = 2. The percentage loss on the labelled price is determined by the formula (Loss)/(Labelled Price) × 100. Substituting the assumed values, we have ²⁄₁₀₀ × 100 = 2%.