SSC CGL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

To find the greatest number that divides 19, 35, and 59 leaving the same remainder in each case, calculate the Highest Common Factor (HCF) of the absolute differences between the numbers. The required number is HCF(|35 - 19|, |59 - 35|, |59 - 19|).

Calculating the differences gives 35 - 19 = 16, 59 - 35 = 24, and 59 - 19 = 40. The HCF of 16, 24, and 40 is determined by identifying the largest integer that divides all three. Factoring the numbers, 16 = 8 × 2, 24 = 8 × 3, and 40 = 8 × 5.

The highest common factor among these values is 8. Dividing each of the original numbers by 8 results in the same remainder of 3, as 19 = (8 × 2) + 3, 35 = (8 × 4) + 3, and 59 = (8 × 7) + 3.

Explanation

Let M, T, W, and Th represent the temperatures of Monday, Tuesday, Wednesday, and Thursday respectively. Based on the definition of an average, the sum of the temperatures for the first three days is M + T + W = 3 × 30°C = 90°C. Similarly, the sum of the temperatures for Tuesday, Wednesday, and Thursday is T + W + Th = 3 × 33°C = 99°C.

To find the difference between the temperatures of Thursday and Monday, subtract the first sum from the second: (T + W + Th) - (M + T + W) = 99°C - 90°C Th - M = 9°C

Given that the temperature on Monday was 32°C, substitute this value into the equation: Th - 32°C = 9°C Th = 32°C + 9°C = 41°C

Explanation

Given the values of a and b, we observe that they are reciprocals of each other, meaning ab = 1. To find their sum, we rationalize the denominators: a = √3 - √2{√3 + √2} × √3 - √2{√3 - √2} = (√3 - √2)²{3 - 2} = 3 + 2 - 2√6 = 5 - 2√6 b = √3 + √2{√3 - √2} × √3 + √2{√3 + √2} = (√3 + √2)²{3 - 2} = 3 + 2 + 2√6 = 5 + 2√6

The sum a + b = (5 - 2√6) + (5 + 2√6) = 10.

The expression to evaluate is (a²)/(b) + (b²)/(a). By taking a common denominator, this becomes (a³ + b³)/(ab). Since ab = 1, the expression simplifies to a³ + b³. Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), we substitute the calculated values: (10)³ - 3(1)(10) = 1000 - 30 = 970

Explanation

The series follows the mathematical pattern Tₙ = n³ - 2, where n represents the position of the term in the sequence. We can verify this pattern by substituting the values of n from 1 to 6: For n = 1: 1³ - 2 = 1 - 2 = -1 For n = 2: 2³ - 2 = 8 - 2 = 6 For n = 3: 3³ - 2 = 27 - 2 = 25 For n = 4: 4³ - 2 = 64 - 2 = 62 For n = 5: 5³ - 2 = 125 - 2 = 123 For n = 6: 6³ - 2 = 216 - 2 = 214

To find the next term in the series, we calculate the value for n = 7. Using the same logic, we take the cube of 7 and subtract 2, which gives 7³ - 2 = 343 - 2 = 341.

Explanation

In a triangle, the circumcenter O is the point where the perpendicular bisectors of the sides intersect. Since OD is perpendicular to BC, it must be the perpendicular bisector of side BC. Therefore, D is the midpoint of BC, which gives BD = DC = (BC)/(2) = ²⁴⁄₂ = 12 cm.

The circumradius is the distance from the circumcenter to any vertex of the triangle, so OB = 13 cm. In the right-angled triangle ODB, we apply the Pythagorean theorem:

OB² = OD² + BD²

Substituting the known values:

13² = OD² + 12² 169 = OD² + 144 OD² = 169 - 144 OD² = 25 OD = 5 cm

The length of OD is 5 cm, which corresponds to the Pythagorean triplet (5, 12, 13).

Explanation

To find the value of a² + b², use the standard algebraic identity (ax + by)² + (bx - ay)² = (a² + b²)(x² + y²).

Square both given equations and add them together: (ax + by)² + (bx - ay)² = 6² + 2² a²x² + b²y² + 2abxy + b²x² + a²y² - 2abxy = 36 + 4

The 2abxy and -2abxy terms cancel each other out. Group the remaining terms by factoring a² and b²: a²(x² + y²) + b²(x² + y²) = 40 (a² + b²)(x² + y²) = 40

Substitute the given condition x² + y² = 4 into the equation: (a² + b²)(4) = 40 a² + b² = 40 / 4 = 10

Explanation

The area of a triangle is calculated using the formula A = ½ × base × height. To maximize the area of a triangle inscribed in a semicircle of radius x, we must maximize both the base and the height. The longest possible base for such a triangle is the diameter of the semicircle, which is 2x. The maximum possible height from this base to the perimeter of the semicircle is the radius x, which occurs when the vertex is at the midpoint of the arc. Substituting these values into the formula gives A = ½ × 2x × x = x².

Explanation

By 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 its length. In Δ ABC, since D and E are the midpoints of sides AB and AC, it follows that DE ∥ BC and DE = ½ BC. Let H be the altitude of Δ ABC drawn from vertex A to the base BC. Because DE connects the midpoints, the perpendicular distance from A to DE is H/2, and the perpendicular distance between the parallel lines DE and BC is also H/2.

The area of Δ ABC is given by the formula: Area (Δ ABC) = ½ × BC × H

Triangle PED is formed with base DE and vertex P. Since P lies on the line BC produced, the altitude of Δ PED with respect to the base DE is the constant perpendicular distance between the parallel lines DE and BC, which is H/2. The area of Δ PED is: Area (Δ PED) = ½ × DE × (H)/(2)

Substituting DE = ½ BC into the equation: Area (Δ PED) = ½ × ( ½ BC ) × (H)/(2) = ⅛ × BC × H

Comparing this to the area of the original triangle: Area (Δ PED) = ¼ × ( ½ × BC × H ) = ¼ × Area (Δ ABC)

Explanation

To find the value of a² + (1)/(a²) from the given equation a + ¹⁄ₐ = 1, we apply the algebraic identity (x + y)² = x² + y² + 2xy. Letting x = a and y = ¹⁄ₐ, the identity becomes:

(a + ¹⁄ₐ)² = a² + (1)/(a²) + 2 · a · ¹⁄ₐ (a + ¹⁄ₐ)² = a² + (1)/(a²) + 2

Substituting the given value a + ¹⁄ₐ = 1 into this identity:

1² = a² + (1)/(a²) + 2 1 = a² + (1)/(a²) + 2

By rearranging the equation to solve for the required expression:

a² + (1)/(a²) = 1 - 2 a² + (1)/(a²) = -1

This approach uses the standard quadratic relationship for reciprocal sums to reach the result efficiently.

Explanation

The sum of all 19 observations is calculated as 19 × 24 = 456. The sum of the first 10 observations is 10 × 17 = 170, and the sum of the last 10 observations is 10 × 24 = 240. When we add the sum of the first 10 observations and the last 10 observations, the 10th observation is counted twice because it is the concluding element of the first set and the starting element of the second set.

To find the 10th observation, we use the relation: 10th observation = (Sum of first 10 observations + Sum of last 10 observations) - Sum of 19 observations Substituting the values: 10th observation = (170 + 240) - 456 10th observation = 410 - 456 = -46

Explanation

Let the cost price of the watch be CP. A profit of 30% implies that the initial selling price is 130% of the CP. A loss of 10% implies that the alternative selling price is 90% of the CP. The numerical difference between these two price points is given as Rs. 80.

The percentage difference between a 30% profit and a 10% loss is 30% - (-10%) = 40%. We can establish the relationship: 40% of CP = 80 ⁴⁰⁄₁₀₀ × CP = 80 CP = (80 × 100)/(40) CP = 2 × 100 = 200

Explanation

Let the length of the train be L metres and its speed be V kmph. When a train overtakes persons moving in the same direction, the relative speed is the difference between their speeds. Since distance equals speed multiplied by time, and 1 kmph = ⁵⁄₁₈ m/s, we can set up two equations for the length of the train:

For the first person at 2 kmph: L = (V - 2) × ⁵⁄₁₈ × 9 = ⁵⁄₂(V - 2)

For the second person at 4 kmph: L = (V - 4) × ⁵⁄₁₈ × 10 = ²⁵⁄₉(V - 4)

Equating the two expressions for L: ⁵⁄₂(V - 2) = ²⁵⁄₉(V - 4) 9(V - 2) = 10(V - 4) 9V - 18 = 10V - 40 V = 22 kmph

Substituting V = 22 back into the first equation: L = ⁵⁄₂(22 - 2) = ⁵⁄₂ × 20 = 50 metres

Explanation

The radii of the two circles are r₁ = 15 cm and r₂ = 20 cm, and the distance between their centers is d = 25 cm. These values satisfy the Pythagorean triple 15² + 20² = 225 + 400 = 625 = 25². This confirms that the triangle formed by the centers of the two circles and one of the intersection points of the circles is a right-angled triangle, where the distance between the centers is the hypotenuse.

The common chord is perpendicular to the line joining the centers and is bisected by it. Let h be the distance from an intersection point to the line joining the centers (the altitude of the right triangle to the hypotenuse). The length of the common chord is 2h. The area of the right triangle can be calculated using the legs: Area = ½ × 15 × 20 = 150 cm². The area can also be expressed using the hypotenuse and the altitude: Area = ½ × 25 × h. Equating the two expressions for area gives 150 = ²⁵ʰ⁄₂, which simplifies to 300 = 25h, resulting in h = 12 cm. Therefore, the length of the common chord is 2 × 12 = 24 cm.

Explanation

The relationship between the marked price (MP) and the cost price (CP) is given by the formula MP/CP = (100 + Gain%) / (100 - Commission%). Substituting the initial values where gain is 20% and commission is 10%:

MP/CP = (100 + 20) / (100 - 10) = 120 / 90 = 4 / 3

This implies MP = (4/3) CP. When the commission is increased to 15%, the new selling price (SP) becomes:

SP = MP × (100 - 15) / 100 = (4/3) CP × 85 / 100 SP = (4/3) × (17/20) CP = 17/15 CP

The gain percent is calculated as ((SP - CP) / CP) × 100:

Gain % = (17/15 - 1) × 100 = (2/15) × 100 = 40/3 = 13 1/3%

Explanation

To find the value of √x + 1/√x, we first determine the value of 1/x. Given x = 2 + √3, we rationalize 1/x by multiplying the numerator and denominator by the conjugate 2 - √3:

1/x = 1 / (2 + √3) × (2 - √3) / (2 - √3) = (2 - √3) / (2² - (√3)²) = (2 - √3) / (4 - 3) = 2 - √3

Now, we calculate the sum x + 1/x:

x + 1/x = (2 + √3) + (2 - √3) = 4

The expression we want to find is related to this sum by the algebraic identity (a + b)² = a² + b² + 2ab. Let a = √x and b = 1/√x:

(√x + 1/√x)² = (√x)² + (1/√x)² + 2(√x)(1/√x) (√x + 1/√x)² = x + 1/x + 2

Substituting the value of x + 1/x = 4 into the equation:

(√x + 1/√x)² = 4 + 2 = 6

Taking the square root of both sides gives the final result:

√x + 1/√x = √6

Explanation

The efficiency relationship established by the problem is that 12 men do the same amount of work as 18 women, which implies 1 man is equivalent to 18/12 = 1.5 women. For the second group, the total labor force of 8 men and 16 women can be expressed entirely in terms of women as (8 × 1.5) + 16 = 12 + 16 = 28 women. Using the work-rate formula M₁D₁ = M₂D₂, where M₁ = 18 women and D₁ = 14 days, the calculation for the new duration D₂ with M₂ = 28 women becomes 18 × 14 = 28 × D₂. Solving for D₂, we get D₂ = (18 × 14) / 28 = 252 / 28 = 9 days.

A faster shortcut for such "And/Or" problems is the formula Time = Days / ((m)/(M) + (w)/(W)), where m, w are the "And" values and M, W are the "Or" values. Substituting the given numbers: 14 / (8/12 + 16/18) = 14 / (2/3 + 8/9) = 14 / (6/9 + 8/9) = 14 / (14/9) = 9 days.

Explanation

For a fixed total transaction amount (1 rupee in both cases), the number of articles is inversely proportional to the selling price expressed as a percentage of the cost price (CP).

In the first case, a loss of 4% means the selling price is 100% - 4% = 96% of the CP. In the second case, a desired gain of 44% means the selling price must be 100% + 44% = 144% of the CP.

Let n₁ be the initial number of articles and n₂ be the required number of articles. The inverse relationship is given by the formula: n₁ × Percentage₁ = n₂ × Percentage₂

Substituting the given values: 9 × 96 = n₂ × 144 n₂ = (9 × 96)/(144) n₂ = ⁸⁶⁴⁄₁₄₄ = 6

To realize a profit of 44%, the man must sell 6 articles for a rupee.

Explanation

The algebraic identity for the difference of two cubes is given by the formula a³ - b³ = (a - b)(a² + b² + ab).

Given the values a³ - b³ = 56 and a - b = 2, we can substitute them directly into the identity:

56 = 2 × (a² + b² + ab)

Dividing both sides by 2 yields:

a² + b² + ab = ⁵⁶⁄₂ = 28

Explanation

Given the equations a = tan θ - cot θ and b = cos θ - sin θ, we can simplify each term individually.

First, express a in terms of sine and cosine: a = (sin θ)/(cos θ) - (cos θ)/(sin θ) = (sin² θ - cos² θ)/(sin θ cos θ)

Now, calculate a² + 4: a² + 4 = ( (sin² θ - cos² θ)/(sin θ cos θ) )² + 4 = ((sin² θ - cos² θ)² + 4 sin² θ cos² θ)/(sin² θ cos² θ)

Using the algebraic identity (x - y)² + 4xy = (x + y)², where x = sin² θ and y = cos² θ: a² + 4 = ((sin² θ + cos² θ)²)/(sin² θ cos² θ) = (1)/(sin² θ cos² θ)

Next, simplify the term involving b: b = cos θ - sin θ b² = (cos θ - sin θ)² = cos² θ + sin² θ - 2 sin θ cos θ = 1 - 2 sin θ cos θ b² - 1 = - 2 sin θ cos θ (b² - 1)² = (- 2 sin θ cos θ)² = 4 sin² θ cos² θ

Finally, multiply the two results: (a² + 4)(b² - 1)² = ( (1)/(sin² θ cos² θ) ) · (4 sin² θ cos² θ) = 4

Explanation

The region bounded by the x-axis, y-axis, and the two parallel lines 3x + 4y = 12 and 6x + 8y = 60 forms a trapezium in the first quadrant. This area can be calculated by finding the difference between the areas of the two right-angled triangles formed by these lines with the coordinate axes.

For the first line 3x + 4y = 12, the x-intercept is found by setting y = 0, giving 3x = 12 or x = 4. The y-intercept is found by setting x = 0, giving 4y = 12 or y = 3. The area of the smaller triangle is A₁ = ½ × base × height = ½ × 4 × 3 = 6 square units.

For the second line 6x + 8y = 60, which simplifies to 3x + 4y = 30, the x-intercept is found by setting y = 0, giving 3x = 30 or x = 10. The y-intercept is found by setting x = 0, giving 4y = 30 or y = 7.5. The area of the larger triangle is A₂ = ½ × 10 × 7.5 = 37.5 square units.

The area of the trapezium is the difference between these two areas: A₂ - A₁ = 37.5 - 6 = 31.5 square units.

Explanation

The area of a triangle formed by the line ax + by + c = 0 and the coordinate axes is given by the formula (c²)/(2|ab|). For the line 2x - 3y + 6 = 0, the coefficients are a = 2, b = -3, and c = 6. Substituting these values into the formula:

Area = (6²)/(2 × |2 × -3|) = (36)/(2 × 6) = ³⁶⁄₁₂ = 3 sq. units.

Alternatively, finding the intercepts of the line provides the base and height of the right-angled triangle. Setting y = 0 gives the x-intercept 2x = -6, so x = -3. Setting x = 0 gives the y-intercept -3y = -6, so y = 2. The lengths of the base and height are the absolute values of these intercepts, which are 3 and 2 units respectively. The area is ½ × base × height = ½ × 3 × 2 = 3 sq. units.

Explanation

The given equation (a² - b²)sin Θ + 2abcos Θ = a² + b² can be compared to the fundamental trigonometric identity by dividing the entire equation by the term on the right-hand side, a² + b². This yields:

(a² - b²)/(a² + b²) sin Θ + (2ab)/(a² + b²) cos Θ = 1

We know that for any angle Θ, the identity sin² Θ + cos² Θ = 1 holds true. By comparing the coefficients, we can identify:

sin Θ = (a² - b²)/(a² + b²) cos Θ = (2ab)/(a² + b²)

This identification is valid because the sum of the squares of these coefficients equals 1, satisfying the Pythagorean triplet condition: (a² - b²)² + (2ab)² = a⁴ - 2a²b² + b⁴ + 4a²b² = a⁴ + 2a²b² + b⁴ = (a² + b²)².

To find the value of tan Θ, we use the ratio of sin Θ to cos Θ:

tan Θ = (sin Θ)/(cos Θ) = ((a² - b²) / (a² + b²))/((2ab) / (a² + b²))

The common denominator a² + b² cancels out, resulting in:

tan Θ = (a² - b²)/(2ab)

Explanation

The profit earned by Prabhat is the difference between the interest he receives from Ashish and the interest he pays to the bank. Since the principal amount P and the time period T are identical for both transactions, the net profit is determined by the difference in the annual interest rates.

Let the interest rate paid to the bank be R₁ = 8% and the interest rate received from Ashish be R₂ = 12%. The total profit over T = 12 years is given by the formula: Profit = (P × T × (R₂ - R₁))/(100)

Substituting the known values into the equation: 960 = (P × 12 × (12 - 8))/(100) 960 = (P × 12 × 4)/(100) 960 = (48P)/(100)

To find the principal amount P, rearrange the equation: P = (960 × 100)/(48) P = 20 × 100 P = 2000

Explanation

Let r be the radius of the circle. Since the chord CD is equal to the radius, OC = OD = CD = r. This makes △ OCD an equilateral triangle, so the angle subtended by the chord CD at the center is ∠ COD = 60°. The angle subtended by a chord at any point on the circumference is half the angle subtended at the center. Therefore, ∠ CBD = ½ ∠ COD = (60°)/(2) = 30°.

Since AB is the diameter, the angle in the semicircle is ∠ ACB = 90°. When AC is produced to P, the angle ∠ BCP is supplementary to ∠ ACB, so ∠ BCP = 180° - 90° = 90°. Now consider △ BCP. The sum of the interior angles of a triangle is 180°. We have ∠ BCP = 90° and ∠ CBP is the same as ∠ CBD, which is 30°. Thus, ∠ CPB = 180° - (90° + 30°) = 60°. Because P is the intersection of the produced lines AC and BD, ∠ APB is the same as ∠ CPB, giving a measure of 60°.

Explanation

The Power of a Point theorem states that for a point P located outside a circle with center O and radius r, if a secant line from P intersects the circle at points A and B, the product PA × PB is equal to OP² - r².

From the given data, PA = 9 cm and AB = 7 cm. The total length from point P to the second intersection point B is PB = PA + AB = 9 + 7 = 16 cm. The distance from the center is OP = 13 cm.

Applying the values to the formula: PA × PB = OP² - r² 9 × 16 = 13² - r² 144 = 169 - r² r² = 169 - 144 r² = 25 r = √(25) = 5 cm

This calculation determines the radius by equating the power of the point relative to the circle's geometry with the distance from the center.

Explanation

Given the equation a + ¹⁄ₐ₊₂ = 0, add 2 to both sides to make the terms consistent: (a + 2) + ¹⁄ₐ₊₂ = 2

Let x = a + 2. The equation becomes x + ¹⁄ₓ = 2. For any real number x, the sum of x and its reciprocal equals 2 if and only if x = 1. This can be shown by solving the quadratic equation x² - 2x + 1 = 0, which is (x - 1)² = 0, giving x = 1.

The expression to be evaluated is (a + 2)² + (1)/((a + 2)³). Substituting x = a + 2 into this expression, we get x² + (1)/(x³). Since x = 1, the value is: 1² + (1)/(1³) = 1 + 1 = 2

Explanation

Using the trigonometric identity cos² α = 1 - sin² α, the equation 2 sin α + 15 cos² α = 7 can be expressed as a quadratic equation in terms of sin α:

2 sin α + 15(1 - sin² α) = 7 2 sin α + 15 - 15 sin² α = 7 15 sin² α - 2 sin α - 8 = 0

Solving the quadratic equation by splitting the middle term: 15 sin² α - 12 sin α + 10 sin α - 8 = 0 3 sin α (5 sin α - 4) + 2 (5 sin α - 4) = 0 (3 sin α + 2)(5 sin α - 4) = 0

From this, we get sin α = 4/5 or sin α = -2/3. Given that α is a positive acute angle (0 < α < π/2), the value of sin α must be positive. Therefore, sin α = 4/5.

In a right-angled triangle with sin α = 4/5, the opposite side is 4 and the hypotenuse is 5. Using the Pythagorean theorem, the adjacent side is √(5² - 4²) = 3. The value of cot α is defined as the ratio of the adjacent side to the opposite side:

cot α = (Adjacent)/(Opposite) = ¾

Explanation

Let x = 876543. The expression can be simplified using the algebraic identity (a - 1)(a + 1) = a² - 1. By substituting x into the expression, we can represent the terms as follows:

Numerator: 1 + 876542 × 876544 = 1 + (x - 1)(x + 1) Applying the identity: 1 + (x² - 1) = x²

Denominator: 876543 × 876543 = x²

The expression then becomes:

(x²)/(x²) = 1

This substitution method resolves the calculation by reducing the large numerical values to a simple algebraic ratio.

Explanation

For similar triangles, the ratio of their perimeters is equal to the ratio of any two corresponding sides. In the triangles Δ ABC and Δ PQR, the relationship is defined by the following formula:

(Perimeter of Δ ABC)/(Perimeter of Δ PQR) = (AB)/(PQ)

Given that the perimeter of Δ ABC is 36 cm and the perimeter of the second triangle (Δ PQR) is 24 cm, and the length of the side PQ is 10 cm, we substitute these values into the equation:

³⁶⁄₂₄ = (AB)/(10)

Reducing the fraction ³⁶⁄₂₄ by dividing both the numerator and the denominator by their greatest common divisor, 12, we get:

³⁄₂ = (AB)/(10)

To solve for AB, cross-multiply:

2 × AB = 3 × 10

2 × AB = 30

AB = ³⁰⁄₂ = 15 cm

Explanation

The given sequence represents the squares of the first ten natural numbers, 1², 2², 3², 4², 5², 6², 7², 8², 9², 10². The sum of the squares of the first n natural numbers is calculated using the formula Sₙ = ⁿ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆. To find the average, we divide the sum by the total number of terms n, resulting in the formula:

Average = ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆

Substituting n = 10 into this formula:

Average = ((10 + 1)(2 × 10 + 1))/(6) Average = (11 × 21)/(6) Average = (11 × 7)/(2) Average = ⁷⁷⁄₂ = 38.5

Alternatively, calculating the sum directly: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385. Dividing the sum by the number of observations, 385 / 10, gives the average of 38.5.

Explanation

The intersection point of the vertical line x = 3 and the horizontal line y = 4 is precisely the point (3, 4). The distance d between the origin (0, 0) and any point (x, y) in the Cartesian plane is given by the formula d = √(x² + y²). Substituting the coordinates of the intersection point into the formula, we calculate d = √(3² + 4²) = √(9 + 16) = √(25). This yields a distance of 5 units, which also represents the hypotenuse of a standard 3-4-5 Pythagorean triple.

Explanation

The number N can be expressed in the form N = 899k + 65, where k is the quotient. When we divide this number by 31, we look at the divisibility of each term. Since 899 = 31 × 29, the term 899k is perfectly divisible by 31 and leaves no remainder. Consequently, the remainder when N is divided by 31 is the same as the remainder when 65 is divided by 31. Performing the division 65 = (31 × 2) + 3, we find that the remainder is 3.

Explanation

In any triangle, the three medians divide the triangle into six smaller triangles of equal area. Since AD, BE, and CF are the medians and G is the centroid, the six triangles formed are △ AFG, △ BFG, △ BDG, △ CDG, △ CEG, and △ AEG.

The area of each of these smaller triangles is exactly 1/6 of the area of the original triangle ABC. Therefore, the area of △ BDG is calculated as:

Area of △ BDG = ⅙ × Area of △ ABC

Area of △ BDG = ⅙ × 72 = 12 sq cm

Explanation

Using the standard algebraic identity (a + b)² + (a - b)² = 2(a² + b²), we can find the required value directly by substituting the given variables. Given a + b = 6 and a - b = 2, the expression becomes 6² + 2² = 2(a² + b²). Evaluating the squares gives 36 + 4 = 2(a² + b²), which results in 40 = 2(a² + b²).

Alternatively, by solving the system of linear equations, adding a + b = 6 and a - b = 2 yields 2a = 8, hence a = 4. Subtracting the second equation from the first yields 2b = 4, hence b = 2. Substituting these specific values into the target expression, we calculate 2(4² + 2²) = 2(16 + 4) = 2(20) = 40.

Explanation

The sequence follows a pattern where the difference between consecutive terms increases by 2, creating a progression of consecutive even numbers. Calculating the differences between the terms provided: 7 - 3 = 4 13 - 7 = 6 21 - 13 = 8 Following this arithmetic progression of differences, the next difference should be 10. Adding 10 to the fourth term results in 21 + 10 = 31. To verify the rest of the sequence, adding 12 to 31 gives 31 + 12 = 43, and adding 14 to 43 gives 43 + 14 = 57. Both 43 and 57 match the subsequent terms in the provided sequence. Thus, the fifth term 33 is the odd element as it should have been 31. Alternatively, the sequence corresponds to the general formula Tₙ = n² + n + 1. For the first four terms, n = 1 gives 3, n = 2 gives 7, n = 3 gives 13, and n = 4 gives 21. For n = 5, the formula yields 5² + 5 + 1 = 31, confirming that 33 is incorrect.

Explanation

In a circle with center O, let P be an external point from which tangents PA and PB are drawn. The radii OA and OB are perpendicular to the tangents at the points of contact, making ∠ OAP = ∠ OBP = 90°. In the quadrilateral OAPB, the sum of interior angles is 360°, leading to the relation ∠ APB = 180° - ∠ AOB. Given ∠ AOB = 120°, we calculate ∠ APB = 180° - 120° = 60°.

The triangles △ OAP and △ OBP are congruent by the RHS (Right-angle, Hypotenuse, Side) criterion because they share the hypotenuse OP, have equal radii OA = OB, and right angles at A and B. Consequently, OP bisects ∠ APB. This means ∠ APO = ½ ∠ APB = ½ × 60° = 30°. The required ratio ∠ APB : ∠ APO is 60° : 30°, which simplifies to 2 : 1.

Explanation

Let H = 90 m be the height of the cliff and h be the height of the tower. In problems involving angles of depression from the top of a height H to the top (₁) and bottom (₂) of an object, the height h can be determined using the formula: h = H ( 1 - tan ₁{tan ₂} )

Substituting the given values H = 90, ₁ = 30°, and ₂ = 60° into the formula: h = 90 ( 1 - (tan 30°)/(tan 60°) )

Using the standard values tan 30° = 1/√3 and tan 60° = √3: h = 90 ( 1 - 1/√3{√3} ) h = 90 ( 1 - ⅓ ) h = 90 × ⅔ h = 60 m.

Explanation

To find the least positive integer that is a perfect square and divisible by 21, 36, and 66, we first determine the prime factorization of each number: 21 = 3 × 7 36 = 2² × 3² 66 = 2 × 3 × 11

The Least Common Multiple (LCM) is found by taking the highest power of each prime factor present in these numbers: LCM = 2² × 3² × 7¹ × 11¹ = 2772

For a number to be a perfect square, the exponent of every prime factor in its prime factorization must be even. In the LCM calculated above, the prime factors 7 and 11 have odd exponents (7¹ and 11¹). To obtain the smallest perfect square divisible by these numbers, we must multiply the LCM by the missing prime factors to make all exponents even.

Required perfect square = (2² × 3² × 7¹ × 11¹) × (7¹ × 11¹) Required perfect square = 2² × 3² × 7² × 11² Required perfect square = (2 × 3 × 7 × 11)² Required perfect square = (462)² (462)² = 213444

Explanation

The relationship provided can be expressed as a linear equation where 5% of the term √(2x) equals 0.01. We start by setting up the equation:

⁵⁄₁₀₀ · √(2x) = 0.01

To simplify the equation, multiply both sides by 100:

5 · √(2x) = 1

Divide both sides by 5 to isolate the radical:

√(2x) = ⅕

√(2x) = 0.2

To solve for x, square both sides of the equation to remove the square root:

(√(2x))² = (0.2)²

2x = 0.04

Finally, divide by 2 to find the value of x:

x = (0.04)/(2)

x = 0.02

Explanation

The equation is structured such that rearranging the terms leads to a direct solution for x. By grouping the radical terms and the fractional terms, we have: √((x - a)/(x - b)) - √((x - b)/(x - a)) = (b - a)/(x)

Let y = √((x - a)/(x - b)). The equation becomes: y - (1)/(y) = (b - a)/(x) (y² - 1)/(y) = (b - a)/(x)

Substituting y² = (x - a)/(x - b) into the numerator: y² - 1 = (x - a)/(x - b) - 1 = (x - a - (x - b))/(x - b) = (b - a)/(x - b)

Substituting this back into the rearranged equation: (b - a)/(y(x - b)) = (b - a)/(x)

Since b ≠ a, we can divide both sides by (b - a), giving: x = y(x - b)

Squaring both sides to eliminate the radical: x² = y²(x - b)² x² = (x - a)/(x - b) · (x - b)² x² = (x - a)(x - b) x² = x² - ax - bx + ab

Canceling x² from both sides: ax + bx = ab x(a + b) = ab x = (ab)/(a + b)

Explanation

The given expression for x can be simplified as x = 2√(4 × 6){√3 + √2} = 4√6{√3 + √2}. This matches the algebraic form x = (4ab)/(a+b), where a = √3 and b = √2. For any such relation, the value of ˣ⁺²ᵃ⁄ₓ₋₂ₐ + (x+2b)/(x-2b) is always 2.

Substituting the values, we have 2a = 2√3 = √(12) and 2b = 2√2 = √8. The expression becomes x + √8{x - √8} + x + √(12){x - √(12)}.

To solve using the Componendo and Dividendo rule, first rewrite x for the first term: (x)/(√(8)) = (x)/(2√(2)) = 2√3{√3 + √2} Applying Componendo and Dividendo: x + √8{x - √8} = 2√3 + √3 + √2{2√3 - (√3 + √2)} = 3√3 + √2{√3 - √2}

Next, rewrite x for the second term: (x)/(√(12)) = (x)/(2√(3)) = 2√2{√3 + √2} Applying Componendo and Dividendo: x + √(12){x - √(12)} = 2√2 + √3 + √2{2√2 - (√3 + √2)} = 3√2 + √3{√2 - √3}

Adding the two results: 3√3 + √2{√3 - √2} + 3√2 + √3{√2 - √3} = 3√3 + √2 - (3√2 + √3){√3 - √2} = 2√3 - 2√2{√3 - √2} = 2(√3 - √2){√3 - √2} = 2

Explanation

Rationalizing a and b by multiplying the numerator and denominator by the conjugate of the denominator: a = 2+√3{2-√3} × 2+√3{2+√3} = (2+√3)²{4-3} = 7+4√3 b = 2-√3{2+√3} × 2-√3{2-√3} = (2-√3)²{4-3} = 7-4√3

Calculating the sum a+b and product ab: a+b = (7+4√3) + (7-4√3) = 14 ab = 2+√3{2-√3} × 2-√3{2+√3} = 1

The required expression is a² + b² + ab. This can be rewritten using the algebraic identity (a+b)² = a² + b² + 2ab: a² + b² + ab = (a+b)² - ab Substituting the values of a+b and ab: 14² - 1 = 196 - 1 = 195

Explanation

The given equation is sec Θ - csc Θ = 0, which can be rearranged as sec Θ = csc Θ. Converting these into their reciprocal trigonometric functions, we get (1)/(cos Θ) = (1)/(sin Θ). This simplifies to sin Θ = cos Θ, or (sin Θ)/(cos Θ) = 1, which means tan Θ = 1. For an acute angle, this occurs when Θ = 45°.

To find the value of sec Θ + csc Θ, we substitute Θ = 45° into the expression: sec 45° + csc 45° = √2 + √2 = 2√2

Explanation

Let the total work be the Least Common Multiple (LCM) of 20 and 15, which is 60 units. Based on this, the efficiency of A is 60 / 20 = 3 units per day, and the efficiency of B is 60 / 15 = 4 units per day.

During the first 6 days, A and B work together and complete 6 × (3 + 4) = 42 units of work. The remaining work is 60 - 42 = 18 units. This remaining work is finished by A and C in the next 4 days. In these 4 days, A completes 4 × 3 = 12 units of work. This implies that C completes the remaining 18 - 12 = 6 units of work in those 4 days.

The efficiency of C is calculated as 6 units / 4 days = 1.5 units per day. The number of days C alone would take to complete the total work of 60 units is 60 / 1.5 = 40 days.

Explanation

The given equation x = 2√6{√3 + √2} can be written in the form x = (2ab)/(a+b) by letting a = √2 and b = √3, which satisfies ab = √6. For any variable x defined as the harmonic mean of a and b in the form x = (2ab)/(a+b), the value of the expression ˣ⁺ᵃ⁄ₓ₋ₐ + (x+b)/(x-b) is always 2.

This can be derived using the Componendo and Dividendo rule. First, rewrite the equation as ˣ⁄ₐ = (2b)/(a+b). Applying the rule gives: ˣ⁺ᵃ⁄ₓ₋ₐ = (2b + (a+b))/(2b - (a+b)) = (3b+a)/(b-a)

Next, rewrite the original equation as (x)/(b) = (2a)/(a+b). Applying the rule again gives: (x+b)/(x-b) = (2a + (a+b))/(2a - (a+b)) = (3a+b)/(a-b)

Adding the two components: (3b+a)/(b-a) + (3a+b)/(a-b) = (3b+a)/(b-a) - (3a+b)/(b-a) = (3b + a - 3a - b)/(b-a) = (2b-2a)/(b-a) = (2(b-a))/(b-a) = 2

By substituting a = √2 and b = √3 into this identity, the value is determined to be 2.

Explanation

Let the length of side AB of rectangle ABCD be L and the width BC be W. The area of rectangle ABCD is given by L × W. Since the side AB is divided into four equal parts by points X, Y, and Z, the length of the segment XY is equal to one-fourth of the total length AB, which is (L)/(4).

The triangle XYC has its base XY on side AB and its third vertex C on the opposite side CD. The height of triangle XYC with respect to base XY is the perpendicular distance from vertex C to the side AB, which is equal to the width BC of the rectangle, or W.

The area of triangle XYC is calculated using the formula ½ × base × height: Area of △ XYC = ½ × XY × BC = ½ × (L)/(4) × W = (LW)/(8)

The ratio of the area of triangle XYC to the area of rectangle ABCD is: (Area of △ XYC)/(Area of ABCD) = (LW)/(8){LW} = ⅛

Explanation

According to the Alternate Segment Theorem, the angle subtended by the chord in the alternate segment of the circle is equal to the angle between the tangent and the chord, which is 45°. The angle subtended by a chord at the center of a circle is twice the angle subtended by it at any point on the remaining part of the circumference. Therefore, the central angle subtended by this chord is 2 × 45° = 90°.

In the triangle formed by the center of the circle and the endpoints of the chord, the two sides are radii r, and the included angle is 90°. Using the properties of a right-angled triangle or the chord length formula c = 2r sin(θ) where θ is the angle between the chord and the tangent:

6 = 2r sin 45° 6 = 2r × (1)/(√(2)) 6 = r√2 r = (6)/(√(2)) r = 3√2 cm

Explanation

Let C be the cost price of the chair and T be the cost price of the table. We are given that C + T = 500. The net gain on the total transaction is Rs. 10, which can be expressed as an overall gain percentage: Overall gain % = (10 / 500) × 100 = 2%

Using the rule of alligation to find the ratio of the cost prices of the table and the chair: Loss on table = -10% Gain on chair = +10% Mean gain = +2%

The ratio of the cost price of the table to the cost price of the chair is calculated by the numerical difference between these values and the mean: CP_Tₐ_bₗₑ : CP_Cₕₐᵢᵣ = (10 - 2) : (2 - (-10)) CP_Tₐ_bₗₑ : CP_Cₕₐᵢᵣ = 8 : 12 = 2 : 3

The total value of the ratios is 2 + 3 = 5 parts, which equals the total cost price of Rs. 500. This means each part represents 500 / 5 = 100. The cost price of the chair corresponds to 3 parts: CP_Cₕₐᵢᵣ = 3 × 100 = 300

The cost price of the chair is Rs. 300.

Explanation

The three medians AD, BE, and CF of triangle ABC intersect at the centroid G. A fundamental property of the centroid is that it divides the triangle into six smaller triangles of equal area: △ GAF, △ GBF, △ GBD, △ GCD, △ GCE, and △ GAE.

Given that the area of △ ABC is 60 sq.cm, the area of each of these six small triangles is: Area = ⅙ × Area(△ ABC) = ⁶⁰⁄₆ = 10 sq.cm

The quadrilateral CDGE (often the intended figure in such problems where the area is one-third of the total) is formed by the union of two of these small triangles, △ GCD and △ GCE. Similarly, the quadrilateral BDGF is formed by △ GBD and △ GBF. The area of such a quadrilateral is: Area = 10 + 10 = 20 sq.cm

In any triangle, the quadrilateral formed by the centroid, the midpoints of two sides, and the vertex common to those sides (or the equivalent region comprising two small median-bounded triangles) always has an area equal to exactly one-third of the total area of the triangle. Area = ⅓ × 60 = 20 sq.cm

Explanation

When a sum of money P triples in value, the final amount A becomes 3P. The total Simple Interest (SI) earned over the period is the difference between the amount and the principal: SI = A - P = 3P - P = 2P.

The standard formula for Simple Interest is SI = (P × R × T)/(100), where R is the annual rate of interest and T is the time in years. Substituting the given values SI = 2P and T = 25:

2P = (P × R × 25)/(100)

By dividing both sides by P and simplifying the equation:

2 = (25R)/(100) 2 = (R)/(4) R = 2 × 4 = 8

Alternatively, for any sum to become n times itself in T years, the rate of simple interest R can be found using the direct formula R = (100(n - 1))/(T). Substituting n = 3 and T = 25:

R = (100(3 - 1))/(25) R = (100 × 2)/(25) R = 4 × 2 = 8%