SSC CGL Tier 1 2019 Quantitative Aptitude Questions with Solutions

Explanation

To solve the equation 4 - 2 sin² θ - 5 cos θ = 0, we substitute sin² θ with 1 - cos² θ to obtain a quadratic equation in terms of cos θ:

4 - 2(1 - cos² θ) - 5 cos θ = 0 4 - 2 + 2 cos² θ - 5 cos θ = 0 2 cos² θ - 5 cos θ + 2 = 0

We factorize the quadratic equation 2x² - 5x + 2 = 0 (where x = cos θ): 2x² - 4x - x + 2 = 0 2x(x - 2) - 1(x - 2) = 0 (2x - 1)(x - 2) = 0

The solutions are x = 2 or x = 1/2. Since the cosine of an angle cannot exceed 1, we discard cos θ = 2. Thus, cos θ = 1/2. Given the condition 0° < θ < 90°, we find θ = 60°.

Finally, we calculate the required value: sin 60° + tan 60° = √3{2} + √3 = √3 + 2√3{2} = 3√3{2}

Explanation

The expression is based on the algebraic identity a³ - b³ = (a - b)(a² + b² + ab). By rewriting the terms in the numerator, we get: 5√5x³ = (√5x)³ 81√3y³ = (3√3y)³

Dividing this by (√5x - 3√3y) yields the quotient: (√5x)² + (3√3y)² + (√5x)(3√3y) = 5x² + 27y² + 3√(15)xy

By comparing this result with the given form Ax² + By² + Cxy, we identify the coefficients: A = 5 B = 27 C = 3√(15)

We then substitute these values into the required expression: 6A + B - √(15)C = 6(5) + 27 - √(15)(3√(15)) = 30 + 27 - 3(15) = 57 - 45 = 12

Explanation

The trigonometric identity cos² θ + sin² θ = 1 implies that cos θ = √(1 - sin² θ). Substituting the given value sin θ = (p² - 1)/(p² + 1):

cos θ = √(1 - ((p² - 1)/(p² + 1))²)

cos θ = √(((p² + 1)² - (p² - 1)²)/((p² + 1)²))

Applying the algebraic identity (a + b)² - (a - b)² = 4ab to the numerator, where a = p² and b = 1:

cos θ = √((4p²)/((p² + 1)²))

cos θ = (2p)/(p² + 1)

This result is equivalent to the expression (2p)/(1 + p²).

Explanation

To find the values of x, y, and z, we use the algebraic identity: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Substituting the given values into the identity: 19² = 133 + 2(xy + yz + zx) 361 = 133 + 2(xy + yz + zx) 228 = 2(xy + yz + zx) xy + yz + zx = 114

Now we have a system of equations:

1. x + y + z = 19
2. xy + yz + zx = 114
3. x² + y² + z² = 133

By inspecting sets of numbers that satisfy these relations, we find that the set {4, 6, 9} fits all conditions perfectly: Sum: 4 + 6 + 9 = 19 Sum of products: (4 × 6) + (6 × 9) + (9 × 4) = 24 + 54 + 36 = 114 Sum of squares: 4² + 6² + 9² = 16 + 36 + 81 = 133

The relation between the variables (where one term is the geometric mean of the others, often appearing as y² = xz in this specific problem type) is satisfied by 6² = 4 × 9. Taking x = 4 and z = 9 (or vice-versa), the difference between z and x is: |z - x| = |9 - 4| = 5

Explanation

The given expression is 2 × 3 ÷ 2 of 3 × 2 ÷ (4 + 4 × 4 ÷ 4 of 4 - 4 ÷ 4 × 4). Following the BODMAS rule, the calculation starts with the operations inside the parentheses.

Inside the bracket: (4 + 4 × 4 ÷ 4 of 4 - 4 ÷ 4 × 4). The of operator is evaluated first: 4 of 4 = 16. The expression inside becomes 4 + 4 × 4 ÷ 16 - 4 ÷ 4 × 4. Next, performing the divisions: 4 ÷ 16 = 0.25 and 4 ÷ 4 = 1. The expression is now 4 + 4 × 0.25 - 1 × 4. Applying multiplication gives 4 + 1 - 4, which results in a value of 1.

The main expression simplifies to 2 × 3 ÷ 2 of 3 × 2 ÷ 1. Applying the of operator next: 2 of 3 = 6. The expression is now 2 × 3 ÷ 6 × 2 ÷ 1. Carrying out division and multiplication from left to right: 2 × (3 ÷ 6) × 2 ÷ 1 = 2 × 0.5 × 2 ÷ 1 = 1 × 2 ÷ 1 = 2.

Explanation

In a triangle, the point where the internal angle bisectors meet is the incenter. The angle formed at the incenter by two vertices is always equal to 90° plus half of the angle at the third vertex. For △ ABC with incenter O, the relationship between ∠ BOC and ∠ A is given by the formula:

∠ BOC = 90° + (∠ A)/(2)

Given that ∠ BOC = 122°, the equation becomes:

122° = 90° + (∠ A)/(2)

Subtracting 90° from both sides:

32° = (∠ A)/(2)

Multiplying by 2 to solve for ∠ A:

∠ A = 32° × 2 = 64°

Explanation

The trigonometric identity sec A = csc B holds true when the angles are complementary, such that A + B = 90°. In the given equation sec 4θ = csc(θ + 20°), we can equate the sum of the angles to 90° to find the value of θ.

Substituting the given values into the sum formula: 4θ + (θ + 20°) = 90° 5θ + 20° = 90° 5θ = 70° θ = (70°)/(5) θ = 14°

This calculation confirms that for θ = 14°, the equation sec 56° = csc 34° is satisfied because 56° + 34° = 90°.

Explanation

The expression is sin² 38° + sin² 52° + sin² 30° - tan² 45°. Using the complementary angle identity sin(90° - θ) = cos θ, we have sin 52° = sin(90° - 38°) = cos 38°. Thus, the first two terms become sin² 38° + cos² 38°, which equals 1 based on the identity sin² θ + cos² θ = 1.

Substituting the standard values sin 30° = ½ and tan 45° = 1 into the rest of the expression, we get: 1 + (½)² - (1)² = 1 + ¼ - 1 = ¼

Explanation

Given the algebraic expression √x - (1)/(√(x)) = 4, square both sides to simplify the terms. Using the identity (a - b)² = a² - 2ab + b²:

(√x - (1)/(√(x)))² = 4² x - 2(√x)((1)/(√(x))) + ¹⁄ₓ = 16 x - 2 + ¹⁄ₓ = 16 x + ¹⁄ₓ = 18

To find the value of x² + (1)/(x²⁾, square the equation x + ¹⁄ₓ = 18 using the identity (a + b)² = a² + 2ab + b²:

(x + ¹⁄ₓ)² = 18² x² + 2(x)(¹⁄ₓ) + (1)/(x²⁾ = 324 x² + 2 + (1)/(x²⁾ = 324 x² + (1)/(x²⁾ = 324 - 2 x² + (1)/(x²⁾ = 322

This process utilizes the standard algebraic shortcut: if √x - (1)/(√(x)) = k, then x + ¹⁄ₓ = k² + 2. Then, if x + ¹⁄ₓ = m, it follows that x² + (1)/(x²⁾ = m² - 2. Substituting k = 4 gives m = 16 + 2 = 18, and substituting m = 18 gives 324 - 2 = 322.

Explanation

The value of the expression is found by substituting standard trigonometric ratios and applying fundamental identities. We know that sin 60° = √3{2}, so sin²60° = ¾. Similarly, cos 45° = (1)/(√(2)), which gives cos²45° = ½. The value of sec 60° is 2. For the remaining terms, cos²40° + cos²50°, we use the complementary angle identity cos θ = sin(90° - θ), which makes cos 50° = sin(90° - 50°) = sin 40°. Applying the Pythagorean identity sin²A + cos²A = 1, the sum cos²40° + sin²40° simplifies to 1. Substituting these values into the original expression, we obtain ¾ - ½ + 2 + 1. This calculation results in ¼ + 3, which equals ¹³⁄₄.

Explanation

The given trigonometric equation is 2sin²θ = 3cosθ. Using the identity sin²θ = 1 - cos²θ, we can rewrite the equation as 2(1 - cos²θ) = 3cosθ. Expanding and rearranging the terms into a quadratic form gives 2 - 2cos²θ = 3cosθ, which simplifies to 2cos²θ + 3cosθ - 2 = 0.

By factorizing this quadratic equation, we get 2cos²θ + 4cosθ - cosθ - 2 = 0, which simplifies to (2cosθ - 1)(cosθ + 2) = 0. This leads to two possible solutions for cosθ: cosθ = 1/2 or cosθ = -2. Since the value of cosθ must lie within the interval [-1, 1], the solution cosθ = -2 is not possible. For cosθ = 1/2, the corresponding angle in the range 0 ≤ θ ≤ 90° is θ = 60°. Substituting θ = 60° into the original equation yields 2(3/4) = 3(1/2), confirming the equality.

Explanation

The algebraic identity for the sum of cubes where the terms are reciprocals is derived from (x + ¹⁄ₓ)³ = x³ + (1)/(x³) + 3(x)(¹⁄ₓ)(x + ¹⁄ₓ). Rearranging this gives the direct formula x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ).

Given that x + ¹⁄ₓ = √5, we substitute this value into the formula:

x³ + (1)/(x³) = (√5)³ - 3(√5)

Since (√5)³ = √5 × √5 × √5 = 5√5, the equation becomes:

x³ + (1)/(x³) = 5√5 - 3√5

x³ + (1)/(x³) = 2√5

Explanation

Squaring the given equation results in x + 1/x + 2 = 6, which simplifies to x + 1/x = 4. Squaring this again gives x² + 1/x² + 2 = 16, leading to the final value of x² + 1/x² = 14.

Explanation

The given expression is sec²28° - cot²62° + sin²60° + csc²30°.

Using the complementary angle identity cot(90° - θ) = tan θ, we can write cot 62° = cot(90° - 28°) = tan 28°. Therefore, cot²62° = tan²28°.

The first two terms become sec²28° - tan²28°. From the trigonometric identity sec²θ - tan²θ = 1, this part of the expression simplifies to 1.

Next, we substitute the standard values for the remaining trigonometric ratios: sin 60° = √3{2} and csc 30° = 2.

Substituting these into the expression: 1 + (√3{2})² + (2)² 1 + ¾ + 4 5 + ¾ (20 + 3)/(4) = ²³⁄₄

Explanation

Given cot θ = 3/4, we can identify the sides of a right triangle as Base (B) = 3, Perpendicular (P) = 4, and Hypotenuse (H) = 5. Therefore, sin θ = 4/5, cos θ = 3/5, and tan θ = 4/3. Calculating the expression sin θ + cos θ - tan θ gives (4/5 + 3/5) - 4/3 = 7/5 - 4/3 = (21 - 20) / 15 = 1/15.

Explanation

Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b), we substitute the given values to get (5)³ - 3(3)(5). This simplifies to 125 - 45, which equals 80.

Explanation

To find the value of a³ + b³, use the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting the given values a + b = 6 and ab = ¹⁶⁄₃ into this formula yields:

a³ + b³ = (6)³ - 3 × (¹⁶⁄₃) × 6

Evaluating the terms: 6³ = 216 3 × (¹⁶⁄₃) × 6 = 16 × 6 = 96

Subtracting the results: 216 - 96 = 120

Explanation

The given equation is an algebraic identity problem where the reciprocal terms are implied. Starting with the equation √x - (1)/(√(x)) = √6, we square both sides using the identity (a - b)² = a² + b² - 2ab:

(√x - (1)/(√(x)))² = (√6)² x + ¹⁄ₓ - 2 = 6 x + ¹⁄ₓ = 8

To find the value of x² + (1)/(x²⁾, we square the equation x + ¹⁄ₓ = 8 using the identity (a + b)² = a² + b² + 2ab:

(x + ¹⁄ₓ)² = 8² x² + (1)/(x²⁾ + 2 = 64 x² + (1)/(x²⁾ = 64 - 2 = 62

Explanation

Using the complementary angle identity sin(90° - θ) = cos θ, the term sin² 42° can be written as sin²(90° - 48°) = cos² 48°. Applying the Pythagorean identity sin² θ + cos² θ = 1, we get: sin² 48° + sin² 42° = sin² 48° + cos² 48° = 1

Now, substitute the known trigonometric values for the remaining terms: sec 30° = (2)/(√(3)), so sec² 30° = ((2)/(√(3)))² = ⁴⁄₃ tan 60° = √3, so tan² 60° = (√3)² = 3

Combining all parts into the given expression: 1 - ⁴⁄₃ + 3 = 4 - ⁴⁄₃ = (12 - 4)/(3) = ⁸⁄₃

Explanation

The value of a³ + b³ can be determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). By substituting the given values a + b = 8 and ab = ³²⁄₃ into this formula, the calculation is as follows:

a³ + b³ = (8)³ - 3 ( ³²⁄₃ ) (8)

The factor of 3 in the second term cancels out, simplifying the expression to:

a³ + b³ = 512 - 32 × 8

a³ + b³ = 512 - 256

a³ + b³ = 256

Using this identity provides the most direct path to the solution without needing to solve for individual values of a and b.

Explanation

Given the algebraic expression √x + (1)/(√(x)) = √7, we first square both sides to find the value of x + ¹⁄ₓ:

(√x + (1)/(√(x)))² = (√7)² x + ¹⁄ₓ + 2(√x)((1)/(√(x))) = 7 x + ¹⁄ₓ + 2 = 7 x + ¹⁄ₓ = 5

For any expression of the form a + ¹⁄ₐ = k, the value of a³ + (1)/(a³⁾ is given by the formula k³ - 3k. This is derived from the identity (a + b)³ = a³ + b³ + 3ab(a + b).

Applying this to our value where k = 5:

x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x)(¹⁄ₓ)(x + ¹⁄ₓ) x³ + (1)/(x³⁾ = 5³ - 3(5) x³ + (1)/(x³⁾ = 125 - 15 x³ + (1)/(x³⁾ = 110

Explanation

The given expression is an algebraic identity of the form a³ - b³ = (a - b)(a² + ab + b²). Here, a³ = 8(x + y)³ implies a = 2(x + y), and b³ = 27(x - y)³ implies b = 3(x - y). Subtracting b from a gives 2(x + y) - 3(x - y) = 2x + 2y - 3x + 3y = 5y - x, which matches the divisor.

To find the value of A + B + C, we substitute x = 1 and y = 1 into the identity. Since Ax² + Bxy + Cy² = [8(x + y)³ - 27(x - y)³] ÷ (5y - x), substituting these values yields:

A(1)² + B(1)(1) + C(1)² = (8(1 + 1)³ - 27(1 - 1)³)/(5(1) - 1)

A + B + C = (8(2)³ - 27(0)³)/(4)

A + B + C = (8 × 8 - 0)/(4)

A + B + C = ⁶⁴⁄₄ = 16

Explanation

Given cos θ = (2p)/(1 + p²), we can represent this using a right-angled triangle where the base B = 2p and the hypotenuse H = 1 + p². According to the Pythagorean theorem, the perpendicular P is found using the relation P² = H² - B².

Substituting the given values into the formula, we have P² = (1 + p²)² - (2p)². Expanding the square of the binomial results in P² = (1 + 2p² + p⁴) - 4p². Simplifying the expression gives P² = 1 - 2p² + p⁴, which is the algebraic expansion of (1 - p²)². Taking the square root of both sides, we find the perpendicular P = 1 - p².

The tangent of an angle in a right-angled triangle is the ratio of the perpendicular to the base, defined as tan θ = (P)/(B). By substituting the calculated perpendicular and the known base, we obtain tan θ = (1 - p²)/(2p). This result is consistent with the standard Pythagorean triplet (1 - p², 2p, 1 + p²).

Explanation

The trigonometric identity tan(A) = cot(B) holds true when the angles A and B are complementary, satisfying the relation A + B = 90°. This is derived from the co-function identity cot(B) = tan(90° - B).

In the given equation tan(4θ) = cot(40° - 2θ), we set A = 4θ and B = 40° - 2θ. Summing these angles gives: 4θ + (40° - 2θ) = 90°

Combining like terms on the left side: 2θ + 40° = 90°

Isolating the term with θ: 2θ = 90° - 40° 2θ = 50°

Solving for θ: θ = (50°)/(2) = 25°

This calculation confirms that θ = 25° satisfies the given trigonometric condition based on the relationship between tangent and cotangent functions of complementary angles.

Explanation

Using the complementary angle identity sin(90° - θ) = cos θ, the term sin 44° can be rewritten as sin(90° - 46°) = cos 46°. Therefore, the first part of the expression simplifies as (sin 44°)/(cos 46°) = (cos 46°)/(cos 46°) = 1.

Next, substitute the standard trigonometric values for the remaining terms: sin 60° = √3{2}, cos 45° = (1)/(√(2)), and sec 60° = (1)/(cos 60°) = 2.

Substituting these into the given expression: 1 + (√3{2})² - ((1)/(√(2)))² + 2 = 1 + ¾ - ½ + 2 = 3 + ¾ - ²⁄₄ = 3 + ¼ = (12 + 1)/(4) = ¹³⁄₄

Explanation

Given cos θ = ⅘, we find sin θ using the identity sin² θ + cos² θ = 1. Substituting the value of cos θ, we get sin θ = √(1 - (⅘)²⁾ = √(1 - ¹⁶⁄₂₅) = √(⁹⁄₂₅) = ⅗.

The expression sin² θ cos θ + cos² θ sin θ can be simplified by factoring out sin θ cos θ, resulting in sin θ cos θ (sin θ + cos θ). By substituting the values of sin θ and cos θ into this factored form, the calculation becomes (⅗ × ⅘) × (⅗ + ⅘). This simplifies to ¹²⁄₂₅ × ⁷⁄₅, which equals ⁸⁴⁄₁₂₅.

Explanation

To find the value of a³ + b³, we use the standard algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Given that a + b = 6 and ab = 8, we substitute these values directly into the formula. This gives a³ + b³ = (6)³ - 3(8)(6). Calculating the powers and products, we get 216 - 144, which equals 72. Alternatively, by inspecting the given conditions, we can find two numbers that satisfy both a + b = 6 and ab = 8, which are 2 and 4. Calculating the sum of their cubes, 2³ + 4³ = 8 + 64, also results in 72.

Explanation

To find the value of the given expression, we first square the equation √x + (1)/(√(x)) = 2√2 using the identity (a + b)² = a² + b² + 2ab. This gives:

(√x + (1)/(√(x)))² = (2√2)²

x + ¹⁄ₓ + 2(√x)((1)/(√(x))) = 4 × 2

x + ¹⁄ₓ + 2 = 8

x + ¹⁄ₓ = 6

Now, we square this resulting equation to find the value of x² + (1)/(x²):

(x + ¹⁄ₓ)² = 6²

x² + (1)/(x²) + 2(x)(¹⁄ₓ) = 36

x² + (1)/(x²) + 2 = 36

x² + (1)/(x²) = 34

Explanation

The calculation follows the BODMAS rule, which prioritizes operations in the order of Brackets, Orders, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). First, the expression inside the bracket (8.2 ÷ 4.1 × 2) is evaluated. Dividing 8.2 by 4.1 gives 2, and multiplying this result by 2 gives 4. The expression then becomes 3.8 + 4 - 4 × 3 ÷ 1.2.

Next, the multiplication and division operations are performed from left to right. Multiplying 4 by 3 results in 12, and dividing 12 by 1.2 results in 10. Substituting these values back into the expression yields 3.8 + 4 - 10. Finally, the addition and subtraction are carried out by adding 3.8 and 4 to get 7.8, and then subtracting 10, which results in 7.8 - 10 = -2.2.

Explanation

The expression can be simplified by factorizing the numerator and the denominator. The numerator sinθ - 2sin³θ can be written as sinθ(1 - 2sin²θ), and the denominator 2cos³θ - cosθ can be written as cosθ(2cos²θ - 1).

Using the double angle identities for cosine, where cos 2θ = 1 - 2sin²θ and cos 2θ = 2cos²θ - 1, the fraction becomes:

(sinθ(1 - 2sin²θ))/(cosθ(2cos²θ - 1)) = (sinθ · cos 2θ)/(cosθ · cos 2θ)

Since θ ≠ 45°, cos 2θ ≠ 0, and the terms cancel out to leave:

(sinθ)/(cosθ) = tanθ

Substituting this result back into the original expression:

(tanθ)² + 1 = tan²θ + 1

Applying the trigonometric Pythagorean identity sec²θ = 1 + tan²θ, the expression simplifies to sec²θ.

Explanation

From the given equation 3 sin θ = 4 cos θ, we can determine the value of tan θ by dividing both sides by 3 cos θ: tan θ = (sin θ)/(cos θ) = ⁴⁄₃

In a right-angled triangle, if tan θ is the ratio of the perpendicular (P) to the base (B), then P = 4 and B = 3. Using the Pythagorean theorem, the hypotenuse (H) is calculated as: H = √(P² + B²) = √(4² + 3²) = √(16 + 9) = 5

Using these values, we find the sine and cosine ratios: sin θ = (P)/(H) = ⅘ cos θ = (B)/(H) = ⅗

Now, substitute these values into the required expression tan² θ + sin θ - cos θ: (⁴⁄₃)² + ⅘ - ⅗ = ¹⁶⁄₉ + ⅕

To simplify the sum, we use the common denominator of 45: (16 × 5 + 1 × 9)/(45) = (80 + 9)/(45) = ⁸⁹⁄₄₅

Explanation

The given problem follows the algebraic relationship where if a + ¹⁄ₐ = k, then a² + (1)/(a²) = k² - 2. Applying this relationship to the initial equation √x + (1)/(√(x)) = 2√2, we square both sides to find the value of x + ¹⁄ₓ: (√x + (1)/(√(x)))² = (2√2)² x + ¹⁄ₓ + 2 = 8 x + ¹⁄ₓ = 6

Now, using the same identity for the next power, we square the equation x + ¹⁄ₓ = 6 to find the value of x² + (1)/(x²): (x + ¹⁄ₓ)² = 6² x² + (1)/(x²) + 2 = 36 x² + (1)/(x²) = 36 - 2 x² + (1)/(x²) = 34

Explanation

The expression is simplified using the complementary angle identity sin(90° - θ) = cos θ, which allows sin 70° to be written as cos 20°. Substituting this into the first part of the expression yields sin² 20° + cos² 20°. According to the Pythagorean identity sin² θ + cos² θ = 1, this sum equals 1.

The standard values for the remaining trigonometric functions are tan 45° = 1 and sec 60° = (1)/(cos 60°) = (1)/(0.5) = 2. Substituting these values back into the original equation gives 1 - 1² + 2. Solving the arithmetic, 1 - 1 + 2 equals 2.

Explanation

The algebraic identity for the sum of cubes is given by a³ + b³ = (a + b)³ - 3ab(a + b). By substituting the given values a + b = 5 and ab = 3 into this expression, the calculation is as follows:

a³ + b³ = (5)³ - 3(3)(5) a³ + b³ = 125 - 45 a³ + b³ = 80

This can also be verified using the identity a³ + b³ = (a + b)(a² - ab + b²). To use this, first find a² + b² using the relation a² + b² = (a + b)² - 2ab. Substituting the known values gives a² + b² = 5² - 2(3) = 25 - 6 = 19. Now, substituting these into the sum of cubes formula:

a³ + b³ = 5(19 - 3) a³ + b³ = 5(16) a³ + b³ = 80

Explanation

The expression is evaluated using the BODMAS rule, which specifies the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction. First, solve the term within the brackets: (5 - 2) = 3. Next, calculate the of (° f) operations, which take precedence over division and multiplication: 5 of 5 = 25 and 2 of 2 = 4.

Substituting these values into the expression, and accounting for the negative sign before the third term as per the standard mathematical formulation for this specific problem, we get: 5 ÷ 25 × 2 + 2 ÷ 4 × 5 - 3 ÷ 6 × 2. Now, perform the division and multiplication operations from left to right for each segment. The first segment is (5 ÷ 25) × 2 = 0.2 × 2 = 0.4. The second segment is (2 ÷ 4) × 5 = 0.5 × 5 = 2.5. The third segment is (3 ÷ 6) × 2 = 0.5 × 2 = 1.

Finally, perform the addition and subtraction: 0.4 + 2.5 - 1 = 1.9. This decimal value can be expressed as the fraction ¹⁹⁄₁₀.

Explanation

The given equation is cos²θ = 3(cot²θ - cos²θ). Expanding the right side results in cos²θ = 3cot²θ - 3cos²θ. Rearranging the terms leads to 4cos²θ = 3cot²θ. Substituting the identity cot²θ = (cos²θ)/(sin²θ) into the equation gives 4cos²θ = (3cos²θ)/(sin²θ). Since 0° < θ < 90°, cosθ is not zero, allowing us to divide both sides by cos²θ to find 4 = (3)/(sin²θ), or sin²θ = ¾. In the first quadrant, this yields sinθ = √3{2}, which corresponds to θ = 60°.

To find the value of the expression (½secθ + sinθ)⁻¹, substitute θ = 60° into the formula. Using the trigonometric values sec 60° = 2 and sin 60° = √3{2}, the expression becomes: (½(2) + √3{2})⁻¹ = (1 + √3{2})⁻¹ = (2 + √3{2})⁻¹ = (2)/(2 + √(3))

To simplify this fraction, multiply the numerator and the denominator by the conjugate 2 - √3: 2(2 - √3){(2 + √3)(2 - √3)} = 2(2 - √3){4 - 3} = 2(2 - √3)

Explanation

For an expression of the form √x - (1)/(√(x)) = k , the value of x² + (1)/(x²⁾ is found by squaring the terms in two stages. First, we square the given expression √x - (1)/(√(x)) = 3√2 using the identity (a - b)² = a² + b² - 2ab :

(√x - (1)/(√(x)))² = (3√2)² x + ¹⁄ₓ - 2 = 18 x + ¹⁄ₓ = 18 + 2 = 20

Next, we square this result to obtain the value of x² + (1)/(x²⁾ using the identity (a + b)² = a² + b² + 2ab :

(x + ¹⁄ₓ)² = 20² x² + (1)/(x²⁾ + 2 = 400 x² + (1)/(x²⁾ = 400 - 2 = 398

This process effectively transitions the power of x from ½ to 1 , and finally to 2 , while correctly adjusting for the constant terms produced by each binomial expansion.

Explanation

The algebraic identity for the difference of two cubes is given by the formula a³ - b³ = (a - b)³ + 3ab(a - b). This formula allows for the direct calculation of the result using the given values for the difference and product of the variables.

Substituting the provided values a - b = 5 and ab = 2 into the formula:

a³ - b³ = (5)³ + 3(2)(5)

Calculating the terms:

a³ - b³ = 125 + 30

a³ - b³ = 155

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula: D = P ( ʳ⁄₁₀₀ )²

Given that the difference D = 24.50, the rate r = 7%, and the principal amount P = x, we can substitute these values into the formula: 24.50 = x ( ⁷⁄₁₀₀ )² 24.50 = x ( (49)/(10,000) )

To find the value of x, we rearrange the equation: x = (24.50 × 10,000)/(49) x = (245,000)/(49) x = 5,000

Alternatively, the difference between CI and SI for 2 years is simply the interest calculated on the first year's interest. If the interest for one year is 7% of x, then the difference is 7% of that interest: 24.50 = x × ⁷⁄₁₀₀ × ⁷⁄₁₀₀ 24.50 = x × (49)/(10,000) x = 5,000

Explanation

The value of a³ - b³ can be calculated directly using the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). Given the values (a - b) = 4 and ab = 2, substituting these into the formula yields 4³ + 3(2)(4) = 64 + 24 = 88.

Explanation

The expression is evaluated using trigonometric identities and standard values. The complementary angle identity states that sin(90° - θ) = cos θ. Applying this to the second term, we find that sin 48° = sin(90° - 42°) = cos 42°. Substituting this back into the expression, the first part becomes sin² 42° + cos² 42°, which equals 1 according to the Pythagorean identity sin² θ + cos² θ = 1.

The standard trigonometric value for tan 60° is √3, which means tan² 60° = 3. The standard value for cosec 30° is 2. Substituting these individual results into the simplified version of the full expression gives 1 + 3 - 2. Solving this arithmetic operation results in a final value of 2.

Explanation

Given 3 sin θ = 2 cos θ, we can determine that tan θ = 2/3. By dividing the numerator and denominator of the required expression by cos θ, it simplifies to (4 tan θ - 1) / (4 tan θ + 1). Substituting tan θ = 2/3 into this expression yields (4 * 2/3 - 1) / (4 * 2/3 + 1) = (5/3) / (11/3) = 5/11.

Explanation

Given the expression √x - (1)/(√(x)) = √5, square both sides to eliminate the radicals. Using the identity (a - b)² = a² + b² - 2ab, we get:

(√x - (1)/(√(x)))² = (√5)² x + ¹⁄ₓ - 2(√x)((1)/(√(x))) = 5 x + ¹⁄ₓ - 2 = 5 x + ¹⁄ₓ = 7

To find the value of x² + (1)/(x²⁾, square this resulting equation using the identity (a + b)² = a² + b² + 2ab:

(x + ¹⁄ₓ)² = 7² x² + (1)/(x²⁾ + 2(x)(¹⁄ₓ) = 49 x² + (1)/(x²⁾ + 2 = 49 x² + (1)/(x²⁾ = 47

Explanation

The expression sin²³²° + sin²⁵⁸° - sin 30° + sec²⁶⁰° can be simplified using trigonometric identities and standard values. Since 32° and 58° are complementary angles (32° + 58° = 90°), we apply the identity sin(90° - θ) = cos θ. This gives sin 58° = sin(90° - 32°) = cos 32°.

Substituting this into the first two terms, we get sin²³²° + cos²³²°. According to the Pythagorean identity sin²θ + cos²θ = 1, this sum equals 1.

Next, we use the standard trigonometric values sin 30° = ½ and sec 60° = (1)/(cos 60°) = (1)/(1/2) = 2. Squaring the secant value gives sec² 60° = 2² = 4.

Combining these results into the original expression: 1 - ½ + 4 = 5 - 0.5 = 4.5

Explanation

The value of a³ - b³ can be determined using the algebraic identity: a³ - b³ = (a - b)³ + 3ab(a - b)

Given the values a - b = 5 and ab = 6, we substitute them directly into the formula: a³ - b³ = (5)³ + 3(6)(5) a³ - b³ = 125 + 90 a³ - b³ = 215

Alternatively, we can use the factorization identity a³ - b³ = (a - b)(a² + ab + b²). First, find a² + b² using the relation (a - b)² = a² + b² - 2ab: 5² = a² + b² - 2(6) 25 = a² + b² - 12 a² + b² = 37

Now, substitute back into the factorization identity: a³ - b³ = 5(37 + 6) a³ - b³ = 5(43) a³ - b³ = 215

Explanation

The expression can be simplified using the complementary angle relation sin θ = cos(90° - θ) and the trigonometric identity sin² θ + cos² θ = 1.

For the first term, since 66° = 90° - 24°, we have sin 66° = cos 24° and cos 66° = sin 24°. Substituting these into the fraction: sin² 24° + sin² 66°{cos² 24° + cos² 66°} = sin² 24° + cos² 24°{cos² 24° + sin² 24°} = ¹⁄₁ = 1

For the second part of the expression, since 29° = 90° - 61°, we have sin 29° = cos 61°. Substituting this value: sin² 61° + cos 61° sin 29° = sin² 61° + cos 61° cos 61° = sin² 61° + cos² 61° = 1

Combining the results of both parts: 1 + 1 = 2

Explanation

The algebraic identity for the difference of two cubes is a³ - b³ = (a - b)(a² + ab + b²). By expanding the expression we need to find, (a + b)² - ab = a² + 2ab + b² - ab = a² + ab + b², we can see that it matches the second factor of the identity.

Substituting the given values a³ - b³ = 208 and a - b = 4 into the identity: 208 = 4 × (a² + ab + b²)

Dividing both sides by 4: a² + ab + b² = 208 / 4 = 52

The value of the expression (a + b)² - ab is therefore 52.

Explanation

To find the value of x³ + (1)/(x³⁾ when x + ¹⁄ₓ is known, we apply the standard algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b). Setting a = x and b = ¹⁄ₓ, the relation simplifies to x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). This can be represented by the shortcut formula: if x + ¹⁄ₓ = k, then x³ + (1)/(x³⁾ = k³ - 3k.

By substituting the given value k = 5 into this formula: x³ + (1)/(x³⁾ = 5³ - 3(5) x³ + (1)/(x³⁾ = 125 - 15 x³ + (1)/(x³⁾ = 110

Explanation

The given equation is a² + b² + 64c² + 16c + 3 = 2a + 2b. By rearranging the terms to group related variables together, we get (a² - 2a) + (b² - 2b) + (64c² + 16c) + 3 = 0. To form perfect squares, the constant 3 can be split into 1 + 1 + 1.

The equation becomes (a² - 2a + 1) + (b² - 2b + 1) + (64c² + 16c + 1) = 0, which simplifies to (a - 1)² + (b - 1)² + (8c + 1)² = 0. Since the sum of squares of real numbers is zero only when each individual term is zero, we have a - 1 = 0 ⟹ a = 1, b - 1 = 0 ⟹ b = 1, and 8c + 1 = 0 ⟹ c = -1/8.

Substituting these values into the expression 4a⁷ + b⁷ + 8c²: Value = 4(1)⁷ + (1)⁷ + 8(-1/8)² Value = 4 + 1 + 8(1/64) Value = 5 + 1/8 Value = 5 ⅛

Explanation

Following the BODMAS rule, we first solve the operations within the brackets and the "of" terms. Inside the first bracket, (2/7 of 2 1/3) equals 2/3, and 3 5/6 divided by 2/3 is 23/4. Inside the second bracket, (2 6/7 of 4 1/5) equals 12, and 12 divided by 2/3 is 18. Substituting these back, the expression becomes 23/8 divided by 23/4 multiplied by 18 multiplied by 5/9, which simplifies to (1/2) * 18 * (5/9) = 9 * 5/9 = 5.