SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

We use the trigonometric identity sec² a - tan² a = 1, which can be factored as (sec a + tan a)(sec a - tan a) = 1. Given that sec a + tan a = 2, we can determine that sec a - tan a = 1 / 2 = 0.5.

By adding these two equations: (sec a + tan a) + (sec a - tan a) = 2 + 0.5 2 sec a = 2.5 sec a = 1.25

By subtracting the second equation from the first: (sec a + tan a) - (sec a - tan a) = 2 - 0.5 2 tan a = 1.5 tan a = 0.75

The value of sin a is obtained by the relation sin a = (tan a)/(sec a). Substituting the values, we get: sin a = (0.75)/(1.25) = ⁷⁵⁄₁₂₅ = ⅗ = 0.6

Explanation

For any expressions of the form a sin Θ + b cos Θ = c and b sin Θ - a cos Θ = x, we can use the identity derived from squaring and adding both equations: (a sin Θ + b cos Θ)² + (b sin Θ - a cos Θ)² = a² (sin² Θ + cos² Θ) + b² (cos² Θ + sin² Θ) = a² + b².

Substituting the given values a = 3, b = 5, and c = 5 into the relation c² + x² = a² + b², we get 5² + x² = 3² + 5². Simplifying this equation leads to 25 + x² = 9 + 25, which results in x² = 9. Taking the square root of both sides yields x = ± 3.

Explanation

The given equation is tan Θ + cot Θ = 2. Since cot Θ is the reciprocal of tan Θ, this can be written as tan Θ + (1)/(tan Θ) = 2. Multiplying by tan Θ yields the quadratic equation tan² Θ - 2 tan Θ + 1 = 0, which simplifies to (tan Θ - 1)² = 0. Solving this gives tan Θ = 1. For an acute angle Θ, this corresponds to Θ = 45°. Since tan 45° = 1 and cot 45° = 1, the value of any expression in the form tanⁿ Θ + cotⁿ Θ for any positive integer n will be 1ⁿ + 1ⁿ = 2. Therefore, tan⁵ Θ + cot⁵ Θ = 1⁵ + 1⁵ = 1 + 1 = 2.

Explanation

In △ AOD, E and F are the midpoints of AO and DO respectively. According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of its length. Therefore, EF = ½ AD.

Similarly, in △ DOC, F and G are the midpoints of DO and CO, which gives FG = ½ DC. In △ COB, G and H are the midpoints of CO and BO, so GH = ½ CB. Finally, in △ BOA, H and E are the midpoints of BO and AO, implying HE = ½ AB.

The perimeter of quadrilateral EFGH is the sum of its sides: Perimeter(EFGH) = EF + FG + GH + HE Substituting the values derived from the Midpoint Theorem: Perimeter(EFGH) = ½ AD + ½ DC + ½ CB + ½ AB Perimeter(EFGH) = ½ (AD + DC + CB + AB) Perimeter(EFGH) = ½ × Perimeter(ABCD)

The ratio of the perimeter of EFGH to the perimeter of ABCD is ½, which is 1:2.

Explanation

The expression is the product tan 1° · tan 2° · tan 3° … tan 89°. Using the trigonometric identity tan(90° - θ) = cot θ, we can see that tan 89° = tan(90° - 1°) = cot 1°, tan 88° = cot 2°, and so on. Substituting these into the product, we can pair the terms as (tan 1° · cot 1°) · (tan 2° · cot 2°) … (tan 44° · cot 44°) · tan 45°. Since the product tan θ · cot θ = 1 for any angle θ, each paired set simplifies to 1. The middle term tan 45° is also equal to 1. Therefore, the entire product simplifies to 1 · 1 · … · 1 = 1.

Explanation

Let O be the center of the circle and P be the external point. Let T be the point of contact of the tangent drawn from point P to the circle. In geometry, the radius of a circle is always perpendicular to the tangent at the point of contact. Therefore, the triangle OTP is a right-angled triangle with the right angle at vertex T.

By applying the Pythagoras theorem to triangle OTP, the square of the distance from the center OP is equal to the sum of the squares of the radius OT and the tangent length PT. The relationship is expressed as OP² = OT² + PT².

Substituting the given values, where the radius OT = 5 units and the tangent length PT = 5√3 units: OP² = 5² + (5√3)² OP² = 25 + 75 OP² = 100 OP = √(100) = 10 units

The distance of the point from the center of the circle is 10 units.

Explanation

Let the interior angles of Δ ABC be A, B, and C. The exterior angle at vertex A is 180° - A. Since AD is the bisector of this exterior angle, the angle ∠ DAC is equal to (180° - A)/(2) = 90° - (A)/(2). In Δ ABD, the total angle at vertex A is ∠ DAB = ∠ DAC + ∠ CAB = (90° - (A)/(2)) + A = 90° + (A)/(2).

Given that AB = AD, the triangle ABD is isosceles with base BD, meaning the base angles ∠ ABD and ∠ ADB are equal. Since D lies on BC produced, ∠ ABD is the same as the interior angle B of Δ ABC. The sum of angles in Δ ABD is 180°, so B + B + (90° + (A)/(2)) = 180°, which simplifies to 4B + A = 180°.

Similarly, for the exterior angle bisector BE at vertex B where AB = BE, the angle ∠ EBA in Δ BAE is 90° + (B)/(2). Since AB = BE, the base angles ∠ BAE and ∠ BEA are equal, thus ∠ BAE = A. The sum of angles in Δ BAE is 180°, giving A + A + (90° + (B)/(2)) = 180°, which simplifies to 4A + B = 180°.

Solving the system of equations A + 4B = 180° and 4A + B = 180° yields A = B = 36°. Consequently, the interior angle ∠ ACB = 180° - (A + B) = 180° - 72° = 108°.

Explanation

In an equilateral triangle ABC, all interior angles are equal to 60°, therefore ∠ ABC = 60°. The circumcentre O is the center of the circle that passes through all three vertices of the triangle. According to the geometric property of circles, the angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle. In this configuration, the arc AC subtends ∠ AOC at the center O and ∠ ABC at the point B on the circumference. Applying the theorem, ∠ AOC = 2 × ∠ ABC. Substituting the known value of the angle in an equilateral triangle, ∠ AOC = 2 × 60° = 120°.

Explanation

In a cyclic quadrilateral, the sum of opposite angles is 180°. Therefore, ∠ BCD = 180° - ∠ BAD = 180° - 95° = 85° and ∠ ABC = 180° - ∠ ADC = 180° - 70° = 110°.

Since ABE is a straight line, the exterior angle ∠ CBE = 180° - ∠ ABC = 180° - 110° = 70°. In △ BCE, it is given that BE = BC, which implies that the triangle is isosceles with ∠ BCE = ∠ BEC.

Applying the angle sum property to △ BCE: ∠ CBE + ∠ BCE + ∠ BEC = 180° 70° + 2 × ∠ BCE = 180° 2 × ∠ BCE = 110° ⟹ ∠ BCE = 55°

The total angle ∠ DCE is the sum of ∠ BCD and ∠ BCE: ∠ DCE = 85° + 55° = 140°

Explanation

For a regular polygon with n sides, the sum of the interior angles is calculated using the formula (n - 2) × 180°. The sum of the exterior angles for any convex polygon is always a constant 360°. Based on the problem statement, the sum of the interior angles is equal to twice the sum of the exterior angles.

We can represent this relationship with the following equation: (n - 2) × 180° = 2 × 360°

Dividing both sides of the equation by 180° simplifies the expression: n - 2 = 2 × 2 n - 2 = 4 n = 4 + 2 n = 6

Explanation

In △ PST and △ PQR, ∠ P is common to both triangles and it is given that ∠ PST = ∠ PQR. According to the AA (Angle-Angle) similarity criterion, △ PST ∼ △ PQR.

By the property of similar triangles, the ratios of their corresponding sides are equal: (PS)/(PQ) = (PT)/(PR)

The length of side PQ is the sum of PT and TQ, so PQ = 5 + 3 = 8 cm. Let the length of SR be x, so PR = PS + SR = 3 + x. Substituting the known values into the ratio: ⅜ = (5)/(3 + x)

Cross-multiplying to solve for x: 3(3 + x) = 40 9 + 3x = 40 3x = 31 x = ³¹⁄₃

Thus, the length of SR is 31/3 cm.

Explanation

The length of a direct common tangent to two circles with radii r₁ and r₂ and distance d between their centres is given by the formula L = √(d²⁾⁻⁽ʳ^_^¹ ⁻ ʳ^₂⁾^². In this problem, the radii are r₁ = 7 cm and r₂ = 2 cm, and the distance between the centres is d = 13 cm.

Substituting these values into the formula: L = √(13²⁾⁻⁽⁷⁻²⁾^^^² L = √(169 - 5²⁾ L = √(169 - 25) L = √(144) L = 12 cm

Explanation

Using the rule of alligation, the ratio in which two components must be mixed to achieve a desired mean concentration is found by calculating the differences between the mean and the individual strengths. Let C₁ = 25% be the lower concentration, C₂ = 50% be the higher concentration, and M = 40% be the target mean concentration.

The ratio of the quantity of 25% alcohol (q₁) to the quantity of 50% alcohol (q₂) is given by the formula:

q₁ / q₂ = (C₂ - M) / (M - C₁)

Substituting the given values:

q₁ / q₂ = (50 - 40) / (40 - 25) q₁ / q₂ = 10 / 15

Reducing this ratio to its simplest form by dividing both terms by 5 yields 2/3. Thus, the alcohol solutions must be mixed in the ratio 2:3.

Explanation

If x and y are two mean proportionals between a and b, then the sequence a, x, y, b forms a geometric progression. Let r be the common ratio. The terms can be represented as x = ar, y = ar², and b = ar³.

In this problem, a = 2 and b = 54. To find the common ratio r, we use the formula for the fourth term of the progression: ar³ = 54 2 × r³ = 54 r³ = 27 r = 3

Using the common ratio r = 3, we calculate the two mean proportionals: The first mean proportional is x = ar = 2 × 3 = 6. The second mean proportional is y = ar² = 2 × 3² = 2 × 9 = 18.

The values 6 and 18 satisfy the condition because 2, 6, 18, 54 is a geometric progression where each term is multiplied by 3 to get the next term.

Explanation

The total age of the 7 children is found by multiplying the number of children by their average age: 7 × 12 = 84 years. When the child aged 6 years dies, the total sum of the ages of the remaining children is 84 - 6 = 78 years. Since there are now 6 children remaining, their average age is calculated by dividing the new total age by the new count: 78 / 6 = 13 years.

Explanation

The total marks obtained by all 22 candidates is calculated using the formula Total = Average × Number of candidates. Thus, the sum of marks for all 22 candidates is 22 × 45 = 990.

The sum of the marks for the first 10 candidates is 10 × 55 = 550, and the sum of the marks for the last 11 candidates is 11 × 40 = 440. The 11th candidate is the only one not included in these two groups. To find the marks of the 11th candidate, subtract the sum of the marks of these 21 candidates from the total marks of all 22 candidates:

Marks of 11th candidate = 990 - (550 + 440) Marks of 11th candidate = 990 - 990 = 0

Explanation

The relationship between two selling prices given different profit/loss percentages on the same cost price is determined by the formula SPₙₑ_w = SPₒₗ_d × 100 ± Gain/Lossₙₑ_w%{100 ± Gain/Lossₒₗ_d%}.

In this case, the initial selling price SP₁ is 1,080 with a loss of 10%, and the desired gain is 10%. Applying the values to the ratio:

SP₂ = 1080 × (100 + 10)/(100 - 10)

SP₂ = 1080 × ¹¹⁰⁄₉₀

SP₂ = 1080 × ¹¹⁄₉

Dividing 1,080 by 9 gives 120. Multiplying 120 by 11 results in 1,320. Alternatively, calculating the Cost Price first: CP = (1080)/(0.9) = 1200. To gain 10%, the shopkeeper must sell it at 1200 × 1.1 = 1320.

Explanation

Yield percentage is the annual income earned on the actual investment made. It is calculated using the formula: Yield % = (Dividend Rate × Face Value)/(Market Price) × 100. For a standard face value of 100, the annual dividend is 12½ or 12.5. Given that the stock is purchased at a market price of 108, the yield is calculated as follows:

Yield % = (12.5)/(108) × 100

Yield % = ¹²⁵⁰⁄₁₀₈ = ⁶²⁵⁄₅₄

Dividing 625 by 54 yields a quotient of 11 and a remainder of 31, which results in the fraction 11³¹⁄₅₄. The total investment of 27,000 is not necessary for this calculation because the yield percentage remains constant regardless of the total number of shares purchased.

Explanation

When two objects start from the same point and move in opposite directions, their relative speed is the sum of their individual speeds. This is because every hour they move further apart by the distance each travels individually. The relative speed of Santa and Julie is 2.5 km/hr + 2 km/hr = 4.5 km/hr. To find the time required to reach a specific distance, we use the formula t = D / Vᵣₑₗ, where D is the distance and Vᵣₑₗ is the relative speed. Substituting the given values, we get t = 18 / 4.5. Simplified, this is 180 / 45, which equals 4 hours.

Explanation

When two objects move towards each other, their relative speed is the sum of their individual speeds. Here, the relative speed is 33 + 39 = 72 km/hr. To convert this speed into metres per second, we multiply by ⁵⁄₁₈, giving 72 × ⁵⁄₁₈ = 20 m/s.

The total distance to be covered for the trains to pass each other completely is the sum of their lengths. Thus, distance D = 125 + 115 = 240 metres. The time taken to pass each other is calculated using the formula Time = (Distance)/(Relative Speed). Substituting the values, we get Time = ²⁴⁰⁄₂₀ = 12 seconds.

Explanation

In compound interest, the amounts at the end of successive years form a geometric progression. This means the ratio between the principal P and the amount after the first year A₁ is the same as the ratio between the amount after the first year A₁ and the amount after the second year A₂.

P / A₁ = A₁ / A₂

Rearranging to solve for the principal: P = A₁² / A₂

Substitute the given values A₁ = 650 and A₂ = 676 into the formula: P = (650 × 650) / 676

To simplify the calculation, note that 676 = 26 × 26 and 650 = 26 × 25: P = (26 × 25 × 26 × 25) / (26 × 26) P = 25 × 25 P = 625

Explanation

To simplify the expression, we factorize the individual terms and perform the operations. Assuming the operations are division, which is consistent with the provided answer, the expression is (p² - p)/(2p³ + 6p²) ÷ (p² - 1)/(p² + 3p) ÷ (p²)/(p + 1).

First, factorize each numerator and denominator: (p² - p)/(2p³ + 6p²) = (p(p - 1))/(2p²(p + 3)) = (p - 1)/(2p(p + 3)) (p² - 1)/(p² + 3p) = ((p - 1)(p + 1))/(p(p + 3)) (p²)/(p + 1) remains as it is.

Now, substitute these factorized forms back into the expression and convert the division operations into multiplication by taking the reciprocals: (p - 1)/(2p(p + 3)) × (p(p + 3))/((p - 1)(p + 1)) × (p + 1)/(p²)

Cancel the common terms in the numerator and denominator: The term (p - 1) cancels out. The term (p + 3) cancels out. The term (p + 1) cancels out. The term p in the numerator of the second fraction cancels the p in the denominator of the first fraction.

After cancellation, we are left with: ½ × ¹⁄₁ × (1)/(p²) = (1)/(2p²)

Explanation

Using the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), we can substitute the given values a + b + c = 4√3 and a² + b² + c² = 16. This gives:

(4√3)² = 16 + 2(ab + bc + ca) 48 = 16 + 2(ab + bc + ca) 32 = 2(ab + bc + ca) ab + bc + ca = 16

Since a² + b² + c² = 16 and ab + bc + ca = 16, it follows that a² + b² + c² = ab + bc + ca. Rearranging this equation leads to:

a² + b² + c² - ab - bc - ca = 0

Multiplying by 2, we get:

2a² + 2b² + 2c² - 2ab - 2bc - 2ca = 0 (a - b)² + (b - c)² + (c - a)² = 0

In the set of real numbers, the sum of squares is zero if and only if each individual term is zero. Therefore, a - b = 0, b - c = 0, and c - a = 0, which means a = b = c. The ratio a:b:c is thus 1:1:1.

Explanation

The given equation is x + 1/x = 2. Multiplying both sides by x gives x² + 1 = 2x, which can be rewritten as x² - 2x + 1 = 0. This is the expansion of the identity (x - 1)² = 0, which implies x = 1.

By substituting x = 1 into the components of the required expression: x² + 1/x² = 1² + 1/1² = 1 + 1 = 2 x³ + 1/x³ = 1³ + 1/1³ = 1 + 1 = 2

The value of the product (x² + 1/x²)(x³ + 1/x³) is therefore 2 × 2 = 4. This result can also be verified using algebraic identities where if x + 1/x = k, then xⁿ + 1/xⁿ = 2 for all n when k = 2.

Explanation

The expression a³ + b³ + c³ - 3abc can be written using the algebraic identity: a³ + b³ + c³ - 3abc = ½(a + b + c)[(a - b)² + (b - c)² + (c - a)²]

Given that a, b, c are positive integers, the term (a + b + c) will always be a positive value (at least 1+1+1=3). The second part of the expression, [(a - b)² + (b - c)² + (c - a)²], consists of the sum of squares, which means its value is always greater than or equal to zero.

To find the least value of this expression, we minimize the sum of the squared differences. The minimum possible value for a square is zero, which occurs when the terms being subtracted are equal. By setting a = b = c (for instance, a = b = c = 1, which are positive integers), we get: (a - b)² = 0, (b - c)² = 0, and (c - a)² = 0

Substituting these into the identity: Value = ½(1 + 1 + 1)[0 + 0 + 0] = ½(3)[0] = 0

If a, b, c are not all equal, the sum of squares becomes at least 2 (e.g., for 1, 1, 2), making the total value of the expression significantly higher. Therefore, 0 is the least value the expression can take for positive integers.

Explanation

Given the equations a + b = 1 and a - b = d/c, we can multiply them to obtain: (a + b)(a - b) = 1 · (d/c) a² - b² = d/c

From the second given condition c + d = 1, we can express 1/c as: 1/c = (c + d)/c = 1 + d/c

Now, adding the two equations a + b = 1 and a - b = d/c gives: 2a = 1 + d/c Substituting the relation 1 + d/c = 1/c into this, we get: 2a = 1/c ⟹ c = 1/(2a)

Subtracting the two given equations a + b = 1 and a - b = d/c gives: 2b = 1 - d/c = (c - d)/c This implies c - d = 2bc. Substituting the value of c = 1/(2a) into this expression: c - d = 2b · ¹⁄₂ₐ = b/a

Since c + d = 1, the expression c² - d² can be evaluated as: c² - d² = (c + d)(c - d) = 1 · (b/a) = b/a

Explanation

According to the Remainder Theorem, when a polynomial f(x) is divided by a linear divisor ax + b, the remainder is given by f(-ᵇ⁄ₐ). Setting the divisor 3x + 2 = 0 gives the substitution value x = -⅔.

Substituting x = -⅔ into the polynomial f(x) = 12x³ - 13x² - 5x + 7:

R = 12(-⅔)³ - 13(-⅔)² - 5(-⅔) + 7

R = 12(-⁸⁄₂₇) - 13(⁴⁄₉) + ¹⁰⁄₃ + 7

Simplifying the terms using a common denominator of 9:

R = -³²⁄₉ - ⁵²⁄₉ + ³⁰⁄₉ + ⁶³⁄₉

R = (-32 - 52 + 30 + 63)/(9)

R = (-84 + 93)/(9) = ⁹⁄₉ = 1

Explanation

Given the equations x = 3t and y = ½(t + 1), we substitute these expressions into the condition x = 2y to solve for t.

The equation becomes: 3t = 2 · [ ½(t + 1) ]

By simplifying the right side, the factor of 2 cancels out with the denominator: 3t = t + 1

To isolate t, subtract t from both sides of the equation: 3t - t = 1 2t = 1

Dividing both sides by 2 yields the value of t: t = ½

This value satisfies the relationship x = 2y for the given parametric equations.

Explanation

For a quadratic expression x² + bx + c to be a perfect square, it must follow the algebraic identity (x + k)² = x² + 2kx + k². By comparing the coefficients of the given expression x² + ⅕x + a² with this standard form, we can solve for a.

Equating the middle term coefficient: 2k = ⅕ k = ⅒

Equating the constant term: a² = k² a² = (⅒)² a = ⅒

With a = ⅒, the expression becomes x² + ⅕x + ¹⁄₁₀₀, which is exactly (x + ⅒)². Thus, the value of a required to make the expression a perfect square is 1/10.

Explanation

The sum of the interior angles of any polygon with n sides is given by the formula (n - 2) × 180°. For a quadrilateral, whether it is cyclic or not, the number of sides n is equal to 4. Substituting this value into the formula yields (4 - 2) × 180° = 2 × 180° = 360°.

Furthermore, a specific property of a cyclic quadrilateral is that its opposite angles are supplementary, meaning m∠ A + m∠ C = 180° and m∠ B + m∠ D = 180°. Adding these two equations together results in m∠ A + m∠ B + m∠ C + m∠ D = 180° + 180° = 360°.

Explanation

In a right-angled triangle, the circumcenter is the point equidistant from all three vertices. According to Thales' Theorem, an angle inscribed in a semicircle is a right angle. This geometric principle implies that the hypotenuse of a right-angled triangle functions as the diameter of its circumcircle. Since the center of any circle is the midpoint of its diameter, the circumcenter of a right-angled triangle necessarily lies at the midpoint of the hypotenuse. The distance from this midpoint to each of the three vertices is equal to the circumradius R, which is half the length of the hypotenuse.

Explanation

The sum of the interior angles of a triangle is always 180° according to the angle sum property. Given two angles of 21° and 38°, the third angle is calculated as:

180° - (21° + 38°) = 180° - 59° = 121°

A triangle is classified as an obtuse-angled triangle if one of its interior angles measures more than 90°. Since the third angle in this triangle is 121°, which is greater than 90°, it is an obtuse-angled triangle.

Explanation

The angle formed by the intersection of the internal bisectors of two angles of a triangle is given by the formula 90° + (A)/(2), where A is the third angle of the triangle.

Given that the angle between the internal bisectors of ∠ B and ∠ C is 120°, we can set up the equation:

120° = 90° + (∠ A)/(2)

Subtracting 90° from both sides:

30° = (∠ A)/(2)

Multiplying by 2:

∠ A = 60°

Explanation

In Δ ABC, since F and E are the midpoints of AB and AC respectively, the line segment FE is parallel to the base BC according to the Midpoint Theorem. Let AD be the third median of the triangle, passing through the centroid G and meeting BC at its midpoint D. Since FE ∥ BC and F is the midpoint of AB, the line FE must bisect any segment drawn from A to BC. Thus, O is the midpoint of the median AD, implying AO = OD = ½ AD.

The centroid G divides the median AD in the ratio 2:1 from the vertex, which means AG = ⅔ AD. The length of OG is the difference between AG and AO:

OG = AG - AO = ⅔ AD - ½ AD = ⅙ AD

Given OG = 2 cm, we can solve for AD:

2 = ⅙ AD ⟹ AD = 12 cm

Using the relation AO = ½ AD:

AO = ½ × 12 = 6 cm

Explanation

In an equilateral triangle, all three sides are equal in length, such that AB = BC = CA = s. The altitude of a triangle is the perpendicular distance from a vertex to the opposite side. For an equilateral triangle, the length of the altitude h is given by the formula h = √3{2}s. Since all three sides are equal, the altitudes drawn from vertices A, B, and C to their respective opposite sides BC, AC, and AB are identical in length. Specifically, the altitude AX = √3{2}BC, the altitude BY = √3{2}AC, and the altitude CZ = √3{2}AB. Because BC = AC = AB, the values of these three altitudes are mathematically equivalent, leading to the relation AX = BY = CZ.

Explanation

In Δ ABC, the sum of the interior angles is 180°. Given ∠ BAC = 85° and ∠ BCA = 55°, we calculate the third angle as ∠ ABC = 180° - (85° + 55°) = 40°. According to the central angle theorem, the angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining part of the circle. Therefore, ∠ AOC = 2 × ∠ ABC = 2 × 40° = 80°.

Since O is the circumcenter, OA and OC are the radii of the circumcircle, meaning OA = OC. This makes Δ OAC an isosceles triangle where the angles opposite to the equal sides are equal, so ∠ OAC = ∠ OCA. In Δ OAC, the sum of the angles is 180°, so ∠ OAC + ∠ OCA + ∠ AOC = 180°. Substituting the known values gives 2 ∠ OAC + 80° = 180°. Solving this equation results in 2 ∠ OAC = 100°, which gives ∠ OAC = 50°.

Explanation

The sum of the interior angles of a triangle is always 180°. Given the ratio of the angles is 2:3:7, let the measures of the angles be 2x, 3x, and 7x respectively.

According to the angle sum property of a triangle: 2x + 3x + 7x = 180° 12x = 180° x = (180°)/(12) = 15°

The smallest angle corresponds to the smallest part of the ratio, which is 2x. Substituting the value of x: Smallest angle = 2 × 15° = 30°

Explanation

The values of the trigonometric functions at the given angles are: sin 60° = √3{2} tan 30° = (1)/(√(3)) sec 60° = (1)/(cos 60°) = (1)/(1/2) = 2 cot 45° = (1)/(tan 45°) = ¹⁄₁ = 1

Substituting these values into the equation x sin 60° tan 30° = sec 60° cot 45°: x · (√3{2}) · ((1)/(√(3))) = 2 · 1

Simplifying the left side by canceling √3: x · ½ = 2

Solving for x: x = 2 · 2 x = 4

Explanation

Using the identity 1 ± sin θ = cos²(θ/2) + sin²(θ/2) ± 2 sin(θ/2) cos(θ/2) = (cos(θ/2) ± sin(θ/2))², the square roots in the expression can be simplified. Given θ = 60°, we have θ/2 = 30°. At this angle, cos 30° = √3/2 and sin 30° = ½. Since cos 30° > sin 30° and both values are positive, the terms simplify as: √(1 + sin θ) = cos(θ/2) + sin(θ/2) √(1 - sin θ) = cos(θ/2) - sin(θ/2)

Now, substitute these into the original expression: ½(cos(θ/2) + sin(θ/2)) + ½(cos(θ/2) - sin(θ/2)) = ½ cos(θ/2) + ½ sin(θ/2) + ½ cos(θ/2) - ½ sin(θ/2) = cos(θ/2)

Explanation

The series 2² + 4² + 6² + … + 20² can be simplified by factoring out 2² from each term. Using the algebraic property (a × b)ⁿ = aⁿ × bⁿ, every term (2n)² in the series is equal to 2² × n², which is 4n².

2² + 4² + 6² + … + 20² = (2 × 1)² + (2 × 2)² + (2 × 3)² + … + (2 × 10)² = 2²(1²) + 2²(2²) + 2²(3²) + … + 2²(10²) = 2²(1² + 2² + 3² + … + 10²)

Substituting the given value 1² + 2² + 3² + … + 10² = 385 into the expression: = 4 × 385 = 1540

Explanation

To find the approximate value, follow the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

First, simplify the expression within the parentheses: 10 + ³⁄₁ - ⅕ = 13 - ⅕ = (65 - 1)/(5) = ⁶⁴⁄₅

Next, convert the mixed fractions to improper fractions: 5 ⅓ = ¹⁶⁄₃ 1 ²⁄₉ = ¹¹⁄₉

Now, substitute these values into the main expression and perform multiplication: ¹⁶⁄₃ + ¹¹⁄₉ × ¼ × ⁶⁴⁄₅ = ¹⁶⁄₃ + (¹¹⁄₉ × ¹⁶⁄₅) = ¹⁶⁄₃ + ¹⁷⁶⁄₄₅

To add the fractions, use the least common multiple of the denominators (3 and 45), which is 45: (16 × 15)/(45) + ¹⁷⁶⁄₄₅ = (240 + 176)/(45) = ⁴¹⁶⁄₄₅

Calculating the decimal value: ⁴¹⁶⁄₄₅ ≈ 9.244

Among the given options, 10 is the closest approximate integer value to 9.244.

Explanation

Using the efficiency method, let the total work to be done be the Least Common Multiple (LCM) of the given time periods. The LCM of 12 and 28 is 84 units.

The combined efficiency of A and B is 84 / 12 = 7 units per day. The efficiency of A alone is 84 / 28 = 3 units per day. The efficiency of B is calculated by subtracting A's efficiency from the combined efficiency: 7 - 3 = 4 units per day.

The time taken by B alone to dig the trench is the total work divided by B's daily efficiency: 84 / 4 = 21 days.

Explanation

The rate of the filling pipe is 1/9 of the cistern per hour. When the leak is operating alongside the pipe, the net rate of filling is 1/10 of the cistern per hour. The rate at which the leak empties the cistern is the difference between the filling rate without the leak and the net filling rate with the leak.

Leak rate per hour = ⅑ - ⅒

To solve this subtraction, find a common denominator: Leak rate per hour = (10 - 9)/(90) = ¹⁄₉₀

This means the leak empties 1/90 of the cistern in one hour. To find the total time T required to empty the full cistern, calculate the reciprocal of the hourly leak rate: T = (1)/(1/90) = 90 hours

Using the standard formula for this relationship where x is the filling time and y is the time with the leak: Time to empty = (xy)/(y - x) = (9 × 10)/(10 - 9) = 90 hours

Explanation

The ratio of daily work for the skilled, half-skilled, and unskilled laborers is given as 1/3 : 1/4 : 1/6. To simplify this ratio, multiply each term by the least common multiple of the denominators, which is 12. This results in a daily wage ratio of 4 : 3 : 2.

Total earnings are determined by multiplying the daily wage by the number of days worked for each category. Earnings of skilled laborer ∝ 4 × 7 = 28 Earnings of half-skilled laborer ∝ 3 × 8 = 24 Earnings of unskilled laborer ∝ 2 × 10 = 20

The ratio of their total earnings is 28 : 24 : 20, which simplifies to 7 : 6 : 5. The sum of the ratio parts is 7 + 6 + 5 = 18. Given the total collective earning is 369, the share of the skilled (trained) laborer is calculated as: Earnings = ⁷⁄₁₈ × 369 Earnings = 7 × 20.5 = 143.50

Explanation

The volume of a cube is given by the formula V = a³, where a represents the side length. Given that the ratio of the volumes of two cubes is 27:64, the ratio of their side lengths is found by taking the cube root of the volume ratio: a₁{a₂} = ³√(V₁₎{V₂} = ³√(²⁷⁄₆₄) = ¾

The surface area of a cube is calculated using the formula A = 6a². Since the constant factor 6 cancels out in a ratio, the ratio of the surface areas of two cubes is equal to the square of the ratio of their side lengths: A₁{A₂} = (a₁{a₂})² = (¾)² = ⁹⁄₁₆

Explanation

The slant height l of the conical tent is calculated using the Pythagorean theorem because the vertical height h and the radius r form a right-angled triangle with the slant height. l = √(r² + h²) = √(12² + 9²) = √(144 + 81) = √(225) = 15 m.

The quantity of canvas required to make the tent is equal to the curved surface area of the cone. The formula for the curved surface area is π r l. Substituting the given values: Curved Surface Area = 3.14 × 12 × 15 = 3.14 × 180 = 565.2 m².

The total cost of the canvas is determined by multiplying the total area by the cost per square metre: Total Cost = 565.2 × 120 = 67824.

Explanation

The area of a circle is calculated using the formula A = π r². Let R be the radius of the larger circle and r be the radius of the smaller circle. From the problem, we have two conditions: the difference between the radii is R - r = 14 cm, and the difference between their areas is π R² - π r² = 1056 cm².

Using the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), the area equation can be rewritten as π (R - r)(R + r) = 1056. Substituting the value of R - r = 14 and taking π = 22/7, the equation becomes:

(22/7) × 14 × (R + r) = 1056 44 × (R + r) = 1056 R + r = 1056 / 44 = 24

Now we have two simple linear equations: R + r = 24 and R - r = 14. To find the radius of the smaller circle r, subtract the second equation from the first:

(R + r) - (R - r) = 24 - 14 2r = 10 r = 5 cm

Explanation

Let the marked price (MP) be 100 units. The discount is one fifth of the marked price, so Discount D = 100 / 5 = 20 units. The selling price (SP) is the marked price minus the discount, which is 100 - 20 = 80 units.

The loss is half of the discount, so Loss L = 20 / 2 = 10 units. The cost price (CP) is the sum of the selling price and the loss, resulting in CP = 80 + 10 = 90 units.

The percentage of loss is calculated on the cost price using the formula: Loss Percentage = (L / CP) × 100 Loss Percentage = (10 / 90) × 100 = 100 / 9 = 11 ⅑%

Explanation

The net amount payable after successive discounts is calculated by applying each discount percentage sequentially to the remaining balance. For an initial bill of 110, the first discount of 10% reduces the price to 90% of its original value.

110 × (1 - 10/100) = 110 × 0.9 = 99

The second discount of 5% is then applied to this new balance of 99.

99 × (1 - 5/100) = 99 × 0.95 = 94.05

Rounding the final value of 94.05 to the nearest rupee gives 94.

Explanation

The Present Worth (PW) is the value which, with its interest for the given time at the given rate, equals the Amount (A). It is calculated as PW = A - TD. Given the Amount A = 1860 and the True Discount TD = 60, the Present Worth is PW = 1860 - 60 = 1800.

The True Discount is the simple interest on the Present Worth for the specified duration. Using the simple interest formula TD = (PW × R × T)/(100), where R = 5%, we substitute the known values to solve for the time T in years:

60 = (1800 × 5 × T)/(100) 60 = 90 × T T = ⁶⁰⁄₉₀ = ⅔ years

To convert the time from years to months, multiply the result by 12: T = ⅔ × 12 = 8 months