RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

To find the overall percentage, sum the marks obtained in all 5 subjects for each student:

Justin: 88 + 85 + 80 + 72 + 90 = 415 marks Jeeva: 76 + 82 + 69 + 70 + 86 = 383 marks Kuber: 72 + 64 + 80 + 83 + 80 = 379 marks Alia: 90 + 60 + 75 + 80 + 73 = 378 marks Hetal: 54 + 78 + 82 + 76 + 85 = 375 marks

Since the total maximum marks are equal for all students, the highest total marks correspond to the highest percentage. Justin achieved the highest total of 415 marks (83%), while Jeeva achieved the second highest total of 383 marks (76.6%).

Explanation

Using the BODMAS rule, simplify the numerator and denominator separately:

Numerator = 46 + (3/4 × 32) - 6 = 46 + 24 - 6 = 64

Denominator = 37 - 3/4 × (34 - 6) = 37 - (3/4 × 28) = 37 - 21 = 16

Dividing the numerator by the denominator: 64 / 16 = 4

Explanation

Let total work = LCM(20, 30, 60) = 60 units.

Individual efficiencies per day: A = 60 / 20 = 3 units B = 60 / 30 = 2 units C = 60 / 60 = 1 unit

On odd-numbered days, A and B work together: Work done = 3 + 2 = 5 units

On even-numbered days, A and C work together: Work done = 3 + 1 = 4 units

Work done in a 2-day cycle = 5 + 4 = 9 units

In 6 complete cycles (12 days): Work completed = 6 × 9 = 54 units Remaining work = 60 − 54 = 6 units

On the 13th day (odd-numbered day): A and B complete 5 units of work. Remaining work = 6 − 5 = 1 unit Total days elapsed = 13 days

On the 14th day (even-numbered day): A and C work together at a rate of 4 units per day. Time required for the remaining 1 unit = ¼ day

Total time taken = 13 + ¼ = 13 ¼ days

Explanation

Let the total work be 30 units (LCM of 10, 15, and 5).

Daily work rate of A = 30 / 10 = 3 units Daily work rate of B = 30 / 15 = 2 units Combined daily work rate of A, B, and C = 30 / 5 = 6 units

Daily work rate of C = 6 - (3 + 2) = 1 unit

Ratio of efficiency of A : B : C = 3 : 2 : 1.

Since all three worked together for the entire duration, payment is divided according to their efficiency ratio.

Amount paid to C = (1 / 6) × ₹4,800 = ₹800

Explanation

Using the successive percentage change relation:

New Price = Original Price × (1 - Discount/100) × (1 + Increase/100)

Substituting the given values: New Price = 9,000 × (1 - 0.20) × (1 + 0.10) = 9,000 × 0.80 × 1.10 = 7,920

Explanation

Let the ratio of 1 rupee, 50 paise, and 25 paise coins be x : 2x : 3x.

Given 2x = 80, we get x = 40.

Number of 1 rupee coins = 40 Number of 50 paise coins = 80 Number of 25 paise coins = 3 × 40 = 120

Total money in the box = (40 × ₹1) + (80 × ₹0.50) + (120 × ₹0.25) = ₹40 + ₹40 + ₹30 = ₹110

Explanation

To factor the quadratic polynomial x² - x - 20, find two numbers whose product is -20 and whose sum is -1. The required numbers are -5 and 4.

x² - x - 20 = x² - 5x + 4x - 20 = x(x - 5) + 4(x - 5) = (x - 5)(x + 4)

Hence, x - 5 is a linear factor of the polynomial.

Explanation

Using the compound interest formula: Amount = Principal × (1 + r/100)ᵗ

Given that the sum becomes 8 times of itself in 3 years: 8P = P(1 + r/100)³ 8 = (1 + r/100)³

Taking the cube root on both sides: 2 = 1 + r/100 r/100 = 1 r = 100%

Explanation

If x is the fourth proportional to three numbers a, b, and c, then a : b = c : x, which gives x = (b × c) / a.

Substituting a = 4, b = 9, and c = 12: x = (9 × 12) / 4 = 9 × 3 = 27.

Explanation

Expressing X, Y, and Z in prime factor form: X = 2¹ × 3² Y = 2³ × 3¹ Z = 2² × 3²

Multiplying these expressions in pairs gives: XY = 2⁴ × 3³ YZ = 2⁵ × 3³ ZX = 2³ × 3⁴

The L.C.M. of a set of numbers expressed in prime factorization is obtained by taking the highest power of each prime factor present: Highest power of 2 = 2⁵ Highest power of 3 = 3⁴

L.C.M.(XY, YZ, ZX) = 2⁵ × 3⁴ = 32 × 81 = 2592.

Explanation

Let x be the distance travelled on foot in km. The distance travelled on bicycle is (61 - x) km.

Since Time = Distance / Speed: Time on foot = x / 4 Time on bicycle = (61 - x) / 9

Total time taken is 9 hours: (x / 4) + ((61 - x) / 9) = 9

Multiplying both sides by 36: 9x + 4(61 - x) = 324 9x + 244 - 4x = 324 5x = 80 x = 16 km

Explanation

Prime factorization of the given numbers: 12 = 2² × 3 15 = 3 × 5 21 = 3 × 7

HCF is the product of the lowest powers of common prime factors: HCF = 3

LCM is the product of the highest powers of all prime factors involved: LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420

Explanation

Using the BODMAS rule, evaluate the expression step-by-step from the innermost brackets outward:

(6 + 3) = 9 (8 - 9) = -1 {4 - (-1)} = 4 + 1 = 5 [6 - 5] = 1

Substituting this result into the outer expression gives: 18 - 1 = 17

Explanation

Time taken by B alone = (xy) / (y - x), where x = 35 days (combined time of A and B) and y = 60 days (time taken by A alone).

Substituting the given values: Time taken by B alone = (35 × 60) / (60 - 35) = 2100 / 25 = 84 days.

Alternatively, taking the total work as LCM(35, 60) = 420 units: Daily work rate of A and B together = 420 / 35 = 12 units/day. Daily work rate of A alone = 420 / 60 = 7 units/day. Daily work rate of B alone = 12 - 7 = 5 units/day. Time taken by B alone = 420 / 5 = 84 days.

Explanation

Cross-multiplying the given equation gives: x² = √128 × √162

Simplifying the square roots: √128 = √(64 × 2) = 8√2 √162 = √(81 × 2) = 9√2

Substituting these back into the equation: x² = (8√2) × (9√2) = 8 × 9 × 2 = 144

Taking the square root on both sides: x = √144 = 12

Explanation

Using the standard algebraic identity (a - b)² = a² + b² - 2ab:

Substitute the given values (a - b) = 5 and (a² + b²) = 41:

5² = 41 - 2ab 25 = 41 - 2ab 2ab = 41 - 25 2ab = 16 ab = 8

Explanation

When a quantity is reduced by r%, the percentage increase required to return to the original value is given by the formula [r / (100 - r)] × 100%.

Substituting r = 20%:

Percentage increase = [20 / (100 - 20)] × 100% = (20 / 80) × 100% = 25%

Alternatively, assuming B's salary is ₹100, A's salary becomes ₹80. The difference in salary is ₹20. Expressed as a percentage of A's salary, this is (20 / 80) × 100% = 25%.

Explanation

Side of square = √(Area) = √4900 = 70 m.

Radius of circle = Side of square = 70 m.

Length of rectangle = (3/5) × 70 = 42 m.

Area of rectangle = Length × Breadth = 42 × 20 = 840 m².

Explanation

Selling Price = Cost Price × (100 + Gain%) / 100

Given Cost Price = ₹650 and Gain = 6%:

Selling Price = 650 × (100 + 6) / 100 Selling Price = 650 × 1.06 = ₹689

Explanation

For any number of the form (n + 0.5), its square is given by n(n + 1) + 0.25. Factoring the integer part 42 into consecutive integers gives 6 × 7 = 42. Therefore, (6.5)² = 6 × 7 + 0.25 = 42.25, which means √(42.25) = 6.5.

Explanation

By the Law of Cosines, cos C = (a² + b² - c²)/(2ab) and cos B = (c² + a² - b²)/(2ca).

Substituting these into the given expression:

a(b cos C - c cos B) = a [b(a² + b² - c²)/(2ab) - c(c² + a² - b²)/(2ca)] = a [(a² + b² - c²)/(2a) - (c² + a² - b²)/(2a)] = ½ [(a² + b² - c²) - (c² + a² - b²)] = ½ (2b² - 2c²) = b² - c²

Explanation

The number of pipes and the time taken to fill the tank are inversely proportional. Using the inverse proportion formula:

P₁ × T₁ = P₂ × T₂

Substituting the given values: 5 × 80 = 8 × T₂ 400 = 8 × T₂ T₂ = 400 / 8 = 50 minutes

Explanation

Total pizza bought = 3

Total pizza eaten by the four friends: John = 2/4 of a pizza Sarah, Tom, and Joane = 3 × 3/4 = 9/4 of a pizza

Total pizza eaten = 2/4 + 9/4 = 11/4 pizzas

Pizza remaining = 3 - 11/4 = 12/4 - 11/4 = 1/4 of a pizza

Explanation

To find the least number to be added to 4707, we first determine the least common multiple (LCM) of 4, 5, 6, and 8.

LCM(4, 5, 6, 8) = 120

Dividing 4707 by 120 gives a quotient of 39 and a remainder of 27: 4707 ÷ 120 = 39 with remainder 27

The smallest multiple of 120 greater than 4707 is: 120 × 40 = 4800

The least number that must be added to 4707 to make it divisible by 120 is: 4800 - 4707 = 93

Explanation

Using the algebraic identity x² - y² = (x + y)(x - y), where x and y are the two numbers:

243 = 27 × (x - y)

x - y = 243 / 27 = 9

Explanation

Simplify the left-hand side of the equation by using a common denominator of 10: 4/5 + (-3/10) = 8/10 - 3/10 = 5/10 = ½

Convert the mixed fraction on the right-hand side to an improper fraction: 1½ = 3/2

Substitute these values back into the equation: ½ = x + 3/2

Solve for x: x = ½ - 3/2 = -2/2 = -1

Explanation

Area of a triangle = ½ × base × height

Substituting the given values: Area = ½ × 6 cm × 8 cm = 24 cm²

Explanation

Convert each fraction into its decimal equivalent:

5/100 = 0.05 2/5 = 0.40 6/25 = 24/100 = 0.24

Substitute these values into the given expression:

0.05 + 0.40 - 0.24 = 0.45 - 0.24 = 0.21

Explanation

The standard intercept form of a straight line is x/a + y/b = 1, where a and b are the x-intercept and y-intercept respectively.

Dividing the line 5x + 7y = 35 by 35 gives: x/7 + y/5 = 1

Thus, the x-intercept a = 7 and the y-intercept b = 5.

The area of the triangle bounded by the line and the coordinate axes is: Area = ½ × a × b = ½ × 7 × 5 = 35/2 square units.

Explanation

The position of a point (x₁, y₁) with respect to a circle S ≡ x² + y² + 2gx + 2fy + c = 0 is determined by evaluating S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c. If S₁ > 0, the point lies outside the circle; if S₁ = 0, it lies on the circle; and if S₁ < 0, it lies inside the circle.

Substituting the point (3, 4) into the circle equation x² + y² - 3x - 4y + 1 = 0:

S₁ = (3)² + (4)² - 3(3) - 4(4) + 1 S₁ = 9 + 16 - 9 - 16 + 1 = 1

Since S₁ = 1 > 0, the point lies outside the circle.

Explanation

Following the BODMAS rule:

Evaluate the innermost expression in parentheses: 27 - 3 = 24 Evaluate the expression inside the square brackets: 24 ÷ 6 - 4 = 4 - 4 = 0 Perform multiplication: 17 × 8 = 136 Complete the remaining operations: 136 - 6 + 0 = 130

Explanation

The empirical relationship between mean, median, and mode is given by the formula:

Mode = 3 × Median - 2 × Mean

Substituting the given values into the formula: Mode = 3 × 5 - 2 × 4 Mode = 15 - 8 = 7

Explanation

Sum of the ratio terms = 11 + 16 + 23 = 50 units.

Given that 50 units = 150, 1 unit = 150 ÷ 50 = 3.

The greatest number corresponds to the largest ratio term 23: Greatest number = 23 × 3 = 69.

Explanation

Perimeter of a rectangle = 2(Length + Breadth)

Given that Length = 3 × Breadth and Perimeter = 240 m: 2(3 × Breadth + Breadth) = 240 2(4 × Breadth) = 240 8 × Breadth = 240 Breadth = 30 m Length = 3 × 30 = 90 m

Area of the rectangle = Length × Breadth = 90 × 30 = 2700 m²

Explanation

The sum of all exterior angles of any convex polygon is 360°. For a regular polygon with n sides, the measure of each exterior angle is given by the formula 360° / n. For a regular octagon (n = 8), each exterior angle is calculated as 360° / 8 = 45°.

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding altitudes.

Ratio of areas = (h₁ / h₂)² = (8 / 11)² = 64 / 121 = 64:121

Explanation

The lowest common multiple (LCM) of 4, 8, 12, and 16 is 48.

The smallest six-digit number is 100000. Dividing 100000 by 48 yields a quotient of 2083 with a remainder of 16.

To find the smallest six-digit number completely divisible by 48, add the difference between the divisor and the remainder to 100000: 100000 + (48 − 16) = 100000 + 32 = 100032.

Explanation

Using the trigonometric reduction identities sec(180° - θ) = -sec θ and cosec(180° - θ) = cosec θ:

sec 143° = sec(180° - 37°) = -sec 37° = -1 / cos 37° cosec 146° = cosec(180° - 34°) = cosec 34° = 1 / sin 34°

Substituting these into the given expression: cos 37° sec 143° + sin 34° cosec 146° = cos 37° × (-1 / cos 37°) + sin 34° × (1 / sin 34°) = -1 + 1 = 0

Explanation

To find the number of prime numbers between 109 and 121 (inclusive), check the primality of each integer in the range by testing divisibility by prime factors up to √121 = 11:

109 is not divisible by 2, 3, or 7, making it prime. 110 is divisible by 2. 111 is divisible by 3 (1 + 1 + 1 = 3). 112 is divisible by 2. 113 is not divisible by 2, 3, or 7, making it prime. 114 is divisible by 2. 115 is divisible by 5. 116 is divisible by 2. 117 is divisible by 3 (1 + 1 + 7 = 9). 118 is divisible by 2. 119 is divisible by 7 (7 × 17 = 119). 120 is divisible by 2. 121 is divisible by 11 (11 × 11 = 121).

Thus, there are exactly 2 prime numbers (109 and 113) in the given range.

Explanation

Number of cubes = Volume of cuboid / Volume of one cube

Number of cubes = (24 × 16 × 16) / (8 × 8 × 8) = 3 × 2 × 2 = 12

Explanation

Let the two numbers be x and y.

Given: x + y = 30 1/x + 1/y = 6/25

Since (1/x + 1/y) = (x + y) / xy, substituting x + y = 30 gives: 30 / xy = 6 / 25 xy = (30 × 25) / 6 = 125

The numbers x and y are roots of the quadratic equation: t² - (sum of roots)t + (product of roots) = 0 t² - 30t + 125 = 0 (t - 5)(t - 25) = 0 t = 5 or t = 25

The two numbers are 5 and 25.

Explanation

Let the initial number of male and female members be 4x and 5x respectively.

After increasing the male members by 80% and female members by 60%: New number of male members = 4x × (180 / 100) = 7.2x New number of female members = 5x × (160 / 100) = 8x

New ratio of male members to female members = 7.2x : 8x = 72 : 80 = 9 : 10

Explanation

The future population is calculated using the compound growth formula: P = P₀ × (1 + r/100)ⁿ, where P₀ is the present population, r is the annual growth rate, and n is the time in years.

Given: P₀ = 8,47,000, r = 20%, n = 2 years

P = 8,47,000 × (1 + 20/100)² P = 8,47,000 × (6/5)² P = 8,47,000 × 36 / 25 P = 33,880 × 36 = 12,19,680

Explanation

To minimize the total number of rows, each row must contain the maximum possible number of trees. Since each row must consist of trees of only one variety, the maximum number of trees per row is given by the Highest Common Factor (HCF) of 18, 21, and 39.

HCF(18, 21, 39) = 3

Minimum number of rows = (18 ÷ 3) + (21 ÷ 3) + (39 ÷ 3) = 6 + 7 + 13 = 26

Explanation

Let the interior angle be I and the exterior angle be E. At any vertex of a polygon, the interior and exterior angles form a linear pair, so I + E = 180°.

Given that I - E = 140°, subtracting this from I + E = 180° gives: 2E = 180° - 140° = 40° E = 20°

The number of sides n of a regular polygon is related to its exterior angle by n = 360° / E = 360° / 20° = 18.

Explanation

Present average age of the husband and wife = 42 + 10 = 52 years. Sum of their present ages = 52 × 2 = 104 years.

Sum of present ages of the husband, wife, and son = 39 × 3 = 117 years.

Present age of the son = 117 - 104 = 13 years.

Explanation

Let the length and breadth of the rectangle be 13x and 11x respectively.

Area = length × breadth 13x × 11x = 1287 143x² = 1287 x² = 9 x = 3

Perimeter = 2 × (length + breadth) Perimeter = 2 × (13x + 11x) = 48x Perimeter = 48 × 3 = 144 cm

Explanation

Using modular arithmetic, when 7 is divided by 6, the remainder is 1 (7 ≡ 1 mod 6).

Therefore, 7⁶ ≡ 1⁶ ≡ 1 mod 6.

Adding 5 to this gives (7⁶ + 5) ≡ (1 + 5) ≡ 6 mod 6.

Since 6 is completely divisible by 6, the final remainder is 0.

Explanation

The greatest six-digit number is 999999 and the smallest six-digit number is 100000.

Sum = 999999 + 100000 = 1099999

Explanation

Efficiency ratio of Rinku to Deepak = 2 : 1 Combined efficiency = 2 + 1 = 3 units per day

Total work = Combined efficiency × Days taken together = 3 × 28 = 84 units

Time taken by Rinku alone = Total work / Rinku's efficiency = 84 / 2 = 42 days