Mathematics and Science PYQs

Calculation:

According to option 1,

A positive number = 2, and a negative number = -3, but sum of these 2 + (-3) = -1 ≠ 0, [ to hold option 1, two numbers must be same] Option 1 is false

According to option 2,

The sum of two odd numbers is (3 + 3) = 6 which is an even number [ Option 2 is false ]

According to option 3,

The sum of two even numbers is (4 +6) = 10 is an even number (Option 3 is true)

 According to option 4,

Product of two prime numbers (2 × 5) = 10 not a prime number (Option 4 is false)

∴ The correct answer is option 3

Given:

The numbers 0.515, 0.5, 0.06, 0.52, 0.053

Calculation:

According to the question, ascending order means small to large

So,  0.053 < 0.06 < 0.5 < 0.515 < 0.52 

∴ The correct answer is option 2

Shortcut Trick

Count the decimal places in 0.000081, which is 6.

The square root of a number with 2n decimal places will have exactly n decimal places.

Here, half of 6 is 3, so the result must have 3 decimal places.

Since √81 = 9, the final answer with 3 decimal places is 0.009.

∴ The correct answer is 0.009.

Alternate Method

Given: √0.000081

Formula Used: √(a ÷ b) = √a ÷ √b

⇒ 0.000081 = 81 ÷ 1,000,000

⇒ √0.000081 = √(81 ÷ 1,000,000)

⇒ √81 ÷ √1,000,000

⇒ 9 ÷ 1,000

⇒ 0.009

∴ The correct answer is 0.009.

Additional Information

Square Root of Decimals

When finding the square root of a decimal, the number of digits after the decimal point in the answer is always half of those in the original number.

Powers of 10

For any even power of 10, √10²ⁿ = 10ⁿ. For example, √10,000 = √10⁴ = 10² = 100.

Perfect Squares

Memorizing perfect squares from 1 to 30 (e.g., 9² = 81, 25² = 625) significantly speeds up simplification tasks.

Concept used:

We know, a^-1 = \(\frac{1}{a}\), where a be any integer

Calculation:

(6-1 - 7-1)-1 - (2-1 - 3-1)-1 ÷ (-6)-1

⇒ \((\frac{1}{6} - \frac{1}{7})^{-1} - (\frac{1}{2} - \frac{1}{3})^{-1} ÷ (-6)^{-1}\) 

⇒ \((\frac{1}{42})^{-1} - (\frac{1}{6})^{-1} ÷(-6)^{-1}\)

⇒ 42 - 6 ÷ \((-\frac{1}{6})\)   [ BODMAS rule ]

⇒ 42 -  6 × (- 6)

⇒ 42 - (- 36)

⇒ 42 + 36

⇒ 78

∴ The correct answer is 78

Concept used:

If A be the HCF of any two numbers, then the numbers are Ax and Ay and then LCM = Axy, it means LCM is a multiple of HCF and LCM > HCF

Calculation:

According to option 2, the number 342 is completely divisible by 38, and LCM = 342 is greater than HCF = 38 [ so the option 2 is true ]

∴ The correct answer is option 2

Calculation:

60% of 400 + k% of 280 = 296

⇒ (6 × 40) + (\(\frac{k}{100}\) × 280) = 296

⇒ \(\frac{k}{10}\) × 28 = 296 - 240

⇒ k = 560/28 = 20

∴ The correct answer is 20

Calculation:

According to the question,

[(4a  - 7ab - 4b - 5) + (a + 3) (b - 4)] - (12a - 9ab - 2)

⇒ [4a  - 7ab - 4b - 5 + ab - 4a + 3b - 12] - (12a - 9ab - 2)

⇒ - 6ab - b - 17 - 12a + 9ab + 2

⇒ -12a + 3ab - b - 15

∴ The correct answer is option 3

Shortcut Trick

Convert Time to minutes: 2 hr 15 min = (2 × 60) + 15 = 135 min.

Apply Inverse Proportion: P₁ × T₁ = P₂ × T₂

4 × 135 = 5 × T₂

T₂ = (4 × 135) ÷ 5 = 108 min.

Convert 108 min back to hours: 108 min = 1 hr 48 min.

∴ The correct answer is 1 hr 48 min.

Alternate Method

Given:

Initial number of pipes (P₁) = 4

Initial time taken (T₁) = 2 hr 15 min

Final number of pipes (P₂) = 5

Formula Used:

For identical pipes filling the same tank, Total Work = Number of Pipes × Time Taken.

Calculations:

⇒ T₁ = 2 × 60 + 15 = 135 minutes

⇒ Constant Work = P₁ × T₁ = 4 × 135 = 540 pipe-minutes

⇒ Time for 5 pipes (T₂) = Total Work ÷ P₂

⇒ T₂ = 540 ÷ 5

⇒ T₂ = 108 minutes

⇒ T₂ in hours = 108 ÷ 60 = 1 hr 48 min

∴ The correct answer is 1 hr 48 min.

Additional Information

Inverse Proportion Rule

If two quantities are such that an increase in one leads to a proportional decrease in the other, they are inversely proportional. The relation is expressed as x₁y₁ = x₂y₂.

Pipes and Efficiency

The rate of filling a tank is directly proportional to the number of pipes (of same type) and inversely proportional to the time taken to fill it.

Time Conversion

To convert minutes to hours and minutes, divide the total minutes by 60. The quotient is the hour part and the remainder is the minutes part.

Given:

Initially in the class Boys : Girls = 2 : 3,

Calculation:

Let's assume that the no of boys and girls are 2k and 3k respectively,

Now, according to the question,

\(\frac{2x - 3}{3x + 3} = \frac{3}{7}\)

⇒ 14x - 21 = 9x + 9

⇒ 5x = 30

⇒ x = 6

Then initially the no of girls is (3 × 6) = 18

∴ The correct answer is 18

Given:

\(\rm x^4+\frac{1}{x^4}=727\)

Concept used:

• If \(x^4+\frac{1}{x^4}\) = a, then \(x^2+\frac{1}{x^2} = √{a +2}\)
• If \(x+\frac{1}{x}\) = a, the \(\rm \left(x-\frac{1}{x}\right)\) = √(a² - 4)

Calculation:

\(\rm x^4+\frac{1}{x^4}=727\) then \(x^2+\frac{1}{x^2} = √{(727 +2)}\) = 27

Similarly, \(x^2+\frac{1}{x^2} = 27\) then \(x+\frac{1}{x}\) = √(27 + 2) = √29

Now, \(x+\frac{1}{x}\) = √29, then \(\rm \left(x-\frac{1}{x}\right)\) = √(29 - 4) = √25 = 5

∴ The correct answer is 5

Calculation:

As we know the sum of the three angles of a triangle is 180° 

Now, sum of the given angles = (30 + 40 + 110) = 180°

We can take the sides as a, b, c in ratio and Angles given as the same 

now take another triangle whose sides are 2a, 2b, 2c in the ratio and Angles given as same in this way we can make infinitely many triangles 

∴ The correct answer is infinitely many.

Given:

AB = 3.1 cm AC = 4.5 cm PR = 3.1 cm QR = 4.5 cm ∠PRQ = 60° ∠BAC = 60°

Concept used:

Two triangles are congruent if their corresponding sides and angles are equal.

Calculation:

Comparing the corresponding sides:

AB = PR (3.1 cm = 3.1 cm)

AC = QR (4.5 cm = 4.5 cm)

Comparing the corresponding angles:

∠BAC = ∠RPQ (60° = 60°)

Using the Side-Angle-Side (SAS) congruence criterion:

Based on the information given, we have two pairs of corresponding sides and one pair of corresponding angles that are equal.

Therefore, we can conclude that ΔABC is congruent to ΔRPQ.

Hence, the correct statement is: ΔABC ≅ ΔRPQ (Option 2).

Given:

Side BC is produced to point D and side CA is produced to point E.

If ∠BAE = 115° and ∠ACD = 104°

Calculation:

According to quetion,

⇒ ∠ACB =180° - 104°  = 76° 

⇒ ∠BAC = 180° - 115° = 65° 

Now by using angle sum property : 

⇒ ∠A + ∠B + ∠C = 180° 

⇒  ∠B + 76° +  65° = 180°

⇒ ∠B = 39° 

Out of ∠ABC and ∠BAC second angle is greater

Difference =  65°  - 39° = 26° 

∴ The correct answer is 26°

Given:

The sides of the triangle are 3 cm, 4 cm, and (x + 1) cm.

Concept used:

For a triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side.

Calculation:

 According to the triangle inequality, for a triangle with sides a, b, and c, the following conditions must be satisfied:

⇒ a + b > c ; b + c > a ; c + a > b

Let's apply this to the given triangle:

⇒ 3 + 4 > (x + 1)

=> 7 > (x + 1) 4 + (x + 1) > 3

=> x + 5 > 3 (x + 1) + 3 > 4

=> x + 4 > 4

 From the first inequality:  : 7 > (x + 1) => 6 > x => x < 6

From the second inequality: x + 5 > 3 => x > -2

From the third inequality: x + 4 > 4 => x > 0

Combining the results:

x must satisfy the following conditions: -2 < x < 6

x can take the values from 1 cm to 5 cm.

Therefore, x can take the value/values from 1 cm to 5 cm. (Option 3)

Given:

The ratios of the angles in the quadrilateral are 3:5:4:6.

Concept used:

The sum of the angles in a quadrilateral is always equal to 360 degrees.

Calculation:

Finding the angles:

Let the angles be 3x, 5x, 4x, and 6x, where x is a common multiplier.

Sum of the angles = 3x + 5x + 4x + 6x = 18x.

Equating the sum of the angles to 360 degrees:

⇒ 18x = 360

⇒ x = 360 / 18

⇒ x = 20

Finding the values of the angles:

Angle 1 = 3x = 3 * 20 = 60 degrees

Angle 2 = 5x = 5 * 20 = 100 degrees

Angle 3 = 4x = 4 * 20 = 80 degrees

Angle 4 = 6x = 6 * 20 = 120 degrees

Finding the sum of the smallest and greatest angles:

Sum = Angle 1 + Angle 4

⇒ 60 + 120

⇒ 180 degrees.

Therefore, the sum of the smallest and greatest angles of the quadrilateral is 180 degrees.

Given:

Length of the hall = 60 m Breadth of the hall = 32 m Height of the hall = 8 m

Concept used:

The area of a rectangular surface, such as a wall, can be calculated using the formula: Area = Length * Height.

Calculation:

Finding the area of one of the longer walls:

The length of the longer wall is the same as the length of the hall, which is 60 m.

The height of the wall is the same as the height of the hall, which is 8 m.

Area of one of the longer walls = Length * Height

⇒  60 m * 8 m

⇒ 480 m².

Therefore, the area of one of the longer walls is 480 m².

Given:

Base diameter is 8 cm.

Height is \(\frac{7}{8}\) of the base radius

Formula used:

Voulme of cylinder = πr²h

Calculation:

According to question,

r = 8/2 = 4 cm

h = 7/8 × 4 = 7/2

Volume of cylinder = 22/7 × 4 × 4 × 7/2

⇒ 22 × 16/2

⇒ 22 × 8 = 176

∴ The correct answer is 176 cm³.

Given:

Cost of painting a cube on all the faces at the rate of Rs. 3 per cm2 is Rs. 1,152.

Formula used:

Surface area of cube = 6a²

Calculation:

According to question:

Surface area of cube = 1152/3 = 384

⇒ 384 = 6a²

⇒ a² = 64

⇒ a = +8, -8

∴ The correct answer is 8.

Given:

Numbers: 10, 8, 2, 7, 3, 8, 5, 1

Concept used:

To find the median of a set of numbers, we arrange the numbers in ascending order and find the middle value. If the total number of values is odd, the median is the middle value. If the total number of values is even, the median is the average of the two middle values.

Calculation:

Arranging the numbers in ascending order:

⇒ 1, 2, 3, 5, 7, 8, 8, 10

F inding the median (k): Since the total number of values is 8, the median is the average of the two middle values:

⇒ (5 + 7) / 2 = 6.

Replacing 10 with 1 and arranging the new set of numbers in ascending order:

⇒ 1, 1, 2, 3, 5, 7, 8, 8

Finding the new median (r): Since the total number of values is still 8, the median is the average of the two middle values:

⇒ (3 + 5) / 2 = 4.

Calculating (k - r): (k - r)

⇒ 6 - 4 = 2.

Therefore, the value of (k - r) is 2.

Calculation:

Numbers divisible by 7 will be : 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98

Probability = Number of favorable outcomes/total number of outcomes

Probability(Numbers not divisible by 7)  = 1 - 14/100

⇒ 86/100 = 43/50

∴ The correct answer is 43/50.