Mathematics and Science PYQs

Calculation:

1/75 = 0.01333

2/149 = 0.01342

3/224 = 0.01339

4/299 = 0.01337

6/449 = 0.01336

So, the largest and the smallest among the fractions \(\frac{2}{149} \) and \(\frac{1}{75} \)

∴ The correct answer is option 2

Calculations:

767 = 13 × 59 

2773 = 47 × 59 

LCM of 767 & 2773 = 13 × 47 × 59 = 36049 

HCF of 767 & 2773 = 59

According to Question Statement 

36049 = 59 × k 

k = 36049/59 = 611

Hence, The Correct Answer is 3.

Calculation:

\(\left(-\frac{3}{4}\right)^{-3}\div\left(-\frac{4}{3}\right)^7=\left[\left(-\frac{4}{3}\right)^2\right]^4×\left(\frac{3}{4}\right)^{5x+2} \)

⇒ \(-\frac{4^3}{3^3}\) × \(- \frac{3^7}{4^7}\) = \( \frac{4^8}{3^8}\) × \(\frac{3^{5x + 2}}{4^{5x + 2}}\) 

⇒ \(\frac{3^4}{4^4}\) = \(\frac{3^{5x + 2 - 8}}{4^{5x + 2 - 8}}\) 

Comparing we get,

(5x + 2 - 8) = 4

⇒ 5x - 6 = 4

⇒ x = 10/5 = 2

∴ The correct answer is 2

Calculation:

2.142 + 1.102 + 0.21 - 4.124 = k - 3.26

⇒ 3.454 - 4.124 = k - 3.26

⇒ - 0.67 = k - 3.26

⇒ k = 3.26 - 0.67

⇒ k = 2.59

Let's assume that x is subtracted from k, to make its value become 2.5

2.59 − x = 2.5

⇒ x = 2.59 - 2.5

⇒ x = 0.09

∴ The correct answer is 0.09

Calculation:

\(√[3]{16\times 1372}÷√{8\times288} \)

⇒ \(√[3]{21952}\)  ÷ √2304

⇒ 28 ÷ 48

⇒ 7/12

∴ The correct answer is \(\frac{7}{12}\)

Shortcut Trick

Numerator simplification: 2(3y − 4) + 6 = 6y − 2

Denominator simplification: 3y − (4 + 5y) = −2y − 4

Equation becomes: (6y − 2) / (−2y − 4) = 0.6 ⇒ 6y − 2 = −1.2y − 2.4

Solving for y: 7.2y = −0.4 → y = −1/18

Target value: 9 × (−1/18) + 2 = −0.5 + 2 = 1.5

∴ The correct answer is 1.5.

Alternate Method

Given: (2(3y − 4) + 6) / (3y − (4 + 5y)) = 0.6

Formula Used: Algebraic simplification and transposition.

⇒ (6y − 8 + 6) / (3y − 4 − 5y) = 0.6

⇒ (6y − 2) / (−2y − 4) = 0.6

⇒ 6y − 2 = 0.6 × (−2y − 4)

⇒ 6y − 2 = −1.2y − 2.4

⇒ 6y + 1.2y = −2.4 + 2

⇒ 7.2y = −0.4

⇒ y = −0.4 / 7.2 = −4 / 72 = −1/18

⇒ Required value = 9y + 2

⇒ 9 × (−1/18) + 2 = −1/2 + 2 = −0.5 + 2 = 1.5

∴ The correct answer is 1.5.

Additional Information

BODMAS Rule

Simplification follows the order: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Transposition Method

While moving terms across the equal sign, addition becomes subtraction, and multiplication becomes division.

Linear Equations

An equation with a single variable and a maximum power of 1, typically solved by isolating the variable on one side.

Calculation:

\(\rm \left(2x^2-3x-\frac{9}{x}\right)\left(-x+\frac{7}{x}\right) \)

⇒ - 2x³ + 3x² + 9 + 14x - 21 - 63/x²

⇒ - 2x³ +  3x2 + 9 + 14x - 21 - 63x^-2

Here, coefficients of x2 and x  are 3 and 14 respectively,

Their product = 3 × 14 = 42

∴ The correct answer is 42

Calculation:

Factors of x2 - 13x + 40 = (x - 8)(x - 5)

Factors of x2 + 3x - 40 = (x + 8)(x - 5)

Factors of x2 - 9x +20 = (x - 4)(x - 5)

So, the common factor among the factored expressions is (x - 5)

∴ The correct answer is (x - 5)

Concept used:

a² + b² = (a - b)² + 2ab

Calculation:

\(\rm \frac{25x}{y}+\frac{9y}{x}\)

⇒ (25x² + 9y²)/xy 

⇒ [(5x)² + (3y)²]/xy

⇒ [(5x - 3y)² + 30xy]/xy

⇒ [(35)² + 30 × (- 20)]/-20

⇒ [1225 - 600]/-20

⇒ 625/-20

⇒ \(-\frac{125}{4}\) 

∴ The correct answer is \(-\frac{125}{4}\)

Calculation:

The shopkeeper sells a toy by giving 20% discount, so MP : SP = 5 : 4

If he gains 25% profit, then CP : SP = 4 : 5

So, CP : SP : MP = 16 : 20 : 25

Now, 16 unit → 160, then 25 unit → 160/16 × 25 = 250

∴ The correct answer is Rs. 250

Calculation:

According to the given picture, BD and CD are the angle bisectors, 

So, ∠BDC = 90 + 1/2∠A

⇒ 108 = 90 + 1/2x

⇒ 1/2x= 108 - 90

⇒ x = 18 × 2

⇒ x = 36° 

∴ The correct answer is 36°

Calculation : 

In Option (4), we have AB + CD < AC.

This is not possible because the sum of the lengths of two sides of a triangle must be greater than the length of the third side.

Therefore, the construction of quadrilateral ABCD is not possible in this case.

In Option (1), we have AB + BC > CD, and ∠B + ∠C < 180°.

In Option (2), we have AB + BC > AC, and ∠A + ∠B + ∠C = 180°.

In Option (3), we have AD + CD > BC, and AB + AC > BD.

Therefore, the answer is (4).

Calculations:

In Δ ACB  & Δ DBC

BC = BC (Common)

∠ACB = ∠DBC (40°)

∠ABC = ∠DCB (60°)

Hence, Δ ACB ≅ Δ DBC (By ASA Angle Side Angle Criteria)

Concept used:

According to Pythagoras' theorem,

Hypotenuse² = Height² + Base²

Calculation:

From the given options, according to option 3

47² = 2209, 35² = 1225 and 12² = 144

So, 47² ≠ (35² + 12²)

∴ 12 cm, 35 cm and 47 cm cannot be the lengths of the sides of a right triangle

Concept use:

Euler's formula can be written as F + V = E + 2, where F is equal to the number of faces, V is equal to the number of vertices, and E is equal to the number of edges.

Calculations:

take option no 3 

F = 7, E = 14 and V = 10

Put the Values in the Equations 

F + V = E + 2

7 + 10 ≠ 14 + 2 satisfied

Hence, Option 3 is not true.

take option 1

F = 6, E = 10 and V = 6 

Put the Values in the Equations 

F + V = E + 2

6 + 6 = 10 + 2 satisfied

Hence, Option 1 is true

take option 2

F = (n + 2), E = 3n and V = 2n 

Put the Values in the Equations 

F + V = E + 2

n + 2 + 2n = 3n + 2 satisfied

Hence, option 2 is true.

take option 4

F = 11, E = 17 and V = 8 

Put the Values in the Equations 

F + V = E + 2

11 + 8 = 17 + 2 satisfied

Hence, option 4 is true

Hence, The Correct option is 3.

Concept used:

• Area of rectangle = Length × Breadth
• Perimeter of rectangle = 2(Length + Breadth)

Calculation:

The factor pairs of 70 are (Length × Breadth): 1 × 70, 2 × 35, 5 × 14, 7 × 10 

According to the factor pairs, all options are satisfies except option 3 which is 98

∴ The correct answer is 98

Concept used:

• Volume of right circular cylinder = π × r² × h
• Total surface area of right circular cylinder = 2πrh + 2πr²

Calculation:

According to the question,

2π8h + 2π8² = 272π 

⇒ 16h + 128 = 272

⇒ 16h = 272 - 128

⇒ 16h = 144

⇒ h = 144/16 

⇒ h = 9

Now, the Volume of the right circular cylinder = π × 82 × 9 = 576π 

∴ The correct answer is 576π

Given:

The areas of three adjacent faces are 108 cm2, 144 cm2 and 192 cm2

Formula Used:

Volume of cuboid = lbh

l = length, b = breadth and h = height 

Calculation:

According to the question 

lb = 108    -----(i)

bh = 144     -----(ii)

hl = 192    -----(iii)

On multiplying (i), (ii) and (iii), we get 

lb × bh × hl = 108 × 144 × 192 

⇒ (lbh)2 = 108 × 144 × 192 

⇒ (Volume of cuboid)2 = 36 × 3 × 144 × 3 × 64

⇒ (Volume of cuboid)2 = (6 × 3 × 12 × 8)2

⇒ Volume of cuboid = 1728 cm3

According to the question 

Volume of cuboid = 2/3 × 1728 = 1152 cm3

∴  The required answer will be 1152 cm³

Calculation:

If we arrange it in ascending order, we get

2, 3, 5, 9, 11, 13, 15, 17, 21

So, the median of the data set is x = 11

If we replace 3 and 21 by 16 and 14 we get in ascending order,

2, 5, 9, 11, 13, 14, 15, 16, 17

So, median of the data set is y = 13

Then the value of (2x - y) = (2 × 11 - 13) = 9

∴ The correct answer is 9

Calculatioin:

Counting multiples of 2:  26, 28, 30, 32,…,50 (all even numbers.)

Counting multiples of 3: 27, 30, 33, 36, …, 51 

There are some common multiples present in 2 and 3 both,

Total number of multiples of 2 is 13

Total number of multiples of 3 is 9

Total number of multiples of 6 (common multiples): 4

So, the total number of favorable outcomes is 13 + 9 − 4 = 18

The total number of possible outcomes is 51 − 26 + 1 = 26

Required probability = 18/26 = 9/13

∴ The correct answer is 9/13