Mathematics and Science PYQs

Calculation:

The least four-digit number is = 1000

Now, if we subtract 1 from 1000, we get = (1000 - 1) = 999

So, 999 is a three-digit greatest number

∴ The correct answer is option 2

Calculation:

The closure property states that a set of numbers (integers, Whole numbers, etc.) is closed under some operation (such as addition, subtraction, multiplication, etc.)

So, he/she explains closure property

∴ The correct answer is option 3

Calculation:

As n is a natural number, we have to make the smallest number in the form n(n + 1) (n + 3)

So, we will check by putting n = 1, 2,... [ as 1 is the smallest natural number ]

If we put n = 1, then the value of  n(n + 1) (n + 3) ⇒ 1 × 2 × 4 = 8,

So 8 is the least number can be written in the form n(n + 1) (n + 3)

∴ The correct answer is 8

Calculation:

Let's assume that the fraction is x, then its reciprocal is 1/x

Now, according to the question,

⇒ \(\frac{x \times x}{\frac{1}{x}}\) = \(18\frac{26}{27}\) 

⇒ x³ = \(\frac{512}{27}\)

⇒ x = \(\sqrt[3]{\frac{512}{27}} \) 

⇒ x = \(\frac{8}{3}\) = \(2\frac{2}{3}\) 

∴ The correct answer is \(2\frac{2}{3}\)

Calculation;

According to the question,

⇒ \(\frac{4}{3}\) × (\(2\frac{1}{4}\) - \(1\frac{2}{3}\)) + \(\frac{1}{2}\) × (\(2\frac{1}{3}\) - \(1\frac{2}{7}\) ) 

⇒ \(\frac{4}{3}\) × (\(\frac{9}{4}\) - \(\frac{5}{3}\)) + \(\frac{1}{2}\) × (\(\frac{7}{3}\) - \(\frac{9}{7}\) )

⇒ (\(\frac{4}{3}\) × \(\frac{7}{12}\))  + (\(\frac{1}{2}\) × \(\frac{22}{21}\)) 

⇒ \(\frac{7}{9}\)  + \(\frac{11}{21}\) 

⇒ \(\frac{82}{63}\)

∴ The correct answer is \(\frac{82}{63}\)

Concept used:

• The curved surface area of cylinder = 2πrh
• Volume  of cylinder = πr²h
• Total surface area of cylinder = 2πrh + 2πr²

Where, r= radius, h = height

Calculation:

According to the question,

⇒ 2πrh = 1/3 ×  (2πrh + 2πr²)

⇒ 6πrh = 2πrh + 2πr2)

⇒ 4πrh = 2πr2

⇒ h = r/2 = 8/2 = 4 cm

Then volume of the cylinder = π × 8² × 4 = 256 π

∴ The correct answer is 256 π

Concept used:

Circumference of a circle = 2πr, where r = radius and π = 22/7

Calculation:

According to the question,

⇒ 2πr = 440

⇒ r = 440 × 7/22 × 1/2

⇒ r = 70 m

The circular race track is 7 m wide, then the outer radius = (70 + 7) = 77 m

∴ The correct answer is 77 m

Concept used:

• The curved surface area of cylinder = 2πrh
• The area of the base of cylinder = πr2

Where, r= radius, h = height

Calculation:

According to the question,

⇒ πr2 = b ⇒ r = \(\sqrt\frac{b}{π}\)    … (i)

Again 

⇒ 2πrh = a 

⇒ 2 × π × \(\sqrt\frac{b}{π}\) × h = a

⇒ h = \(\frac{a}{2π}\) × \(\sqrt\frac{π}{b}\)   [ rationalizing π we get ]

⇒ h =  \( \frac{a}{2 \sqrt{π b}} \)

∴ The correct answer is \( \frac{a}{2 \sqrt{π b}} \) cm

Concept used:

We know, tri means 3 and nomial means term, then trinomial has three terms

Calculation:

Now, from the given options 3x2 + 3x + 7 is the trinomial

∴ The correct answer is 3x2 + 3x + 7

Shortcut Trick

Check options where the denominator is 7 more than the numerator: 44/51, 13/20, and 4/11 fit.

Apply the second condition to 44/51: Increase numerator by 20 and decrease denominator by 3.

New numerator = 44 + 20 = 64; New denominator = 51 − 3 = 48.

Resulting fraction = 64 / 48 = 4 / 3 (after dividing by 16).

Since 44/51 satisfies both conditions, it is the rational number.

∴ The correct answer is 44/51.

Alternate Method

Given:

Let the numerator be x.

The denominator is x + 7.

Formula Used:

New Numerator / New Denominator = Resulting Value

Calculations:

⇒ Original Number = x / (x + 7)

⇒ (x + 20) / ((x + 7) − 3) = 4 / 3

⇒ (x + 20) / (x + 4) = 4 / 3

⇒ 3 × (x + 20) = 4 × (x + 4)

⇒ 3x + 60 = 4x + 16

⇒ 60 − 16 = 4x − 3x

⇒ 44 = x

⇒ Numerator = 44

⇒ Denominator = 44 + 7 = 51

⇒ Rational Number = 44 / 51

∴ The correct answer is 44/51.

Additional Information

Rational Numbers

A rational number is defined as a number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

Cross-Multiplication Property

For any two fractions a/b and c/d, if a/b = c/d, then a × d = b × c. This property is vital for solving linear equations involving fractions.

Fraction Simplification

A fraction is in its simplest form if the numerator and denominator have no common factors other than 1. For example, 64/48 = (16 × 4) / (16 × 3) = 4/3.

Concept used:

We know, 6% = 3/50, so 6% loss means CP : SP = 50 : 47

Calculation:

Here 6% loss means CP : SP = 50 : 47

According to the question,

⇒ SP = 47 unit → 3290 then

⇒ CP = 50 unit → 3290/47 × 50 = Rs. 3500

∴ The correct answer is Rs. 3500

Calculation:

According to the question,

⇒ x3y2 + x4 + x2y3 + y⁴ - [(- 3x3y2 + 2x2y3) + (- 5y4 - 3x2y³)]

⇒ x3y2 + x4 + x2y3 + y4 - [- 3x3y2 + 2x2y3 - 5y4 - 3x2y³]

⇒ x3y2 + x4 + x2y3 + y4 - [- 3x3y2 - 5y4 - x2y³]

⇒ x3y2 + x4 + x2y3 + y4 + 3x3y² + 5y4 + x2y3

⇒ 4x3y2 + 2x2y3 + 6y4  + x⁴

Here the coefficient of y4 is 6

∴ The correct answer is 6

Shortcut Trick

Substitute an easy value for x, say x = 0, to quickly find the correct option.

Original expression at x = 0: [15 × (−15)] / [25 × 3 × (−25)] = −225 / −1875 = 3 / 25

Check the given options at x = 0: Option 4 yields 3 / [5(0 + 5)] = 3 / 25

Since the evaluated values match, Option 4 is the correct simplified expression.

∴ The correct answer is 3 / [5(x + 5)].

Alternate Method

Given:

Expression = 15(x² − 2x − 15) / [25(x + 3)(x² − 25)]

Formula Used:

a² − b² = (a − b)(a + b)

Calculations:

⇒ Factorise the quadratic expression in the numerator:

⇒ x² − 2x − 15 = x² − 5x + 3x − 15 = x(x − 5) + 3(x − 5) = (x − 5)(x + 3)

⇒ Factorise the term in the denominator:

⇒ x² − 25 = x² − 5² = (x − 5)(x + 5)

⇒ Substitute these back into the original fraction:

⇒ [15 × (x − 5)(x + 3)] / [25 × (x + 3)(x − 5)(x + 5)]

⇒ Cancel the common terms (x − 5) and (x + 3) from numerator and denominator:

⇒ 15 / [25 × (x + 5)]

⇒ Divide the constant terms by 5:

⇒ 3 / [5(x + 5)]

∴ The correct answer is 3 / [5(x + 5)].

Additional Information

Factorising Quadratic Polynomials

To factorise ax² + bx + c, find two numbers that multiply to ac and add to b, then split the middle term.

Difference of Squares

The algebraic identity a² − b² = (a − b)(a + b) is frequently used to simplify fractions involving squared terms.

Value Substitution Method

For multiple-choice questions involving algebraic expressions, substituting a convenient numerical value (like 0 or 1) can rapidly identify the correct option without full simplification.

Concept used:

Three numbers will be Pythagorean triplet when, (Greatest number)² = sum of the square of smaller two numbers

Calculation:

According to option 2, 

⇒ (12)² + (35)² 

⇒ 144 + 1225

⇒ 1369

⇒ (37)²

∴ The correct answer is 12, 35, 37

Calculation:

A polyhedron means a hexagonal prism with 8 faces and 12 vertices have 18 edges

∴ The correct answer is 18

Calculation:

According to the picture, ABCD is an isosceles trapezium where side AD and BC are parallel and AB = DC

Then Only non-parallel sides are of equal length.

∴ The correct answer is option 3

Calculation:

 1 right-angle = 90° 

Then, (3 + \(\frac{1}{2}\)) = 7/2 right-angle = 90 × 7/2 = 315° 

∴ The answer is 315

Shortcut Trick

Sum of ratios = 1 + 2 + 3 + 4 = 10 units.

Total interior angles of a quadrilateral = 360°.

10 units → 360°

1 unit → 36°

Greatest angle = 4 units = 4 × 36° = 144°.

∴ The correct answer is 144°.

Alternate Method

Given:

Ratio of angles = 1 : 2 : 3 : 4

Formula Used:

The sum of all interior angles of a quadrilateral is always 360°.

 

Let the angles of the quadrilateral be 1x, 2x, 3x, and 4x.

⇒ 1x + 2x + 3x + 4x = 360°

⇒ 10x = 360°

⇒ x = 360° ÷ 10

⇒ x = 36°

⇒ Smallest angle = 1x = 36°

⇒ Second angle = 2x = 72°

⇒ Third angle = 3x = 108°

⇒ Greatest angle = 4x = 4 × 36° = 144°

∴ The correct answer is 144°.

Additional Information

Sum of Interior Angles of a Polygon

The sum of the interior angles of any n-sided polygon is given by the formula (n − 2) × 180°.

Cyclic Quadrilateral Property

In a cyclic quadrilateral, the sum of each pair of opposite angles is 180°.

Regular Polygon Angles

For a regular polygon, each interior angle is [(n − 2) × 180°] ÷ n, where n is the number of sides.

Calculation:

Here, the mean of 10 numbers is 96,

Then, total of the 10 numbers = 96 × 10 = 960

If one member is 150, then the total of  remaining 9 members = (960 - 150) = 810

Now, the mean of the remaining nine numbers = 810/9 = 90

∴ The correct answer is 90

Concept used:

Probability of an event P(A) = (number of favourable events)/(total number of events)

Calculation:

If a dice is thrown, the total number of events (1, 2, 3, 4, 5, 6) is 6

So, the total number of events that have prime number is (2, 3, 5) is 3

So, the required  probability = 3/6 = 1/2

∴ The correct answer is 1/2