Mathematics and Science PYQs

Given:

Digits available: 2, 4, 7, 9, 8

To form the greatest 5-digit number, we need to place the largest digit in the leftmost position.

The greatest 5 - digit Number will be 98742

The largest digit available is 9. Therefore, the leftmost digit will be 9.

Now, we need to determine the place value of the third digit from the Left, which is 7.

The place value of each digit in a 5-digit number is as follows:

Ten-thousands place (10,000)

Thousands place (1,000)

Hundreds place (100)

Tens place (10)

Units place (1)

Since we are looking for the place value of 7, it will be in the hundreds place (100).

Therefore, the place value of 7 in the greatest 5-digit number formed using the digits 2, 4, 7, 9, and 8 is 700.

The correct answer is 700.

Calculation:

\(\left(-2 \frac{1}{3} \div \frac{7}{3} × \frac{2}{7}\right) \)

⇒ -7/3 × 3/7 × 2/7

⇒ -2/7

Reciprocal of -2/7

⇒ -7/2

⇒ \(-3\frac{1}{2}\)

∴ The correct answer is \(-3\frac{1}{2}\).

Calculation:

Here for a number to be divide by 99 it must be divided by 11 and 9.

By going through the options,

⇒ 2707056

Sum = 27 and difference of digit at odd and even place = -11

So it is divided by 99

⇒ 2703459

Sum = 30  and difference of digit at odd and even place = 0

So it is not divided by 99

⇒ 2112462

Sum = 18 and  difference of digit at odd and even place = 0

So it is divided by 99

⇒ 2409759

Sum = 36 and difference of digit at odd and even place = 0

So it is divided by 99

∴The correct answer is option 2.

Calculation:

\(\left(a^{-1}+b^{-1}\right) ×(a+b)^{-1}\)

⇒ (1/a + 1/b)[1/(a + b)]

⇒ (9/7 + 13/5)[1/(7/9 + 5/13)]

⇒ [(45 + 91)/35][117/136]

⇒ 136/35 × 117/136

⇒ 117/35

∴ The correct answer is 117/35.

Shortcut Trick

Prime factorization of 196 = 14 × 14 = 2² × 7².

For a perfect cube, every prime factor power must be a multiple of 3.

Required factors = 2⁽³⁻²⁾ × 7⁽³⁻²⁾ = 2¹ × 7¹ = 14.

∴ The correct answer is 14.

Alternate Method

Given: Number = 196

Method: Prime Factorization

⇒ 196 = 2 × 98

⇒ 196 = 2 × 2 × 49

⇒ 196 = 2 × 2 × 7 × 7

⇒ To make it a perfect cube, we need triplets of each prime factor.

⇒ We have two 2s, so we need one more 2 (2 × 2 × 2).

⇒ We have two 7s, so we need one more 7 (7 × 7 × 7).

⇒ Least number to be multiplied = 2 × 7 = 14.

∴ The correct answer is 14.

Additional Information

Perfect Cube Definition

A perfect cube is an integer that can be expressed as the cube of another integer (n³). In its prime factorization, all exponents must be divisible by 3.

Perfect Square Property

For a number to be a perfect square, all exponents in its prime factorization must be even (divisible by 2). For example, 196 = 2² × 7² is a perfect square of 14.

Smallest Multiplier for nth Power

To make a number a perfect k^th power, multiply by factors such that each prime's exponent p becomes the smallest multiple of k ≥ p.

Calculation:

⇒ 3x - 2(2x - 5) = 2(x + 3) - 8

⇒ 3x - 4x + 10 = 2x + 6 - 8

⇒ -x + 10 = 2x - 2

⇒ 3x = 12

⇒ x = 4

Now putting the value in \(x^2+6 x-5 \)

⇒ 16 + 24 - 5

⇒ 40 - 5 

⇒ 35

∴ The correct answer is 35.

Given:

The initial amount of money: ₹1350

Number of tickets he can buy with the initial amount: 5

Number of tickets he wants to buy: 7

Calculation:

Cost of one ticket = Total amount of money / Number of tickets

Cost of one ticket = ₹1350 / 5 = ₹270

Additional money required to buy 7 tickets = Cost of one ticket × (Number of tickets he wants to buy - Number of tickets he can buy)

= ₹270 × (7 - 5)

= ₹270 × 2 = ₹540

Therefore, the person will need ₹540 more if he buys 7 tickets.

The correct answer is ₹540.

Given:

x⁴ - (x - y)⁴

Concept used:

We will expand the expression using the binomial expansion formula.

Calculation:

x4 - (x - y)4

⇒ (x²)² - [(x - y)²]²

⇒ [x² - (x - y)²] [x² + (x - y)²]

⇒ [x² - (x² + y² - 2xy)] [x² + x² + y² - 2xy]

⇒ [- y² + 2xy] [2x² + y² - 2xy]

⇒ y(2x - y) [2x2 + y2 - 2xy]

∴ The correct answer is y(2x - y) [2x2 + y2 - 2xy].

Given:

(x + y)(2x + y) + (x + 2y)(x - y)

Concept used:

We will expand the given expression and identify the coefficient of y2.

Calculation:

(x + y)(2x + y) + (x + 2y)(x - y)

= (2x² + xy + 2xy + y²) + (x² - xy + 2xy - 2y²)

= 2x² + 3xy + y² + x² + xy - 2y²

Combining like terms:

= 3x² + 4xy - y²

The coefficient of y² is -1.

Therefore, the correct answer is -1.

Concept used:

• (x² - y²) = (x + y)(x - y)

Calculation:

\(\frac{2 x^3-2 x y^2+x^2 y-y^3}{(x-y)} \)

⇒ \(\frac{2x(x^2 - y^2) + y(x^2 - y^2)}{(x-y)} \) 

⇒ \(\frac{(x^2 - y^2)(2x +y)}{(x - y)} \)

⇒ \(\frac{(x +y)(x - y)(2x +y)}{(x - y)} \)

⇒ (x + y)(2x + y)

∴ The correct answer is (x + y)(2x + y)

Shortcut Trick

In ΔABD, AD = BD ⇒ ∠ABD = ∠BAD = 30°.

Exterior ∠BDC = ∠BAD + ∠ABD = 30° + 30° = 60°.

In ΔBCD, BC = CD and ∠BDC = 60°, so it is an equilateral triangle (∠CBD = ∠BCD = 60°).

∠ABC = ∠ABD + ∠CBD = 30° + 60° = 90°.

Total Sum = ∠ABC + ∠BDC = 90° + 60° = 150°.

∴ The correct answer is 150°.

Alternate Method

Given: ΔABC with D on AC. AD = BD, CD = BC, and ∠BAC = 30°.

Formula Used:

1. Isosceles triangle property: Angles opposite to equal sides are equal.
2. Exterior angle theorem: Exterior angle of a triangle is equal to the sum of two opposite interior angles.
3. The sum of the three angles of a triangle is 180°

Calculations:

In ΔABD, since ∠BAC = 30°, therefore ∠ABD = 30° [AD = BD]

Therefore, in ΔABD, ∠ADB = 180 - (30 + 30) = 120°

Therefore, ∠BDC = 180 - 120 = 60° [since ADC is a straight line]

Now, in ΔDBC, since ∠BDC = 60°, therefore ∠DBC = 60°

Then, ∠ABC + ∠BDC

⇒ (∠ABD + ∠DBC) + ∠BDC

⇒ (30 + 60) + 60

⇒ 150°

∴ The correct answer is 150°

Shortcut Trick

Properties of parallel lines intersected by a transversal:

1. Corresponding angles: Lie on the same side of the transversal and are equal.
2. Alternate interior/exterior angles: Lie on opposite sides of the transversal and are equal.
3. Consecutive interior angles (Same side interior): Lie on the same side and are supplementary (sum = 180°).

Checking

Statements:

(A) is false (same side), (B) is true (supplementary), (C) is true (opposite sides), (D) is false (they are equal, not supplementary).

∴ The correct answer is Only (B) and (C).

Alternate Method

Given: Two parallel lines (say l and m) and a transversal line t.

Formula Used: Geometric properties of angles formed by a transversal intersecting parallel lines.

Calculation:

In this figure, l and m are parallel lines and t is a transversal.

Here, ∠6 corresponds to ∠4. They lie on the same side of the transversal. Therefore, statement (A) is false.

Also, ∠2 and ∠4 are co-interior angles on the same side of the transversal, and

⇒ ∠2 + ∠4 = 180°

Therefore, statement (B) is true.

Also, alternate interior angles ∠2 and ∠1 lie on opposite sides of the transversal. Therefore, statement (C) is true.

∴ The correct answer is option 2.

Additional Information

Interior Angles Sum

The sum of consecutive interior angles on the same side of the transversal is always 180° if the lines are parallel.

Alternate Angles Equality

Alternate interior angles are equal in measure. They are located between the two lines but on opposite sides of the transversal.

Vertically Opposite Angles

Angles opposite to each other at the intersection point of two lines are always equal, regardless of whether the lines are parallel.

Linear Pair Postulate

Two adjacent angles forming a straight line are supplementary, meaning their measures sum to 180°.

Given:

Three angles of a quadrilateral is 3 ∶ 6 ∶ 5 and its fourth angle is of 80°

Concept used:

Sum of four angles of a quadrilateral is 360°

Calculation:

Let's assume the three angles are 3x, 6x and 5x

According to the question,

⇒ (3x + 6x + 5x) + 80° = 360 ° 

⇒ 14x = 280

⇒ x = 20° 

Now, the difference between the greatest and the smallest angles

⇒ (6x - 3x) = 3x = 3 × 20 = 60°

Given:

F, V and E be the number of faces, vertices and edges of a pentagonal pyramid

Concept used:

A pentagonal pyramid has edges = 10, vertices = 6 and faces = 6

Calculation:

Here, F = 6, V = 6 and E = 10, then

⇒ 4E + 2F - 3V

⇒ 4 × 10 + 2 × 6 - (3 × 6)

⇒ 52 - 18

⇒ 34

∴ The correct answer is 34

Concept used:

In two triangles, two pairs of corresponding angles and the corresponding sides between them are equal then the triangles are congruent by ASA

Calculation:

In ΔABC,  ∠A = 50°, ∠C = 75° then ∠B = 55° and AB = 5

In ΔPQR,  ∠Q = 55°, ∠R = 50° then ∠P = 75° and QR = 5

In ΔABC and ΔPQR,  AB = QR and ∠A = ∠R, ∠Q = ∠B 

Then by ASA rule,\(\Delta \mathrm{PQR} \equiv \Delta \mathrm{CBA}\)

∴ The correct answer is option 4

Concept used:

• Area of parallelogram = Base × Height
• Area of rectangle = Length × Breadth
• Perimeter of the rectangle = 2( Length + Breadth)

Calculation:

Area of the parallelogram = 6 × (6 + 5) = 66 cm²

Now, according to the question,

Area of rectangle = Area of parallelogram

⇒ 22 × B = 66

⇒ B = 3 cm

Now, the perimeter of the rectangle = 2(L + B) =  2(22 + 3) = 50 cm

∴ The correct answer is 50 cm

Given:

The radius of the base of cylindrical = 35 cm and height = 2 m = 200 cm

Concept used:

The capacity of the cylinder = Volume of the cylinder = π × r² × h

Calculation:

Here, r = 35, h = 200 and π = 22/7

The capacity of the cylinder

⇒ 22/7 × 35² × 200

⇒ 22 × 5 × 35 × 200

⇒ 770000 cm³ = 770 m³ (litres)

∴ The correct answer is 770 litres

Shortcut Trick

Diagonal d₂ = (2 × Area) ÷ d₁ = (2 × 240) ÷ 16 = 30 m.

Half-diagonals are 8 m and 15 m. Using Pythagorean triplet (8, 15, 17), side = 17 m.

Perimeter = 4 × Side = 4 × 17 = 68 m.

Total Cost = 68 × 21 = ₹ 1428.

∴ The correct answer is ₹ 1428.

Alternate Method

Given: Area = 240 m², Diagonal d₁ = 16 m, Rate of fencing = ₹ 21/m.

Formula Used: Area = (1/2) × d₁ × d₂ | Side (a) = √[(d₁/2)² + (d₂/2)²]

d1/2 = 8 m
d2/2 = 15 m
Side (a)
90°
Area = 240 m²
Perimeter = 4 × Side

⇒ Area = (1/2) × d₁ × d₂

⇒ 240 = (1/2) × 16 × d₂

⇒ 240 = 8 × d₂

⇒ d₂ = 30 m

⇒ Side a = √[(16/2)² + (30/2)²]

⇒ Side a = √[8² + 15²] = √[64 + 225]

⇒ Side a = √289 = 17 m

⇒ Perimeter of field = 4 × Side = 4 × 17 = 68 m

⇒ Cost of fencing = Perimeter × Rate = 68 × 21

⇒ Cost of fencing = ₹ 1428

∴ The correct answer is ₹ 1428.

Additional Information

Properties of Rhombus

A rhombus is a quadrilateral with all sides equal. Its diagonals bisect each other at right angles (90°). Relation: 4a² = d₁² + d₂².

Perimeter and Costing

Perimeter of a rhombus = 4 × side. Fencing cost is calculated by multiplying the perimeter by the rate per unit length.

Pythagorean Triplets in Rhombus

Since diagonals bisect at 90°, each small triangle formed is a right-angled triangle. Common triplets like (8, 15, 17) and (5, 12, 13) are frequently used to find sides from diagonals.

Given:

The observations = 45, 50, 96, 68, 58, 96 and 94

Concept used:

• Median is the middle value of a data set
• Mode is the value that occurs most often
• Mean is the average

Calculation:

1^st we arrange the observations in ascending order i.e 45, 50, 58, 68, 94, 96, 96

The median will be the 4^th data of the seven observations, so Median = 68

Mode = 96 [ as it occurs 2 times]

So, Mean of the two values 68 and 96 is (68 + 96)/2 = 82

∴ The correct answer is 82

Concept used:

P(A) = (Number of favorable outcomes)/(Total no of outcomes)

Calculation:

Total prime numbers from 23 to 49 is 7 and total numbers are 27

Then probability of getting a prime number = 7/27 

∴ The correct answer is 7/27