Mathematics and Science PYQs

Given:

The ratio of these numbers is 11 ∶ 17

Calculation:

Let's assume that the two numbers are 11k and 17k

So, LCM of (11k, 17k) = 187k

Now according to the question,

187k = 4675

⇒ k = 4675/187

⇒ k = 25

So, two numbers are (11 × 25) = 275 and (17 × 25) = 425

∴ The correct answer is option 2

Calculation:

Let's assume that if we added x to \(\frac{19}{27}\), it becomes \(-\frac{13}{15}\) 

⇒ x + \(\frac{19}{27}\) = \(-\frac{13}{15}\) 

⇒ 27x + 19 = \(-\frac{13}{15}\) × 27

⇒ 27x + 19 =  \(-\frac{13}{5}\) × 9

⇒ 135x + 95 = - 117

⇒ 135x = - 212

⇒ x = \(-​​\frac{212}{135}\)

∴ The correct answer is \(-​​\frac{212}{135}\)

Concept used:

In the Roman numeral system, we cannot have two of the same letter in a row, unless that letter is I, X, C, or M.

Calculation:

The Roman numeral L represents 50.

It is not possible that two L's in a row, because that would represent 100, which is greater than 50.

So, LLXXX is not a valid Roman numeral.

∴ The correct answer is LLXXX

Calculation:

\(​​\left(\frac{16}{81}\right)^{\frac{5}{4}-\frac{3}{4}}=\left(\frac{2}{3}\right)^x \)

⇒ \(​​\left(\frac{16}{81}\right)^{\frac{2}{4}}=\left(\frac{2}{3}\right)^x \) 

⇒ \(\sqrt{​​\left(\frac{16}{81}\right)}= \left(\frac{2}{3}\right)^x \)

⇒ \(\frac{4}{9}\) = \(\frac{2}{3}^x \) 

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

So, comparing we get, x = 2

Now, the value of 3x + 4 = (3 × 2 + 4) = 10

∴ The correct answer is 10

Calculation:

√2352 = √(4 × 588) = 2√588

√9408 = √(16 × 588) = 4√588

√49 = 7

Now, √2352 × √9408 ÷ √49

⇒ 2√588 × 4√588 ÷ 7 

⇒ \(\frac{8 × 588}{7}\) 

⇒ 8 × 84

⇒ 672

∴ The correct answer is 672

Shortcut Trick

Substitute x = −1 in the equation: (3 − 1) × (1 + 1/3 + 13/3) + (19/3 + 1/3) = A − B + C + D

⇒ 2 × (17/3) + 20/3 = 54/3 = 18 ⇒ A − B + C + D = 18

Substitute x = 0 in the equation: (−1) × (13/3) + 19/3 = D ⇒ D = 2

⇒ Required value = (A − B + C − D) = (A − B + C + D) − 2D = 18 − 4 = 14

∴ The correct answer is 14.

Alternate Method

Given:

(3x² − 1)(x² − x/3 + 13/3) + (19/3 − x/3) = Ax⁴ + Bx³ + Cx² + D

Method Used:

Expand the Left Hand Side (LHS) and compare the coefficients of like powers of x with the Right Hand Side (RHS).

Calculations:

⇒ LHS = 3x²(x² − x/3 + 13/3) − 1(x² − x/3 + 13/3) + 19/3 − x/3

⇒ LHS = 3x⁴ − x³ + 13x² − x² + x/3 − 13/3 + 19/3 − x/3

⇒ LHS = 3x⁴ − x³ + 12x² + (x/3 − x/3) + (19/3 − 13/3)

⇒ LHS = 3x⁴ − x³ + 12x² + 6/3

⇒ LHS = 3x⁴ − x³ + 12x² + 2

Comparing this with the RHS (Ax⁴ + Bx³ + Cx² + D), we get:

⇒ A = 3, B = −1, C = 12, D = 2

We need to find the value of (A − B + C − D):

⇒ A − B + C − D = 3 − (−1) + 12 − 2

⇒ 3 + 1 + 12 − 2 = 14

∴ The correct answer is 14.

Additional Information

Polynomial Identity

If two polynomials are identically equal for all values of x, then the coefficients of their corresponding terms are exactly equal.

Substitution Method

By substituting convenient values like x = 0, x = 1, or x = −1 into an identity, we can directly find the constant term, sum of coefficients, or alternating sum of coefficients without fully expanding the entire expression.

Shortcut Trick

To get the terms 2x and 1/(3x), multiply the given equation by 2/3.

⇒ (2/3) × [3x − 1/(2x)] = 4 × (2/3) ⇒ 2x − 1/(3x) = 8/3

Direct formula: If ax − 1/(bx) = k, then a²x² + 1/(b²x²) = k² + 2a/b

⇒ 4x² + 1/(9x²) = (8/3)² + 2 × (2/3) = 64/9 + 4/3 = 64/9 + 12/9

⇒ 4x² + 1/(9x²) = 76/9

∴ The correct answer is 76/9.

Alternate Method

Given:

3x − 1/(2x) = 4

Formula Used:

(a − b)² = a² + b² − 2ab

Calculations:

We need to find the value of 4x² + 1/(9x²), which can be written as (2x)² + 1/(3x)².

Multiply the given equation by 2/3 on both sides to match the coefficients:

⇒ (2/3) × [3x − 1/(2x)] = 4 × (2/3)

⇒ 2x − 1/(3x) = 8/3

Squaring both sides, we get:

⇒ [2x − 1/(3x)]² = (8/3)²

⇒ (2x)² + 1/(3x)² − 2 × (2x) × [1/(3x)] = 64/9

⇒ 4x² + 1/(9x²) − 4/3 = 64/9

⇒ 4x² + 1/(9x²) = 64/9 + 4/3

⇒ 4x² + 1/(9x²) = (64 + 12) / 9

⇒ 4x² + 1/(9x²) = 76/9

∴ The correct answer is 76/9.

Additional Information

Algebraic Identities

(a + b)² = a² + 2ab + b²

Important Deductions

If x + 1/x = k, then x² + 1/x² = k² − 2

Inverse Term Squares

If ax − 1/(bx) = k, then a²x² + 1/(b²x²) = k² + 2a/b

Calculation:

Factors of x2 + 4x - 60 = (x + 10)(x - 6)

Factors of x2 - x - 30 = (x - 6)(x + 5)

Factors of x2 - 11x + 30 = (x - 6)(x - 5)

Factors of x2 - 16x + 60 = (x - 10)(x - 6)

The common factors among all four expressions are (x - 6)

∴ The correct answer is (x - 6)

Calculation:

\(\frac{6 x+1}{3}=\frac{7 x-3}{2}-1\)

⇒ \(\frac{6 x+1}{3}=\frac{7 x-3-2}{2}\) 

⇒ 12x + 2 = 21x - 9 - 6

⇒ 21x - 12x = 17

⇒ 9x = 17

⇒ x = 17/9

So, \(\frac{9 x-1}{9 x+1}\) = \(\frac{9 \times \frac{17}{9}-1}{9 \times \frac{17}{9}+1}\) = 16/18 = \(\frac{8}{9}\) 

∴ The correct answer is \(\frac{8}{9}\)

Concept used:

4% = 1/25 and \(6\frac{2}{3}\)% = 1/15

Calculation:

The shopkeeper loses 4%, so CP : SP = 25 : 24

According to the question, 24 unit → 230.40, then 25 unit → 230.40/24 × 25 = 240, so, CP = 240

Now, to gain \(6\frac{2}{3}\)%, CP : SP = 15 : 16

So, 15 unit → 240, then 16 unit → 240/15 × 16 = 256

∴ The correct answer is ₹256

Shortcut Trick

Identify the Right angle, Hypotenuse, and Side (RHS) for both triangles.

In ΔPQR and ΔABC: ∠P = ∠A = 90°, Hypotenuse QR = BC = 6.5 cm, and Side RP = AB = 2.5 cm.

Match vertices: Since ∠P = ∠A, P corresponds to A. Since RP = AB and P → A, then R corresponds to B.

The remaining vertex Q corresponds to C.

Therefore, the congruence relation is ΔPQR ≅ ΔACB by the RHS rule.

∴ The correct answer is Δ PQR ≅ Δ ACB, by RHS.

Alternate Method

Given:

In ΔPQR: ∠P = 90°, RP = 2.5 cm, QR = 6.5 cm

In ΔABC: ∠A = 90°, AB = 2.5 cm, BC = 6.5 cm

Formula Used:

RHS Congruence Rule: Two right-angled triangles are congruent if the hypotenuse and one side of one triangle are equal to the corresponding hypotenuse and side of the other triangle.

P
R
Q
2.5 cm
6.5 cm

Explanations:

In a kite, the diagonals do not bisect each other or the opposite angles. However, in a rhombus, the diagonals bisect each other at right angles, forming congruent opposite angles.

So, if the diagonal of a quadrilateral bisects the opposite angles or bisects each other then it is rhombus.

Rhombus:

Kite:

Hence, The Correct Answer is option 3.

Calculation:

Using Euler's formula to find the value of x

F + V - E = 2

⇒ 17 + x - 30 = 2

⇒ x = 32 - 17

⇒ x = 15

Now, the value of 3x - 10 = (3 × 15 - 10) = 35

∴ The correct answer is 35

Concept used:

If the sum of two angles is 180°, then the angles are supplementary angles

Calculation:

According to the question, the smaller angle is x, then other angle is 3x

(3x + x) = 180° 

⇒ 4x = 180°

⇒ x = 180/4

⇒ x = 45° 

Then the value of 5x - 25° = (5 × 45 - 25) = 200°

∴ The correct answer is 200°

Concept used:

To construct a triangle the sum of the length of any two sides must be greater than the other.

Calculation:

According to the given options, 

In the Option 2, (AB + BC) = 10 = CA which is not possible as (AB + BC) must be > CA

∴ The correct answer is option 2

Concept used:

• The total surface area of cylinder = 2πrh + 2πr²

• Volume of cylinder = πr²h

Calculation:

According to the question,

2πrh + 2πr² = 1628   

⇒ 2πr(h + r) = 1628  

⇒ 2πr × 37 = 1628

⇒ r = 1628/37 × 1/2 × 7/22

⇒ r = 7

So, height = h = (37 - 7) = 30

The volume of the cylinder = πr2h = 22/7 × 7 × 7 × 30 = 4620 cm3

∴ The correct answer is 4620 cm3

Concept used:

Area of triangle = 1/2 × Base × Height

Calculation:

Let's assume altitude = x, then base = 3x

According to the question,

⇒ 1/2 × 3x × x = 600

⇒ x² = 600 × 2/3

⇒ x² = 400

⇒ x = 20

So, length of the base = (3 × 20) = 60 cm

∴ The correct answer is 60 cm

Concept used:

• Volume of cube = side³

• Lateral surface area = 4 × side²

Calculation:

According to the question,

side³ = (3³ + 4³ + 5³)

⇒ side3 = 216

⇒ side = 6

So, the lateral surface area of the new cube = 4 × 6² = 144

∴ The correct answer is 144

Shortcut Trick

Sorted Data: 5, 7, 8, 10, 22, 24, 25, 27, 27, 55

Mean = 210 ÷ 10 = 21

Median = (22 + 24) ÷ 2 = 23

Mode = 27 (most frequent value)

Sum = 21 + 23 + 27 = 71

∴ The correct answer is 71.

Alternate Method

Given: Data = 7, 5, 8, 10, 22, 27, 24, 25, 27, 55

Formula: Mean = Σx ÷ n; Median = (n/2)^th + (n/2+1)^th / 2; Mode = Most frequent value.

⇒ Ascending Order: 5, 7, 8, 10, 22, 24, 25, 27, 27, 55

⇒ Mean = (5 + 7 + 8 + 10 + 22 + 24 + 25 + 27 + 27 + 55) ÷ 10 = 210 ÷ 10 = 21

⇒ Median = (22 + 24) ÷ 2 = 23 (Average of 5^th and 6^th terms)

⇒ Mode = 27 (Value with highest frequency)

⇒ Sum = 21 + 23 + 27 = 71

∴ The correct answer is 71.

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Shortcut Trick

Shortcut for sum on 2 dice: The number of ways to get a sum S is (S − 1) for S ≤ 7.

Ways to get sum 3 = 3 − 1 = 2

Ways to get sum 5 = 5 − 1 = 4

Total favourable ways = 2 + 4 = 6

Total possible outcomes = 6 × 6 = 36 → Probability = 6 ÷ 36 = 1/6

∴ The correct answer is 1/6.

Alternate Method

Given:

Two dice are thrown once by Mohit and Rohit.

Formula Used:

Probability P(E) = Number of favorable outcomes n(E) ÷ Total number of possible outcomes n(S)

Calculation:

⇒ Total possible outcomes when 2 dice are thrown, n(S) = 6 × 6 = 36

⇒ Let E be the event that the sum of the numbers appearing on the tops is 3 or 5.

⇒ Favorable outcomes to get a sum of 3 = (1, 2) and (2, 1) → 2 cases

⇒ Favorable outcomes to get a sum of 5 = (1, 4), (2, 3), (3, 2), and (4, 1) → 4 cases

⇒ Total favorable outcomes, n(E) = 2 + 4 = 6

⇒ Required Probability P(E) = 6 ÷ 36

⇒ P(E) = 1/6

∴ The correct answer is 1/6.

Additional Information

Total Outcomes for Dice

When 'n' dice are thrown simultaneously, the total number of possible outcomes is given by 6ⁿ.

Sum of Probabilities

The sum of the probabilities of an event happening and not happening is always equal to 1: P(E) + P(not E) = 1.

Mutually Exclusive Events

When two events cannot occur at the same time, the probability of either event occurring is the sum of their individual probabilities: P(A or B) = P(A) + P(B).