CDS-I 2025 (Elementary Mathematics) Official Paper (Held On: 13 Apr, 2025) (Previous Year Paper)

Given:

A real number M is squared to give the value N. Find the minimum value of (M + N).

Formula used:

N = M²

Minimize (M + N) = M + M²

Calculation:

Let f(M) = M + M²

To find the minimum value, take the derivative and set it to zero:

f'(M) = 1 + 2M

⇒ 1 + 2M = 0

⇒ M = -0.5

Substitute M = -0.5 into f(M):

f(M) = M + M²

⇒ f(-0.5) = (-0.5) + (-0.5)²

⇒ f(-0.5) = -0.5 + 0.25

⇒ f(-0.5) = -0.25

∴ The minimum value of (M + N) is -0.25, and the correct answer is option 1.

Given:

We need to find the sum of all 3-digit numbers that give a remainder of 5 when divided by 50.

Formula used:

Sum of arithmetic progression (AP): S_n = (n/2) × (a + l)

Where,

a = First term, l = Last term, n = Number of terms

Calculation:

The 3-digit numbers that give a remainder of 5 when divided by 50 are: 105, 155, 205, ..., 955.

a = 105, l = 955

Common difference (d) = 50

n = (l - a)/d + 1

⇒ n = (955 - 105)/50 + 1

⇒ n = 850/50 + 1

⇒ n = 17 + 1

⇒ n = 18

Sum of all terms:

S_n = (n/2) × (a + l)

⇒ S₁₈ = (18/2) × (105 + 955)

⇒ S₁₈ = 9 × 1060

⇒ S₁₈ = 9540

∴ The correct answer is option (2).

Given:

The average of 64, 69, 72, 75, x lies between 62 and 76 (excluding 62 and 76).

Formula used:

Average = $\dfrac{\text{Sum of all numbers}}{\text{Total numbers}}$

Calculation:

Let the sum of the given numbers = 64 + 69 + 72 + 75 + x = 280 + x

The average = $\dfrac{280 + x}{5}$

Since the average lies between 62 and 76:

⇒ $62 < \dfrac{280 + x}{5} < 76$

⇒ 310 < 280 + x < 380

⇒ 30 < x < 100

The possible integer values of x are from 31 to 99 (both inclusive).

Total number of integers = 99 - 31 + 1 = 69

∴ The correct answer is option (2).

Given:

(x + y + z) = k

(x + z – y) × (x – z + y) ∝ yz

Formula used:

Algebraic manipulation and properties of proportionality.

From (x + y + z) = k, we can derive:

x + z – y = (x + y + z) – 2y = k – 2y

x – z + y = (x + y + z) – 2z = k – 2z

Calculations:

Given that (x + z – y) × (x – z + y) ∝ yz

⇒ (k – 2y) × (k – 2z) = C × yz (where C is the constant of proportionality)

⇒ k² – 2kz – 2ky + 4yz = C × yz

⇒ k² – 2k(z + y) + 4yz = C × yz

⇒ k² – 2k(y + z) = C × yz – 4yz

⇒ k² – 2k(y + z) = (C – 4)yz

Now, let's rearrange to find what (y + z – x) is proportional to.

From (x + y + z) = k, we have y + z = k – x.

Substitute (y + z) in the equation k² – 2k(y + z) = (C – 4)yz:

⇒ k² – 2k(k – x) = (C – 4)yz

⇒ k² – 2k² + 2kx = (C – 4)yz

⇒ –k² + 2kx = (C – 4)yz

⇒ k(2x – k) = (C – 4)yz

We are interested in (y + z – x).

We know (y + z – x) = (k – x) – x = k – 2x

From k(2x – k) = (C – 4)yz, we can write:

⇒ –k(k – 2x) = (C – 4)yz

⇒ k – 2x = $\dfrac{(C – 4)yz}{-k}$

⇒ k – 2x = $\dfrac{(4 – C)yz}{k}$

Since k, C are constants, $\dfrac{(4 – C)}{k}$ is also a constant.

Therefore, (k – 2x) is proportional to yz.

Since (y + z – x) = k – 2x,

∴ (y + z – x) is proportional to yz.

Given:

p = remainder when 7⁸⁴ is divided by 342

q = remainder when 7⁸⁴ is divided by 344

Formula used:

Modular arithmetic and properties of remainders.

If a ≡ b (mod m), then aⁿ ≡ bⁿ (mod m)

aⁿ - bⁿ is divisible by a - b

Calculations:

For p:

We need to find the remainder when 7⁸⁴ is divided by 342.

Notice that 342 = 7³ - 1 (since 7³ = 343).

We know that aⁿ - bⁿ is divisible by a - b.

Consider 7³ ≡ 1 (mod 342) because 7³ - 1 = 343 - 1 = 342, which is divisible by 342.

Now, we have 7⁸⁴ = (7³)²⁸.

⇒ 7⁸⁴ ≡ 1²⁸ (mod 342)

⇒ 7⁸⁴ ≡ 1 (mod 342)

So, p = 1.

For q:

We need to find the remainder when 7⁸⁴ is divided by 344.

Notice that 344 = 7³ + 1 (since 7³ = 343).

We know that aⁿ + bⁿ is divisible by a + b when n is odd.

We also know that aⁿ - bⁿ is divisible by a + b when n is even.

Consider 7³ ≡ -1 (mod 344) because 7³ + 1 = 343 + 1 = 344, which is divisible by 344.

Now, we have 7⁸⁴ = (7³)²⁸.

⇒ 7⁸⁴ ≡ (-1)²⁸ (mod 344)

⇒ 7⁸⁴ ≡ 1 (mod 344)

So, q = 1.

Value of (p – q):

⇒ p – q = 1 – 1

⇒ p – q = 0

∴ (p – q) is equal to 0.

Given:

N is the smallest 5-digit number which leaves a remainder of 1 when divided by 2, 2², 2³, 2⁴, ..., 2ⁿ

Formula used:

N ≡ 1 (mod 2^k) for all k = 1 to n

⇒ N - 1 is divisible by LCM of all 2^k for k = 1 to n

LCM(2, 4, 8, ..., 2ⁿ) = 2ⁿ

Calculations:

N - 1 = multiple of 2ⁿ

N = 2ⁿ × k + 1

We want the smallest N ≥ 10000

Try values of n:

n = 13 ⇒ 2¹³ = 8192

⇒ N = 8192 × 2 + 1 = 16385

n = 14 ⇒ 2¹⁴ = 16384

⇒ N = 16384 × 1 + 1 = 16385

n = 14 follows the condition.

∴ The value of n is 14.

Given:

The expression is:

$\frac{1}{532900} + \frac{p^{2}}{266450} + \frac{p^{4}}{532900}$

Calculation:

$\frac{1}{532900} + \frac{p^{2}}{266450} + \frac{p^{4}}{532900}$

⇒ $\frac{1}{532900} + \frac{p^2}{266450} + \frac{p^4}{532900} = \frac{1}{532900} + \frac{2p^2}{532900} + \frac{p^4}{532900}$

⇒ $\frac{1 + 2p^2 + p^4}{532900}$

⇒ $\frac{(p^2 + 1)^2}{(730)^2}$

Since both numerator and denominator both are perfect square and the result is an integer, minimum possible value of expression is 1

So. p² + 1 = 730

p2 = 730 - 1 = 729

p = 27

The minimum value of p for which the expression is an integer is 27.

Given:

Equation: x + a + b = $(\dfrac{abx}{ab + ax + bx})$

Roots of the equation are α and β.

Formula used:

For a quadratic equation of the form Ax2 + Bx + C = 0, with roots α and β:

Sum of roots: α + β = $-\dfrac{B}{A}$

Product of roots: αβ = $\dfrac{C}{A}$

Calculations:

First, rearrange the given equation into the standard quadratic form Ax2 + Bx + C = 0.

x + a + b = $(\dfrac{abx}{ab + ax + bx})$

Multiply both sides by (ab + ax + bx):

⇒ (x + a + b)(ab + ax + bx) = abx

⇒ x(ab) + x(ax) + x(bx) + a(ab) + a(ax) + a(bx) + b(ab) + b(ax) + b(bx) = abx

⇒ abx + ax² + bx² + a²b + a²x + abx + ab² + abx + b²x = abx

⇒ ax² + bx² + abx + a²x + abx + abx + b²x − abx + a²b + ab² = 0

⇒ (a + b)x² + (a²x + b²x + 2abx) + (a²b + ab²) = 0

⇒ (a + b)x² + (a² + b² + 2ab)x + ab(a + b) = 0

Recognize that (a² + b² + 2ab) = (a + b)²:

⇒ (a + b)x² + (a + b)²x + ab(a + b) = 0

⇒ x² + (a + b)x + ab = 0

Sum and product of the roots (α and β):

Sum of roots: α + β = $(-\dfrac{B}{A}) = -\dfrac{(a+b)}{1}$ = −(a + b)

Product of roots: αβ = $\dfrac{C}{A} = \dfrac{ab}{1}$ = ab

Finally, calculate (αβ + α + β):

⇒ (αβ + α + β) = ab + (−(a + b))

⇒ (αβ + α + β) = ab − a − b

∴ The correct answer is option 2.

Given:

Let the 2-digit number be N = 10×a + b, where:

a = digit in tens place

b = digit in units place

P = a × b

Formula used:

Condition 1: P + a² = 84

Condition 2: P + b² = 60

Calculations:

From (1): P = 84 − a²

From (2): P = 60 − b²

⇒ 84 − a² = 60 − b²

⇒ 24 = a² − b²

⇒ (a − b)(a + b) = 24

Try factor pairs of 24: (1,24), (2,12), (3,8), (4,6)

Try a − b = 2 and a + b = 12

⇒ Add: 2a = 14 ⇒ a = 7, b = 5

Check P = a × b = 7 × 5 = 35

P + a² = 35 + 49 = 84 

P + b² = 35 + 25 = 60

N = 10×7 + 5 = 75

P = 35

P + N = 35 + 75 = 110

∴ The correct answer is: 110

Given:

The initial mixture contains kerosene and turpentine oil in the ratio 3:2.

Total mixture = 100 L

Kerosene = (3/5) × 100 = 60 L

Turpentine oil = (2/5) × 100 = 40 L

Final kerosene percentage = 20%

Formula used:

Final kerosene in the mixture = Total mixture × (Final kerosene percentage / 100)

Additional kerosene to be added = Final kerosene - Initial kerosene

Calculation:

Let the additional kerosene to be added = x L.

Total mixture after adding kerosene = 100 + x L

Kerosene in the new mixture = 60 + x L

For 20% kerosene in the final mixture:

⇒ (60 + x) / (100 + x) = 20 / 100

⇒ (60 + x) = (20 / 100) × (100 + x)

⇒ 60 + x = 20 × (100 + x) / 100

⇒ 60 + x = (2000 + 20x) / 100

⇒ 100(60 + x) = 2000 + 20x

⇒ 6000 + 100x = 2000 + 20x

⇒ 6000 - 2000 = 20x - 100x

⇒ 4000 = -80x

⇒ x = -4000 / 80

⇒ x = -50

∴ It is not possible to achieve 20% kerosene in the final mixture. The correct answer is option (4).

Given:

Height of pole = PQ = $\dfrac{5\sqrt{3}}{2}$ m.

BQ = 0.7m

CQ = 2.4m

We need to find length AP.

Formula used:

Distance from the corner point = $\sqrt{(x^2 + y^2 + h^2)}$

Where, x = 0.7 m, y = 2.4 m, and h = height of the lamp = $\dfrac{5\sqrt{3}}{2}$ m.

Calculation:

We can see AQ² = BQ² + AB² and AB = CQ (Shown in fig.)

AP² = AQ² + PQ²

AP² = BQ2 + AB2 + PQ2

AP = $\sqrt{BQ^2 + AB^2 + PQ^2}$

The distance of the top of the lamp from the corner point (AP) is:

⇒ Distance = $\sqrt{(0.7^2 + 2.4^2 + \left(\dfrac{5\sqrt{3}}{2}\right)^2)}$

⇒ Distance = $\sqrt{(0.49 + 5.76 + \dfrac{75}{4})}$

⇒ Distance = $\sqrt{(0.49 + 5.76 + 18.75)}$

⇒ Distance = $\sqrt{25}$

⇒ Distance = 5 m.

∴ The distance of the top of the lamp from the corner point is 5 meters.

Given:

C is the center of a circle with radius = 20 cm.

AB is a chord of length = 32 cm.

CE = 13 cm, E is a point on AB.

We need to find AE × EB.

Formula used:

In a circle, for any chord AB and point E on AB:

AE × EB = (Radius² - Distance from center to chord²)

Calculation:

Radius = 20 cm, so Radius² = 20² = 400

Distance from center to chord (CE) = 13 cm, so CE² = 13² = 169

⇒ AE × EB = Radius² - CE²

⇒ AE × EB = 400 - 169

⇒ AE × EB = 231

∴ The correct answer is option (1).

Given:

The inside of the bowl is part of a sphere.

Depth of water (d) = d

Radius of the water surface = 2d

Formula used:

Using the geometry of spheres, the radius of the sphere (R) is given by:

R = $\frac{(r^2 + d^2)}{2d}$

Where r = radius of the water surface = 2d

Calculation:

R = $\frac{(r^2 + d^2)}{2d}$

⇒ R = $\frac{((2d)^2 + d^2)}{2d}$

⇒ R = $\frac{(4d^2 + d^2)}{2d}$

⇒ R = $\frac{5d^2}{2d}$

⇒ R = 2.5d

∴ The correct answer is option (1).

Alternate Method

Given:

Depth of water = d

Radius of water surface = 2d

Let Radius of sphere = R

Formula used:

By geometry of sphere cross-section:

$(R - d)^2 + (2d)^2 = R^2$

Calculations:

⇒ (R - d)² + (2d)² = R²

⇒ (R² - 2Rd + d²) + 4d² = R²

⇒ R² - 2Rd + 5d² = R²

⇒ -2Rd + 5d² = 0

⇒ 2Rd = 5d²

⇒ R = (5d / 2) = 2.5d

∴ Radius of sphere = 2.5d

Given:

AB = 2 cm, BC = 4 cm, AC = 3 cm

Formula used:

Angle bisector theorem: If a bisector of an angle divides the opposite side of a triangle, the ratio of the divided parts equals the ratio of the other two sides.

BD/DC = AB/AC

Calculations:

In Δ ABC,

AD is the bisector of ∠A

So, BD/DC = AB/AC = 2/3

Since, BC = 4 cm

So, BD = 2/5 × 4 = 8/5 cm

Now, In Δ ABD

BE is the bisector of ∠B

So, BD/AB = ED/AE

⇒ (8/5) / 2 = ED/AE

⇒ 4/5 = ED/AE

⇒ 5/4 = AE / ED

So, AE : ED = 5 : 4

Hence, the correct answer is option (1).

Given:

AB + AC = 10 cm

BD : DC = 3 : 1

Formula used:

Angle bisector theorem: If a bisector of an angle divides the opposite side of a triangle, the ratio of the divided parts equals the ratio of the other two sides.

BD/DC = AB/AC 

Calculation:

Since AD is the angle bisector of ∠A,

So, AB/AC = BD/DC 

⇒ AB/AC = 3/1

Since AB + AC = 10 cm

So, AC = 1/(1 + 3) × 10 = 10/4 = 2.5 cm

Thus, the length of AC is 2.5 cm.

The correct answer is option (1).

Given:

AB + BC = 7.1 cm

BC + CA = 12.1 cm

CA + AB = 7.2 cm

Calculation:

Let the sides of the triangle be:

AB = a

BC = b

CA = c

We are given the following system of equations:

a + b = 7.1

b + c = 12.1

c + a = 7.2

(a + b) + (b + c) + (c + a) = 7.1 + 12.1 + 7.2

2a + 2b + 2c = 26.4

a + b + c = 13.2

c = 13.2 - 7.1 = 6.1

a = 13.2 - 12.1 = 1.1

b = 13.2 - 7.2 = 6.0

Heron’s formula for the area of a triangle is given by:

$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$

Where s is the semi-perimeter of the triangle, given by:

s = $\frac{a + b + c}{2} = \frac{1.1 + 6.0 + 6.1}{2} = \frac{13.2}{2} = 6.6$ cm

$\text{Area} = \sqrt{6.6(6.6 - 1.1)(6.6 - 6.0)(6.6 - 6.1)}$

$\text{Area} = \sqrt{6.6 \times 5.5 \times 0.6 \times 0.5}$

$\text{Area} = \sqrt{6.6 \times 5.5 \times 0.3}$

$\text{Area} = \sqrt{10.89}$

$\text{Area} = 3.3 , \text{cm}^2$

Therefore, the area of the triangle is: 3.3 cm²

Given:

Adjacent sides of a parallelogram: 10 cm and 8 cm

Angle between them: 150^∘

Formula used:

Area of parallelogram = a × b × sin(θ)

Where:

a = one side, b = adjacent side, θ = angle between sides

Calculation:

Area = 10 × 8 × sin(150^∘)

⇒ sin(150^∘) = sin(30^∘) = 1/2

⇒ Area = 10 × 8 × (1/2)

⇒ Area = 40 cm²

∴ The correct answer is option (2).

Given:

Triangle ABC

Bisectors of angles A and C intersect and form an angle = 130°

Find angle B

Formula used:

Angle between bisectors of A and C inside triangle = 90° + B/2

Calculations:

Given, angle between bisectors = 130°

Using formula: 90 + B/2 = 130

⇒ B/2 = 130 - 90

⇒ B/2 = 40

⇒ B = 80°

∴ Angle B = 80°.

Given:

$\log_{10}2000 + \log_{10}400 + 4\log_{10}25 + 5\log_{10}20$

Formula used:

$\log a + \log b = \log(ab)$

$n\log a = \log(a^n)$

Calculation:

$\log_{10}2000 + \log_{10}400 + 4\log_{10}25 + 5\log_{10}20$

= $\log_{10}(2000 \times 400 \times 25^4 \times 20^5)$

⇒ $2000 = 2^4 \times 5^3$,   $400 = 2^4 \times 5^2$,   $25^4 = (5^2)^4 = 5^8$,   $20^5 = (2^2 \times 5)^5 = 2^{10} \times 5^5$

⇒ Product = $(2^4 \times 5^3) \times (2^4 \times 5^2) \times (5^8) \times (2^{10} \times 5^5)$

⇒ $= 2^{4+4+10} \times 5^{3+2+8+5}$

⇒ $= 2^{18} \times 5^{18}$

⇒ $= (2 \times 5)^{18} = 10^{18}$

Therefore,

$\log_{10}(10^{18}) = 18$

∴ The value is 18.

Given:

$\frac{\log_{10}(100001 - 4^x)}{5 - x} = 1$

Formula used:

log a = b ⇒ a = 10^b

Calculation:

⇒ log₁₀(100001 - 4^x) = 5 - x

⇒ 100001 - 4^x = 10^5-x

⇒ 100001 - 4^x = 10^5-x

Check all the options:

x = 0

⇒ 100001 - 40 = 105-0

⇒ 100001 - 1 = 100000

x = 0 satisfies the condition.

x = 1

⇒ 100001 - 41 = 105 - 1

⇒ 100001 - 4 = 10000

⇒ 99997 ≠ 10000

x = 1 does not satisfy the condition.

x = 10

⇒ 100001 - 410 = 105 - 10

⇒ 100001 - 410 ≠ 10- 5

x = 10 does not satisfy the condition.

∴ The correct answer is option 1.

Given:

2 sin⁴(α) + 2 cos⁴(α) - 1 = 0, where 0 < α < π/2

Formula used:

sin⁴(α) + cos⁴(α) = (sin²(α) + cos²(α))² - 2sin²(α) cos²(α)

We know that sin²(α) + cos²(α) = 1.

Calculation:

The given equation is:

2 sin⁴(α) + 2 cos⁴(α) - 1 = 0

⇒ 2 (sin⁴(α) + cos⁴(α)) = 1

⇒ sin⁴(α) + cos⁴(α) = 1/2

Now, using the identity:

sin⁴(α) + cos⁴(α) = (sin²(α) + cos²(α))² - 2sin²(α) cos²(α)

⇒ (1)² - 2sin²(α) cos²(α) = 1/2

⇒ 1 - 2sin²(α) cos²(α) = 1/2

⇒ 2sin²(α) cos²(α) = 1/2

⇒ sin²(α) cos²(α) = 1/4

Now, take the square root of both sides:

⇒ sin(α) cos(α) = 1/2

Using the identity sin(2α) = 2 sin(α) cos(α):

⇒ sin(2α) = 2 × 1/2 = 1

sin(2α) = 1 implies that: 2α = 90°

Therefore: α = 45°

Now, find sin(2α) + cos(2α):

Since α = 45°:

sin(2α) = sin(90°) = 1

cos(2α) = cos(90°) = 0

Thus: sin(2α) + cos(2α) = 1 + 0 = 1.

sin(2α) + cos(2α) = 1.

∴ The correct answer is option 2.

Given:

(I): 1 − sin⁶α = cos²α (cos⁴α − 3cos²α + 3)

(II): cos⁸α − sin⁸α = 2sin²α (1 − cos⁴α + sin²α cos²α)

Check (I):

LHS = 1 − sin⁶α = (1 − sin²α)(1 + sin²α + sin⁴α)

⇒ = cos²α (1 + (1 − cos²α) + (1 − cos²α)²)

⇒ = cos²α (3 − 3cos²α + cos⁴α)

⇒ RHS ✔

So, (I) is true.

Check (II):

LHS = cos⁸α − sin⁸α = (cos⁴α − sin⁴α)(cos⁴α + sin⁴α)

⇒ (cos²α − sin²α)(cos²α + sin²α)(cos⁴α + sin⁴α)

⇒ (cos 2α)(1)(cos⁴α + sin⁴α)

RHS = 2sin²α(1 − cos⁴α + sin²α cos²α)

This form does not simplify to LHS.

So, (II) is false.

Final Answer:

Only (I) is an identity.

Given:

p = 1/(cosecθ + cotθ)

q = cosecθ

Calculations:

We have,

p = 1/(cosecθ + cotθ) = 1/ (1/sinθ + cosθ/sinθ)

⇒ 1/{(1 + cosθ)/sinθ}

⇒ sinθ / (1 + cos θ)

q = cosec θ = 1/sinθ

So, p² - 2pq

⇒ {sinθ / (1 + cos θ)}² - 2 × {sinθ / (1 + cos θ)} × {1/sinθ}

⇒ {sin²θ / (1 + cos θ)2} - 2 / (1 + cos θ)

⇒ {sin²θ / (1 + cos θ)2 - 2 / (1 + cos θ)

⇒ (1 - cos²θ) /(1 + cos θ)2 - 2 / (1 + cos θ)

⇒ (1 - cosθ)(1 + cosθ) /(1 + cos θ)2 - 2 / (1 + cos θ)

⇒ (1 - cosθ) / (1 + cos θ) - 2 / (1 + cos θ)

⇒ (1 - cosθ - 2) / (1 + cos θ)

⇒ (-1 - cosθ) / (1 + cos θ)

⇒ - (1 + cosθ) / (1 + cos θ)

⇒ - 1

∴ The correct answer is option (1)

Given:

Statement I: (cosec α - sec α) is always positive in the first quadrant.

Statement II: (tan α - cot α) is always negative in the first quadrant.

Formula used:

1. In the first quadrant, all trigonometric functions are positive.
2. cosec α = 1/sin α, sec α = 1/cos α, tan α = sin α/cos α, cot α = cos α/sin α.

Calculation:

For Statement I:

(cosec α - sec α) = 1/sin α - 1/cos α

⇒ Taking LCM:

(cosec α - sec α) = (cos α - sin α)/(sin α × cos α)

Since cos α and sin α are both positive in the first quadrant, the numerator (cos α - sin α) can be positive, zero, or negative depending on the specific value of α. Thus, Statement I is not always true.

For Statement II:

(tan α - cot α) = (sin α/cos α) - (cos α/sin α)

⇒ Taking LCM:

(tan α - cot α) = (sin² α - cos² α)/(sin α × cos α)

In the first quadrant, sin α and cos α are positive. However, sin² α - cos² α is negative because cos α > sin α for α in the first quadrant. Hence, (tan α - cot α) is always negative in the first quadrant.

∴ The correct answer is option (4).

Given:

AB = h

Calculation:

Let OP be the height of the tower,

In Δ ABP,

tan 30° = p/b = AB / AP

⇒ 1/√3 = h / AP

⇒ AP = h √3

Now, In Δ AOP,

tan 60° = p/b = OP/AP

⇒ √3 = OP / h √3

⇒ OP = h √3 × √3

⇒ OP = 3 h

Hence, the height of the tower is 3h.

Given:

Expression = (sin θ) / (1 − cot θ) + (cos θ) / (1 − tan θ), where θ ≠ π/4.

Formula used:

cot θ = cos θ / sin θ, tan θ = sin θ / cos θ.

Simplify each term separately and then add.

Calculation:

First term:

(sin θ) / (1 − cot θ) = (sin θ) / (1 − cos θ/sin θ)

= (sin θ) / ((sin θ − cos θ)/sin θ)

= (sin²θ) / (sin θ − cos θ)

Second term:

(cos θ) / (1 − tan θ) = (cos θ) / (1 − sin θ/cos θ)

= (cos θ) / ((cos θ − sin θ)/cos θ)

= (cos²θ) / (cos θ − sin θ)

= − (cos²θ) / (sin θ − cos θ)

Now, add both terms:

(sin²θ)/(sin θ − cos θ) − (cos²θ)/(sin θ − cos θ)

= (sin²θ − cos²θ)/(sin θ − cos θ)

= −(cos²θ − sin²θ)/(sin θ − cos θ)

= −(cos θ − sin θ)(cos θ + sin θ)/(sin θ − cos θ)

= −(−1)(cos θ + sin θ)

= cos θ + sin θ

∴ The correct answer is (sin θ + cos θ).

Given:

Radius (r) = 4 cm

Length of arc (l) = π cm

Formula used:

Length of arc (l) = r × θ

Where θ = Angle subtended by the arc at the centre (in radians)

Calculation:

l = r × θ

π = 4 × θ

⇒ θ = π / 4

Hence, the magnitude of the angle subtended by the arc at the center is π/4.

The correct answer is option (4).

Given:

cot2θ - $3\sqrt{3}$ cotθ + 6 = 0

Formula used:

cot2θ - $3\sqrt{3}$ cotθ + 6 = 0

put cotθ = x

So, x² - $3\sqrt{3}$x + 6 = 0

⇒ x² - 2$\sqrt3$x - $\sqrt3$x + 6 = 0

⇒ x(x - 2$\sqrt3$) - $\sqrt3$(x - 2$\sqrt3$) = 0

⇒ (x - 2$\sqrt3$) (x - $\sqrt3$) = 0

so, x = $\sqrt3$ & 2$\sqrt3$

i.e. cot θ = $\sqrt3$ = cot 30°

So, θ = 30°

Now,

sin θ + cos 2θ

⇒ sin 30 + cos 60

⇒ 1/2 + 1/2 = 1

∴ The correct answer is option (2).

Given:

I. sin²θ = ((x + y)²) / (4xy), where x, y are positive unequal real numbers.

II. sin θ + cos θ = x + 1/x, where x is a positive real number.

Formula used:

For any real angle θ, sin²θ ≤ 1 and |sin θ + cos θ| ≤ √2.

Check whether the given equations satisfy these constraints.

Calculation:

For I: sin²θ = ((x + y)²) / (4xy)

Since x, y > 0 and x ≠ y, (x + y)² > 4xy ⇒ ((x + y)²)/(4xy) > 1

But sin²θ ≤ 1 always.

Hence equation I is not possible.

For II: sin θ + cos θ = x + 1/x

By AM–GM inequality, x + 1/x ≥ 2 for x > 0

Also, |sin θ + cos θ| ≤ √2 ≈ 1.414

So x + 1/x ≥ 2 > √2 ⇒ equation II is also not possible.

∴ Neither I nor II is possible.

Given:

Equation: m²(sin θ - 1) + n²(sin θ + 1) = 0

Range: 0 ≤ θ ≤ π/2

Expression to find: (m² + n²) cos θ - (m² - n²) cot θ

Formula used:

Basic trigonometric identities:

cot θ = cos θ / sin θ

Calculation:

From the given equation:

⇒ m²(sin θ - 1) + n²(sin θ + 1) = 0

⇒ m² sin θ - m² + n² sin θ + n² = 0

⇒ (m² + n²) sin θ + (-m² + n²) = 0

⇒ (m² + n²) sin θ = (m² - n²)

⇒ sin θ = (m² - n²) / (m² + n²)

Now, substitute sin θ into the given expression:

(m² + n²) cos θ - (m² - n²) cot θ

⇒ (m² + n²) cos θ - (m² - n²) × (cos θ / sin θ)

⇒ (m² + n²) cos θ - (m² - n²) × (cos θ / [(m² - n²) / (m² + n²)])

⇒ (m² + n²) cos θ - (m² - n²) × (cos θ × (m² + n²) / (m² - n²))

⇒ (m² + n²) cos θ - (m² + n²) cos θ

⇒ 0

∴ The correct answer is option (4).

Shortcut Trick

sin α + cos α = $\sqrt{2}$

It is valid when α = π/4

As, sin π/4 + cos π/4 = 1/$\sqrt{2}$ + 1/$\sqrt{2}$ = 2/$\sqrt{2}$ = $\sqrt{2}$

So, sin α - cos α = sin π/4 - cos π/4 = 1/$\sqrt{2}$ - 1/$\sqrt{2}$ = 0

So, sin3α − cos3α

⇒ (sin α - cos α) (sin² α + cos²α - sinα cosα)

⇒ 0 (sin2 α + cos2α - sinα cosα)

⇒ 0

Hence, the correct answer is 0.

Alternate Method

Given:

sin α + cos α = √2, where 0 < α < π/2

Formula used:

sin³α - cos³α = (sin α - cos α)(sin²α + sin α cos α + cos²α)

Calculation:

sin α + cos α = √2

⇒ (sin α + cos α)² = (√2)²

⇒ sin²α + cos²α + 2sin α cos α = 2

⇒ 1 + 2sin α cos α = 2

⇒ 2sin α cos α = 1

⇒ sin α cos α = 1/2

Now, sin α - cos α = √(sin²α + cos²α - 2sin α cos α)

⇒ sin α - cos α = √(1 - 1)

⇒ sin α - cos α = 0

Plugging into the formula:

sin³α - cos³α = (sin α - cos α)(sin²α + sin α cos α + cos²α)

⇒ sin³α - cos³α = 0 × (1 + 1/2)

⇒ sin³α - cos³α = 0

∴ The correct answer is option (4).

Given:

Expression: (1 + cot α − cosec α)(1 + tan α + sec α)

Formula used:

cot α = cos α / sin α

cosec α = 1 / sin α

tan α = sin α / cos α

sec α = 1 / cos α

(x − 1)(x + 1) = x² − 1

sin²α + cos²α = 1

Calculations:

(1 + cot α − cosec α)(1 + tan α + sec α)

⇒ (1 + cos α / sin α − 1 / sin α)(1 + sin α / cos α + 1 / cos α)

⇒ ((sin α + cos α − 1) / sin α) × ((cos α + sin α + 1) / cos α)

Multiply numerators:

⇒ ((sin α + cos α − 1)(sin α + cos α + 1)) / (sin α × cos α)

Use difference of squares:

⇒ ((sin α + cos α)² − 1) / (sin α × cos α)

Expand and apply identity:

⇒ (sin²α + cos²α + 2 sin α cos α − 1) / (sin α × cos α)

⇒ (1 + 2 sin α cos α − 1) / (sin α × cos α)

⇒ 2 sin α cos α / (sin α × cos α)

⇒ 2

∴ (1 + cot α − cosec α)(1 + tan α + sec α) = 2

Given:

tan θ = (sin α - cos α) / (sin α + cos α)

θ and α are acute angles, α ≠ π/4.

Formula used:

√(2)sin θ = ?

Calculation:

tan θ = (sin α - cos α) / (sin α + cos α)

⇒ θ = tan^-1[(sin α - cos α) / (sin α + cos α)]

Using trigonometric identity: sin θ = tan θ / √(1 + tan²θ)

⇒ sin θ = [(sin α - cos α) / (sin α + cos α)] / √(1 + [(sin α - cos α) / (sin α + cos α)]²)

Let k = sin α - cos α and m = sin α + cos α.

⇒ sin θ = k / m / √[1 + (k / m)²]

⇒ sin θ = k / √(m² + k²)

⇒ √(2)sin θ = √(2) × (sin α - cos α) / √[(sin α + cos α)² + (sin α - cos α)²]

⇒ √(2)sin θ = √(2) × (sin α - cos α) / √[2(sin²α + cos²α)]

⇒ √(2)sin θ = √(2) × (sin α - cos α) / √(2)

⇒ √(2)sin θ = sin α - cos α

∴ The correct answer is option (1).

Given:

(sin α + 2)(sin α + 4)(sin α − 2)(sin α − 4) = 0

Formula used:

Expression = 0 ⇔ at least one factor = 0. But sin α ∈ [−1, 1]

Calculations:

⇒ sin α + 2 = 0 ⇒ sin α = −2 (not possible)

⇒ sin α + 4 = 0 ⇒ sin α = −4 (not possible)

⇒ sin α − 2 = 0 ⇒ sin α = 2 (not possible)

⇒ sin α − 4 = 0 ⇒ sin α = 4 (not possible)

Number of real values of α = 0.

∴ The correct answer is option 1.

Given:

tan3x × tan6x = 1, 0 ≤ x ≤ 30°

Formula used:

tanA × tanB = 1 ⇒ tanA = cotB = tan(90° − B + k × 180°)

Calculations:

⇒ tan3x = tan(90° − 6x + n × 180°)

⇒ 3x = 90° − 6x + n × 180°

⇒ 9x = 90° + n × 180°

⇒ x = 10° + n × 20°

Check n values for 0 ≤ x ≤ 30°:

n = 0 → x = 10° (valid)

n = 1 → x = 30° (but 3x = 90° → tan3x undefined → reject)

n = −1 → x = −10° (out of range)

∴ x = 10°.

Given:

Expression to evaluate:

$\frac{(a - b)^2}{(b - c)(c - a)} + \frac{(b - c)^2}{(c - a)(a - b)} + \frac{(c - a)^2}{(a - b)(b - c)} - 3$

Step 1: Let

x = $\frac{(a - b)^2}{(b - c)(c - a)}$

y = $\frac{(b - c)^2}{(c - a)(a - b)}$

z = $\frac{(c - a)^2}{(a - b)(b - c)}$

So expression becomes: x + y + z − 3

Step 2: Take LCM of all 3 terms:

LCM = (a − b)(b − c)(c − a)

Write all terms with common denominator:

x = $\frac{(a - b)^2}{(b - c)(c - a)} = \frac{(a - b)^3}{(a - b)(b - c)(c - a)}$

y = $\frac{(b - c)^2}{(c - a)(a - b)} = \frac{(b - c)^3}{(b - c)(c - a)(a - b)}$

z = $\frac{(c - a)^2}{(a - b)(b - c)} = \frac{(c - a)^3}{(c - a)(a - b)(b - c)}$

Numerators:

⇒ x + y + z = $\frac{(a - b)^3 + (b - c)^3 + (c - a)^3}{(a - b)(b - c)(c - a)}$

Use identity:

If x + y + z = 0 ⇒ x³ + y³ + z³ = 3xyz

Here, let: u = (a − b), v = (b − c), w = (c − a)

Then: u + v + w = 0

So: u³ + v³ + w³ = 3uvw

Apply:

Numerator = (a − b)³ + (b − c)³ + (c − a)³ = 3(a − b)(b − c)(c − a)

Denominator = (a − b)(b − c)(c − a)

So: x + y + z = $\frac{3(a - b)(b - c)(c - a)}{(a - b)(b - c)(c - a)} = 3$

Final Expression = x + y + z − 3 = 3 − 3 = 0

∴ The correct answer is: 0

Given:

Expression: $\dfrac{100!}{100^n}$ is an integer

Formula used:

We need to find the largest n such that 100ⁿ divides 100!

100 = 2² × 5² ⇒ 100ⁿ = 2²ⁿ × 5²ⁿ

So, we must find maximum n such that both 2²ⁿ and 5²ⁿ divide 100!

Calculations:

Powers of 2 in 100!

⇒ [100/2] + [100/4] + [100/8] + [100/16] + [100/32] + [100/64]

⇒ 50 + 25 + 12 + 6 + 3 + 1 = 97

Powers of 5 in 100!

⇒ [100/5] + [100/25] + [100/125]

⇒ 20 + 4 + 0 = 24

Now: 2²ⁿ must ≤ 2⁹⁷ ⇒ 2n ≤ 97 ⇒ n ≤ 48.5

And: 5²ⁿ must ≤ 5²⁴ ⇒ 2n ≤ 24 ⇒ n ≤ 12

So, the limiting factor is powers of 5 ⇒ n ≤ 12

The largest value of n is 12.

∴ The correct answer is option 4.

Given:

Distance traveled East = x meters

Distance traveled South = 40 meters

Distance traveled West = 60 meters

Distance traveled North = 200 meters

PR = 200 meters

Formula used:

Pythagoras theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Calculation:

PR = 200 meters

Horizontal distance between P and R = x - 60

Vertical distance between P and R = 200 - 40 = 160

Using Pythagoras theorem:

⇒ (PR)² = (Horizontal distance)² + (Vertical distance)²

⇒ 200² = (x - 60)² + 160²

⇒ 40000 = (x - 60)² + 25600

⇒ (x - 60)² = 40000 - 25600

⇒ (x - 60)² = 14400

⇒ x - 60 = √14400

⇒ x - 60 = 120

⇒ x = 120 + 60

⇒ x = 180

∴ The correct answer is option (2).

Given:

Equation: x² + y² + z² = 3

x, y, and z are integers.

Formula used:

The algebraic identity: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

Calculations:

We are given x² + y² + z² = 3, where x, y, and z are integers.

Since x, y, and z are integers, their squares (x², y², z²) must be non-negative integers (0, 1, 4, 9, ...).

The only way to sum three non-negative integer squares to get 3 is if each square is 1

⇒ x² = 1, y² = 1, z² = 1

This implies that x, y, and z can only be either 1 or -1.

Now, let P = (xy + yz + zx) and S = (x + y + z).

Using the algebraic identity:

(x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

⇒ S² = 3 + 2P

Rearrange the equation to solve for P:

⇒ 2P = S² − 3

⇒ P = $(\dfrac{S^2 - 3}{2})$

Now, we will consider all possible combinations of x, y, and z from {1, -1} and calculate S and P for each case:

Case 1: All three integers are 1.

(x, y, z) = (1, 1, 1)

S = 1 + 1 + 1 = 3

P = $(\dfrac{3^2 - 3}{2})$ = $(\dfrac{9 - 3}{2})$ = $(\dfrac{6}{2})$ = 3

Case 2: Two integers are 1, and one is -1.

Possible combinations: (1, 1, -1), (1, -1, 1), (-1, 1, 1)

For any of these combinations, S = 1 + 1 − 1 = 1

P = $(\dfrac{1^2 - 3}{2})$ = $(\dfrac{1 - 3}{2})$ = $(\dfrac{-2}{2})$ = -1

Case 3: One integer is 1, and two are -1.

Possible combinations: (1, -1, -1), (-1, 1, -1), (-1, -1, 1)

For any of these combinations, S = 1 − 1 − 1 = -1

P = $(\dfrac{(-1)^2 - 3}{2})$ = $(\dfrac{1 - 3}{2})$ = $(\dfrac{-2}{2})$ = -1

Case 4: All three integers are -1.

(x, y, z) = (-1, -1, -1)

S = -1 − 1 − 1 = -3

P = $(\dfrac{(-3)^2 - 3}{2})$ = $(\dfrac{9 - 3}{2})$ = $(\dfrac{6}{2})$ = 3

The distinct values that (xy + yz + zx) can have are 3 and -1.

Therefore, there are 2 possible values for (xy + yz + zx).

Given:

x + y + z = 10

xy + yz + zx = 18

Formula used:

The expression for the sum of cubes of three numbers can be expressed as:

x³ + y³ + z³ - 3xyz = (x + y + z) [(x + y + z)² - 3(xy + yz + zx)]

Calculation:

Substituting the given values:

x³ + y³ + z³ - 3xyz = (10) [(10)² - 3(18)]

⇒ x³ + y³ + z³ - 3xyz = 10 [100 - 54]

⇒ x³ + y³ + z³ - 3xyz = 10 × 46

⇒ x³ + y³ + z³ - 3xyz = 460

∴ The correct answer is option (3).

Given:

The required expression: $\sqrt{17 - 4\sqrt{15}} + \sqrt{8 - 2\sqrt{15}}$

Calculation:

$\sqrt{17 - 4\sqrt{15}}$ = $\sqrt{(2\sqrt3)^2 + (\sqrt5)^2 - 2 \times 2\sqrt{3}\times \sqrt5}$ = $\sqrt{(2\sqrt3 -\sqrt 5)^2}$ = $2\sqrt3 - \sqrt5$

$\sqrt{8 - 2\sqrt{15}}$ = $\sqrt{(\sqrt{3})^2 + (\sqrt{5})^2 - 2 \times \sqrt{3} \times \sqrt{5}}$ = $\sqrt{(\sqrt{5} -\sqrt{3})^2 }$ = $\sqrt{5} - \sqrt{3}$

Now, adding both terms:

$2\sqrt3 - \sqrt5$ + $\sqrt{5} - \sqrt{3}$ = $\sqrt3$

Therefore, the final answer is option 1.

Given:

The sequence is: 36, 33, 30, 27, 24, ...

We need to find the maximum value of the sum of the numbers.

This is an arithmetic progression (AP) where:

First term (a) = 36

Common difference (d) = 33 - 36 = -3

The sum of n terms of an AP is given by:

Formula used:

Sum (S_n) = $\dfrac{n}{2}[2a + (n-1)d]$

Calculation:

We need to find the maximum value of the sum. The AP ends when the last term becomes ≥ 0.

Last term (l) = a + (n-1)d

For the last term to be ≥ 0:

⇒ 36 + (n-1)(-3) ≥ 0

⇒ 36 - 3(n-1) ≥ 0

⇒ 36 - 3n + 3 ≥ 0

⇒ 39 - 3n ≥ 0

⇒ 3n ≤ 39

⇒ n ≤ 13

Thus, the sequence has a maximum of n = 13 terms.

Now, calculate the sum of these 13 terms:

S₁₃ = $\dfrac{13}{2}[2(36) + (13-1)(-3)]$

⇒ S₁₃ = $\dfrac{13}{2}[72 + 12(-3)]$

⇒ S₁₃ = $\dfrac{13}{2}[72 - 36]$

⇒ S₁₃ = $\dfrac{13}{2}[36]$

⇒ S₁₃ = 13 × 18

⇒ S₁₃ = 234

∴ The correct answer is option (3).

Given:

m ≡ 4 (mod 12)

n ≡ 6 (mod 12)

Formula used:

If a ≡ r₁ (mod m) and b ≡ r₂ (mod m), then:

a ± b ≡ r₁ ± r₂ (mod m)

Calculations:

Statement I: (m + n) mod 12

⇒ m + n ≡ 4 + 6 ≡ 10 (mod 12)

The statement I is correct.

Statement II: (m − n) mod 12

⇒ m − n ≡ 4 − 6 ≡ −2 ≡ 10 (mod 12)

The statement II is correct.

∴ The correct answer is: Both I and II

Given:

(x + y) : (y + z) : (z + x) = 3 : 5 : 6

x + y + z = 14

Formula used:

x² + y² + z² = (x + y + z)² − 2(xy + yz + zx)

Calculations:

Let (x + y) = 3k, (y + z) = 5k, (z + x) = 6k

⇒ (x + y) + (y + z) + (z + x) = 3k + 5k + 6k = 14k

But (x + y) + (y + z) + (z + x) = 2(x + y + z) = 2 × 14 = 28

⇒ 14k = 28 ⇒ k = 2

So, x + y = 6, y + z = 10, z + x = 12

Adding all: 2(x + y + z) = 28 ⇒ x + y + z = 14 ✔

Now, solve individually:

x + y = 6, x + y + z = 14 ⇒ z = 8

y + z = 10 ⇒ y + 8 = 10 ⇒ y = 2

x + z = 12 ⇒ x + 8 = 12 ⇒ x = 4

Now, x² + y² + z² = 4² + 2² + 8² = 16 + 4 + 64 = 84

∴ x² + y² + z² = 84

Given:

The ratio of the sum of two numbers to their difference is 5 : 1.

Formula used:

If the ratio of the sum of two numbers (a + b) to their difference (a - b) is given as 5 : 1, then:

(a + b)/(a - b) = 5/1

The ratio of the sum of their squares to the difference of their squares can be calculated using:

(a² + b²) / (a² - b²) = [(a + b)² - 2ab] / [(a - b)² + 2ab]

Calculation:

From the ratio (a + b)/(a - b) = 5/1:

⇒ a + b = 5k and a - b = k

⇒ a = (5k + k)/2 = 3k, b = (5k - k)/2 = 2k

Now calculate the ratio (a² + b²) / (a² - b²):

⇒ a² + b² = (3k)² + (2k)² = 9k² + 4k² = 13k²

⇒ a² - b² = (3k)² - (2k)² = 9k² - 4k² = 5k²

⇒ (a² + b²) / (a² - b²) = 13k² / 5k²

⇒ Ratio = 13 : 5

∴ The correct answer is option (1).

Given:

A man is traveling at 3/5^th of his usual speed and is late by 20 minutes.

Formula used:

Time = Distance ÷ Speed

Let the usual time taken by the man be T minutes.

When traveling at 3/5^th of the usual speed, time taken = T × (5/3).

Difference in time: T × (5/3) - T = 20 minutes.

Calculation:

⇒ T × (5/3) - T = 20

⇒ (5T/3) - T = 20

⇒ (5T - 3T)/3 = 20

⇒ 2T/3 = 20

⇒ T = 20 × (3/2)

⇒ T = 30 minutes

∴ The correct answer is option (2).

Given:

Expression: 2^p−1

Divisor: p

Calculations:

We need to find the remainder when 2^p−1 is divided by p.

Since p is a prime number greater than 5: Direct putting value p = 7.

Substitute p = 7 into the expression:

Expression = 2⁷ − 1

2⁷ = 128

Expression = 128 − 1

⇒ Expression = 127

Now, we need to divide 127 by p = 7 and find the remainder.

127 ÷ 7:

127 = 7 × 18 + 1

The remainder when 2⁷−1 is divided by 7 is 1.

∴ The correct answer is option 1.

Given:

Expression: 24³ − 16³ − 8³

Calculations:

⇒ 8³(27 − 8 − 1)

⇒ 8³ × 18

⇒ 512 × 18 = 9216

Prime factorization of 9216:

9216 = 2¹⁰ × 3²

Number of factors = (10 + 1)(2 + 1) = 11 × 3 = 33

∴ The correct answer is: 33

Given:

Rate of Interest (r) = 20%

Amount (A) = 2 × Principal (P)

Formula used:

A = P(1 + r/100)^t

Calculation:

2P = P(1 + 20/100)^t

⇒ 2 = (1 + 0.2)^t

⇒ 2 = (1.2)^t

Now, we calculate t for different values:

For t = 2, (1.2)² = 1.44

For t = 3, (1.2)³ = 1.728

For t = 4, (1.2)⁴ = 2.0736

⇒ At t = 4, the amount exceeds double the principal.

∴ The least number of complete years is 4, and the correct answer is option (3).

Given:

A train of length L passes a station of length x in time t.

The same train with same speed passes another station of length y in time 2t.

We need to find the time taken to pass a station of length (x + y).

Formula used:

Time = (Distance covered) / (Speed)

When a train passes a station, distance covered = (Length of train + Length of station).

Calculation:

Case 1: Through station of length x

(L + x) / v = t … (i)

Case 2: Through station of length y

(L + y) / v = 2t … (ii)

From (i): v = (L + x)/t

From (ii): v = (L + y)/(2t)

Equating both values of v:

(L + x)/t = (L + y)/(2t)

2(L + x) = (L + y)

2L + 2x = L + y

L = y − 2x … (iii)

Now, time to pass station of length (x + y):

Required time = (L + (x + y)) / v

Using (i): v = (L + x)/t

So, Required time = (L + x + y) / ((L + x)/t)

= [(L + x + y) × t] / (L + x)

Substitute L = y − 2x:

= [(y − 2x + x + y) × t] / (y − 2x + x)

= [(2y − x) × t] / (y − x)

∴ The correct answer is (2yt - xt) / (y - x).

Given:

Top diameter of the frustum (d₁) = 2k

Bottom diameter of the frustum (d₂) = 2.5k

Height of the frustum (h) = k

Formula used:

Radius of top (r₁) = d₁ / 2

Radius of bottom (r₂) = d₂ / 2

Slant height of frustum (l) = $\sqrt{h^2 + (r_2 - r_1)^2}$

Whole Surface Area of Frustum = Area of top base + Area of bottom base + Lateral Surface Area

Area of top base = $\pi r_1^2$

Area of bottom base = $\pi r_2^2$

Lateral Surface Area = $\pi(r_1 + r_2)l$

Calculations:

Calculate radii:

r₁ = 2k / 2 = k

r₂ = 2.5k / 2 = 1.25k

Calculate the difference in radii:

r₂ - r₁ = 1.25k - k = 0.25k

Calculate the slant height (l):

l = $\sqrt{k^2 + (0.25k)^2}$

⇒ l = $\sqrt{k^2 + 0.0625k^2}$

⇒ l = $\sqrt{1.0625k^2}$

⇒ l = k$\sqrt{1.0625}$

Now, calculate the surface areas:

Area of top base = $\pi r_1^2$ = $\pi (k)^2$ = $\pi k^2$

Area of bottom base = $\pi r_2^2$ = $\pi (1.25k)^2$ = $\pi (1.5625k^2)$ = 1.5625$\pi k^2$

Lateral Surface Area = $\pi(r_1 + r_2)l$

⇒ Lateral Surface Area = $\pi(k + 1.25k)k\sqrt{1.0625}$

⇒ Lateral Surface Area = $\pi(2.25k)k\sqrt{1.0625}$

⇒ Lateral Surface Area = 2.25$\pi k^2\sqrt{1.0625}$

Whole Surface Area = $\pi k^2$ + 1.5625$\pi k^2$ + 2.25$\pi k^2\sqrt{1.0625}$

⇒ Whole Surface Area = $\pi k^2 (1 + 1.5625 + 2.25\sqrt{1.0625})$

⇒ Whole Surface Area = $\pi k^2 (2.5625 + 2.25 \times 1.030776...)$

⇒ Whole Surface Area = $\pi k^2 (2.5625 + 2.319246...)$

⇒ Whole Surface Area = $\pi k^2 (4.881746...)$

Let's use the fraction form for sqrt(1.0625) for precision:

1.0625 = 10625 / 10000 = 425 / 400 = 85 / 80 = 17/16 (Error in calculation here, 10625/10000 = 17/16 was for 1.0625, but 0.25k is (1/4)k)

0.25k = (1/4)k

(0.25k)² = (1/16)k²

l = $\sqrt{k^2 + \frac{1}{16}k^2}$ = $\sqrt{\frac{16k^2 + k^2}{16}}$ = $\sqrt{\frac{17k^2}{16}}$ = $\frac{k\sqrt{17}}{4}$

Lateral Surface Area = $\pi(k + 1.25k)\frac{k\sqrt{17}}{4}$

⇒ Lateral Surface Area = $\pi(2.25k)\frac{k\sqrt{17}}{4}$

⇒ Lateral Surface Area = $\pi(\frac{9}{4}k)\frac{k\sqrt{17}}{4}$

⇒ Lateral Surface Area = $\frac{9\pi k^2\sqrt{17}}{16}$

Whole Surface Area = $\pi k^2$ + $\pi (\frac{5}{4}k)^2$ + $\frac{9\pi k^2\sqrt{17}}{16}$

⇒ Whole Surface Area = $\pi k^2$ + $\frac{25\pi k^2}{16}$ + $\frac{9\pi k^2\sqrt{17}}{16}$

⇒ Whole Surface Area = $\pi k^2 \left(1 + \frac{25}{16} + \frac{9\sqrt{17}}{16}\right)$

⇒ Whole Surface Area = $\pi k^2 \left(\frac{16}{16} + \frac{25}{16} + \frac{9\sqrt{17}}{16}\right)$

⇒ Whole Surface Area = $\frac{\pi k^2}{16} (16 + 25 + 9\sqrt{17})$

⇒ Whole Surface Area = $\frac{\pi k^2}{16} (41 + 9\sqrt{17})$

Using $\sqrt{17} \approx 4.123$

⇒ Whole Surface Area = $\frac{\pi k^2}{16} (41 + 9 \times 4.123)$

⇒ Whole Surface Area = $\frac{\pi k^2}{16} (41 + 37.107)$

⇒ Whole Surface Area = $\frac{\pi k^2}{16} (78.107)$

⇒ Whole Surface Area ≈ 4.8816 $\pi k^2$

The exact expression is $\frac{\pi k^2}{16} (41 + 9\sqrt{17})$

∴ The whole surface area of the frustum is $\frac{\pi k^2}{16} (41 + 9\sqrt{17})$.