Search Previous Year Questions

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).