Algebra (SSC) (Part-4)

Total Questions: 50

1. Simplify the following expression. 49a² + 70ab + 25b² [Higher Secondary 26/06/2024 (Shift - 2)]

Correct Answer: (d) (7a + 5b)²
Solution:We know
(a + b)² = a² + b² + 2ab
Hence,
49a² + 70ab + 25b² = (7a + 5b)²

2. If x + y = 10 and xy = 12, find the value of 𝓍² - 𝓍y + y² [Graduate Level 21/06/2024 (Shift - 3)]

Correct Answer: (c) 64
Solution:(x + y)² = x² + y² + 2xy
Subtracting both side by 3xy, we get ;
(x + y)² − 3xy = x² + y² − xy
10² − 3 × 12 = x² + y² − xy
100 − 36 = x² + y² − xy ⇒ x² + y² − xy = 64

3. Solve the following equation: [Graduate Level 24/06/2024 (Shift - 3)]

Correct Answer: (c) 6
Solution:

(x − 1/(x − 1)) = 3
Subtracting both side by 1, we have ;
(x − 1 − 1/(x − 1)) = 2

Let , (x − 1) = k ⇒ k − 1/k = 2

k² + 1/k² = 2² + 2 = 6

So, (x − 1)² + 1/(x − 1)² = 6

4. If p = 106, q = 105 and r = 104, then value of p² + q² + r² - pq - qr - rp is: [Graduate Level 25/06/2024 (Shift - 3)]

Correct Answer: (d) 3
Solution:

Formula used :
a² + b² + c² − ab − bc − ca
= 1/2 [(a − b)² + (b − c)² + (c − a)²]

Then, p² + q² + r² − pq − qr − rp
= 1/2 [(106 − 105)² + (105 − 104)² + (104 − 106)²]

= 1/2 [1² + 1² + (−2)²] ⇒ 1/2 × 6 = 3

5. Solve the following equation: [Graduate Level 26/06/2024 (Shift - 3)]

Correct Answer: (c) 6
Solution:

√(1 + x³/9) = 5

1 + x³/9 = 25 ⇒ x³/9 = 24

⇒ x³ = 24 × 9 = 216 ⇒ x = 6

6. Solve the following equation: [Graduate Level 26/06/2024 (Shift - 3)]

Correct Answer: (b) 223
Solution:

x + 1/x = 15

On squaring both side, we get;

x² + 1/x² + 2 = 225

⇒ x² + 1/x² = 225 − 2 = 223

7. If (a − 2b) = 8 and 2ab = 9, then find the value of (a + 2b). [SSC CHSL Tier II (02/11/2023)]

Correct Answer: (c) 10
Solution:(a − 2b) = 8, 2ab = 9

∵ (a − 2b)² = (a + 2b)² − 8ab
(a + 2b)² = 8² + 8 × 9/2
(a + 2b)² = 64 + 36 = 100
(a + 2b) = 10

8. A and B each have a certain number of oranges. A says to B, “If you give me 10 of your oranges, I will have twice the number of oranges left with you.” B replies, “If you give me 10 of your oranges, I will have the same number of oranges left with you.” Find the number of oranges that A has. [SSC CHSL Tier II (02/11/2023)]

Correct Answer: (d) 70
Solution:

According to question,
A + 10 = (B − 10) × 2
A − 2B = −30 ...(i)
A − 10 = B + 10
A − B = 20 ...(ii)

From equation (ii)×2 − (i), we get, A = 70

9. Solve the following equation: [SSC CPO 03/10/2023 (1st Shift)]

Correct Answer: (a) 0
Solution:

Given,
x + y = 2 and x/y + y/x = 1

⇒ x² + y² = xy

Formula:
x³ + y³ = (x + y)(x² + y² − xy)

⇒ x³ + y³ = 2 × 0 = 0

10. Solve the following equation: [SSC CPO 03/10/2023 (2nd Shift)]

Correct Answer: (a) 1
Solution:

According to the question,
A/L = 1 − M/B = (B − M)/B
L/A = B/M and C/N = 1 − B/M = (M − B)/M

⇒ L/A = B/M and C/N = (M − B)/M

⇒ L/A + C/N = B/(B − M) − M/(B − M) = (B − M)/(B − M) = 1