Test: SSC CGL Previous Year Questions: Algebra (2023-18) - 2 - SSC CGL MCQ

A number is the factor of its square. 
Squaring the identity (mx + n)
(mx + n)² = (m² x² + 2mnx + n²)

x³ + y³ + z³ - 3xyz = (x + y + z)
x² + y² + z² - xy - yz - zx
When (x + y + z) = 0, then x³ + y³ + z³ - 3xyz ...eq .(1)
(x + y +z) = 0 …… (given)
 

x⁴ − a⁴ = 1 ⇒ x⁴ = a⁴ + 1

Statement 1.
100² − 99² + 98² − 97² ........ + 22² − 21²
⇒ (100 + 99)(100 - 99) + (98 + 97)(98 - 97) ……… + (22 + 21)(22 - 21)
⇒ 100 + 99 + 98 + 97…….22 + 21
Sum = n/2 x (a + l)
= 80/2(100 + 21) = 40 × 121 = 4840
Statement 2.


Now,  …..e.q .(2)
Where, e.q.(1) ≠ e.q.(2)

xy = 8 ….(given)
(4x - 7y) = 11
On squaring, we get
16x² + 49y² - 56xy = 121
16x² + 49y² = 121 + 56 x 8 = 569

x + 1/x = 2, 
it is only possible when x = 1
 1 + 1 = 2

I. K + 1/K = 12 then K² + 1/(K²) = 144 − 2 = 142(True)
II. The value of 

So, only 1 statement is correct.

(a + b)² – (a – b)²
= (a² + b² + 2ab) - (a² + b² - 2ab)
= a² + b² + 2ab - a² - b² + 2ab = 4ab

9m² + n² = 40 and mn = 4
⇒ 9m² + n² + 6mn = 40 + 6mn
⇒ (3m + n)² = 40 + 24 = 64
⇒ 3m + n = 8

x² + 4y² + 4y - 4xy - 2x - 8
= (x² - 4xy + 4y²) + 4y - 2x - 8
= (x − 2y)² − 2 (x − 2y) − 8
(x − 2y)² − 4 (x − 2y) + 2 (x − 2y) − 8
= (x − 2y){(x − 2y) − 4} + 2 {(x − 2y) − 4}
= {x − 2y − 4 }{x − 2y + 2}

On checking all the options for the values a and b one by one, option (c) gets satisfied.
a² = b² + (ab)²
cot²x = cos²x + cos²xcot²x
cot²x = cos²x(1 + cot²x) = cos²x.cosec²x
cot²x = cot²x
LHS = RHS (satisfied)

4(z + 7) (2z – 1) = (4z + 28)(2z - 1)
= 8z² - 4z + 56z - 28 = 8z² + 52z - 28
Now, 8z² + 52z - 28 = Az² + Bz + C
On comparing both side, we have :
A = 8, B = 52 , C = - 28
So, A + B + C = 8 + 52 - 28 = 32

p³ + 27 = p³ + 3³
= (p + 3)(p² + 3² - 3p) = (p + 3)(p² + 9 - 3p)

Formula used :
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
a² + b² + c² = 9² - 2 x 23 = 81 - 46 = 35

If x² + 1/(x²) = 66
(x - 1/x)² = 66 - 2 = 64 ⇒ x - 1/x = √64 = 8

m + 1/(m-2) = 4
Subtracting both side by 2. We get :
m - 2 + 1/(m-2) = 2
So, (m − 2)² +  2² - 2 = 2

By applying formula, [a³ + b³ = (a + b){a² - ab + b²}] we get,

x = - 8 So, x³ = (-8)³ = -512

Let y = x + 2/x.

Step 1: Square both sides of the equation to simplify it:

y² = (x + 2/x)²
y² = x² + 2 X x X (2/x) + (2/x)²
y² = x² + 4 + 4/x²

Step 2: Use the given equation x⁴ + 16/x⁴ = 15617. We can express x⁴ + 16/x⁴ as:

(x²)² + (4/x²)² = 15617

Let z = x². Then the equation becomes:

z² + 16/z² = 15617

Step 3: Now, subtract 4 from both sides of the equation for y²:

y² = x² + 4 + 4/x²
y² = (x² + 4 + 4/x²)
y² = 129 (after solving the above equation).

Step 4: Solve for y:

y = √129

Thus, the value of x + 2/x is √129.

Formula used ; (x + y + z)²
= x² + y² + z² + 2(xy + yz + zx)
0 = 40 + 2(xy + yz + zx)
-40 = 2(xy + yz + zx)
xy + yz + zx = (-40)/2  = -20

I. x³ + y³ + z³ = 12³ - 2³ - 10³
= 1728 - 8 - 1000 = 720
II. 4 (x² + y²) = 4 (x + y)² [−2xy]
= 4[48² - 64] = 4[2304 - 64]
= 4 x 2240 = 8960
Clearly, we can see that neither statement I nor II is correct.

K + 1/K + 2 = 0
K + 1/K = -2 ⇒ k = -1
So, (-1)¹¹ + (-1)⁴ = (-1) + 1 = 0

On cubing above equation, we get

So, x⁶ + x^-6 = -2

a² + b² + 49c² + 18
= 2 (b − 28c − a)
a² + b² + 49c² + 18
= 2b − 56c − 2a
a² + 2a + 1 + b² − 2b + 1 + 49c² + 56c + 16 = 0
(a + 1)² + (b - 1)² + (7c + 4)² = 0
So, a = -1, b = 1 and c = - (4/7)
a + b − 7c = -1 + 1 -7(-4/7) = 0 + 4 = 4.

On squaring, we get

On squaring, we get

On squaring, we get

(x + y)³ − (x − y)³ − 3y(2x² - 3y²) = ky³
⇒ x³ + y³ + 3x²y + 3xy² – (x³ – 3x²y + 3xy² –y³) – 6x²y + 9y³ = ky³
⇒ x³ + y³ + 3x²y + 3xy² – x³ + 3x²y – 3xy² + y³ – 6x²y + 9y³ = ky³
⇒ 11y³ = ky³ ⇒ k = 11

so, 
= 0.27

x³ + y³ + z³ − 3xyz = 0.98k
⇒ 1/2 (x + y + z)[(x − y)² + (y − z)² + (z − x)²] = 0.98k
⇒ 1/2(32.5 + 34.6 + 30.9)[(32.5 − 34.6)² + (34.6 − 30.9)² + (30.9 − 32.5)²] = 0.98k

⇒ (49)[(4.41) + (13.69) + (2.56)] = 0.98k
⇒ (49)[20.66] = 0.98k
⇒ 1012.34 = 0. 98k ⇒ k = 1033

x + 1/x = 3 ...(1)
On squaring eqn (1), we get

On squaring eqn (2), we get

On cubing eqn (1), we get
 (3)³ − (3)(3) 27 − 9 = 18
So, 
On multiplying (3) and (4), we get

16x² + y² = 48
adding 8xy on both the sides
16x² + y² + 8xy = 48 + 16 = 64
(4x + y)² = 8²
4x + y = 8 taking cube on both the side
⇒ 64x³ + y³ + 3 x 4x x y(4x + y) = 512
⇒ 64x³ + y³ = 512 - 192 = 320

If x - 1/x = 5 then

 dividing the numerator and denominator by x³