Infinitely Many Solutions MCQ Quiz in मल्याळम - Objective Question with Answer for Infinitely Many Solutions - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

Last updated on Mar 28, 2025

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Latest Infinitely Many Solutions MCQ Objective Questions

Top Infinitely Many Solutions MCQ Objective Questions

Infinitely Many Solutions Question 1:

Given simultaneous equations have infinitely many solution.

2x + 3y + z = 0

4x + 6y + 6z = 0

8x + 12ay + 2z = 0

Which of the following is true for a?

  1. a = 1
  2. a ≠ 1
  3. a = -1
  4. a = 0

Answer (Detailed Solution Below)

Option 1 : a = 1

Infinitely Many Solutions Question 1 Detailed Solution

Since given system of equations is homogeneous and has infinitely many solutions.

Therefore, given matrix is consistent and determinant of the coefficient matrix should be equal to 0

|231466812a2|=0

2×(1272a)3×(848)+1(48a48)=0

24 – 144a – 24 + 144 + 48a – 48 = 0

- 96a + 96 = 0

∴ a = 1

Infinitely Many Solutions Question 2:

Let λ be a real number for which the system of linear equations

x+y+z=6

4x+λyλz=λ2

3x+2y4z=5

has many infinitely many solutions. Then λ is a root of the quadratic equation

  1. λ23λ4=0
  2. λ2λ6=0
  3. λ2+3λ4=0
  4. λ2+λ6=0

Answer (Detailed Solution Below)

Option 2 : λ2λ6=0

Infinitely Many Solutions Question 2 Detailed Solution

D=0

|111 4λλ 324|=0λ=3

Infinitely Many Solutions Question 3:

For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b- 15)y = 8b has infinitely many solutions for x and y. Then, the maximum possible value of ab is

  1. 33
  2. 55
  3. 15
  4. 25

Answer (Detailed Solution Below)

Option 1 : 33

Infinitely Many Solutions Question 3 Detailed Solution

It is given that for some real numbers 'a' and 'b' , the system of equations x+y=4 and (a+5)x+(b215)y=8b has infinitely many solutions for 'x' and 'y'.

when two system equations a1x+b1y=c1 and a2x+b2y=c2 then

⇒ a1a2=b1b2=c1c2

Here,

⇒ a+51=b2151=8b4    ⇒ (a+5)=(b215)=(2b) 

This equation can be used to find the value of a, and b.

Firstly, we will determine the value of b.

⇒ (b215)=(2b) 

⇒ b22b15=0 

⇒ b22b15=0

⇒ (b5)(b+3)=0 

∴ b = 5 or b = -3

Now, we will determine the value of a.

⇒ (a+5)=(b215)  ⇒ a=b220

when b = 5, → a=(5)220=2520=5

thus, ab = (5)(5) = 25

when b = -3, → a=(3)220=920=11

thus, ab = (-3)(-11) = 33

 The maximum value of 𝑎𝑏 = (−3)(−11) = 33.

 

Infinitely Many Solutions Question 4:

The value of k for which kx + 3y - k + 3 = 0 and 12x + ky = k have infinite solution is:

  1. -6
  2. 0
  3. 6
  4. 1
  5. Not Attempted

Answer (Detailed Solution Below)

Option 3 : 6

Infinitely Many Solutions Question 4 Detailed Solution

Concept:

Let the two equations be:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Then,

  • For unique solutiona1a2b1b2
  • For infinitely many solutionsa1a2=b1b2=c1c2
  • For no solutiona1a2=b1b2c1c2

Formula used:

For any quadratic equation, ax2 + bx + c = 0, the quadratic formula is:

x=b±b24ac2a

Calculation:

Given equations are:

kx + 3y - k + 3 = 0 and 12x + ky = k 

On comparing the equations with ax + by + c = 0, we get

a1 = k, b1 = 3, c1 = -k + 3

a2 = 12, b2 = k, c2 = -k

So, for infinitely many solutions,

a1a2=b1b2=c1c2

k(12)=3k=k+3k

Solving, k12=3k

k2 = 36

⇒ k = 6

Solving, 3k=k+3k

⇒ -3k = -k2 + 3k

k2 - 6k = 0

⇒ k2 = 6k

k = 6

Hence, k = 6.

Infinitely Many Solutions Question 5:

If two vectors of X1, X2 and X3 are linearly independent and A3×3 is the coefficient matrix in the homogenous system whose column vectors are X1, X2 and X3, then the system has

  1. Only trivial solution
  2. Zero solution
  3. trivial solution and non-trivial solution
  4. Non-trivial solution

Answer (Detailed Solution Below)

Option 4 : Non-trivial solution

Infinitely Many Solutions Question 5 Detailed Solution

Explanation:

Given system is homogeneous,

AX = 0

Where A = [X1  X2  X3]3×3 whose column are linearly independent.

But given that two vectors of X1, X2 and X3 are linearly independent.

ρ(A3×3) = 2 < number of variables (3)

∴ System has many solution i.e. Non-trivial solutions

Infinitely Many Solutions Question 6:

The values of x, y, z will satisfy the following system of linear equation.

x + y + z = 6

x + 2 y + 3 z = 14

x + 4 y + 7 z = 30

  1. 7, 3, -4
  2. -3, 10, -1
  3. 3, 8, -5
  4. None

Answer (Detailed Solution Below)

Option 2 : -3, 10, -1

Infinitely Many Solutions Question 6 Detailed Solution

Obtain the Matrix representation for the system of equations we get

[A:B]=[11161231414730]R2R2R1,R3R3R1[A:B]=[111:6012:8036:24]R3R33R2[A:B]=[111:6012:8000:0]

ρ[A:B]=ρ(A)=2<3System has infinite no. of solutions with one independent solution (i.e. n – r = 1)

Assume z = K then y = 8 - 2K

x+y+z=6x+82K+K=6x=K2X=[xyz]=[K282KK]

Put K=1X=[xyz]=[3101]
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