PYQs Homework Notes Mocks Board Daily Homework JEE Mains Maths Question type Which statement about unique, infinite or no solutions of a system with parameters is (not) correct This type of question, asked 9 times: 2021 Paper 2, 2021 Paper 26, 25 Jan 2023, 29 Jan 2023, 31 Jan 2023, 6 Apr 2023, 10 Apr 2023, 13 Apr 2023, 30 Jan 2024. All 10 questions below, oldest first.
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Question 1 2021 Paper 2
72 For the system of linear equations: x − 2 y = 1 , x − y + k z = − 2 , k y + 4 z = 6 , k ∈ R x- 2y= 1,x-y+kz= - 2,ky+ 4z= 6,k\in R x − 2 y = 1 , x − y + k z = − 2 , k y + 4 z = 6 , k ∈ R consider the following statements: (A) The system has unique solution if k ≠ 2 , k ≠ − 2 k\neq 2,k\neq - 2 k = 2 , k = − 2 . (B) The system has unique solution if k = − 2 k= - 2 k = − 2 . (C) The system has unique solution if k = 2 k= 2 k = 2 . (D) The system has no-solution if k = 2 k= 2 k = 2 . (E) The system has infinite number of solutions if k ≠ − 2 k\neq - 2 k = − 2 . Which of the following statements are correct? A. (B) and (E) only
B. (C) and (D) only
C. (A) and (D) only
D. (A) and (E) only
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Question 2 2021 Paper 26
70 Consider the following system of equations: x + 2 y − 3 z = a x + 2y - 3z = a x + 2 y − 3 z = a 2 x + 6 y − 11 z = b 2x + 6y - 11z = b 2 x + 6 y − 11 z = b x − 2 y + 7 z = c x - 2y + 7z = c x − 2 y + 7 z = c Where a , b a,b a , b and c c c are real constants. Then the system of equations: A. has a unique solution when 5 a = 2 b + c 5a = 2b + c 5 a = 2 b + c B. has infinite number of solutions when 5 a = 2 b + c 5a = 2b + c 5 a = 2 b + c C. has no solution for all a , b a,b a , b and c c c D. has a unique solution for all a , b a,b a , b and c c c Tap an option to check your answer.
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Question 3 25 Jan 2023
67 Let S 1 S_{1} S 1 and S 2 S_{2} S 2 be respectively the sets of all a ∈ R − { 0 } \mathbf{a \in R - \{ 0\}} a ∈ R − { 0 } for which the system of linear equations a x + 2 a y − 3 a z = 1 ( 2 a + 1 ) x + ( 2 a + 3 ) y + ( a + 1 ) z = 2 ( 3 a + 5 ) x + ( a + 5 ) y + ( a + 2 ) z = 3 \begin{matrix} & ax + 2ay - 3az = 1 \\ & \ (2a + 1)x + (2a + 3)y + (a + 1)z = 2 \\ & \ (3a + 5)x + (a + 5)y + (a + 2)z = 3 \end{matrix} a x + 2 a y − 3 a z = 1 ( 2 a + 1 ) x + ( 2 a + 3 ) y + ( a + 1 ) z = 2 ( 3 a + 5 ) x + ( a + 5 ) y + ( a + 2 ) z = 3 has unique solution and infinitely many solutions respectively. Then A. S 1 S_{1} S 1 is an infinite set and n ( S 2 ) = 2 n(S_{2})=2 n ( S 2 ) = 2 B. S 1 = Φ S_{1}=\Phi S 1 = Φ and S 2 = R − { 0 } S_{2}=R-\{0\} S 2 = R − { 0 } C. n ( S 1 ) = 2 n(S_{1})=2 n ( S 1 ) = 2 and S 2 S_{2} S 2 is an infinite set D. S 1 = R − { 0 } S_{1}=R-\{0\} S 1 = R − { 0 } and S 2 = Φ S_{2}=\Phi S 2 = Φ Tap an option to check your answer.
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Question 4 29 Jan 2023
64 Consider the following system of equations α x + 2 y + z = 1 \alpha x + 2y + z = 1 α x + 2 y + z = 1 2 α x + 3 y + z = 1 2\alpha x + 3y + z = 1 2 α x + 3 y + z = 1 3 x + α y + 2 z = β 3x + \alpha y + 2z = \beta 3 x + α y + 2 z = β for some α , β ∈ R \alpha,\beta \in R α , β ∈ R . Then which of the following is NOT correct. A. It has a solution if α = − 1 \alpha = - 1 α = − 1 and β ≠ 2 \beta \neq 2 β = 2 B. It has a solution for all α ≠ − 1 \alpha \neq - 1 α = − 1 and β = 2 \beta = 2 β = 2 C. It has no solution for α = 3 \alpha = 3 α = 3 and for all β ≠ 2 \beta \neq 2 β = 2 D. It has no solution for α = − 1 \alpha = - 1 α = − 1 and for all β ∈ R \beta \in R β ∈ R Tap an option to check your answer.
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Question 5 31 Jan 2023
72 For the system of linear equations x + y + z = 6 x + y + z = 6 x + y + z = 6 α x + β y + 7 z = 3 \alpha x + \beta y + 7z = 3 α x + β y + 7 z = 3 x + 2 y + 3 z = 14 x + 2y + 3z = 14 x + 2 y + 3 z = 14 which of the following is NOT true? A. If α = β \alpha = \beta α = β and α ≠ 7 \alpha \neq 7 α = 7 , then the system has a unique solution B. If α = β = 7 \alpha = \beta = 7 α = β = 7 , then the system has no solution C. For every point ( α , β ) ≠ ( 7 , 7 ) (\alpha,\beta) \neq (7,7) ( α , β ) = ( 7 , 7 ) on the line x − 2 y + 7 = 0 x - 2y + 7 = 0 x − 2 y + 7 = 0 , the system has infinitely many solutions D. There is a unique point ( α , β ) (\alpha,\beta) ( α , β ) on the line x + 2 y + 18 = 0 x + 2y + 18 = 0 x + 2 y + 18 = 0 for which the system has infinitely many solutions Tap an option to check your answer.
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Question 6 6 Apr 2023
158 For the system of equations x + y + z = 6 x + y + z = 6 x + y + z = 6 x + 2 y + α z = 10 x + 2y + \alpha z = 10 x + 2 y + α z = 10 x + 3 y + 5 z = β x + 3y + 5z = \beta x + 3 y + 5 z = β , which one of the following is NOT true? A. System has a unique solution for α = 3 , β ≠ 14 \alpha = 3,\beta \neq 14 α = 3 , β = 14 . B. System has no solution for α = 3 , β = 24 \alpha = 3,\beta = 24 α = 3 , β = 24 . C. System has a unique solution for α = − 3 , β = 14 \alpha = - 3,\beta = 14 α = − 3 , β = 14 . D. System has infinitely many solutions for α = 3 \alpha = 3 α = 3 , β = 14 \beta = 14 β = 14 . Tap an option to check your answer.
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Question 7 10 Apr 2023
71 For the system of linear equations 2 x − y + 3 z = 5 2x - y + 3z = 5 2 x − y + 3 z = 5 3 x + 2 y − z = 7 3x + 2y - z = 7 3 x + 2 y − z = 7 4 x + 5 y + α z = β 4x + 5y + \alpha z = \beta 4 x + 5 y + α z = β Which of the following is NOT correct? A. The system has infinitely many solutions for α = − 5 \alpha = - 5 α = − 5 and β = 9 \beta = 9 β = 9 B. The system has a unique solution for α ≠ − 5 \alpha \neq - 5 α = − 5 and β = 8 \beta = 8 β = 8 C. The system has infinitely many solutions for α = − 6 \alpha = - 6 α = − 6 and β = 9 \beta = 9 β = 9 D. The system is inconsistent for α = − 5 \alpha = - 5 α = − 5 and β = 8 \beta = 8 β = 8 Tap an option to check your answer.
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Question 8 13 Apr 2023
67 For the system of linear equations 2 x + 4 y + 2 a z = b 2x + 4y + 2az = b 2 x + 4 y + 2 a z = b x + 2 y + 3 z = 4 x + 2y + 3z = 4 x + 2 y + 3 z = 4 2 x − 5 y + 2 z = 8 2x - 5y + 2z = 8 2 x − 5 y + 2 z = 8 which of the following is NOT correct? A. It has infinitely many solutions if a = 3 , b = 6 a = 3,b = 6 a = 3 , b = 6 B. It has unique solution if a = b = 6 a = b = 6 a = b = 6 C. It has unique solution if a = b = 8 a = b = 8 a = b = 8 D. It has infinitely many solution if a = 3 , b = 8 a = 3,b = 8 a = 3 , b = 8 Tap an option to check your answer.
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Question 9 30 Jan 2024
73 Consider the system of linear equations x + y + z = 4 μ , x + 2 y + 2 λ z = 10 μ , x + 3 y + 4 λ 2 z = μ 2 + 15 x+y+z=4\mu,\ x+2y+2\lambda z=10\mu,\ x+3y+4\lambda^{2}z=\mu^{2}+15 x + y + z = 4 μ , x + 2 y + 2 λ z = 10 μ , x + 3 y + 4 λ 2 z = μ 2 + 15 , where λ , μ ∈ R \lambda,\mu\in R λ , μ ∈ R . Which one of the following statements is NOT correct? A. The system has unique solution if λ ≠ 1 2 \lambda \neq \frac{1}{2} λ = 2 1 and μ ≠ 1 , 15 \mu \neq 1,15 μ = 1 , 15 B. The system is inconsistent if λ = 1 2 \lambda = \frac{1}{2} λ = 2 1 and μ ≠ 1 \mu \neq 1 μ = 1 C. The system has infinite number of solutions if λ = 1 2 \lambda = \frac{1}{2} λ = 2 1 and μ = 15 \mu = 15 μ = 15 D. The system is consistent if λ ≠ 1 2 \lambda \neq \frac{1}{2} λ = 2 1 Tap an option to check your answer.
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Question 10 30 Jan 2024
151 Consider the system of linear equations x + y + z = 5 , x + 2 y + λ 2 z = 9 x + y + z = 5,x + 2y +\lambda^{2}z = 9 x + y + z = 5 , x + 2 y + λ 2 z = 9 , x + 3 y + λ z = μ x + 3y + \lambda z = \mu x + 3 y + λ z = μ , where λ , μ ∈ R \lambda,\mu \in R λ , μ ∈ R . Then, which of the following statement is NOT correct? A. System has infinite number of solution if λ = 1 \lambda = 1 λ = 1 and μ = 13 \mu = 13 μ = 13 B. System is inconsistent if λ = 1 \lambda = 1 λ = 1 and μ ≠ 13 \mu \neq 13 μ = 13 C. System is consistent if λ ≠ 1 \lambda \neq 1 λ = 1 and μ = 13 \mu = 13 μ = 13 D. System has unique solution if λ ≠ 1 \lambda \neq 1 λ = 1 and μ ≠ 13 \mu \neq 13 μ = 13 Tap an option to check your answer.
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