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Q. Let $a, b, c$ be such that $\mathrm{b}(\mathrm{a}+\mathrm{c}) \neq 0 . \mathrm{If}$ If $\left|\begin{array}{ccc}{a} & {a+1} & {a-1} \\ {-b} & {b+1} & {b-1} \\ {c} & {c-1} & {c+1}\end{array}\right|+\left|\begin{array}{ccc}{a+1} & {b+1} & {c-1} \\ {a-1} & {b-1} & {c+1} \\ {-1} & {a} & {(-1)^{n+1} b} & {(-1)^{n} c}\end{array}\right|=0$ then the value of n is :-
(1) Any odd integer
(2) Any integer
(3) Zero
(4) Any even integer
[AIEEE - 2009]
Ans. (1)
Q. Consider the system of linear equations :
$\mathrm{x}_{1}+2 \mathrm{x}_{2}+\mathrm{x}_{3}=3$
$2 \mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}=3$
$3 x_{1}+5 x_{2}+2 x_{3}=1$
The system has
(1) Infinite number of solutions
(2) Exactly 3 solutions
(3) A unique solution
(4) No solution
[AIEEE - 2010]
Ans. (4)
Here $D=0$ $\& \quad D_{1} \neq 0$ so we can say
no solution
Q. The number of values of k for which the linear equations
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
possess a non-zero solution is :-
(1) 1 (2) zero (3) 3 (4) 2
[AIEEE - 2011]
Ans. (4)
Q. If the trivial solution is the only solution of the system of equations
x – ky + z = 0
kx + 3y – kz = 0
3x + y – z = 0
Then the set of all values of k is:
(1) {2, –3} (2) R – {2, –3} (3) R – {2} (4) R – {–3}
[AIEEE - 2011]
Ans. (2)
Here for trival solution $D \neq 0$
So $\mathrm{D}=\left|\begin{array}{ccc}{1} & {-\mathrm{k}} & {1} \\ {\mathrm{k}} & {3} & {-\mathrm{k}} \\ {3} & {1} & {-1}\end{array}\right|=0$
$\Rightarrow \mathrm{D}=2 \mathrm{k}^{2}-12+2 \mathrm{k}=0 \Rightarrow \mathrm{k}=-3,2$ so $\mathrm{R}-\{-3,2\}$
Q. The number of values of k, for which the system of equations :
(k + 1)x + 8y = 4k
kx + (k + 3)y = 3k – 1
has no solution, is –
(1) infinite (2) 1 (3) 2 (4) 3
[JEE(Main)-2013]
Ans. (2)
$\frac{k+1}{k}=\frac{8}{k+3}=\frac{4 k}{3 k-1}$
(1) = (2)
$\Rightarrow \quad k^{2}-4 k+3=0$
k = 1, 3
for k = 1 (2) = (3)
for $\mathrm{k}=3 \quad(2) \neq(3)$
k = 3
Q. If $\alpha, \beta \neq 0,$ and $f(\mathrm{n})=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$ and $\left|\begin{array}{ccc}{3} & {1+f(1)} & {1+f(2)} \\ {1+f(1)} & {1+f(2)} & {1+f(3)} \\ {1+f(2)} & {1+f(3)} & {1+f(4)}\end{array}\right|=\mathrm{K}(1-\alpha)^{2}(1-\beta)^{2}(\alpha-\beta)^{2}$ then K is equal to :
(1) $\alpha \beta$
(2) $\frac{1}{\alpha \beta}$
(3) 1
(4) –1
[JEE(Main)-2014]
Ans. (3)
$\therefore k=1$
$\therefore k=1$
Q. The set of all values of $\lambda$ for which the system of linear equations :
$2 \mathrm{x}_{1}-2 \mathrm{x}_{2}+\mathrm{x}_{3}=\lambda \mathrm{x}_{1}$
$2 \mathrm{x}_{1}-3 \mathrm{x}_{2}+2 \mathrm{x}_{3}=\lambda \mathrm{x}_{2}$
$-\mathrm{x}_{1}+2 \mathrm{x}_{2}=\lambda \mathrm{x}_{3}$
has a non-trivial solution
(1) contains two elements
(2) contains more than two elements
(3) is an empty set
(4) is a singleton
[JEE(Main)-2015]
Ans. (1)
Q. The system of linear equations
$\mathrm{x}+\lambda \mathrm{y}-\mathrm{z}=0$
$\lambda \mathrm{x}-\mathrm{y}-\mathrm{z}=0$
$\mathrm{x}+\mathrm{y}-\lambda \mathrm{z}=0$
has a non-trivial solution for :
(1) exactly three values of $\lambda$
(2) infinitely many values of $\lambda$
(3) exactly one value of $\lambda$
(4) exactly two values of $\lambda$
[JEE(Main)-2016]
Ans. (1)
$\left|\begin{array}{ccc}{1} & {\lambda} & {-1} \\ {\lambda} & {-1} & {-1} \\ {1} & {1} & {-\lambda}\end{array}\right|=0 \quad \Rightarrow \quad \lambda=0,1,-1$
Q. If S is the set of distinct values of 'b' for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is :
(1) a singleton
(2) an empty set
(3) an infinite set
(4) a finite set containing two or more elements
[JEE(Main)-2017]
Ans. (1)
$D=\left|\begin{array}{lll}{1} & {1} & {1} \\ {1} & {a} & {1} \\ {a} & {b} & {1}\end{array}\right|=0 \Rightarrow a=1$
and at a = 1
$\mathrm{D}_{1}=\mathrm{D}_{2}=\mathrm{D}_{3}=0$
But at a = 1 and b = 1
$\left.\begin{array}{ll}{\text { First two equations are }} & {x+y+z=1} \\ {\text { and third equation is }} & {x+y+z=0}\end{array}\right] \Rightarrow$ There is nosolution.
$\mathrm{b}=\{1\} \Rightarrow$ it is a singleton set
Q. If $\left|\begin{array}{ccc}{x-4} & {2 x} & {2 x} \\ {2 x} & {x-4} & {2 x} \\ {2 x} & {2 x} & {x-4}\end{array}\right|=(A+B x)(x-A)^{2},$ then the ordered pair $(A, B)$ is equal to :
(1) (–4, 3) (2) (–4, 5) (3) (4, 5) (4) (–4, –5)
[JEE(Main)-2018]
Ans. (2)
$\left|\begin{array}{ccc}{\mathrm{x}-4} & {2 \mathrm{x}} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {\mathrm{x}-4} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {2 \mathrm{x}} & {\mathrm{x}-4}\end{array}\right|=(\mathrm{A}+\mathrm{Bx})(\mathrm{x}-\mathrm{A})^{2}$
Put $x=0 \Rightarrow\left|\begin{array}{ccc}{-4} & {0} & {0} \\ {0} & {-4} & {0} \\ {0} & {0} & {-4}\end{array}\right|=A^{3} \Rightarrow A=-4$
$\left|\begin{array}{ccc}{\mathrm{x}-4} & {2 \mathrm{x}} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {\mathrm{x}-4} & {2 \mathrm{x}} \\ {2 \mathrm{x}} & {2 \mathrm{x}} & {\mathrm{x}-4}\end{array}\right|=(\mathrm{Bx}-4)(\mathrm{x}+4)^{2}$
$\left|\begin{array}{ccc}{1-\frac{4}{\mathrm{x}}} & {2} & {2} \\ {2} & {1-\frac{4}{\mathrm{x}}} & {2} \\ {2} & {2} & {1-\frac{4}{\mathrm{x}}}\end{array}\right|=\left(\mathrm{B}-\frac{4}{\mathrm{x}}\right)\left(1+\frac{4}{\mathrm{x}}\right)^{2}$
Put $\mathrm{x} \rightarrow \infty \quad \Rightarrow \quad\left|\begin{array}{lll}{1} & {2} & {2} \\ {2} & {1} & {2} \\ {2} & {2} & {1}\end{array}\right|=\mathrm{B} \Rightarrow \mathrm{B}=5$
ordered pair (A, B) is (–4, 5)
Q. If the system of linear equations $x+k y+3 z=0$
$3 x+k y-2 z=0$
$2 x+4 y-3 z=0$
has a non-zero solution $(\mathrm{x}, \mathrm{y}, \mathrm{z}),$ then $\frac{\mathrm{xz}}{\mathrm{y}^{2}}$ is equal to :
(1) 10 (2) – 30 (3) 30 (4) –10
[JEE(Main)-2018]
Ans. (1)
Q. If the system of linear equations :
$x+a y+z=3$
$\mathrm{x}+2 \mathrm{y}+2 \mathrm{z}=6$
$x+5 y+3 z=b$
has no solution, then :-
(1) $a=-1, b=9$
(2) $a \neq-1, b=9$
(3) $a=1, b \neq 9$
(4) $a=-1, b \neq 9$
[JEE(Main)-2018]
Ans. (4)
Q. The number of values of k for which the system of linear equations,
(k+2)x + 10y = k
kx + (k+3) y = k – 1
has no solution is :
(1) infinitely many (2) 1 (3) 2 (4) 3
[JEE(Main)-2018]
Ans. (2)
Comments
Riya
Aug. 30, 2023, 6:35 a.m.
It's very helpful thank you but I think the last question option should be 3 bcz values of K are 2 what do u think