The equation- - 1 x y - - 1 3 1 0 2 -1 0 0 1 - - 1 x y - - 0 - has fori y-0- p rational rootsii y-1 q irrational roots r integral roots

The correct option is B (i) (p) (ii) (q) [ 1 amp; x amp; y ][ 1 amp; 3 amp; 1; 0 amp; 2 amp; 1; 0 amp; 0 amp; 1 ][ 1; x; y ]=[ ...

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Byju's Answer

Standard XII

Mathematics

Determinant

The equation,...

Question

The equation, $[\begin{matrix} 1 & x & y \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 1 0 & 2 & - 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 x y \end{matrix} ⎤ ⎥ ⎦ = [0]$ has for
(i) $y = 0$ , (p) rational roots
(ii) $y = - 1$ (q) irrational roots
(r) integral roots

(i) (p) (ii) (r)

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(i) (q) (ii) (p)

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(i) (p) (ii) (q)

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(i) (r) (ii) (p)

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Solution

The correct option is B (i) (p) (ii) (q)
$[\begin{matrix} 1 & x & y \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 1 0 & 2 & - 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 x y \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 0 \end{matrix}]$
$(i)$ if $y = 0$
$[\begin{matrix} 1 & x & 0 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 1 0 & 2 & - 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 x 0 \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 0 \end{matrix}] [\begin{matrix} 1 & x & 0 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 + 3 x 2 x 0 \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 0 \end{matrix}] [2 x^{2} + 3 x + 1] = [\begin{matrix} 0 \end{matrix}]$
By equality
$2 x^{2} + 3 x + 1 = 0 x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot 1}}{4} x = \frac{- 3 \pm \sqrt{1}}{4} x = \frac{- 3 + 1}{4} = - \frac{1}{2} x = \frac{- 4}{4} = - 1$
Ans: $(p)$ rational roots
$(i i)$ if $y = - 1$
$[\begin{matrix} 1 & x & - 1 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 1 0 & 2 & - 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 x - 1 \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 0 \end{matrix}] [\begin{matrix} 1 & x & - 1 \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 + 3 x - 1 2 x + 1 - 1 \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 0 \end{matrix}] [3 x + x (2 x + 1) + 1] = [\begin{matrix} 0 \end{matrix}] 2 x^{2} + 4 x + 1 = 0 x = \frac{- 4 \pm \sqrt{16 - 8}}{4} = \frac{- 4 \pm 2 \sqrt{2}}{4} = \frac{- 2 \pm \sqrt{2}}{2}$
Ans: $(q)$ irrational roots
Ans: $(i) (p) (i i) (q)$

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Which of the following is the only CORRECT combination?

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Q. Match the following:

\begin{matrix} C o l u m n 1 & C o l u m n 2 (p) & S c r u b b e r & (i) & 1000 v o l t s e l e c t r o d e s (q) & C a t a l y t i c c o n v e r t e r & (i i) & f l u e g a s c o l u m n (r) & E l e c t r o s t a t i c p r e c i p i t a t o r & (i i i) & r e d o x r e a c t i o n s \end{matrix}

Match the following:
$\begin{matrix} C o l u m n 1 & C o l u m n 2 (p) & S c r u b b e r & (i) & 1000 v o l t s e l e c t r o d e s (q) & C a t a l y t i c c o n v e r t e r & (i i) & f l u e g a s c o l u m n (r) & E l e c t r o s t a t i c p r e c i p i t a t o r & (i i i) & r e d o x r e a c t i o n s \end{matrix}$

Q. Two quadratic equations

p x^{2} - 2 q x + p = 0 \dots (i)

and

q x^{2} - 2 p x + q = 0 \dots (i i) \forall p, q \in R .

If the roots of the equation

(i)

are real and unequal, then equation

(i i)

will have:

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The equation, [1xy]⎡⎢⎣13102−1001⎤⎥⎦⎡⎢⎣1xy⎤⎥⎦=[0] has for(i) y=0, (p) rational roots(ii) y=−1 (q) irrational roots (r) integral roots

The equation, $[\begin{matrix} 1 & x & y \end{matrix}] ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 & 1 0 & 2 & - 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 1 x y \end{matrix} ⎤ ⎥ ⎦ = [0]$ has for
(i) $y = 0$ , (p) rational roots
(ii) $y = - 1$ (q) irrational roots
(r) integral roots