Assertion -consider the determinant - - a 1 - b 1 x 2 a 1 x 2 - b 1 c 1 a 2 - b 2 x 2 a 2 x 2 - b 2 c 2 a 3 - b 3 x 2 a 3 x 2 - b 3 c 3 -0- where x i - y i - z i - R i-1-2-3- x- R- The values of x satisfying -0 are x-1-1- Reason- If a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 -0- then -0-

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Properties of Determinants

Assertion :co...

Question

Assertion :consider the determinant $Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} + b_{1} x^{2} & a_{1} x^{2} + b_{1} & c_{1} a_{2} + b_{2} x^{2} & a_{2} x^{2} + b_{2} & c_{2} a_{3} + b_{3} x^{2} & a_{3} x^{2} + b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$ , where $x_{i}, y_{i}, z_{i} \in R (i = 1, 2, 3), x \in R$ . The values of $x$ satisfying $Δ = 0$ are $x = 1, - 1$ . Reason: If $∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$ , then $Δ = 0$ .

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Assertion is correct but Reason is incorrect

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Assertion is incorrect but Reason is correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} + b_{1} x^{2} & a_{1} x^{2} + b_{1} & c_{1} a_{2} + b_{2} x^{2} & a_{2} x^{2} + b_{2} & c_{2} a_{3} + b_{3} x^{2} & a_{3} x^{2} + b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{1} x^{2} + b_{1} & c_{1} a_{2} & a_{2} x^{2} + b_{2} & c_{2} a_{3} & a_{3} x^{2} + b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} b_{1} x^{2} & a_{1} x^{2} + b_{1} & c_{1} b_{2} x^{2} & a_{2} x^{2} + b_{2} & c_{2} b_{3} x^{2} & a_{3} x^{2} + b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{1} x^{2} & c_{1} a_{2} & a_{2} x^{2} & c_{2} a_{3} & a_{3} x^{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} b_{1} x^{2} & a_{1} x^{2} & c_{1} b_{2} x^{2} & a_{2} x^{2} & c_{2} b_{3} x^{2} & a_{3} x^{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ ∣ \begin{matrix} b_{1} x^{2} & b_{1} & c_{1} b_{2} x^{2} & b_{2} & c_{2} b_{3} x^{2} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow x^{2} ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{1} & c_{1} a_{2} & a_{2} & c_{2} a_{3} & a_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ + ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ + x^{4} ∣ ∣ ∣ ∣ \begin{matrix} b_{1} & a_{1} & c_{1} b_{2} & a_{2} & c_{2} b_{3} & a_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ + x^{2} ∣ ∣ ∣ ∣ \begin{matrix} b_{1} & b_{1} & c_{1} b_{2} & b_{2} & c_{2} b_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow (1 - x^{4}) ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow x = 1, - 1$
Also, if $∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0$
Then $Δ = 0$

Suggest Corrections

Similar questions

Assertion: All the points (1, 0), (-1, 0), (2, 0) and (5, 0) lie on the x – axis.

Reason: Equation of the x – axis is y = 0
Now, choose the correct option.

Q. Assertion: The compressibility factor for

H_{2}

and

H e

(1 + \frac{P b}{R T})

Reason: The compressibility factor for

H_{2}

and

H e

can be derived from Van der Waal's equation.

Q. Assertion: In the equation,

y = A sin (k x - ω t)

, particle starts from its mean position

x = 0

and moves towards negative

y -

axis.
Reason: The wave corresponding to given equation is travelling towards positive

x -

direction.

Q. Assertion (A)
The equation x² + x + 1 = 0 has both real roots.

Reason (R)
The equation ax2 + bx + c = 0, (a ≠ 0) has both real roots, if (b2 − 4ac) > 0.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Assertion (A) Order and molecularity are same.

Reason (R) Order is determined experimentally and molecularity is the sum of the stoichimetric coeffcient of rate determining elementary step.

(a) Both assertion and reason are correct and the reason is correct explanation of assertion
(b) Both assertion and reason are correct, but reason does not explain assertion.
(c) Assertion is correct, but reason is incorrect
(d) Both assertion and reason are incorrect
(e) Assertion is incorrect but reason is correct.

Join BYJU'S Learning Program

Assertion :consider the determinant Δ=∣∣ ∣ ∣∣a1+b1x2a1x2+b1c1a2+b2x2a2x2+b2c2a3+b3x2a3x2+b3c3∣∣ ∣ ∣∣=0, where xi,yi,zi∈R(i=1,2,3),x∈R. The values of x satisfying Δ=0 are x=1,−1. Reason: If ∣∣ ∣∣a1b1c1a2b2c2a3b3c3∣∣ ∣∣=0, then Δ=0.