Mindyard | Let  R 2  denote  R × R . Let S = { ( a , b , c ) : a , b , c ∈ R and  a x 2 + 2 b x y + c y 2 > 0 for all  ( x , y ) ∈ R 2 − { ( 0 , 0 ) } } . Then which of the following statements is (are) TRUE? ( 2 , 7 2 , 6 ) ∈ S , ( 3 , b , 1 12 ) ∈ S , then | 2 b | < 1 , For any given  ( a , b , c ) ∈ S , the system of linear equations  a x + b y = 1 b x + c y = − 1 has a unique solution., For any given  ( a , b , c ) ∈ S , the system of linear equations a + 1 ) x + b y = 0 b x + ( c + 1 ) y = 0 has a unique solution.

Let $R^{2}$ denote $R \times R$ . Let $S = {(a, b, c) : a, b, c \in R$ and $a x^{2} + 2 b x y + c y^{2} > 0$ for all $(x, y) \in R^{2} - {(0, 0)}}$ . Then which of the following statements is (are) TRUE?

Options

$(2, \frac{7}{2}, 6) \in S$

$(3, b, \frac{1}{12}) \in S$ , then $| 2 b | < 1$

For any given $(a, b, c) \in S,$ the system of linear equations $\begin{aligned} a x + b y = 1 \\ b x + c y = - 1 \end{aligned}$ has a unique solution.

For any given $(a, b, c) \in S,$ the system of linear equations $\begin{aligned} a + 1) x + b y = 0 \\ b x + (c + 1) y = 0 \end{aligned}$ has a unique solution.

Correct Answer :

Option D

For any given $(a, b, c) \in S,$ the system of linear equations $\begin{aligned} a + 1) x + b y = 0 \\ b x + (c + 1) y = 0 \end{aligned}$ has a unique solution.

Option C

For any given $(a, b, c) \in S,$ the system of linear equations $\begin{aligned} a x + b y = 1 \\ b x + c y = - 1 \end{aligned}$ has a unique solution.

Option B

$(3, b, \frac{1}{12}) \in S$ , then $| 2 b | < 1$

Option 2, Option 3, and Option 4 are true.

Solution :

To determine which statements are true, let us first understand the condition defining the set $S$ .
The set is defined as:

$S = {(a, b, c) : a, b, c \in R and a x^{2} + 2 b x y + c y^{2} > 0 for all (x, y) \in R^{2} - {(0, 0)}}$

The quadratic form

a x^{2} + 2 b x y + c y^{2}

is strictly positive for all non-zero vectors if and only if the symmetric matrix corresponding to the quadratic form:

$A = [\begin{matrix} a & b \\ b & c \end{matrix}]$

is positive definite. By Sylvester's criterion, a symmetric 2 × 2 matrix is positive definite if and only if its leading principal minors are positive:
1.

a > 0

\det (A) = a c - b^{2} > 0

(which also implies

c > 0

Now, let us analyze each option based on these two criteria:

Analysis of Option 1:
Let $(a, b, c) = (2, \frac{7}{2}, 6)$ .
Here, $a = 2 > 0$ . Let us check the determinant condition:

$a c - b^{2} = (2) (6) - {(\frac{7}{2})}^{2} = 12 - \frac{49}{4} = 12 - 12.25 = - 0.25$

Since

a c - b^{2} < 0

, the triplet

(2, \frac{7}{2}, 6) \notin S

. Therefore, Option 1 is FALSE.

Analysis of Option 2:
Let $(3, b, \frac{1}{12}) \in S$ .
Since the triplet belongs to $S$ , we must have:

$a c - b^{2} > 0 \Rightarrow (3) (\frac{1}{12}) - b^{2} > 0$

Simplifying the inequality:

$\frac{1}{4} - b^{2} > 0 \Rightarrow b^{2} < \frac{1}{4} \Rightarrow | b | < \frac{1}{2} Multiplying by 2: | 2 b | < 1$

Therefore, Option 2 is TRUE.

Analysis of Option 3:
For any given $(a, b, c) \in S$ , the system of linear equations is:
$a x + b y = 1$
$b x + c y = - 1$
The determinant of the coefficient matrix is:

$D = | \begin{matrix} a & b \\ b & c \end{matrix} | = a c - b^{2}$

Since

(a, b, c) \in S

, we have

a c - b^{2} > 0

. Since the determinant

D \neq 0

, the system of linear equations has a unique solution. Therefore, Option 3 is TRUE.

Analysis of Option 4:
For any given $(a, b, c) \in S$ , the homogeneous system of linear equations is:
$(a + 1) x + b y = 0$
$b x + (c + 1) y = 0$
The determinant of this system's coefficient matrix is:

$D^{'} = | \begin{matrix} a + 1 & b \\ b & c + 1 \end{matrix} | = (a + 1) (c + 1) - b^{2} = a c + a + c + 1 - b^{2} = (a c - b^{2}) + a + c + 1$

Since

(a, b, c) \in S

, we know that:
-

a c - b^{2} > 0

a > 0

c > 0

1 > 0

Since

D^{'}

is the sum of positive values, it must be strictly positive:

$D^{'} > 0$

Since the determinant

D^{'} \neq 0

, the system has a unique solution (the trivial solution

x = 0, y = 0

). Therefore, Option 4 is TRUE.

Get 3 Months FREE trial

Create, conduct, and manage professional online assessments with Mindyard. Perfect for teachers and institutes.

Let $R^{2}$ denote $R \times R$ . Let $S = {(a, b, c) : a, b, c \in R$ and $a x^{2} + 2 b x y + c y^{2} > 0$ for all $(x, y) \in R^{2} - {(0, 0)}}$ . Then which of the following statements is (are) TRUE?

Unlock Our Free Library

You may also like

Study Material

CTET Paper 2 Set E 20th Aug 2023 Question Paper with Solutions

CTET Paper 2 Set M 31st Jan 2021 Question Paper with Solutions

Mock Tests

CTET Paper 2 Set E 20th Aug 2023

CTET Paper 2 Set M 31st Jan 2021

Ask AI Tutor