11th Maths Monthly Test- 4 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test- 4 (Matrices and Determinants)-Aug 2020

Time : 2.00 Hours

Part - A

Generic 5 - Answer All Question

Prove that \(\left| \begin{matrix} { a }^{ 2 }+\lambda & ab & ac \\ ab & { b }^{ 2 }+\lambda & bc \\ ac & bc & { c }^{ 2 }+\lambda \end{matrix} \right| ={ \lambda }^{ 2 }\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+\lambda \right) \)

Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k,0) is a point such that area of \(\triangle\)ABC is 3 sq. units.

A shopkeeper in a Nuts and Spices shop makes gift packs of cashew nuts, raisins, and almonds.
Pack I contains 100 gm of cashew nuts, 100 gm of raisins and 50 gm of almonds.
Pack-II contains 200 gm of cashew nuts, 100 gm of raisins and 100 gm of almonds.
Pack-III contains 250 gm of cashew nuts, 250 gm of raisins and 150 gm of almonds.
The cost of 50 gm of cashew nuts is Rs.50, 50 gm of raisins is Rs.10, and 50 gm of almonds is Rs.60. What is the cost of each gift pack?

Let a, b, c denote the sides BC, CA and AB respectively of \(\triangle\)ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| \)=0 then find the value of sin²A+sin²B+sin²C.

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.

If A(x₁,y₁), B(x₂,y₂) and C(x₃,y₃) are vertices of an equilateral triangle whose each side is equal to a, then prove that \(\left| \begin{matrix} { x }_{ 1 } & { y }_{ 1 } & 2 \\ { x }_{ 2 } & { y }_{ 2 } & 2 \\ { x }_{ 3 } & { y }_{ 3 } & 2 \end{matrix} \right| \)=3a⁴

If \(A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right] \) and \(B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right] \) and (A + B)²=A² + B², find X and.

Solve : \(X+2Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \) ;\(X-Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \)

If A=\(\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}\) is a matrix such that AA^T =9I, find the values of x and y.

Show that \(\begin{vmatrix} 2bc-a^2 & c^2 &b^2 \\ c^2 &2ca-b^2 &a^2 \\ b^2 & a^2 &2ab-c^2 \end{vmatrix}=\begin{vmatrix} a &b &c \\ b & c & a \\ c &a &b \end{vmatrix}^2\) .

Using Factor Theorem, prove that \(\begin{vmatrix} x+1 & 3 & 5 \\ 2 & x+2 &5 \\ 2 & 3 & x+4 \end{vmatrix}\)=(x-1)²(x+9)

In the set Z of integers, define mRn if m-n is divisible by 7.Prove that R is an equivalence relation.

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11th Maths Monthly Test- 4 (Matrices and Determinants)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test- 4 (Matrices and Determinants)-Aug 2020

Part - A

Generic 5 - Answer All Question

Prove that \(\left| \begin{matrix} { a }^{ 2 }+\lambda & ab & ac \\ ab & { b }^{ 2 }+\lambda & bc \\ ac & bc & { c }^{ 2 }+\lambda \end{matrix} \right| ={ \lambda }^{ 2 }\left( { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }+\lambda \right) \)

Find the equation of the line joining A(1,3) and B(0,0) using determinants and find k if D(k,0) is a point such that area of \(\triangle\)ABC is 3 sq. units.

Let a, b, c denote the sides BC, CA and AB respectively of \(\triangle\)ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| \)=0 then find the value of sin2A+sin2B+sin2C.

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.

If A(x1,y1), B(x2,y2) and C(x3,y3) are vertices of an equilateral triangle whose each side is equal to a, then prove that \(\left| \begin{matrix} { x }_{ 1 } & { y }_{ 1 } & 2 \\ { x }_{ 2 } & { y }_{ 2 } & 2 \\ { x }_{ 3 } & { y }_{ 3 } & 2 \end{matrix} \right| \)=3a4

If \(A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right] \) and \(B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right] \) and (A + B)2=A2 + B2, find X and.

Solve : \(X+2Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \) ;\(X-Y=\left[ \begin{matrix} 4 & 6 \\ -8 & 10 \end{matrix} \right] \)

If A=\(\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}\) is a matrix such that AAT =9I, find the values of x and y.

Show that \(\begin{vmatrix} 2bc-a^2 & c^2 &b^2 \\ c^2 &2ca-b^2 &a^2 \\ b^2 & a^2 &2ab-c^2 \end{vmatrix}=\begin{vmatrix} a &b &c \\ b & c & a \\ c &a &b \end{vmatrix}^2\) .

Using Factor Theorem, prove that \(\begin{vmatrix} x+1 & 3 & 5 \\ 2 & x+2 &5 \\ 2 & 3 & x+4 \end{vmatrix}\)=(x-1)2(x+9)

In the set Z of integers, define mRn if m-n is divisible by 7.Prove that R is an equivalence relation.

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Let a, b, c denote the sides BC, CA and AB respectively of \(\triangle\)ABC. If \(\left| \begin{matrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{matrix} \right| \)=0 then find the value of sin²A+sin²B+sin²C.

If A(x₁,y₁), B(x₂,y₂) and C(x₃,y₃) are vertices of an equilateral triangle whose each side is equal to a, then prove that \(\left| \begin{matrix} { x }_{ 1 } & { y }_{ 1 } & 2 \\ { x }_{ 2 } & { y }_{ 2 } & 2 \\ { x }_{ 3 } & { y }_{ 3 } & 2 \end{matrix} \right| \)=3a⁴

If \(A=\left[ \begin{matrix} 1 & -1 \\ 2 & -1 \end{matrix} \right] \) and \(B=\left[ \begin{matrix} x & 1 \\ y & -1 \end{matrix} \right] \) and (A + B)²=A² + B², find X and.

If A=\(\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}\) is a matrix such that AA^T =9I, find the values of x and y.

Using Factor Theorem, prove that \(\begin{vmatrix} x+1 & 3 & 5 \\ 2 & x+2 &5 \\ 2 & 3 & x+4 \end{vmatrix}\)=(x-1)²(x+9)