DETERMINANTS Maths - DETERMINANTS

1. 20
2. 100
3. 10
4. 0

1. 0
2. -16
3. 16
4. None of these.

1. 3
2. 9
3. 1
4. 2

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.
5. Both A and R are false.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.
5. Both A and R are false.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.
5. Both A and R are false.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.
5. Both A and R are false.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.
5. Both A and R are false.

1. $\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}$
   
2. $\begin{vmatrix}x^2-x+1&x-1\\x+1&x+1\end{vmatrix}$
   

$ \begin{vmatrix} \text{a + b} & \text{b + c} & \text{c + a} \\ \text{b + c} & \text{c + a} & \text{a + b} \\ \text{c + a} & \text{a + b} & \text{b + c} \end{vmatrix}=2 \begin{vmatrix} \text{a} & \text{b} & \text{c} \\ \text{b} & \text{c} & \text{a} \\ \text{c} & \text{a} & \text{b} \end{vmatrix}$.

1. The statement is true for n = 1
2. When the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1.

1. If $\text{A}=\begin{bmatrix}3&-4\\1&-1\end{bmatrix},$ then for any positive integer n,
   

1. $\text{A}^\text{n}=\begin{bmatrix}3\text{n}&-4\text{n}\\\text{n}&-\text{n}\end{bmatrix}$
   
2. $\text{A}^\text{n}=\begin{bmatrix}1+2\text{n}&-4\text{n}\\\text{n}&1-2\text{n}\end{bmatrix}$
   
3. $\text{A}^\text{n}=\begin{bmatrix}3\text{n}&-8\text{n}\\1&-\text{n}\end{bmatrix}$
   
4. $\text{A}^\text{n}=\begin{bmatrix}1+3\text{n}&-4\text{n}\\\text{n}&1-3\text{n}\end{bmatrix}$
   

2. If $\text{A}=\begin{bmatrix}1&2\\0&1\end{bmatrix},$ then |Aⁿ|, where $\text{n}\in\text{ N},$ is equal to:
   

1. 2ⁿ
2. 3ⁿ
3. n
4. 1

3. If $\text{A}=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ and $\text{I}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$ then which of the following holds for all natural numbers $\text{n}\geq1?$
   

1. Aⁿ = nA - (n - 1)I
2. Aⁿ = 2^n-1 A - (n - 1)I
3. Aⁿ = nA + (n - 1)I
4. Aⁿ = 2^n-1 A + (n - 1)I

4. Let $\text{A}=\begin{bmatrix}\text{a}&0&0\\0&\text{a}&0\\0&0&\text{a}\end{bmatrix}$ and $\text{A}^\text{n}=[\text{a}_{\text{ij}}]_{3\times3}$ for some positive integer n, then the cofactor of a₁₃ is:
   

1. aⁿ
2. -aⁿ
3. 2aⁿ
4. 0

5. If A is a square matrix such that |A| = 2, then for any positive integer n, |Aⁿ| is equal to:

1. 0
2. 2n
3. 2ⁿ
4. n²

1. Find the area of the triangle whose vertices are (-2, 6), (3, -6) and (1, 5).

1. 30 sq. units
2. 35 sq. units
3. 40 sq. units
4. 15.5 sq. units

2. If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k.

1. 4
2. $\frac{7}{140}$
   
3. 47
4. $\frac{40}{7}$
   

3. If the area of a triangle ABC, with vertices A(1, 3), B(0, 0) and C(k, 0) is 3 sq. units, then a value of k is:

1. 2
2. 3
3. 4
4. 5

4. Using determinants, find the equation of the tine joining the points A(1, 2) and B(3, 6).

1. y = 2x
2. x = 3y
3. y = x
4. 4x - y = 5

5. If $\text{A}\equiv(11,7),\text{B}\equiv(5,5)$ and $\text{C}\equiv(-1,3),$ then:
   

1. $\Delta\text{ABC}$ is scalene triangle.
   
2. $\Delta\text{ABC}$ is equilateral triangle.
   
3. A, B and C are collinear.
4. None of these.

1. The matrix U₁ + U₂ is equal to:

1. $\begin{bmatrix}1\\-1\end{bmatrix}$
   
2. $\begin{bmatrix}2\\-2\end{bmatrix}$
   
3. $\begin{bmatrix}3\\-3\end{bmatrix}$
   
4. $\begin{bmatrix}4\\-4\end{bmatrix}$
   

2. The value of |U| is:

1. 2
2. -2
3. 3
4. -3

3. If $\text{X}=\begin{bmatrix}3&2\end{bmatrix}\text{U}\begin{bmatrix}3\\2\end{bmatrix},$ then the value of |X| =
   

1. 3
2. -3
3. -5
4. 5

4. The minor of element at the position a₂₂ in U is:

1. 1
2. 2
3. -2
4. -1

5. If $\text{U}=[\text{a}_\text{ij}]_{2\times2},$ then the value of a₁₁A₁₁ + a₁₂A₁₂, where A_ij denotes the cofactor of a_ij, is:
   

1. 1
2. 2
3. -3
4. 3

1. If the vertices ofoneof the smaller equilateral triangle are (0, 0), $(3,\sqrt{3})$ and $(3,-\sqrt{3}),$ then the area of such triangle is:
   

1. $\sqrt{3}\text{ sq}.\text{units}$
   
2. $2\sqrt{3}\text{ sq}.\text{units}$
   
3. $3\sqrt{3}\text{ sq}.\text{units}$
   
4. None of these.

2. The area of a face of the Pyramid is:

1. $25\sqrt{3}\text{ sq}.\text{units}$
   
2. $50\sqrt{3}\text{ sq}.\text{units}$
   
3. $75\sqrt{3}\text{ sq}.\text{units}$
   
4. $35\sqrt{3}\text{ sq}.\text{units}$
   

3. The length of a altitude of a smaller equilateral triangle is:

1. 2 units
2. 3 units
3. $\sqrt{3}\text{ units}$
   
4. 4 units

4. If (2, 4), (2, 6) are two vertices of a smaller equilateral triangle, then the third vertex will lie on the line represented by:

1. $\text{x}+\text{y}=5$
   
2. $\text{x}=1+\sqrt3$
   
3. $\text{x}=2+\sqrt3$
   
4. $2\text{x}+\text{y}=5$
   

5. Let A(a, 0), B(0, b) and C(1, 1) be three points. If $\frac{1}{\text{a}}+\frac{1}{\text{b}}=1,$ then the three points are:
   

1. Vertices of an equilateral triangle.
2. Vertices of a right angled triangle.
3. Collinear.
4. Vertices of an isosceles triangle.

1. Find the sum of the cofactors of all the elements of $\begin{vmatrix}1&-2\\4&3\end{vmatrix}.$
   

1. 2
2. -2
3. 4
4. 1

2. Find the minor of a₂₁ of $\begin{vmatrix}5&6&-3\\-4&3&2\\-4&-7&3\end{vmatrix}.$
   

1. 3
2. -3
3. 39
4. -39

3. In the determinant $\begin{vmatrix}2&-3&5\\6&0&4\\1&5&-7\end{vmatrix},$ find the value of a₃₂.A₃₂.
   

1. 27
2. -110
3. 110
4. -27

4. If $\Delta=\begin{vmatrix}5&3&8\\2&0&1\\1&2&3\end{vmatrix},$ then write the minor of a₂₃.
   

1. -10
2. -7
3. 10
4. 7

5. If $\Delta=\begin{vmatrix}2&-3&5\\6&0&4\\1&5&-7\end{vmatrix},$ then find the value of $|\Delta|.$
   

1. 26
2. 28
3. 72
4. 46