Determinants are one of the most important and widely used topics in mathematics. They play a crucial role not only in school-level mathematics but also in engineering mathematics and higher mathematics, especially in areas such as linear algebra, coordinate geometry, calculus, vector spaces, and systems of linear equations. Determinants are extensively applied in solving real-world problems, finding areas of geometric figures, checking collinearity of points, and understanding advanced concepts used in engineering and scientific computations. In this post, we provide a carefully designed set of 150 practice questions on determinants, ranging from basic to advanced levels. These questions are highly useful for board classes (CBSE, NCERT, state boards) as well as for competitive examinations, helping students build strong conceptual clarity, speed, and confidence in this essential topic.
What is a Determinant?
A determinant is a single numerical value associated with a square matrix. It is widely used in algebra and coordinate geometry to:
- solve linear equations,
- find the area of a triangle,
- check whether points are collinear,
- determine invertibility of a matrix.
- and many other fields of mathematics
A determinant is written using vertical bars.
Determinant of a 2 × 2 Matrix
For a 2 × 2 matrix
Evaluate \[ \begin{vmatrix} 3 & 5\\ 2 & 4 \end{vmatrix} \]
Solution:
\[ = (3 \times 4) - (5 \times 2) \] \[ = 12 - 10 = 2 \]
Determinant of a 3 × 3 Matrix
For a 3 × 3 matrix
Expand along the first row:
Evaluate \[ \begin{vmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{vmatrix} \]
Solution:
\[ =1(45-48)-2(36-42)+3(32-35) \] \[ =-3+12-9=0 \]
Shortcut for Diagonal Matrix
If a matrix is diagonal, its determinant is the product of diagonal elements.
\[ \begin{vmatrix} 2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 5 \end{vmatrix} = 2 \times 3 \times 5 = 30 \]
Condition for Collinearity of Points
Three points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) are collinear if
Check collinearity of \((1,2),(2,4),(3,6)\).
\[ \begin{vmatrix} 1 & 2 & 1\\ 2 & 4 & 1\\ 3 & 6 & 1 \end{vmatrix} = -2+2=0 \]
Hence, the points are collinear.
Area of a Triangle Using Determinants
Area of triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\):
Find area of triangle with vertices \((1,2),(3,6),(5,2)\).
\[ \text{Area}=\frac12| -16 |=8 \]
Important Exam Shortcuts
- If two rows or columns are identical → determinant = 0
- If one row is a multiple of another → determinant = 0
- Diagonal matrix → multiply diagonal elements
- Collinear points → determinant = 0
- Triangle area → half of absolute determinant
Q.1
Evaluate the determinant: \(\begin{vmatrix} 3 & 5 \\ 2 & 4 \end{vmatrix}\).
Q.2
Evaluate the determinant: \(\begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix}\).
Q.3
Find the value of: \(\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}\).
Q.4
If \(\begin{vmatrix} x & 1 \\ 4 & x \end{vmatrix} = 0\), find the value of \(x\).
Q.5
Evaluate: \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\).
Q.6
Find the value of \(k\) such that \(\begin{vmatrix} 1 & 2 & 3 \\ 2 & k & 6 \\ 3 & 6 & 9 \end{vmatrix} = 0\).
Q.7
Evaluate: \(\begin{vmatrix} 3 & 1 \\ 2 & 4 \end{vmatrix}\).
Q.8
Find the value of: \(\begin{vmatrix} 1 & 2 \\ 2 & 4 \end{vmatrix}\).
Q.9
Evaluate: \(\begin{vmatrix} -1 & 3 \\ 4 & 2 \end{vmatrix}\).
Q.10
If \(\begin{vmatrix} x & 2 \\ 3 & x \end{vmatrix} = 5\), find \(x\).
Q.11
Evaluate: \(\begin{vmatrix} 1 & 0 & 0 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{vmatrix}\).
Q.12
Evaluate: \(\begin{vmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 4 & 1 & 8 \end{vmatrix}\).
Q.13
Find the value of \(x\) if \(\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix} = 0\).
Q.14
Evaluate: \(\begin{vmatrix} 2 & -1 & 0 \\ 1 & 3 & 2 \\ 4 & 1 & 5 \end{vmatrix}\).
Q.15
If \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 4\), find \(\begin{vmatrix} 2a & 2b \\ c & d \end{vmatrix}\).
Q.16
Evaluate: \(\begin{vmatrix} 1 & 2 & 1 \\ 3 & 6 & 3 \\ 2 & 4 & 2 \end{vmatrix}\).
Q.17
Find the value of: \(\begin{vmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{vmatrix}\).
Q.18
Evaluate: \(\begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 5 \end{vmatrix}\).
Q.19
Find the value of \(k\) if \(\begin{vmatrix} 1 & k \\ k & 1 \end{vmatrix} = 0\).
Q.20
Evaluate: \(\begin{vmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 1 & 4 & 0 \end{vmatrix}\).
Q.21
If two rows of a determinant are identical, then the value of the determinant is:
Q.22
Evaluate: \(\begin{vmatrix} a & a & a \\ b & b & b \\ c & c & c \end{vmatrix}\).
Q.23
Find the value of: \(\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}\).
Q.24
If \(\begin{vmatrix} x & y \\ y & x \end{vmatrix} = 0\), find the relation between \(x\) and \(y\).
Q.25
Evaluate: \(\begin{vmatrix} 0 & 1 & 2 \\ 3 & 4 & 5 \\ 6 & 7 & 8 \end{vmatrix}\).
Q.26
If \(\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = -2\), find \(\begin{vmatrix} 4 & 3 \\ 2 & 1 \end{vmatrix}\).
Q.27
Evaluate: \(\begin{vmatrix} 4 & 5 \\ 6 & 7 \end{vmatrix}\).
Q.28
Find the value of: \(\begin{vmatrix} 1 & -1 \\ -1 & 1 \end{vmatrix}\).
Q.29
Evaluate: \(\begin{vmatrix} 2 & 3 & 4 \\ 1 & 2 & 3 \\ 0 & 1 & 2 \end{vmatrix}\).
Q.30
Find the value of \(x\) if \(\begin{vmatrix} x & 1 \\ 1 & x \end{vmatrix} = 3\).
Q.31
Evaluate: \(\begin{vmatrix} 1 & 2 & 3 \\ 3 & 6 & 9 \\ 2 & 4 & 6 \end{vmatrix}\).
Q.32
If \(\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 7\), find \(\begin{vmatrix} a & b \\ 2c & 2d \end{vmatrix}\).
Q.33
Evaluate: \(\begin{vmatrix} 2 & 0 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 5 \end{vmatrix}\).
Q.34
Find the value of: \(\begin{vmatrix} x & x \\ x & x \end{vmatrix}\).
Q.35
Evaluate: \(\begin{vmatrix} 1 & 1 & 0 \\ 2 & 3 & 1 \\ 4 & 5 & 2 \end{vmatrix}\).
Q.36
If \(\begin{vmatrix} 1 & k \\ k & 1 \end{vmatrix} = 0\), find \(k\).
Q.37
Evaluate: \(\begin{vmatrix} \sin\theta & \cos\theta \\ -\cos\theta & \sin\theta \end{vmatrix}\).
Q.38
Evaluate: \(\begin{vmatrix} a & b & c \\ a & b & c \\ x & y & z \end{vmatrix}\).
Q.39
Find the value of: \(\begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix}\).
Q.40
If \(\begin{vmatrix} a & b \\ b & a \end{vmatrix} = 0\), find the relation between \(a\) and \(b\).
Q.41
Evaluate: \(\begin{vmatrix} 2 & 1 & 3 \\ 1 & 1 & 2 \\ 3 & 2 & 4 \end{vmatrix}\).
Q.42
Evaluate: \(\begin{vmatrix} 0 & 2 & 1 \\ 3 & 0 & 4 \\ 5 & 6 & 0 \end{vmatrix}\).
Q.43
If \(\begin{vmatrix} x & 2 \\ 3 & x \end{vmatrix} = 7\), find \(x\).
Q.44
Evaluate: \(\begin{vmatrix} 1 & 2 & 1 \\ 0 & 1 & 3 \\ 0 & 0 & 4 \end{vmatrix}\).
Q.45
Find the value of: \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\).
Q.46
Evaluate: \(\begin{vmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \\ 3 & 5 & 7 \end{vmatrix}\).
Q.47
Evaluate the determinant of the diagonal matrix: \(\begin{vmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{vmatrix}\).
Q.48
Find the value of: \(\begin{vmatrix} 7 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 4 \end{vmatrix}\).
Q.49
Evaluate the determinant of the anti-diagonal matrix: \(\begin{vmatrix} 0 & 0 & 3 \\ 0 & 2 & 0 \\ 1 & 0 & 0 \end{vmatrix}\).
Q.50
Evaluate: \(\begin{vmatrix} 0 & 0 & 5 \\ 0 & 4 & 0 \\ 3 & 0 & 0 \end{vmatrix}\).
Q.51
Find the determinant: \(\begin{vmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{vmatrix}\).
Q.52
Evaluate: \(\begin{vmatrix} 0 & 0 & a \\ 0 & b & 0 \\ c & 0 & 0 \end{vmatrix}\).
Q.53
Find the value of: \(\begin{vmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \end{vmatrix}\).
Q.54
Evaluate the anti-diagonal determinant: \(\begin{vmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 3 & 0 & 0 \\ 4 & 0 & 0 & 0 \end{vmatrix}\).
Q.55
Find the determinant: \(\begin{vmatrix} x & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & x \end{vmatrix}\).
Q.56
Evaluate: \(\begin{vmatrix} 0 & 0 & x \\ 0 & x & 0 \\ x & 0 & 0 \end{vmatrix}\).
Q.57
If \(\begin{vmatrix} 3 & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & 2 \end{vmatrix} = 24\), find \(k\).
Q.58
Find the value of \(a\) if \(\begin{vmatrix} 0 & 0 & a \\ 0 & 5 & 0 \\ 2 & 0 & 0 \end{vmatrix} = -40\).
Q.59
Evaluate: \(\begin{vmatrix} 5 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{vmatrix}\).
Q.60
Evaluate: \(\begin{vmatrix} 0 & 0 & 4 \\ 0 & -3 & 0 \\ 2 & 0 & 0 \end{vmatrix}\).
Q.61
Find the determinant: \(\begin{vmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{vmatrix}\).
Q.62
Evaluate the anti-diagonal determinant: \(\begin{vmatrix} 0 & 0 & 1 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \end{vmatrix}\).
Q.63
If \(\begin{vmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & x \end{vmatrix} = 30\), find \(x\).
Q.64
Evaluate: \(\begin{vmatrix} 0 & 0 & 2 \\ 0 & y & 0 \\ 4 & 0 & 0 \end{vmatrix}\).
Q.65
Find the value of: \(\begin{vmatrix} p & 0 & 0 \\ 0 & q & 0 \\ 0 & 0 & r \end{vmatrix}\).
Q.66
Evaluate the anti-diagonal determinant: \(\begin{vmatrix} 0 & 0 & a \\ 0 & b & 0 \\ c & 0 & 0 \end{vmatrix}\).
Q.67
Evaluate: \(\begin{vmatrix} \sin\theta & \cos\theta \\ \cos\theta & -\sin\theta \end{vmatrix}\).
Q.68
Evaluate: \(\begin{vmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{vmatrix}\).
Q.69
Find the value of: \(\begin{vmatrix} \sin A & \cos A & 0 \\ \cos A & -\sin A & 0 \\ 0 & 0 & 1 \end{vmatrix}\).
Q.70
Evaluate: \(\begin{vmatrix} \sin x & \cos x & \sin x \\ \cos x & -\sin x & \cos x \\ 0 & 0 & 1 \end{vmatrix}\).
Q.71
Find the value of \(x\) if \(\begin{vmatrix} x-1 & 2 \\ 3 & x-1 \end{vmatrix} = 0\).
Q.72
Find the value of \(x\) if \(\begin{vmatrix} x-2 & 1 \\ 4 & x-2 \end{vmatrix} = 0\).
Q.73
Evaluate: \(\begin{vmatrix} x-a & 0 \\ 0 & x-a \end{vmatrix}\).
Q.74
Evaluate: \(\begin{vmatrix} x-a & x-a \\ x-a & x-a \end{vmatrix}\).
Q.75
Find the value of: \(\begin{vmatrix} x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1 \end{vmatrix}\).
Q.76
Find the value of \(x\) if \(\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix} = 0\).
Q.77
Evaluate: \(\begin{vmatrix} \cos x & \sin x & 0 \\ -\sin x & \cos x & 0 \\ 0 & 0 & x \end{vmatrix}\).
Q.78
Evaluate: \(\begin{vmatrix} x & \sin x \\ \sin x & x \end{vmatrix}\).
Q.79
Find the value of: \(\begin{vmatrix} x-a & b \\ b & x-a \end{vmatrix}\).
Q.80
Find the value of \(x\) if \(\begin{vmatrix} x-3 & 4 \\ 4 & x-3 \end{vmatrix} = 0\).
Q.81
Evaluate: \(\begin{vmatrix} \sin^2 x & \sin x \cos x \\ \sin x \cos x & \cos^2 x \end{vmatrix}\).
Q.82
Evaluate: \(\begin{vmatrix} 1 & \cos x \\ \cos x & 1 \end{vmatrix}\).
Q.83
Find the value of: \(\begin{vmatrix} x-1 & x-1 & x-1 \\ x-1 & x-1 & x-1 \\ 1 & 1 & 1 \end{vmatrix}\).
Q.84
Evaluate: \(\begin{vmatrix} x & 0 & \sin x \\ 0 & x & \cos x \\ \sin x & \cos x & x \end{vmatrix}\).
Q.85
Find the value of \(x\) if \(\begin{vmatrix} x & \sin x \\ \sin x & x \end{vmatrix} = 0\).
Q.86
Evaluate: \(\begin{vmatrix} x-a & 1 & 1 \\ 1 & x-a & 1 \\ 1 & 1 & x-a \end{vmatrix}\).
Q.87
Find the value of \(x\) if \(\begin{vmatrix} x & 2 \\ 3 & x \end{vmatrix} = 10\).
Q.88
If \(\begin{vmatrix} a & 1 \\ 2 & b \end{vmatrix} = 7\) and \(a+b=9\), find \(a\) and \(b\).
Q.89
Find the value of \(a\) if \(\begin{vmatrix} a & 3 \\ 2 & a \end{vmatrix} = 1\).
Q.90
If \(\begin{vmatrix} a & b \\ b & a \end{vmatrix} = 0\) and \(a \neq 0\), find \(\frac{b}{a}\).
Q.91
Find the value of \(k\) if \(\begin{vmatrix} 1 & k \\ k & 4 \end{vmatrix} = 3\).
Q.92
Evaluate: \(\begin{vmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{vmatrix}\).
Q.93
Evaluate: \(\begin{vmatrix} a+b & b+c & c+a \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}\).
Q.94
Find the value of: \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\).
Q.95
Evaluate: \(\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix}\).
Q.96
Find the value of \(x\) if \(\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix} = 0\).
Q.97
Evaluate: \(\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}\).
Q.98
Evaluate: \(\begin{vmatrix} a & a^2 & 1 \\ b & b^2 & 1 \\ c & c^2 & 1 \end{vmatrix}\).
Q.99
If \(\begin{vmatrix} x-1 & 2 \\ 3 & x-1 \end{vmatrix} = 4\), find \(x\).
Q.100
Find the value of \(a\) if \(\begin{vmatrix} a & 4 \\ 1 & a \end{vmatrix} = 5\).
Q.101
Evaluate: \(\begin{vmatrix} x & y & z \\ z & x & y \\ y & z & x \end{vmatrix}\).
Q.102
If \(\begin{vmatrix} x & y & 1 \\ y & x & 1 \\ 1 & 1 & 1 \end{vmatrix} = 0\), find the relation between \(x\) and \(y\).
Q.103
Evaluate: \(\begin{vmatrix} 1 & a & b \\ a & 1 & b \\ a & a & 1 \end{vmatrix}\).
Q.104
Find the value of \(k\) if \(\begin{vmatrix} 1 & 1 & 1 \\ 1 & k & k^2 \\ 1 & k^2 & k \end{vmatrix} = 0\).
Q.105
Evaluate: \(\begin{vmatrix} x & x+1 & x+2 \\ 1 & 1 & 1 \\ 1 & 2 & 3 \end{vmatrix}\).
Q.106
Find the value of \(x\) if \(\begin{vmatrix} x & x & 1 \\ x & x & 1 \\ 1 & 1 & x \end{vmatrix} = 0\).
Q.107
Evaluate: \(\begin{vmatrix} 1+i & 1-i \\ 1-i & 1+i \end{vmatrix}\).
Q.108
Find the value of: \(\begin{vmatrix} i & 1 \\ -1 & i \end{vmatrix}\).
Q.109
Evaluate: \(\begin{vmatrix} 1 & i \\ -i & 1 \end{vmatrix}\).
Q.110
Find the value of \(x\) if \(\begin{vmatrix} x+i & 1 \\ 1 & x-i \end{vmatrix} = 0\).
Q.111
Evaluate: \(\begin{vmatrix} i & 1 & 0 \\ 1 & -i & 0 \\ 0 & 0 & 1 \end{vmatrix}\).
Q.112
Evaluate: \(\begin{vmatrix} 1 & i & i \\ -i & 1 & i \\ -i & -i & 1 \end{vmatrix}\).
Q.113
Find the value of: \(\begin{vmatrix} z & \bar z \\ \bar z & z \end{vmatrix}\), where \(z=a+ib\).
Q.114
Evaluate: \(\begin{vmatrix} 1+i & 2i \\ -2i & 1-i \end{vmatrix}\).
Q.115
Find the value of \(a\) if \(\begin{vmatrix} a & i \\ -i & a \end{vmatrix} = 5\).
Q.116
Evaluate: \(\begin{vmatrix} i & 1 & 1 \\ 1 & i & 1 \\ 1 & 1 & i \end{vmatrix}\).
Q.117
Find the value of \(x\) if \(\begin{vmatrix} x+i & 1 \\ 1 & x+i \end{vmatrix} = 0\).
Q.118
Evaluate: \(\begin{vmatrix} 1 & i & 0 \\ -i & 1 & 0 \\ 0 & 0 & i \end{vmatrix}\).
Q.119
Evaluate: \(\begin{vmatrix} 1+i & 1 & 1 \\ 1 & 1+i & 1 \\ 1 & 1 & 1+i \end{vmatrix}\).
Q.120
Find the value of: \(\begin{vmatrix} i & i & i \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{vmatrix}\).
Q.121
Evaluate: \(\begin{vmatrix} 1 & i & -i \\ -i & 1 & i \\ i & -i & 1 \end{vmatrix}\).
Q.122
Find the value of \(a\) if \(\begin{vmatrix} a+i & 1 \\ 1 & a-i \end{vmatrix} = 1\).
Q.123
Evaluate: \(\begin{vmatrix} z & i \\ -i & \bar z \end{vmatrix}\), where \(z=a+ib\).
Q.124
Find the value of: \(\begin{vmatrix} 1+i & 0 & 0 \\ 0 & 1-i & 0 \\ 0 & 0 & i \end{vmatrix}\).
Q.125
Evaluate: \(\begin{vmatrix} i & 1 & 1 \\ 1 & i & -1 \\ 1 & -1 & i \end{vmatrix}\).
Q.126
Find the value of \(x\) if \(\begin{vmatrix} x+i & i \\ -i & x-i \end{vmatrix} = 0\).
Q.127
Find the area of the triangle with vertices \(A(1,2)\), \(B(3,6)\), and \(C(5,2)\) using determinants.
Q.128
Find the area of the triangle formed by the points \((0,0)\), \((4,0)\), and \((0,6)\).
Q.129
Using determinants, check whether the points \((1,2)\), \((2,4)\), and \((3,6)\) are collinear.
Q.130
Find the value of \(k\) such that the area of the triangle formed by \((1,1)\), \((2,3)\), and \((3,k)\) is zero.
Q.131
Find the area of the triangle with vertices \((-1,2)\), \((3,4)\), and \((2,-2)\).
Q.132
Show that the points \((2,1)\), \((4,3)\), and \((6,5)\) lie on a straight line using determinants.
Q.133
Find the value of \(k\) if the area of the triangle formed by \((0,0)\), \((k,4)\), and \((2,6)\) is 10 square units.
Q.134
Find the area of the triangle formed by the intercepts of the line \(2x+3y=6\) with the coordinate axes.
Q.135
Using determinants, check whether the points \((1,-1)\), \((3,1)\), and \((5,3)\) are collinear.
Q.136
Find the area of the triangle formed by the points \((2,3)\), \((4,7)\), and \((6,3)\).
Q.137
Find the value of \(k\) if the points \((1,2)\), \((3,k)\), and \((5,8)\) are collinear.
Q.138
Find the area of the triangle formed by the points \((0,4)\), \((3,0)\), and \((6,4)\).
Q.139
Using determinants, check whether the points \((2,5)\), \((4,9)\), and \((6,13)\) lie on the same straight line.
Q.140
Find the area of the triangle with vertices \((1,1)\), \((2,4)\), and \((5,1)\).
Q.141
Find the value of \(k\) if the area of the triangle formed by \((1,k)\), \((2,3)\), and \((4,7)\) is zero.
Q.142
Find the area of the triangle formed by the points \((-2,0)\), \((2,0)\), and \((0,4)\).
Q.143
Using determinants, check whether the points \((0,0)\), \((1,2)\), and \((2,5)\) are collinear.
Q.144
Find the area of the triangle formed by the points \((1,3)\), \((3,1)\), and \((5,3)\).
Q.145
Find the value of \(k\) if the points \((2,k)\), \((4,6)\), and \((6,10)\) are collinear.
Q.146
Find the area of the triangle formed by the points \((0,2)\), \((3,6)\), and \((6,2)\).
Q.147
Using determinants, show that the points \((1,4)\), \((2,6)\), and \((3,8)\) lie on a straight line.
Q.148
Find the area of the triangle formed by the points \((2,1)\), \((4,5)\), and \((6,1)\).
Q.149
Find the value of \(k\) if the area of the triangle formed by \((k,1)\), \((2,3)\), and \((4,5)\) is zero.
Q.150
Find the area of the triangle formed by the points \((1,0)\), \((4,4)\), and \((7,0)\) using determinants.
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