By using properties of determinants, show that:
(i)\(\begin{vmatrix} x+4 & 2x & 2x\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)=(5x+4)(4-x)2
(II)\(\begin{vmatrix} y+k & y & y\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)=k2(3y+k)
(i)\(\begin{vmatrix} x+4 & 2x & 2x\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)=(5x+4)(4-x)2
(II)\(\begin{vmatrix} y+k & y & y\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)=k2(3y+k)
Solution and Explanation
(i)△=\(\begin{vmatrix} x+4 & 2x & 2x\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)
Applying R1 → R1 + R2 + R3, we have:
△=\(\begin{vmatrix} 5x+4 & 5x+4 & 5x+4\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)
=(5x+4)\(\begin{vmatrix} 1 & 1 & 1\\ 2x & x+4 & 2x \\2x &2x&x+4\end{vmatrix}\)
Applying C2 → C2 − C1, C3 → C3 − C1, we have:
△=(5x+4)I\(\begin{vmatrix} 1 & 0 & 0\\ 2x & -x+4 & 0 \\2x &0&-x+4\end{vmatrix}\)
=(5x+4)(4-x)(4-x)\(\begin{vmatrix} 1 & 0 & 0\\ 2x & 1 & 0 \\2x &0&1\end{vmatrix}\)
Expanding along C3, we have:
△=(5x+4)(4-x)2\(\begin{vmatrix} 1 & 0 \\ 2x & 1 \end{vmatrix}\)
=(5x+4)(4-x)2
(ii)△=\(\begin{vmatrix} y+k & y & y\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)
Applying R1 → R1 + R2 + R3, we have:
△=\(\begin{vmatrix} 3y+k & 3y+k & 3y+k\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)
=(3y+k)\(\begin{vmatrix}1 & 1 & 1\\ y & y+k & y \\y &y&y+k\end{vmatrix}\)
Applying C2 → C2 − C1 and C3 → C3 − C1, we have:
△=(3y+k)I\(\begin{vmatrix}1 & 0 & 0\\ y & k & 0 \\y &0&k\end{vmatrix}\)
=k2(3y+k)I\(\begin{vmatrix}1 & 0 & 0\\ y & 1 & 0 \\y &0&1\end{vmatrix}\)
Expanding along C3, we have:
△=k2(3y+k)\(\begin{vmatrix} 1 & 0 \\ y & 1 \end{vmatrix}\)=k2(3y+k)
Hence, the given result is proved.
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Determinants Questions
Evaluate the determinants in Exercises 1 and 2.
\(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)
Evaluate the determinants in Exercises 1 and 2.
(i) \(\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}\)(ii) \(\begin{vmatrix}x^2&-x+1&x-1\\& x+1&x+1\end{vmatrix}\)
Using properties of determinants,prove that:
\(\begin{vmatrix} x & x^2 & 1+px^3\\ y & y^2 & 1+py^3\\z&z^2&1+pz^3 \end{vmatrix}\)\(=(1+pxyz)(x-y)(y-z)(z-x)\)- Using properties of determinants,prove that:
\(\begin{vmatrix}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{vmatrix}=1\) Using properties of determinants,prove that:
\(\begin{vmatrix} 3a& -a+b & -a+c\\ -b+a & 3b & -b+c \\-c+a&-c+b&3c\end{vmatrix}\)\(=3(a+b+c)(ab+bc+ca)\)
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
Concepts Used:
Properties of Determinants
Properties of Determinants:
- Reflection Property: The value of the determinant remains unchanged if its rows and columns are interchanged.
- Switching Property: The interchanging of any two rows (or columns) of the Determinant changes its signs.
- All- Zero Property: The Determinants will be equivalent to zero if each term of rows and columns is zero.
- Proportionality (Repetition) Property: If all elements of a row (or column) are proportional or identical to the elements of some other row (or column), then the determinant is zero.
- Scalar Multiple Property: If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
- Sum Property: If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
- Property of Invariance: If to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + kCj
- Factor Property: If a Determinant Δ becomes 0 while considering the value of x = α, then (x - α) is considered as a factor of Δ
- Triangle Property: If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.
Read More: Properties of Determinants