Define a binary operation * on the set 0, 1, 2, 3, 4, 5 as: a = a + b,&if a + b<6 + b-6,&if a + b 6 Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.

Let X = 0, 1, 2, 3, 4, 5 . The operation * on X is defined as: a = a + b,&if a + b<6 + b-6,&if a + b 6 An element e is the identity element for the operation *, if a * e = a = e * a a . For a , we observed that: a * 0 = a + 0 = a [a +0<6] 0 * a = a = 0 * a [a 0+a<6] a * 0 = a = 0 * a a Thus, 0 is the identity element for the given operation *. An element a is invertible if there exists be X such that a * b = 0 = b * a. i.e., a + b=0=b + a,&if a + b<6 + b-6=0=b + a-6,&if a + b 6 i.e., a = -b or b = 6 - a But, X = 0, 1, 2, 3, 4, 5 and a, b . Then, a -b. Therefore, b = 6 - a is the inverse of a . Hence, the inverse of an element a ,a 0 is 6 - a i.e., a^ -1 = 6 - a.

Define a binary operation * on the set 0, 1, 2, 3, 4, 5 as: a = a…

If $A=\left[\begin{array}{cc}0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0\end{array}\right]$ and I is the identity matrix of order 2 , show that
$I+A=(I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$

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Find the equations of the tangent and the normal to the following curves at the indicated points.
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A library has to accommodate two different types of books on a shelf. The books are 6 cm and 4 cm thick and weigh 1kg and $1\frac{1}{2}$ kg each respectively. The shelf is 96 cm long and at most can support a weight of 21kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an LPP and solve it graphically.

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Find the integrals of the functions in Exercises:
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Prove that $2\tan^{-1}\bigg(\sqrt{\frac{\text{a}-\text{b}}{\text{a}+\text{b}}}\tan\frac{\theta}{2}\bigg)=\cos^{-1}\Big(\frac{\text{a}\cos\theta+b}{\text{a}+\text{b}\cos\theta}\Big)$

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Maximise Z = x + y subject to $\text{x}+4\text{y}\leq8,2\text{x}+3\text{y}\leq12,3\text{x}+\text{y}\leq9,\text{x}\geq0,\text{y}\geq0.$

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The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

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If $\text{A}=\begin{bmatrix}3&-5\\-4&2\end{bmatrix},$ find $A^2 - 5A - 14$.

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