Arts (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

if `A = [(0, -tan  alpha/2), (tan  alpha/2, 0)]` and I is the identity matrix of order 2, show that I + A = `(I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`

Let A = `[(0,1),(0,0)]`show that (aI+bA)ⁿ  = aⁿI + na^n-1 bA , where I is the identity matrix of order 2 and n ∈ N

if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`

if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where n is any positive integer

Find the matrix X so that  X`[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]`

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N

If A = `[(alpha, beta),(gamma, -alpha)]` is such that A2 = I then ______.

If A is a square matrix such that A² = A, then (I + A)³ – 7 A is equal to ______.

Write Minors and Cofactors of the elements of the following determinant:

`|(2,-4),(0,3)|`

Write Minors and Cofactors of the elements of the following determinant:

`|(a,c),(b,d)|`

Write Minors and Cofactors of the elements of the following determinant:

`|(1,0,0),(0,1,0),(0,0,1)|`

Write Minors and Cofactors of the elements of the following determinant:

`|(1,0,4),(3,5,-1),(0,1,2)|`

Using Cofactors of elements of second row, evaluate Δ = `|(5,3,8),(2,0,1),(1,2, 3)|`.

Using Cofactors of elements of third column, evaluate Δ = `|(1,x,yz),(1,y,zx),(1,z,xy)|`.

If Δ = `|(a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)|` and A_ij is Cofactors of a_ij, then the value of Δ is given by ______.

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units /kg of vitamin A and 5 units /kg of vitamin B while food Q contains 4 units /kg of vitamin A and 2 units /kg of vitamin B. Determine the minimum cost of the mixture?

One kind of cake requires 200g flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes?

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

What number of rackets and bats must be made if the factory is to work at full capacity?

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

A manufacturer produces nuts ad bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit, of Rs 17.50 per package on nuts and Rs. 7.00 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 12 hours a day?