If `|[2x,5],[8,x]|=|[6,-2],[7,3]|`, write the value of *x*.

Concept: Applications of Determinants and Matrices

Show that the lines `(x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5` intersect. Also find their point of intersection

Concept: Three - Dimensional Geometry Examples and Solutions

Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`

Concept: Three - Dimensional Geometry Examples and Solutions

Find the value of *a* if `[[a-b,2a+c],[2a-b,3c+d]]=[[-1,5],[0,13]]`

Concept: Applications of Determinants and Matrices

Differentiate x^{sinx}+(sinx)^{cosx} with respect to x.

Concept: Derivative - Exponential and Log

If `A=[[2,0,1],[2,1,3],[1,-1,0]]` , find A^{2} − 5 A + 16 I.

Concept: Introduction of Operations on Matrices

Find the the differential equation for all the straight lines, which are at a unit distance from the origin.

Concept: Methods of Solving First Order, First Degree Differential Equations > Linear Differential Equations

Write the element a_{12} of the matrix A = [a_{ij}]_{2 × 2}, whose elements aij are given by a_{ij} = e^{2ix }sin jx.

Concept: Introduction of Operations on Matrices

Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).

Concept: Area of the Region Bounded by a Curve and a Line

Write the element a_{23} of a 3 ✕ 3 matrix A = (*a*_{ij}) whose elements *a*_{ij} are given by `a_(ij)=∣(i−j)/2∣`

Concept: Introduction of Operations on Matrices

If `[[3x,7],[-2,4]]=[[8,7],[6,4]]`, find the value of x

Concept: Introduction of Operations on Matrices

Find the equation of the curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Concept: Methods of Solving First Order, First Degree Differential Equations > Linear Differential Equations

If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements?

Concept: Introduction of Operations on Matrices

If *A* = [*a _{ij}*] =`[[2,3,-5],[1,4,9],[0,7,-2]]`and

*B*= [

*b*] `[[2,-1],[-3,4],[1,2]]`

_{ij}then find (i) *a*_{22} + *b*_{21} (ii) *a*_{11} *b*_{11} + *a*_{22} *b*_{22}

Concept: Introduction of Operations on Matrices

Let A be a matrix of order 3 × 4. If R_{1} denotes the first row of A and C_{2} denotes its second column, then determine the orders of matrices R_{1} and C_{2}

Concept: Introduction of Operations on Matrices

Construct a 2 × 2 matrix whose elements `a_(ij)`

are given by: `(i+j)^2/2`

Concept: Introduction of Operations on Matrices

Construct a 2 × 2 matrix whose elements *a _{ij}* are given by:

`aij=(i-j)^2/2`

Concept: Introduction of Operations on Matrices

Construct a 2 × 2 matrix whose elements *a _{ij}* are given by:

`a_(ij)=(i-2_j)^2/2`

Concept: Introduction of Operations on Matrices

Construct a 2 × 2 matrix whose elements *a _{ij}* are given by:

`a_(ij)= (2i +j)^2/2`

Concept: Introduction of Operations on Matrices

Construct a 2 × 2 matrix whose elements *a _{ij}* are given by:

`a_(ij)=|2_i - 3_i|/2`

Concept: Introduction of Operations on Matrices