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

Write the value of `tan(2tan^(-1)(1/5))`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

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

Concept: Applications of Determinants and Matrices

If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of *x*.

Concept: Applications of Determinants and Matrices

Find the value of the following: `tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))],|x| <1,y>0 and xy <1`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Using integration, find the area bounded by the curve *x*^{2} = 4y and the line *x* = 4y − 2.

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

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

Concept: Derivative - Exponential and Log

Solve the equation for x:sin^{−1}x+sin^{−1}(1−x)=cos^{−1}x

Concept: Inverse Trigonometric Functions (Simplification and Examples)

If `cos^-1( x/a) +cos^-1 (y/b)=alpha` , prove that `x^2/a^2-2(xy)/(ab) cos alpha +y^2/b^2=sin^2alpha`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Prove that the curves y^{2 }= 4x and x^{2} = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

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

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

Concept: Introduction of Operations on Matrices

Prove that :

`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Solve the following for *x* :

`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`

Concept: Inverse Trigonometric Functions (Simplification and Examples)

Using integration, find the area of the region bounded by the lines *y *= 2 + *x*, *y *= 2 – *x *and *x *= 2.

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

Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle π/4 with the plane *x* + *y* = 3. Also find the equation of the plane

Concept: Three - Dimensional Geometry Examples and Solutions

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

Find the coordinates of the foot of perpendicular drawn from the point A (-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

Concept: Three - Dimensional Geometry Examples and Solutions

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