Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

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

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

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`

Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.

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

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`

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector

`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.

Write the direction ratios of the following line :

`x = −3, (y−4)/3 =( 2 −z)/1`

Prove that :

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

Solve the following for x :

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

The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10. According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively. A man receives Rs 225 a day and a woman receives Rs 200 a day. How many men and women helpers should be hired to keep the pay-roll at a minimum ? Formulate an LPP and solve it graphically.

Find :`int(x^2+x+1)/((x^2+1)(x+2))dx`

Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1

Solve for x:

`2tan^(-1)(cosx)=tan^(-1)(2"cosec" x)`

Solve the following for x:

`sin^(-1)(1-x)-2sin^-1 x=pi/2`

Show that:

`2 sin^-1 (3/5)-tan^-1 (17/31)=pi/4`

If sin [cot⁻¹ (x+1)] = cos(tan⁻¹x), then find x.

If (tan⁻¹x)² + (cot⁻¹x)² = 5π²/8, then find x.

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.