Science (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Evaluate: `int(5x-2)/(1+2x+3x^2)dx`

Evaluate : ` int x^2/((x^2+4)(x^2+9))dx`

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

Show that the following two lines are coplanar:

`(x−a+d)/(α−δ)= (y−a)/α=(z−a−d)/(α+δ) and (x−b+c)/(β−γ)=(y−b)/β=(z−b−c)/(β+γ)`

Find:

`int(x^3-1)/(x^3+x)dx`

If the function f(x)=2x³−9mx²+12m²x+1, where m>0 attains its maximum and minimum at p and q respectively such that p²=q, then find the value of m.

Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`

Prove that `y=(4sintheta)/(2+costheta)-theta `

Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b

Evaluate:

`int((x+3)e^x)/((x+5)^3)dx`

Show that lines: 

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar 

Also, find the equation of the plane containing these lines.

Let f : N→N be a function defined as f(x)=`9x^2`+6x−5. Show that f : N→S, where S is the range of f, is invertible. Find the inverse of f and hence find `f^-1`(43) and` f^−1`(163).

If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

If the function f : R → R be given by f[x] = x² + 2 and g : R ​→ R be given by  `g(x)=x/(x−1)` , x≠1, find fog and gof and hence find fog (2) and gof (−3).

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.

Find the distance between the planes 2x - y +  2z = 5 and 5x - 2.5y + 5z = 20

If A is a skew symmetric matric of order 3, then prove that det A  = 0

If `A = [(-1,2,3),(5,7,9),(-2,1,1)]  "and"  B = [(-4,1,-5),(1,2,0),(1,3,1)]` then verify that (A+ B)' = A' + B'