Commerce (English Medium) Class 12 - CBSE Question Bank Solutions

The sum of the surface areas of a rectangular parallelopiped with sides x, 2x and `x/3` and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.

If x is real, the minimum value of x² – 8x + 17 is ______.

The smallest value of the polynomial x³ – 18x² + 96x in [0, 9] is ______.

The function f(x) = 2x³ – 3x² – 12x + 4, has ______.

The maximum value of sin x . cos x is ______.

Maximum slope of the curve y = –x³ + 3x² + 9x – 27 is ______.

The maximum value of `(1/x)^x` is ______.

The curves y = 4x² + 2x – 8 and y = x³ – x + 13 touch each other at the point ______.

Evaluate the following:

`int x/(sqrt(x) + 1) "d"x`  (Hint: Put  `sqrt(x)` = z)

Evaluate the following:

`int sqrt(("a" + x)/("a" - x)) "d"x`

Evaluate the following:

`int x^(1/2)/(1 + x^(3/4)) "d"x`   (Hint: Put `sqrt(x)` = z⁴)

If A = `[(0,0,0),(0,0,0),(0,1,0)]` then A is ____________.

A square matrix A = [a_ij]_nxn is called a diagonal matrix if a_ij = 0 for ____________.

If `[("a","b"),("c", "-a")]`is a square root of the 2 x 2 identity matrix, then a, b, c satisfy the relation ____________.

If k be an integer, then `lim_("x" -> "k") ("x" - ["x"])` ____________.

If A is a square matrix, then A – A’ is a ____________.

For any square matrix A, AA^T is a ____________.

If A `= [("cos x", - "sin x"),("sin x", "cos x")]`, find AAT.

If A `= [(0,-1,2),(1,0,3),(-2,-3,0)],` then A + 2A^T equals

If the matrix A `= [(5,2,"x"),("y",2,-3),(4, "t",-7)]` is a symmetric matrix, then find the value of x, y and t respectively.