Science (English Medium) कक्षा १२ - CBSE Important Questions

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

The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?

If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.

Using properties of determinants, prove that 

`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`

Show that all the diagonal elements of a skew symmetric matrix are zero.

Using properties of determinants, prove that `|(x,x+y,x+2y),(x+2y, x,x+y),(x+y, x+2y, x)| = 9y^2(x + y)`

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).

If for a square matrix A, A² – A + I = 0, then A^–1 equals ______.

Read the following passage:

Based on the above information, answer the following questions:

1. Convert the given above situation into a matrix equation of the form AX = B. (1)
2. Find | A |. (1)
3. Find A^–1. (2)
   OR
   Determine P = A² – 5A. (2)

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  `dy/dx `at t = `pi/4`

If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x²≤1, then find dy/dx.

Differentiate the function with respect to x.

`(sin x)^x + sin^(-1) sqrtx`

if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`

If y = sin (sin x), prove that `(d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0`

If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find `dy/dx` when `theta = pi/3`

If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.

If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.

Find the intervals in which the function f(x) = 3x⁴ − 4x³ − 12x² + 5 is

(a) strictly increasing

(b) strictly decreasing