Arts (English Medium) Class 12 - CBSE Important Questions

Write the domain and range (principle value branch) of the following functions:

f(x) = tan^–1 x.

Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)³ (x – 3)², then

Assertion (A): f(x) has a minimum at x = 1.

Reason (R): When `d/dx (f(x)) < 0, ∀  x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀  x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.

ASSERTION (A): The relation f : {1, 2, 3, 4} `rightarrow` {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.

REASON (R): The function f : {1, 2, 3} `rightarrow` {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.

Find the domain of sin^–1 (x² – 4).

Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].

Write the principal value of `tan^(-1)+cos^(-1)(-1/2)`

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

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`

Prove that: `tan^(-1)(1/2)+tan^(-1)(1/5)+tan^(-1)(1/8)=pi/4`

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`

Solve for x : tan^-1 (x - 1) + tan^-1x + tan^-1 (x + 1) = tan^-1 3x

Prove that `tan^(-1)((6x-8x^3)/(1-12x^2))-tan^(-1)((4x)/(1-4x^2))=tan^(-1)2x;|2x|<1/sqrt3`

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`

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(sin^(-1)  1/5 + cos^(-1) x)  = 1` then find the value of x

Solve the following equation for x:  `cos (tan^(-1) x) = sin (cot^(-1)  3/4)`

For the principal value, evaluate of the following:

`cos^-1  1/2 + 2 sin^-1 (1/2)`