Commerce (English Medium) कक्षा १२ - CBSE Important Questions

Explain the arguments given in favour of strong Centre in the Constituent Assembly. 

Match the following:

Consider the given statements regarding Constituent Assembly and select the correct from the following options:

1. Motilal Nehru moved resolution of National flag in the Constituent Assembly.
2. G.B. Pant was the Legal Advisor.
3. Sardar Patel was the Constitutional Advisor.
4. K.M. Munshi was called as Frontier Gandhi.

"One of the topics most vigorously debated in the Constituent Assembly was the respective rights of the Central and State governments." Analyse the statement with supporting arguments.

Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].

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).

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.

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).

If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3`, show that fof (x) = x for all `x ≠ 2/3`. Also, find the inverse of f.

The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.

Read the following passage:

Based on the above information, answer the following questions:

1. How many relations are possible from B to G? (1)
2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
3. Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
   OR
   A function f : B `rightarrow` G be defined by f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}. Check if f is bijective. Justify your answer. (2)

Prove that `cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4) `

Prove that:

`tan^(-1)""1/5+tan^(-1)""1/7+tan^(-1)""1/3+tan^(-1)""1/8=pi/4`

Solve for x:

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

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

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

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

If tan^-1x+tan^-1y=π/4,xy<1, then write the value of x+y+xy.

Prove that

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`