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

If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)ⁿ on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?

Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?

If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].

If 4a + 2b + c = 0, then the equation 3ax² + 2bx + c = 0 has at least one real root lying in the interval

For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is 

If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]

Rolle's theorem is applicable in case of ϕ (x) = a^sin x, a > a in

The value of c in Rolle's theorem when
f (x) = 2x³ − 5x² − 4x + 3, x ∈ [1/3, 3] is

When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is ______.

The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is

The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is

The value of c in Rolle's theorem for the function f (x) = x³ − 3x in the interval [0,\[\sqrt{3}\]] is 

If f (x) = e^x sin x in [0, π], then c in Rolle's theorem is

Find the points on the curve x² + y² − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f⁻¹(x) = 4
(ii) f⁻¹(7)

Solve for x : \[\tan^{- 1} \left( \frac{x - 2}{x - 1} \right) + \tan^{- 1} \left( \frac{x + 2}{x + 1} \right) = \frac{\pi}{4}\] .

If \[A = \begin{bmatrix}5 & 6 & - 3 \\ - 4 & 3 & 2 \\ - 4 & - 7 & 3\end{bmatrix}\] , then write the cofactor of the element a₂₁ of its 2^nd row.

Three machines E₁, E₂ and E₃ in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E₁ and E₂are defective and that 5% of those produced by machine E₃ are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.

Prove that

\[2 \tan^{- 1} \left( \frac{1}{5} \right) + \sec^{- 1} \left( \frac{5\sqrt{2}}{7} \right) + 2 \tan^{- 1} \left( \frac{1}{8} \right) = \frac{\pi}{4}\] .

If \[\begin{pmatrix}a + 4 & 3b \\ 8 & - 6\end{pmatrix} = \begin{pmatrix}2a + 2 & b + 2 \\ 8 & a - 8b\end{pmatrix},\] ,write the value of a − 2b.