List of top Questions asked in CBSE CLASS XII

A ladder 13 m long is leaning against the wall. The bottom of the ladder is pulled along the ground away from the wall at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 12 m away from the wall?
If \( \hat{a}, \hat{b} \) and \( \hat{c} \) are unit vectors such that \[ \hat{a} \cdot \hat{b} = \hat{a} \cdot \hat{c} = 0 \] and the angle between \( \hat{b} \) and \( \hat{c} \) is \( \frac{\pi}{6} \), then prove that: \[ \hat{a} = \pm 2 (\hat{b} \times \hat{c}) \]

Determine the vector equation of the line that passes through the point \( (1, 2, -3) \) and is perpendicular to both of the following lines:

\[ \frac{x - 8}{3} = \frac{y + 16}{7} = \frac{z - 10}{-16} \quad \text{and} \quad \frac{x - 15}{3} = \frac{y - 29}{-8} = \frac{z - 5}{-5} \]

Find the value of λ, if the points A(−1,−1,2), B(2,8,λ), C(3,11,6) are collinear.

Prove that:
\( \tan^{-1}(\sqrt{x}) = \frac{1}{2} \cos^{-1}\left( \frac{1 - x}{1 + x} \right), \quad x \in [0, 1] \)

Let \( \vec{a} \) and \( \vec{b} \) be two co-initial vectors forming adjacent sides of a parallelogram such that:
\[ |\vec{a}| = 10, \quad |\vec{b}| = 2, \quad \vec{a} \cdot \vec{b} = 12 \] Find the area of the parallelogram.

Find the principal value of:
\( \cos^{-1}\left(-\frac{1}{2}\right) + 2\sin^{-1}(1) \)

Find the particular solution of the differential equation \[ \frac{dy}{dx} - y \cot x = \sin 2x, \quad \text{given that } y = 2 \text{ when } x = \frac{\pi}{2}. \]

For the curve \( \sqrt{x} + \sqrt{y} = 1 \), find the value of \( \frac{dy}{dx} \) at the point \( \left(\frac{1}{9}, \frac{1}{9}\right) \).

Assertion (A): \( f(x) = [x] \), \( x \in \mathbb{R} \), the greatest integer function is not differentiable at \( x = 2 \). Reason (R): The greatest integer function is not continuous at any integral value.
Consider the function \( f : \mathbb{R} \rightarrow \mathbb{R} \), defined as \( f(x) = x^3 \). Assertion (A): \( f(x) \) is a one-one function. Reason (R): \( f(x) \) is a one-one function, if co-domain = range.
Find the maximum slope of the curve \( y = x^3 + 3x^2 + 9x - 30 \).
Find the general solution of the differential equation \[ x^2 \frac{dy}{dx} = x^2 + xy + y^2 \] OR
Two vectors \( \vec{a} \) and \( \vec{b} \) are such that \( |\vec{a} \times \vec{b}| = \vec{a} \cdot \vec{b} \). The angle between the two vectors is:
A coin is tossed three times. The probability of getting at least two heads is:
If \( m' \) and \( n' \) are the degree and order respectively of the differential equation \( 1 + \left( \frac{dy}{dx} \right)^3 = \frac{d^2 y{dx^2} \), then the value of \( (m + n) \) is: }
If \( |\vec{a}|^2 = 1 \), \( |\vec{b}| = 2 \) and \( \vec{a} \cdot \vec{b} = 2 \), then the value of \( |\vec{a} + \vec{b}| \) is:
If the rate of change of volume of a sphere is twice the rate of change of its radius, then the surface area of the sphere is:
The general solution of the differential equation \( \frac{dy}{dx} = 2x \cdot e^{x^2 + y} \) is:
In an LPP, corner points of the feasible region determined by the system of linear constraints are \((1,1), (3,0), (0,3)\). If \(Z = ax + by\), where \(a>0, b>0\) is to be minimized, the condition on \(a\) and \(b\) so that the minimum of \(Z\) occurs at \((3, 0)\) and \((1, 1)\) will be:
If \(\dfrac{d}{dx}f(x) = 3x^2 - \dfrac{3}{x^4}\), and \(f(1) = 0\), then \(f(x)\) is:
The area bounded by the curve \(x = y^2\), the y-axis, and the lines \(y = 3\) and \(y = 4\) is:
The maximum value of \(Z = 3x + 4y\) subject to the constraints \(x + y \leq 1\), \(x \geq 0\), \(y \geq 0\) is:
If \(A\) is an identity matrix of order \(n\), then \(A (\text{Adj } A)\) is a/an:
\[ \int \frac{\tan^2 \sqrt{x}}{\sqrt{x}} \, dx \text{ is equal to:} \]