List of top Mathematics Questions asked in CBSE CLASS XII

Let two rods placed on the ground be represented by vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $-2\hat{i} + \hat{j} - 2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with the $x$-axis, $\frac{\pi}{3}$ with the $y$-axis and an acute angle $\theta$ with the $z$-axis. Find $\theta$ and the components of $\vec{a}$.

Find the interval(s) in which the function $f(x) = \frac{x}{\log x}$, where $x \in (0, 1) \cup (1, \infty)$, is increasing.

Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : A function f : N $\to$ N given by f(x) = $x^3$ + 2, $\forall$ x $\in$ N is one-one but not onto.
Reason (R) : Since $\forall$ y $\in$ N (Codomain), there does not exist x = $(y - 2)^{1/3}$ in N (Domain) such that f(x) = $x^3$ + 2 = y.

If $(2\hat{i} + 3\hat{j} + \hat{k})$ and $(2\hat{i} + \hat{j} - \hat{k})$ represent the sides $\vec{AB}$ and $\vec{AC}$ respectively of $\Delta ABC$, find the vector representing the median through A.

Evaluate : $\tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) + \cot^{-1}\left(\frac{1}{\sqrt{3}}\right) + \tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right) + \tan^{-1}\left(\tan\left(\frac{2\pi}{3}\right)\right)$}

The probability that it will rain tomorrow in cities A, B and C is \(60\%\), \(70\%\) and \(80\%\) respectively. Assuming these events are independent, the probability that it will rain tomorrow in at least one of the cities is:

The region represented by the system of linear inequalities \( 3x + y \ge 3 \), \( 2x - y \ge -5 \), and \( x, y \ge 0 \) is:

Which of the following equations is NOT a Linear Differential Equation?

The greatest integer function, \(f(x) = [x]\), evaluated across the open domain boundary \(0 < x < 3\), is not differentiable at how many points?

Three points \( A(0, 1, 1) \), \( B(2, 0, -1) \) and \( C(1, 0, 3) \) form \(\Delta ABC\). The area of \(\Delta ABC\) is:

The sum of the order and the degree of the differential equation \(\left[\frac{d^2y}{dx^2}\right]^2 + \left[\frac{dy}{dx}\right]^4 + 3\left(\frac{dy}{dx}\right) + 1 - y^2 = 0\) is:

The area of the region bounded by the curve \(y = x\) and the x-axis, between the boundaries \(x = 0\) and \(x = 2\) is:

In a linear programming problem, the feasible region representing the maximization of the objective function \( Z = px + qy \), where \( p, q > 0 \), is a shaded polygonal region. If all points on the line segment joining the corner points \( A(0, 3) \) and \( B(3, 2) \) give the maximum value of \( Z \), then which of the following is true?

The differential of \(e^{e^x}\) with respect to \(x\) is:

If the distance travelled by a particle in \(t\) seconds is given by \(S = 72t + 3t^2 - t^3\), then the time taken by the particle to come to rest is:

The indefinite integral \(\int \frac{dx}{\sqrt{25 - 16x^2}}\) is equal to:

If \(\int_{0}^{1} \frac{dx}{e^x + e^{-x}} = \tan^{-1}e + k\), then the constant value of \(k\) is:

The graph of an inverse trigonometric function passes through the point $(0, \frac{\pi}{2})$ with asymptotes at $y = 0$ and $y = \pi$. This graph represents the function:

If the points \((2, 3)\), \((0, 4)\) and \((p, 2)\) are collinear, then the value of \(p\) is:

If \[ 3P(A)=P(B)=\frac{3}{5} \] and \[ P(A\mid B)=\frac{2}{3}, \] then \[ P(A\cup B) \] is:

There are three types of vaccines $A_1$, $A_2$, $A_3$, available in the market to protect the population of the country from the spread of a certain infection. According to a survey conducted, it was found that $25\%$ of the population was given Vaccine $A_1$, $35\%$ of the population was given Vaccine $A_2$ and $40\%$ of the population was given Vaccine $A_3$. The survey also stated that the probabilities that Vaccines $A_1$, $A_2$ and $A_3$ would protect against the infection were $60\%$, $55\%$ and $50\%$ respectively. Based on the above information, answer the following questions:

A company produces cylindrical tumblers, open from the top. Since they want uniformity in the product, they fix the surface area of the tumblers produced. If for a tumbler, $V$ is its volume, $h$ is the height and $r$ is the radius of the circular base, then:

A school wants the students of class XII to do a project on ‘Sustainability’ keeping the world environment in mind. They select the student participants on the basis of an essay writing competition. 7 students out of 80 are selected for the project and are categorized into two sets such that:
Girl students belong to Set A = $\{G_1, G_2, G_3, G_4\}$,
Boy students belong to Set B = $\{B_1, B_2, B_3\}$.
Based on the above information, answer the following questions:

Find the value of $p$ if the shortest distance between the lines \[ \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \quad \text{and} \quad \vec{r} = (p\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k}) \] is exactly $\frac{3}{\sqrt{2}}$ units.