List of top Mathematics Questions asked in CBSE CLASS XII

The traffic police has installed Over Speed Violation Detection (OSVD) system at various locations in a city. These cameras can capture a speeding vehicle from a distance of 300 m and even function in the dark. A camera is installed on a pole at the height of 5 m. It detects a car travelling away from the pole at the speed of 20 m/s. At any point, \(x\) m away from the base of the pole, the angle of elevation of the speed camera from the car C is \(\theta\). 
On the basis of the above information, answer the following questions: 
(i)Express \(\theta\) in terms of the height of the camera installed on the pole and x.
(ii) Find \(\frac{d\theta}{dx}\).
(iii) (a) Find the rate of change of angle of elevation with respect to time at an instant when the car is 50 m away from the pole.
(iii) (b) If the rate of change of angle of elevation with respect to time of another car at a distance of 50 m from the base of the pole is \(\frac{3}{101} \, \text{rad/s}\), then find the speed of the car.

According to recent research, air turbulence has increased in various regions around the world due to climate change. Turbulence makes flights bumpy and often delays the flights. Assume that an airplane observes severe turbulence, moderate turbulence or light turbulence with equal probabilities. Further, the chance of an airplane reaching late to the destination are \(55\%\), \(37\%\) and \(17\%\) due to severe, moderate and light turbulence respectively. 
On the basis of the above information, answer the following questions: 
(i) Find the probability that an airplane reached its destination 
(ii)If the airplane reached its destination late, find the probability that it was due to moderate turbulence.

If a function \(f : X \to Y\) defined as \(f(x) = y\) is one-one and onto, then we can define a unique function \(f(x) = y\) such that \(f(x) = y\), where \(f(x) = y\) and \(f(x) = y\), \(f(x) = y\). Function \(g\) is called the inverse of function \(f\). 

The domain of sine function is \(\mathbb{R}\) and function sine : \(\mathbb{R} \to \mathbb{R}\) is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set \(A\) to \([-1, 1]\) such that inverse of sine function exists, i.e., \(\sin^{-1} x\) is defined from \([-1, 1]\) to \(A\). 

On the basis of the above information, answer the following questions: 
(i) If \(A\) is the interval other than principal value branch, give an example of one such interval.
(ii) If \(\sin^{-1}(x)\) is defined from \([-1, 1]\) to its principal value branch, find the value of \(\sin^{-1}\left(-\frac{1}{2}\right) - \sin^{-1}(1)\).  
(iii) Draw the graph of \(\sin^{-1} x\) from \([-1, 1]\) to its principal value branch. 
(iv) Find the domain and range of \(f(x) = 2 \sin^{-1}(1 - x)\).

Let A=\(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\),show that(aI+bA)ⁿ=aⁿI+na^n-1bA,where I is the identity matrix of order 2 and n∈N

Evaluate \(\begin{vmatrix} cos\alpha cos\beta &cos\alpha sin\beta  &-sin\alpha \\   -sin\beta&cos\beta  &0 \\   sin\alpha cos\beta&sin\alpha\sin\beta  &cos\alpha  \end{vmatrix}\)