Area of the region bounded by the curve \(y=\cos (x)\) and x -axis between \(x=0\) and \(x=\pi\) is :

A fair coin is tossed 10 times. The probability of obtaining exactly six heads is:

The corner points of the feasible region determined by the system of linear constraints are \((0,0),(0,40),(20,40),(60,20),(60,0)\).
The objective function is \(z=4 x+3 y\).
Compare the quantity in Column - A and Column - B.

Column - A	Column - B
Maximum value of z	350

A manufacturing unit makes two models, 'classic' and 'supreme' of the scooter.
Each piece of the classic model requires 9 labour hours for assembling and 1 labour hour for finishing.
Each piece of supreme model requires 12 labour hours for assembling and 3 labour hour for finishing.
Maximum labour hours available for assembling and finishing are 180 and 30 respectively.
Profit is Rs. 10000 on each classic piece and Rs. 15000 on each supreme piece.
Which option describes the LPP to maximize profit Z (where x and y are classic and supreme pieces)?

In the adjoining figure, the inequality representing AB is :

If A and B are two events such that \(P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\) and \(P(A \cap B)=\frac{1}{4}\), then which of the following statements are true?
(A) A and B are independent events
(B) \(P(A \mid B)=\frac{3}{4}\)
(C) \(P\left(A^{\prime} \mid B^{\prime}\right)=\frac{5}{8}\)
(D) \(P\left(A^{\prime} \mid B\right)=\frac{1}{4}\)
Choose the correct answer from the options given below:

The area (in sq. units) of the region bounded by the line \(y=x+2, x=0, x=1\) and \(y=0\) is:

Answer the question on the basis of passage given below:
In the periodic table, the d-block contains the elements of group \(3\) to \(12\). The d-orbitals are progressively filled in each of the four long periods. The elements of d block referred as transition metals have partly filled d orbitals and exhibit certain characteristic properties such as variety of oxidation states, formation of coloured ions, act as catalyst and show paramagnetic behaviour.
The two inner transition metals series \(4f\) and \(5f\) are known as Lanthanoids and Actinoids respectively. The lanthanoids resemble one another more closely as compared to ordinary transition elements in any series.
In context of the Lanthanoids, which of the following statements is/are NOT correct?
A. There is a gradual decrease in the radii of the members with increasing atomic number in the series
B. Availability of \(4f\) electrons results in the formation of compounds in \(+4\) oxidation state for all the members of the series
C. Because of similar properties, the separation of lanthanoids is not easy
D. \(La(OH)_3\) is least basic among hydroxides of lanthanoids
E. Ce can act as an oxidising agent
Choose the correct answer from the options given below:

A plane wave is incident on a concave mirror of radius R. The shape of the reflected wavefront will be

In a 10 km race, \(A, B\) and \(C\) each run at uniform speed and get the first, second and third positions respectively.
If \(A\) beats \(B\) by 1 km and \(B\) beats \(C\) by 1 km , then by how much distance does \(A\) beat \(C\) ?

Glucose gets oxidised to gluconic acid on reaction with bromine water. This indicates the presence of :

Out of students in a school, it is known that \(65 \%\) reside in hostel and \(35 \%\) are day scholars. Previous year results report that \(40 \%\) of the students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination.At the end of the year, one student is chosen at random from the class.
If the selected student attains grade A, find the probability that the student is a hosteller.

If \(\left[\begin{array}{lll}1 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=0\), then value of x is