List of top Questions asked in CBSE CLASS XII

Determine P(E|F):Two coins are tossed once.(i)E:tail appears on one coin,F:one coin shows head.(ii)E:no tail appears,F:no head appears.

Using properties of determinants,prove that:
\(\begin{vmatrix} 3a& -a+b & -a+c\\ -b+a & 3b & -b+c \\-c+a&-c+b&3c\end{vmatrix}\)\(=3(a+b+c)(ab+bc+ca)\)

Using properties of determinants,prove that:
\(\begin{vmatrix}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{vmatrix}=1\)

A toy company manufactures two types of dolls,A and B.Market tests and available resources have indicate that the combined production lebel should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A.Further the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600units.If the company makes profit of Rs12 and Rs16 per doll respectively on dolls A and B,how many of each should be produced weekly in order to maximize the profit?

Using properties of determinants,prove that:
\(\begin{vmatrix} x & x^2 & 1+px^3\\ y & y^2 & 1+py^3\\z&z^2&1+pz^3 \end{vmatrix}\)\(=(1+pxyz)(x-y)(y-z)(z-x)\)

If A and B are two events such that \(A⊂B\) and \(P(B)≠0\),then which of the following is correct?

A card from a pasck of 52 card is lost.From the remaining cards of the pack,two cards are drawn and are found to be both diamonds.Find the probability of the lost card being a diamond.

Find a particular solution of the differential equation\((x+1)\frac{dy}{dx}=2e^{-y}-1\),given that \(y=0\), when \(x=0\)

State which of the following are not the probability distributions of a random variable.Give reasons for your answer.
(i)

X	0	1	2
P(X)	0.4	0.4	0.2

(ii)

X	0	1	2	3	4
P(X)	0.1	0.5	0.2	-0.1	0.3

(iii)

Y	-1	0	1
P(Y)	0.6	0.1	0.2

(iv)

Z	3	2	1	0	-1
P(Z)	0.3	0.2	0.4	0.1	0.05

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Find a particular solution of the differential equation \(\frac{dy}{dx}+ycotx=4x\,cosec\, x(x≠0),\) given that \(y=0\) when \(x=\frac{π}{2}\)

Solve the differential equation \([\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}]\frac{dx}{dy}=1(x≠0)\)

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is

An oil company has two depots A and B with capacities of 7000 L and 4000 L respectively.The company is to supply oil to three petrol pumps D, E and F whose requirements are 4500 L, 3000 L and 3500 L respectively. The distance (in km) between the depots and the petrol pumps is given in the following table:

Distance (in km)
From/To	A	B
D	7	3
E	6	4
F	3	2

Assuming that the transportation cost of 10litres of oil is Re1 per km,how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?

Find a particular solution of the differential equation \((x-y)(dx+dy)=dx-dy\), given that \(y=-1\), when \(x=0\) (Hint:put \(x-y=t\))

Probability that A speaks truth is \(\frac{4}{5}\).A coin is tossed.A report that a head appears.The probability that actually there was head is:

Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively.They supply to 3 ration shops,D,E and F whose requirements are 60,50 and 40 quintals respectively.The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in Rs)
From/To	A	B
D	6	4
E	3	2
F	2.50	3

How should the supplies be transported in order that the transportation cost is minimum?What is the minimum cost?

In a meeting 70% of the members favour a certain proposal,30%being opposed.A member is selected at a random and we let X=0 if he is in favour.Find E(X) and Var (X).

Sketch the graph of y=|x+3|and evaluate \(\int_{-6}^{0} |x+3| \,dx\)

Find the area of the region bounded by the ellipse \(\frac{x^2}{16}+\frac{y^2}{9}=1\)

Find the mean number of heads in three tosses of fair coin.

Find the area of the region bounded by x²=4y,y=2,y=4 and the x-axis in the first quadrant.

A manufacturer has three machine operators A,B and C.The first operator A produces 1% defective items,whereas the other two B and C produces 5%and 7% defective items respectively.A is on the job for 50%of the time,B on the job for 30%of the time and C on the job for 20% of the time.A defective item is produced,what is the probability that it was produced by A?

If the matrix \(A\) is both symmetric and skew symmetric,then

Find the area of the region bounded by y²=9x, x=2, x=4 and the x-axis in the first quadrant.