CBSE 12 Maths Question Paper-2022 Set-4

If the distance of the point (1, 1, 1) from the plane x-y+z+λ=0 is 5/√3, find the value(s) of λ.

Write the projection of the vector (b+c) on the vector a where a=2i-2j+k, b=i+2j-2k and c=2i-j+4k.

Find the general solution of the differential equation: dy/dx = (3e^(2x)+3e^(4x))/(e^(x)+e^(-x))

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spade cards.

Find: ∫(dx)/(x²-6x+13)

A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be 7. Find the probability that the number 5 has appeared on atleast one die.

The probability that A hits the target is 2/5 and the probability that B hits it, is 1/3. If both try to hit the target independently, find the probability that the target is hit.

Find the shortest distance between the following lines: r=3i+5j+7k+λ(i-2j+k) and r=(-i-j-k)+μ(7i-6j+k).

The two adjacent sides of a parallelogram are represented by vectors 2i-4j+5k and i-2j-3k. Find the unit vector parallel to one of its diagonals. Also, find the area of the parallelogram.

If a=2i+2j+3k, b=-i+2j+k and c=3i+j are such vectors that vector (a+λb) is perpendicular to vector c, then find the value of λ.

A bag contains 3 red and 4 white balls. Three balls are drawn one by one, randomly and without replacement. If the first ball is red, find the probability that the remaining two balls are also red.

A coin is tossed twice. The following table shows the probability distribution of the number of tails:
X|0|1|2
P(X)|K|6K|9K
(a) Find the value of K. (b) Is this coin biased or unbiased? Justify your answer.

The coordinates of the foot of the perpendicular from the point (-2, -1, -3) to a plane are (1, -3, 3). Find the equation of the plane.

If |a|=4 and |b|=5, find the maximum value of |a x b|.

Evaluate: ∫₀¹ x²eˣdx

Three friends A, B and C had their photo taken. Find the probability that B stands in the middle, if A stands at the left corner.

Using integration, find the area of the triangular region whose vertices are (1, 0), (2, 2) and (3, 1).

Find the general solution of the differential equation (x²+y²)dy = xy dx.

Find: ∫(2x-3)/((x²-1)(2x+3)) dx

If a and b are unit vectors and θ is the angle between them, prove that tan(θ/2) = |a-b| / |a+b|.

Find the general solution of the differential equation: (dy/dx) = e^(x-y) + x²e^(-y).

Find the area of the region bounded by the curve y² = 4x and the line x = 3.

Evaluate: ∫(from 0 to 1) tan⁻¹(x) dx.

A and B are two events such that P(A)=0.5, P(B)=0.6 and P(A∪B)=0.8. Find P(A|B).

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to the planes x+2y+3z=5 and 3x+3y+z=0.

Find the shortest distance between the lines r = (2i - j - k) + λ(i + 2j + 3k) and r = (i + j + 2k) + μ(4i + 5j + 6k).

Find the area of the region bounded by the parabola y = x² and the line y = x.

A die is thrown 6 times. If 'getting an odd number' is a success, what is the probability of 5 successes?

Find the value of 'a' for which the vectors 2i - j + k and i + aj - 3k are coplanar with the vector 3i + 2j + k.

Evaluate: ∫(sin²x)/(1+cos x) dx.

Find the value of k for which the function f(x) = {kx+1, if x≤5; 3x-5, if x>5} is continuous at x=5.

Find the equation of the line passing through the point (2, 3, 2) and parallel to the line (x-2)/3 = (y+1)/2 = (z-1)/5.