Mathematics 65/1/1 2018-2019 Commerce (English Medium) Class 12 Question Paper Solution

If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.

If f(x) = x + 1, find `d/dx (fof) (x)`

Find the order and the degree of the differential equation `x^2 (d^2y)/(dx^2) = { 1 + (dy/dx)^2}^4`

If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.

Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.

Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?

Find a matrix A such that 2A − 3B + 5C = 0, where B =`[(-2, 2, 0), (3, 1, 4)] and  "C" = [(2, 0, -2),(7, 1, 6)]`.

Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`

Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`

Find: `int sin^-1 (2x) dx.`

Form the differential equation representing the family of curves y = e²x (a + bx), where 'a' and 'b' are arbitrary constants.

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.

if  `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?

The random variable X has probability distribution P(X) of the following form, where k is some number:

`P(X = x) {(k, if x = 0),(2k, if x = 1),(3k, if x = 2),(0, "otherwise"):}`

1. Determine the value of 'k'.
2. Find P(X < 2), P(X ≥ 2), P(X ≤ 2).

Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.

Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Solve: tan^-1 4 x + tan^-1 6x `= π/(4)`.

Using properties of determinants, prove that 

`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`

If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`

If x^y - y^x = a^b, find `(dy)/(dx)`.

If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`

Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.

Find: `int (3x +5)/(x^2+3x-18)dx.`

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.

Solve the differential equation: `(1 + x^2) dy/dx + 2xy - 4x^2 = 0,` subject to the initial condition y(0) = 0.

if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and  hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.

Find the value of λ, so that the lines `(1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5)` are at right angles. Also, find whether the lines are intersecting or not.

If `"A" = [(1,1,1),(1,0,2),(3,1,1)]`, find A^-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

Find the inverse of the following matrix using elementary operations.

`"A" = [(1,2,-2), (-1,3,0),(0,-2,1)]`

A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m³. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?

Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).

Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.

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

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.