Date & Time: 21st March 2019, 10:30 am

Duration: 2h30m

- This question paper contains 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.
- All questions in Section A are to be answered in one word, one sentence, or as per the exact requirement of the question.
- There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C, and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions.

If `3"A" - "B" = [(5,0),(1,1)] and "B" = [(4,3),(2,5)]`, then find the martix A.

Chapter: [2.02] Matrices

Write the order and the degree of the following differential equation: `"x"^3 ((d^2"y")/(d"x"^2))^2 + "x" ((d"y")/(d"x"))^4 = 0`

Chapter: [3.04] Differential Equations

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

Chapter: [3.01] Continuity and Differentiability

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

Chapter: [4.01] Three - Dimensional Geometry

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

Chapter: [4.01] Three - Dimensional Geometry

Find: ∫ sin x · log cos x dx

Chapter: [1.01] Inverse Trigonometric Functions

Evaluate: `int_-π^π (1 - "x"^2) sin "x" cos^2 "x" d"x"`.

Chapter: [3.05] Integrals

Evaluate: `int_-1^2 (|"x"|)/"x"d"x"`.

Chapter: [3.05] Integrals

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?

Chapter: [1.02] Relations and Functions

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)]`.

Chapter: [2.02] Matrices

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.

Chapter: [6.01] Probability

Form the differential equation representing the family of curves y = e^{2}x (a + bx), where 'a' and 'b' are arbitrary constants.

Chapter: [3.04] Differential Equations

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?

Chapter: [6.01] Probability

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"):}`

Determine the value of 'k'.

Chapter: [6.01] Probability

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

Chapter: [4.02] Vectors

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"]`

Chapter: [4.02] Vectors

Using properties of determinants, prove the following:

`|(a, b,c),(a-b, b-c, c-a),(b+c, c+a, a+b)| = a^3 + b^3 + c^3 - 3abc`.

Chapter: [2.01] Determinants

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

Chapter: [1.01] Inverse Trigonometric Functions

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

Chapter: [1.02] Relations and Functions

Prove that the function f : N → N, defined by f(x) = x^{2} + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Chapter: [3.02] Applications of Derivatives

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.

Chapter: [3.02] Applications of Derivatives

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

Chapter: [3.01] Continuity and Differentiability

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

Chapter: [3.01] Continuity and Differentiability

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

Chapter: [3.01] Continuity and Differentiability

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"`

Chapter: [3.05] Integrals

Find: `int_ (cos"x")/((1 + sin "x") (2+ sin"x")) "dx"`

Chapter: [3.05] Integrals

Solve the differential equation: `(d"y")/(d"x") - (2"x")/(1+"x"^2) "y" = "x"^2 + 2`

Chapter: [3.04] Differential Equations

Solve the differential equation: ` ("x" + 1) (d"y")/(d"x") = 2e^-"y" - 1; y(0) = 0.`

Chapter: [3.04] Differential Equations

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.

Chapter: [4.02] Vectors

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.

Chapter: [4.01] Three - Dimensional Geometry

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 m^{3}. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

Chapter: [3.02] Applications of Derivatives

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.

Chapter: [2.01] Determinants

Find the inverse of the following matrix using elementary operations.

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

Chapter: [2.02] Matrices

Prove that the curves y^{2 }= 4x and x^{2} = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

Chapter: [3.03] Applications of the Integrals

Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4).

Chapter: [2.01] Determinants

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.

Chapter: [5.01] Linear Programming

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.

Chapter: [4.01] Three - Dimensional Geometry

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.

Chapter: [4.01] Three - Dimensional Geometry

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.

Chapter: [6.01] Probability

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