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

Duration: 2h30m

(i) **All **questions are compulsory.

(ii) 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.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

(iv) 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.

(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.

Find the acute angle between the planes `vec"r". (hat"i" - 2hat"j" - 2hat"k") = 1` and `vec"r". (3hat"i" - 6hat"j" - 2hat "k") = 0`

Chapter: [4.02] Vectors

Chapter: [4.01] Three - Dimensional Geometry

If y = log (cos e^{x}) then find `"dy"/"dx".`

Chapter: [3.01] Continuity and Differentiability

A is a square matrix with ∣A∣ = 4. then find the value of ∣A. (adj A)∣.

Chapter: [2.01] Determinants

Form the differential equation representing the family of curves y = A sin x, by eliminating the arbitrary constant A.

Chapter: [3.04] Differential Equations

**Find:**

`int"x".tan^-1 "x" "dx"`

Chapter: [3.05] Integrals

**Find:**`int"dx"/sqrt(5-4"x" - 2"x"^2)`

Chapter: [3.05] Integrals

**Solve the following differential equation :**

`"dy"/"dx" + "y" = cos"x" - sin"x"`

Chapter: [3.04] Differential Equations

**Find:**

`int_(-pi/4)^0 (1+tan"x")/(1-tan"x") "dx"`

Chapter: [3.05] Integrals

Let * be an operation defined as *: **R × R ⟶ R**, a * b = 2a + b, a, b ∈ **R**. Check if * is a binary operation. If yes, find if it is associative too.

Chapter: [1.02] Relations and Functions

X and Y are two points with position vectors `3vec("a") + vec("b")` and `vec("a")-3vec("b")`respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally.

Chapter: [4.02] Vectors

Let `vec("a") = hat"i" + 2hat"j" - 3hat"k"` and `vec("b") = 3hat"i" -"j" +2hat("k")` be two vectors. Show that the vectors `(vec("a")+vec("b"))` and `(vec("a")-vec("b"))`are perpendicular to each other.

Chapter: [4.02] Vectors

Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected.

Chapter: [6.01] Probability

Chapter: [6.01] Probability

The probabilities of solving a specific problem independently by A and B are `1/3` and `1/5` respectively. If both try to solve the problem independently, find the probability that the problem is solved.

Chapter: [6.01] Probability

For the matrix A = `[(2,3),(5,7)]`, find (A + A') and verify that it is a symmetric matrix.

Chapter: [1.02] Relations and Functions

A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall.

Chapter: [3.02] Applications of Derivatives

**Prove that :**

`cos^-1 (12/13) + sin^-1(3/5) = sin^-1(56/65)`

Chapter: [3.01] Continuity and Differentiability

Prove that `int_0^"a" "f(x)" "dx" = int_0^"a" "f"("a"-"x")"dx"` ,and hence evaluate `int_0^1 "x"^2(1 - "x")^"n""dx"`.

Chapter: [3.05] Integrals

If x = sin t, y = sin pt, prove that`(1-"x"^2)("d"^2"y")/"dx"^2 - "x" "dy"/"dx" + "p"^2"y" = 0`

Chapter: [3.01] Continuity and Differentiability

Differentiate `tan^-1[(sqrt(1+"x"^2)-sqrt(1-"x"^2))/(sqrt(1+"x"^2) + sqrt(1-"x"^2))]`with respect to cos^{−1}x^{2}.

Chapter: [3.05] Integrals

Integrate the function `cos("x + a")/sin("x + b")` w.r.t. x.

Chapter: [3.05] Integrals

Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f^{−1}.

Chapter: [1.02] Relations and Functions

Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.

Chapter: [1.02] Relations and Functions

Solve the differential equation `"dy"/"dx" = 1 + "x"^2 + "y"^2 +"x"^2"y"^2`, given that y = 1 when x = 0.

Chapter: [3.04] Differential Equations

Find the particular solution of the differential equation `"dy"/"dx" = "xy"/("x"^2+"y"^2),`given that y = 1 when x = 0

Chapter: [3.04] Differential Equations

Using properties of determinants, find the value of x for which

`|(4-"x",4+"x",4+"x"),(4+"x",4-"x",4+"x"),(4+"x",4+"x",4-"x")|= 0`

Chapter: [2.01] Determinants

Find the vector equation of the plane which contains the line of intersection of the planes `vec("r").(hat"i"+2hat"j"+3hat"k"),-4=0, vec("r").(2hat"i"+hat"j"-hat"k")+5=0`and which is perpendicular to the plane`vec("r").(5hat"i"+3hat"j"-6hat"k"),+8=0`

Chapter: [4.01] Three - Dimensional Geometry

Find the value of x such that the four-point with position vectors,

`"A"(3hat"i"+2hat"j"+hat"k"),"B" (4hat"i"+"x"hat"j"+5hat"k"),"c" (4hat"i"+2hat"j"-2hat"k")`and`"D"(6hat"i"+5hat"j"-hat"k")`are coplaner.

Chapter: [4.02] Vectors

If y = (log x)^{x} + x^{log x}, find `"dy"/"dx".`

Chapter: [3.01] Continuity and Differentiability

Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line `vec("r") = (-2hat"i"+3hat"j") +lambda(2hat"i"-3hat"j"+6hat"k").`Also, find the distance between these two lines.

Chapter: [4.01] Three - Dimensional Geometry

Chapter: [4.01] Three - Dimensional Geometry

Using elementary row transformation, find the inverse of the matrix

`[(2,-3,5),(3,2,-4),(1,1,-2)]`

Chapter: [2.02] Matrices

**Using matrices, solve the following system of linear equations :**

x + 2y − 3z = −4

2x + 3y + 2z = 2

3x − 3y − 4z = 11

Chapter: [2.01] Determinants

Using integration, find the area of the region bounded by the parabola y^{2 }= 4x and the circle 4x^{2} + 4y^{2} = 9.

Chapter: [3.03] Applications of the Integrals

Using the method of integration, find the area of the region bounded by the lines 3x − 2y + 1 = 0, 2x + 3y − 21 = 0 and x − 5y + 9 = 0

Chapter: [3.03] Applications of the Integrals

An insurance company insured 3000 cyclists, 6000 scooter drivers, and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver, and a car driver are 0⋅3, 0⋅05 and 0⋅02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist?

Chapter: [6.01] Probability

Using integration, find the area of the smaller region bounded by the ellipse `"x"^2/9+"y"^2/4=1`and the line `"x"/3+"y"/2=1.`

Chapter: [3.03] Applications of the Integrals

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ₹ 35 per package of nuts and ₹ 14 per package of bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates each machine for almost 12 hours a day? convert it into an LPP and solve graphically.

Chapter: [5.01] Linear Programming

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