Mathematics Delhi Set 3 2014-2015 Commerce (English Medium) Class 12 Question Paper Solution

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Mathematics [Delhi Set 3]
Marks: 100 Academic Year: 2014-2015
Date: March 2015
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[1] 1

Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
[1] 2

Find the integrating factor of the differential equation.


Concept: Solutions of Linear Differential Equation
Chapter: [0.09] Differential Equations
[1] 3

If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[1] 4

Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk`  are coplanar.

Concept: Scalar Triple Product of Vectors
Chapter: [0.1] Vectors
[1] 5

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[1] 6

Write the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by `a_(ij)=∣(i−j)/2∣`

Concept: Introduction of Operations on Matrices
Chapter: [0.03] Matrices
[4] 7

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`

Concept: Second Order Derivative
Chapter: [0.05] Continuity and Differentiability
[4] 8

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[4] 9

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 10

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:

Hand-fans 40 25 35
Mats 50 40 50
Plates 20 30 40

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

Write one value generated by the above situation.

Concept: Multiplication of Two Matrices
Chapter: [0.03] Matrices
[4] 11 | Attempt any one
[4] 11.1

If `A=([2,0,1],[2,1,3],[1,-1,0])` find A2 - 5A + 4I and hence find a matrix X such that  A2 - 5A + 4I + X = 0

Concept: Algebraic Operations on Matrices - Addition of Matrices
Chapter: [0.03] Matrices
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[4] 11.2

If A = `[[1,-2,3],[0,-1,4],[-2,2,1]]` ,find (A')-1

Concept: Inverse of Matrix - Inverse of a Matrix by Elementary Transformation
Chapter: [0.03] Matrices
[4] 12

If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).

Concept: Properties of Determinants
Chapter: [0.04] Determinants
[4] 13 | Attempt any one
[4] 13.1

Find: `I=intdx/(sinx+sin2x)`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.07] Integrals
[4] 13.2

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

Concept: Evaluation of Simple Integrals of the Following Types and Problems
Chapter: [0.07] Integrals
[4] 14

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[4] 15 | Attempt any one
[4] 15.1

A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
[4] 15.2

An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.

Concept: Mean of a Random Variable
Chapter: [0.13] Probability
[4] 16

If `vecr=xhati+yhatj+zhatk` ,find `(vecrxxhati).(vecrxxhatj)+xy`

Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[4] 17

Find the distance between the point (−1, −5, −10) and the point of intersection of the line `(x-2)/3=(y+1)/4=(z-2)/12` and the plane x-y+z=5

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
[4] 18 | Attempt any one
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[4] 18.1

If sin [cot−1 (x+1)] = cos(tan1x), then find x.

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 18.2

If (tan1x)2 + (cot−1x)2 = 5π2/8, then find x.

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 19

If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x21, then find dy/dx.

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [0.05] Continuity and Differentiability
[6] 20

If A and B are two independent events such that `P(barA∩ B) =2/15 and P(A ∩ barB) = 1/6`, then find P(A) and P(B).

Concept: Independent Events
Chapter: [0.13] Probability
[6] 21

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

Concept: Maximum and Minimum Values of a Function in a Closed Interval
Chapter: [0.06] Applications of Derivatives
[6] 22

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :

2x + 4y  83

x + y  6

x + y  4

x  0, y 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 23

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
[6] 24 | Attempt any one
[6] 24.1

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

Concept: Area Under Simple Curves
Chapter: [0.08] Applications of the Integrals
[6] 24.2

Evaluate `int_1^3(e^(2-3x)+x^2+1)dx`  as a limit of sum.

Concept: Definite Integral as the Limit of a Sum
Chapter: [0.07] Integrals
[6] 25 | Attempt any one
[6] 25.1

Solve the differential equation : (tan1yx)dy=(1+y2)dx.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[6] 25.2

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

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[6] 26

If lines `(x−1)/2=(y+1)/3=(z−1)/4 and  (x−3)/1=(y−k)/2=z/1` intersect, then find the value of k and hence find the equation of the plane containing these lines.

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry

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