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Mathematics Panchkula Set 1 2014-2015 CBSE (Commerce) Class 12 Question Paper Solution

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

If `veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk` ,then find ` |veca xx vecb|`

 
Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[1] 2

Find the angle between the vectors `hati-hatj and hatj-hatk`

Concept: Introduction of Product of Two Vectors
Chapter: [0.1] Vectors
[1] 3

Find the distance of a point (2, 5, −3) from the plane `vec r.(6hati-3hatj+2 hatk)=4`

Concept: Distance of a Point from a Plane
Chapter: [0.11] Three - Dimensional Geometry
[1] 4

Write the element a12 of the matrix A = [aij]2 × 2, whose elements aij are given by aij = e2ix sin jx.

Concept: Introduction of Operations on Matrices
Chapter: [0.03] Matrices
[1] 5

Find the differential equation of the family of lines passing through the origin.

Concept: Formation of a Differential Equation Whose General Solution is Given
Chapter: [0.09] Differential Equations
[1] 6

Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`

Concept: Solutions of Linear Differential Equation
Chapter: [0.09] Differential Equations
[4] 7 | Attempt any one :
[4] 7.1

If `A=[[1,2,2],[2,1,2],[2,2,1]]` ,then show that `A^2-4A-5I=0` and hence find A-1.

Concept: Operations on Matrices - Addition of Matrices
Chapter: [0.03] Matrices
[4] 7.2

If `A=|[2,0,-1],[5,1,0],[0,1,3]|` , then find A-1 using elementary row operations

Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
[4] 8

Using the properties of determinants, solve the following for x:

`|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=0`

Concept: Elementary Transformations
Chapter: [0.03] Matrices [0.04] Determinants
[4] 9 | Attempt any one:
[4] 9.1
 
 

Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`

 
 
Concept: Fundamental Theorem of Calculus
Chapter: [0.07] Integrals
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[4] 9.2

Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums

Concept: Definite Integral as the Limit of a Sum
Chapter: [0.07] Integrals
[4] 10

Evaluate:

`int x^2/(x^4+x^2-2)dx`

Concept: Methods of Integration: Integration Using Partial Fractions
Chapter: [0.07] Integrals
[4] 11 | Attempt any one
[4] 11.1

In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
[4] 11.2

How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?

Concept: Probability Examples and Solutions
Chapter: [0.13] Probability
[4] 12

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[4] 13
 

A line passing through the point A with position vector `veca=4hati+2hatj+2hatk` is parallel to the vector `vecb=2hati+3hatj+6hatk` . Find the length of the perpendicular drawn on this line from a point P with vector `vecr_1=hati+2hatj+3hatk`

 
Concept: Vectors Examples and Solutions
Chapter: [0.1] Vectors
[4] 14 | Attempt any one
[4] 14.1

Solve the following for x:

`sin^(-1)(1-x)-2sin^-1 x=pi/2`

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 14.2
 

Show that:

`2 sin^-1 (3/5)-tan^-1 (17/31)=pi/4`

 

 
Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [0.02] Inverse Trigonometric Functions
[4] 15

If y = eax. cos bx, then prove that

`(d^2y)/(dx^2)-2ady/dx+(a^2+b^2)y=0`

Concept: Derivatives of Composite Functions - Chain Rule
Chapter: [0.05] Continuity and Differentiability
[4] 16
 

if xx+xy+yx=ab, then find `dy/dx`.

Concept: Logarithmic Differentiation
Chapter: [0.05] Continuity and Differentiability
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[4] 17
 

 If x=a sin 2t(1+cos 2t) and y=b cos 2t(1cos 2t), find `dy/dx `

 
Concept: Derivatives of Functions in Parametric Forms
Chapter: [0.05] Continuity and Differentiability
[4] 18

Evaluate:

`int((x+3)e^x)/((x+5)^3)dx`

Concept: Integrals of Some Particular Functions
Chapter: [0.07] Integrals
[4] 19

Three schools X, Y, and Z organized a fete (mela) for collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the number of articles of each type sold:

School/Article School X School Y School z
Hand-held fans 30 40 35
Mats 12 15 20
toys 70 55 75

Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation.

Concept: Multiplication of Two Matrices
Chapter: [0.03] Matrices
[6] 20 | Attempt any one :
[6] 20.1

Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) =  (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A.

(iii)and hence write the inverse of elements (5, 3) and (1/2,4)

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[6] 20.2

Let f : W → W be defined as

`f(n)={(n-1, " if n is odd"),(n+1, "if n is even") :}`

Show that f is invertible a nd find the inverse of f. Here, W is the set of all whole
numbers.

Concept: Composition of Functions and Invertible Function
Chapter: [0.01] Relations and Functions
[6] 21

Sketch the region bounded by the curves `y=sqrt(5-x^2)` and y=|x-1| and find its area using integration.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[6] 22 | Attempt any one :
[6] 22.1

Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[6] 22.2

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

Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[6] 23

Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin2x-cosx,x ∈ (0,π)

Concept: Maximum and Minimum Values of a Function in a Closed Interval
Chapter: [0.06] Applications of Derivatives
[6] 24
 

Show that lines: 

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar 

Also, find the equation of the plane containing these lines.

 
Concept: Shortest Distance Between Two Lines
Chapter: [0.11] Three - Dimensional Geometry
[6] 25

Minimum and maximum z = 5x + 2y subject to the following constraints:

x-2y ≤ 2

3x+2y ≤ 12

-3x+2y ≤ 3

x ≥ 0,y ≥ 0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 26

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

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

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