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Question Paper Solutions - Mathematics 2014 - 2015-CBSE 12th-Class 12 CBSE (Central Board of Secondary Education)

Alternate Sets

     

Marks: 100
[1]1

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

Chapter: [5] Vectors (Section B)
Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
[1]2

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

Chapter: [7] Vectors
Concept: Scalar Triple Product of Vectors
[1]3

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

Chapter: [3] Trigonometric Functions
Concept: Properties of Inverse Trigonometric Functions
[1]4

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

Chapter: [2.01] Matrices
Concept: Introduction of Operations on Matrices
[1]5

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

Chapter: [3.04] Differential Equations
Concept: Formation of a Differential Equation Whose General Solution is Given
[1]6

Find the integrating factor of the differential equation.

`((e^(-2^sqrtx))/sqrtx-y/sqrtx)dy/dx=1`

Chapter: [3.05] Differential Equations
Concept: Solutions of Linear Differential Equation
[4]7 | Attempt any one
[4]7.1

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=O

Chapter: [2.01] Matrices and Determinants
Concept: Operations on Matrices - Addition of Matrices
[4]7.2
 

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

 
Chapter: [2.01] Matrices
Concept: Inverse of a Matrix by Elementary Operations
[4]8
 

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).

 
Chapter: [2.02] Determinants
Concept: Properties of Determinants
[4]9 | Attempt any one
[4]9.1

Find: `I=intdx/(sinx+sin2x)`

Chapter: [15] Integration
Concept: Methods of Integration - Integration Using Partial Fractions
[4]9.2

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

Chapter: [3.03] Integrals
Concept: Evaluation of Simple Integrals of the Following Types and Problems
[4]10

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

Chapter: [3.03] Integrals
Concept: Evaluation of Definite Integrals by Substitution
[4]11 | Attempt any one:
[4]11.1

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.

Chapter: [6.01] Probability
Concept: Probability Examples and Solutions
[4]11.2

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

Chapter: [4] Probability (Section A)
Concept: Mean of a Random Variable
[4]12

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

Chapter: [4.01] Vectors
Concept: Vectors Examples and Solutions
[4]13

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

Chapter: [4.02] Three - Dimensional Geometry
Concept: Three - Dimensional Geometry Examples and Solutions
[4]14 | Attempt any one
[4]14.1

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

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions (Simplification and Examples)
[4]14.2

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

Chapter: [1.02] Inverse Trigonometric Functions
Concept: Inverse Trigonometric Functions (Simplification and Examples)
[4]15

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.

Chapter: [13] Differentiation
Concept: Derivatives of Inverse Trigonometric Functions
[4]16

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`

Chapter: [4] Differentiation
Concept: Second Order Derivative
[4]17

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 ?

Chapter: [5] Applications of Derivative
Concept: Increasing and Decreasing Functions
[4]18

Find : `int(x+3)sqrt(3-4x-x^2dx)`

Chapter: [3.03] Integrals
Concept: Methods of Integration - Integration by Substitution
[4]19

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:

SchoolArticle      
A B C
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.

Chapter: [2.01] Matrices and Determinants
Concept: Multiplication of Two Matrices
[6]20

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.

Chapter: [1] Relations and Functions (Section A)
Concept: Types of Relations
[6]21 | Attempt any one :
[6]21.1

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)`

Chapter: [3.04] Applications of the Integrals
Concept: Area Under Simple Curves
[6]21.2

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

Chapter: [15] Integration
Concept: Definite Integral as the Limit of a Sum
[6]22 | Attempt any one
[6]22.1

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

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[6]22.2

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

Chapter: [3.05] Differential Equations
Concept: General and Particular Solutions of a Differential Equation
[6]23

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.

Chapter: [4.02] Three - Dimensional Geometry
Concept: Three - Dimensional Geometry Examples and Solutions
[6]24

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).

Chapter: [6.01] Probability
Concept: Independent Events
[6]25

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

Chapter: [3.02] Applications of Derivatives
Concept: Maximum and Minimum Values of a Function in a Closed Interval
[6]26

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

Chapter: [11] Linear Programming Problems
Concept: Graphical Method of Solving Linear Programming Problems
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