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Mathematics and Statistics 2013-2014 HSC Science (General) 12th Board Exam Question Paper Solution

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Mathematics and Statistics
Marks: 80Date: 2013-2014 October

[12]1
[6]1.1 | Select and write the most appropriate answer from the given alternatives in each of the following sub-questions:
[2]1.1.1

If`[ bara  bar b barc ] ≠ 0  and  barp = [ barb xx barc ]/([ bara  bar b barc  ]), barq = [ barc xx bara ]/([ bara  bar b barc  ]), barr = [ bara xx barb ]/[ bara  bar b barc ]`

then `bara . barp + barb . barq + barc . barr` is equal to ______.

0

1

2

3

Concept: Vectors - Linear Combination of Vectors
Chapter: [7] Vectors
[2]1.1.2

The inverse of the matrix `[[2,0,0],[0,1,0],[0,0,-1]]`is --------

(a) `[[1/2,0,0],[0,1,0],[0,0,-1]]`

(b) `[[-1/2,0,0],[0,-1,0],[0,0,1]]`

(c) `[[-1,0,0],[0,-1/2,0],[0,0,1/2]]`

(d) `1/2[[-1/2,0,0],[0,-1,0],[0,0,-1]]`

Concept: Matrices - Inverse of a Matrix Existance
Chapter: [2] Matrices
[2]1.1.3

Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........

`+-1/sqrt51,+-5/sqrt51,+-1/sqrt51`

`+-5/sqrt51, +-1/sqrt51, +- (-5)/sqrt51`

`+-sqrt5,+-1,+-5`

`+-sqrt51,+-sqrt51+-sqrt51`

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [8] Three Dimensional Geometry
[6]1.2 | Attempt any THREE of the following:
[2]1.2.1

Write truth values of the following statements :`sqrt5` is an irrational number but 3 +`sqrt 5` is a complex number.

True

False

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [1] Mathematical Logic

Write truth values of the following statements : ∃ n ∈ N such that n + 5 > 10.

True

False

Concept: Mathematical Logic - Truth Value of Statement in Logic
Chapter: [1] Mathematical Logic
[2]1.2.2

If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`

Concept: Scalar Triple Product of Vectors
Chapter: [7] Vectors
[2]1.2.3

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector  `2hati + hatj + 2hatk.`

Concept: Vector and Cartesian Equation of a Plane
Chapter: [10] Plane
[2]1.2.4

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

 

Concept: Equation of a Line in Space
Chapter: [9] Line
[2]1.2.5

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

Concept: Basic Concepts of Vector Algebra
Chapter: [7] Vectors
[12]2
[6]2.1 | Attempt any TWO of the following:
[3]2.1.1

Using truth table, prove the following logical equivalence :

(p ∧ q)→r = p → (q→r)

Concept: Mathematical Logic - Truth Tables of Compound Statements
Chapter: [1] Mathematical Logic
[3]2.1.2

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Chapter: [4] Pair of Straight Lines
[3]2.1.3

Show that: `2sin^-1(3/5)=tan^-1(24/7)`

Concept: Basic Concepts of Trigonometric Functions
Chapter: [3] Trigonometric Functions
[8]2.2 | Attempt any TWO of the following:
[4]2.2.1

Solve the following equations by the method of reduction :

2x-y + z=1,  x + 2y +3z = 8, 3x + y-4z=1.

Concept: Elementary Operation (Transformation) of a Matrix
Chapter: [2] Matrices
[4]2.2.2

Prove that the volume of a parallelopiped with coterminal edges as  ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges  `bar i+barj, barj+bark `

Concept: Scalar Triple Product of Vectors
Chapter: [7] Vectors
[4]2.2.3

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤  3, -2x + y ≤  1, x ≥  0, y ≥ 0.

Also find maximum value of Z.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [11] Linear Programming Problems
[14]3
[6]3.1 | Attempt any TWO of the following
[3]3.1.1
 

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a +   c - b)

 
Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
[3]3.1.2

If p : It is a day time, q : It is warm, write the compound statements in verbal form
denoted by -
(a) p ∧ ~ q  
(b)  ~ p  → q   
(c)  q  ↔  p

Concept: Mathematical Logic - Compound Statement in Logic
Chapter: [1] Mathematical Logic
[3]3.1.3

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Concept: Line - Distance of a Point from a Line
Chapter: [9] Line
[8]3.2 | Attempt any TWO of the following:
[4]3.2.1

Parametric form of the equation of the plane is `bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk)` λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

Concept: Vector and Cartesian Equation of a Plane
Chapter: [10] Plane
[4]3.2.2

If the angle between the lines represented by ax2 + 2hxy + by2 = 0 is equal to the angle between the lines 2x2 - 5xy + 3y2 =0,

then show that 100(h2 - ab) = (a + b)2

Concept: Angle Between Two Lines
Chapter: [8] Three Dimensional Geometry
[4]3.2.3

Find the general solution of :  sinx · tanx = tanx - sinx + 1

Concept: Trigonometric Functions - General Solution of Trigonometric Equation of the Type
Chapter: [3] Trigonometric Functions
[12]4
[6]4.1 |  Select and write the most appropriate answer from the given alternatives in each of the following sub-questions:
[2]4.1.1

The differential equation of the family of curves y=c1ex+c2e-x is......

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

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

(c)`(d^2y)/dx^2+1=0`

(d)`(d^2y)/dx^2-1=0`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[2]4.1.2

If X is a random variable with probability mass function

P(x) = kx ,  x=1,2,3

      = 0 ,     otherwise

then , k=..............

(a) 1/5

(b) 1/4

(c) 1/6

(d) 2/3

 

Concept: Probability Distribution - Probability Mass Function (P.M.F.)
Chapter: [19] Probability Distribution
[2]4.1.3

If `sec((x+y)/(x-y))=a^2. " then " (d^2y)/dx^2=........`

(a) y

(b) x

(c) y/x

(d) 0

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [13] Differentiation
[6]4.2 | Attempt any THREE of the following:
[2]4.2.1

If `y=sin^-1(3x)+sec^-1(1/(3x)), `  find dy/dx

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [13] Differentiation
[2]4.2.2

Evaluate :`intxlogxdx`

Concept: Methods of Integration - Integration by Substitution
Chapter: [15] Integration
[2]4.2.3

If `int_0^h1/(2+8x^2)dx=pi/16 `then find the value of h.

Concept: Fundamental Theorem of Calculus
Chapter: [15] Integration
[2]4.2.4

The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive.

Concept: Conditional Probability
Chapter: [19] Probability Distribution
[2]4.2.5

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [16] Applications of Definite Integral
[14]5
[6]5.1 | Attempt any TWO of the following:
[3]5.1.1

Examine the continuity of the following function at given point:

`f(x)=(logx-log8)/(x-8) , `

` =8, `

Concept: Continuity - Discontinuity of a Function
Chapter: [12] Continuity
[3]5.1.2

If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[3]5.1.3

Solve : 3ex tanydx + (1 +ex) sec2 ydy = 0

Also, find the particular solution when x = 0 and y = π.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[8]5.2 | Attempt any TWO of the following:
[4]5.2.1

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Concept: Rate of Change of Bodies Or Quantities
Chapter: [14] Applications of Derivative
[4]5.2.2
 
 

Evaluate : `intlogx/(1+logx)^2dx`

 
 
Concept: Properties of Definite Integrals
Chapter: [15] Integration
[4]5.2.3

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and  hence, find dy/dx if x=a cost, y=a sint

Concept: Derivatives of Functions in Parametric Forms
Chapter: [13] Differentiation
[14]6
[6]6.1 | Attempt any TWO of the following:
[3]6.1.1

Show that the function defined by f(x) =|cosx| is continuous function.

Concept: Introduction of Continuity
Chapter: [12] Continuity
[3]6.1.2

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

Concept: General and Particular Solutions of a Differential Equation
Chapter: [17] Differential Equation
[3]6.1.3

Given X ~ B(n, p). If n = 20, E(X) = 10, find p, Var. (X) and   S.D. (X).

Concept: Bernoulli Trials and Binomial Distribution - Normal Distribution (P.D.F)
Chapter: [20] Bernoulli Trials and Binomial Distribution
[8]6.2 | Attempt any TWO of the following:
[4]6.2.1

A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?

Concept: Statistics - Bivariate Frequency Distribution
Chapter: [18] Statistics
[4]6.2.2

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

Concept: Mean Value Theorem
Chapter: [14] Applications of Derivative
[4]6.2.3

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`

Concept: Methods of Integration - Integration by Substitution
Chapter: [15] Integration

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