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Mathematics and Statistics Set 1 2018-2019 HSC Science (Electronics) 12th Board Exam Question Paper Solution

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Mathematics and Statistics
Set 1
2018-2019 February
Marks: 80

SECTION -A
[1]1 | Select and write the most appropriate answer from the given alternatives for each question: [6]

The principal solutions of cot x = -`sqrt3`  are .................

`pi/6 ,(5pi)/6`

`(5pi)/6 , (7pi)/6`

`(5pi)/6,(11pi)/6`

`pi/6,(11pi)/6`

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
[1]2

The acute angle between the two planes x+y+2z = 3 and 3x -2y +2z = 7 ________.

`"sin"^-1( 5/sqrt102)`

`"cos"^-1(5/sqrt102)`

`"sin"^-1(15/sqrt102)`

`"cos"^-1(15/sqrt102)`

Concept: Angle Between Two Planes
Chapter: [10] Plane
[1]3

The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are _______.

–2, –1, –2

2, 1, 2

2, –1, –2

–2, 1, –2

Concept: Direction Cosines and Direction Ratios of a Line
Chapter: [8] Three Dimensional Geometry
[1]4

If f(x) = `(1+2"x")^(1/"x)"` for x≠0 is continuous at x = 0 then f(0) _______.

e

e2

0

2

Concept: Introduction of Continuity
Chapter: [12] Continuity
[1]5

`int "dx"/(9"x"^2 + 1)= ______. `

`1/3 "tan"^-1(2"x") +"c"`

`1/3 "tan"^-1"x" +"c"`

`1/3 "tan"^-1(3"x") +"c"`

`1/3 "tan"^-1(6"x") +"c"`

Concept: Methods of Integration - Integration by Substitution
Chapter: [15] Integration
[1]6

If y = ae5x + 5e-5x then the differential equation is _________.

`("d"^2"y")/"dx"^2 = 25"y"`

`("d"^2"y")/"dx"^2 = -25"y"`

`("d"^2"y")/"dx"^2 = -5"y"`

`("d"^2"y")/"dx"^2 = 5"y"`

Concept: Formation of Differential Equation by Eliminating Arbitary Constant
Chapter: [17] Differential Equation
SECTION -B (16)
[2]7

Write the truth values of the following statements:
(1) 2 is a rational number and`sqrt2` is an irrational number. .
(2) 2 + 3 = 5 or `sqrt 2 +sqrt3 = sqrt5`

Concept: Mathematical Logic - Statements - Introduction in Logic
Chapter: [1] Mathematical Logic
[2]8

Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.                               

Concept: Scalar Triple Product of Vectors
Chapter: [7] Vectors
OR
[2]8.A

Find the value of p, if the vectors `hat"i" + 2hat"j" + hat"k", 2hat"i" +5hat"j"+"p" hat "k" , 5hat"i" +9hat"j" - 4 hat"k"`   are coplanar.

Concept: Scalar Triple Product of Vectors
Chapter: [7] Vectors
[2]9

 Show that the points A (-7 , 4 , -2),B (-2 , 1 , 0)and C (3 ,-2 ,2) are collinear.

Concept: Vectors - Collinearity and Coplanarity of Vectors
Chapter: [7] Vectors
[2]10

 Write the equation of the plane 3x + 4y -2z = 5 in the vector form.

Concept: Plane - Equation of Plane in Normal Form
Chapter: [10] Plane
[2]11

 If `"y" ="x"^"x" , "find"  "dy"/"dx"`.

Concept: Exponential and Logarithmic Functions
Chapter: [12] Continuity [13] Differentiation
[2]12

 Find the equation of tangent to the curve y = x2 +4x + 1 at (-1 , -2).

Concept: Tangents and Normals
Chapter: [14] Applications of Derivative
[2]13

Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`

Concept: Methods of Integration - Integration by Substitution
Chapter: [15] Integration
[2]14

Evaluate : `int _0^(pi/2) "sin"^ 2  "x"  "dx"`

Concept: Properties of Definite Integrals
Chapter: [15] Integration
SECTION -C (18)
[3]15

 In , ΔABC prove that 

`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`                               

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
OR
[3]15.A

Show that `"sin"^-1(5/13) + "cos"^-1(3/5) = "tan"^-1(63/16)`

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter: [3] Trigonometric Functions
[3]16

If `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that

 `bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

Concept: Section formula
Chapter: [7] Vectors
[3]17

 The equation of a line is 2x -2 = 3y +1 = 6z -2 find the direction ratios and also find the vector equation of the line. 

Concept: Introduction of Three Dimensional Geometry
Chapter: [8] Three Dimensional Geometry
[3]18

Discuss the continuity of the function 

f(x) = `("log"(2+"x") - "log"(2-"x"))/("tan""x")` , for x ≠ 0

 = 1       for x = 0

at the point x =0

Concept: Continuity - Continuity of a Function at a Point
Chapter: [12] Continuity
[3]19
x 0 1 2 3 4
P(X = x) 0.45 0.35 0.15 0.03 0.02

Find the variance of X.                                  

Concept: Probability Distribution - Discrete and Continuous Random Variable
Chapter: [19] Probability Distribution
OR
[3]19.A

For  the following probability density function (p. d. f) of X, find : (1) P ( X<1) , (2) P |x| < 1

if f(x) =`"x"^2/18 , -3<"x" <3`

= 0 , otherwise

Concept: Random Variables and Its Probability Distributions
Chapter: [19] Probability Distribution
[3]20

Given is X ~ B (n,p)

If E(X) = 6,Var(X) = 4.2 , find n and p.

Concept: Variance of Binomial Distribution (P.M.F.)
Chapter: [20] Bernoulli Trials and Binomial Distribution
SECTION -D
[4]21

 Find the symbolic form of the given switching circuit. Construct its switching table and interpret yout result. diagram 

Concept: Mathematical Logic - Statements - Introduction in Logic
Chapter: [1] Mathematical Logic
[4]22

If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices. 

Concept: Elementary Operation (Transformation) of a Matrix
Chapter: [2] Matrices
[4]23

 In ,Δ ABC with usual notations prove that 
b2 = c2 +a2 - 2 cacos B

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
OR
[4]23.A

 In , ΔABC with usual notations prove that

(a-b)2 cos2 `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`

Concept: Trigonometric Functions - Solution of a Triangle
Chapter: [3] Trigonometric Functions
[4]24

 Find 'p' and 'q' if the equation px2 - 8xy +3y2 +14 x +2y +q = 0 represents a pair of perpendicular lines.

Concept: Pair of Straight Lines - Pair of Lines Passing Through Origin - Combined Equation
Chapter: [4] Pair of Straight Lines
[4]25

 Maximize: z = 3x + 5y  Subject to

x +4y ≤ 24                3x + y  ≤ 21 

x + y ≤ 9                     x ≥ 0 , y ≥0

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [11] Linear Programming Problems
[4]26

If X = f(t) and Y = g(t) Are Differentiable Functions of t ,  then prove that y is a differentiable function of
x and

`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`

Hence find `"dy"/"dx"` if x = a cos2 t and y = a sin2 t.

Concept: Derivatives of Functions in Parametric Forms
Chapter: [13] Differentiation
[4]27

f(x) = (x-1)(x-2)(x-3) , x ε[0,4], find if 'c' LMVT can be applied

Concept: Mean Value Theorem
Chapter: [14] Applications of Derivative
OR
[4]27.A

 A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle. 

Concept: Maxima and Minima
Chapter: [14] Applications of Derivative
[4]28

 Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log  |"x" +sqrt("x"^2 +"a"^2) | + "c"`

Concept: Methods of Integration - Integration by Substitution
Chapter: [15] Integration
[4]29

 Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`

Concept: Methods of Integration - Integration by Substitution
Chapter: [15] Integration
[4]30

 Solve the differential equation: 

`"dy"/"dx" + "y"  "sec"  "x" = "tan""x"`

Concept: Derivative - Revision of Derivative
Chapter: [13] Differentiation
[4]30.A

 Solve the differential equation:

`("x" + "y") "dy"/"dx" = 1`

Concept: Derivative - Revision of Derivative
Chapter: [13] Differentiation

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