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

Mathematics and Statistics [Set 1]
Date & Time: 2nd March 2019, 11:00 am
Duration: 3h

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

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: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
 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: [0.1] Plane
 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: [0.08] Three Dimensional Geometry
 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: [0.12] Continuity
 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: [0.023] Indefinite Integration [0.15] Integration
 6

If y = ae5x + be-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: [0.17] Differential Equation
SECTION - B
 7

Write the truth values of the following statements:
(1) 2 is a rational number andsqrt2 is an irrational number. .
(2) 2 + 3 = 5 or sqrt 2 +sqrt3 = sqrt5

Concept: Mathematical Logic - Statements - Introduction in Logic
Chapter: [0.01] Mathematical Logic
 8
 8.A

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: [0.015] Vectors [0.07] Vectors
OR
 8.B

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: [0.015] Vectors [0.07] Vectors
 9

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

Concept: Collinearity and Coplanarity of Vectors
Chapter: [0.07] Vectors
 10

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

Concept: Plane - Equation of Plane in Normal Form
Chapter: [0.1] Plane
 11

If "y" ="x"^"x" , "find"  "dy"/"dx".

Concept: Exponential and Logarithmic Functions
Chapter: [0.12] Continuity [0.13] Differentiation
 12

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

Concept: Tangents and Normals
Chapter: [0.14] Applications of Derivative
 13

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

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
 14

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

Concept: Properties of Definite Integrals
Chapter: [0.15] Integration
SECTION - C
 15

In , ΔABC prove that

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

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
OR
 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: [0.03] Trigonometric Functions
 16

Let "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: [0.015] Vectors [0.07] Vectors
 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: Equation of a Line in Space
Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line
 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: Definition of Continuity - Continuity of a Function at a Point
Chapter: [0.12] Continuity
 19
 19.A
 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: [0.19] Probability Distribution
OR
 19.B

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

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

= 0,             otherwise

Concept: Random Variables and Its Probability Distributions
Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution
 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: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution
SECTION - D
 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: [0.01] Mathematical Logic
 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 Transformations
Chapter: [0.02] Matrices
 23

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

Concept: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
OR
 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: Solutions of Triangle
Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions
 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: [0.04] Pair of Straight Lines
 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: [0.017] Linear Programming [0.11] Linear Programming Problems
 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: [0.13] Differentiation
 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: [0.14] Applications of Derivative
OR
 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: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative
 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: [0.023] Indefinite Integration [0.15] Integration
 29

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

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.023] Indefinite Integration [0.15] Integration
 30

Solve the differential equation:

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

Concept: The Concept of Derivative - Revision of Derivative
Chapter: [0.13] Differentiation
 30.A

Solve the differential equation:

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

Concept: The Concept of Derivative - Revision of Derivative
Chapter: [0.13] Differentiation

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