This document provides a 75 question multiple choice test paper with questions ranging in topics from algebra, trigonometry, geometry, and calculus. The test is allotted 90 minutes and covers concepts such as functions, equations, properties of circles, ellipses, parabolas, and hyperbolas, trigonometric identities, and geometric shapes. Several questions involve finding lengths, areas, angles between lines, points of intersection, tangents, and properties of conic sections.
This document contains a 75 question test paper covering topics in mathematics including algebra, trigonometry, coordinate geometry, and calculus. The test has 90 minutes allotted and covers topics such as functions, quadratic equations, trigonometric identities, binomial coefficients, and geometric concepts like angles, triangles, and coordinate planes.
Aieee 2003 maths solved paper by fiitjeeMr_KevinShah
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -i.
The document contains a mathematics exam with three groups of questions testing different concepts:
Group A contains 10 multiple choice questions covering domains of functions, trigonometric functions, derivatives, integrals, determinants, and properties related to maxima and minima of functions.
Group B contains another 10 multiple choice questions testing concepts like distance between parallel lines, matrix operations, complex numbers, solving equations, properties of concurrent lines, integrals involving logarithms, and solving inequalities.
Group C contains 2 problems to be solved in detail, the first finding the length of a perpendicular from a point to a line, and the second evaluating a definite integral.
This document contains 54 multiple choice questions related to polynomials and their properties. Some key questions asked about:
- Finding the degree of polynomials
- Identifying the number of real zeros of polynomials
- Factoring polynomials
- Evaluating polynomials for given values
- Identifying coefficients and constants in polynomial expressions
- Relating the zeros of a polynomial to its factors
The questions cover topics like polynomial definitions, operations, factorization, finding zeros, and other properties of polynomials.
1. The document is a model question paper with 3 sections containing multiple choice and long answer questions on mathematics.
2. Section A contains 15 multiple choice questions worth 1 mark each. Section B contains 10 long answer questions worth 2 marks each. Section C contains 9 long answer questions worth 5 marks each and 1 compulsory question.
3. The questions cover topics in algebra, trigonometry, geometry, sequences and series, and probability.
This document provides a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. The paper is for 3 hours and carries a total of 100 marks. Some questions provide internal choices. Calculators are not permitted. Sample questions include finding inverse functions, evaluating integrals, solving differential equations, and probability questions.
This document contains 26 multiple choice questions about quadratic equations. The questions cover a range of topics including finding the roots of quadratic equations, determining the nature of the roots based on coefficients, and other properties of quadratic equations. Sample answers are provided but no full solutions are shown.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
This document contains a 75 question test paper covering topics in mathematics including algebra, trigonometry, coordinate geometry, and calculus. The test has 90 minutes allotted and covers topics such as functions, quadratic equations, trigonometric identities, binomial coefficients, and geometric concepts like angles, triangles, and coordinate planes.
Aieee 2003 maths solved paper by fiitjeeMr_KevinShah
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -i.
The document contains a mathematics exam with three groups of questions testing different concepts:
Group A contains 10 multiple choice questions covering domains of functions, trigonometric functions, derivatives, integrals, determinants, and properties related to maxima and minima of functions.
Group B contains another 10 multiple choice questions testing concepts like distance between parallel lines, matrix operations, complex numbers, solving equations, properties of concurrent lines, integrals involving logarithms, and solving inequalities.
Group C contains 2 problems to be solved in detail, the first finding the length of a perpendicular from a point to a line, and the second evaluating a definite integral.
This document contains 54 multiple choice questions related to polynomials and their properties. Some key questions asked about:
- Finding the degree of polynomials
- Identifying the number of real zeros of polynomials
- Factoring polynomials
- Evaluating polynomials for given values
- Identifying coefficients and constants in polynomial expressions
- Relating the zeros of a polynomial to its factors
The questions cover topics like polynomial definitions, operations, factorization, finding zeros, and other properties of polynomials.
1. The document is a model question paper with 3 sections containing multiple choice and long answer questions on mathematics.
2. Section A contains 15 multiple choice questions worth 1 mark each. Section B contains 10 long answer questions worth 2 marks each. Section C contains 9 long answer questions worth 5 marks each and 1 compulsory question.
3. The questions cover topics in algebra, trigonometry, geometry, sequences and series, and probability.
This document provides a sample question paper for Class XII Mathematics. It consists of 3 sections - Section A has 10 one-mark questions, Section B has 12 four-mark questions, and Section C has 7 six-mark questions. The paper is for 3 hours and carries a total of 100 marks. Some questions provide internal choices. Calculators are not permitted. Sample questions include finding inverse functions, evaluating integrals, solving differential equations, and probability questions.
This document contains 26 multiple choice questions about quadratic equations. The questions cover a range of topics including finding the roots of quadratic equations, determining the nature of the roots based on coefficients, and other properties of quadratic equations. Sample answers are provided but no full solutions are shown.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
The document discusses expanding and factorizing algebraic expressions. It provides examples of expanding expressions using the distributive property, such as expanding (a + b)(c + d) to get ac + ad + bc + bd. It also discusses factorizing expressions by finding common factors, such as factorizing a2 + 2ab + b2 to get (a + b)2. Tips and techniques are presented for expanding, factorizing, finding common factors, and using the distributive property to manipulate algebraic expressions.
This document contains 38 multiple choice questions related to polynomials and their properties. Specifically, it covers topics like:
- Finding zeroes of polynomials
- Relationships between zeroes and coefficients
- Determining the degree and number of zeroes of polynomials
- Factoring polynomials
- Identifying quadratic polynomials based on their zeroes
The document contains 46 mathematical formulae related to algebra, quadratic equations, arithmetic progressions, geometric progressions, factorials, and binomial expansions. Some key formulae include:
1) (a + b)2 = a2 + 2ab + b2 for expanding a binomial square.
2) The quadratic formula for solving ax2 + bx + c = 0 is x = (-b ± √(b2 - 4ac))/2a.
3) The nth term of an arithmetic progression with first term a and common difference d is an = a + (n - 1)d.
4) The nth term of a geometric progression with first term a and common ratio
This document contains solutions to several inequality problems:
1) It proves three inequalities using Cauchy-Schwarz inequality and other techniques.
2) It proves another inequality involving four positive real numbers using Cauchy-Schwarz inequality twice.
3) It proves an inequality about positive real numbers a0, a1, ..., an where ak+1 - ak ≥ 1 for all k using mathematical induction.
1. The given relation R defines a line with points (2,3), (4,2), and (6,1). The range of y-values is {1,2,3}.
2. The two trigonometric equations are equal when xy < 1.
3. The expressions 7A - (I + A)3 and -I are equal after expanding and simplifying the terms.
This document contains questions from assignments in differential calculus, continuity and differentiation, rate of change of quantities, increasing and decreasing functions, tangents and normals, and approximation. It also includes word problems involving optimization such as finding dimensions that result in maximum area, volume, or other quantities. There are over 25 questions in total across these calculus topics.
This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.
IIT JAM MATH 2021 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2021 Question Paper
IIT JAM Preparation Strategy
For any query about exams feel free to contact us
Call - 9836793076
The document contains 30 multiple choice questions related to various mathematical concepts such as relations, functions, matrices, determinants, and trigonometric functions. The questions test knowledge of properties such as equivalence, injectivity, surjectivity, inverse functions, composition of functions, matrix operations, and evaluating determinants.
This document contains 5 questions regarding a mathematics exam. It covers topics like algebra, geometry, calculus, differential equations, and matrices. Some key details:
- The exam has 5 questions worth a total of 80 marks.
- Question 1 has 8 short answer parts worth 16 marks total.
- Questions 2-4 have 4 medium length parts each worth 16 marks total.
- Question 5 has 2 long answer parts worth 16 marks total.
- The questions cover topics such as finding GCDs, eigenvalues, limits, differential equations, and geometry concepts.
Given a document describing functions and relations:
- It defines four types of relations: one-to-one, many-to-one, one-to-many, and many-to-many. Relations can be represented using arrow diagrams, ordered pairs, or graphs.
- An example question is worked through to identify the domain, codomain, range, and type of relation for a given set of ordered pairs.
- Functions are represented by lowercase letters like f, g, h. The value of a function f at x is written as f(x).
- Examples demonstrate evaluating functions by substituting values into the function equation and solving composite functions like fg(x).
1. The document contains 10 questions assessing knowledge of complex numbers and quadratic equations.
2. The questions cover topics like solving complex quadratic equations, operations on complex numbers, expressing complex numbers in polar form, and properties of complex functions and arguments.
3. The answers provided solve each question and explain steps like rationalizing expressions, taking conjugates, using trigonometric identities, and applying definitions of absolute value and argument.
The document provides solutions to problems from an IIT-JEE 2004 mathematics exam. Problem 1 asks the student to find the center and radius of a circle defined by a complex number relation. The solution shows that the center is the midpoint of points dividing the join of the constants in the ratio k:1, and gives the radius. Problem 2 asks the student to prove an inequality relating dot products of four vectors satisfying certain conditions. The solution shows that the vectors must be parallel or antiparallel.
The document provides information about sets and operations on sets such as union, intersection, complement, difference, properties of these operations, counting theorems for finite sets, and the number of elements in power sets. It defines key terms like union, intersection, complement, difference of sets. It lists properties of union, intersection, and complement. It presents counting theorems for finite sets involving union, intersection. It states that the number of elements in the power set of a set with n elements is 2n and the number of proper subsets is 2n-2.
The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability distributions, matrices, and linear transformations.
The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability, matrices, vectors, and linear transformations.
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -1.
The document contains a multiple choice test on polynomials with 50 questions. The questions cover topics such as identifying polynomials, finding zeroes of polynomials, factorizing polynomials, and evaluating polynomials. The test also includes some word problems involving polynomials. The answers to all 50 questions are provided at the end.
This document contains 30 multiple choice questions related to mathematics. Some of the concepts covered include: properties of complex numbers, sets, arithmetic progressions, trigonometric functions, derivatives, integrals, lines, conic sections, inequalities and vector projections. The questions have 4 possible answer choices labeled a, b, c or d for each problem.
This document provides a review sheet with multiple math problems involving algebra, trigonometry, and calculus concepts. There are 23 problems covering topics such as simplifying rational expressions, solving equations, graphing functions, working with complex numbers, trigonometric identities, and applying trigonometric functions. Answers are provided for selected problems. The review sheet is intended to help students practice and review a wide range of mathematics topics.
The document contains instructions for a math exam for Class XII. It states that the paper contains 50 multiple choice questions (MCQs), and to darken the appropriate circle on the answer sheet. Each correct answer receives 4 marks, incorrect answers receive -1 mark, and unattempted questions receive 0 marks. It wishes the student good luck for their bright future.
The document discusses expanding and factorizing algebraic expressions. It provides examples of expanding expressions using the distributive property, such as expanding (a + b)(c + d) to get ac + ad + bc + bd. It also discusses factorizing expressions by finding common factors, such as factorizing a2 + 2ab + b2 to get (a + b)2. Tips and techniques are presented for expanding, factorizing, finding common factors, and using the distributive property to manipulate algebraic expressions.
This document contains 38 multiple choice questions related to polynomials and their properties. Specifically, it covers topics like:
- Finding zeroes of polynomials
- Relationships between zeroes and coefficients
- Determining the degree and number of zeroes of polynomials
- Factoring polynomials
- Identifying quadratic polynomials based on their zeroes
The document contains 46 mathematical formulae related to algebra, quadratic equations, arithmetic progressions, geometric progressions, factorials, and binomial expansions. Some key formulae include:
1) (a + b)2 = a2 + 2ab + b2 for expanding a binomial square.
2) The quadratic formula for solving ax2 + bx + c = 0 is x = (-b ± √(b2 - 4ac))/2a.
3) The nth term of an arithmetic progression with first term a and common difference d is an = a + (n - 1)d.
4) The nth term of a geometric progression with first term a and common ratio
This document contains solutions to several inequality problems:
1) It proves three inequalities using Cauchy-Schwarz inequality and other techniques.
2) It proves another inequality involving four positive real numbers using Cauchy-Schwarz inequality twice.
3) It proves an inequality about positive real numbers a0, a1, ..., an where ak+1 - ak ≥ 1 for all k using mathematical induction.
1. The given relation R defines a line with points (2,3), (4,2), and (6,1). The range of y-values is {1,2,3}.
2. The two trigonometric equations are equal when xy < 1.
3. The expressions 7A - (I + A)3 and -I are equal after expanding and simplifying the terms.
This document contains questions from assignments in differential calculus, continuity and differentiation, rate of change of quantities, increasing and decreasing functions, tangents and normals, and approximation. It also includes word problems involving optimization such as finding dimensions that result in maximum area, volume, or other quantities. There are over 25 questions in total across these calculus topics.
This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.
IIT JAM MATH 2021 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2021 Question Paper
IIT JAM Preparation Strategy
For any query about exams feel free to contact us
Call - 9836793076
The document contains 30 multiple choice questions related to various mathematical concepts such as relations, functions, matrices, determinants, and trigonometric functions. The questions test knowledge of properties such as equivalence, injectivity, surjectivity, inverse functions, composition of functions, matrix operations, and evaluating determinants.
This document contains 5 questions regarding a mathematics exam. It covers topics like algebra, geometry, calculus, differential equations, and matrices. Some key details:
- The exam has 5 questions worth a total of 80 marks.
- Question 1 has 8 short answer parts worth 16 marks total.
- Questions 2-4 have 4 medium length parts each worth 16 marks total.
- Question 5 has 2 long answer parts worth 16 marks total.
- The questions cover topics such as finding GCDs, eigenvalues, limits, differential equations, and geometry concepts.
Given a document describing functions and relations:
- It defines four types of relations: one-to-one, many-to-one, one-to-many, and many-to-many. Relations can be represented using arrow diagrams, ordered pairs, or graphs.
- An example question is worked through to identify the domain, codomain, range, and type of relation for a given set of ordered pairs.
- Functions are represented by lowercase letters like f, g, h. The value of a function f at x is written as f(x).
- Examples demonstrate evaluating functions by substituting values into the function equation and solving composite functions like fg(x).
1. The document contains 10 questions assessing knowledge of complex numbers and quadratic equations.
2. The questions cover topics like solving complex quadratic equations, operations on complex numbers, expressing complex numbers in polar form, and properties of complex functions and arguments.
3. The answers provided solve each question and explain steps like rationalizing expressions, taking conjugates, using trigonometric identities, and applying definitions of absolute value and argument.
The document provides solutions to problems from an IIT-JEE 2004 mathematics exam. Problem 1 asks the student to find the center and radius of a circle defined by a complex number relation. The solution shows that the center is the midpoint of points dividing the join of the constants in the ratio k:1, and gives the radius. Problem 2 asks the student to prove an inequality relating dot products of four vectors satisfying certain conditions. The solution shows that the vectors must be parallel or antiparallel.
The document provides information about sets and operations on sets such as union, intersection, complement, difference, properties of these operations, counting theorems for finite sets, and the number of elements in power sets. It defines key terms like union, intersection, complement, difference of sets. It lists properties of union, intersection, and complement. It presents counting theorems for finite sets involving union, intersection. It states that the number of elements in the power set of a set with n elements is 2n and the number of proper subsets is 2n-2.
The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability distributions, matrices, and linear transformations.
The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability, matrices, vectors, and linear transformations.
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -1.
The document contains a multiple choice test on polynomials with 50 questions. The questions cover topics such as identifying polynomials, finding zeroes of polynomials, factorizing polynomials, and evaluating polynomials. The test also includes some word problems involving polynomials. The answers to all 50 questions are provided at the end.
This document contains 30 multiple choice questions related to mathematics. Some of the concepts covered include: properties of complex numbers, sets, arithmetic progressions, trigonometric functions, derivatives, integrals, lines, conic sections, inequalities and vector projections. The questions have 4 possible answer choices labeled a, b, c or d for each problem.
This document provides a review sheet with multiple math problems involving algebra, trigonometry, and calculus concepts. There are 23 problems covering topics such as simplifying rational expressions, solving equations, graphing functions, working with complex numbers, trigonometric identities, and applying trigonometric functions. Answers are provided for selected problems. The review sheet is intended to help students practice and review a wide range of mathematics topics.
The document contains instructions for a math exam for Class XII. It states that the paper contains 50 multiple choice questions (MCQs), and to darken the appropriate circle on the answer sheet. Each correct answer receives 4 marks, incorrect answers receive -1 mark, and unattempted questions receive 0 marks. It wishes the student good luck for their bright future.
1. The height of the hill is equal to 1/2tanα, where α is the angle of elevation of the top of the hill from each of the vertices A, B, and C of a horizontal triangle.
2. The solutions of the equation 4cos^2x + 6sin^2x = 5 are x = π/4 + nπ/2, where n is any integer.
3. A four-digit number formed using the digits 1, 2, 3, 4, 5 without repetition has a 1/3 probability of being divisible by 3.
This document contains a 30 question objective test on various mathematics topics. The questions cover topics like trigonometry, algebra, relations, functions, polynomials, and integers. Test takers have 60 minutes to complete the 30 questions online. The questions require calculating values, identifying true statements, finding remainders, and factoring polynomials.
09 p.t (straight line + circle) solutionAnamikaRoy39
This document provides solutions to mathematical problems involving straight lines and circles. Some key details:
- Problem 1 involves solving a system of linear equations to find the value of a variable in the first quadrant.
- Problem 2 describes the family of lines passing through a given point and the equation of the angle bisector.
- Problem 3 discusses how the area between a circle and line is maximized when the line acts as the diameter.
Level # 1 51
Level # 2 30
Level # 3 24
Level # 4 12
Total no. of questions 117
LEVEL-1
Equation and properties of the ellipse
Q.1 The equation to the ellipse (referred to its axes as the axes of x and y respectively) whose foci are(± 2,0) and eccentricity1/2,is-
Q.9 The equation of the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point (– 3, 1) and has eccentricity
2
5 , is-
(A) 3x2 + 6y2 = 33 (B) 5x2 + 3y2 = 48
x 2 y2
x 2 y2
(C) 3x2 + 5y2 –32 = 0 (D) None of these
(A) 12 16 = 1 (B) 16
x 2 y2
12 = 1
Q.10 Latus rectum of ellipse
4x2 + 9 y2 – 8x – 36 y + 4 = 0 is-
(C) 16 8 = 1 (D) None of these 5
Q.2 The eccentricity of the ellipse
(A) 8/3 (B) 4/3 (C) 3
(D) 16/3
9x2 + 5y2 – 30 y = 0 is-
(A) 1/3 (B) 2/3
(C) 3/4 (D) None of these
Q.3 If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is-
(A) 3/2 (B) 3 /2
(C) 2/3 (D) 2 /3
Q.4 If distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is-
(A) 1/2 (B) 2/3
Q.11 The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is-
(A) x2 + 2y2 = 100 (B) x2 + 2 y2 =10
(C) x2 – 2y2 = 100 (D) None of these
Q.12 If the distance between the foci of an ellipse be equal to its minor axis, then its eccentricity is-
(A) 1/2 (B) 1/ (C) 1/3 (D) 1/
Q.13 The equation 2x2 + 3y2 = 30 represents-
(A) A circle (B) An ellipse
(C) 1/
(D) 4/5
(C) A hyperbola (D) A parabola
Q.5 The equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents an ellipse if-
(A) = 0, h2 < ab (B) 0, h2 < ab
(C) 0, h2 > ab (D) 0, h2 = ab
Q.14 The equation of the ellipse whose centre is (2,– 3), one of the foci is (3,– 3) and the corresponding vertex is (4,– 3) is-
Q.6 Equation of the ellipse whose focus is (6,7) directrix is x + y + 2 = 0 and e = 1/ 3 is-
(A) 5x2 + 2xy + 5y2 – 76x – 88y + 506 = 0 (B) 5x2 – 2xy + 5y2 – 76x – 88y + 506 = 0 (C) 5x2 – 2xy + 5y2 + 76x + 88y – 506 = 0
(x 2)2
(A) 3
(x 2)2
(B)
4
(y 3)2
+ = 1
4
(y 3) 2
+ 3 = 1
(D) None of these
Q.7 The eccentricity of an ellipse
x2
a 2 +
y2
b2 = 1 whose
x 2
(C) 3
+ y = 1
4
latus rectum is half of its major axis is-
1
(D) None of these
Q.15 Eccentricity of the ellipse
(A) 2
(B)
4x2 +y2 – 8x + 2y+ 1= 0 is-
(C)
3 (D) None of these
2
(A) 1/ 3 (B) 3/2
(C) 1/2 (D) None of these
Q.8 The equation of the ellipse whose centre is at origin and which passes through the points (– 3,1) and (2,–2) is-
(A) 5x2 + 3y2 = 32 (B)3x2 + 5y2 = 32
(C) 5x2 – 3y2 = 32 (D) 3x2 + 5y2 + 32= 0
Q.16 The equation of ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is-
(A) 5x2 – 9y2 = 180 (B) 9x2 + 5y2 = 180
(C) x2 + 9y2 =180 (D) 5x2 + 9y2 = 180
This document contains questions from 5 units of a mathematics exam for an automotive engineering course. The questions cover topics like differential equations, vector calculus, complex variables, and Laplace transforms. Some sample questions include solving differential equations using variation of parameters, finding divergence and curl of vector fields, determining if functions are analytic or harmonic, and calculating Laplace transforms. The document provides 10 multiple choice questions for each of the 5 units to total 50 questions for the exam.
The document contains 49 multiple choice questions related to number systems in mathematics for Class IX. The questions cover topics like rational and irrational numbers, operations on fractions and decimals, and properties of integers. A YouTube channel for online math lectures is provided. The questions are divided into two levels, with Level II containing more complex questions.
x
y
2.5 3.0 3.5
-1.0 6 7 8
1.0 0 1 2
3.0 -6 -5 -4
MATH 223
FINAL EXAM REVIEW PACKET ANSWERS
(Fall 2012)
1. (a) increasing (b) decreasing
2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis with radius 5.
(b) 3x = , 3x = − . These are vertical planes parallel to the yz-plane.
(c) 2 2 2z x y= + . This is a cone (one opening up and one opening down) centered on the z-axis.
3. There are many possible answers.
(a) 0x = produces the curve 23y z= − .
(b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − − .
(c)
2
x
π
= produces the curves 3z = and 3z = − .
4. (a) (b) (i) 1 (ii) Increase (iii) Decrease
5. (a) Paraboloids centered on the x-axis, opening up in the positive x direction. 2 2x y z c= + +
(b) Spheres centered at the origin with radius 1 ln c− for 0 c e< ≤ . 2 2 2 1 lnx y z c+ + = −
6. (a) 6 am 11:30 am
(b) Temperature as a function of time at a depth of 20 cm.
(c) Temperature as a function of depth at noon.
7. ( , ) 2 3 2z f x y x y= = − −
8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI (e) I, V
9. (a)
12
4 12
5
z x y= − + (b) There are many possible answers.
12
4
5
i j k+ −
(c)
3 569
2
10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f) i, ix
11. There are many possible answers.
(a) ( )5 4 3
26
i j k− +
or ( )5 4 3
26
i j k− − +
(b) 2 3i j− +
(c)
4
cos
442
θ = , 1.38θ ≈ radians (d) ( )4 4 3
26
i j k− +
(e) 4 11 17i j k− − −
12. (a)
3
5
a = − (b)
1
3
a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)
1 2 , 2 , 3 3x t y t z t= + = − − = +
13. 6 39i
or 6 39i−
14. (a)
( )
2
23 2 2
3 2
3 1
z x y x
x x y x y
∂
= −
∂ + + +
(b)
( )4
10 4 3
5
H
H T
f
H
+ +
=
−
(c)
2
2 2
1 1z
x y y x
∂
= − −
∂ ∂
15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3) 8( 3) 6( 6) 0x y z− + − + − =
16. (a)
2sin(2 ) cos(2 )
5 5
v v
ds dv d
α α
α= +
(b) The distance s decreases if the angle α increases and the initial speed v remains constant.
(c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.
17. (a) The water is getting shallower.
4
( 1, 2)
17
uh − = −
(b) There are many possible answers. 3i j+
(c) 72 ft/min
18. (a)
( )
2 2 2
22 2 22
2 2
1 1 11
yz xyz z yz
grad i j k
x x xx
= − + + + + + +
(b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = + −
(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k x x x y y e+ + = − + +
(d)
37
3
(e) ( , , ) sin zg x y z xy e c= + +
19. ( , ) 4 3vG a b = −
20. (a) positive (b) negative (c) negative (d) negative (e) positive (f) zero
21. (a)
(.
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1. TEST PAPER 1
Total Questions: 75 Time allotted 90 minutes
1. The set of all integers x such that |x – 3| < 2 is equal to
(a) {1, 2, 3, 4, 5} (b) {1, 2, 3, 4}
(c) {2, 3, 4} (d) {-4, -3, -2}
2. The Range of the function f(x) = x 2
−
−
2 x
is
(a) R (b) R – {1}
(c) (-1) (d) R – {-1}
3. The value of (i)i is
(a) ω (b) ω2
(c) e-π/2 (d) 2√2
4. ( )
( )
4
5
cos isin
icos sin
θ + θ
θ + θ
is equal to
(a) cos− isin θ (b) cos9θ − isin9θ
(c) sin θ − icosθ (d) sin 9θ − icos9θ
5. The roots of the quadratic equation ax2 + bx + c = 0 will be reciprocal to each other if
(a) a = 1/c (b) a = c
(c) b = ac (d) a = b
6. If α, β are the roots of ax2 − 2bx + c = 0 then α3β3 + α2β3 + α3β2 is
2 ( )
(a) c c 2b
+ (b)
3
a
3
3
bc
a
(c)
2
3
c
a
(d) None of these
7. The sixth term of a HP is 1/61 and the 10th term is 1/105. The first term of the H.P. is
(a) 1/39 (b) 1/28
(c) 1/17 (d) 1/6
8. Let Sn denote the sum of first n terms of an A.P.. If S2n = 3Sn, then the ratio S3n / 5n is equal to
(a) 4 (b) 6
(c) 8 (d) 10
9. Solution of |3 – x| = x – 3 is
(a) x < 3 (b) x > 3
(c) x > 3 (d) x < 3
10. If the product of n positive numbers in 1, then their sum is
(a) a positive integer (b) divisible by n
(c) equal to n 1
+ (d) never less than n
n
11. A lady gives a dinner party to six quests. The number of ways in which they may be selected from
among ten friends, if two of the friends will not attend the party together is
2. (a) 112 (b) 140
(c) 164 (d) None of these
12. For 1 ≤ r ≤ n, the value of n 1 n 2 r
r r r nCr + − C + − C + _ _ _ + C is
(a) r 1 nC + (b) n 1
r + C
(c) n 1
+ (d) None of them.
r 1 + C
13. 2.42n+1 + 33n+1 is divisible by
(a) 2 (b) 9
(c) 11 (d) 27
P
P
14. If Pn denotes the product of the binomial coefficients in the expansions of (1 + x)n, the n 1
+ equals
n
+ (b)
(a) n !
n!
nn
n!
(c) ( )n 1 n 1
+ + (d) ( )
n!
n 1
+ +
+
( )
!
n 1
n 1
15. If x is very large and n is a negative integer or a proper fraction, then an approximate value of
n 1 x
x
⎛ + ⎞
⎜ ⎟
⎝ ⎠
is
(a) 1 x
+ (b) 1 n
n
x
+
+ (d) n 1 1
(c) 1 1
x
⎛ + ⎞ ⎜ ⎝ x
⎟
⎠
16. If 4 log93 + 9 log24 = 10log x 83, (x ∈ R)
(a) 4 (b) 9
(c) 10 (d) None of these
17. The sum of the series 2 2 2
4 8 16 log − log + log _ _ _ _ to∞is
(a) e2 (b) loge2 + 1
(c) loge3 – 2 (d) 1 – loge2
18. tan 5x tan 3x tan2x is equal to
(a) tan5x − tan3x − tan 2x (b) sin5x sin3x sin 2x
− −
− −
cos5x cos3x cos2x
(c) 0 (d) None of these
19. If a = tan60 tan 420 and B = cot660 cot 780
(a) A = 2B (b) A 1 B
3
=
(c) A = B (d) 3A = 2B.
20. The value of cos 2 cos 4 cos 6
π π π
+ + is
7 7 7
(a) 1 (b) -1
(c) 1/2 (d) -1/2
21. If tan 1 andsin 1 ,
α = β = where 0 ,
7 10
π
< α β < , then 2β is equal to
2
3. (a)
π
− α (b) 3
4
π
− α
4
(c)
π
− α (d) 3
8
π π
−
8 2
22. If sin θ + cosθ = 2 sin θ , then
(a) 2 cosθ (b) − 2 sin θ
(c) − 2 cosθ (d) None of these
23. Value of
2 0 4 0
4 0 2 0
sin 20 +
cos 20
sin 20 +
cos 20
is
(a) 1 (b) 2
(c) ½ (d) None of these
24. Value of 32cos6 200 − 48cos4 200 +18cos2 200 −1 is
(a) 1
− (b) 1
2
2
(c) 3
2 (d) None of these
25. If sinθ + cosecθ = 2 , then value of sin3 θ + cosec3θ is
(a) 2 (b) 4
(c) 6 (d) 8
26. If cosecθ + cot θ = 5 2 , then the value of tanθ is
(a) 15
16 (b) 21
20
(c) 15
21 (d) 20
21
27. General value of x satisfying the equation 3sin x + cos x = 3 is given by
(a) n
π
π ± (b) ( )n n 1
6
π π
π + − +
4 3
(c) n
π
π ± (d) ( )n n 1
3
π π
π + − −
3 6
28. If length of the sides AB, BC and CA of a triangle are 8cm, 15 cm and 17 cm respectively, then
length of the angle bisector of ∠ABC is
(a) 120 2 cm
23
(b) 60 2 cm
23
(c) 30 2cm
23
(d) None of these
29. A man from the top of a 100 metre high tower sees a car moving towards the tower at an angle of
depression of 300. After sometimes, the angle of depression becomes 600. The distance (in metres)
traveled by the car during this time is
(a) 100 3 (b) 200 3
3
(c) 100 3
3
(d) 200 3
4. 30. The shadow of a tower of height (1+ 3) metre standing on the ground is found to be 2 metre
longer when the sun’s elevation is 300, then when the sun’s elevation was
(a) 300 (b) 450
(c) 600 (d) 750
− ⎛ π ⎞
⎜ ⎟
⎝ ⎠
31. cos 1 cos 5
4
is equal to
−π (b) 4
(a) 4
π
(c) 3
π (d) 5
4
4
π
32. If cos 1 x cos 1 y
− − π
+ = , then value of
2 3 6
x2 xy y2
4 2 3 9
− + is
(a) 3
4 (b) 1
2
(c) 1
4 (d) None of these
33. The distance between the lines 4x + 3y = 11 and 8x + 6y = 15 is
(a) 7/2 (b) 7/3
(c) 7/5 (d) 7/10
34. The straight lines x + y – 4 = 0, 3x + y – 4 = 0, x + 3y – 4 = 0 form a traigle which is
(a) isosceles (b) right angled
(c) equilateral (d) None of these
35. Incentre of the triangle whose vertices are (6, 0) (0, 6) and (7, 7) is
(a) 9 , 9
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
(b) 7 , 7
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
(c) 11,11
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
(d) None of these
36. The area bounded by the curves y = |x| − 1 and y = − |x| + 1 is
(a) 1 (b) 2
(c) 2 2 (d) 4
37. The coordinates of foot of the perpendicular drawn from the point (2, 4) on the line x + y = 1 are
(a) 1 , 3
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
(b) 1, 3
⎛ − ⎞
⎜ ⎝ 2 2
⎟
⎠
(c) 3 , 1
⎛ − ⎞
⎜ ⎝ 2 2
⎟
⎠
(d) 1, 3
⎛ − − ⎞
⎜ ⎝ 2 2
⎟
⎠
38. Three lines 3x + 4y + 6 = 0, 2x + 3y + 2 2 = 0 and 4x + 7y + 8 = 0 are
(a) Parallel (b) Sides of a triangles
(c) Concurrent (d) None of these
39. Angle between the pair of straight lines x2 – xy – 6y2 – 2x + 11y – 3 = 0 is
(a) 450, 1350
(b) tan-1 2, π = tan-1 2
(c) tan-1 3, π = tan-1 3
(d) None of these
5. 40. If a circle passes through the point (a, b) and cuts the circle x2 + y2 = 4 orthogonally, then locus of
its centre is
(a) 2ax + 2by + (a2 + b2 + 4) = 0
(b) 2ax + 2by − (a2 + b2 + 4) = 0
(c) 2ax − 2by + (a2 + b2 + 4) = 0
(d) 2ax − 2by − (a2 + b2 + 4) = 0
41. Centre of circle whose normals are x2 − 2xy − 3x + 6y = 0 is
(a) 3, 3
⎛ ⎞
⎜ ⎝ 2
⎟
⎠
(b) 3 ,3
⎛ ⎞
⎜ ⎝ 2
⎟
⎠
(c) 3, 3
⎜ ⎛− ⎞ ⎝ 2
⎟
⎠
(d) 3, 3
⎛ − ⎜− ⎞ ⎝ 2
⎟
⎠
42. Centre of a circle is (2, 3). If the line x + y = 1 touches, its equation is
(a) x2 + y2 − 4x − 6y + 4 = 0
(b) x2 + y2 − 4x − 6y + 5 = 0
(c) x2 + y2 − 4x − 6y − 5 = 0
(d) None of these
43. The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle x2 + y2 = 9 is
(a) 3 , 1
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
(b) 1 , 3
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
(c) 1 , 1
⎛ ⎞
⎜ ⎟
⎝ 2 2
⎠
2 1, 2
2
⎛ − ⎞ ⎜ ⎟
⎝ ⎠
(d) 1
44. The line y = mx + 1 is a tangent to the parabola y2 = 4x if
(a) m = 1 (b) m = 2
(c) m = 3 (d) m = 4
45. The angle between the tangents drawn from the origin to the parabola y2 = 4a (x – a) is
(a) 900 (b) 300
(c) tan 1 1
− ⎛ ⎞
⎜ ⎝ 2
⎟
⎠
(d) 450
46. The area of the triangle formed by the tangent and the normal to the parabola y2 = 4ax, both drawn
at the same end of the latus rectum and the axis of the parabola is
(a) 2 2 a2 (b) 2a2
(c) 4a2 (d) None of these
47. The eccentricity of the eclipse 16x2 + 7y2 = 112 is
(a) 4/3 (b) 7/16
(c) 3/ 17 (d) 3/4
48. A common tangent to the circle x2 + y2 = 16 and an ellipse
x2 y2 1
49 4
+ = is
(a) y = x + 4 5 (b) y = x + 53
(c) y 2 x 4 4
= + (d) None of these
11 11
6. 49. If the hyperbolas x2 − y2 = a2 and xy = c2 are of equal size, then
(a) c2 = 2a2 (b) c = 2a
(c) 2c2 = a2 (d) none of these
50. If a circle cuts rectangles hyperbola xy = 1 in the point (xi, yi), i = 1, 2, 3, 4 then
(a) 1 2 3 4 x x x x = 0 (b) 1 2 3 4 y y y y =1
(c) 1 2 3 4 y y y y = 0 (d) 1 2 3 4 x x x x = −1
51. If
a b 0
0 a b
b 0 a
= 0 then
(a) a is a cube root of 1 (b) b is a cube root of 1
(c) a/b is a cube root of 1 (d) a/b is a cube roots of -1
52. If 1 1 1 0
+ + = , then
a b c
1 +
a 1 1
1 1 +
b 1
1 1 1 +
c
is equal to
(a) 0 (b) abc
(c) –abc (d) None of these
53. The determinant
cos( ) sin( ) cos2 B
sin cos sin
α + β − α + β
α α β
− α α β
cos sin cos
is independent of
(a) α (b) β
(c) α and β (D) Neither α nor β
54. If 3 4
A
⎡ − ⎤
= ⎢ ⎣ 1 − 1
⎥ ⎦
, the value of An
(a) 3n 4n
⎡ − ⎤
⎢ ⎣ n n
⎥
⎦
(b) 2 n 5 n
⎡ + − ⎤
⎢ ⎣ n − n
⎥ ⎦
(c)
( )
( )
⎡ 3 n − 4
n
⎤
⎢ ⎥
⎢⎣ 1 − 1
n
⎥⎦
(d) None of these
f x 1
55. The domain of the function ( ) 2
x 3x 2
=
− +
is
(a) (−∞,1)∪(2,∞) (b) (−∞,1]∪[2,∞)
(c) [−∞,1)∪(2,∞] (d) (1, 2)
sin ( π⎡⎣ x 2 + 1
⎤⎦
)
56. Range of function 4
x +
1
is
(a) 0 (b) {0}
(c) [-1, 1] (d) (0, 1)
57.
3
−
lim 1 cot x
→π 2 cot x cot x
x 3 4
− −
is
(a) 11
4 (b) 3
4
(c) 1
2 (d) None of these
7. limsec sin x
58. 1
x 0
x
−
→
⎛ ⎞ = ⎜ ⎟
⎝ ⎠
(a) 1 (b) 0
(c) 2
π (d) Does not exist
59. The function y = 3 x − x −1 is continuous
(a) x < 0 (b) x > 1
(c) no point (d) None of these
60. The function ( ) 0,x is irrational is
f x
1,x is rational
⎛
=⎜⎝
(a) continuous at x = 1 (b) discontinuous only at 0
(c) discontinuous only at 0, 1
(d) discontinuous everywhere
61. Let f : R → R be a function defined by f(x) = max. {x, x3}. The set of all points where f(x) is not
differentiable is
(a) {-1, 1} (b) {-1, 0}
(c) {0, 1} (d) {-1, 0, 1}
⎧⎪ ≠ ⎫⎪ =⎨ ⎬
⎩⎪ = ⎪⎭
62. If the function ( ) ( )1
cos x x ,x 0 f x
K x 0
is continuous of x = 0 then value of k is
(a) 1 (b) -1
(c) 0 (d) e
63.
1 +
x5dx
1 x
∫
=
+ (a) 1− x + x2 − x3 + x4 + c (b)
x − x2 + x3 − x4 + x5 +
x
2 3 4 5
(c) ( )5 1+ x + C (d) None of these
64. ∫ x x dx
(a)
x3
3
(b)
x2 x
3
(c)
x2 x
2
(d) None of these
65.
1
∫
+ 1
(a) 1 (b) 2
(c) 0 (d) -1
x 2
dx
+
x 2 −
=
π
2
∫
(a) 4
66. ( )
0
log tan x dx
π (b) π
2
(c) 0 (d) 1
8. 67. If a < 0 < b, then
b
∫
x
dx
x a
(a) a – b (b) b – a
(c) a + b (d) –a – b
68.
2
∫ x 2
x dx
0
(a) 5/3 (b) 7/3
(c) 8/3 (d) 4/3
π
+ ∫
(a) 2
69. 2
0
xsin x dx
1 cos x
π (b) 2
8
4
π
(c) 3
π (d) 4
8
8
π
70. The area bounded by curve y = 4x – x2 and x – axis is
(a) 30 sq. units.
7
(b) 31 sq. units.
7
(c) 32 sq. units.
3
(d) 34 sq. units.
3
71. The area bounded by the curves y = |x| - 1 and y = -|x| + 1 is
(a) 1 (b) 2
(c) 2 2 (d) 4
72. The area bounded by the curves y = x4 − 2x3 + x2 − 3 , the x-axis and the two ordinates
corresponding to the points of minimum of this Function is
(a) 91/15 (b) 91/30
(c) 19/30 (d) None of these
73. Degree of the differential equation
2 3
⎛ ⎞
4 d y
⎛ 2 ⎞ 5 ⎜ ⎝ 2 ⎟ ⎠ 3
⎜ ⎟
+ + = − ⎝ ⎠
d y dx d y x 2
1
dx 2 d 3 y dx
3
3
dx
, then
(a) m = 3, n = 3 (b) m = 3, n = 2
(c) m = 3, n = 5 (d) m = 3, n = 1
74. A solution of the differential equation
2 dy x. dy y 0
dx dx
⎛ ⎞ − + = ⎜ ⎟
⎝ ⎠
is
(a) y = 2 (b) y = 2x
(c) 4y = x2 + c (d) y = 2x2 – 4
r r
75. The area (in square units) of the parallelogram whose diagonals are a = ˆi + ˆj− 2kˆ and b = ˆi − 3ˆj+ 4kˆ
(a) 14 (b) 2 14
(c) 2 6 (d) 38