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Algebra Questions with Answers and Solutions - Grade 12

Grade 12 algebra questions with answers and solutions are presented. Some of these questions may be challenging; you need to spend time on them as these are the ones that make you think and learn how to solve problems. Also group work on challenging questions is an excellent opportunity to interact with others and learn from them. Let me know of any other possible solutions to any of the questions below.

• Show that the 3 by 3 system of equations given below has no solutions. 2 x + y - 3z = 5 -5 x + 3 y + 2 z = 7 3 x - 4 y + z = 8

Solutions to the Above Problems

• 25 100 2 300 = (2 3 ) 100 = 8 100 3 400 = (3 4 ) 100 = 81 100 4 200 = (4 2 ) 100 = 16 100 2 600 = (2 6 ) 100 = 64 100 from greatest to least: 3 400 , 2 600 , 25 100 , 4 200 , 2 300
• P(x) = x 3 - 7x + 6 : given leading coefficient 1 and its factors are : +1,-1 constant term is 6 and its factors are : +1,-1,+2,-2,+3,-3,+6,-6 possible rational zeros : +1,-1,+2,-2,+3,-3,+6,-6 test : P(1) = 0, P(2) = 0 and P(-3) = 0 x = 1, x = 2 and x = -3 are the zeros of P(x).
• From the graph, x = -3 is a zero of multiplicity 2, x = 0 is a zero of multiplicity 1 and x = 2 is a zero of multiplicity 2. P(x) = -x (x + 3) 2 (x - 2) 2 : polynomial with real zeros hence with lowest degree.
• ) if 2 - i is a zero and the coefficients of the polynomial are real then 2 + i (the conjugate) is also a solution. P(x) = (x - (2 - i))(x - (2 + i))×q(x) = ((x - 2) 2 + 1)×q(x) q(x) = P(x)/((x - 2) 2 + 1) = (x 2 - 2) x = 2 - i , x = 2 + i , x = √2 and x = - √2 are the 4 zeros of P(x).
• f(x) = a (x + 2) 2 + 1 : equation of parabola in vertex form f(0) = -3 = 4 a + 1 a = -1 : solve for a f(x) = -(x + 2) 2 + 1 = -x 2 - 4x - 3 a = -1 , b = -4 and c = -3 : identify coefficients
• f(x) = a x 2 + b x + c f(1) = 3 which give 3 = a + b + c f(5) = 3 which gives 3 = 25a + 5b + c 24 a + 4 b = 0 : subtract equation B from equation C x coordinate of vertex = -b/2a = 3 : from above equation
• The oblique asymptote is the quotient resulting from the long division of ax 4 + bx 3 + 3 by x 3 - 2 The quotient obtained is ax + b a x + b = 2 x - 3 a = 2 and b = -3 : for two polynomials to be equal, the corresponding coefficients has to be equal.
• log 9 (x 3 ) = log 2 (8) : given log 2 (2 3 ) = 3 : simplify right hand side of given equation. log 9 (x 3 ) = 3 : rewrite the above equation log 9 (x 3 ) = log 9 (9 3 ) : rewite 3 as a log base 9. x 3 = 9 3 : obtain algebraic equation from eqaution D. x = 9 : solve above for x.
• log x (y 3 ) = 2 : given x 2 = y 3 : rewrite in exponential form x 4 = y 6 : square both sides x 4 = y 6 : rewrite the above using the log base y log y (x 4 ) = log y (y 6 ) = 6
• log x (8 e 3 ) = 3 : given x 3 = 8 e 3 = (2e) 3 x = 2 e
• 16 x + 16 x - 1 = 10 : given 4 2x + 4 2x / 16 = 10 4 2x = 160/17 : solve for 4 2x 4 x = 4 √(10) / √(17) : extract the square root 2 2x = 4 x = 4 √(10) / √(17)
• a 2 - b 2 = 8 : given a 4 + b 4 - 2a 2 b 2 = 8 2 : square both sides and expand. a×b = 2 : given a 2 b 2 = 2 2 : square both sides. a 4 + b 4 - 2(4) = 8 2 : substitute a 4 + b 4 = 72
• -1 ? sin(2x - ?/3) ? 1 : range of a sine function -2 ? 2sin(2x - ?/3) ? 2 : multiply all terms of the double inequality by 2 -2 - 5 ? 2sin(2x - ?/3) - 5 ? 2 - 5 : add -5 to all terms of the inequality. -7 ? 2sin(2x - ?/3) - 5 ? -3 3 ? |2sin(2x - ?/3) - 5| ? 7 : change the above using absolute value. 3 + 3 ? |2sin(2x - ?/3) - 5| + 3 ? 7 + 3 : add 3 to all terms of the double inequality. The maximum value of f(x) is equal to 10 and the minimum value of f(x) is equal to 6.
• If x < -7 then x < - 3 and x + 3 < 0 and |3 + x| = -(3 + x) |4 - |3 + x|| = |4 + 3 + x| = |x + 7| = - (x + 7) = - x - 7 : since x + 7 < 0
• Let d be the distance between A and B T1 = d / 50 : travel time from A to B Let S be the speed from B to A T2 = d/S : travel time from B to A 60 = 2d/(T1 + T2) : average speed for the whole trip 60 = 2d/(d/50 + d/S) : substitute T1 and T2 S = 75 km/hour : solve the above equation for S.
• x 2 - y 2 = (x - y)(x + y) = -12 : given 6(x - y) = -12 : substitute x + y by 6 (x - y) = -2 : solve for x - y (x - y) = -2 and x + y = 6 : 2 by 2 system. x = 2 , y = 4 : solve above system.
• f(x) + 3f(8 - x) = x : given f(2) + 3f(6) = 2 : x = 2 above f(6) + 3f(2) = 6 : x = 6 above f(6) = 6 - 3f(2) : solve equation C for f(6) f(2) + 3(6 - 3f(2)) = 2 : substitute f(2) = 2 : solve above equation.
• f(2x + 1) = 2f(x) + 1 : given f(3) = 2f(1) + 1 : x = 1 in A f(1) = 2f(0) + 1 : x = 0 in A f(3) = 11 : substitute
• x 2 + y 2 = 4 : given x 2 + (2x + b) 2 = 4 : substitute y by 2x + b 5x 2 + 4bx + b 2 - 4 = 0 The number of points of intersection is given by the number of solutions of the above equation. The line and circle are tangent if the above quadratic equation has only one solution which means that the discriminant is equal to zero. Find the discriminant as a function of b and solve. b = √2 and b = -√2 : 2 solutions.
• (x 100 - x 99 - x + 1) / (x 2 - 3x + 2) Let P(x) = x 100 - x 99 - x + 1 , D(x) = x 2 - 3x + 2 The division of the two polynomials may be written as P(x) = D(x) Q(x) + r(x) , where Q(x) is the quotient and r(x) is the remainder that will have a degree equal to one or lower. r(x) = a x + b We now need to find a and b that define the remainder. Note that D(x) may be factored as follows: D(x) = x 2 - 3x + 2 = (x - 1)(x - 2) Hence: P(x) = (x - 1)(x - 2) Q(x) + a x + b Using the zeros of D(x) to write: P(1) = (1 - 1)(1 - 2) Q(1) + a (1) + b gives a + b = P(1) P(2) = (2 - 1)(2 - 2) Q(2) + a (2) + b gives 2 a + b = P(2) We now need to evaluate P(1) and P(2) P(1) = 1 100 - 1 99 - (1) + 1 = 0 First rewrite P(x) = x 99 (x - 1) - x + 1 ; Hence P(2) = 2 99 (2 - 1) - 2 + 1 = 2 99 - 1 We now have a system of equations to solve and find a and b. a + b = 0 and 2 a + b = 2 99 - 1 a = 2 99 - 1 and b = 1 - 2 99 remainder: r(x) = (2 99 - 1) x + 1 - 2 99
• Let y = √(1/3 + √(1/3 + √(1/3 + ...))). square both sides to obtain: y 2 = 1/3 + √(1/3 + √(1/3 + √(1/3 + ...))) We can write: y 2 = 1/3 + y Solve the above quadratic equation to obtain: y = (3 + √(21)) / 6 and y = (3 - √21) / 6 y is positive hence the solution: √(1/3 + √(1/3 + √(1/3 + ...))) = y = (3 + √(21)) / 6

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12th Grade Math Worksheets

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Easily download and print our 12th grade math worksheets. Click on the free 12th grade math worksheet you would like to print or download. This will take you to the individual page of the worksheet. You will then have two choices. You can either print the screen utilizing the large image loaded on the web page or you can download the professional print ready PDF file. The quality of your printable 12th grade math worksheet will be pristine with the PDF version of the worksheet. We recommend downloading the PDF file.

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CBSE Worksheets for Class 12 Maths

CBSE Worksheets for Class 12 Maths: One of the best teaching strategies employed in most classrooms today is Worksheets. CBSE Class 12 Maths Worksheet for students has been used by teachers & students to develop logical, lingual, analytical, and problem-solving capabilities. So in order to help you with that, we at WorksheetsBuddy have come up with Kendriya Vidyalaya Class 12 Maths Worksheets for the students of Class 12. All our CBSE NCERT Class 12 Maths practice worksheets are designed for helping students to understand various topics, practice skills and improve their subject knowledge which in turn helps students to improve their academic performance. These chapter wise test papers for Class 12 Maths will be useful to test your conceptual understanding.

Board: Central Board of Secondary Education(www.cbse.nic.in) Subject: Class 12 Maths Number of Worksheets: 42

CBSE Class 12 Maths Worksheets PDF

All the CBSE Worksheets for Class 12 Maths provided in this page are provided for free which can be downloaded by students, teachers as well as by parents. We have covered all the Class 12 Maths important questions and answers in the worksheets which are included in CBSE NCERT Syllabus. Just click on the following link and download the CBSE Class 12 Maths Worksheet. CBSE Worksheets for Class 12 Maths can also use like assignments for Class 12 Maths students.

CBSE Class 12 Definite Integrals Worksheets

• CBSE Class 12 Definite integrals Assignment 1
• CBSE Class 12 Definite integrals Assignment 2
• CBSE Class 12 Definite integrals Assignment 3
• Class 12 Maths CBSE Worksheets for Definite integrals Assignment 4
• CBSE Class 12 Maths Worksheets Definite integrals Assignment 5
• CBSE Worksheets for Class 12 Maths Definite integrals Assignment 6
• CBSE Maths Worksheets for Class 12 Definite integrals Assignment 7

CBSE Determinants Class 12 Worksheets

• CBSE Worksheets for Class 12 Maths Determinants Assignment 1
• CBSE Class 12 Maths Determinants Assignment 2
• CBSE Class 12 Maths Determinants Assignment 3
• CBSE Class 12 Maths Determinants Assignment 4
• CBSE Class 12 Maths Determinants Assignment 5
• CBSE Class 12 Maths Worksheets for Determinants Assignment 6
• CBSE  Class 12 Maths Worksheets for Determinants Assignment 7

CBSE Class 12 Integration Worksheets

• CBSE Worksheets for Class 12 Maths Integration Assignment 1
• CBSE Worksheets for Class 12 Maths Integration Assignment 2
• CBSE Worksheets for Class 12 Maths Integration Assignment 3
• CBSE Worksheets for Class 12 Maths Integration Assignment 4
• CBSE Class 12 Maths Integration Assignment 5
• CBSE Class 12 Maths Integration Assignment 6
• CBSE Class 12 Maths Integration Assignment 7

CBSE Class 12 Relations and Functions Worksheets

• CBSE Class 12 Maths Relations and Functions Assignment 1
• CBSE Class 12 Maths Relations and Functions Assignment 2
• CBSE Class 12 Maths Relations and Functions Assignment 3
• CBSE Class 12 Maths Relations and Functions Assignment 4
• CBSE Class 12 Maths Relations and Functions Assignment 5
• CBSE Class 12 Maths Relations and Functions Assignment 6

CBSE Class 12 Maths Linear Programming Worksheets

• CBSE Class 12 Maths Linear Programming Assignment 1
• CBSE Class 12 Maths Linear Programming Assignment 2
• CBSE Class 12 Maths Linear Programming Assignment 3
• CBSE Class 12 Maths Linear Programming Assignment 4

Application of Derivatives CBSE Class 12 Maths Worksheets

• CBSE Worksheets for Class 12 Maths Application of Derivatives

Application of Integrals CBSE Class 12 Maths Worksheets

• CBSE Worksheets for Class 12 Maths Application of Integrals

Continuity and Differentiation Class 12 Worksheets

• CBSE Worksheets for Class 12 Maths Continuity and Differentiability Assignment 01
• CBSE Class 12 Maths Continuity and Differentiability Assignment 02

Differential Equations CBSE Class 12 Mathematics Worksheets

• CBSE Class 12 Differential Equations Assignment 01
• CBSE Class 12 Differential Equations Assignment 02

CBSE Class 12 Inverse Trigonometric Functions Worksheets

• CBSE Class 12 Maths Inverse Trigonometric Functions Assignment 01
• CBSE Class 12 Maths Inverse Trigonometric Functions Assignment 02

Continuity Class 12 Maths Worksheets

• CBSE Class 12 Maths Continuity Assignment

Class 12 CBSE Maths Matrices Worksheets

• CBSE Class 12 Maths Matrices
• CBSE Class 12 Maths Matrices and determinants

Class 12 Vector Algebra CBSE Maths Worksheets

• CBSE Class 12 Maths Vector Algebra Assignment

Advantages of CBSE Class 12 Maths Worksheets

• By practising NCERT CBSE Class 12 Maths Worksheet , students can improve their problem solving skills.
• Helps to develop the subject knowledge in a simple, fun and interactive way.
• No need for tuition or attend extra classes if students practise on worksheets daily.
• Working on CBSE worksheets are time-saving.
• Helps students to promote hands-on learning.
• One of the helpful resources used in classroom revision.
• CBSE Class 12 Maths Workbook Helps to improve subject-knowledge.
• CBSE Class 12 Maths Worksheets encourages classroom activities.

Worksheets of CBSE Class 12 Maths are devised by experts of WorksheetsBuddy experts who have great experience and expertise in teaching Maths. So practising these worksheets will promote students problem-solving skills and subject knowledge in an interactive method. Students can also download CBSE Class 12 Maths Chapter wise question bank pdf and access it anytime, anywhere for free. Browse further to download free CBSE Class 12 Maths Worksheets PDF .

Now that you are provided all the necessary information regarding CBSE Class 12 Maths Worksheet and we hope this detailed article is helpful. So Students who are preparing for the exams must need to have great solving skills. And in order to have these skills, one must practice enough of Class 12 Maths revision worksheets . And more importantly, students should need to follow through the worksheets after completing their syllabus.  Working on CBSE Class 12 Maths Worksheets will be a great help to secure good marks in the examination. So start working on Class 12 Maths Worksheets to secure good score.

CBSE Worksheets For Class 12

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Category Archives: Grade 12

Worksheets for grade 12 students free to teachers to download once they are registered on the site. There are also fully worked out memos. All worksheets are free and follow the South African curriculum. They cover all the major topics in the South African NCS curriculum such as calculus, algebra, functions, trigonometry. New CAPS worksheets to arrive in 2014.

Finance , Math Literacy

Worksheet – running a business.

Euclidean Geometry , General , Technical Maths

Euclidean geometry for technical maths.

This grade 12 worksheet on Euclidean Geometry for technical maths students focuses on similarity, congruency and triangles. It pays particular attention to proportion as well as proving similarity and congruency.   Download here: Worksheet 12: Euclidean Geometry (Similarity) Worksheet 12 Memorandum: Euclidean Geometry (Similarity). 

Analytical Geometry , Maths

Worksheet 12 analytical geometry – grade 12 maths.

Chapters , Maths , News , Patterns , Patterns, Functions and Algebra

Revamped maths study guide chapters.

Calculus , General , Grade 12 , Maths

Worksheet 11: calculus part 2.

Data Handling , General , Grade 12 , Grades , Math Literacy , Statistics

Worksheet 5: data handling (2).

This grade 12 mathematical literacy worksheet gives students practice question on data handling including analysing bar graphs, and pie charts, drawing their own graphs and answering questions about the data and scale of the graphs.   Worksheet 5: Data Handling (2) Worksheet 5 Memorandum: Data Handling (2)

Finance , General , Grade 12 , Grades , Math Literacy

Worksheet 4: finance – financial documents and tariff systems, exams , general , grade 12 , grades , maths, grade 12 june exam paper 1.

This grade 12 June exam paper tests the skills learnt in grade 12 terms 1 and 2, and all work covered in grade 11 according to the CAPS curriculum.   Download here: Paper 1 June Exam Paper 1 June Exam Memorandum  

General , Grade 12 , Grades , Math Literacy , Measurement

Worksheet 3: measurement.

This worksheet looks at converting of various measurements as well as calculations involving cost for grade 12 maths literacy students.   Download here: Worksheets 3: Measurement Worksheet 3 Memorandum: Measurement

Algebraic Expressions, Equations and Inequalities , General , Grade 12 , Grades , NCS Mathematics

Worksheet 4 – solving for x.

Use this worksheet to practice factorising, completing the square, inequalities, simultaneous equations, fraction questions and exponential questions. Use the fully worked out memo to check that the answers are correct.   Download here: Worksheet 4 – Solving For x Woksheet 4 – Memo

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Free Printable Grade 12 worksheets

Grade 12 worksheets: Discover a vast collection of free printable resources, tailored for educators teaching high school students. Enhance learning experiences and explore diverse subjects with Quizizz.

• Complex Numbers
• Graphs & Functions
• Conic Sections
• Sequences and Series
• Probability & Combinatorics
• Trigonometry
• arithmetic and number theory
• probability and statistics

10th - 12th

11th - 12th

• High School Science

• social studies

• Text Analysis
• Literature - Books, Stories
• Informational Stories and Texts
• Compare and Contrast

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Explore Grade 12 Worksheets by Subjects

Grade 12 worksheets are an essential tool for teachers looking to provide their students with a comprehensive and engaging learning experience. These worksheets cover a wide range of subjects, including math, science, language arts, and social studies, ensuring that students have access to the resources they need to succeed in their final year of high school. Teachers can use these worksheets to supplement their lesson plans, provide additional practice opportunities, or even as a means of assessment. With a variety of formats and difficulty levels available, Grade 12 worksheets cater to the diverse learning needs of students, making them an invaluable resource for educators.

Quizizz is a fantastic platform that offers more than just Grade 12 worksheets for teachers. This interactive learning tool allows educators to create fun and engaging quizzes, polls, and activities that can be used in conjunction with worksheets to provide a well-rounded learning experience. Teachers can choose from a vast library of pre-made quizzes or create their own, tailoring the content to suit their specific curriculum and learning objectives. Quizizz also offers real-time feedback and analytics, enabling teachers to monitor student progress and identify areas where additional support may be needed. By incorporating Quizizz into their teaching strategies, educators can enhance the effectiveness of Grade 12 worksheets and ensure that their students are well-prepared for success in their academic pursuits.