Class 10th Competency based questions vol2, Exercises of Mathematics

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• Mathematics (Volume 2) | Grade

Assessments are an important tool that help gauge learning. They provide valuable feedback about

the effectiveness of instructional methods; about what students have actually understood and also

provide actionable insights. The National Education Policy, 2020 has outlined the importance of

competency‐based assessments in classrooms as a means to reform curriculum and pedagogical

methodologies. The policy emphasizes on the development of higher order skills such as analysis,

critical thinking and problem solving through classroom instructions and aligned assessments.

Central Board of Secondary Education (CBSE) has been collaborating with Educational Initiatives (Ei)

in the area of assessment. Through resources like the Essential Concepts document and A‐ Question‐

A‐Day (AQAD), high quality questions and concepts critical to learning have been shared with schools

and teachers.

Continuing with the vision to ensure that every student is learning with understanding, Question

Booklets have been created for subjects for Grade 10th and 12th. These booklets contain

competency‐based items, designed specifically to test conceptual understanding and application of

concepts.

Process of creating competency‐based items

All items in these booklets are aligned to the NCERT curriculum and have been created keeping in

mind the learning outcomes that are important for students to understand and master. Items are a

mix of Free Response Questions (FRQs) and Multiple‐Choice Questions (MCQs). In case of MCQs, the

options (correct answer and distractors) are specifically created to test for understanding and

capturing specific errors/misconceptions that students may harbour. Each incorrect option can

thereby inform teachers on specific gaps that may exist in student learning. In case of subjective

questions, each question also has a detailed scoring rubric to guide evaluation of students’

responses.

Each item has been reviewed by experts, to check for appropriateness of the item, validity of the

item, conceptual correctness, language accuracy and other nuances.

How can these item booklets be used?

There are 197 questions in this booklet.

The purpose of these item booklets is to provide samples of high‐quality competency‐based items to

teachers. The items can be used to–

● get an understanding of what good competency‐based questions could look like

● give exposure to students to competency‐based items

● assist in classroom teaching and learning

● get inspiration to create more such competency‐based items

Students can also use this document to understand different kinds of questions and practice specific

concepts and competencies. There will be further additions in the future to provide competency

focused questions on all chapters.

The item booklets are aligned with the 2022‐23 curriculum. However, a few questions from topic

which got rationalized in 2023‐24 syllabus are also there in the booklet which may be used as a

reference for teachers and students.

Please write back to us to give your feedback.

Team CBSE

Preface

Chapter - 1

Polynomials

Polynomials CLASS 10

p ( x ) is a polynomial given by:

p ( x ) = -2 x + 8 x^2 - 1

At which of the following points will the graph of p ( x ) intersect the positive x -axis?

(i) (^12)

(ii) (^14) 1 only (i) 2 only (ii) 3 both (i) and (ii) 4 (none, it never intersects positive x -axis)

Q: 1

Which of these are the zeros of the polynomial x ( x - 7)? 1 only 0 2 only 7 3 both 0 and 7 4 (the polynomial does not have any zero)

Q: 2

Which of these are the quotient and the remainder when (2 x^3 - 9 x + 3x 2 +12) is divided by ( x - 1)? 1 quotient = (2 x^2 - 7 x - 4) and remainder = 8. 2 quotient = (2 x^2 + 7 x + 4) and remainder = 16. 3 quotient = (2 x^2 + 5 x - 4) and remainder = 8. 4 quotient = (2 x^2 + 5 x + 4) and remainder = 16.

Q: 3

Which of these is the coefficient of x^2 in the quotient when ( x^4 + x^3 + x + 1) is divided by ( x - 4)? 1 0 2 -3 3 5 4 1

Q: 4

(3 a^3 - 2 a^2 - 9 a + 17) is divided by ( a - 2). What is the coefficient of a in the quotient? 1 -2 2 3 3 -9 4 4

Q: 5

P( t ) is a polynomial in t such that,

P( t ) = ( t^2 + 5t - 14)( t^2 - 7t + 10)( t^2 + 2 t - 35)

Which of these is the square root of P( t )? 1 ( t + 2)( t - 5)( t +7) 2 ( t - 2)( t - 5)( t +7) 3 ( t + 2)( t + 5)( t -7) 4 ( t - 2)( t - 5)( t -7)

Q: 6

Multiple Choice Questions

Shown below is an expression:

At how many points does the graph of the above expression intersect the x -axis? Show your work.

Polynomials CLASS 10

Q: 16 [2]

When a polynomial is divided by (2 x - 1), the quotient is (3 x - 2) and the remainder is ( x - 3).

Find the polynomial. Show your work.

Q: 17 [2]

p ( x ) is a polynomial given by a x^2 - 4 x + 3 , where a is a non-zero real number. One of the zeroes of p ( x ) is 3 times the other zero.

i) Find the value of a****. Show your work. ii) Based on the value of a, what would be the shape of the graph of p ( x )? Give a reason for your answer.

Q: 18 [3]

A polynomial is given by p ( x ) = x^3 + 3 x^2 - 4 x + c , where c is a constant.

The sum of two zeroes of p ( x ) is zero.

Using the relationship between the zeroes and coefficients of a polynomial, find the:

i) zeroes of p ( x ). ii) value of c****.

Show your steps.

Q: 19 [3]

Anand multiplied a variable with 6, subtracted 27 and added the square of the original variable. He expressed the final expression as a product of 2 factors.

His friend, Amit, said that the factors will always have a difference of 6.

Is Amit right? Show your work.

Q: 20 [3]

The graph of the polynomial f ( x ) = x^4 + 4 x^3 + x^2 - 6 x is shown below.

Identify all the zeroes of the polynomial from the graph. Verify your answer.

Polynomials CLASS 10

Q: 21 [3]

p(x) = x^3 + ( k - 3) x^2 - ( k + 4) x - 6, where k is a non-zero real number and ( x + 2) is a factor of p(x).

Find the zeroes of p ( x ). Show your work.

Q: 22 [5]

f ( x ) = ax^2 + bx + 325 is a polynomial where a and b are real numbers. The zeroes of f ( x ) are distinct prime numbers. Find the:

i) zeroes of f ( x ). ii) values of a and b****.

Show your work and give valid reasons.

Q: 23 [5]

Polynomials CLASS 10

Q.No Correct Answers

1 1 2 3 3 3 4 3 5 4 6 2 7 1 8 2

Answer key

Polynomials CLASS 10

Q.No What to look for Marks

9 Writes False(F). 0.

Justifies the answer. For example, substituting x = -2 5 in the given polynomial does not 0. yield 0, so -2 5 is not a zero.

10 Finds the value of f (2) as: 1 (2) 3 + 7(2) 2 + 3(2) - 12 = 30

11 Divides x^4 + x^2 + 4 by x^2 - 1 using the long division method to get quotient as 1 x^2 + 2 and the remainder as 6.

12 Factorizes the given polynomial and finds the roots as (-3) and (-7). The working may 1 look as follows:

f ( x ) = x^2 + 10 x + 21 = x^2 + 3 x + 7 x +21 = 0 => x ( x + 3) + 7( x + 3) = 0 => ( x + 7)( x + 3) = 0 => ( x + 7) = 0 or ( x + 3) = 0 => x = (-7) or x = (-3)

(Note: Award full marks if the correct roots are obtained by any alternative approach.)

13 Writes that P( x ) = 0 at x = 6 or P(6) = 0 and hence ( x - 6) is a factor of the 1 polynomial.

(Award 0.5 marks if only the factor is written.)

Answer key

Polynomials CLASS 10

Q.No What to look for Marks

Simplifies the above expression to find the polynomial as 6 x^2 - 6 x - 1. 1

18 i) Assumes the roots of p ( x ) to be m and n to write the relation as m = 3 n. 0.

Finds the relation between β and a using the sum of the roots as: 0.

m + n = 3n + n = 4n = (^4) a

=> n = (^1) a

Finds the value of a using the product of the roots as: 1

m.n = 3 n^2 = (^3) a

=> a = 1.

ii) Writes that, since a is positive, the graph of p ( x ) is an open upward parabola or 1 open upwards like U.

(Note: Award half mark if the student just writes parabola instead of upward parabola.)

19 i) Assumes the values of zeroes of p ( x ) as (-α), α and β. 0.

Writes the sum of zeroes as: 0.

• α + α + β = -

Finds β as -3.

Writes the equation for the sum of the products of zeroes taken two at a time as: 1

• α^2 - αβ + βα = -

Finds α^2 as 4.

Finds the zeroes of p ( x ) as (-2), 2 and (-3). 0.

Answer key

Polynomials CLASS 10

Q.No What to look for Marks

ii) Writes the equation for the product of zeroes as (- α^2 β ) = (- c ) and finds the 0. value of c as (-12).

20 Assumes the original variable as x and frames the expression as 6 x - 27 + x^2. 1

Factorises the above expression as ( x - 3)( x + 9). 1

Concludes that Amit was wrong as the above factors have a difference of 12. 1

21 Identifies all the zeroes of the polynomial from the graph as: (-3), (-2), 0 and 1. 1

Verifies f (-3) = 0 as: 2

f (-3) = (-3) 4 + 4(-3) 3 + (-3) 2 - 6(-3) = 81 - 108 + 9 + 18 = 0

Similarly verifies for the rest of the three roots.

(Note: Award half mark for each correct verification.)

22 Writes that, since p ( x ) is divisible by ( x + 2), p (-2) = 0 and finds the value of k as 1

Uses the above step and writes p ( x ) as x^3 - 7 x - 6. 1

Answer key

Polynomials CLASS 10

Q.No What to look for Marks

24 i) Finds the zeroes of the polynomial as (-1) , 0 and 2 since the y -coordinate = 0 at x 1 = -1, 0 and 2.

ii) Uses the three zeroes to form the polynomial as x ( x + 1)( x - 2) = x^3 - x^2 - 2 x. 1

iii) Compares the polynomial in the question and that obtained in (ii) and finds the 1 values of a = 1, b = -1, c = -2 and d = 0.

iv) Finds the values of p and q using the relationship between the roots and the 1. coefficients as: -p 1 = d => p = 0 and q 1 =^

16

=> q = 16 -

Finds the polynomial as x^2 - 161 by substituting the above values in the given 0. polynomial expression.

25 Subtracts E( x ) from R( x ) to find P( x ) as 2 x^3 + 4 x^2 - 2 x + 8. 1

26 Finds the revenue made by the company from 100 products as: 1

R(100) = 5(100) 3 + 4(100) 2 + 7

=> R(100) = 5000000 + 40000 + 7 = Rs 50,40,007.

Finds the profit made by the company from 100 products as: 1

P(100) = 2(100) 3 + 4(100) 2 - 2(100) + 8

=> P(100) = 2000000 + 40000 - 200 + 8 = Rs 20,39,

27 Finds profit for 10 items as: 1

P(10) = 2(10) 3 + 4(10) 2 - 2(10) + 8

=> P(10) = 2000 + 400 - 20 + 8 = Rs 2388.

Answer key

Polynomials CLASS 10

Q.No What to look for Marks

Finds tax as: 1

T(2388) = 0.3(2388) + 100 = Rs 816.4.

Answer key

Probability CLASS 10

A library receives a shipment for a series of encyclopedias. The shipment includes volumes 31 - 40. These encyclopedias arrived in a box and are not ordered.

One encyclopedia is picked at random from the box without looking into it.

What is the probability that the volume of the encyclopedia picked is a multiple of 2 OR 5? (^1) 101 (^2) 105 (^3) 106 (^4) 107

Q: 1

In basketball, different shots have varying point values – a two-point shot is taken from inside the three-point line, while a three-point shot is taken from outside the three-point line.

In a basketball match, a player shot 5 three-point shots and 9 two-point shots out of the 35 shots he made.

A particular shot he took was chosen at random. What is the probability that the shot that was chosen was NEITHER a three-point shot NOR a two-point shot? 1^17 2^25 3^35 4^67

Q: 2

Jyoti and Dara are playing a game of tic-tac-toe. The probability of Jyoti winning the game is 0.7.

What is the probability that Jyoti NOT winning the game? 1 0.7 2 0. 3 0.3 4 (cannot be determined)

Q: 3

A card is drawn at random from a well shuffled standard deck of 52 cards.

What is the probability that the card drawn is NEITHER a black card NOR a three?

( Note: A deck of cards is divided into four suits - 2 black and 2 red. Each suit contains 13 ranks including numbered cards 2 through 10, and the face cards (jack, queen, king), along with the ace.) 1^2252 2^2452 3^2652 4^2852

Q: 4

Matilda made the following pattern during art class.

If she colours a shape at random, what is the probability that she will colour a circle? 1^13 (^2) 103 (^3) 133 (^4) 131

Q: 5

Multiple Choice Questions

An apartment complex has 20 apartments of different sizes - 2BHK, 3BHK, 4BHK. The probability of a randomly picked apartment being a 3BHK is 25.

How many 3BHK apartments are in the apartment complex? 1 2 2 4 3 5 4 8

Probability CLASS 10

Q: 6

A box contains some new and weathered cricket balls of two colours. This data is shown in the table below.

Pratik picks a new, white ball and puts it back in the box.

If a ball is then picked randomly from the box, what is the probability that it is NOT the same variety as Pratik picked? (^1) 151 (^2) 205 (^3) 155 4^1520

Q: 7

In a school, each student is assigned to one of the three houses- Honesty, Integrity and Courage. In a class of 43 students, 13 students are in Honesty house, 16 students are in Integrity house, and rest are in Courage house.

If a student from this class is selected at random as a class representative, what is the probability that they belong to EITHER Integrity or Courage house? (^1) 431 2^2943 3^2043 4^3043

Q: 8

Pritam is throwing a fair 6-sided die, with faces numbered from 1 to 6. Shown below are the outcomes of his first 4 throws:

Throw 1 Throw 2 Throw 3 Throw 4 6 6 6 6

Pritam says, "The probability of getting a 6 in my next throw is higher than that of getting a different number on the die."

Is Pritam's statement true or false? Give a valid reason.

Q: 9 [1]

Free Response Questions