Deleted portion of class 12 maths exercise-wise

Chapter 1 – Relations and Functions

Deleted topics – Composite functions, Inverse of a function.

Exercise 1.3 & 1.4 (Deleted)

Miscellaneous – Q1, Q3, Q9, Q11, Q12, Q13, Q14, Q15, Q18, Q19 (Deleted)

Chapter 2 – Inverse Trigonometric Functions

Exercise 2.2 – Q1 to Q4, and Q12 to Q15 (Deleted)

Miscellaneous – Q3 to Q8, Q12, 13, 14, 16 (Deleted)

Chapter 3 – Matrices

Exercise – 3.4 (Deleted)

Chapter 4 – Determinants

Exercise – 4.2 (Deleted)

Miscellaneous – Q2, 4, 6, 11, 12, 13, 15, 17 (Deleted)

Chapter 5 – Continuity and Differentiability

Exercise – 5.8 (Deleted)

Chapter 6 – Applications of Derivatives

Exercise – 6.3, 6.4 (Deleted)

Miscellaneous – Q1, 4, 5, 20, 21, 22, 23, 24 (Deleted)

Chapter 7 – Integrals

Exercise – 7.8 (Deleted)

Miscellaneous – Q40 (Deleted)

Chapter 8 – Applications of the Integrals

Exercise – 8.2 (Deleted)

Miscellaneous – Q2, 6, 7, 10, 12, 13, 14, 15, 18, 19

Chapter 9 – Differential Equations

Exercise – 9.3 (Deleted)

Miscellaneous – Q3, 5

Chapter 10 – Vectors (No Deletion)

Chapter 11 – Three-dimensional Geometry

Exercise – 11.3 (Deleted)

Miscellaneous – Q8 to Q 23 (Except Q9 and Q 20) except 2 questions, Q8 to 23 are deleted.

Chapter 12 – Linear Programming

Exercise – 12.2 (Deleted)

Miscellaneous Exercise (Deleted)

Chapter 13 – Probability

Exercise – 13.4 (Questions are mixed) We don’t have to calculate variance.

Exercise – 13.5 (Deleted)

Miscellaneous – Q4, 5, 6,9, 10

problems on probability

 OBJECTIVE 

1. If there coins are tossed simultaneously, then the probability of getting at least two heads, is

(A) 1/4 (B) 3/8 (C) 1/2 (D) 1/4

2. A bag contains three green marbles four blue marbles, and two orange marbles. If marble is picked at random, then the probability that it is not a orange marble is

(A) 1/4  (B) 1/3 (C) 4/9  (D) 7/9

3. A number is selected from number 1 to 27. The probability that it is prime is

(A) 2/3  (B) 1/6 (C) 1/3 (D) 2/9

4. IF (P(E) = 0.05, then P (not E) =

(A) -0.05 (B) 0.5 (C) 0.9 (D) 0.95

5. A bulb is taken out at random from a box of 600 electric bulbs that contains 12 defective bulbs. Then the probability of a non-defective bulb is

(A) 0.02 (B) 0.98 (C) 0.50 (D) None

SUBJECTIVE 

1. To dice are thrown simultaneously. Find the probability of getting :

(i) An even number of the sum

(ii) The sum as a prime number

(iii) A total of at least 10

(iv) A multiple of 2 on one dice and a multiple of 3 on the other.

2. Find the probability that a leap year selected at random will contain 53 Tuesdays.

3. A bag contains 12 balls out of which x are white.(i) If one ball is drawn at random, what is the probability it will be a white ball ? (ii) If 6 more white balls are put in the box. The probability of drawing a white ball will be double than that is (i). Find x.

4. In a class, there are 18 girls and 16 boys. The class teacher wants to choose one pupil for class monitor. What she does, she writes the name of each pupil a card and puts them into a basket and mixes thoroughly. A child is asked to pick one card from the basket. What is the probability that the name written on the card is: (i) The name of a girl (ii) The name of boy ?

5. The probability of selecting a green marble at random from a jar that contains only green, white and yellow marbles is 1/4. The probability of selecting a white marble from the same jar is 1/3. If this jar contains 10 yellow marbles. What is the total number of marbles in the jar ?

6. A card is drawn at random from a well suffled desk of playing cards. Find the probability that the card drawn is (i) A card of spade or an ace (ii) A red king (iii) Neither a king nor a queen (iv) Either a king or a queen

7. There are 30 cards of same size in a bag on which number 1 to 30 are written. One card is taken out of the bag at random. Find the probability that the number of the selected card is not divisible by 3.

8. In figure points A,B,C and D are the centers of four circles that each have a radius of length on unit. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region ?

9. A bag contains 5 white balls, 6 red balls, 6 black balls and 8 green balls. One ball is drawn at random from the bag. Find the probability that the ball drawn is (i) White (ii) Red or black (iii) Note green (iv) Neither white nor black  [CBSE - 2006]

10. A bag contains 4 red and 6 black balls. A ball is taken out of the bag at random. Find the probability of getting a black ball. [CBSE - 2008]

11. Cards. marked with number 5 to 50, are placed in a box and mixed thoroughly. A card is drawn from the box at random. Find the probability that the number on the taken out card is (i) a prime number less than 10. (ii) a number which is a perfect square.

#jee #math #probability 

Probability

1.1 EXPERIMENT : The word experiment means an operation, which can produce well defined outcomes. 

The are two types of experiment : 

                   (i) Deterministic experiment 

                    (ii) Probabilistic or Random experiment 

(i) Deterministic Experiment : Those experiment which when repeated under identical conditions, produced the same results or outcome are known as deterministic experiment. For example, Physics or Chemistry experiments performed under identical conditions.

 (ii) Probabilistic or Random Experiment :- In an experiment, when repeated under identical conditions donot produce the same outcomes every time. For example, in tossing a coin, one is not sure that if a head or tail will be obtained. So it is a random experiment. 

Sample space : The set of all possible out comes of a random experiment is called a sample space associated with it and is generally denoted by S. 

For example, When a dice is tossed then S = {1, 2, 3, 4, 5, 6}.

 Event : A subset of sample space associated with a random experiment is called an event. For example, In tossing a dive getting an even no is an event. 

Favorable Event : Let S be a sample space associated with a random experiment and A be event associated with the random experiment. The elementary events belonging to A are know as favorable events to the event A. 

For example, in throwing a pair of dive, A is defined by “Getting 8 as the sum”. Then following elementary events are as out comes : (2, 6), (3, 5), (4, 4) (5, 3) (6, 2). So, there are 5 elementary events favorable to event A. 

1.2 PROBABILITY : If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of happening or occurrence of event A is denoted by P(A)

 Thus,

    

P(A)     =   (Total number of favourable outcomes)/(Total number of possible outcomes)

                   =  m/n 

And 0 £ P(A) £ 1

 If, P(A) = 0, then A is called impossible event

If, P(A) = 1, then A is called sure event

                                P(A) + P (A") = 1

Where P(A) = probability of occurrence of A. 

            P (A') = probability of non - occurrence of A.

 

Ex.1 A box contains 5 red balls, 4 green balls and 7 white balls. A ball is drawn at random from the box. Find the probability that the ball drawn is (i) white (ii) neither red nor white

Sol. Total number of balls in the bag = 5 + 4 + 7 = 16

\ Total number of elementary events =16

(i)                There are 7 white balls in the bag.

\ Favorable number of elementary events = 7

P(Getting a white ball ) = (Total No. of elementaryevents) /(Total No. favourable elementaryevents)

                    = 7 /16

 (ii) There are 4 balls that are neither red nor white 

 Favorable number of elementary events = 4 

Hence, P (Getting neither red not white ball) = 4/16 =1/4

 Ex.2 All the three face cards of spades are removed from a well-shuffled pack of 52 cards. A card is then drawn at random from the remaining pack. Find the probability of getting  [CBSE - 2007] (i) black face card                     (ii) a queen             (iii) a black card.

 Sol. After removing three face cards of spades (king, queen, jack) from a deck of 52 playing cards, there are 49 cards left in the pack. 

Out of these 49 cards one card can be chosen in 49 ways. 

Therefore, Total number of elementary events = 49 

(i) There are 6 black face cards out of which 3 face cards of spades are already removed. 

So, out of remaining 3 black face cards one black face card ban be chosen in 3 ways. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a black face card ) = 3/49

 (ii) There are 3 queens in the remaining 49 cards.

 So, out of these three queens, on queen can be chosen in 3 ways 

Therefore, Favorable number of elementary events = 3

 Hence P (Getting a queen) = 3/49

 (iii) There are 23 black cards in the remaining 49 cards, 

So, out to these 23 black card, one black card can be chosen in 23 ways 

Therefore, Favorable number of elementary events = 23 

Hence, P (Getting a black card) =  23/49

 Ex.3 A die is thrown, Find the probability of (i) prime number (ii) multiple of 2 or 3 (iii) a number greater than 3.

Sol.

In a single throw of die any one of six numbers 1,2,3,4,5,6 can be obtained. 

Therefore, the tome number of elementary events associated with the random experiment of throwing a die is 6.

 (i) Let A denote the event “Getting a prime no”. 

Clearly, event A occurs if any one of 2,3,5 comes as out come. 

Therefore, Favorable number of elementary events = 3 

Hence, P (Getting a prime no.) = 3/6=1/2

(ii) An multiple of 2 or 3 is obtained if we obtain one of the numbers 2,3,4,6 as out comes.

Therefore, Favorable number of elementary events = 4 

Hence, P (Getting multiple of 2 or 3) = 4/6=2/3

 (iii) The event “Getting a number greater than 3” will occur, if we obtain one of number 4,5,6 as an out come. 

Therefore, Favorable number of out comes = 3

 Hence, required probability = 3/6=1/2

 Ex.4 Two unbiased coins are tossed simultaneously. Find the probability of getting (i) two heads (ii) at least one head (iii) at most one head. 

Sol. 

If two unbiased coins are tossed simultaneously, 

we obtain any one of the following as an out come : HH, HT, TH, TT 

Therefore, Total number of elementary events = 4

 (i) Two heads are obtained if elementary event HH occurs. 

Therefore, Favorable number of events = 1 Hence, P (Two heads) =  1/4

 (ii) At least one head is obtained if any one of the following elementary events happen : HH, HT, TH 

Therefore, favorable number of events = 3 

Hence P (At least one head) = 3/4

 (iii) If one of the elementary events HT, TH, TT occurs, than at most one head is obtained 

Therefore, favorable number of events = 3 

Hence, P (At most one head) = 3/3

Ex.5 A box contains 20 balls bearing numbers, 1,2,3,4…...20. A ball is drawn at random from the box. What is the probability that the number of the ball is (i) an odd number (ii) divisible by 2 or 3 (iii) prime number 

Sol. Here, total numbers are 20. 

 Total number of elementary events = 20 

(i) The number selected will be odd number, if it is elected from 1,3,5,7,9,11,13,15,17,19 

 Therefore, Favorable number of elementary events = 10 

Hence, P (An odd number ) = 10/20=1/2

 (ii) Number divisible by 2 or 3 are 2,3,4,6,8,9,10,12,14,15,16,18,20 

Therefore, Favorable number of elementary events = 13

 P (Number divisible by 2 or 3) =  13/20

 (iii) There are 8 prime number from 1 to 20 i.e., 2,3,5,7,11,13,17,19

Therefore, Favorable number of elementary events = 8 

P (prime number ) = 8/20=2/5

 Ex.6 A die is drop at random on the rectangular region as shown in figure. What is the probability that it will land inside the circle with diameter 1m ? 

comment answer of this question

New syllabus of class 12 session 2021-2022 MathematicsCOURSE STRUCTURE MATHEMATICS (class XII) (Code No. 041) Session – 2021-22

CLASS-XII

 MATHEMATICS (2021-22)

 TERM - I

Unit-I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto

functions.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch.

Unit-II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix,

symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and

multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Non-

commutativity of multiplication of matrices, Invertible matrices; (Here all matrices will have real entries)

2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of

determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Solving system

of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse

trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions

expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives

Applications of derivatives: increasing/decreasing functions, tangents and normals, maxima and

minima (first derivative test motivated geometrically and second derivative test given as a provable

tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-

life situations).

Unit-V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, different types of

linear programming (L.P.) problems. Graphical method of solution for problems in two variables, feasible

and infeasible regions (bounded), feasible and infeasible solutions, optimal feasible solutions (up to

three non-trivial constraints).

TERM - II

Unit-III: Calculus

1. Integrals

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by

partial fractions and by parts, Evaluation of simple integrals of the following types and problems based

on them.

Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation

of definite integrals.

2. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, parabolas; area of circles /ellipses

(in standard form only) (the region should be clearly identifiable).

3. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of

differential equations by method of separation of variables, solutions of homogeneous differential

equations of first order and first degree of the type: 𝑑𝑦/𝑑𝑥= 𝑓(y/x). 

Solutions of linear differential equation

of the type:

dy/dx + py = q, where p and q are functions of x or constant.

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a

vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point,

negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar,

position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation,

properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three - dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation

of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation

of a plane. Distance of a point from a plane.

Unit-VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability,

Bayes’ theorem, Random variable and its probability distribution.

Note: For activities NCERT Lab Manual may be referred

Assessment of Activity Work:

In first term any 4 activities and in second term any 4 activities shall be performed by the

student from the activities given in the NCERT Laboratory Manual for the respective class

(XI or XII) which is available on the link :

http://www.ncert.nic.in/exemplar/labmanuals.html 

Record of the same may be kept by the

student. A term end test on the activity is to be conducted.

The weightage are as under:

 The activities performed by the student in each term and record keeping

: 3 marks

 Assessment of the activity performed during the term end test and Viva-voce

: 2 marks

New syllabus of class 11 session 2021-2022 MathematicsCOURSE STRUCTURE MATHEMATICS (class XI) (Code No. 041) Session – 2021-22

TERM 1

Unit-I: Sets and Functions

1. Sets

Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set

of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and

Intersection of sets.

2. Relations & Functions

Ordered pairs. Cartesian product of sets. Number of elements in the Cartesian product of two finite sets.

Cartesian product of the set of reals with itself ( R x R only).Definition of relation, pictorial diagrams, domain,

co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a

function, domain, co-domain and range of a function. Real valued functions, domain and range of these

functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest

integer functions, with their graphs.

Unit-II: Algebra

1. Complex Numbers and Quadratic Equations

Need for complex numbers, especially√−1, to be motivated by inability to solve some of the quardratic

equations. Algebraic properties of complex numbers. Argand plane. Statement of Fundamental Theorem

of Algebra, solution of quadratic equations (with real coefficients) in the complex number system.

2. Sequence and Series

Sequence and Series. Arithmetic Progression (A. P.). Arithmetic Mean (A.M.) Geometric Progression

(G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.),

relation between A.M. and G.M.

Unit-III: Coordinate Geometry

1. Straight Lines

Brief recall of two dimensional geometry from earlier classes. Slope of a line and angle between two lines.

Various forms of equations of a line: parallel to axis, point -slope form, slope-intercept form, two-point form,

intercept form and normal form. General equation of a line. Distance of a point from a line.

Unit-IV: Calculus

1. Limits

Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and

logarithmic functions

Unit-V: Statistics and Probability

1. Statistics

Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped

data.

TERM - II

Unit-I: Sets and Functions

1. Trigonometric Functions

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one

measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity

sin2x + cos2x = 1, for all x. Signs of trigonometric functions. Domain and range of trigonometric functions

and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple

applications. Deducing identities like the following:

Identities related to sin2x, cos2x, tan2 x, sin3x, cos3x and tan3x.

Unit-II: Algebra

1. Linear Inequalities

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the

number line. Graphical solution of linear inequalities in two variables. Graphical method of finding a solution

of system of linear inequalities in two variables.

2. Permutations and Combinations

Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, formula

for nPr and nCr, simple applications.

Unit-III: Coordinate Geometry

1. Conic Sections

Sections of a cone: circles, ellipse, parabola, hyperbola. Standard equations and simple properties of

parabola, ellipse and hyperbola. Standard equation of a circle.

2. Introduction to Three-dimensional Geometry

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two

points and section formula.

Unit-IV: Calculus

1. Derivatives

Derivative introduced as rate of change both as that of distance function and geometrically. Definition of

Derivative, relate it to scope of tangent of the curve, derivative of sum, difference, product and quotient of

functions. Derivatives of polynomial and trigonometric functions.

Unit-V: Statistics and Probability

1. Probability

Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, ‘not’,

‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Probability of an event, probability of

‘not’, ‘and’ and ‘or’ events

Linear programming problem class 12

Q1: Solve the following LPP graphically:

Maximise Z = 2x + 3y, subject to x + y ≤ 4, x ≥ 0, y ≥ 0

Q2. Solve the following linear programming problem graphically:

Minimise Z = 200 x + 500 y subject to the constraints:

x + 2y ≥ 10

3x + 4y ≤ 24

x ≥ 0, y ≥ 0

Q3.  A manufacturing company makes two types of television sets; one is black and white and the other is colour. The company has resources to make at most 300 sets a week. It takes Rs 1800 to make a black and white set and Rs 2700 to make a coloured set. The company can spend not more than Rs 648000 a week to make television sets. If it makes a profit of Rs 510 per black and white set and Rs 675 per coloured set, how many sets of each type should be produced so that the company has a maximum profit? Formulate this problem as a LPP given that the objective is to maximise the profit.

Q4. A dietician wishes to mix two types of foods in such a way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’. Formulate this problem as a linear programming problem to minimise the cost of such a mixture.

Q5. Consider the linear programming problem;

Maximise; Z = x +y , 2x + y – 3< 0, x – 2y + 1 < 0, y < 3, x < 0, y < 0

(i) Draw its feasible region.

(ii) Find the corner points of the feasible region.

(iii) Find the corner at which Z attains its maximum. (March – 2012)

Consider the LPP Minimise; Z = 200 k + 500y, x + 2y > 10, 3x + 4y < 24, x > 0, y > 0

(i) Draw the feasible region.

(ii) Find the co-ordinates of the comer points of the feasible region.

(iii) Solve the LPP. (May – 2012)

Q6. Consider the LPP

Maximise; Z = 5x + 3y

Subject to; 3x + 5y < 15, 5x + 2y < 10x, y > 0

(i) Draw the feasible region.

(ii) Find the corner points of the feasible region.

(iii) Find the corner at which Z attains its maximum. (March – 2013)

Q7. Consider the linear programming problem: Minimize Z = 3x + 9y Subject to the constraints: x + 3y < 60 x + y > 10, x < y, x > 0, y > 0

(i) Draw its feasible region.

(ii) Find the vertices of the feasible region

(iii) Find the minimum value of Z subject to the given constraints. (March-2014, MAY-2016) NCERT example 3

Q8.  Consider the linear inequalities 2x + 3y < 6; 2x + y < 4; x, y < 0

(a) Mark the feasible region.

(b) Maximise the function z = 4x + 5y subject to the given constraints. (March – 2014)

Q9. Consider the linear programming problem: Minimise Z = 4x + 4y Subject to x + 2y < 8; 3x + 2y < 12x, y<0

(a) Mark its feasible region.

(b) Find the comer points of the feasible region.

(c) Find the corner at which Z attain its minimum. (May – 2014)

Q10. Consider the linear programming problem: Maximum z = 4x + y

Subject to constraints: x + y < 50, 3x + y < 9x, y < 0

(a) Draw the feasible region

(b) Find the corner points of the feasible region

(c) Find the corner at which ‘z’ aftains its maximum value. (May – 2015)

Q11. Consider the LPP

Maximise z = 3x + 2y

Subject to the constraints: x + 2y < 10, 3x + y < 15; x, y < 0

(a) Draw its feasible region

(b) Find the corner points of the feasible region

(c) Find the maximum value of Z. (March – 2016)

Q12. Consider the linear programming problem: Maximum z = 50x + 40y

Subject to constraints:

x + 2y < 10; 3x + 4y < 24; x, y < 0

(i) Draw the feasible region

(ii) Find the comer points of the feasible region

(iii) Find the maximum value of z. (March – 2017)

Q13.  A  furniture dealer sells only tables and chairs. He has Rs. 12,000 to invest and a space to store 90 pieces. A table costs him Rs. 400 and a chair Rs. 100. He can sell a table at a profit of Rs. 75 and a chair at a profit of Rs. 25. Assume that he can sell all the items. The dealer wants to get maximum profit.

(i) By defining suitable variables, write the objective function.

(ii) Write the constraints.

(iii) Maximise the objective function graphically. (March – 2010)

Q14. A company produces two types of cricket balls A and B. The production time of one ball of type B is double the type A (time in units). The company has the time to produce a maximum of 2000 balls per day. The supply of raw materials is sufficient for the production of 1500 balls (both A and B) per day. The company wants to make maximum profit by making profit of Rs. 3 from a ball of type A and Rs. 5 from type B. Then,

(i) By defining suitable variables write the objective function.

(ii) Write the constraints.

(iii) How many balls should be produced in each type per day in order to get maximum profit? (May – 2010)

Q15.