Chapter 3 : PAIR OF LINEAR EQUATIONS IN TWO VARIABLES What is Linear Equation? A linear equation is an algebraic equation in which the highest power of the variable(s) is 1, and its graph always forms a straight line. Example : 3x + 4 = 0 , 3x + 4y = 6 Linear Equation in Two variable: A linear equation in two variables is an equation of the form ax+by+c=0, where a,b,c are real numbers and x,y are variables. Its graph is always a straight line in the coordinate plane. Example : a) 2x + 3y = 6 , b) 4x + 5y = 8. Solution of pair of linear equation in two variable: A pair of linear equations in two variables can be represented, and solved, by the: (i) graphical method (ii) algebraic method a) Graphical method 1. General Form A pair of linear equations in two variables can be written as: a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 Here, x and y are variables, and (a 1 , b 1 , c 1 , a 2 , b 2 , c 2 ) are constants. 2. Steps in the...

Chapter 2 : Polynomial What is polynomial? A polynomial is an expression made up of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Example : 3x 4 -2x 2 +7x-5. Key Features of Polynomial : let's discuss some features of polynomial: Take a polynomial : 3x 4 -2x 2 +7x-5. a) Degree: The highest power of the variable(x)(e.g. degree of 3x 4 -2x 2 +7x-5 is 4). b) Terms: Each part separated by + or – (e.g., 3x 4 , -2x 2 , 7x, -5). c) Coefficients: Numbers multiplying the variables (e.g., 3, -2, 7). d) Constant term: The standalone number (e.g. -5). Types of polynomial: a) Linear polynomial: A polynomial of degree 1.     Example : 2x – 3 , u + 5 , 3u+ 5 b) Quadratic polynomial : A polynomial of degree 2     Example : 3x 2 +2x – 3 ,3u 2 - u + 5 c) Cubic polynomial : A polynomial of degree 3     Example : 2...

Physics Question for revision 1. State any two properties of magnetic field lines. 2. What is a magnetic field ? How can the direction of magnetic field lines at a place be determined ? 3. Explain why, two magnetic field lines do not intersect each other. 4. State and explain Maxwell’s right-hand thumb rule. 5. Draw the magnetic lines of force due to a circular wire carrying current. 6. In the straight wire A, current is flowing in the vertically downward direction whereas in wire B the current is flowing in the vertically upward direction. What is the direction of magnetic field : (a) in wire A ? (b) in wire B ? Name the rule which you have used to get the answer. 7. A thick wire is hanging from a wooden table. An anticlockwise magnetic field is to be produced around the wire by passing current through this wire by using a battery. Which terminal of the battery should be connected to the : (a) top end of wire ? (b) bottom end of wire ? Giv...

Mathematics Question for revision | Area Related to circle There will be 10 questions and timing will be 1 hour. 1. Find the area of a quadrant of a circle whose circumference is 22 cm. 2. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. 3. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. 4. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades. 5. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned.(Use π = 3.14) 6. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see Fig below). Find (i...

Mathematics Question for revision | Coordinate Geometry There will be 10 questions and timing will be 1 hour. 1. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order 2. Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts. 3. Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4). 4. If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = (3/7)AB and P lies on the line segment AB. 5. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y 6. If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p 7. In what ratio does the point (– 4, 6) divide the line segment joining the points A(– 6, 10) and B(3, – 8)? 8. Determine if the point...

Mathematics Question for revision | Triangle Question There will be 10 questions and timing will be 1 hour. 1. Fill the follwoing with right answer : a) All circles are _____________. (congruent, similar) b) All ___________ triangles are similar. (isosceles, equilateral) c) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ______________ and (b) their corresponding sides are ______________ proportional) 2. To prove : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 3. To Prove : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 4. ABCD is a trapezium with AB || DC. E and F are points on non-parallel sides AD and BC respectively such that EF is parallel to AB.Show that AE/ED =BF/FC. . 5. The diagonals of a quadrilateral ABCD intersect eac...

Physics Question for revision Give The reason For following Question : 1. Why is the series arrangement not used for domestic circuits? 2. How does the resistance of a wire vary with its area of cross-section? 3. Why are copper and aluminium wires usually employed for electricity transmission? 4. Explain why, filament type electric bulbs are not power efficient. 5. State whether an electric heater will consume more electrical energy or less electrical energy per second when the length of its heating element is reduced. Give reasons for your answer. Numericals : 1. A wire that has resistance R is cut into two equal pieces. The two parts are joined in parallel. What is the resistance of the combination ? 2. An electric motor takes 5 amperes current from a 220 volt supply line. Calculate the power of the motor and electrical energy consumed by it in 2 hours. 3. Two lamps, one rated 40 W at 220 V and the other 60 W at 220 V, are connected in pa...

Physics Question for revision Give The reason For following Question : 1. Why is the tungsten used almost exclusively for filament of electric lamps? 2. Why are the conductors of electric heating devices, such as bread-toasters and electric irons, made of an alloy rather than a pure metal? 3. Why does the cord of an electric heater not glow while the heating element does? 4. What are the advantages of connecting electrical devices in parallel with the battery instead of connecting them in series? 5. Will current flow more easily through a thick wire or a thin wire of the same material, when connected to the same source? Why? Numericals : 1. An electric lamp, whose resistance is 20 Ω, and a conductor of 4 Ω resistance are connected to a 6 V battery. Calculate (a) the total resistance of the circuit, (b) the current through the circuit, and (c) the potential difference across the electric lamp and conductor. 2. A wire of given material ha...

Mathematics Question for revision There will be 10 questions and timing will be 1 hour. 1. The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP. 2. Find the sum of the first 22 terms of an AP in which d = 7 and 22nd term is 149. 3. The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. 4. How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78? 5. The sum of 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP. 6. How many multiples of 4 lie between 10 and 250? 7. If the 3rd and the 9th terms of an AP are 4 and -8, respectively, then which term of this AP is zero. 8. Check whether – 150 is a term of the AP: 11, 8, 5, 2 . . . 9. Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0? Give reason for yo...

Mathematics Question for revision There will be 10 questions and timing will be 1 hour. 1. On comparing the ratios a1/a2 , b1/b2 , c1/c2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 18x + 6y + 24 = 0 (iii) 6x – 3y + 10 = 0 2x – y + 9 = 0 2. Solve the following pair of linear equations by the substitution method : (i) x + y = 14 x – y = 4 (ii) s – t = 3 (s/3) + (t/2) = 6 (iii) 3x – y = 3 9x – 3y = 9 (iv) 0.2x + 0.3y = 1.3 0.4x + 0.5y = 2.3 (v) √2x + √3y = 0 √3x - √8y = 0 (vi) (3x/2) – (5y/3) = -2 (x/3) + (y/2) = (13/6) 3. The coach of a cricket team buys 7 bats and 6 balls for Rs.3800. Later, she buys 3 bats and 5 balls for Rs.1750. Find the cost of each bat and each ball. 4. A fraction becomes 9/11 , if 2 is added to both the numerator and the denominator. I...

Mathematics Question for revision There will be 10 questions and timing will be 1 hour. 1. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the Length of PQ. 2. In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to 3. If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, What is the value of ∠ POA ? 4. Prove that the tangents drawn at the ends of a diameter of a circle are parallel. 5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. 6. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle. 7. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of ...

Mathematics Question for revision There will be 10 questions and timing will be 1 hour. 1. Find the roots of the following quadratic equations by factorisation: i) 100x 2 – 20x + 1 = 0 ii) 2x 2 + x – 6 = 0 iii) 2x 2 – x +1/8 = 0 2. Find the roots of the following quadratic equations, if they exist, by the method of completing the square: (i) 2x 2 – 7x +3 = 0 (ii) 2x 2 + x – 4 = 0 (iii) 4x 2 + 4√3x + 3 = 0 3. Find the roots of the following equations: (i) x-1/x = 3, x ≠ 0 (ii) 1/x+4 – 1/x-7 = 11/30, x ≠ -4, 7 4. Prove the quadratic formula for finding the roots using the completing the square method 5. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them. i) 2x 2 – 3x + 5 = 0 (ii) 3x 2 – 4√3x + 4 = 0 (iii) 2x 2 – 6x + 3 = 0 6. Find the values of k for each of the following quadratic equations so that they have two equal roots. (i) 2x 2 + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0...

Statistics - Mathematics Question Paper There will be 9 question and 1 hr time 1. Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in Rs.) 500-520 520-540 540-560 560-580 580-600 Number of workers 12 14 8 6 10 Find the mean daily wages of the workers of the factory by using an appropriate method. 2. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency f. Daily Pocket Allowance(in c) 11-13 13-15 15-17 17-19 19-21 21-23 23-25 Number of children 7 6 9 13 f 5 4 3. The following data gives the information on the observed lifetimes (in hours) of 225 electrical components: Lifetime (in hours) 0-20 20-40 40-60 60-80 80-100 100-120 Frequency 10 35 52 61 38 29 Determine...

Mathematics Question for revision There will be 15 questions and timing will be 1.5 hour. 1. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically. 2. On comparing the ratios a1/a2 , b1/b2 , c1/c2 find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0 7x + 6y – 9 = 0 (ii) 9x + 3y + 12 = 0 18x + 6y + 24 = 0 (iii) 6x – 3y + 10 = 0 2x – y + 9 = 0 3. Find the 31st term of an A.P. whose 11th term is 38 and the 16th term is 73. 4. If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively. Which term of this A.P. is zero. 5. Which term of the A.P. 3, 15, 27, 39,.. will be 132 more than its 54th term? 6. Determine if the points (1, 5), (2, 3) and (-2, -11) are c...

Probability and Application of Trigonometry Question There will be 15 questions and timing will be 1.5 hour. 1. A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red ? (ii) not red? 2. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face card (iii) a red face card (iv) the jack of hearts (v) a spade (vi) the queen of diamonds 3. 12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one. 4. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divis...

Trigonometry Test There will be 25 questions and timing will be 2 hour. 1. In ∆ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine: (i) sin A, cos A (ii) sin C, cos C 2. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠ A = ∠ B. 3. If 3 cot A = 4, check whether (1 – tan 2 A)/(1 + tan 2 A) = cos 2 A – sin 2 A or not. 4. In triangle PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P. 5. Show that : (i) tan 48° tan 23° tan 42° tan 67° = 1 (ii) cos 38° cos 52° – sin 38° sin 52° = 0 6. Prove the identities: (i) √[1 + sinA/1 – sinA] = sec A + tan A (ii) (1 + tan 2 A/1 + cot 2 A) = (1 – tan A/1 – cot A) 2 = tan 2 A 7. If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1. 8. Prove that (sin A – 2 sin 3 A)/(2 cos 3 A – cos A) = tan A. 9. Prove that (1 – cos 2 A) cosec 2 A = 1 10. Prove that (sec 2 θ − 1)(cosec 2 θ − 1) = 1 11. Prove that (1 – sin θ) / (1 + sin θ) = (sec θ...

Difference between Environment , Ecology and Ecosystem Here's a simple breakdown of ecosystem, ecology, and environment: 1.Ecosystem An ecosystem is a specific area where living organisms (plants, animals, microorganisms) interact with each other and with their non-living surroundings (like soil, water, air, and sunlight).It's a system of interconnected relationships. Examples: A forest, a pond, or even a desert can be ecosystems. 2. Ecology Ecology is the scientific study of how living organisms interact with each other and their environment.It focuses on understanding relationships and processes within ecosystems. It includes topics like food chains, species distribution, and population dynamics. 3.Environment The environment refers to all the external conditions and factors—both biotic (living) and abiotic (non-living)—that influence an organism's life.It’s everything around us.The air you breathe, the water you drink, and the trees i...

Start the test Question no. 1 Suppose, ω and ω 2 are two distinct cube roots of unity different from 1. Then, what is (ω 1 - ω 2 ) 2 equal to? 3   1   -1   -3 Question no. 2 What is the argument of the complex number (−1 − i) , where i = √(−1)?   5π/4   -5π/4   3π/4 None of these Question no. 3 What is one of the square roots of 3 + 4i, where i = √(−1)? 2+i   2-i   -2 + i   -3 - i Question no. 4 Consider the following statements I. ( ω 10 +1) 7 + ω = 0 II. ( ω 105 +1) 10 = p 10 for some prime number p, where, ω ≠ 1 is a cubic root of unity. Which of the above statement(s) is/are correct? Only I   Only II   Both I and II   Neither I nor II Question no. 5 The origin and the roots of the equation z 2 + pz + q = 0 form an equilateral triangle, if p 2 = q ...

Biochemical oxygen demand Biochemical Oxygen Demand (BOD) is a way to measure how much oxygen is needed by microorganisms to break down organic material in water. Think of it as a test to see how "dirty" the water is with organic waste. The higher the BOD, the more polluted the water is, because it means there's a lot of organic matter that needs to be broken down. How does BOD affect aquatic life? High Biochemical Oxygen Demand (BOD) levels can significantly impact aquatic life. The following are: Oxygen Depletion: When BOD is high, it means there’s a lot of organic matter in the water that microorganisms need to break down. This process consumes a lot of oxygen, reducing the amount available for fish and other aquatic organisms. Stress on Aquatic Life: Low oxygen levels can stress aquatic organisms, making it difficult for them to survive. Fish, in particular, need a certain amount of dissolved oxygen to breathe. Death of Aquatic Organisms: If ...

Probability ⚫What is Probability ? ⚪ Probability is a measure of chance that tells us how likely an event is to happen. It’s always represented as a number between 0 and 1, inclusive. ⚪ In probability , Experiment is refered as action and event can be termed as the required action. ⚪ Suppose Event = E and Sample Space = S , the probability is : ⚪ If P(E) = 0 , it means it is null , happening of this event is impossible. Similarly, If P(E) = 1 , it is sure that event will happen (100% chance). ⚪ ⚪                     ⚪ Independent Probability: Probability of A when A event is independent of B (doesn't depend on B) ⚪ Conditional Probability : The probability of B event , when A events already happens (Event A and B are dependent) ⚪ Probability of happening at least one of independent event : Let P 1 , P 2 ....... P n are the probability ...