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APlusTopper.com provides ICSE Class 10 Chemistry Previous Year Board Question Papers Solved Pdf Free Download with Solutions, Answers and Marking Scheme. Here we have given ICSE Class 10 Chemistry Solved Question Papers Last Ten Years. Students can view or download the ICSE Board 10th Chemistry Previous Year Question Papers with Solutions for their upcoming examination.

These ICSE Class 10 Chemistry Previous 10 Years Board Question Papers with Answers are useful to understand the pattern of questions asked in the board exam. Know about the important concepts to be prepared for ICSE Class 10 Board Exam and Score More marks.

Board – Indian Certificate of Secondary Education (ICSE), www.cisce.org
Class – Class 10
Subject – Chemistry
Year of Examination – 2020, 2019, 2018, 2017.

ICSE Class 10 Chemistry Solved Question Papers Last Ten Years

ICSE Chemistry Class 10 Solved Question Papers Last 10 Years 

• ICSE 2019 Chemistry Question Paper Solved
• ICSE 2018 Chemistry Question Paper Solved
• ICSE 2017 Chemistry Question Paper Solved
• ICSE 2016 Chemistry Question Paper Solved
• ICSE 2015 Chemistry Question Paper Solved
• ICSE 2014 Chemistry Question Paper Solved
• ICSE 2013 Chemistry Question Paper Solved
• ICSE 2012 Chemistry Question Paper Solved
• ICSE 2011 Chemistry Question Paper Solved
• ICSE 2010 Chemistry Question Paper Solved
• ICSE 2009 Chemistry Question Paper Solved
• ICSE 2008 Chemistry Question Paper Solved

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ICSE Previous Year Question Papers Class 10

Get ICSE Solutions for Class 10 Mathematics Chapter 9 Factorization for ICSE Board Examinations on ICSESolutions.com. We provide step by step Solutions for ICSE Mathematics Class 10 Solutions Pdf. You can download the Class 10 Maths ICSE Textbook Solutions with Free PDF download option.

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Factorization Class 10 Maths ICSE Solutions

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Formulae

1. Factor Theorem: If f(x) is a polynomial and a is a real number, then (x – a) is a factor of f(x) if f(α) = 0.
2. Remainder Theorem: If a polynomial f(x) is divided by (x – a), then remainder =f(x).

Determine the Following

Question 1. Use remainder theorem and find the remainder when the polynomial g(x) = x³ + x² – 2x + 1 is divided by x – 3.

Question 2. (i) When x³ + 3x² – kx + 4 is divided by (x – 2), the remainder is k. Find the value of k.

 

Question 11. In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:

Question 12. In the following two polynomials, find the value of ‘a’ if x – a is a factor of each of the two:

Question 15. In the following problems use the factor theorem to find if g(x) is a factor of p(x):

Question 16. If x – 2 is a factor of each of the following three polynomials. Find the value of ‘a’ in each case:

Question 17. Find the value of the constant a and b, if (x – 2) and (x + 3) are both factors of

Question 19. If x – 2 is a factor of

Prove the Following 

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Get ICSE Solutions for Class 10 Mathematics Chapter 6 Quadratic Equation for ICSE Board Examinations on ICSESolutions.com. We provide step by step Solutions for ICSE Mathematics Class 10 Solutions Pdf. You can download the Class 10 Maths ICSE Textbook Solutions with Free PDF download option.

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Quadratic Equation Class 10 Maths ICSE Solutions

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Formulae

1. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b and c are all real numbers and a ≠ 0.
   e.g., equation 4x² + 5x – 6 = 0 is a quadratic equation in standard form.
2. Every quadratic equation gives two values of the unknown variable and these values are called roots of the equation.
3. Zero Product Rule: Whenever the product of two expressions is zero; at least one of the expressions is zero.
   
4. Solving quadratic equations using the formula:
   The roots of the quadratic equation ax² + bx + c = 0; where a ≠ 0 can be obtained by using the formula:
   
5. To examine the nature of the roots:
   Examining the roots of a quadratic equation means to see the type of its roots i.e., whether they are real or imaginary, rational or irrational, equal or unequal.
   The nature of the roots of a quadratic equation depends entirely on the value of its discriminant b² – 4ac.
   Case I: If a, b and c are real numbers and a ≠ 0, then discriminant:
   (i) b² – 4ac = 0 ⇒ the roots are real and equal.
   (ii) b² – 4a c > 0 ⇒ the roots are real and unequal.
   (iii) b² – 4ac < 0 ⇒ the roots are imaginary (not real).
   Case II: If a, b and c are rational numbers and a ≠ 0, then discriminant.
   (i) b² – 4ac = 0 ⇒ the roots are rational and equal.
   (ii) b² – 4ac > 0 and b² – 4ac is a perfect square ⇒ the roots are rational and unequal.
   (iii) b² – 4ac > 0 and b² – 4ac is not a perfect square ⇒ the roots are irrational and unequal.
   (iv) b² – 4ac < 0 ⇒ the roots are imaginary.
6. Sum and product of the roots: If a and P are the roots of quadratic equation ax² + bx + c = 0 then
   
7. To form a quadratic equation with given roots: Let α, β be the roots of the required quadratic equation, then
   

Determine the Following

Question 1. Which of the following are quadratic equation:

Question 2. Determine, if 3 is a root of the given equation

Question 3. Examine whether the equation 5x² -6x + 7 = 2x² – 4x + 5 can be put in the form of a quadratic equation.

Question 4. Find if x = – 1 is a root of the equation 2x² – 3x + 1 = 0.

Question 6. 48x² – 13x -1 = 0

Question 12. Without solving the following quadratic equation, find the value of ‘p’ for which the given equation has real and equal roots: x² + (p – 3) x + p = 0

Question 13. Find the value of k for which the following equation has equal roots:

Question 14. If one root of the equation 2x² – px + 4 = 0 is 2, find the other root. Also find the value of p.

Question 16. The sum of two numbers is 15. If the sum of reciprocals is 3/10, find the numbers.

Question 17. Find two consecutive natural numbers whose squares have the sum 221.

Question 18. The sum of the squares of three consecutive natural numbers is 110. Determine the numbers.

Question 19. If an integer is added to its square the sum is 90. Find the integer with the help of a quadratic equation.

Question 20. Find two consecutive positive even integers whose squares have the sum 340.

Question 21. Divide 29 into two parts so that the sum of the square of the parts is 425.


Question 22. In a two digit number, the unit’s digit is twice the ten’s digit. If 27 is added to the number, the digit interchange their places. Find the number.

Question 23. A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.

Question 24. Five years ago, a woman’s age was the square of her son’s age. Ten years later her age will be twice that of her son’s age. Find:
(i) The age of the son five years ago.
(ii) The present age of the woman.

Question 25. The length of verandah is 3m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter.
(i) Taking x, breadth of the verandah write an equation in ‘x’ that represents the above statement.
(ii) Solve the equation obtained in above and hence find the dimension of verandah.

Question 26. A two digit number is such that the product of its digit is 14. When 45 is added to the number, then the digit interchange their places. Find the number.

Question 27. In each of the following determine the; value of k for which the given value is a solution of the equation:

Question 28. If x = 2 and x = 3 are roots of the equation 3x² – 2kx + 2m = 0. Find the values of k and m.

Question 29. Solve the following equation and give your answer up to two decimal places:

Question 30. Determine whether the given values of x is the solution of the given quadratic equation below:

Question 32. Solve using the quadratic formula x² – 4x + 1 = 0

Question 33. Solve the quadratic equation:

Question 36. Form the quadratic equation whose roots are:

Question 37. Find the value of k for which the given equation has real roots:

Question 38. Without actually determining the roots comment upon the nature of the roots of each of the following equations:

Question 39. Solve the equation 3x² – x – 7 = 0 and give your answer correct to two decimal places.

Question 40. Solve for x using the quadratic formula. Write your answer correct to two significant figures (x -1)² – 3x + 4 = 0.

Question 41. Without solving the following quadratic equation, find the value of ‘m’ for which the given equation has real and equal roots.

Question 42. Solve the following by reducing them to quadratic equations:

Question 44. Solve for x:

Question 45. Solve the following equation by reducing it to quadratic equation:

Question 46. Solve:

Question 47. A two digit number is such that the product of the digits is 12. When 36 is added to this number the digits interchange their places. Determine the number.

Question 48. The side (in cm) of a triangle containing the right angle are 5x and 3x – 1. If the area of the triangle is 60 cm². Find the sides of the triangle.

Question 49. Rs. 480 is divided equally among ‘x’ children. If the number of children were 20 more then each would have got Rs.  12 less. Find ‘x’.


Question 50. By increasing the speed of a car by 10 km/hr, the time of journey for a distance of 72 km. is reduced by 36 minutes. Find the original speed of the car.

Question 51. A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/hr more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

Question 52. The speed of an express train is x km/hr arid the speed of an ordinary train is 12 km/hr less than that of the express train. If the ordinary train takes one hour longer than the express train to cover a distance of 240 km, find the speed of the express train.

Question 53. Some students planned a picnic. The budget for the food was Rs. 480. As eight of them failed to join the party, the cost of the food for each member increased by Rs. 10. Find how many students went for the picnic.

Question 54. Two pipes flowing together can fill a cistern in 6 minutes. If one pipe takes 5 minutes more than the other to fill the cistern, find the time in which each pipe would fill the cistern.

Question 55. One fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.

Question 56. An aeroplane travelled a distance of 400 km at an average speed of x km/hr. On the return journey the speed was increased by 40 km/hr. Write down the expression for the time taken for
(i) The outward journey (ii) the return Journey. If the return journey took 30 minutes less than the onward journey write down an equation in x and find its value.

Question 57. Car A travels x km for every litre of petrol, while car B travels (x + 5) km for every litre of petrol.
(i) Write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.
(ii) If car A use 4 litre of petrol more than car B in covering the 400 km, write down and equation in x and solve it to determine the number of litre of petrol used by car B for the journey.

Solution: Given Distance = 400 km

Question 58. A shopkeeper purchases a certain number of books for Rs. 960. If the cost per book was Rs. 8 less, the number of books that could be purchased for Rs. 960 would be 4 more. Write an equation, taking the original cost of each book to be Rs. x, and Solve it to find the original cost of the books.

Question 59. Two pipes running together can 1 fill a cistern in 11 1/9 minutes. If one pipe takes 5 minutes more than the other to fill the cistern find the time when each pipe would fill the cistern.

Question 60. In each of the following find the values of k of which the given value is a solution of the given equation:

Question 61. Solve the following quadratic equation by factorisation:

Question 62. Solve the following quadratic equation by factorisation method:

Question 63. Solve the following quadratic equation:

Question 64. Determine whether the given quadratic equations have equal roots and if so, find the roots:

Question 65. Find the value of k so that sum of the roots of the quadratic equation is equal to the product of the roots:

Question 66. Find the values of k so that the sum of tire roots of the quadratic equation is equal to the product of the roots in each of the following:

Question 67. Solve the following by reducing them to quadratic equations:





Question 69. Solve the following by reducing them to quadratic form:

Question 70. Solve: x(x + 1) (x + 3) (x + 4) = 180.

Question 71. Solve the equation:

Question 72. Solve for x:

Question 73. Solve the equation

Prove the Following 

Question 1. Given that one root of the quadratic equation ax² + bx + c = 0 is three times the other, show that 3b² – 16ac.

Question 2. If one root of the quadratic equation ax² + bx + c = 0 is double the other, prove that 2b² = 9 ac.

Question 3. If the ratio of the roots of the equation

Question 4. In each of the following determine whether the given values are solutions of the equation or not.

Question 5. In each of the following, determine whether the given values are solution of the given equation or not:

Question 7. Show, that a, b, c, d are in proportion if:

Concept Based Questions

Question 1. The hypotenuse of a right angled triangle is 3√5. If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15 cm. Find the length of each side.

For More Resources

Get ICSE Solutions for Class 10 Mathematics Chapter 2 Sales Tax and Value Added Tax for ICSE Board Examinations on APlusTopper.com. We provide step by step Solutions for ICSE Mathematics Class 10 Solutions Pdf. You can download the Class 10 Maths ICSE Textbook Solutions with Free PDF download option.

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Sales Tax and Value Added Tax Class 10 Maths ICSE Solutions

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Formulae

The amount of money paid by a customer for an article = The sale price of the article + Sales Tax on it, if any.
VAT (Value Added Tax):

1. Unlike sales tax, VAT is also collected by the state government.
2. It is not in addition to the existing Sales Tax, but is the replacement of Sales Tax. Presently, a majority of state governments have accepted the VAT system.
3. It is tax on the value added at each transfer of goods, from the original manufacturer to the retailer.
   Assuming that the rate of tax is 10% and a trader purchases an article for Rs. 800, the tax he pays = 10% Rs. 800 = Rs. 80.
   Now, if he sells the same article for Rs. 1,150.
   The tax he recovers (gets) = 10% of Rs. 1,150 = 115
   ∴ VAT = Tax recovered on the sale – Tax he paid on the purchase
   = Rs. 115 – Rs. 80 = 35.
4. The difference of tax recovered on the sale value and paid on the purchase value is deposited with the government as VAT.

Formulae Based Questions

Question 1. Sheela bought a V.C.R., at the list price of Rs. 13,500. If the rate of sale tax was 8%. Find the amount she had to pay for it.

Question 2. Rani purchases a pair of shoe whose sale price is Rs. 175. If she pays sales tax at the rate of 7%, how much amount does she pay as sales tax? Also find the net values of the pair of shoe.

Question 3. Sunita purchases a bicycle for Rs. 660. She has paid a sale tax of 10%. Find the list price of the bicycle.

Question 4. The price of a washing machine, inclusive of sales tax i, Rs. 13,530. If the sales tax is 10% find its list (or basic) price.

Question 5. Savita purchased an almirah for Rs. 4536 including sale tax. If the list price of the almirah is Rs. 4,200, find the rate of sale tax charged.

Question 6. The sale price of a television, inclusive of sales tax is Rs. 13,500. If sales tax is charged at the rate of 8% of the list price, find the list price of the television.

Question 7. The price of a washing machine inclusive of sale tax is Rs. 13,530. If the sale tax is 10% find its basic price.

Question 8. A shopkeeper buys on article whose list price is Rs. 450 at some rate of discount from a wholesaler. He sells the article to a consumer at the list price and charges sales tax at the rate of 6%. If the shopkeeper has to pay a VAT of Rs. 2.70, find the rate of discount at which he bought the article from the wholesaler.

Concept Based Questions

Question 1. Samir bought the following articles from a departmental store:

Question 2. A shopkeeper bought a washing machine at a discount of 20% from a wholesaler, the printed price of the washing machine being Rs. 18,000. The shopkeeper sells it to a consumer at a discount of 10% on the printed price. If the rate of sales tax is 8%, find:
(i) the VAT paid by the shopkeeper.
(ii) the total amount that the consumer pays for the washing machine.

Question 3. Find the tax paid by (i) the manufacturer, (ii) the whole saler, (iii) the retailer, (iv) the customer. If the rate of sales tax be 10%.

Question 4. A shopkeeper sells an article at its marked price (Rs. 7,500) and charges sales-tax at the rate of 12% from the customer. If the shopkeeper pays a VAT of Rs. 180; calculate the price (inclusive of tax) paid by the shopkeeper.

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Get ICSE Solutions for Class 10 Mathematics Chapter 15 Circles for ICSE Board Examinations on ICSESolutions.com. We provide step by step Solutions for ICSE Mathematics Class 10 Solutions Pdf. You can download the Class 10 Maths ICSE Textbook Solutions with Free PDF download option.

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Circles Class 10 Maths ICSE Solutions

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Formulae

Theorems based on chord properties:

1. Theorem: A straight line drawn from the centre of the circle to bisect a chord, which is not a diameter, is at right angles to the chord.
   Conversely, the perpendicular to a chord, from the centre of the circle, bisects the chord.
2. Theorem: There is one circle, and only one, which passes through three given points not in a straight line.
3. Theorem: Equal chords of a circle are equidistant from the centre.
   Conversely, chords of a circle, equidistant from the centre of the circle, are equal.
   

Theorems based on Arc and Chord properties:

1. Theorem: The angle which an arc of a circle subtends at the centre is double, that which it subtends at any point on the remaining part of the circumference.
2. Theorem: Angles in the same segment of a circle are equal.
3. Theorem: The angle in a semicircle is a right angle.
4. Theorem: In equal circles (or, in the same circle), if two arcs subtend equal angles at the centre, they are equal.
   Conversely, in equal circles (or, in the same circle), if two arcs are equal, they subtend equal angles at the centre.
5. Theorem: In equal circles (or, in the same circle), if two-chord are equal, they cut off equal arcs.
   Conversely, in equal circles (or, in the same circle, if two arcs are equal the chords of the arcs are also equal.

Theorems based on Cyclic properties: ABCD is a cyclic quadrilateral.

1. Theorem: The opposite angles of a cyclic quadrilateral (quadrilateral inscribed in a circle) are supplementary.
2. Theorem: The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.

Theorems based on Tangent Properties:

1. Theorem: The tangent at any point of a circle and the radius through this point are perpendicular to each other.
2. Theorem: If two circles touch each other, the point of contact lies on the straight line through the centres.
3. Theorem: From any point outside a circle two tangents can be drawn and they are equal in length.
4. Theorem: If a chord and a tangent intersect externally, then the product of the lengths of segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection.
5. Theorem: If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments.

Prove the Following 

Question 1. If a diameter of a circle bisect each of the two chords of a circle, prove that the chords are parallel.

Question 2. If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.

Question 3. In the given figure, OD is perpendicular to the chord AB of a circle whose centre is O. If BC is a diameter, show that CA = 2 OD.

Question 4. In Fig., l is a line intersecting the two concentric circles, whose common centre is O, at the points A, B, C and D. Show that AB = CD.

Question 6. ABCD is a cyclic quadrilateral AB and DC are produced to meet in E. Prove that

Question 7. In an isosceles triangle ABC with AB = AC, a circle passing through B and C intersects the sides. AB, and AC at D and E respectively. Prove that DE || BC.

Question 8. ABCD is quadrilateral inscribed in circle, having ∠A = 60°, O is the centre of the circle, show that

Question 9. Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB

Question 11. Two circles are drawn with sides AB, AC of a triangle ABC as diameters. The circles intersect at a point D. Prove that D lies on BC.

Question 12. In the given figure, PT touches a circle with centre O at R. Diameter SQ when

Question 14. Prove that the line segment joining the midpoints of two equal chords of a circle substends equal angles with the chord.

Question 15. In an equilateral triangle, prove that the centroid and centre of the circum-circle (circumcentre) coincide.


Question 16. In Fig. AB and CD are two chords of a circle intersecting each other at P such that AP = CP. Show that AB = CD.

Question 17. Prove that the angle bisectors of the angles formed by producing opposite sides of a cyclic quadrilateral (Provided they are not parallel) intersect at right angle.

Question 18. In Fig. P is any point on the chord BC of a circle such that AB = AP. Prove that CP = CQ.

Question 19. The diagonals of a cyclic quadrilateral are at right angles. Prove that the perpendicular from the point of their intersection on any side when produced backward bisects the opposite side.

Question 20. In a circle with centre O, chords AB and CD intersect inside the circumference at E. Prove that ∠AOC + ∠BOD = 2∠AEC.

Question 21. In Fig. ABC is a triangle in which ∠BAC = 30°. Show that BC is the radius of the circum circle of A ABC, whose centre is O.

Question 22. Prove that the circle drawn on any one of the equal sides of an isosceles triangles as diameter bisects the base.

Question 23. In Fig. ABCD is a cyclic quadrilateral. A circle passing through A and B meets AD and BC in the points E and F respectively. Prove that EF || DC.

Question 25. If PA and PB are two tangent drawn from a point P to a circle with centre C touching it A and B, prove that CP is the perpendicular bisector of AB.

Question 26. Two circle with radii r₁ and r₂ touch each other externally. Let r be the radius of a circle which touches these two circle as well as a common tangent to the two circles, Prove that:

Question 27. If AB and CD are two chords which when produced meet at P and if AP = CP, show that AB = CD.

Question 28. In the figure, PM is a tangent to the circle and PA = AM. Prove that:

Question 29. In Fig. the incircle of ΔABC, touches the sides BC, CA and AB at D, E respectively. Show that:

Question 30. In Fig. TA is a tangent to a circle from the point T and TBC is a secant to the circle. If AD is the bisector of ∠BAC, prove that ΔADT is isosceles.

Question 32. In Fig. l and m are two parallel tangents at A and B. The tangent at C makes an intercept DE between n and m. Prove that ∠ DFE = 90⁰

Question 33. A circle touches the sides of a quadrilateral ABCD at P, Q, R, S respectively. Show that the angles subtended at the centre by a pair of opposite sides are supplementary.

Question 34. Two equal chords AB and CD of a circle with centre O, when produced meet at a point E, as shown in Fig. Prove that BE = DE and AE = CE.

Question 35. Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral ABCD is also cyclic.

Figure Based Questions

Question 1. Two concentric circles with centre 0 have A, B, C, D as the points of intersection with the lines L shown in figure. If AD = 12 cm and BC s = 8 cm, find the lengths of AB, CD, AC and BD.

Question 2. In the given circle with diameter AB, find the value of x.

Question 3. In the given figure, the area enclosed between the two concentric circles is 770 cm². If the radius of outer circle is 21 cm, calculate the radius of the inner circle.

Question 4. Two chords AB and CD of a circle are parallel and a line L is the perpendicular bisector of AB. Show that L bisects CD.

Question 5. In the adjoining figure, AB is the diameter of the circle with centre O. If ∠BCD = 120°, calculate:

Question 6. Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 13 cm.

Question 7. The radius of a circle is 13 cm and the length of one of its chord is 10 cm. Find the distance of the chord from the centre.

Question 9. AB is a diameter of a circle with centre C = (- 2, 5). If A = (3, – 7). Find
(i) the length of radius AC
(ii) the coordinates of B.

Question 10. AB is a diameter of a circle with centre O and radius OD is perpendicular to AB. If C is any point on arc DB, find ∠ BAD and ∠ ACD.

Question 11. In the given below figure,

Question 14. In the given figure O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm. Calculate the radius of the circle.

Question 15. In Fig., chords AB and CD of the circle intersect at O. AO = 5 cm, BO = 3 cm and CO = 2.5 cm. Determine the length of DO.

Question 16. In the figure given below, PT is a tangent to the circle. Find PT if AT = 16 cm and AB = 12 cm.

Question 18. A, B and C are three points on a circle. The tangent at C meets BN produced at T. Given that ∠ ATC = 36° and ∠ ACT = 48°, calculate the angle subtended by AB at the centre of the circle.

Question 19. In the given figure, O is the centre of the circle and ∠PBA = 45°. Calculate the value of ∠PQB.

Question 20. Two chords AB, CD of lengths 16 cm and 30 cm, are parallel. If the distance between AB and CD is 23 cm. Find the radius of the circle.

Question 21. Two circles of radii 10 cm and 8 cm intersect and the length of the common chord is 12 cm. Find the distance between their centres.

Question 22. AB and CD are two chords of a circle such that AB = 6 cm, CD = 12 cm and AB || CD. If the distance between AB and CD is 3 cm, find the radius of the circle.

Question 24. AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm. If the chords are on the opposite sides of the centre and the distance between them is 17 cm, find the radius of the circle.

Question 26. In the figure given below,O is the centre of the circle. AB and CD are two chords of the circle. OM is perpendicular to AB and ON is perpendicular to CD.

Question 27. If O is the centre of the circle, find the value of x in each of the following figures

Question 29. In the given figure, AB is the diameter of a circle with centre O.

Question 30. In ABCD is a cyclic quadrilateral; O is the centre of the circle. If BOD = 160°, find the measure of BPD.

Question 32. In the given figure O is the centre of the circle, ∠ BAD = 75° and chord BC = chord CD. Find:

Question 37. ABC is a triangle with AB = 10 cm, BC = 8 cm and AC = 6 cm (not drawn to scale). Three circles are drawn touching each other with the vertices as their centres. Find the radii of the three circles.

Question 40. P and Q are the centre of circles of radius 9 cm and 2 cm respectively; PQ = 17 cm. R is the centre of circle of radius x cm, which touches the above circles externally, given that ∠ PRQ = 90°. Write an equation in x and solve it.

For More Resources

Get ICSE Solutions for Class 10 Mathematics Chapter 11 Coordinate Geometry for ICSE Board Examinations on ICSESolutions.com. We provide step by step Solutions for ICSE Mathematics Class 10 Solutions Pdf. You can download the Class 10 Maths ICSE Textbook Solutions with Free PDF download option.

ICSE Solutions  Selina ICSE Solutions

Coordinate Geometry Class 10 Maths ICSE Solutions

Download Formulae Handbook For ICSE Class 9 and 10

Formulae

Formulae Based Questions

Question 1. Find the distance of the following points from origin.
(i) (5, 6)   (ii) (a+b, a-b)    (iii) (a cos θ, a sin θ).

Question 2. Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Solution: Here B is (11, 0)

Question 3. KM is a straight line of 13 units If K has the coordinate (2, 5) and M has the coordinates (x, – 7) find the possible value of x.

Question 4. The midpoint of the line segment joining (2a, 4) and (-2, 2b) is (1, 2a+1). Find the value of a and b.
show that the points A(- 1, 2), B(2, 5) and C(- 5, – 2) are collinear.

Question 5. Use distance formula to show that the points A(-1, 2), B(2, 5) and C(-5, -2) are collinear.

Determine the Following

Question 1. PQ is straight line of 13 units. If P has coordinate (2, 5) and Q has coordinate (x, – 7) find the possible values of x.

Question 2. Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).

Question 3. The line segment joining A (2, 3) and B (6, – 5) is intersected by the X axis at the point K. Write the ordinate of the point K. Hence find the ratio in which K divides AB.
Solution: A (2, 3) and B (6,- 5)

Question 4. Find the value of x so that the line passing through (3, 4) and (x, 5) makes an angle 135° with positive direction of X-axis.

Question 5. Find the value, of k, if the line represented by kx – 5y + 4 = 0 and 4x – 2y + 5 = 0 are perpendicular to each other.

Question 6. Find the equation of a line which is inclined to x axis at an angle of 60° and its y – intercept = 2.

Question 7. Find the equation of a line with slope 1 and cutting off an intercept of 5 units on Y-axis.
Solution: We have
Slope of the line m = 1

Question 8. Find the equations of a line passing through the point (2, 3) and having the x – interecpt of 4 units.

Question 9. The line through A (- 2, 3) and B (4, b) is perpendicular to the line 2a – 4y = 5. Find the value of b.

Question 11. Find the equation of a straight line which cuts an intercept of 5 units on Y-axis and is parallel to the line joining the points (3, – 2) and (1, 4).

Question 12. Find the equation of a line that has Y-intercept 3 units and is perpendicular to the line joining (2, – 3) and (4, 2).

Question 13. Find a general equation of a line which passes through:
(i) (0, -5) and (3, 0) (ii) (2, 3) and (-1, 2).

Question 14. Find the equation of the line passing through (0, 4) and parallel to the line 3x + 5y + 15 = 0.

Question 15. Find the equation of a line passing through (3, – 2) and perpendicular to the line.

Question 16. Find the equation of the straight line which has Y-intercept equal to 4/3 and is perpendicular to 3x – 4y + 11 = 0.

Question 17. Find the equation of the straight line perpendicular to 5x – 2y = 8 and which passes through the mid-point of the line segment joining (2, 3) and (4, 5).

Question 18. A line passing through the points (a, 2a) and (- 2, 3) is perpendicular to the line 4a + 3y + 5 = 0. Find the value of a.

Prove the Following 

Question 1. A line is of length 10 units and one end is at the point (2, – 3). If the abscissa of the other end be 10, prove that its ordinate must be 3 or – 9.

Question 2. Show that the line joining (2, – 3) and (- 5, 1) is:
(i) Parallel to line joining (7, -1) and (0, 3).
(ii) Perpendicular to the line joining (4, 5) and (0, -2).

Question 3. With out Pythagoras theorem, show that A(4, 4), B(3, 5) and C(-1, -1) are the vertices of a right angled.

Question 4. Show that the points A(- 2, 5), B(2, – 3) and C(0, 1) are collinear.

Question 5. By using the distance formula prove that each of the following sets of points are the vertices of a right angled triangle.
(i) (6, 2), (3, -1) and (- 2, 4)
(ii) (-2, 2), (8, -2) and (-4, -3).

Question 6. Show that each of the triangles whose vertices are given below are isosceles :
(i) (8, 2), (5,-3) and (0,0)
(ii) (0,6), (-5, 3) and (3,1).

Question 7. Show that the quadrilateral with vertices (3, 2), (0, 5), (- 3, 2) and (0, -1) is a square.

Question 9. If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.

Question 10. Prove that A(4, 3), B(6, 4), C(5, 6) and D(3, 5) are the angular points of a square.

Question 14. Show that the points A(1, 3), B(2, 6), C(5, 7) and D(4, 4) are the vertices of a rhombus.

Figure Based Questions

Question 2. Determine the ratio in which the line 3x + y – 9 = 0 divides the line joining (1, 3) and (2, 7).

Question 3. The midpoint of the line segment AB shown in the diagram is (4, – 3). Write down the coordinates of A and B.
Solution: Let the coordinates of A and Bare (x, 0) and (0, y).

Question 4. The centre ‘O’ of a circle has the coordinates (4, 5) and one point on the circumference is (8, 10). Find the coordinates of the other end of the diameter of the circle through this point.

Hence (0, 0) be the coordinates of the other end.

Question 5. In the following figure line APB meets the X-axis at A, Y-axis at B. P is the point (4, -2) and AP : PB = 1 : 2. Write down the coordinates of A and B.

Question 6. The three vertices of a parallelogram taken in order are (-1, 0), (3, 1) and (2, 2) respectively. Find the coordinates of the fourth vertex.

Question 7. Find the equation of a straight line which cuts an intercept – 2 units from Y-axis and being equally inclined to the axis.

Question 8. In ΔABC, A (3, 5), B (7, 8) and C (1, – 10). Find the equation of the median through A.

Question 10. In the adjoining figure, write

Question 11. In the figure given below, the line segment AB meets X-axis at A and Y-axis at B. The point P (- 3, 4) on AB divides it in the ratio 2 : 3. Find the coordinates of A and B.

Question 12. Determine the centre of the circle on which the points (1, 7), (7 – 1), and (8, 6) lie. What is the radius of the circle ?

Question 13. Find the image of a point (-1, 2) in the line joining (2, 1) and (- 3, 2).

Question 14. The line through P (5, 3) intersects Y axis at Q.
(i) Write the slope of the line.
(ii) Write the equation of the line.
(iii) Find the coordinates of Q.

Question 15. Find the value of ‘a’ for which the following points A (a, 3), B (2, 1) and C (5, a) are collinear. Hence find the equation of the line.

Question 16. If the image of the point (2,1) with respect to the line mirror be (5, 2). Find the equation of the mirror.

Question 17. The vertices of a triangle are A(10, 4), B(- 4, 9) and C(- 2, -1). Find the

Question 18. Given equation of line L₁ is y = 4.
(i) Write the slope of line, if L₂ is the bisector of angle O.
(ii) Write the coordinates of point P.
(iii) Find the equation of L₂

Question 19. From the adjacent figure:
(i) Write the coordinates of the points A, B, and

Graphical Depiction

Question 1. Given a line segment AB joining the points A (- 4, 6) and B (8, – 3). Find:
(i) the ratio in which AB is divided by the y- axis.
(ii) find the ordinates of the point of intersection.
(iii) the length of AB.

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Ratio and Proportion Class 10 Maths ICSE Solutions

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Formulae

1. When two or more ratios are multiplied together, they are said to be compounded.
   
2. A ratio compounded with itself is called duplicate ratio of the given ratio.
   
3. The reciprocal ratio a : b is b : a.
4. Proportion: An equality of two ratios is called a proportion.
   
5. The quantities a, b, c and d are called the terms of the proportion; a, b, c and d are the first, second, third and fourth terms respectively. First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms).
   
   ⇒ product of extreme terms = product of middle terms.
   Thus, if the four quantities are in proportion then the product of extreme terms = product of middle terms. This is called cross product rule.
6. Fourth proportional: If a, b, c and d are in proportion then d is called the fourth proportional.
   
7. In particular, three (non-zero) quantities of the same kind, a, b and c are said to be in continued proportion iff the ratio of a to b is equal to the ratio of b to c.
   
   
8. First proportional: If a, b are c are in continued proportion, then a is called the first proportional.
   Third proportional: If a, b and c are in continued proportion, then c is called the third proportional.
   Mean proportional: If a, b and c are in continued proportion, then b is called the mean proportional of a and c.
   Thus, if b is the mean proportional of a and c, then
   
   Hence, the mean proportion between two numbers is the positive square root of their product.

Properties of Ratio & Proportion:

Determine the Following

Question 1. Which is greater 4 : 5 or 19 : 25.

Question 2. Arrange 5 : 6, 8 : 9, 13 : 18 and 7 : 12 in ascending order of magnitude.

Question 3. Find:

Question 4. If 3x – 2y – 7z = 0 and 2x + 3y – 5z = 0 find x : y : z.

Question 5. Two numbers are in the ratio of 3 : 5. If 8 is added to each number, the ratio becomes 2 : 3. Find the numbers.

Question 6. If x : y = 2 : 3, find the value of (3x + 2y) : (2x + 5y).

Question 7. Divide Rs. 720 between Sunil, Sbhil and Akhil. So that Sunil gets 4/5 of Sohil’s and Akhil’s share together and Sohil gets 2/3 of Akhil’s share.

Question 8. The ratio between two numbers is 3 : 4. If their L.C.M., is 180. Find the numbers.

Question 10. Find the compound ratio of the following:
(i) If A : B = 4 : 5, B : C = 6 : 7 and C : D = 14 : 15. Find A : D.
(ii) If P : Q = 6 : 7, Q : R = 8 : 9 find P : Q : R.

Question 11. Find:
(i) The duplicate ratio of 7 : 9
(ii) The triplicate ratio of 3 : 7
(iii) The sub-duplicate ratio of 256 : 625
(iv) The sub-triplicate ratio of 216 : 343
(v) The reciprocal ratio of 8 : 15.

Question 13. (i) What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional ?
(ii) What least number must be added to each of the numbers 5, 11, 19 and 37, so that they are in proportion ?

Question 15. The work done by (x – 3) men in (2x + 1) days and the work done by (2x + 1) men in (x + 4) days are in the ratio of 3 : 10. Find the value of x.

Question 16. Find the third proportional to:

Question 17. Find the fourth proportional to:

Question 18. Find the two numbers such that their mean proprtional is 24 and the third proportinal is 1,536.

Question 19. Using componendo and idendo, find the value of x

Question 20. Solve for x:

Question 21. Using the properties of proportion, solve for x, given.

Question 25. Find the value of

Prove the Following 

Question 1. If (a – x) : (b – x) be the duplicate ratio of a : b show that:

Question 2. If a : b with a ≠ b is the duplicate ratio of a + c : b + c, show that c² = ab.

Question 3. If a : b = 5 : 3, show that (5a + 8b) : (6a – 7b) = 49 : 9.

Question 7. If x and y be unequal and x : y is the duplicate ratio of (x + z) and (y + z) prove that z is mean proportional between x and y.

Question 8. If ax = by = cz, prove that

Question 15. If a, b, c, d are in continued proportion, prove that

Question 16. If q is the mean proportional between p and r, prove that

Question 21. If a, b, c, d are in continued proportion, prove that:

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Compound Interest Class 10 Maths ICSE Solutions

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Formulae

Formulae Based Questions

Question 1. Find the Compound Interest on Rs. 2,000 for 3 years at 15% per annum Compounded annually.

Question 2. If the interest is compounded half yearly, calculate the amount when the Principal is Rs. 7,400, the rate of interest is 5% per annum and the duration is one year.

Question 3. In how many years will Rs. 15,625 amount to Rs. 17,576 at 4% p.a., compound interest?

Question 4. The population of a city is 1,25,000. If the annual birth rate and death rate are 5.5% and 3.5% respectively. Calculate the population of the city after 3 years.

Question 5. There is a continuous growth in population of a village at the rate of 5% per annum. If its present population is 9,261, what population was 3 years ago?

Question 6. The population of a town 2 years ago was 62,500. Due to migration to cities, it decreases at the rate of 4% per annum. Find its present population.

Question. 7. The total number of industries in a particular portion of the country is approximately 1,600. If the government has decided to increase the number of industries in the area by 20% every year; find the approximate number of industries after 2 years.

Question 8. The cost of a machine depreciates by 10% every year. If its present worth is Rs.18,000; what will be its value after three years?

Question 9. 6000 workers were employed to construct a river bridge in four years. At the end of first year, 20% workers were retrenched; At the end of second year 5% of the workers at that time were retrenched. However, to complete the project in time, the number of workers was increased by 15% at the end of third year. How many workers were working during the fourth year?

Concept Based Questions

Question 1. The S.I. and C.I. on a sum of money for 2 years is Rs. 200 and 210 respectively. If the rate of interest is the same. Find the sum and rate.

Question 2. Find the difference between the simple interest and compound interest on 2,500 for 2 years at 4% p.a., compound interest being reckoned semi-annualy.

Question 3. A sum of money is lent out at compound interest for two years at 20% p.a., being reckoned yearly. If the same sum of the money was lent Gut at compound interest of the same rate of percent per annum C.I., being reckoned half yearly would have fetched Rs. 482 more by way of interest. Calculate the sum of money lent out.

Question 5. Mr. Kumar borrowed Rs. 15,000 for two years. The rate of interest for the two successive years are 8% and 10% respectively. If he repays Rs. 6,200 at the end of the first year, find the outstanding amount at the end of the second year.

Question 6. The compound interest, calculated yearly, on a certain sum of money for the second year is Rs. 1320 and for the third year is Rs. 1452. Calculate the rate of interest and the original sum of money.

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Matrices Class 10 Maths ICSE Solutions

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Formulae

Addition of Matrices: Let A and B be two matrices each of order m × n. Then their sum A + B is a matrix of order m × n and is obtained by adding the corresponding elements of A and B.

Properties of Matrix Addition:

1. Matrix addition is commutative
   i.e., A + B = B + A
2. Matrix addition is associative for any three matrices A, B and C.
   A + (B + C) = (A + B) + C.
3. Existence of identity.
   A null matrix is identity element for addition.
   i.e., A + 0 = A = 0 + A.
4. Cancelation laws hold good in case of matrices.
   A + B = A + C ⇒ B = C.

Subtraction of Matrices:
For two matrices A and B of the same order, we define
A – B = A + (- B).

Properties of Matrices Multiplication

1. Matrix multiplication is not commutative in general for any two matrices AB ≠ BA.
2. Matrix multiplication is associative
   i.e., (AB) C = A (BC) when both sides are defined.
3. Matrix multiplication is distributed over matrix addition
   i.e., A (B + C) = AB + AC
   (A + B) C = AC + BC.
4. If A is an n × n matrix then
   I_nA = A = AI_n
5. The product of two matrices can be the null matrix while neither of them is the null matrix.

Determine the Following

Prove the Following 

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