PRACTICSE QUESTION FOR 10th MATHS
PRACTICSE QUESTION FOR 10th MATHS
Polynomials
1. Find the value of k if (x – 2) is a factor of 2x3 – 6x2 + 5x + k.
2. For what value of m is 2x3 + mx2 + 11x + m + 3 exactly divisible by (2x – 1)
3. If x + 1 and x – 1 are factors of mx3 + x2 – 2x + n, find the values of m and n.
4. Find p and k such that x + 2 and x – 2 are factors of the polynomials px4 + 2x3 – 3x2 + kx – 4.
5. What must be subtracted from 4x4 – 2x3 – 6x2 + x – 5 so that the result is exactly divisible by
2x2 + x – 1?
6. Using factor theorem, factorise the polynomial
(a) x4 – 2x3 – 7x2 + 8x + 12 (b) x4 + x3 – 7x2 – x + 6
7. Find the values of m and n so that y4 + my3 + 2y2 – 3y + n is divisible by y2 – 1.
8. If mx3 + nx2 + x – 6 has x + 2 as a factor and leaves a remainder 4 when divided by (x – 2), find the values of m and n.
9. Let A and B are the remainders when the polynomial y3 + 2y2 – 5ay – 7 and y3 + ay2 – 12y + 6 are divided by y + 1 and y – 2 respectively. If 2A + B = 6, find the value of a.
Factorizing a Polynomial
Factorize each of the following expressions : (Q.No. 1 to Q.No. 6)
1. p4 – 81q4. 2. 7 √2 x2 – 10x – 4√2. 3. 6√3a2 – 47a + 5√3.
4. x4 – (2y – 3z)2 5. x3 + x – 3x2 – 3. 6. 8x3 + 3√3y3
7. If one of the factors of x2 + x – 20 is (x + 5), find other factor.
Factorize each of the following expressions : (Q.No. 8 to Q.No. 15)
8. 4(m + n)2 – 28p(m + n) + 49p2 9. (x2 + 4y)2 + 21(x2 + 4y) + 98
10. 25a2 – 10a + 1 – 36b2 11. 12(x2 + 7x2) – 8(x2 + 7x)(2x – 1) – 15(2x – 1)2
12. x8 – 11x4y4 – 80y8 13. 8x3 + 125y3 + 60x2y + 150xy2
14. x3 – 8y3 – 6x2y + 12xy2 15. (2a – 3b)3 + (3b – 5c)3 + (5c – 2a)3
Linear Equations in Two Variables
1. Express y in terms of x in the equation 2x + 3y = 11. Find the point where the line represented by the equation 2x + 3y = 11 cuts y-axis.
2. Solve for x and y : 2/x +2/3y= 1/6,3/x+2/y = 0 and hence find ‘a’ for which y = ax – 4.
3. Solve 4x +6y= 15 and 6x – 8/y = 14 and hence find ‘p’ if y = px – 2.
4. For what value of k will the system of linear equations have infinite number of solutions :
kx + 4y = k – 4, 16x + ky = k ?
5. Determine the value of k for which the following system of linear equations has infinite number of solutions : (k – 3)x + 3y = k; kx + ky = 12.
6. For what value of k, will the following system of equations have infinite solutions :
2x + 3y = 4, (k + 2)x + 6y = 3k + 2 ?
7. For what value of k, will the following system of equations have infinite solutions :
2x – 3y = 7, (k + 2)x – (2k + 1)y = 3(2k – 1) ?
8. A person invested some amount at the rate of 10% simple interest and some other amount at the rate of 12% simple interest. He received yearly interest of Rs. 130. But if he had interchanged the amount invested, he would have received Rs. 4 more as interest. How much amount did he invest at different rates?
9. Solve the system of equations graphically : 2(x – 1) = y; x + 3y = 15. Also, find the co-ordinates of the points where the lines meet the axis of y.
10. Solve graphically the system of linear equations : 2x + 3y = 12; 2y – 1 = x. Also find the co-ordinates of the points where the lines meet the y-axis.
11. Ramesh travels 760 km to his home partly by train and partly by car. He takes 8 hrs, if he travels 160 km by train and the rest by car. He takes 12 minutes more if he travels 240 km by train and the rest by car. Find the speed of the train and the car separately.
12. Solve for x, y and z : x + y + z = 9; 2y – z = 2; z – x = 2.
13. Solve for x and y : 631x + 279y = 910; 279x + 631y = 910.
14. Solve for x and y : 6x + 3y = 8x + 9y – 5 = 10x + 12y – 8.
15. Solve the following equation by using the method of cross multiplication:
x/a+y/b= a + b;x/a2+y/b2=2.
16. Solve the following equation by using the method of cross multiplication:
a(x + y) + b(x – y) = a2 – ab + b2
a(x + y) – b(x – y) = a2 + ab + b2
17. Ratio between the girls and boys in a class of 40 students is 2 : 3. Five new students joined the class. How many of them must be boys so that the ratio between girls and boys becomes 4 : 5 ?
18. If you travel by an autorickshaw the fare for the first kilometre is different from the rate per km for the remaining distance. The total fare for a distance of 20 km is Rs. 37.70 and that for a distance of 26 km is Rs. 48.50. Find the auto fare for the first kilometre and for each successive kilometre.
19. Ten years ago, the sum of the ages of two sons was one third of their father’s age. One son is two years older than the other and sum of their present ages is 14 years less than the father’s present age. Find the present ages of all.
20. Solve for x and y :
bx + ay = a + b
ax (1/a-b-1/a+b)+by(1/b-a-1/b+a)= 2a/a+b
21. A villager went to a hotel in a town with his big family. They consumed 23 idlies, 18 pooris, 7 dosas and 19 vadas. The bill come to Rs. 108. On next day, they consumed 34 idlies, 8 vadas, 22 pooris and 7 dosas. The bill came to Rs. 114. If an idli costs the same as a vada, what is the cost of one poori?
22. A boat goes 24 km upstream and 28 km downstream in 6 hrs. It goes 30 km upstream and 21 km downstream in 6½ hrs. Find the speed of the boat in still water and also speed of stream.
23. A dealer sold a TV and VCR for Rs. 25,820 making a profit of 15% on TV and 10% on VCR. By selling them for Rs. 25,930, he would have realised a profit of 10% on TV and 15% on VCR. Find cost price of each.
24. A man sold a chair and a table together for Rs. 1520 there by making a profit of 25% on chair and 10% on table. By selling them together for Rs. 1535 he would have made a profit of 10% on the chair and 25% on the table. Find the cost price of each.
25. Solve for x and y :
x- y2/y –x+y/14 =x-y-1/8 –y+12/4;x+7/3+y-5/10= 1 – x –5(y-1)/7
26. Solve for a and b : 2a + 3b = 17 and 2a + 2 – 3b + 1 = 5.
27. For what value of k, will the following system of equations have infinite solutions : 2x – 3y = 7, (k + 2)x – (2k + 1)y = 3(2k – 1)?
Quadratic Equations
1 The sum of the reciprocals of two consecutive even numbers is 7/24. One number is 6. What is the other number?
2. If a, b are roots of the quadratic equation x2 – 6x + k = 0, find the value of k such that
a2 + b2 = 40.
3. If a, b are roots of quadratic equation 2x2 + 5x + k = 0, find the value of k if a2 + b2 + ab = 21/4.
4. If a, b are roots of the quadratic equation kx2 + 4x + 4 = 0, find the values of k such that
a2 + b2 = 24.
5. Find the value of k such that the quadratic equation x2 – (k + 6)x + 2(2k – 1) = 0 has sum of the roots equal to half their products.
6. If –4 is a root of the quadratic equation x2 + px – 4 = 0 and the quadratic equation x2 + px
+ k = 0 has equal roots, find the value of k.
7. One root of the equation 2x2 – 8x – m = 0 is 5/2. Find the other root and the value of m.
8. If one root of the quadratic equation 2x2 + ax + 3 = 0 is 1, find the other root and the value of a.
9. If a and b are the roots of x2 – 5x + 4 = 0, find the value of 1/ a + 1/ b – 2ab.
10. Rs. 250 are divided equally among a certain number of children. If there were 25 children more, each would have received 50 paise less. Find the number of children.
11. Two circle touch internally. The sum of their areas is 116 p sq. cm and the distance between their centres is 6 cm. Find the radii of the circles.
12. If a, b are the roots of the quadratic equation 2x2 + 5x + 1 = 0, form an equation whose roots are
a + a/b and b + b/a .
13. Solve for x : 9x+ 2 – 6.3x + 1 + 1 = 0.
14. If a, b are the roots of the quadratic equation 3x2 – 6x + 4 = 0, find the value of
(a/b + b/a ) + 3(1/a +1/b) + 4ab
15. In a flight of 3000 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 100 km/hour and time increased by one hour. Find the original duration of flight.
16. Find a and b such that x + 1 and x + 2 are factors of the polynomial x3 + ax2 – bx + 10.
17. Solve for x : 6 (x2 +1/x2 )– 25 ( x –1/x)+ 12 = 0.
18. Solve for x : (x + 1/x)2 -- 3/2(x – 1/x) – 4 = 0.
21. Solve for x :
(x – 1/x) 2+ 8 (x +1/x)= 29 , x ¹ 0.
23. X and Y are centres of circles of radius 9 cm and 2 cm and XY = 17 cm. Z is the centre of a circle of radius r cm, which touches the above circles externally. Given that ÐXYZ = 90°, write an equation in r and solve it for r.
24. For what value of k, (4 – k)x2 + (2k + 4)x + (8k + 1) = 0 is a perfect square?
25. Find the value of p so that the roots of the equation 2x2 + 6x + p = 0 differ by 1.
26. Write a rational expression whose numerator is a quad. polynomial with zeros 2 and –3 and whose denominator is a polynomial with zeros –2, 1 and 4.
27. Find the value of k so that one root of the equation 8x2 + kx + 1 = 0 may be double of the other.
28. The difference of mother’s age and her daughter’s age is 21 years and the twelfth part of the product of their ages is less than the mother’s age by 18 years. Find their ages.
29. If a, b are the roots of the equation 3x2 – 4x + 1 = 0, form an equation whose roots are 
and 
30. Solve : x2/3 + x1/3 – 2 = 0.
31. Some students planned a picnic. The budget for food was Rs. 480. But eight of these failed to go and thus the cost of food for each member increased by Rs. 10. How many students attended the picnic?
32. Solve the equation :
– 
= 
33. Solve the equation : 2(x – 3)2 + 3(x – 2)(2x – 3) = 8(x + 4)(x – 4) – 1.
34. If one root of 3x2 = 8x + (2k + 1) is seven times the other, then find the roots and value of k.
35. Solve : (x2 + 3x + 2)2 – 8(x2 + 3x) – 4 = 0.
36. Solve : (x – 1)(x – 2)(3x – 2)(3x + 1) = 21.
37. A fox and an eagle lived at the top of a cliff of height 6 m whose base was at a distance of 10m from a point A on the ground. The fox descends the cliff and went straight to the point A. The eagle flew vertically up to a height x and then flew in a straight line to point A, the distance, travelled by each being the same. Find the value of x.
38. Solve for x : (x2 + x – 6)(x2 – 3x – 4) = 24.
39. If one root of the quadratic equation 2x2 – 3x + p = 0 is 3, find the other root of the quadratic equation.
40. If ax2 – 7x + c = 0 has 14 as the sum of the roots and also as the product of the roots, find the values of a and c.
41. Rs. 6,500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs. 30 less. Find the original number of persons.
42. For what value of 'p' the equation (1 + p)x2 + 2(1 + 2p)x + (1 + p) = 0 has coincident roots ?
8. Mensuration A
1. Find the area of a right ∆ABC at B if the radius of its circumcircle is 3 cm and altitude drawn to the hypotenuse is 2 cm..tr abc with hypothnuse AC=6cm,BD ┴ AC,BD=2cm
2. If the diameter of a semicircular protractor is 14 cm, then find its perimeter. (Take p = 22/7)
3. The diagonals of a rhombus are 15 cm and 36 cm long. Find its perimeter.
4. The length of a rectangle is twice its breadth. Find the dimensions of the rectangle, if its area is 288 sq. cm.
5. Find the volume and the total surface area of a hemisphere of radius 2 cm.
6. A wire is looped in the form of a circle of radius 28 cm. It is reverted into a square form. Determine the side of the square. (Use p = 22/7)
7. In the given figure, OPQR is a rhombus, three of whose vetices lie on a circle with centre O. If the area of the rhombus is 32 √3 cm2, find the radius of the circle. OR= r
8. A trapezium PBCQ, with its parallel sides QC and PB in the ratio 7 : 5, is cut off from a rectangle ABCD. If the area of the trapezium is 4/7 part of the area of the rectangle, find the length of QC and PB.
9. A sector is cut off from a circle of radius 21 cm. The angle of the sector is 120°. Find the length of its arc and the area. (Take p = 22/7)
10. AB is chord of circle of radius 10 cm. The chord subtends a right angle at the centre of the circle. Find the area of the minor segment. (Take p = 3.14)
11. The difference between outside and inside surfaces of a cylindrical metallic pipe 14 cm long is 44 sq. cm. If the pipe is made of 99 cubic cm. of metal, find the outer and inner radii of the pipe. (Take p = 22/7)
12. How many metres of cloth 5 m wide will be required to make a conical tent, the radius of whose base is 7 m and whose height is 24 m. (Take p = 22/7)
13. The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of the distances travelled by their tips in two days. (Take p = 22/7)
14. A road which is 7 meters wide surrounds a circular park whose circumference 352 meters. Find the area of the road. (Take p = 22/7)
5. If h, c and V respectively are the height, the curved surface and volume of a cone, prove that
3pVh3 – c2 h2 + 9V2 = 0.
16. A right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled in cones of height 12 cm and diameter 6 cm having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream.
17. A cylindrical jar of radius 6 cm contains oil. Iron spheres each of radius 1.5 cm are immersed in the oil. How many spheres are necessary to raise the level of the oil by two centimetres?
18. A circus tent is in the form of a right circular cylinder and a right circular cone above it. The diameter and the height of the cylindrical part of the tent are 126 m and 5 m respectively. The total height of the tent is 21 m. Find the total surface area of the tent. Also find the cost of the tent if the canvas used costs Rs. 12 per square metre. (Take p = 22/7)
19. The radii of the internal and external surfaces of a hollow spherical shell are 3 cm and 5 cm respectively. If it is melted and recast into a solid cylinder of diameter 14 cm, find the height of the cylinder.
20. A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 2.1 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical tub, full of water, in such a way that the whole solid is submerged in water. If the radius of the cylinder is 5 cm and its height is 9.8 cm, find the volume of the water left in the cylindrical tub. (Take p = 22/7)
21. The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle.
22. Find the area of the shaded Region in the given figure 
.
23. The diameter of the driving wheel of a bus is 140 cm.
How many revolutions per minute must the wheel make
in order to keep a speed of 66 km/hr?
24. In the figure OE = 20 cm. In the sector OSFT, square OEFG is inscribed. Find the area of the shaded region.
25. The area of the bottom of a rectangular pit is 8.75 m2. If its depth is 1.8 m, determine its volume.
26. A well is 6 m deep. The cost of cementing its inner surface at 50 paise per dm2 is Rs. 1320. Determine the diameter of the well.
27. The total surface area of a solid right circular cylinder is 231 cm2. Its curved surface is 2/3rd of the total surface. Determine the radius of its base and height.
28. If V is the volume of cuboid of dimensions a, b, c and S is the surface area, then prove that
= 
(
+
+
)
29. The perimeter of a triangular field is 450 mts. and its sides are in the ratio 13 : 12 : 5. Find the area of triangle.
30. The sides of a quadrangular field, taken in order are 26 m, 27 m, 7 m and 24 m respectively. The angle contained by the last two sides is a right angle. Find its area.
31. A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60 cm, calculate the speed per hour with which the boy is cycling.
32. A chord AB of a circle of radius 15 cm makes an angle of 60° at the centre of the circle. Find the area of the major and minor segments.
33. Water in a canal 30 dm wide and 12 dm deep is flowing with a velocity of 20 km/hr. How much area will it irrigate in 30 min if 9 cm of standing water is desired?
34. A field is in the form of a rectangle of length 18 m and width 15 m. A pit 7.5 m long, 6 m broad and 0.8 deep is dug in a corner of the field and the earth taken out is spread over the remaining area of the field. Find out the extent to which the level of the field has been raised.
35. The diameter of a roller 120 cm long is 84 cm. If it takes 500 complete revolutions to level a playground, determine the cost of levelling it at the rate of 30 paise per square metre.
36. The circumference of the base of 10 m high conical tent is 44 m. Calculate the length of canvas used in making the tent if width of canvas is 2 m.
37. A heap of wheat is in the form of a cone of diameter 9 m and height 3.5 m. Find its volume. How much canvas cloth is required to just cover the heap?
38. A cylindrical tub of radius 12 cm contains water to a depth of 20 cm. A spherical ball is dropped into the tub and thus the level of water is raised by 6.75 cm. What is the radius of the ball?
39. A rectangular sheet of aluminium foil is 44 m long and 20 cm wide. A cylinder is made out of it, by rolling the foil along its length. Find the volume of the cylinder.
40. A hollow spherical shell is made of metal of density 4.9 g/km3. If the internal and external radii are 10 cm and 12 cm respectively. Find the weight of the shell.
41. The length of the sides forming right angle of a right angled triangle are 5x cm and (3x – 1) cm. If the area of the triangle is 60 cm2, find its hypotenuse.
42. Find the weight of a lead pipe 3.5 m long if the external diameter of the pipe is 2.4 cm and the thickness of the lead is 2 mm and 1 cubic cm of lead weighs 11 g.
43. An iron pillar consists of a cylindrical portion 2.8 m high and 20 cm in diameter and a cone
42 cm high is surmounting it. Find the weight of the pillar, given that 1 cubic cm of iron weighs 7.5 g.
44. The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27 of the volume of the given cone, at what height above the base, the section has been made?
45. A circle park is surrounded by a road 21 m wide. If the radius of the park is 105 m, find the area of the road.
46. An agricultural field is in the form of a rectangle of length 20 m and width 14 m. A pit 6 m long, 3 m wide and 2.5 m deep is dug in the corners of the field and the earth taken out of the pit is spread uniformly ever the remaining area of the field. Find the extent to which the level of the field has been raised?
Measures of Central Tendency
1 If the mean of the following data is 21, find the value of p :
x : 10 15 20 25 35
y : 6 10 p 10 8
2. There are 50 students in a class of which 40 are boys and the rest girls. The average weight of the class is 44 kg and the average weight of the girls is 40 kg. Find the average weight of the boys.
3. There are 120 students in a class in which 20 of them are girls and the rest boys. If the average marks in mathematics of the boys is 65% and that of girls is 80%, find the average marks of the class.
4. The median of the following of observations, arranged in ascending order is 24. Find x :
11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41
5. If the mean of n observations x1, x2, x3, ..........., xn is x, prove that the mean of the observations x1 + a, x2 + a, x3 + a, ..........., xn + a, is x + a.
6. If the mean of the following distribution is 6, find the value of p :
x : 2 4 6 10 p + 5
y : 3 2 3 1 2
7. If x is the mean of the ten natural numbers x1, x2, .........., x10, show that
(x1 – x) + (x2 – x) + (x3 – x) + ............ (x10 – x) = 0.
8. The aggregate expenditure of a family on certain number of units of different household commodities in 1987 was Rs. 7,200 and the cost of living index number of 1996 on the basis of 1987 was Rs. 180.50. How much did the family spend in 1996 on the same units of commodities?
9. If the mean of n observations x1, x2, x3, .........., xn is x, prove that
(x1 – x) + (x2 – x) + (x3 – x) + ............ (xn – x) = 0.
10. The median of following observations, arranged in ascending order, is 25. Find x.
11, 13, 15, 19, x + 2, x + 4, 30, 35, 39, 46
11. Determine the median of 24, 23, a, a – 1, 12, 16, where a is the mean of 10, 20, 30, 40, 50.
12. The average score of boys of boys in an examination of a school is 71 and that of the girls is 73. The average score of the school examination is 71.8. Find the ratio of the number of boys to the number of girls that appeared in the examination.
13. Assuming that the consumption remains the same, find the cost of living index number (to the nearest integer) for the year 1981 (using 1975 as a base year) for the data of items used in a family which is given in the table :
S. No. Items Consumption Rate per kg (in Rs.)
(in kg) 1975 1981
1 A 30 2.50 3.50
2 B 15 10.00 11.00
3 C 7 7.00 9.00
4 D 12 3.50 4.50
5 E 5 40.00 45.00
14. Find the value of p, if the mean of the following distribution is 18 :
x : 13 15 17 19 20 + p 23
f : 8 2 3 4 5p 6
15. Calculate the arithmetic mean for the following frequency distribution :
Class interval 0-80 80-160 160-240 240-320 320-400
Frequency 22 35 44 25 24
16. If the arithmetic mean of 6, 8, 5, 7, x and 4 is 7, then find the value of x.
17. If the mean of the marks of five students is 33 and that of the marks of four of them is 32.5, then find the makes obtained by fifth student.
18. If the median of 6, 7, x – 2, x, 17 and 20 written in ascending order, is 16, find the value of x.
19. 20 years ago, when my parents got married, their average age was 23 years, now the average age of my family consisting of myself and my parents is 34 years. What is my present age?
20. The mean of 6, y, 7, x and 14 is 8. Express y in terms of x.
21. Find the median of the following data : 46, 64, 87, 41, 58, 77, 35, 90, 55, 33, 92.
If the observation 92 is replaced by 19 and 41 by 43, determine the new median.
22. The mean heights of 10 students was 153 cm. But later on it was discovered that 151 cm was wrongly read as 141 cm. Find the corrected mean.
23. Complete the following table and find Crude Death Rate (CDR) for the following :
Age group Population Number of deaths
0-10 23000 300
10-25 ....... 110
25-45 37000 100
45-70 25000 .......
above 70 15000 400
Total 125000 1230
24. The mean monthly salary of the 12 employees of a firm is Rs. 1450. If one more person joins the firm who gets Rs. 1600 per month, what will be the mean monthly salary now?
25. The average weight of A, B, C is 45 kg. If the average weight of A and B be 40 kg and that of B and C be 43 kg, find the weight of B.
26. The mean of the following frequency table is 50. But the frequencies f1 and f2 in classes 20-40 and 60-80 are missing. Find the missing frequencies.
Class 0-20 20-40 40-60 60-80 80-100 Total
Frequency 17 f1 32 f2 19 100
27. Find the average marks of students from the following table :
Marks No. of Students Marks No. of Students
Above 0 80 Above 60 23
Above 10 77 Above 70 16
Above 20 72 Above 80 10
Above 30 65 Above 90 8
Above 40 55 Above 100 0
Above 50 43
28. A candidate obtains the following percentages in an examination, English 46%, Mathematics 67%, Sanskrit 72%, Economics 58%, Political Science 53%. It is agreed to give double weights to marks in English and Mathematics as compared to other subjects. What is the weight mean?
29. The mean and median of the numbers 1, 2, 3, 4, y, 8, 9, 10, 12 and x written in increasing order, are both 7. Find the values of x and y.
30. The mean of 200 items was 50. Later on it was discovered that two items were misread as 92 and 8 instead of 192 and 88. Find the correct mean.
31. Calculate the cost of living index by aggregate expenditure method :
| Commodity | Base Year Quantity Price | Current Year Quantity Price | ||
| | ||||
| A | 12 | 10 | 15 | 12 |
| B | 15 | 7 | 20 | 5 |
| C | 24 | 5 | 20 | 9 |
| D | 5 | 16 | 5 | 14 |
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