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1
SUMMER BREAK
–
ACADEMIC ENGAGEMENT
–
2019 GRADE 10 MATHEMATICS
REAL NUMBERS
1 Find the decimal expansion of
931500
2 Show that any number of the form
15
n
can
not have 0 in its one’s place, where n is a natural
number. 3 Find the LCM of the numbers given below: m, 2m, 3m, 4m and 5m, where m is any positive integer. 4 5 Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q. 6 Three alarm clocks ring at intervals of 4, 12 and 20 minutes respectively. If they start ringing together, after how much time will they next ring together? 7 If
3
2782
m
has a terminating decimal expansion and m is a positive integer such that 2 < m < 9, then find the value of m. 8 A charitable trust donates 28 different books of Mathematics, 16 different books of Science and 12 different books of Social Science to students. Each student is given maximum number of books of only one subject of their interest and each student got equal number of books. (a) Find the number of books each student got. (b) Find the total number of students who got the books. 9
Using Euclid’s
division algorithm, find the largest number which divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3 respectively. 10 Prove that the product of three consecutive positive integers is divisible by 6. 11 What will be the least possible number of planks, if three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length? 12 The H.C.F and L.C.M. of two numbers are 33 and 264 respectively. When the first number is completely divided by 2, the quotient is 33. Find the other number.
POLYNOMIALS
1 Find a quadratic polynomial whose sum of the zeroes is 6 and one of the zero is 3 -
√5
. 2 On dividing 2x
4
–
4x
3
–
4x
2
+ 7x
–
2 by a polynomial g(x), the quotient and the remainder were 2x
2
–
4x and (
–
x
–
2) respectively. Find g(x).
3
If α and β are the two zeroes of 25p
2
–
15p + 2 find a quadratic polynomial whose zeroes are 1/
2α and 1/2β
4 If one zero of the polynomial 3x
2
–
8x - (2k +1) is seven times the other, find the zeroes of the polynomial and the value of k 5 If the polynomial f(x) = 3x
4
–
9x
3
+ x
2
+15x +k is completely divisible by 3x
2
–
5 find the value of k. 6 If one of the zero of the quadratic polynomial 4x
2
–
8kx +8x- 9 is additive inverse of the other, then find the zeroes of kx
2
+3kx +2.
2
7
α and β are the zeroes of the quadratic polynomial 2x
2
–
5x +7. Find a polynomial whose
zeroes are 2α+3β and 3α +2β
. 8 Find a quadratic polynomial, the sum and product
of whose zeroes are √2 and
32
respectively. Also find its zeroes. 9 If the remainder on division of x
3
+ 2x
2
+ kx +3 by x
–
3 is 21, find the quotient and the value of k. 10
Given that √2 is a zero of the cubic polynomial 6x
3
+ √2 x
2
–
10x
–
4 √2 , find its other two
zeroes. 11 If
α and β are the zeroes of the quadratic polynomial x
2
–
x
–
k and 3 α +2
β = 20 find k.
12 On dividing 9x
4
–
4x
2
+ 4 by 3x
2
+ x
–
1 the remainder is ax
–
b find a and b
LINEAR EQUATIONS
1 Draw the graphs of the equations
x
–
y
+ 1 = 0 and 3
x
+ 2
y
–
12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the
x
-axis, and shade the triangular region. 2 Solve the following pair of linear equations by the cross-multiplication method : 8
x
+ 5
y
= 9, 3
x
+ 2
y
= 4 3 Solve for x and y : (a - b)x + (a + b) y = a
2
–
2ab - b
2
( a + b )(x + y) = a
2
+ b
2
4 Solve for x and y:
0
x ya b
ax + by = a
2
+ b
2
5 Find the values of a and b for which the following system of linear equations has infinite number of solutions: i)
2x
–
y =5; (a- 2b) x
–
(a + b) y = 15. ii) (2a
–
1) x
–
3y = 5 ; 3x + ( b-2)y = 3 iii) 3x
–
y = 14 ; (a+ b) x -2by =5a +2b + 1 6 Determine the value of k for which the following system of linear equations has no solution: a)
2x
–
k y +3 = 0 ; 3x +2y
–
1 = 0 7 A person invested some amount at the rate of 12% simple interest and some other amount at the rate of 10% simple interest. He received yearly interest of Rs 130 .But if he had interchanged the amounts invested, he would have received Rs 4 more as interest. How much amount did he invest at different rates?
3
8 After covering a distance of 30 km with a uniform speed there is some defect in a train engine and therefore its speed is reduced to
45
of its srcinal speed .Consequently the train reaches its destination late by 45 minutes .Had it happened after covering 18 km more the train would have reached 9 minutes earlier .Find the speed of the train and the distance of journey. 9 X takes 3 hours more than Y to walk 30km . But , if X doubles his pace , he is ahead of Y by 1
12
hrs. Find their speed of walking. 10 Solve for x and y :
7 258 715
x y xy x y xy
11 4 men and 6 boys can finish a piece of work in 5 days while 3 men and 4 boys can finish it in 7 days. Find the time taken by 1 man alone or than by 1 boy alone. 12 On selling a tea-set at 5% loss and a lemon-set at 15% gain, a crockery seller gains Rs7.If he sells the tea-set at 5% gain and the lemon-set at 10% gain ,he gains Rs13.Find the actual price of the tea-set and the lemon-set.
QUADRATIC EQUATION
1 If
12
is a root of the equation x
2
+kx -
54
= 0 Find the value of k 2 Find the roots of the following quadratic equations by factorization method a)
3x
2
+ 5
5
x + 10 =0 b) 6 x
2
–
x -2 = 0 ; x
0 b)
3 Solve for x :
16 15- 1=x x+1
; x
≠ 0 ,1
4 Solve for x : 12abx
2
- ( 9a
2
–
8b
2
)x - 6ab = 0 5 Find the value of k for which the equation x
2
+ k ( 2x + k
–
1 ) + 2 = 0 has real and equal roots 6
If ad ≠
bc, then prove that (a
2
+ b
2
) x
2
+2 (ac +bd) x +(c
2
+d
2
) = 0 has no real roots. 7 If the equation ( 1+m
2
)x
2
+ 2mcx +(c
2
–
a
2
) = 0 has equal roots, show that c
2
= a
2
(1+m
2
) 8 Find three consecutive positive integers whose product is equal to sixteen times their sum 9 90% and 97% pure acid solutions are mixed to obtain 21 litres of 95% pure acid solution . Find the quantity of each type of acid to be mixed to form the mixture.
4
10 If zeba is younger by 5 years than what she really is,then the square of her age (in years) would have been 11 more than five times her actual age.What is her age now? 11 A train travelling at a uniform speed for 360km/hr would have taken 48 minutes less to travel the same distance if its speed were 5km/hr more.Find the srcinal speed of the train. 12 Some students planned a picnic.The total budget for food was Rs 2000.But 5 students failed to attend the picnic and thus the cost of food for each member increased by Rs 20.How many students attended the picnic and how much did each student pay for the food?
STATISTICS
1 Find the mean (by step deviation method), median and mode of the following data: Classes 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Frequency 5 10 18 30 20 12 5 2 The mean of the following data is 53, find the missing frequencies. Age(in years) 0-20 20-40 40-60 60-80 80-100 Total Number of people 15 F
1
21 F
2
17 100 3 Find the mean marks of the students from the following frequency distribution. Marks Less than 10 Less than 20 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 Less than 80 Less than 90 Less than 100 No: of students 5 9 17 29 45 60 70 78 83 85 4 The following table shows the marks obtained by 100 students of class X in a school during a particular academic session. Find the mode of this distribution. Marks Less than 10 Less than 10 Less than 10 Less than 10 Less than 10 Less than 10 Less than 10 Less than 10 No: of students 7 21 34 46 66 77 92 100 5 The mode of the following distribution is 43.75. Find the value of p. Class Interval 20-30 30-40 40-50 50-60 60-70 Frequency 25 47 62 p 10 6 Find the median wages for the following frequency distribution. Wages per day(in Rs) 61-70 71-80 81-90 91-100 101-110 111-120 No: of workers 5 15 20 30 20 8

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