Polynomials-linear equation test paper

10TH Mathematics
Section A
Solve any five of the following 5x3=15
1. Find the value of a and b for which the following system of linear equations has infinite numbers of solutions:
2x - 3y = 7 (a + b) x - (a + b - 3)y = 4a + b.) Solve the following system of linear equations

2. For what value of k will the system of equations

x + 2y = 5; 3x + ky + 15 = 0 has (i) unique solution (ii) no solution

3. solve the following system of linear equations: ax + by = c, b x + ay = 1 + c 4

4. Find a quadratic polynomial whose the sum and product of its zeros respectively. √3,2

5. If a and b are the zeros of a given quadratic polynomial f(x) = x2 + x - 2, find the value of 1/a + 1/b

6. If 2, ½ are the zeros of px2+5x+r, prove that p= r.

Section B       Solve any five questions 5x4=20

1. Find the zeros of the quadratic polynomial x2+ 9x + 20 , and verify the basic relationships between the zeros and the coefficients.

2. Five years ago, Neeta was trice as old as Reeta. Ten years later, Neeta will be twice as old as Reeta, How old are Neeta and Reeta now?

3. Person can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find man’s speed of rowing is still water and the speed of the current.

4. The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.

5. There are two class rooms A and B containing students. If 5 students are shifted from room A to room B, the resulting number of students in the two rooms become equal. If 5 student are shifted from room B to room A, the resulting number of student's in room A becomes double the number of student left in room B. find the original number of student in the two rooms separately.

6. Places A and B are 80 km apart from each other on a highway. A car starts from A and another from B at the same time . If they moves in the same direction, they meet in 8 hours and if they move in opposite directions, they meet in 1 hours and 20 minutes. Find the speed of the cars.

7. If a & ß are the zeroes of the polynomial 2x2 _ 4x + 5, then find the value of a3 + ß3

10th Probability Test sample paper

Probability

1.     A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545 compute the probability for each event.

2.     Two coins are tossed simultaneously 500 times, and we get- Two heads: 105 times,
One head: 275 times, No head: 120 times, find the probability of occurrence of each of these events.

3.     A die is thrown 1000 times with the frequencies for the outcome 1, 2, 3, 4, 5 and 6 as given in the following table:

Outcome	1	2	3	4	5	6
Frequency	179	150	157	149	1175	190

Find the probability of getting each outcome.

4.     The record of a weather station shows that out of the past 250 consecutive days, its weather forecasts were correct 175 times.
(i) What is the probability that on a given day it was correct?
(ii) What is the probability that it was not correct on a given day?

5.     The percentage (%) of the marks obtained by a student in the monthly unit test are given below:

Unit test	I	II	III	IV	V
Percentage (%) of the marks obtained	69	71	73	68	74

Based on this data, find the probability that the student gets more than 70% marks in a unit test.

6.     An insurance company selected 2000 drivers at random (i.e. without any preference of one driver over another) in a particular city to find a relationship between age and accidents. The data obtained are given in the following table:

Age of  drivers (in years	Accidents in one year
	0	1	2	3	Over 3
18-29	440	160	110	61	35
30-50	505	125	60	22	18
Above 50	360	45	35	15	9

Find the probabilities of the following events for a driver chosen at random from the city:
(i) Being 18-29 years of age and having exactly 3 accidents in one year.
(ii) Being 30-50 years of age and having one or more accidents in a year.
(iii) Having no accidents in one year.

7.     In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary.

8.     1500 families with 2 children were selected randomly, and the following data were recorded:

Numbers of girls in a family	2	1	0
Number of families	475	814	211

Compute the probability of a family, chosen at random, having
(i) 2 girls
(ii) 1 girl
(iii) No girl.

9.     Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

Outcome	3 heads	2 heads	1 head	No head
Frequency	23	72	77	28

If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.

10.   An organization selected 2400families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family, the information gathered is listed in the table below:

Monthly in come  (in Rs)	Vehicles per family
	0	1	2	Above 2
Less than 7000	10	160	25	0
7000-10000	0	305	27	2
10000-13000	1	535	29	1
13000-16000	2	469	59	25
16000 or more	1	579	82	88

Suppose a family is chosen. Find the probability that the family chosen is
(i) Earning Rs 10000-13000 per month and owing exactly 2 vehicles.
(ii) Earning Rs 16000 or more per month and owning exactly 1 vehicle.
(iii) Earning less than Rs 7000 per month and does not own any vehicle.
(iv) Earning Rs 13000-16000 per month and owning more than 2 vehicles.
(v) Owning not more than 1 vehicle.
11.   Eleven bags of wheat flour, each marked 5 kg, actually contained following weights of flour (in kg): 4.97 5.05 5.08 5.03 5.00 5.06 5.08 4.98 5.04 5.07 5.00 find the probability that any of these Bags chosen at random contains more than 5 kg of flour.

Table of Trigonometric Identities

Table of Trigonometric Identities

Reciprocal identities



Pythagorean Identities



Quotient Identities



Co-Function Identities



Even-Odd Identities



Sum-Difference Formulas



Double Angle Formulas



Power-Reducing/Half Angle Formulas



Sum-to-Product Formulas



Product-to-Sum Formulas

Polynomials

Mathematics For Class 10
Polynomials

Q 1   If the sum of the squares of the roots of the equation x² + 2x – p = 0 is 8, find the value of p

Q 2   If one root of the equation 6x² + 13x+ m = 0 is reciprocal of the other, find the value of m. 

Q 3 If f(x) = ax³ + bx² + cx + d, a ≠ 0, then what will be the sum of zeros?                              

Q 4 If the sum of the zeroes of the polynomial f(x) = 2x³ - 3kx² + 4x - 5 is 6, then find the value of k. 

Q 5 Find a quadratic polynomial, if the sum and the product of the zeroes are 4 and 4 respectively.

Q 6 The product of two zeroes of the polynomial f(x) = 2x³ + 6x² -4x + 9 is 3, then find its third zero.      

Q 7 If (x + 2)(2x - 1)(3x - 2) = 0, find the zeroes of the polynomial

Q 8 Find a quadratic polynomial, the sum and product of its zeroes are 1 and –6 respectively. 

 Q 9 Find a quadratic equation, whose roots are(1+√5) and (1-√5)  

Q 10 Find a quadratic polynomial, the sum and product of its zeroes are 8 and 15 respectively.
Q 11 Find a quadratic polynomial, the sum and product of whose zeroes are -7 and 7 respectively.
Q 12 Find a quadratic polynomial, the sum and product of whose zeroes are -5 and 4 respectively.           

Q 13 Find a quadratic polynomial, the sum and product of whose zeroes are –3 and 2 respectively.    
Q 14 Find the quadratic polynomials, the sum and product of whose zeroes are 4 and 1 respectively.  
Q 15 If α and β  are the root of  the equation ax² + bx + c= 0  Find the value of 1/α     + 1/β

Q 16 If α and β are the roots of the equation 25x² - 10x + 1= 0, find the value of α² + β².  

Q 17 If the zeroes of the polynomial f(x) = x³ - 3x + x + 1 are  a – b, a and a + b find a, b.
Q 18 If αand β are the zeroes of the polynomials f(x) = x² - px + q, find the value of α²+β².
Q 19 If αand β are the zeroes of a quadratic polynomial such that α+ β = 24 and α- β = 8, then find the quadratic polynomial. 

Q 20  Find a quadratic polynomial , the sum of whose Zero is  -1 and sum of their reciprocal is  1/6 

Q 21 Write a quadratic polynomial , the sum of whose roots  is  2√3and sum of their reciprocal is  2.

Q 22 If α and β are the root ofs of a ax² -  b x + c= 0 , what is the value of √ α/β     + √β/α

  

Q 23 If α and β are the zeroes of polynomial 9x² - 3x - 2, evaluate α^-1 + β^-1. 

Q 24 A quadratic polynomial 2x² - mx + n has α and β as its two zeroes. Evaluate α² + (β)2. 

Q 25 Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time, and the product of its zeroes as 2, - 7 and –14 respectively. 

Q 26 If α, β are the roots of the equation 3x² - 4x + 1 = 0, find the value of  α³ + β³.

Q 27 If the zeroes of the polynomial f(x)  = x³  - 3x² + x + 1 are a – b, a and a + b, find the values of a and b.
Q 28 If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, then evaluate α²β + αβ². 

 Q 29If α and β are the zeroes of the polynomial p(x) = x² - 16x + 63, then find the value of α⁴β³ + α³β⁴.

Q 30 If the sum of the squares of the zeroes of a quadratic polynomial x^­­­2 –18x + p is 180, find the value of p. 

TEST PAPER Real Numbers

Mathematics For Class 10

TEST PAPER Real Numbers

1.     Why is 7x11x13+7 a composite integer?

2.     Without actual division, state whether 13 by 2⁰ x 5⁷ is a terminating or a non terminating rational number.

3.     Identify whether √16 is a rational number. 

4.     what is the conjugate number of 2+√5?

5.     Express 140 as the product of its prime factors.  

6.     Determine whether 875/10³ is a terminating or a non-terminating decimal. 

7.     Q 7 HCF of two integers 26, 91 is 13, what will be its LCM?  

8.     If f(x) is divisble by q(x), what will be the value of r(x), where f(x) = g(x)q(x) + r(x)? 

9.     Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer. 

10.  Use the division algorithm to find the quotient q(y) and remainder r(y), when f(y) = 8y⁴ – 12y³ – 2y² + 15y – 4 is divided by g(y) = 2y² – 3y + 1. 

11.  Show that 3√2 is irrational. 

12.  Show that 5 – √3 is irrational.  

13.  Prove that √2 is irrational number?    

14.  Express 32760 as the product of its prime factors. 

15.  Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.  

16.  Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM. 

17.  Find the LCM and HCF of 6 and 20 by the prime factorisation method. 

18.  A sweet seller has 420 Kaju burfis and 130 Badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the maximum number of burfis that can be placed in each stack for this purpose? 

19.  Use Euclid’s division algorithm to find the HCF of 4052 and 12576. 

20.  Find the largest number that will divide 2053 and 967 and leaves a remainder of 5 and 7 respectively.