Tuesday, September 27, 2011

CBSE MATH X Ch-6 X Trigonometry Introduction and identities

X Trigonometry Introduction and identities Test Paper - 1
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X Trigonometry Introduction and identities Test Paper - 2
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X Trigonometry Introduction and identities Test Paper - 3
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X Trigonometry Introduction and identities Test Paper - 4
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X Trigonometry Introduction and identities Test Paper - 5
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X Trigonometry Introduction and identities Hots Questions
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Sunday, September 25, 2011

10th maths Extra score questions chapter Pair of Linear Equation in Two Variable

Pair of Linear Equation in Two Variable HOTS By JSUNIL TUTORIAL
1 mark questions :- 
Q. 1 Is the pair of linear equations consistent: - x + y = 3 ; 3x – 2y = 4 .

Q. 2 Find out whether the lines representing the following pair of linear equations are parallel or coincide:- 6x – 3y +10 = 0 ; 2x – y + 9 = 0

Q. 3 Write the value of k for which the following pair of linear equations has unique solution:-

x + k y + 6 = 0 ; 2x + 3y + 8 = 0

Q. 4 Write the value of k for which the following pair of linear equations has no solution: -

4x +y =11 ; k x + 3y =5

Q. 5 Write the value of k for which the system of equations has infinite solutions:-

2x – 3y +10 = 0 ; 3x – ky +15 = 0

Q. 6 Does (2, –3) lie on the linear equation 3x – 2y + 5 = 0

Monday, September 19, 2011

10th maths Chapter Polynomial extrascore questions

Math Adda  
Chapter: POLYNOMIALS        
LEVEL-I
1. The zeroes of the polynomial 2x2-3x-2 are
a. 1, 2        b. -1/2,1    
c. ½,-2      d. -1/2,2                       [Ans- (d)]

2. If a and b are zeroes of the polynomial 2x2+7x-3, then the value of a2 +b2 is
a. 49/4                b. 37/4       
c. 61/4                d. 61/2                                                      
Ans-( c ) 

3. If the polynomial 6x3+16x2+px -5 is exactly divisible by 3x+ 5 , then the value of p is
a. -7      b. -5     
c.5        d.7                                                                                  
[Ans- (d )]

4. If 2 is a zero of the polynomials 3x2+ax-14 and 2x3+bx2+x-2, then the value of 2 - 2b is
a. -1       b. 5         
c. 9       d. -9                         
[Ans-( c ) ]

5. A quadratic polynomial whose product and sum of zeroes are 1/3 and √2 respectively is

(a) 3x2 – x +3√ 2 (b) 3x2 + x - 3√2 (c) 3x2 + 3√2x +1 (d) 3x2 – 3√2x +1         
[ Ans: (d)]

LEVEL-II

1. If 1 is a zero of the polynomial p(x) = ax2 -3(a-1) x -1, then find the value of a.       
Ans: a=1

2. For what value of k, (-4) is zero of the polynomial x2 – x – (2k+2)?                          
Ans: k=9

3. Write a quadratic polynomial, the sum and product of whose zeroes are 3 and -2. 
Ans: x2 -3x-2

4. Find the zeroes of the quadratic polynomial 2x2-9-3x and verify the relationship between the zeroes and the coefficients.                                   Ans: 3, -3/2

5. Write the polynomial whose zeroes are 2 +√3 and 2 - √3.          Ans: p(x)=x2-4x+1

LEVEL – III

1. Find all the zeroes of the polynomial 2x3+x2-6x-3, if two of its zeroes are -√3 and √3. 
Ans: x=-1/2

2. If the polynomial x+ 2x3 + 8x2+12x+18 is divided by another polynomial x+ 5, the remainder comes out to be  px+q. 
Find the value of p and q.                                                         Ans:p=2,q=3

3. If the polynomial 6x4+8x3+17x2+21x+7 is divided by another polynomial 3x2+4x+1, the remainder  comes out to  be (ax+b), find a and b.                      Ans:a=1, b=2

4. If two zeroes of the polynomial f(x)= x3-4x2-3x+12 are √3 and -√3, then find its third zero. Ans: 4

5. If a, b are zeroes of the polynomial x2-2x-15 then form a quadratic polynomial whose zeroes are  (2a) and (2b).              Ans: x2-4x-60

LEVEL – IV

1. Find other zeroes of the polynomial p(x)=2x4 +7x319x2-14x +30 if two of its zeroes are √2 and -
√2.

Ans: 3/2 and -5

2. Divide 30x4 +11x3-82x2-12x-48 by (3x2 +2x-4) and verify the result by division algorithm.

3. If the polynomial 6x4 +8x3-5x2+ax+b is exactly divisible by the polynomial 2x2-5, then find the 
value of a and b.     
Ans: a = -2 0 , b = - 2 5

4. Obtain all other zeroes of 3x4 -15x3+13x2+25x-30, if two of its zeroes are and ±√5/3. 
Ans: 
±√5/3,3,2

5. If a, b are zeroes of the quadratic polynomial p(x)=kx2+4x+4 such that a2 +b2=24, find the value 
of k.                                                                        
Ans: k=2/3 or k= -1

SELF EVALUATION

1. If a, b are zeroes of the quadratic polynomial ax2+bx+c then find (a) a/b +b/a (b) a2 +b2

2. If a, b are zeroes of the quadratic polynomial ax2+bx+c then find the value of a2 - b2

3. If a, b are zeroes of the quadratic polynomial ax2+bx+c then find the value of a3 +b3

4. What must be added to 6x5+5x4+11x3-3x2+x+5 so that it may be exactly divisible by 3x2 -2x+4 ?

5. If the square of difference of the zeroes of the quadratic polynomial f(x)= x2+px+45 is equal to 144, 
find the value of p.

Sunday, September 18, 2011

10th Number system solved important question


1. Prove that one of every three consecutive integers is divisible by 3.
Ans:        n,n+1,n+2 be three consecutive positive integers
We know that n is of the form 3q, 3q +1, 3q + 2
So we have the following cases
Case – I when n = 3q
In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3
Case - II When n = 3q + 1
Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not  divisible by 3
Case – III When n = 3q +2
Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3
Hence one of n, n + 1 and n + 2 is divisible by 3
2. If the H C F of 657 and 963 is expressible in the form of 657x + 963x - 15 find x.

Ans: Using Euclid’s Division Lemma
a= bq+r , o £ r < b
963=657×1+306
657=306×2+45
306=45×6+36
45=36×1+9
36=9×4+0
\ HCF (657, 963) = 9
now 9 = 657x + 963× (-15)
657x=9+963×15
=9+14445
657x=14454
x=14454/657
\ x =22
3. If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y.

Ans: Using Euclid’s algorithm, the HCF (30, 72)
72 = 30 × 2 + 12      
30 = 12 × 2 + 6
12 = 6 × 2 + 0
HCF (30,72) = 6
6=30-12×2
6=30-(72-30×2)2
6=30-2×72+30×4
6=30×5+72×-2
\ x = 5, y = -2
Also 6 = 30 ×5 + 72 (-2) + 30 × 72 – 30 × 72
Solve it, to get
x = 77, y = -32
Hence, x and y are not unique

4. Show that for odd positive integer to be a perfect square, it should be of the form
8k +1.
Let a=2m+1
Ans: Squaring both sides we get
a2 = 4m (m +1) + 1
\ product of two consecutive numbers is always even
m(m+1)=2k
a2=4(2k)+1
a2 = 8 k + 1
Hence proved

5.  Use Eculid’s division demma, to show that the cube of any positive integer is of the form  9m, 9m + 1 or 9m + 8.

Solution.  Let x be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.
So, we have the following cases :
Case I : When x = 3q.

then, x3 = (3q)3 = 27q3 = 9 (3q3) = 9m, where m = 3q3
Case II : When x = 3q + 1
then, x3 = (3q + 1)3
= 27q3 + 27q2 + 9q + 1
= 9 q (3q2 + 3q + 1) + 1
= 9m + 1, where m = q (3q2 + 3q + 1)
Case III. When x = 3q + 2
then, x3 = (3q + 2)3
= 27 q3 + 54q2 + 36q + 8
= 9q (3q2 + 6q + 4) + 8
= 9 m + 8, where m = q (3q2 + 6q + 4)
Hence, x3 is either of the form 9 m or 9 m + 1 or, 9 m + 8.

6. Check whether 6n can end with the digit 0 for any natural number n.

Solution. We have, 6n = (2 × 3)n = 2n × 3n. Therefore, prime factorisation of 6n does not contain 5 as a factor.   Hence, 6n can never end with the digit 0 for any natural number n.

7. If p is a prime number, prove that p is irrational.
Solution :  Let p be a prime number and if possible, let p is rational.
Let its simplest form be  p = m/n   
where m and n are integers having no common factor other than 1, and n ¹ 00.
Now, Sq. both side
 p = m2/n2
p n 2  = m2
p divides m2
p divides m. ...(i)
Let m = p q for some integer q.
putting m = pq in (i), we get
pn2 = p2q2
 n2 = pq2
 p divides n2
 p divides n. ...(ii)
from (i) and (ii), we observe that p is a common factor of m and n, which contradicts our
assumption.
Hence, p is irrational.
8. (ii) Show that  √(n -1)  + (n + 1) is irrational for every natural number n.

Monday, September 12, 2011

Thursday, September 08, 2011

Sample Paper – 2012 Class – X Subject – Mathematics

1. A motorboat takes 6 hours to cover 100km down stream and 30km upstream. If the motorboat goes 75km down stream and returns back to its starting point in 8hours, find the speed of the motorboat in still water and the rate of the stream.

2. Solve the following system of linear equations: 3x - 5y = -1, x - y = -1


3. The polynomial x4 +bx³ +59 x² +cx +60 is exactly divisible by x² +4 x +3. Find the values of b and c.


4. If the equation (1 + m2) x2 + 2 mcx + (c2 - a2) = 0 has equal roots, prove that c2 = a2(1 + m2)


5. A polygon has 10 sides .The lengths of the sides starting with the smallest form an AP .If the perimeter of the polygon is 420 Cm and the length of the longest side is twice that of the shortest side

6. A train covers a distance of 90 Km at uniform speed .Had the speed been 15 Km /hr more . it would have taken 30 minutes less for the journey find the original speed of the train

Wednesday, September 07, 2011

CBSE Mathematics for Class 10 chapter Statistics Test Paper



Q1 - If the mean of the following data is 20.6, find the value of p.

x:
10
15
p
25
35
f:
3
10
25
7
5

Q2 - If the mean of the following data is 20, find the value of p
 
x:
15
17
19
21
23
f:
2
3
4
5p
6

Q3 -The distribution below gives the weight of 30 students in a class. Find the median weight of students.

Weight (in Kg.)
40-50
50-60
60-70   
70-80  
F
5
14
9
2
Q4 -If the median of the following frequency distribution is 46. Find the missing frequency.  

Variable 10-20    20-30    30-40    40-50    50-60    60-70   70-80   Total
F:              12          30         P            65           q          25          18      229

Q5 - The total number of marks scored by class in test is given below. Find the mean. 

Below 20
4
Below 40
12
Below 60
30
Below 80
44
Below 100
50

Q6 - Find the median class of the following data:

Marks Obtained
0-10
10-20
20-30
30-40
40-50
50-60
Frequency
8
10
12
22
30
18

Q7 -Find the mean by direct methos for the following data:

Classes
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Frequency
4
8
10
12
10
4
2

Q8 - Find the mean of following frequency distribution:

Class-inetrval:
0-10
10-20
20-30
30-40
40-50
No. of workers:
7
10
15
8
10

Q9- If the mean of the following distribution is 27, find the value of p:

Class-interval  0-10    10-20   20-30     30-40     40-50 
No. of Workers   8          p         12            13           10

Q10 -Find the mean of the following frequency distributions:

Class-interval  0-6    6-12   12-18     18-24    24-30 
No. of Workers   6       8        10            9            7

Q11 - Find the median for the following data:

Classes
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Frequency
4
8
10
12
10
4
2

Q12- Find the mode for the following data:

Classes
10-20
20-30
30-40
40-50
50-60
60-70
70-80
Frequency
4
8
10
12
10
4
2

Q13 -Compute the mode for the following frequency distribution.

Size of items:
0-4
4-8
8-12
12-16
16-20
20-40
24-28
28-32
32-36
36-40
Frequency:
5
7
9
17
12
10
6
3
1
0


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