Number System & Polynomials
What is the HCF (124, 72)
4
HCF can be found using Long division method & Prime factorization method.
For what value of p, system of equations 2x + py = 8 and x + y = 6 have no solution?
p=2
No solution means
a1/a2 =b1/b2
2/1=p/1
p=2
11th term of an A.P. – 3, - 1/ 2 ......... is
22
an = a1 + (n-1)d
Here a1 = -3, n=11, d= -1/2 - (-3)=-1/2+3=5/2
Without actual performing the long division, find if 395/10500 will have terminating or non terminating (repeating) decimal expansion.
Non-terminating repeating
As 395/10500=79/2100
And since 2100 have prime factors (23 * 32 * 52 ) other than 2 or 5. Hence the 395/10500 will have non terminating decimal expansion.
How many solution does the following pair of linear equation has?
3x + 2y = 11 and 2x + 3y = 4
Unique solution
The first three terms of an A.P. respectively are 3y – 1, 3y + 5 and 5y + 1 then y equals:
5
Because when terms are in AP there common difference is same.
Hence, (3y+5) - (3y-1) =(5y+1) - (3y+5)
Find HCF and LCM of two numbers is 1 and 360 respectively & one of the number is 9. What will be the other number?
Other number is 40
HCF × LCM = Product of two given number.
1 × 360=9 × other no.
Other no. =360/9=40
When are pair of linear equations called consistent?
When they have a solution. That is either when they have unique solution(intersecting lines) or infinite solution(Coinciding lines)
Sum of first 20 natural numbers is 410. Tre or False?
False
As Sn = n/2(2a1 + (n-1)d)
Here n=20, a1 = 1, d=1
If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is
-10
As 22 + 3(2)+k=0 (Since 2 is a zero of this quadratic polynomial)
4+6 +k =0
k=-10
Every quadratic equation can have atmost how many roots?
2
What is Basic Proportionality Theorem (BPT)?
Basic Proportionality Theorem : If a line is drawn parallel to one side of a triangle to intersect the other sides in distinct points, the other two sides are divided in the same ratio.
A quadratic polynomial, whose zeros are – 3 and 4, is ?
x2 – x - 12
As sum (S) of zeroes=-3+4=1
Product (P) of zeroes= -12
Hence, quadratic polynomial = x2 -Sx+P
Discriminant of 2x2 + 5/3 x – 2 = 0 is?
169/9
D=b2 - 4ac
(5/3)2 - 4 (2)(-2)=25/9+16=169/9
What are the criterias of Similarity of Triangles?
(i) AAA Similarity : ΔABC ~ ΔDEF when
∠A=∠D, ∠B = ∠E and ∠C = ∠F
(ii) SAS Similarity : ΔABC ~ ΔDEF when
AB/DE = BC/EF and ∠B = ∠E
(iii) SSS Similarity : ΔABC ~ ΔDEF,
AB/DE = AC/DF = BC/EF