CBSE CLASS 10 MATHEMATICS QUESTIONS AND ANSWERS PART 5

CBSE CLASS 10 MATHEMATICS QUESTIONS AND ANSWERS PART 5

ANSWERS ARE GIVEN IN YELLOW

1. What is the nth term of the AP a, 3a, 5a,....?

d= 3a-a= 2a
aₙ= a +(n-1)2a
= a + 2an - 2a
= 2an - a
= (2n-1) a

2. A pair of dice is thrown once. What is the probability of getting a doublet?

Total possible outcomes =36
{(1,1),(1,2)(1,3)(1,4)(1,5)(1,6)
(2,1)...(2,6)
(3,1)...(3,6)
(4,1)...(4,6)
(5,1)...(5,6)
(6,1)..(6,6)}

Number of doublets = 6
{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}
Probability of getting a doublet
=6/36=1/6

3. If the radius of the sphere is increased by 100%, by how much percentage will the volume of the sphere increase?

V=(4/3)πr³
Increased radius= r+100%(r)=r+r=2r
New volume= (4/3)π(2r)³
= (4/3)π(8r³)
=8V
Percentage increase
= ((8V-V)/V )x 100 = 7 x 100 = 700%

4. What is the distance of the point P(3,4) from the x-axis?

The distance of the point P(3,4) from the x-axis is 4.

5. Explain why (11x19x23) + 23 is a composite number.

(11x19x23) + 23
=23 ((11x19) + 1)
Thus it is a multiple of 23 and hence composite. The number has more than 2 factors. 1, 23, and (11x29 +1).

6. A right circular cone of radius 3 cm has a curved surface area of 47.1 sq.cm. Find the volume of the cone. (π=3.14)

πrl= 47.1
3.14 x 3 x l = 47.1
l = 47.1/(3.14x3)
l= 15.7/3.14= 1570/314= 5 cm
h= √(5²-3²)= √16=4 cm
V= (1/3)πr²h
= (1/3) x 3.14 x 3 x 3 x 4
= 3.14 x 3 x 4
= 9.42 x 4
= 37.68 cu.cm

7. A two digit number is such that the product of the digits is 14. If 45 is added to the number, the digits interchange their place. Find the number.

Let units place be x and tens place be y
The number is 10y+x
Given xy=14
Also given 10y+x+45 = 10x+y
9x-9y=45
Dividing by 9,
x-y=5
x-(14/x)=5
x²-14=5x
x²-5x-14=0
(x-7)(x+2)=0
x=7 (as x can only be positive)
y=14/7=2
The number is 27.

8. The median and mode of a frequency distribution are 26 and 29 respectively. Find the mean.

The empirical formula is
3 Median = Mode + 2 Mean
3(26) = 29 + 2Mean
78-29 = 2Mean
Mean = 49/2 = 24.5
The mean is 29.2.

9. If the sum of the zeroes of the quadratic polynomial kx²+2x+3k is equal to their product, then find k.

Sum of the zeroes= -2/k
Product of the zeroes= 3k/k = 3
Given they are equal
-2/k = 3
So k= -2/3

10. Find the centroid of the triangle whose vertices are (3,-7), (-8,6) and (5,10).

Coordinates of the centroid are
( (3-8+5)/3, (-7+6+10)/3 )
(0, 3)

11. Each root of x²-bx+c=0 is decreased by 2. Find b and c if the resulting equation is x²-2x+1=0

Let m,n be the roots
m+ n= -(-b)=b
mn= c
(m-2) + (n-2)=-(-2)=2
m+n=6
Therefore b=6
(m-2)(n-2) = 1
mn-2m-2n+4=1
mn-2(m+n) =-3
c-2(6)=-3
c= -3+12=9

12. The sum of four consecutive numbers in AP is 32 and the ratio of the product of the first and last term to the product of two middle terms is 7:15. Find the numbers.

Let a-3d, a-d, a+d, a+3d be the four consecutive numbers in AP.
Given their sum is 32
a - 3d +a-d +a+d + a+3d = 32
4a= 32
a=8
Also given
(a-3d)(a+3d) : (a-d)(a+d) = 7:15
a² - (3d)² : (a²-d²) = 7:15
64-9d² : 64-d² =7:15
15(64-9d²) = 7(64-d²)
64(15-7)= d²(-7+135)
64x8 = d² x 128
d²= (64x8)/128
d²=4
d=2 or -2

The numbers in AP are 2,6,10,14 or 14,10,6,2

13. If 2x+y= 23, 4x-y=19, find the value of (5y-2x) and ((y/x) - 2)

Adding,
2x + y+ 4x - y= 23 + 19
6x = 42
x= 7
2(7) +y= 23
y= 23-14= 9
(5y-2x) = 45-14= 31
((y/x) - 2) = (9/7)-2=(9-14)/7= -5/7

14. If m,n are the roots of 2x²-4x+5, then find the value of (m-n)².

m+n=-(-4/2)=2
mn=5/2

We can write
(m-n)² = (m+n)² - 4mn
=(2)²-4(5/2)
= 4-10 = -6

15. If sinx+sin²x =1, then find cos²x+cos⁴x.

sinx+sin²x =1
sinx = 1-sin²x = cos²x ---> equation 1
So, we have sin²x= cos⁴x -->equation 2
Now,
cos²x+cos⁴x
= sinx + sin²x (from equations 1 and 2)
=1