10th ncert cbse mathematics chapter 3 miscellaneous / optional exercise
pair of linear equations in two variables
ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,
if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)
We know that the opposite angles of a cyclic quadrilateral are supplementary,
which means they add up to 180 degrees.
A+C=180 degrees
and
B+D = 180 degrees
A+C=180
gives
(4y+20) +(-4x)=180
or
-4x +4y =180-20
or
-4x +4y =160
dividing by 4 makes it
-x+y=40--------------(1)
B+D = 180
gives
(3y-5)+(-7x+5) =180
or
-7x+3y=180-------------(2)
eliminating y
-x+y=40--------------(1) *3
-7x+3y=180-------------(2)
-3x+3y =120
-7x+3y =180
--------------------------subtracting
4x =(-60)
x = (-60)/4
x=(-15)
substitute in
-x+y=40
-(-15) +y =40
15+y-40
y=40-15
y=25
use x=(-15) and y=25
A =(4y+20) =4*25+20=120 degrees
B =(3y-5)=3*25-5=75-5=70 degrees
C=(-4x) =(-4)*(-15) = 60 degrees
D=(-7x+5) =(-7)*(-15) +5 =105+5=110 degrees
exercise 3.6
2
(iii)
Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.
let x = speed of train in km/hr
and y = speed of bus in km/hr
plan 1
60 km by train and remaining 300-60 = 240 km by bus
time = distance / speed
t1=(60/x) in the train
and
t2=(240/y) hours in the bus
total of 4 hours means
t1 +t2 =4 hours
(60/x) +(240/y) =4--------------------(1)
plan 2
100 km by train and remaining 300-100 = 200 km by bus
time = distance / speed
t3=(100/x) in the train
and
t4=(200/y) hours in the bus
10 minutes longer means 4hours 10minutes
This has to be changed in term of hours
4hours 10minutes =4+(10/60)=4+(1/6) = (25/6)hoursso
t3 +t4 = (25/6)hours
(100/x)+(200/y) =(25/6)----------------(2)
equations are
(60/x) +(240/y) =4--------------------(1)
(100/x)+(200/y) =(25/6)----------------(2)
we have to use substitution (1/x)=u , (1/y)=v
equations change to
60u + 240v =4 ----------------------(3) *5
100u + 200v =(25/6)-------------------(4) *3
--------------------------------------------------------------------eliminate u
300u + 1200v =20
300u +600v =(25/2)
------------------------------------------subtracting
600v = 20 -(25/2)
600v = (15/2)
v = [ 15 / (2*600) ]
v = [1/80]
resubstitute v = [1/80] in 60u + 240v =4
60u+{240/80} =4
60u+3=4
60u=4-3
60u=1
u=(1/60)
u=(1/60)
v = [1/80]
take reciprocal
x=60 km/hr
y=80 km/hr
speed of train =60km/hr
speed of bus =80km/hr
=================================================
ncert cbse 10th mathematics chapter 3 optional exercise 3.7
The ages of two friends Ani and Biju differ by 3 years. Ani’s father is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Ani’s father differ by 30 years. Find the ages of Ani and Biju
2. One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital?
3. A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And, if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.
4. The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.
5. In a ∆ ABC, ∠ C = 3 ∠ B = 2 (∠ A + ∠ B). Find the three angles.
Solve the following pair of linear equations:
px + qy = p – q
qx – py = p + q
(ii) ax + by = c
bx + ay = 1 + c
(iii)
(x/a) -(y/b) = 0
ax +by = (a^2) + (b^2)
(iv)
(a – b)x + (a + b) y = (a^2) – 2ab – (b^2)
(a + b)(x + y) = (a^2) + (b^2 )
(v)
152x – 378y = – 74
–378x + 152y = – 604
ABCD is a cyclic quadrilateral Find the angles of the cyclic quadrilateral,
if angles are A =(4y+20) , B =(3y-5) , C=(-4x), D=(-7x+5)
exercise 3.6
2
(iii)
Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.
ncert cbse 10th mathematics chapter 2 optional exercise
If the zeroes of the polynomial (x^3) – 3(x^2) + x + 1 are a – b, a, a + b, find a and b.
solution
2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively
solution
4. If two zeroes of the polynomial (x^4) – 6(x^3) – 26(x^2) + 138x – 35 are
[2 ±sqrt(3) ] , find other zeroes
solution
5. If the polynomial (x^4) – 6(x^3) + 16(x^2) – 25x + 10 is divided by another polynomial (x^2) – 2x + k, the remainder comes out to be x + a, find k and a
solution
exercise 2.3
3. obtain all other zeroes of 3(x^4)+6(x^3)-2(x^2)-10x-5 if two of its zeroes are sqrt(5/3) and [sqrt(5/3)]
solution
4. On dividing (x^3) – 3(x^2) + x + 2 by a polynomial g(x), the quotient and remainder were x – 2
and –2x + 4, respectively. Find g(x).
solution
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