AD, BE, CF are equal altitudes of a a triangle ABC . . .

Question : AD, BE, CF are equal altitudes of a ΔABC. Prove that ΔABC is an equilateral triangle.

Doubt by Suraj 

Solution : 

Given : AD, BE and CF are altitudes of ΔABC.
AD=BE=CF
To Prove : ABC is an equilateral triangle. 

Proof :
Area of ΔABC
= ½×Base×Height
= ½×BC×AD — (1) 
Again, 
Area of ΔABC
= ½×Base×Height
= ½×AB×CF — (2) 

From equation (1) and (2) 
½×BC×AD=½×AB×CF
BC×AD=AB×CF
BC×AD=AB×AD [∵AD=CF]
BC=AB — (3)

Similarly 
BC=AC — (4)

From equation (3) and (4) 

AB=BC=CA

All sides of triangle ABC are equal. 

So, ABC is an equilateral triangle. 

Hence Proved. 

Note : The above question can also be proved by using the concept of congruency. Can you do that by yourself? 

Similar Question : In given figure the altitudes AD, BE and CF, the altitudes of triangle ABC are equal. Prove that ABC is an equilateral triangle.

Similar Question : The altitudes of ΔABC, AD, BE and CF are equal. Prove that ΔABC is an equilateral triangle.

The value of root 32 + root 48 / root 8 + root 12 is equal to . . .

Question : The value of (√32 + √48) / (√8 + √12) is equal to 
(a) √2
(b) 2
(c) 4
(d) 8

Doubt by Suraj 

Solution : 
(√32 + √48) / (√8 + √12)
=(√48 + √32) / (√12 + √8)
=(4√3 + 4√2) / (2√3 + 2√2)

Rationalising the denominator

=(4√3 + 4√2) × (2√3 - 2√2) / (2√3 + 2√2)×(2√3 - 2√2)
=[4√3(2√3 - 2√2) + 4√2(2√3 - 2√2)] / [(2√3)² - (2√2)²]
[∵(a+b)(a-b)=(a²-b²)]
=[8×3-8√6+8√6-8×2] / [12-8]
=[24-0-16] / 4
=[8]/4
=2

Hence, (b) 2, would be the correct option. 

If x3+mx2+nx+6 has (x-2) as a factor & leaves a remainder. . .

Question : If x³+mx²+nx+6 has (x-2) as a factor & leaves a remainder 3 when divided by (x-3), find the values of m and n.

Doubt by Bhumika

Solution : 

Let p(x) = x³+mx²+nx+6

(x-2) is a factor of p(x)

x-2=0
x=2
p(2)=0
(2)³+m(2)²+n(2)+6=0
8+4m+2n+6=0
4m+2n=-14
2(2m+n)=-14
2m+n=-14/2
2m+n=-7 — (1)

Also

When p(x) is divided by (x-3) then remainder is 3.

x-3=0
x=3
p(3)=3
(3)³+m(3)²+n(3)+6=3
27+9m+3n+6=3
9m+3n+33=3
9m+3n=3-33
9m+3n=-30
3(3m+n)=-30
3m+n=-30/3
3m+n=-10 — (2)

Subtracting equation (2) from (1)

2m+n-(3m+n)=-7-(-10 )
2m+n-3m-n=-7+10
-m+0=3

-m=3
m=-3

Putting m=-3 in equation (1)

2(-3)+n=-7 
-6+n=-7
n=-7+6
n=-1

∴ m=-3 & n=-1

If 5^2x-1-25^x-1=2500, find the value . . .

Question : If 5^2x-1-25^x-1=2500, find the value of x.

Doubt by Bhumika

Solution : 

5^2x-1-25^x-1=2500
5^2x-1-(5²)^x-1=2500
5^2x-1-5^2x-2=2500
5^2x-1-5^2x-1-1=2500
5^2x-1-5^2x-1÷5¹=2500
5^2x-1(1-1/5)=2500
5^2x-1(4/5)=2500
5^2x-1=[2500×5]/4
5^2x-1=3125
5^2x-1=5⁵
Comparing the exponents both sides
2x-1=5
2x=5+1
2x=6
x=6/2
x=3

Given below is a circle with O. Find the value . . .

Question : Given below is a circle with O. Find the value of ∠BCD in terms of a and b.

(a) 180°+(a-b/2)
(b) 180°-(a+b/2)
(c) 180°-(a-b/2)
(d) None of these. 

Doubt by Vikrant

Solution : 

∠1+b=180° (Linear Pair)

∠1=180°-b — (1)
In ΔOAD

OA=OD (Radii of the same circle)
∠2=∠3 (Angles opposite to equal sides of a triangle are equal)
Also,

∠1+∠2+∠3=180° (ASP)

∠1+∠3+∠3=180° [∵∠2=∠3]
∠1+2∠3=180°
2∠3=180°-∠1

2∠3=180°-(180°-b) [From equation (1)]
2∠3=180°-180°+b
2∠3=b
∠3=b/2 — (2)

In ΔADB

∠3+∠4=a [Exterior Angle Property of the Triangle]

b/2+∠4=a [Using equation (2)]
∠4=a-b/2 — (3)

Now, 

ABCD is a cyclic Quadrilateral

∠BCD+∠4=180° [Sum of opposite angles of a cyclic quadrilateral is supplementary]

∠BCD=180°-∠4

∠BCD=180°-(a-b/2) [Using equation (3)]

Hence, c) 180°-(a-b/2), would be the correct option.

In the given figure, the perpendicular bisector of a chord . . .

Question : In the given figure, the perpendicular bisector of a chord AB of circle PXAQBY intersects the circle at P and Q. Prove that arc PXA ≅ arc PYB.

Doubt by Bhumika

Solution : 

Given : PQ is a perpendicular bisector of AB i.e. AM=BM & ∠PMA=∠PMB=90°

To prove : arc PXA ≅ arc PYB

Proof : 

In ΔPMA and ΔPMB

PM=PM (Common)
∠PMA=∠PMB (Each 90°)

AM=BM (Given)

ΔPMA ≅ ΔPMB (By SAS)

PA=PB (By CPCT)

arc PXA ≅ arc PYB [If two chords are equal, they subtend equal arcs]

Hence Proved

Express the linear equation 6=4x in . . .

Question : Express the linear equation 6=4x in the form ax + by + c = 0 and indicate the value of a, b and c.

Doubt by Bhumika

Solution : 

6=4x
6=4x+0.y
0=4x+0.y-6
OR

4x+0.y-6=0

On comparing with ax+by+c=0

We get
a=4

b=0

c=-6

Find the area of a triangular field, the length of . . .

Question : Find the area of a triangular field, the length of whose sides are 275 m, 660 m and 715 m. What is the cost of cultivating the field at the rate of Rs 200 per hectare? [1 hectare = 10000 m²]

Doubt by Aastha

Solution : 

Here
a=275 m

b=660 m

c=715 m

s = (a+b+c)/2
s = (275+660+715)/2
s = 1650/2
s = 825 m

Using Heron's Formula
Area of triangular field
= √[s(s-a)(s-b)(s-c)]
= √[825(825-275)(825-660)(825-715)]
= √[825×550×165×110]
= √[3×5×5×11×2×5×5×11×3×5×11×2×5×11]
= √[2×2×3×3×5×5×5×5×5×5×11×11×11×11]
= 2×3×5×5×5×11×11
= 30×25×121
= 750×121
= 90750 m²

= (90750/10000) Hectares

= 9.075 Hectares

Cost of cultivating 1 Hectares of field = Rs 200
Cost of cultivating 9.075 Hectares of field = Rs(200×9.075)
= Rs 1815

Evaluate {root[5+2root6]}+{root[8-2root15]}

Question : Evaluate {√[5+2√6]}+{√[8-2√15]}

Doubt by Bhumika

Solution :
 

= {√[5+2√6]}+{√[8-2√15]}
= {√[3+2+2√6]}+{√[5+3-2√15]}
= {√[(√3)²+(√2)²+2√3.√2]}+{√[(√5)²+(√3)²-2√5.√3]}

= {√[√3+√2]²}+{√[√5-√3]²}
[∵ Using Identities
a²+b²+2ab=(a+b)²
a²+b²-2ab=(a-b)² ]

= {√3+√2}+{√5-√3}
= √3+√2+√5-√3
= √2+√5

Zero of the zero polynomial . . .

Question : Zero of the zero polynomial is 

a) 0

b) 1

c) Any real nymber

d) Not defined

Doubt by Ashira

Solution : 

We know, zero polynomial is a type of constant polynomial in which constant term is itself a zero.

Proper Definition of Zero Polynomial 

A zero polynomial is a polynomial in which all the coefficients are equal to zero.

Here are some examples of zero polynomials:

Example 1 : 0

Example 2 : 0x + 0

Example 3 : 0x² + 0x + 0

Example 4 : 0x³ + 0x² + 0x + 0

As you can see, all of these polynomials have all of their coefficients equal to 0. Therefore, they are all zero polynomials.

Basically, it is a polynomial that is always equal to 0, no matter what value is substituted for the variables. So, the zero of a zero polynomial could be any real number. 

Hence, c) Any real number would be the correct option. 

Addition Knowledge : 

Degree of Zero Polynomial 

The zero polynomial is the only polynomial that has an undefined degree. This is because the degree of a polynomial is the highest power of the variable that appears in the polynomial. Since the zero polynomial has no terms, it has no variable and therefore no highest power.