8th Maths Squares and square roots and Cubes and cube roots

Assignments: 8th Maths Squares and square roots and Cubes and cube roots

Q-1. Show that 63504 is a perfect square.

Q-2. Find the smallest no. by which 25200 should be divided so that result is a perfect square.

Q-3. Find the greatest no. of 6-digit which is a prefect square.

Q-4. Find the least no. of 6-digit which is a prefect square.

Q-5. Write Pythagorean triplet whose one number is 14.

Q-6. Find the square root of 298116 by long division method.

Q-7. Find the square root of 3013696 by long division method.

Q-8. A welfare association collected Rs202500 as donation from the residents. If each paid as many rupees as there were residents, find the no. of residents.

Q-9. Find the square root of 121 and 169 by repeated subtraction method.

Q-10. A teacher wants to arrange maximum possible no. of 6000 students in a field such that the no. of rows is equal is the no. of column. Find the no. of rows if 71 were left out after arrangement.

Q-11. Find the values of

Q-12. Find the square root of 52 +857/2116

Q-13. Simplify  

Q-14. Find the square root of 237.615 correct to 3 places of decimal.

Q-15. What is the smallest no. by which 3087 must be divided so that the quotient is a prefect cube?

Q-16. Evaluate {(242 +72)½}3.

Q-17. Evaluate

Q-18. Find the volume of a cube whose surface area is 384m2 .

Q-19. Find the volume of a cube, one face of which has an area of 64m2 .

Q-20. Show that -17576 is a prefect cube.

Q-21. Find the cube root of 140x2450.

Q-22. Divide the no. 26244 by the smallest no. so that the quotient is a prefect cube. Also, find the cube root of the quotient .

Q-23. Find the smallest square number divisible by each one of the numbers 8, 9 and 10.

Q-24. The area of a square field is 5184 m2. A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area of the rectangular field.

Q-25. Three numbers are to one anther 2:3:4. The sum of their cubes is 33957. Find the numbers.

IX Proof of Heron’s formula

IX Proof of Heron’s formula

Let a, b, c are length of the sides and h is height to side of length c  of ∆  ABC.

We have S = (a + b + c)/2

So, 2s = a + b + c 

 Þ  2(s - a) = - a + b + c 
Þ  2(s - b) = a - b + c 
Þ  2(s - c) = a + b – c

Let p + q = c as indicated.

Then,   h² =  a² - p²       -------------(1)

 Also,   h² = b² - q ²       -------------- (ii)

From (i) and (ii)

  Þ  a² - p^2     =   b² -  q ²       

   Þ  q²   = - a² + p² + b²      

Since,   q = c - p  Þ q² = (c-p)² Þ q²  = c² + p² -2pc

Then, c² + p² -2pc   = - a² + p² + b²      

Þ - 2pc  =  - a² +b² – c² = - ( a² -b² + c²)

Þ   p = ( a² -b² + c²)/2c

Now, Put this value of p in equation (i)

h²    =   a² -  p²     

h²      =   ( a – p ) ( a + p )

h²   =    {a – ( a² -b² + c²)/2c }   {a + ( a² -b² + c²)/2c }

h²   =   {(2ac - a² + b² - c²)/2c}x{(2ac+ a² - b² + c²)/2c}

h²      = {(b² – (a - c)² }{(a + c)² – b²}/4c²

h²     = {(b – a + c) (b + a - c)}{(a + c + b)(a + c – b )

h² = { 2(s - a) x 2(s - c)  x 2s x2(s - b)}/4c²

h² = { 4 s (s - a) x (s - c) x(s - b)}/c²

h = 2/c  √ s (s - a) x (s - b) x(s - c) 

½ h c = √ s (s - a) x (s - b) x(s - c) 

Area of triangle = √ s (s - a) x (s - b) x(s - c)

CBSE Test paper-1

1. Two sides of a triangle are 8cm and 11cm and its perimeter is 32cm.The third side is :

(a) 4cm (b) 13cm (c) 14cm (d) 16cm

2. The base of a triangle is 12cm and height is 8cm .Its area is:

(a) 24cm2 (b) 96cm2 (c) 48cm2 (d) none

3. The sides of a triangular plot are in the ratio 3:5:7 and its perimeter is 300m . The sides of a triangle are.

(a) 60m,100m,40m (b) 50m,80m,60m (c) 45m,75m,95m (d) none

4. What will be the area of quadrilateral ABCD if AB =3cm, BC=4cm, CD=4cm, DA=5cm and AC=5cm.

(a) 12.5cm (b) 15.2cm (c) 18.2cm (d)19.2cm

5. An isosceles triangle has perimeter 30cm and each of equal side is 12cm .Area of triangle is:

(a) 8√15cm2 (b) 7√12cm2 (c) 9√15cm2 (d)none

Complete the following sentences

6. Area of an equilateral triangle with side ‘a’ is _______________.

7. If a, b, and c are the three sides of a triangle then by Hero’s formula area is___________.

8. In Heron’s formula semi perimeter is equal to ____________.

9. Area of a right angled triangle is ________________.

10. The area of a parallelogram is 392m2.If its altitude is twice the corresponding base, determine the base and height.

11. The adjacent sides of a parallelogram are 36cm and 27cm in length .If the distance between the shorter sides is 12cm, find the distance between the longer sides.

12. A rectangular lawn, 75m by 60m, has two roads , each 4m wide, running through the middle of the lawn, one parallel to length and other parallel to breadth. Find the cost of gravelling the roads at Rs 5.50 per m2

13. Using Heron’s formula, find the area of an equilateral triangle if its side is ‘a ‘units.

14. Find the percentage increase in the area of a triangle if its each side is doubled.

15. Find the area of quadrilateral ABCD whose sides in meters are 9, 40, 28 and 15 respectively and the angle between first two sides is a right angle.

16. The difference between the sides containing a right angle in a right angled triangle is 14cm. The area of a triangle is 120cm2.Calculate the perimeter of a triangle.

Factor and reminder theorem :Polynomial class IX

Proof of this factor theorem

Let p(x) be a polynomial of degree greater than or equal to one and a be areal number such that p(a) = 0. Then, we have to show that (x – a) is a factor of p(x).

Let q(x) be the quotient when p(x) is divided by (x – a).

By remainder theorem

Dividend = Divisor x Quotient + Remainder

p(x) = (x – a) x q(x) + p(a) [Remainder theorem]
⇒ p(x) = (x – a) x q(x) [p(a) = 0]
⇒ (x – a) is a factor of p(x)

Conversely, let (x – a) be a factor of p(x). Then we have to prove that p(a) = 0

Now,     (x – a) is a factor of p(x)
⇒ p(x), when divided by (x – a) gives remainder zero. But, by the remainder theorem, p(x) when divided by (x – a) gives the remainder equal to p(a). ∴ p(a) = 0

Proof of remainder theorem.

Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a).

Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant.

Let r(x) = c, where c is some constant. Then

p(x) = (x–a) q(x) + c

Putting x = a in p(x) = (x–a) q(x) + c, we get
p(a) = (a–a) q(a) + c ⇒ p(a) = 0 x q(a) + c ⇒ p(a) = c

This shows that the remainder is p(a) when p(x) is divided by (x–a).

Check Your understanding

1. Factories

(i)   a2-b2-4ac+4c2                              (ii) 7x2 + 2 √14x + 2

(iii) 4a²-4b²+4a+1                                (iv)  x⁴+y⁴-x²y²

(v) x⁶ - x³                                                                       (vi)  x³-5x²-x+5

(vii)  x²+3√3x +6                                 (viii)  a³(b-c)³+b³(c-a)³+c³(a-b)³

(ix)  .8a³-b³ -12a² b +6ab²                            (x) b.4x² +9y ²+ 25z² -12xy - 30yz +20xz

(xi)  x ³ + 4x ² + x – 6                         (xii)  4x⁴ + 7x² – 2

(xii)  x ² - 2√3x – 45                             (xiii)  3 - 12(a - b)2

Q. Find degree of 5x^{3 -}6x³y+10y²+11                        [4]

Q. find the value of k if(x-2) is a factor of p(x)=k x^2 -- √2x +1                       [ (2√2- 1)/4 ]

Q. Find the remainder when x³+3x²+3x+1 when divided by 3x+1.

Q.  if x-1 and x-3 are the factors of p(x) x (raise to the power 3)-a x (raise to the power 2)-13x-b then find the value of a and b

Q. If(X2-1) is a factor of ax⁴+bx³+cx²+dx+e,show that  a + c + e = b + d =0

Q. prove that (x+y)³-(x-y)³-6y(x²-y²)=8y³

Ans:  x³ + 3x²y + 3xy² + y³ - (x³ - 3x ²y + 3xy² - y³) - 6yx² + 6y³           [(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3 ab² - b³]  = x³ + 3x²y + 3xy² + y³ - x³ + 3x²y - 3xy² + y³ - 6yx² + 6y³       = 2y³ + 6 y³        = 8y³

Q. The polynomials {ax³-3x²+4} and {3x²-5x+a} when divided by {x-2} leave remainder "p" and "q" respectively. if p-2q+=a find the values of "a".                          [a is –8.]

Q. If ( x − 4) is a factor of the polynomial 2 x ² + Ax + 12 and ( x − 5) is a factor of the polynomial x ³ − 7 x ² + 11 x + B , then what is the value of ( A − 2 B )?

Q. f x -1/x =3; then find the value of x³ -1/x³                                                       [36]

Q. if a+b+c=7 and ab+bc+ca=20 find the value of (a+b+c)²

Q. The polynomial f(x)=x⁴-2x³+3x²-ax+b when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively. Find the values of a and b. Hence, find the remainder when f(x) is divided by (x-2)

Q. check:  2x +1 is a factor of p(x)=4x³ + 4x² - x -1