দশম শ্ৰেণী গণিত অনুশীলনী (R-5)

দশম শ্ৰেণী গণিত (R-5)

১। সাধাৰণ উত্পাদক উলিওৱা

(i) 14pq, 28p2q2   (ii) 16x3-4x2, 32x  (iii) 20pq, 30qr, 40rp     (iv) 3x2y3, 10x3y2, 6x2y2z

উত্তৰ:(i) 14pq = 2 X 7 pq

28p2q2 = 2 X 2 X 7 X p X q X p X q

এতেকে, সাধাৰণ উত্পাদক= 2 X 7 X p X q

                 = 14pq

(ii) 16x3-4x2 = 2X2X2X2XxXxXx – 2X2XxXx

         = 2X2XxXx (2X2Xx – 1)

    32x  = 2X2X2X2X2Xx

এতেকে, সাধাৰণ উত্পাদক= 2X2Xx

                 = 4x

(iii) 20pq = 2x2x5xpxq

30qr = 2 x 3 x5 x q x r

40rp = 2x2x2x5x r xp

এতেকে, সাধাৰণ উত্পাদক= 2X5

                 = 10

(iv) 3x2y3 = 3XxXxXyXyXy

10x3y2 = 2x5xXxXxXxXyXy

6x2y2z = 2X3XxXxXyXyXz

এতেকে, সাধাৰণ উত্পাদক= xXxXyXy

                 = x2y2

২। উত্পাদক বিশ্লেষণ কৰা

(i) 4a2 + 8a3 (ii) 7x2y – 21xy2   (iii) a2bc + ab2c + abc(iv) a3 – a2b2

উত্তৰ : (i) 4a2 + 8a3

    = 2a2 (2+4a)

    = 4a2 (1+2a)

(ii) 7x2y – 21xy2 = 7xy (x-3y)

(iii) a2bc + ab2c + abc2 = abc (a+b+c)

(iv) a3 – a2b2 = a2 (a-b2)

৩। উত্পাদক বিশ্লেষণ কৰা :

(i) x2 + xy + 6x + 6y

(ii) xy + x + y+ 1

(iii) 24x2y + 12x2 – 12xy – 6x

(iv) z-7+7xy-xyz

উত্তৰ : (i) x2 + xy + 6x + 6y

= x(x+y) + 6(x+y)

= (x+y)(x+6)

(ii) xy + x + y+ 1

= x(y+1) + 1(y+1)

= (y+1)(x+1)

(iii) 24x2y + 12x2 – 12xy – 6x

= 6x [ 4xy + 2x – 2y -1]

=6x [2x (2y+1) -1(2y+1)]

=6x(2y+1)(2x-1)

(iv) z-7+7xy-xyz

= (z-7)-xy(z-7)

= (z-7)(1-xy)

৪। উত্পাদকত প্ৰকাশ কৰা :

(i) 4x2 + 12x + 9

(ii) 25m2 + 30m + 9

(iii) x2 -10x + 25

(iv) 121b2 – 88bc + 16c2

(v) 9p2 – 16q2

(vi) (l+m)2 – (l-m)2

(vii) x2 – 13x – 30

(viii) y2 – 5y – 36

(ix) 4y2 + 25y – 21

(x) 3x6 + 6x2y – 45x2y2

উত্তৰ : (i) 4x2 + 12x + 9

= (2x)2 + 2 X 2x X 3 + 32

=(2x+3)2

(ii) 25m2 + 30m + 9

=(5m)2 + 2X5mX3 + 32

= (5m + 3)2

(iii) x2 -10x + 25

=(x)2 – 2XxX5 + 52

=(x-5)2

(iv) 121b2 – 88bc + 16c2

= (11b)2 – 2X11bX4c + (4c)2

= (11b – 4c)2

(v) 9p2 – 16q2

= (3p)2 – (4q)2

= (3p–4q)(3p+4q)

(vi) (l+m)2 – (l-m)2

= (l+m+l-m)(l+m-l+m)

= 2l X 2m

= 4lm

(vii) x2 – 13x – 30

= x-15x+2x-30

= x(x-15)+2(x-15)

=(x+2)(x-15)

(viii) y2 – 5y – 36

= y2 -9y + 4y -36

= y(y-9) + 4(y-9)

=(y-9)(y+4)

(ix) 4y2 + 25y – 21

= 4y2 + 28y – 3y -21

= 4y(y+7) – 3(y+7)

=(y+7)(4y-3)

(x) 3x6 + 6x2y – 45x2y2

= 3x2 (x4 – 2y – 15y2)

৫। তলৰ বহুপদ ৰাশিক একপদ ৰাশিৰে হৰণ কৰা।

(i) (3y8-4y6+5y4) ᷁ y4   (ii) (p3q3 – p6q3) ᷁ p3q3

উত্তৰ : (i) (3y8-4y6+5y4) ᷁ y4

= (3y8-4y6+5y4)/ y4

= y4(3y4-4y2+5)/ y4

= 3y4-4y2+5

(ii) (p3q3 – p6q3) ᷁ p3q3

= (p3q3 – p6q3)/ p3q3

= p3q3(1– p3)/ p3q3

= 1– p3

৬। হৰণফল নিৰ্ণয় কৰা :

(i) (10x-25) ᷁ (2x-5)  (ii) 20(y+4)(y2+5y+3) ᷁ 5(y+4)

উত্তৰ : (i) (10x-25) ᷁ (2x-5)

= (10x-25)/ (2x-5)

= 5(2x-5) / (2x-5)

= 5

(ii) 20(y+4)(y2+5y+3) ᷁ 5(y+4)

= 20(y+4)(y2+5y+3)/ 5(y+4)

= 4 (y2+5y+3)

= 4y2+20y+12

৭। উত্পাদকত প্ৰকাশ কৰি হৰণফল নিৰ্ণয় কৰা ।

(i) (4u2 + 25u – 21) ᷁ (u+7)

(ii) (m2 – 14m – 32) ᷁ (m+2)

উত্তৰ : (i) (4u2 + 25u – 21) ᷁ (u+7)

= (4u2 + 25u – 21)/ (u+7)

= 4u2 + 25u – 21 / (u+7)

= 4u2 + 28u -3u – 21 / (u+7)

= 4u(u+7) – 3(u+7) / (u+7)

= (4u-3)(u+7) / (u+7)

= 4u-3

(ii) (m2 – 14m – 32) ᷁ (m+2)

= (m2 – 14m – 32)/ (m+2)

= (m2 – 16m +2m – 32)/ (m+2)

= m(m-16) +2(m-16) / (m+2)

= (m-16)(m+2) / (m+2)

= m-16

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