ASSEB HSLC 2025 Mathematics Question paper solutions

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অন&#= 2497;শীলনী 2.5

 

1.তলৰ পূ&#= 2544;ণফল কে&#= 2439;টা নি&#= 2544;্ণয় কৰ&#= 2495;বলৈ উপ&#= 2479;ুক্ত ৰা&#= 2486;ি ব্&#= 2479;ৱহাৰ কৰ&#= 2494;ঃ�

 

(i) (x + 4)(x + 10)

সমাধা= ;নঃ

(x + 4)(x + 10)

=3D x² + (4 + 10)= x + 40�10

=3D x² + 14x + 40

 

(ii) (x + 8)(x � 10)=

সমাধা= ;নঃ

�(x + 8)(x = � 10)

=3D x² + {8 + (� 10)}x + 8�(� 10)

=3D x² + (8 � 10)x � 80

=3D x² � 2x � 80

 

(iii) (3x + 4)(3x � 5)

সমাধা= ;নঃ

�(3x + 4)(3x � 5)

=3D (3x)² + {4 + ( � 5)}3x + 4�( � 5= )

=3D 9x² + (4 � 5)3x � 20

=3D 9x² � 3x � 20

 

(iv)

সমাধা= ;নঃ

 

 

(v) (3 � 2x)( 3 + 2x)

সমাধা= ;নঃ

(3 � 2x)( 3 + 2x)

=3D 3² � (2x)²

=3D 9 � 4x²

 

2. প্ৰত্যক্ষভাৱে পূৰণ নõ= 3;ৰি তø= 2;ৰ পù= 8;ৰণফলসমূহ নù= 5;ৰ্ণয় কþ= 4;াঃ �

&nb= sp;

(i) 103 = � 107

সমাধা= নঃ

103 � 107

=3D (100 + = 3)(100 + 7) 

=3D (100)2 + (3 + 7)100 + 3�7

=3D 10000 += 10�100 + 21

=3D 10000 += 1000 + 21

=3D 11021

 =

(ii) 95�= 96

সমাধা= নঃ

95�96<= /o:p>

=3D (100 � = 5)(100 � 4)

=3D 1002 + {(� 5) + (� 4)}100 + (� 5)(� 4)

=3D 10000 += (� 5 � 4)100 + 20

=3D 10000 += (� 9)100 + 20

=3D 10000 �= 900 + 20

=3D 9120

 =

(iii) 10= 4�96

সমাধা= নঃ

104�96=

=3D (100 + = 4)(100 � 4)

=3D (100)2 � 4²

=3D 10000 �= 16

=3D 9984

 =

 

 

3. উপযুক= ্ত অভেদ ব্= ;যৱহাৰ কৰি উৎপাদ= ক বù= 5;শ্লেষণ কþ= 4;াঃ �

&nb= sp;

(i) 9= x² + 6xy + y²

সমাধা= নঃ

9x2 + 6xy + y²

=3D (3x<= /i>)² + 2�3x � y + y²

=3D (3x<= /i> + y)²

=3D (3x<= /i> + y)(3x + y)

 =

(ii) 4y² � 4y + 1

সমাধা= নঃ

4y2 � 4y + 1

=3D (2y<= /i>)² � 2�2y�1 + 1²

=3D (2y = � 1)²

=3D (2y = � 1)(2y � 1)<= /o:p>

 

(iii)

সমাধা= নঃ

 

4. উপযুক্ত অভেদ বú= 9;যৱহাৰ কৰি তø= 2;ৰ ৰù= 4;শিবোৰ বù= 5;স্তাৰ কþ= 4;া।

&nb= sp;

(i) (= x + 2y + 4z)²

সমাধা= নঃ

(x += 2y + 4z)²

�=3D x² + (2y)2 + (4z)² + 2.x.(2y) + 2(2y)(4z= ) + 2(4z)x

�=3D x² + 4y2 + 16z² + 4xy + 16yz + 8xz=

 

 

(ii) (2<= i>x � y + 2)²

সমাধা= নঃ

(2x = � y + 2)²

�=3D (2x)<= sup>2 + (�y) ² + z² + 2�2x(� y) + 2(� y)z + 2�z�2x

�=3D 4x2 + y² + z² � 4xy � 2yz + 4= xz

 

(iii) (� 2x= + 3y + 2z)²

সমাধা= নঃ

�(� 2x + 3y + 2z)2

�=3D (� 2x= )² + (3y)² + (2z)² + 2( � 2x)3y<= /i> + 2�3y�2z + 2�2z(� 2x)

�=3D 4x2 + 9y² + 4z² � 12xy + 12yz= � 8xz

 =

(iv) (3<= i>a � 7b � c)²

সমাধা= নঃ

(3a = � 7b � c)²

=3D {3a<= /i> + (� 7b) + (� c)}²

=3D (3a<= /i>)² + ( � 7b)² + ( � c)² + 2�3a(� 7<= i>b) + 2(� 7b)(� c) + 2(� c)3a

=3D 9a² + 49b² + c² � 42ab + 14bc= � 6ca

 =

(v) (� 2= x + 5y � 3z)²

সমাধা= নঃ

(� 2x + 5y � 3z)²

=3D {(� 2x) + 5y + (� 3z)}²

�=3D (� 2x)² + (5y)² + (� 3z)² + 2( � 2x)(5y) + 2(5y)(� 3= z) + 2(� 3z)(2x)

�=3D 4x² + 25y2 + 9z² � 20xy � 30yz + 12zx

 =

(vi)

সমাধা= নঃ

5. উৎপাদক বিশ্&#= 2482;েষণ কþ= 4;াঃ

 =

(i) 4= x² + 9y² + 16x² + 12xy � 2yz= � 16xz

সমাধা= নঃ

4x2 + 9y² + 16x² + 12xy � 2yz= � 16xz

=3D (2x<= /i>)² + (3y)² + (� 4z)² + 2(2x).(3y= ) + 2(3y)(� 4z) + 2(2x)(� 4z)

=3D (2x<= /i> + 3y � 4z)²

=3D (2x<= /i> + 3y � 4z)(2x + 3y � 4z)

 =

(ii)

সমাধা= নঃ

6. তলৰ ঘনকেই= টা বিস্তাৰ কৰাঃ

 

(i) (2x + 1)³

সমাধা= নঃ

=3D (2x<= /i>)³ + 3(2x)²�1 + 3�2x�1² + 1³

=3D 8x³ +12x² + 6x + 1

 =

(ii) (2<= i>a � 3b)²

সমাধা= নঃ

(2a = � 3b)²

=3D (2a<= /i>)³ � 3(2a)²( 3b) + 3(2a)( 3b)^{2 � (3b)3}

=3D 8a³ � 36a²b + 54ab² � 27b3

&nb= sp;

&nb= sp;

&nb= sp;

(iii)

সমাধা= নঃ

 =

(iv)

সমাধা= নঃ

 =

7. উপযুক্ত অভেদ ব্= ;যৱহাৰ কৰি তলৰ= ৰাশিৰ মা= ;ন নিৰ্ণয় কৰ= ;াঃ

(i) (99)³

সমাধা= নঃ

 =

 =

 =

(ii) (102)³

সমাধা= নঃ

 

(iii) (998)³

সমাধা= নঃ

 

8. তলৰ প্ৰত&#= 2495;টোৰ উû= 0;পাদক বù= 5;শ্লেষণ কþ= 4;াঃ �

 =

(i) 8= a³ + b³ + 12a²b + 6ab^{2<= /sup>}

সমাধা= নঃ

8a3 + b³ + 12a²b + 6ab^{2<= /sup>}

=3D (2a<= /i>)³ + b³ + 3(2a)²b + 3(2a) b²

=3D (2a<= /i> + b)³

=3D (2a<= /i> + b)(2a + b)(2a + b)

 =

(ii) 8a³ � b³ � 12a²b + 6ab^{2<= /sup>}

সমাধা= নঃ

8a3 � b³ � 12a²b + 6ab^{2<= /sup>}

=3D (2a<= /i>)³ + b³ � 3(2a)²b + 3(2a) b²

=3D (2a<= /i> � b)³

=3D (2a<= /i> � b)(2a � b)(2a � b)

 =

 =

 =

(iii) 27= � 125a³ � 135a + 225a²

সমাধা= নঃ

27 � 125= a³ � 135a + 225a²

=3D (3)3 � (5a)³ � 3�3²�5a + 3�3(5a)^{2<= /sup>}

=3D (3 � 5<= i>a)³

=3D (3 � 5<= i>a)(3 � 5a)(3 � 5a)

 =

(iv) 64<= i>a³ � 27b³ � 144a²b + 108ab2

সমাধা= নঃ

64a<= sup>3 � 27b³ � 144a²b + 108ab2

=3D (4a<= /i>)³ � (3b)³ � 3(4a)²(3b) + 3(4a)(3b)²

=3D (4a<= /i> � 3b)³

=3D (4a<= /i> � 3b)(4a � 3b)(4a � 3b)

 =

(v)

সমাধা= নঃ

 =

9. সত্যাপন কৰাঃ �

(i) x³= + y³ =3D (x + y)(x² � xy + y²⁼ )

সমাধা= নঃ

R.H.S =3D (= x + y)(x² � xy + y²)

=3D x(x² � xy + y²) + y(x² � xy= + y²)

=3D x³ � x²y + xy² + x^{2y � xy2 + y3}

=3D x³ + y³

=3D LHS

 =

 =

 =

(ii) x^{3 � y3 =3D (x � y)(x2 + xy + y2= )}

সমাধা= নঃ

R.H.S =3D (= x � y)(x² + xy + y²)

=3D x(x² + xy + y²) � y(x² + xy= + y²)

=3D x³ + x²y + xy² � x^{2y � xy2 � y3}

=3D x³ � y³

=3D LHS

 =

10. উৎপাদক বিশ্&#= 2482;েষণ কþ= 4;াঃ �

(i) 27y^{3 + 125z3}

সমাধা= নঃ

27y<= sup>3 + 125z³

�=3D (3y)³ + (5z= )³

�=3D (3y + 5z){(3y)<= sup>2 � (3y)(5z) + (5z)²}

�=3D (3y + 5z)(9y² � 15= yz + 25z²)

 =

(ii) 64<= i>m³ � 343n³

সমাধা= নঃ

64m<= sup>3 � 343n³

=3D (4m<= /i>)³ � (7n)³

=3D (4m<= /i> � 7n){(4m)² + (4m)(7n) + (7n)²}

=3D (4m<= /i> � 7n)(16m² + 28mn + 49n²)

 =

11. 27x³ + y³ + z³ � 9xyz <= span lang=3DBN style=3D'font-size:12.0pt;font-family:Kalpurush;mso-ascii-font-fa= mily: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-language: BN'>ৰ উৎপাদক বù= 5;শ্লেষণ কþ= 4;াঃ �

সমাধা= নঃ =

27x³ + y³ + z3 � 9xyz

�=3D (3x)³ + y3 + z³ � 3(3x).y.z

�=3D (3x + y + z){(3= x)² + y² + z² � (3x)(y) � = yz � z(3x)}

�=3D (3x + y + z)(9<= i>x² + y² + z² � 3xy � yz � 3<= i>xz)

 =

 =

 =

 =

 =

 =

 =

 =

12. সত্যাপন কৰা= যে 

সমাধা= নঃ =

 =

13.(i) <= /span>যদি x<= /span> + y + z =3D 0 তেন্তে দেখু&#= 2451;ৱা যে= ; x^{<= span style=3D'font-size:12.0pt'>3} + y³ + z³ =3D 3xyz

সমাধা= নঃ

আমি জা= ;নো যে= ;,=

x= ³ = + y³ + z³ � 3xyz =3D� (x + y + z)(x² + y² = + z² � xy � yz � zx)

⇒ x³ + y³ + <= i>z³ � 3xyz =3D 0(x² + y² + = z² � xy � yz � zx) ���������= �� [⸪x + y + z =3D 0]

⇒ x³ + y³ + <= i>z³ � 3xyz =3D 0

⇒ x³ + y³ + <= i>z³ =3D 3xyz

 =

(ii) যদি = a + b + c =3D 0 তেন্ত= ে দú= 3;খুওৱা যে,

�a²(b + c) + b²(c + a) + c^{2(a + b) + 3abc} =3D 0

সমাধা= নঃ

দিয়া আছ= ;ে,

a= + b + c =3D 0

⇒ a + b =3D � c

 

a= + b + c =3D 0<= o:p>

⇒ b + c =3D � a

a= + b + c =3D 0<= o:p>

⇒ a + c =3D � b

 

L.H.S =3D <= i>a²(b + c) + b²(c + a) + c^{2(a + b) + 3abc}

��������� =3D a²(� a= ) + b² (� b) + c² (� c) + 3abc

��������� =3D � a³ � b³ � c³ + 3abc

��������� =3D � (a³ + <= i>b³ + c³ � 3abc)

��������� =3D � (a + b + c)(a² + b² + c² � ab � bc � = ca)

��������� =3D � 0(a² + = b² + c² � ab � bc � ca)�������������������������������� [⸪= a + b + c =3D 0]

��������� =3D 0

��������� =3D RHS

 

(iii) যদি 2a= � b + c =3D 0 তেন্ত= ে দú= 3;খুওৱা 4a² � b² + <= i>c² + 4ac 

সমাধা= নঃ

দিয়া আছ= ;ে,

2a �= b + c =3D 0

⇒ 2a + c =3D b

এতিয়া,

LHS =3D 4a² �= b² + c² + 4ac

=3D 4a² � (2a + c)² + c² + 4ac��������������� [⸪2a + c<= /i> =3D b]

=3D 4a² � {(2a)² +2(2a)c + c²} + = c² + 4ac

=3D 4a² � (4a² +4ac + c²) + c2 + 4ac

=3D 4a² � 4a² � 4ac � c² + c^{= 2} + 4ac

=3D 0<= /o:p>

=3D RHS

(iv) যদি a + b + c =3D 0, তেন্ত= ে দেখুওৱা যে=

সমাধা= নঃ

দিয়া আছ= ;ে,

a= + b + c =3D 0

⇒ = b + c =3D �a

 =

a= + b + c =3D 0

⇒ = c + a =3D �b

a= + b + c =3D 0

⇒ a + b =3D �c

আমি জা= ;নো যে= ;, যদ= ;ি,

a= + b + c =3D 0

তেনেহলে,

a³ + b³ + c³ =3D 3abc ���������������� (1)=

 

(v) যদি a<= /span>² + b² + c² � ab � bc � ca= =3D 0 তেন্তে= দú= 3;খুওৱা যে a =3D b =3D c

সমাধা= নঃ

দিয়া আছ= ;ে,

�a² = + b² + c² � ab � bc � ca =3D 0=

⇒ 2(a² + b² += c² � ab � bc � ca) =3D 2�0

⇒ 2a² + 2b² += 2c² � 2ab � 2bc � 2ca =3D 0

⇒ a² + a² + <= i>b² + b² + c² + c² � 2= ab � 2bc � 2ca =3D 0

⇒ a² � 2ab + b2 + b² � 2bc + c² + c²⁼ � 2ca + a² =3D 0

⇒ (a � b)² + (b �= c)² + (c � a)² =3D 0

⇒ (a � b)² =3D 0<= span lang=3DAS style=3D'font-size:12.0pt;mso-bidi-language:AS'> আৰ= ;ু �= (b � c)² =3D 0 <= span lang=3DAS style=3D'font-size:12.0pt;mso-bidi-language:AS'>�আৰু �= (c � a)² =3D 0

⇒ a � b =3D 0 আৰ= ;ু b � = c =3D 0 আৰু c<= /i> � a =3D 0

⇒ a =3D b আৰ= ;ু b =3D c আৰু c =3D a

⇒ a =3D b =3D c <= /span>

&nb= sp;

&nb= sp;

&nb= sp;

&nb= sp;

14. ঘনফল প্ৰক&#= 2499;তাৰ্থত নù= 5;ৰ্ণয় নõ= 3;ৰাকৈ তø= 2;ৰ পú= 9;ৰতিটোৰ মান নù= 5;ৰ্ণয় কþ= 4;াঃ �

(i) (� 12)³ += (7) ³ + (5) ³

সমাধা= নঃ

ইয়াত,

�12+7+5 =3D= 0

আমি জা= ;নো যে= ;,

যদি,

a= + b + c =3D 0

তেনেহলে,

a³ + b³ + c³ =3D 3abc

গতিকে,

(�12)³⁼ +(7)³+(5)³

=3D 3�(�12)�7�5

=3D �1260

 =

(ii) (28) ³ += (� 15)³ + (� 13)³

সমাধা= নঃ

ইয়াত,

28 + (�15) = + (�13) =3D 28 �15 �13 =3D 0

আমি জা= ;নো যে= ;,

যদি,

a= + b + c =3D 0

তেনেহলে,

a³ + b³ + c³ =3D 3abc

গতিকে,

(28)^{3<= /sup>+(�15)3+(�13)3}

=3D 3�28(�15)(�13)

=3D 16380

 =

&nb= sp;

&nb= sp;

&nb= sp;

&nb= sp;

15. তলৰ কালি দù= 5;য়া আü= 7;ত বù= 5;লাকৰ দù= 6;ঘ আþ= 4;ু পú= 9;ৰস্থ বù= 4;বে স÷= 8;্ভাব্য ৰাশিবোৰ= ; উø= 2;িওৱাঃ

(i) কালিঃ 25a² � 35a + 12

সমাধা= নঃ

দিয়া আছ= ;ে কা= ;লি =3D= 25a² � = 35a + 12

এতিয়া,

25a<= sup>2 � 35a + 12

=3D 25a² � (15 + 20)a + 12

=3D 25a² � 15a � 20a = + 12

=3D 5a(5a � 3) � = 4(5a � 3)

=3D (5a<= /i> � 4) (5a � 3)

আমি জা= ;নো যে= ;,

আয়তক্ষে= 68;্ৰৰ কালি =3D দৈ= ;ৰ্ঘ্য � প্ৰস্থ

⸫দৈৰ্ঘ্য =3D (5a � 3) আৰু প্ৰস্থ =3D (5a � 4)

 

(ii) কালিঃ� 35y² + 13<= b>y � 12

সমাধা= নঃ

দিয়া আছ= ;ে কা= ;লি �=3D 35y² + = 13y � 12

এতিয়া,

35y<= sup>2 + 13y � 12

=3D 35y<= /i>² + (<= /span>28 � 15)y<= /i> � 12

=3D 35y<= /i>² + 28y � 15y � 12

=3D 7y (5y + 4) � 3 (5y + 4)

=3D (5y<= /i> + 4)(7y � 3)

আমি জা= ;নো যে= ;,

আয়তক্ষে= 68;্ৰ ফল =3D দৈৰ্= ;ঘ্য � প্= ;ৰস্থ

⸫দৈৰ্ঘ্= ;য =3D = 5y + 4 আৰু প্ৰস&#= 2509;থ =3D <= span dir=3DLTR>7<= i>y � 3

 =

&nb= sp;

&nb= sp;

&nb= sp;

&nb= sp;

&nb= sp;

&nb= sp;

16. তলৰ আয়তন দù= 5;য়া ঘ÷= 2;কৰ মù= 4;ত্ৰা তù= 5;নিটাৰ বù= 4;বে স÷= 8;্ভাব্য ৰাশিকেই= ;টা কù= 5; কù= 5; হ�ব<= /span> ?

(i) আয়তন = =3D 3x² � 12x

সমাধা= নঃ

দিয়া আছ= ;ে �আয়তন =3D 3x² � 12x

এতিয়া,

3x2 � 12x

=3D 3x(x � 4)

⸫দীঘ � = প্ৰস্থ � = উচ্চতা = =3D 3<= i>x (x � 4)

⸫আয়তীয় ঘন= ;কটো দী= ;ঘ =3D 3, প্= ;ৰস্থ�� =3D x আৰু উচ= ;্চতা =3D (<= span style=3D'font-size:12.0pt'>x � = 4)

 =

(ii) আয়তন = =3D 12ky² + 8ky � 20k

সমাধা= নঃ

দিয়া আছ= ;ে আয়= ;তীয় ঘনকৰ আয়তন =3D 12ky² + 8ky � 20k

এতিয়া,

12ky= ² + 8ky � 20k

=3D 4k(3y^{2 + 2y � 5)}

=3D 4k{3y^{2 + (5 � 3)y � 5}}

=3D 4k(3y^{2 + 5y � 3y � 5)}

=3D 4k{y (3y + 5) � 1 (3y + 5)} 

=3D 4k(3y + 5)(y � 1)

 

⸫দীঘ � = প্ৰস্থ � = উচ্চতা = =3D 4<= i>k(3y + 5)(y � 1)

⸫আয়তীয় ঘন= ;কটো দী= ;ঘ =3D 4k, প্= ;ৰস্থ�� =3D (3y + 5) আৰ= ;ু উচ= ;্চতা =3D (<= span style=3D'font-size:12.0pt;mso-bidi-language:AS'>y � 1)

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GYAAN KUNJA TUI= TION CENTRE | 86382 13230 IX-M|2.5-1

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