Date And Test Detail (18 Test ).pdf

18 TEST FIRST YEAR FROM OLD PAPER

TEST #

DESCRIPTION

1.

EX # 9.1+9.2+9.3

2.

EX # 9.4+10.1+10.2

3.

EX # 10.3+10.4+11.1

4.

EX # 12.1 TO 12.5

5.

EX # 12.6 TO 12.8

6.

UNIT # 13 + UNIT # 14 FULL

7.

UNIT # 1(SHORT QUESTION)

8.

EX # 2.1 TO 2.5

9.

EX # 2.6 TO 2.8 +EX # 3.1

10.

EX # 3.2+3.4+3.5

11.

EX # 3.3

12.

EX # 4.1 TO 4.3

13.

EX # 4.4 TO 4.7

14.

EX # 4.8 TO 4.10 + UNIT # 5 (SHORT QUESTION)

15.

EX # 6.1 TO 6.5

16.

EX # 6.6 TO 6.11

17.

UNIT # 7 FULL

18.

UNIT # 8 FULL

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 9.1 +9.2+9.3 ) TEST#:1 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1.

Area of sector of circle of radius 𝒓 is: 𝟏 𝟏 (a) 𝒓𝟐 𝜽 (b) 𝒓𝜽𝟐

3. 4. 5. 6. 7.

𝟐

(d)

𝟐

𝟏

𝟐𝒓𝟐 𝜽

60 part of 𝟏° is equal to (a) One second (b) One minute (c) 1 Radian (d) 𝝅 radian Angles with same initial and terminal sides are called: (a) Acute angles(b) Allied Angles (c)Coterminal angles (d) Quadrentel angles 𝐜𝐨𝐬𝐞𝐜 𝟐 𝜽 − 𝐜𝐨𝐭 𝟐 𝜽 is equal to: (a) 0 (b) 1 (c) -1 (d) 2 The point (𝟎, 𝟏) lies on the terminal side of angle: (a) 𝟎° (b) 𝟗𝟎° (c) 𝟏𝟖𝟎° (d) 𝟐𝟕𝟎° th

2.

𝟐

𝟏

(c) (𝒓𝜽)𝟐

(8)

If initial and the terminal side of an angle falls on 𝒙 − 𝒂𝒙𝒊𝒔 𝒐𝒓 𝒚 − 𝒂𝒙𝒊𝒔 then it is called:

(a) Coterminal angle (b) Quadrantal angl (c) Allied angle

(d) None of these

If 𝒔𝒆𝒄𝜽 < 𝟎 and 𝒔𝒊𝒏𝜽 < 𝟎 then the terminal arm of angle lies in ___________ Quad.

(a) I

(b) II

𝟏

𝒔𝒆𝒄𝟒𝟓° + 𝟐 𝑪𝒐𝒔𝒆𝒄𝟒𝟓° =?

8. Q#2

𝟐

(a) √𝟑

𝟑

(b)

√𝟐

(c) III

(d) IV

(c) −𝟏

(d) 1

SHORT QUESTION 𝟏𝟐

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

SECTION(II)

2.

Find 𝑪𝒐𝒔 𝜽 𝒂𝒏𝒅 𝑻𝒂𝒏 𝜽 if 𝑺𝒊𝒏 𝜽 =

3.

What is circular measure of the angle between the hands of a watch at 4 O’clock?

1.

4. 5. 6.

𝟏

(b)

and the terminal arm of the angle is in quad 1st.

Find 𝒓, when 𝒍 = 𝟓 𝒄𝒎 , 𝜽 = 𝒓𝒂𝒅𝒊𝒂𝒏 𝟐

Find the radius of the circle, in which the arms of a central angle of measure 1 radian cut off an arc of length 35cm. Find 𝒙, if 𝒕𝒂𝒏𝟐 𝟒𝟓° − 𝒄𝒐𝒔𝟐 𝟔𝟎° = 𝒙𝒔𝒊𝒏𝟐 𝟒𝟓°𝒄𝒐𝒔𝟒𝟓°𝒕𝒂𝒏𝟒𝟓° 𝝅

𝝅

𝝅

𝝅

Verify 𝒔𝒊𝒏𝟐 : 𝒔𝒊𝒏𝟐 : 𝒔𝒊𝒏𝟐 : 𝒔𝒊𝒏𝟐 = 𝟏: 𝟐: 𝟑: 𝟒

Q#3

(a)

𝟏𝟑

(12)

𝟔

𝟓

𝟒

𝟑

𝟐

LONG QUESTION

SECTION(III)

(10)

If 𝒄𝒐𝒕𝜽 = and the terminal arm of the angle is in 𝒊𝒗 quad., find the value of 𝟐 𝟑𝒔𝒊𝒏𝜽+𝟒𝒄𝒐𝒔𝜽 𝒄𝒐𝒔𝜽−𝒔𝒊𝒏𝜽

.

𝒎𝟐 +𝟏

𝝅

If 𝒄𝒔𝒄𝜽 = 𝒂𝒏𝒅 𝒎 > 𝟎 (𝟎 < 𝜽 < ), find the values of the remaining 𝟐𝒎 𝟐 trigonometric ratios.

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 9.4+10.1+10.2) TEST#:2 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1.

2.

3. 4. 5.

6.

7. 8.

𝝅

𝒄𝒐𝒔 ( − 𝜷) = 𝟐

(a) 𝒄𝒐𝒔𝜷

(b) – 𝒄𝒐𝒔𝜷

(c) 𝒔𝒊𝒏𝜷

𝜶+𝜷

If 𝜶, 𝜷 and 𝜸 are the angles of a triangle ABC then 𝒄𝒐𝒔 ( (a) 𝐬𝐢𝐧

𝜸

(b) – 𝐬𝐢𝐧

𝟐

𝜸

(c) 𝐜𝐨𝐬

𝟐

𝜸 𝟐

𝟐

(8)

(d) – 𝒔𝒊𝒏𝜷

)=

(d) – 𝐜𝐨𝐬

Angles associated with basic angles of measure 𝜽 to a right angle or its multiple are called:

𝜸 𝟐

(a)Coterminal angle (b) angle in standard position (c) Allied angle (d) obtuse angle

𝒄𝒐𝒔𝒆𝒄𝜽𝒔𝒆𝒄𝜽𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽 = (a) 1 (b) 0 (c) 𝒔𝒊𝒏𝜽 (𝒔𝒆𝒄𝜽 + 𝒕𝒂𝒏𝜽)(𝒔𝒆𝒄𝜽 − 𝒕𝒂𝒏𝜽) = (a) 1 (b) 0 𝒔𝒆𝒄𝜽 𝒄𝒐𝒔𝟏𝟏°+𝒔𝒊𝒏𝟏𝟏°

(d) 𝒄𝒐𝒔𝜽

(d) 𝒕𝒂𝒏𝜽

=

𝒄𝒐𝒔𝟏𝟏°−𝒔𝒊𝒏𝟏𝟏° (a) 𝒕𝒂𝒏𝟓𝟔° (b) 𝒕𝒂𝒏𝟑𝟒° (c) 𝒄𝒐𝒕𝟓𝟔° (d) 𝒄𝒐𝒕𝟑𝟒° If 𝑺𝒊𝒏(𝜶 + 𝜷) is – 𝒊𝒗𝒆 and 𝑪𝒐𝒔(𝜶 + 𝜷) is +𝒊𝒗𝒆 then terminal arm of (𝜶 + 𝜷) lies in

(a) I Quad (b) II Quad (𝒔𝒊𝒏𝜶 + 𝒔𝒊𝒏𝜷)(𝒔𝒊𝒏𝜶 − 𝒔𝒊𝒏𝜷) =

(c) III Quad

(a) 𝐬𝐢𝐧𝟐 𝜶 − 𝐬𝐢𝐧𝟐 𝜷 (b) 𝐬𝐢𝐧𝟐 𝜶 − 𝐜𝐨𝐬𝟐 𝜷

Q#2

(d) IV Quad

(c) 𝐜𝐨𝐬𝟐 𝜶 − 𝐬𝐢𝐧𝟐 𝜷 (d) None of these

SHORT QUESTION

SECTION(II)

2.

Prove that 𝑺𝒊𝒏(𝟏𝟖𝟎° + 𝜶)𝑺𝒊𝒏(𝟗𝟎° − 𝜶) = −𝑺𝒊𝒏𝜶𝑪𝒐𝒔𝜶

3.

Prove that

1.

4.

5.

6. Q#3

𝜶+𝜷

If 𝜶, 𝜷, 𝜸 are angles of triangle ABC, then prove that 𝑪𝒐𝒔 ( 𝒄𝒐𝒔𝟖°−𝒔𝒊𝒏𝟖° 𝒄𝒐𝒔𝟖°+𝒔𝒊𝒏𝟖°

= 𝒕𝒂𝒏𝟑𝟕°

𝟐

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(12)

𝜸

) = 𝑺𝒊𝒏 𝟐

Prove that (𝒔𝒊𝒏𝟑 𝜽 − 𝒄𝒐𝒔𝟑 𝜽) = (𝒔𝒊𝒏𝜽 − 𝒄𝒐𝒔𝜽)(𝟏 − 𝒔𝒊𝒏𝟐 𝜽𝒄𝒐𝒔𝟐 𝜽) If 𝜶, 𝜷, 𝜸 are angles of a triangle 𝑨𝑩𝑪 , Show that

𝒄𝒐𝒕

𝜶 𝟐

𝜷

𝜸

𝜶

𝜷

+ 𝒄𝒐𝒕 + 𝒄𝒐𝒕 = 𝒄𝒐𝒕 𝒄𝒐𝒕 𝒄𝒐𝒕 𝟐

𝟐

𝟐

𝟐

𝜸 𝟐

Show that 𝒄𝒐𝒕𝟒 𝜽 + 𝒄𝒐𝒕𝟐 𝜽 = 𝒄𝒐𝒔𝒆𝒄𝟒 𝜽 − 𝒄𝒐𝒔𝒆𝒄𝟐 𝜽

(a)

If 𝒄𝒐𝒔𝜶 = −

(b)

Prove that

𝟐𝟒 𝟐𝟓

, 𝒕𝒂𝒏 𝜷 =

LONG QUESTION 𝟗

𝟒𝟎

SECTION(III)

, then terminal side of the angle of measure of 𝜶 in the II

quadrant and that of 𝜷 is in the III quadtant, find the value of 𝒄𝒐𝒔(𝜶 + 𝜷). 𝒕𝒂𝒏𝜽+𝒔𝒆𝒄𝜽−𝟏 𝒕𝒂𝒏𝜽−𝒔𝒆𝒄𝜽+𝟏

(10)

= 𝒕𝒂𝒏𝜽 + 𝒔𝒆𝒄𝜽

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 10.3+10.4+unit #11 ) TEST#:3 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

Period of 𝒕𝒂𝒏𝟒𝒙 is 1. (a) 𝝅 (b) 𝟐𝝅

(c) −𝟐𝝅

Range of 𝒚 = 𝒔𝒆𝒄𝒙 is 2. (𝒂)𝑹 (b) 𝒚 ≥ 𝟏𝒐𝒓 𝒚 ≤ −𝟏 𝒕𝒂𝒏𝟐𝜶 = 𝟐𝒕𝒂𝒏𝜶

3.

4.

(a) 𝟏+𝒕𝒂𝒏𝟐 𝜶

8.

𝟒

(d) 𝑹 − [−𝟏, 𝟏]

𝟐 𝐭𝐚𝐧𝟐 𝜶

(d)

𝟏−𝐭𝐚𝐧𝟐 𝜶

(c) −𝟐 𝐬𝐢𝐧 (

𝟐

) 𝐬𝐢𝐧 (

𝟐𝒔𝒊𝒏𝟕𝜽𝒄𝒐𝒔𝟑𝜽 =

(a)

𝟐

𝐭𝐚𝐧𝟐 𝜶

𝟏−𝐭𝐚𝐧𝟐 𝜶

𝒙

Period of 𝟑𝒄𝒐𝒔 𝟓 is

1.

I. II.

)

(a) 𝟐𝝅

𝟐

)

𝟐 𝜶−𝜷

) 𝐜𝐨𝐬 (

𝟐

)

(b)

𝝅

(b)

𝝅

(c) −𝟐𝝅

𝟑

(d)

(c) 𝝅

𝟐

SHORT QUESTION

(d) 10𝝅

SECTION(II)

𝝅 𝟒

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(12)

Find the period of 𝟑𝑺𝒊𝒏 𝒙 Define the 𝒑𝒆𝒓𝒊𝒐𝒅 𝒐𝒇 𝒕𝒓𝒊𝒈𝒐𝒏𝒐𝒎𝒆𝒕𝒓𝒊𝒄 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

2.

Prove that

3.

Prove that

𝜽 𝜽 𝟐 𝟐 𝜽 𝜽 𝒄𝒐𝒕 −𝒕𝒂𝒏 𝟐 𝟐

𝒕𝒂𝒏 +𝒄𝒐𝒕

𝝅

= 𝒔𝒆𝒄𝜽

𝟏 + 𝒕𝒂𝒏𝜶 𝒕𝒂𝒏𝟐𝜶 = 𝒔𝒆𝒄𝟐𝜶 𝝅

𝟏

5.

Prove that 𝒔𝒊𝒏 ( − 𝜽) 𝒔𝒊𝒏 ( + 𝜽) = 𝒄𝒐𝒔𝟐𝜽

6.

Prove the identity :

4.

𝟐 𝜶+𝜷

(d) 𝟐 𝐜𝐨𝐬 (

𝜶−𝜷

) 𝐬𝐢𝐧 (

𝒔𝒊𝒏𝟏𝟎𝜽 + 𝒔𝒊𝒏𝟒𝜽 (b) 𝒔𝒊𝒏𝟓𝜽 − 𝒔𝒊𝒏𝟐𝜽 (c) 𝒄𝒐𝒔𝟏𝟎𝜽 + 𝒄𝒐𝒔𝟒𝜽 (d) 𝒄𝒐𝒔𝟓𝜽 − 𝒄𝒐𝒔𝟐𝜽

Period of 𝒄𝒐𝒕𝟑𝒙 is (a) 𝝅

Q#2

𝜶+𝜷

(b) 𝟐 𝐜𝐨𝐬 (

𝜶−𝜷

𝜶+𝜷

7.

𝝅

(𝒂) 𝟑𝒔𝒊𝒏𝜶 − 𝟐 𝐬𝐢𝐧𝟑 𝜶(b) 𝟑𝒔𝒊𝒏𝜶 + 𝟐 𝐬𝐢𝐧𝟑 𝜶 (c) 𝟑𝒔𝒊𝒏𝜶 − 𝟒 𝐬𝐢𝐧𝟑 𝜶 (d) 𝟑𝒄𝒐𝒔𝜶 − 𝟐 𝐬𝐢𝐧𝟑 𝜶

𝒄𝒐𝒔𝜶 − 𝒄𝒐𝒔𝜷 is equal to: 𝜶+𝜷 𝜶−𝜷 5. (a) 𝟐 𝐬𝐢𝐧 ( 𝟐 ) 𝐜𝐨𝐬 ( 𝟐 ) 6.

(c)

𝟏−𝐭𝐚𝐧𝟐 𝜶

𝒔𝒊𝒏𝟑𝜶 =

(d)

(c) −𝟏 ≤ 𝒚 ≤ 𝟏

𝟐𝒕𝒂𝒏𝜶

(b)

(8)

Prove that

Q#3

𝟒

𝟒

𝑺𝒊𝒏𝟖𝒙+𝑪𝒐𝒔𝟑𝒙

𝟐

= 𝒕𝒂𝒏𝟓𝒙

𝑪𝒐𝒔𝟓𝒙+𝑪𝒐𝒔𝟐𝒙 𝒔𝒊𝒏𝜶−𝒔𝒊𝒏𝜷 𝒔𝒊𝒏𝜶+𝒔𝒊𝒏𝜷

= 𝒕𝒂𝒏

𝜶−𝜷

LONG QUESTION

(a)

Prove that

(b)

Prove that

𝒄𝒐𝒔𝒆𝒄𝜽+𝟐𝒄𝒐𝒔𝒆𝒄𝟐𝜽 𝒔𝒆𝒄𝜽

= 𝒄𝒐𝒕

𝟐

𝜽 𝟐

𝒄𝒐𝒕

𝜶+𝜷 𝟐

SECTION(III)

𝑺𝒊𝒏𝟏𝟎°𝑺𝒊𝒏𝟑𝟎°𝑺𝒊𝒏𝟓𝟎°𝑺𝒊𝒏𝟕𝟎° =

𝟏

𝟏𝟔

(10)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 12.1 to 12.5) TEST#:4 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

A “Triangle” has : 1. (a) Two elements (b) 𝟑 elements If 𝒔𝒊𝒏𝒙 = 𝟎. 𝟓𝟏𝟎𝟎 then 𝒙 = 2. (a) 𝟑𝟎°𝟒𝟎′ (b) 𝟑𝟓°𝟒𝟎′ 3.

(c) 𝟒 elements

(c) 𝟒𝟎°𝟒𝟎′

(8)

(d) 𝟔 elments

(d) 𝟒𝟒°𝟒𝟒′

When we look an object below the horizontal ray, the angle formed is called angle of: (c) incidence (d) reflects (a) Elevation (b) depression

To solve an oblique triangle we use: 4. (a) Law of Sine (b) Law of Cosine

(c) Law of Tangents

(d) All of these

In any triangle 𝑨𝑩𝑪, if 𝜷 = 𝟗𝟎° , then 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄𝒄𝒐𝒔𝜷 becomes: 5. (a) Law of sine (b) Law of tangents (c) Law of cosine (d) Pythagoras theorem 6.

In any triangle 𝑨𝑩𝑪, √ (a)

7.

𝜶 𝐬𝐢𝐧 𝟐

In any triangle (a)

𝜸 𝐬𝐢𝐧 𝟐

𝒂𝒄

=

𝜷 (b) 𝐬𝐢𝐧 𝟐 (𝒔−𝒂)(𝒔−𝒃) 𝑨𝑩𝑪, √ 𝒔(𝒔−𝒄) =

8. In any triangle 𝑨𝑩𝑪, (a)𝒄𝒐𝒔𝜶 Q#2

(𝑺−𝒂)(𝒔−𝒄)

(b)

𝒃𝟐 +𝒄𝟐 −𝒂𝟐 𝟐𝒃𝒄

(c)

𝜸 𝐬𝐢𝐧 𝟐

𝜸 𝐜𝐨𝐬 𝟐

=? (b) 𝒔𝒊𝒏𝜶

(c)

2. Find 𝜽, if 𝒔𝒊𝒏𝜽 = 𝟎. 𝟓𝟕𝟗𝟏 and 𝒕𝒂𝒏𝜽 = 𝟏. 𝟕𝟎𝟓

Solve the triangle 𝑨𝑩𝑪 in which :

6. Find the values of 𝒄𝒐𝒔𝟑𝟔°𝟐𝟎′ and 𝒄𝒐𝒕𝟖𝟗°𝟗′ Q#3 (a) (b)

(d)

SECTION(II) ′

𝜸 𝐜𝐨𝐭 𝟐

(d) 𝒄𝒐𝒔𝜸

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(12)

, 𝜷 = 𝟓𝟎°𝟏𝟎 , 𝒄 = 𝟎. 𝟖𝟑𝟐

A vertical pole is 𝟖𝒎 high and length of its shadow is 𝟔𝒎. What is the angle of elevation of the sun at that moment?

4. Solve the triangle 𝑨𝑩𝑪, if 𝒄 = 𝟏𝟔. 𝟏 , 𝜶 = 𝟒𝟐°𝟒𝟓′ 5.

𝜸 𝐭𝐚𝐧 𝟐

(c) 𝒄𝒐𝒔𝜷

SHORT QUESTION

1. Solve the right triangle 𝑨𝑩𝑪, in which 𝜸 = 𝟗𝟎° 3.

(d)

𝜶 𝐜𝐨𝐬 𝟐

A

LONG QUESTION

, 𝜸 = 𝟕𝟒°𝟑𝟐′

𝒃 = 𝟑,

𝒄 = 𝟔 and 𝜷 = 𝟑𝟔°𝟐𝟎′

SECTION(III)

(10)

Solve the triangle using first law of tangents and then law of sines: 𝒃 = 𝟏𝟒. 𝟖 , 𝒄 = 𝟏𝟔. 𝟏 and 𝜶 = 𝟒𝟐°𝟒𝟓′ Solve the triangle 𝑨𝑩𝑪 in which : 𝒂 = √𝟑 − 𝟏 , 𝒃 = √𝟑 + 𝟏 and 𝜸 = 𝟔𝟎°

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 12.6 to 12.8) TEST#:5 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

(8)

To solve an oblique triangles when measure of three sides are given , we can use: 1. (a) Hero’s formula (b) Law of cosine (c) Law of sine (d) Law of tangents In any triangle 𝑨𝑩𝑪 Area if triangle is : 2. 3. 4.

𝟏

𝟏

(a) 𝒃𝒄 𝐬𝐢𝐧 𝜶

(b) 𝒄𝒂 𝒔𝒊𝒏𝜶

(a) 𝑹

(b)

(c) 𝒂𝒃 𝒔𝒊𝒏𝜷

𝟐

𝟐

𝟏

(d)

𝟐

𝒂𝒃𝒔𝒊𝒏𝜸

The smallest angle of ∆𝑨𝑩𝑪, when 𝒂 = 𝟑𝟕. 𝟑𝟒 , 𝒃 = 𝟑. 𝟐𝟒 , 𝒄 = 𝟑𝟓. 𝟎𝟔 is (a) 𝜶 (b) 𝜷 (c) 𝜸 (d) cannot be determined In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔𝒊𝒏 𝜸 = 𝒄

(c)

𝟐𝑹

𝟐𝑹

(d)

𝒄

𝑹 𝟐

The point of intersection of the right bisectors of the sides of the triangle is : 5. (a) Circum centre (b) In-centre (c) Escribed center (d) Diameter In any triangle 𝑨𝑩𝑪, with usual notation , 𝒓: 𝑹: 𝒓𝟏 : 𝒓𝟐 : 𝒓𝟑 = 6. (a) 3:3:3:2:1 (b) 1:2:2:3:3 (c) 1:2:3:3:3 (d) 1:1:1:1:1 In a triangle 𝑨𝑩𝑪, if 𝜷 = 𝟔𝟎° , 𝜸 = 𝟏𝟓° then 𝜶 = 7. (𝒂) 𝟗𝟎° (b) 𝟏𝟖𝟎° (c) 𝟏𝟓𝟎° (d) 𝟏𝟎𝟓° 8.

Q#2

1.

∆

In any triangle 𝑨𝑩𝑪, with usual notations, 𝒔−𝒄 = (𝒂)𝒓𝟑

(b) 𝑹

(c) 𝒓𝟏

SHORT QUESTION

SECTION(II)

(d) 𝒓𝟐

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(12)

Find the area of the triangle 𝑨𝑩𝑪, in which 𝒃 = 𝟐𝟏. 𝟔 , 𝒄 = 𝟑𝟎. 𝟐 and 𝜶 = 𝟓𝟐°𝟒𝟎′

3.

The area of the triangle is 𝟐𝟒𝟑𝟕. If 𝒂 = 𝟕𝟗 and 𝒄 = 𝟗𝟕, then find angle 𝜷.

4.

Solve the triangle , in which

2.

A

Find the measure of the greatest angle , if sides of the angle are 𝟏𝟔, 𝟐𝟎, 𝟑𝟑. 𝒂=𝟕,

𝒃=𝟕

,

𝒄=𝟗

5.

The sides of triangle are 𝒙𝟐 + 𝒙 + 𝟏, 𝟐𝒙 + 𝟏 and 𝒙𝟐 − 𝟏. Prove that the greatest angle of

6.

Show that

the triangle is 𝟏𝟐𝟎°.

Q#3 (a) (b)

𝒓𝟐 = 𝒔 𝒕𝒂𝒏

𝜷 𝟐

LONG QUESTION

Prove that 𝒓𝟏 + 𝒓𝟐 + 𝒓𝟑 − 𝒓 = 𝟒𝑹 Prove that 𝒓 = 𝒔 𝒕𝒂𝒏

𝜶 𝟐

𝒕𝒂𝒏

𝜷 𝟐

𝒕𝒂𝒏

𝜸 𝟐

SECTION(III)

(10)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( UNIT # 13 +14) TEST#:6 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

𝑻𝒂𝒏−𝟏 𝒙 = 𝝅 1. (a) − 𝐬𝐞𝐜 −𝟏 𝒙 𝟐

𝟏 𝑺𝒊𝒏 (𝑺𝒊𝒏−𝟏 𝟐) 𝟏

2.

(a)

𝟐

(b)

=

(b)

𝝅 𝟐

𝟐

− 𝐬𝐢𝐧−𝟏 𝒙

𝟑

𝑪𝒐𝒔−𝟏 (−𝒙) = (a) – 𝑪𝒐𝒔−𝟏 𝒙 (b) 𝑪𝒐𝒔−𝟏 𝒙 𝑺𝒊𝒏−𝟏 𝑨 − 𝑺𝒊𝒏−𝟏 𝑩 = (a) 𝑺𝒊𝒏−𝟏 (𝑨√𝟏 − 𝑩𝟐 − 𝑩√𝟏 − 𝑨𝟐 ) 4. (c) 𝑺𝒊𝒏−𝟏 (𝑩√𝟏 − 𝑨𝟐 + 𝑨√𝟏 − 𝑩𝟐 ) 3.

(c)

𝝅 𝟐

−𝐜𝐨𝐭 −𝟏 𝒙

(d) 𝝅 − 𝑪𝒐𝒔𝒙

(d) 𝑺𝒊𝒏−𝟏 (𝑨𝑩√(𝟏 − 𝑨𝟐 )(𝟏 − 𝑩𝟐 ))

𝝅

𝑻𝒂𝒏−𝟏 (−𝒙) = (a) – 𝑻𝒂𝒏−𝟏 𝒙 (b) 𝑻𝒂𝒏−𝟏 𝒙 𝑻𝒂𝒏−𝟏 𝑨 + 𝑻𝒂𝒏−𝟏 𝑩 = 8. (a) 𝑻𝒂𝒏−𝟏 ( 𝑨−𝑩 ) (b) 𝑻𝒂𝒏−𝟏 ( 𝑨+𝑩 ) 𝟏+𝑨𝑩 𝟏+𝑨𝑩 7.

Q#2

1. Prove that

=

𝟐𝟒 𝒔𝒊𝒏−𝟏 𝟐𝟓

(d) None

𝝅

𝝅

(c) 𝟒

(c) 𝝅 − 𝑻𝒂𝒏−𝟏 𝒙 𝑨+𝑩

(c) 𝑻𝒂𝒏−𝟏 (𝟏−𝑨𝑩)

SHORT QUESTION

𝟒 𝟐𝒄𝒐𝒔−𝟏 𝟓

𝟑

(b) 𝑺𝒊𝒏−𝟏 (𝑨√𝟏 − 𝑨𝟐 − 𝑩√𝟏 − 𝑩𝟐 )

6.

(b) 𝟔

𝟏

(c) 𝝅 − 𝑪𝒐𝒔−𝟏 𝒙

𝒔𝒊𝒏𝒙 = 𝟐 , 𝒙 is equal to: 𝝅

𝟐

− 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙

(d)

Number of solutions of trigonometric function is: (a) Finite (b) Infinite (c) Only one

(a) 𝟐

𝝅

(d)

(c) 2

5.

𝟏

(8)

(d) 𝟑

(d) 𝝅 − 𝑻𝒂𝒏𝒙

𝑨+𝑩

(d) 𝑻𝒂𝒏−𝟏 (𝟏+𝑨𝑩)

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

SECTION(II)

(12)

SECTION(III)

(10)

𝟏

2. Find the value of 𝒕𝒂𝒏 (𝒔𝒊𝒏−𝟏 (− 𝟐)) −𝟏 −𝟏 3. Show that 𝒕𝒂𝒏 (−𝒙) = −𝒕𝒂𝒏 𝒙 𝒙

−𝟏 4. Show that 𝒕𝒂𝒏 (𝒔𝒊𝒏 𝒙) = √𝟏−𝒙𝟐

−𝟏 𝟐 5. Show that 𝒄𝒐𝒔(𝒔𝒊𝒏 𝒙) = √𝟏 − 𝒙

6. Solve 𝟏 + 𝒄𝒐𝒔𝒙 = 𝟎 Q#3 (a) (b)

Prove that Prove that

LONG QUESTION

𝟏𝟐𝟎 𝒕𝒂𝒏−𝟏 𝟏𝟏𝟗 𝟏

=

𝟏𝟐 𝟐𝒄𝒐𝒔−𝟏 𝟏𝟑 𝟏

𝟗

𝒕𝒂𝒏−𝟏 𝟒 + 𝒕𝒂𝒏−𝟏 𝟓 = 𝒕𝒂𝒏−𝟏 𝟏𝟗

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TEST : MATHEMATICS ( unit # 1 )

TIME: 1:10 HOUR

TEST#:7

Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1.

Every recurring decimal is (a)a rational number

(b) an irrational number (c) a prime number

The set {1,-1} possess closure property 𝒘. 𝒓. 𝒕 (a) Addition (b) multiplication (c) division 𝟑 Imaginary part of (−𝟐 + 𝟑𝒊 ) is 3. (a)-2 (b) -3 (c) 26 The product of two conjugate complex numbers is 2.

4.

(𝟎, 𝟑)(𝟎, 𝟓) is equal to: (a) 15 (b) -15 𝟑 (𝟎, 𝟏) is equal to: 6. (a) 1 (b) -1 If 𝒏 is an even integer, then (𝒊)𝒏 is equal to: 8.

(d) a whole number

(d) subtraction (d) -8

(a)A real number (b) an imaginary number (c) may be an irrational number (d) not defined

5.

7.

(10)

(𝒂) 𝒊

(b) – 𝒊

Multiplicative inverse of (−𝟒, 𝟕) 𝟒

𝟕

𝟒

(a) (− 𝟔𝟓 , − 𝟔𝟓)

𝟕

(b) (𝟔𝟓 , − 𝟔𝟓)

(c) (−

(b) 2

(c) -1

(c) −𝟖𝒊

(d) 𝟖𝒊

(c) 𝒊

(c) ±𝟏 𝟒

√𝟔𝟓

,−

(d) – 𝒊 (d) 1

𝟕

)

√𝟔𝟓

(d) (

𝟒

√𝟔𝟓

,−

𝟕

)

√𝟔𝟓

Factors of 𝟑(𝒙𝟐 + 𝒚𝟐 ) are: 9. (𝒂) 𝟑(𝒙 + 𝒚)(𝒙 − 𝒚) (b) 𝟑(𝒙 + 𝒊𝒚)(𝒙 − 𝒊𝒚)(c) √𝟑(𝒙 + 𝒊𝒚)(𝒙 − 𝒊𝒚) (d) None of these Real part of

10.

(a) 1

𝟐+𝒊 𝒊

is:

Q#2

𝒂

=

𝒄

SHORT QUESTION

<=> 𝒂𝒅 = 𝒃𝒄

1.

Prove that

2.

Separate into real and imaginary parts

𝒃

𝒅

𝟏 + √𝟑𝒊

3.

Express the complex number

4.

Find the multiplicative inverse of

5. 6. 7. 8. 9. 10.

Simplify 𝒊𝟏𝟎𝟏

SECTION(II)

and 𝒊−𝟑

𝒊

𝟏+𝒊

into polar form.

(1 , 2)

Define Rational and Irrational numbers. 𝟐

Show that ∀ 𝒛 ∈ 𝑪 , 𝒛𝟐 + 𝒛 𝒊𝒔 𝒂 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓. Simplify (𝒂 − 𝒊𝒃)𝟑 Prove that

𝒛̅ = 𝒛 𝒊𝒇𝒇 𝒛 is real.

Simplify by expressing in the form of 𝒂 + 𝒃𝒊 ,

𝟐

√𝟓+√−𝟖

𝟏

(d) 𝟐

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(20)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 2.1 to 2.5 ) TEST#:8 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1. 2. 3. 4. 5. 6. 7. 8.

Truth set of a tautology is the (a) Power set (b) Subset (c) Universal set (d) Super set The symbol “∀” is called (a) Universal quantifier (b) Existential quantifier (c) Converse(d) Inverse Truth set of a tautology is (a) Universal set (b) True (c) True (d) False If 𝐔 = {𝟏, 𝟐, 𝟑, 𝟒, 𝟓, … … , 𝟐𝟎} and 𝐀 = {𝟏, 𝟑, 𝟓, … . . , 𝟏𝟗} then 𝐀 ∩ 𝐔 = (d) 𝐀′ (a) 𝐀 (b) 𝐔 (c) ∅ If the intersection of two sets is the empty then sets are called (a) Disjoint sets (b) Overlapping Sets (c) Subsets (d) Power sets If 𝑨 and 𝑩 are disjoint sets then : (a) 𝐀 ∩ 𝐁 = 𝛗 (b) 𝐀 ∩ 𝐁 ≠ 𝛗 (c) 𝐀 ⊂ 𝐁 (d) 𝐀 − 𝐁 = 𝛗 The set of odd integers between 2 and 4 is (a) Null set (b) Power set (c) Singleton set (d) Subset Total number of subsets that can be formed from the set {𝒙,𝒚,𝒛} is (c) 5 (d)2 (a) 1 (b) 8

Q#2

1. 2. 3. 4. 5. 6.

(b)

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

SECTION(II)

(12)

Write the following set in “𝒅𝒆𝒔𝒄𝒓𝒊𝒑𝒕𝒊𝒗𝒆 𝒎𝒆𝒕𝒉𝒐𝒅” and “ 𝒕𝒂𝒃𝒖𝒍𝒂𝒓 𝒇𝒐𝒓𝒎” (a) {𝒙|𝒙 ∈ 𝑵 ∧ 𝟒 < 𝑥 < 12} (b) {𝒙|𝒙 ∈ 𝑹 ∧ 𝒙 = 𝒙} Write down the power set of {𝝋} and {+, −, ×,÷}. Let 𝑼 =The set of English alphabet , 𝑨 = {𝒙|𝒙 is a vowel } and 𝑩 = {𝒚|𝒚 is consonant} Verify (𝑨 ∪ 𝑩)′ = 𝑨′ ∩ 𝑩′ Write converse , inverse and contra positive of ~𝒑 → 𝒒 Show that ~(𝒑 → 𝒒) → 𝒑 is tautology. Define Absurdity with example.

Q#3 (a)

SHORT QUESTION

(8)

LONG QUESTION

SECTION(III)

Prove that 𝒑 ∨ (∼ 𝒑 ∧∼ 𝒒) ∨ (𝒑 ∧ 𝒒) = 𝒑 ∨ (∼ 𝒑 ∧∼ 𝒒)

Convert (𝑨 ∪ 𝑩) ∪ 𝑪 = 𝑨 ∪ (𝑩 ∪ 𝑪) into logical form and prove it by constructing the truth table.

(10)

ILM GROUP OF COLLEGES KHANPUR

NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 2.6 TO 2.8+ 3.1 ) TEST#:9 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1. 2. 3. 4. 5. 6. 7. 8. Q#2

The range of {(𝟐, 𝟏), (𝟑, 𝟐), (𝟒, 𝟑), (𝟓, 𝟒), (𝟔, 𝟓)} (a) {𝟐, 𝟑, 𝟒, 𝟓, 𝟔} (b) {𝟏, 𝟐, 𝟑, 𝟒, 𝟓} (c) {𝟐, 𝟏, 𝟑, 𝟐, 𝟒} (d) {𝟏, 𝟐, 𝟑, 𝟓} Cube root of a number is example of (a) Binary operation (b) Unary operation (c) relation (d) function Inverse and identity of a set 𝑺 under binary operation ∗ is (a) Unique (b) Two (c) Three (d) Four A semi-group having an identity is called Group (b) monoid (c) Closed (d) Not closed In a group the inverse is (b) two (d) three (d) four (a) Unique (𝑨𝒕 )𝒕 = (𝒂) 𝑨𝒕 (c) – 𝑨 (d) (𝑨𝒕 )𝒕 (b) 𝑨 Which of the following Sets is a field. (a) R (b) Q (c) C (d) all of these For the square matrix 𝑨 = [𝒂𝒊𝒋 ]𝒏×𝒏 then 𝒂𝟏𝟏 , 𝒂𝟐𝟐 , 𝒂𝟑𝟑… 𝒂𝒏𝒏 are: (a) Main diagonal(b) primary diagonal(c) proceding diagonal(d) secondary diagonal SHORT QUESTION

(8)

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

SECTION(II)

(12)

1. For 𝑨 = {𝟏, 𝟐, 𝟑, 𝟒}, find the following relation in 𝑨. State domain and range of the relation. {(𝒙, 𝒚)| 𝒙 + 𝒚 < 𝟓} 2. 3. 4.

Prepare a table of addition of the elements of the set of residue classes modulo 4. Prove that (𝒂𝒃)−𝟏 = 𝒃−𝟏 𝒂−𝟏 𝒙+𝟑 𝟏 𝟐 𝟏 Find the value of 𝒙 and 𝒚 if ] [ ]=[ −𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐

𝟑 −𝟕 𝟓 −𝟏 Find the matrix 𝑨 if ; [ ] 𝑨 = [ 𝟎 𝟎] 𝟎 𝟎 5. 𝟕 𝟐 𝟑 𝟏 𝟏 −𝟏 𝟏 𝟎 ] and 𝑨𝟐 = [ ], find the values of 𝒂 and 𝒃. 6. If 𝑨 = [ 𝒂 𝒃 𝟎 𝟏 Q#3 (a)

(b)

LONG QUESTION

SECTION(III)

(10)

Prove that all 𝟐 ×𝟐 non-singular matrices over the real field form a non-abelian group under multiplication. 𝟐 𝟎 𝒙 𝟏 𝒙 𝒚 𝟒 −𝟐 𝟑 Find 𝒙 and 𝒚 if [ ]=[ ] ] + 𝟐[ 𝟏 𝒚 𝟑 𝟏 𝟔 𝟏 𝟎 𝟐 −𝟏

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 3.2+3.4+3.5 ) TEST#:10 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1. 2.

In general matrix multiplication is not (a) Commutative (b) Associative A square matrix A is skew- Hermitian if:

(c) Closure

(a) 𝑨𝒕 = 𝑨

(c) (𝑨) = 𝑨

(b) 𝑨𝒕 = −𝑨

(8)

(d) Distributive

𝒕

𝒕

(d) (𝑨) = −𝑨

If A is symmetric (Skew symmetric), then 𝑨𝟐 must be 3. (a) Singular (b) non singular (c) symmetric (d) non trivial solution The main diagonal elements of a skew hermitian matrix must be: 4. (a) 1 (b) 0 (c) any non-zero number (d) any complex number 5.

A square matrix 𝑨 = [𝒂𝒊𝒋 ] for which 𝒂𝒊𝒋 = 𝟎, 𝒊 > 𝒋 then A is called: (a) Upper triangular (b) Lower triangular (c) Symmetric

6.

In a homogeneous system of linear equations , the solution (0,0,0) is: (a) Trivial solution (b) non trivial solution (c) exact solution (d) anti symmetric

7.

(d) Hermitian

If the system 𝒙 + 𝟐𝒚 = 𝟎; 𝟐𝒙 + 𝝀𝒚 = 𝟎 has non-trivial solution, then 𝝀 is: (a) 4 (b) -4 (c) ±𝟒 (d) any real number

If the system of linear equations have no solution at all, then it is called a/an 8. (a) Consistent system (b) Inconsistent system (c) Trivial System (d) Non Trivial System Q#2

SHORT QUESTION

1. Solve the following system of linear equations:

SECTION(II)

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

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𝟑𝒙 − 𝟓𝒚 = 𝟏 ; −𝟐𝒙 + 𝒚 = −𝟑

𝟐 2. If 𝑨 is symmetric or skew-symmetric, show that 𝑨 is symmetric.

𝟏

̅ )𝒕 . 3. If 𝑨 = [𝟏 + 𝒊], find 𝑨(𝑨 𝒊 If = [ 4. 𝟏 5.

𝒊 𝟏+𝒊 ̅ )𝒕 is hermitian. ] , show that 𝑨 + (𝑨 −𝒊

Find the inverse of

𝟐 If 𝑨 = [ 𝟏 6. −𝟑 Q#3 (a)

(b)

𝟐𝒊 [ 𝒊

𝒊 ] −𝒊

−𝟏 𝟑 𝟎 𝟎 𝟒 −𝟐 ] then find 𝑨𝒕 𝑨 𝟓 𝟐 −𝟏

LONG QUESTION

SECTION(III)

Solve by using Cramer’s Rule 𝟐𝒙 + 𝟐𝒚 + 𝒛 = 𝟑 ; 𝟑𝒙 − 𝟐𝒚 − 𝟐𝒛 = 𝟏 ; 𝟓𝒙 + 𝒚 − 𝟑𝒛 = 𝟐

Solve the following matrix equation for 𝑨: [

𝟒 𝟑 𝟐 ]𝑨 − [ 𝟐 𝟐 −𝟏

−𝟏 −𝟒 𝟑 ]=[ ] 𝟑 𝟔 −𝟐

(10)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TEST : MATHEMATICS ( EX# 3.3 )

TIME: 1:10 HOUR

TEST#:11

Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

For any matrix A , it is always true that 1. ̅ (a) 𝑨 = 𝑨𝒕 (b) – 𝑨 = 𝑨 2. 3. 4. 5.

(c) |𝑨| = |𝑨𝒕 |

(8)

If all entries of a square matrix of order 𝟑 is multiplied by 𝒌, then value of |𝒌𝑨| is equal to:

(b) 𝒌𝟐 |𝑨|

(a) 𝒌|𝑨|

(c) 𝒌𝟑 |𝑨|

For a non-singular matrix it is true that : (a) (𝑨−𝟏 )−𝟏 = 𝑨 (b) (𝑨𝒕 )𝒕 = 𝑨

̿=𝑨 (c) 𝑨

𝟏

(d) 𝑨−𝟏 = 𝑨 (d) |𝑨|

(d) all of these

For any non-singular matrices A and B it is true that: (a) (𝑨𝑩)−𝟏 = 𝑩−𝟏 𝑨−𝟏 (b) (𝑨𝑩)𝒕 = 𝑩𝒕 𝑨𝒕 (c) 𝑨𝑩 ≠ 𝑩𝑨 (d) all of these

If a square matrix 𝑨 has two identical rows or two identical columns then (a) 𝑨 = 𝟎 (b) |𝑨| = 𝟎 (c) 𝑨𝒕 = 𝟎 (d) 𝑨 = 𝟏

If a matrix is in triangular form, then its determinant is product of the entries of its

6. (a)Lower triangular matrix

(b) Upper triangular matrix (c) main diagonal (d) none of these

If 𝑨 is non-singular matrix then 𝑨−𝟏 = 7. 𝟏 𝟏 (𝒂) 𝒂𝒅𝒋𝑨 (b) − 𝒂𝒅𝒋𝑨 |𝑨|

|𝑨|

(𝑨−𝟏 )𝒕 = 8. (a) 𝑨−𝟏

(b) (𝑨−𝟏 )𝒕

Q#2

|𝑨|

(c) 𝒂𝒅𝒋𝑨

(c) (𝑨𝒕 )−𝟏

SHORT QUESTION

𝟐𝒂 𝒂 𝒂 1. Evaluate | 𝒃 𝟐𝒃 𝒃 | 𝒄 𝒄 𝟐𝒄

SECTION(II)

𝟏

(d) |𝑨|𝒂𝒅𝒋𝑨

(d) 𝑨𝒕

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(12)

𝟏

𝟐 −𝟐 𝟏 𝟑| 𝟐 𝟒 −𝟏 𝟏 𝟐 𝟑𝒙 3. Without expansion verify that |𝟐 𝟑 𝟔𝒙| = 𝟎 𝟑 𝟓 𝟗𝒙 𝟑 𝟏 𝒙 4. Find the value of 𝒙 if |−𝟏 𝟑 𝟒| = −𝟑𝟎 𝒙 𝟏 𝟎 If 𝑨 and 𝑩 are non-singular matrices, then show that (𝑨−𝟏 )−𝟏 = 𝑨 5. 2. Find the determinant of |−𝟏

6.

𝟐

Show that |𝟑

Q#3

𝟐

𝟑 𝟎 𝟐 𝟗 𝟔| = 𝟗 |𝟏 𝟏𝟓 𝟏 𝟐

𝟐 (a) Find the inverse [𝟏 𝟐

(b)

𝒂+𝝀 Show that | 𝒂 𝒂

𝟏 𝟏 𝟓

𝟎 𝟐|. 𝟏

LONG QUESTION

SECTION(III)

𝟏 𝟗 −𝟏 𝟑] and show that 𝑨−𝟏 𝑨 = 𝒍𝟑 . −𝟒 𝟏

𝒃 𝒄 𝒃+𝝀 𝒄 | = 𝝀𝟐 (𝒂 + 𝒃 + 𝒄 + 𝝀) 𝒃 𝒄+𝝀

(10)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 4.1 to 4.3) TEST#:12 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

(8)

The equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 will be quadratic if: 1. (a) 𝒂 = 𝟎, 𝒃 ≠ 𝟎 (b) 𝒂 ≠ 𝟎 (c) 𝒂 = 𝒃 = 𝟎 (d) 𝒃 = any real number The solution of a quadratic equation are called 2. (a)Roots (b) identity (c) quadratic equation (d) solution Solution set of the equation 𝒙𝟐 − 𝟒𝒙 + 𝟒 = 𝟎 is: 3. (𝒂) {𝟐, −𝟐} (b) {𝟐} (c) {−𝟐}

To convert 𝟒𝟏+𝒙 + 𝟒𝟏−𝒙 = 𝟏𝟎 into quadratic , the substitution is: 4. (𝒂) 𝒚 = 𝒙𝟏−𝒙 (b) 𝒚 = 𝟒𝟏+𝒙 (c) 𝒚 = 𝟒𝒙

(d) {𝟒, −𝟒}

(d) 𝒚 = 𝟒−𝒙

The equation 𝒙𝟒 − 𝟑𝒙𝟑 + 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟏 = 𝟎 is example of 5. (a) Exponential equation (b) Quadratic equation (c) Radical equation (d) Reciprocal equation 6.

To convert 𝒂𝒙𝟐𝒏 + 𝒃𝒙𝒏 + 𝒄 = 𝟎(𝒂 ≠ 𝟎) into quadratic form , the correct substitution is:

(a) 𝒚 = 𝒙𝒏

(b) 𝒙 = 𝒚𝒏

(c) 𝒚 = 𝒙−𝒏

The equation in which variable occurs in exponent , called: (b) Quadratic equation (c) Reciprocal equation (d) Exponential equation The equations involving redical expressions of the variable are called: 8. (a) Reciprocal equations (b) Redical equations (c) Quadratic functions (d) exponential equation 7. (a) Exponential function

Q#2

1. 2.

SHORT QUESTION 𝟐 𝟐

𝟏

(d) 𝒚 = 𝒙

SECTION(II)

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(12)

Solve by quadratic formula 𝟏𝟓𝒙 + 𝟐𝒂𝒙 − 𝒂 = 𝟎 Solve 𝟒𝟏+𝒙 + 𝟒𝟏−𝒙 = 𝟏𝟎 𝟐

𝟏

3. Solve 𝒙𝟓 + 𝟖 = 𝟔𝒙𝟓 4. 5. 6.

Solve

√𝟐𝒙 + 𝟖 + √𝒙 + 𝟓 = 𝟕

Define “𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏”.

Solve √𝒙𝟐 + 𝒙 − 𝟏 = 𝟏 and check

Q#3 (a)

(b)

LONG QUESTION

Solve by factorization

𝒂

𝒂𝒙−𝟏

+

𝒃

𝒃𝒙−𝟏

=𝒂+𝒃

SECTION(III) 𝟏 𝟏

;𝒙≠ ,

𝒂 𝒃

Solve (𝒙 − 𝟏)(𝒙 + 𝟓)(𝒙 + 𝟖)(𝒙 + 𝟐) − 𝟖𝟖𝟎 = 𝟎

(10)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 4.4 to 4.7 ) TEST#:13 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

The complex fourth roots of unity are ……. of each other. (c) square of

1. (a) Additive inverse (b) equal to 2.

The sum of all four fourth roots of 16 is: (a) 16 (b) -16

(8)

(d) None of these

(c) 0

(d) 1

A

B

C

D

A

B

C

D

A

B

C

D

The cube roots of unity are : 3. (a) 𝟏, −𝟏+√𝟑𝒊 , −𝟏−√𝟑𝒊 (b) 𝟏, 𝟏+√𝟑𝒊 , 𝟏+√𝟑𝒊 𝟐

4.

𝟐

𝟐

Product of cube roots of -1 is: (a) 0 (b) -1

𝟐

(c) −𝟏,

𝒙 − 𝟐 is a factor of 𝒙𝟐 − 𝒌𝒙 + 𝟒, if 𝒌 is: 5. (a) 2 (b) 𝟒

−𝟏+√𝟑𝒊 −𝟏+√𝟑𝒊 𝟐

,

𝟐

(d) −𝟏,

𝟏+√𝟑𝒊 𝟏+√𝟑𝒊 𝟐

,

𝟐

(c) 1

(d) None

A

B

C

D

(c) 8

(d) -4

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

If 𝜶 and 𝜷 are the roots of 𝟑𝒙𝟐 − 𝟐𝒙 + 𝟒 = 𝟎, then the value of 𝜶 + 𝜷 is: 𝟐 𝟐 𝟒 𝟒 6. (a) 𝟑 (b) − 𝟑 (c) 𝟑 (d) − 𝟑 If roots of 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, (𝒂 ≠ 𝟎) are equal , then 7. (a) Disc= 𝟎 (b) Disc< 𝟎 (c) Disc≠ 𝟎 The expression 𝒃𝟐 − 𝟒𝒂𝒄 is called: 8. (a) Discriminant (b) Quadratic equation Q#2

(d) None of these

(c) Linear equation

SHORT QUESTION

𝟐

(d) roots

SECTION(II) 𝟐

(12) 𝟑

1. If 𝝎 is cube root of 𝒙 + 𝒙 + 𝟏 = 𝟎, show that its other root is 𝝎 and prove that 𝝎 = 𝟏 𝟐 2. If 𝝎 is cube root of unity , form an equation whose roots are 𝟐𝝎 and 𝟐𝝎 .

𝟐 3. For what values of 𝒎 will the equation (𝒎 + 𝟏)𝒙 + 𝟐(𝒎 + 𝟑)𝒙 + 𝟐𝒎 + 𝟑 = 𝟎 have equal root?

4.

Use synthetic division to find the quotient and the remainder when the polynomial 𝒙𝟒 − 𝟏𝟎𝒙𝟐 − 𝟐𝒙 + 𝟒 is divided by 𝒙 + 𝟑.

𝒏 𝒏 5. Use factor theorem to determine if 𝒙 + 𝒂 is a factor of 𝒙 + 𝒂 , where 𝒏 is odd integer. 𝟐 6. If 𝜶, 𝜷 are the roots of 𝒙 − 𝒑𝒙 − 𝒑 − 𝒄 = 𝟎, prove that (𝟏 + 𝜶)(𝟏 + 𝜷) = 𝟏 − 𝒄

Q#3 (a)

(b)

LONG QUESTION

SECTION(III)

(10)

Use synthetic division to find the values of 𝒑 and 𝒒 if 𝒙 + 𝟏 and 𝒙 − 𝟐 are factors of the polynomial 𝒙𝟑 + 𝒑𝒙𝟐 + 𝒒𝒙 + 𝟔. If 𝜶 and 𝜷 are the roots of 𝒙𝟐 − 𝟑𝒙 + 𝟓 = 𝟎, form the equation whose roots are

𝟏−𝜶

𝟏−𝜷

and

𝟏−𝜷

.

𝟏+𝜷

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 4.8 to 4.10+unit #5(short ) TEST#:14 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

(8)

A mixed form of fraction is : (b) a polynomial+improper fraction 1. (a) An integer+ improper fraction (c) a polynomial+proper fraction (d) a polynomial+rational fraction

2.

𝑷(𝒙)

The quotient of two polynomials 𝑸(𝒙) , 𝑸(𝒙) ≠ 𝟎 is called : (a) Rational fraction

(b) Irrational fraction

(c) Partial fraction

(d) Proper fraction

An open sentence formed by using sign of “ = ” is called a/an 3. (a) Equation (b) Formula (c) Rational fraction

(d) Theorem

When a rational fraction is separated into partial fractions, then result is always : (b) an identity 4. (a) A conditional equations (c) a partial fraction (d) an improper fraction

5.

6.

𝒙𝟓

The number of Partial fraction of 𝒙(𝒙+𝟏)(𝒙𝟐−𝟒)are: (a) 2

(b) 3

𝟗𝒙𝟐 is 𝒙𝟑 −𝟏

an (a) Improper fraction

(b) Proper fraction

(c) 4

(d) 6

(c) Polynomial

(d) equation

A quadratic factor which cannot written as a product of linear factors with real coefficients is called:

7. (a) An irreducible factor (b) reducible factor 8.

Which is a reducible factor: (a) 𝒙𝟑 − 𝟔𝒙𝟐 + 𝟖𝒙

Q#2

1. 2.

(b) 𝒙𝟐 + 𝟏𝟔𝒙

(c) 𝒙𝟐 + 𝟓𝒙 − 𝟔

SHORT QUESTION

(d) all of these

SECTION(II) 𝟐𝟔

The sum of a positive number and its reciprocal is

𝟓

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

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. Find the number.

Solve the following systems of equations. 𝒙𝟐 − 𝒚𝟐 = 𝟏𝟔 ; 𝒙𝒚 = 𝟏𝟓

3. Resolve 4. Resolve 5.

(c) an irrational factor (d) an improper factor

A

Write the

𝟕𝒙+𝟓 (𝒙+𝟑)(𝒙+𝟒) 𝟏

𝒙𝟐 −𝟏

into Partial fraction.

into Partial fraction.

𝟑𝒙+𝟕

(𝒙𝟐 +𝟒)(𝒙+𝟑)

into Partial fraction.

6. Define rational fraction and improper fraction . Q#3 (a) (b)

LONG QUESTION

SECTION(III)

Solve the following systems of equations. 𝒙+𝒚=𝟓 ; 𝒙𝟐 + 𝟐𝒚𝟐 = 𝟏𝟕 The sum of a positive number and its square is 380. Find the number.

(10)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX# 6.1 to 6.5) TEST#:15 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

(8)

The next two terms of the sequence 𝟏, −𝟑, 𝟓, −𝟕, 𝟗, −𝟏𝟏, … are 1. (𝒂) 𝟏𝟑, 𝟏𝟓 (b) −𝟏𝟑, −𝟏𝟓 (c) 𝟏𝟑, −𝟏𝟓 (d) −𝟏𝟑, 𝟏𝟓 𝒏 If 𝒂𝒏 = {𝒏 + (−𝟏) }, then 𝒂𝟏𝟎 = 2. (a) 10 (b) 11 (c) 12 (d) 13 n𝒕𝒉 term of an A.P is 𝟑𝒏 − 𝟏 then 10th term is : 3. (a) 9 (b) 29 (c) 12 4.

(d) cannot determined

The arithmetic mean between √𝟐 and 𝟑√𝟐 is: 𝟒 (b) (c) √𝟐 (a) 𝟒√𝟐

(d) none of these

√𝟐

𝐚𝐧 +𝐛𝐧 𝐧−𝟏 𝐚 +𝐛 𝐧−𝟏

may be the A.M between 𝐚 and 𝐛 if 5. (a) 𝐧 = 𝟏 (b) 𝐧 = 𝟎 (c) 𝐧 > 𝟏 Sum of 𝒏 −term of an Arithmetic series 𝑺𝒏 is equal to: 𝐧 𝐧 𝐧 6. (𝐚) [𝟐𝐚 + (𝐧 − 𝟏)𝐝] (b) [𝐚 + (𝐧 − 𝟏)𝐝] (c) 𝟐 [𝟐𝐚 + (𝐧 + 𝟏)𝐝] 𝟐 𝟐 Forth partial sum of the sequence {𝒏𝟐 } is called: (a) 16 (b) 1+4+9+16 (c) 8 If 𝒂𝒏−𝟏 , 𝒂𝒏 , 𝒂𝒏+𝟏 are in A.P, then 𝒂𝒏 is 8. (a) A.M (b) G.M 7.

Q#2

𝟒 𝟐

𝟕 𝟐

(d) 𝟐 [𝟐𝐚 + 𝐥] (d) 1+2+3

(c) H.M

SHORT QUESTION

−𝟏, 𝟐, 𝟏𝟐, 𝟒𝟎, ….

1. Find the next two terms of

𝐧

(d) 𝐧 < 𝟏

(d) Mid point

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

SECTION(II)

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𝟏𝟎 𝟐

2. Find the 𝒏th term of the sequence , ( ) , ( ) , ( ) , … 𝟑

𝟑

𝟑

3. Find the 𝟏𝟑th term of the sequence 𝒙, 𝟏, 𝟐 − 𝒙, 𝟑 − 𝟐𝒙, …

4. How many terms of the series – 𝟕 + (−𝟓) + (−𝟑) + ⋯ amount to 65? 5. Find the sum of 20 terms of the series whose 𝒓th term is 𝟑𝒓 + 𝟏. 6. Sum the series Q#3 (a)

(b)

𝟑

√𝟐

+ 𝟐√𝟐 +

𝟓

√𝟐

+ ⋯ + 𝒂𝟏𝟑

LONG QUESTION

SECTION(III)

If 𝟓th term of an A.P., is 16 and the 𝟐𝟎th term is 46, what is its 𝟏𝟐th term? If 𝑺𝟐 , 𝑺𝟑 , 𝑺𝟓 are the sums of 𝟐𝒏, 𝟑𝒏, 𝟓𝒏 terms of an A.P., show that 𝑺𝟓 = 𝟓(𝑺𝟑 − 𝑺𝟐 )

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ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TIME: 1:10 HOUR TEST : MATHEMATICS ( EX#6.6 to 6.11 ) TEST#:16 Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

For any 𝑮. 𝑷., the common ratio 𝒓 is equal to: 𝒂 𝒂 1. (a) 𝒂𝒏 (b) 𝒂𝒏−𝟏 (c) 𝒂 𝒏 𝒂 2.

𝒏+𝟏

𝒏

Geometric mean between 4 and 16 is 3. (𝒂) ± 𝟐 (b) ±𝟒

(c) 𝒂𝒏 = 𝒂𝒓𝒏+𝟏

If the reciprocal of the terms a sequence form an 𝑨. 𝑷., then it is called: 4. (a) 𝑯. 𝑷 (b) 𝑮. 𝑷 (c) 𝑨. 𝑷 (c) ±𝟒

𝟓

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

(d) all of these

A

B

C

D

(d) Geometric series

A

B

C

D

A

B

C

D

(d) None of these

(c) ±𝟔

Harmonic mean between 𝟐 and 𝟖 is: 𝟏𝟔 5. (a) 5 (b)

A

(d) 𝒂𝒏+𝟏 − 𝒂𝒏 , 𝒏 ∈ 𝑵, 𝒏 > 𝟏

𝒏−𝟏

The general term of a 𝑮. 𝑷., is : (a) 𝒂𝒏 = 𝒂𝒓𝒏−𝟏 (b) 𝒂𝒏 = 𝒂𝒓𝒏

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(d) ±𝟖 (d) sequence 𝟓

(d) 𝟏𝟔

If 𝑨, 𝑮 and 𝑯 are Arithmetic , Geometric and Harmonic means between two positive numbers then

6. (a) 𝑮𝟐 = 𝑨𝑯

(b) 𝑨, 𝑮, 𝑯 𝒂𝒓𝒆 𝒊𝒏 𝑮. 𝑷 (c) 𝑨 > 𝑮 > 𝑯

If sum of series is not defined then it is called: 7. (a) Convergent series (b) Divergent series (c) finite series No term of a 𝑮. 𝑷., is: 8. (a) 0

(b) 1

(c) negative

Q#2

SHORT QUESTION 𝟏 𝟏

(d) imaginary number SECTION(II)

𝟏

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𝒂

1. If 𝒂 , 𝒃 and 𝒄 are in G.P. show that the common ratio is ±√ 𝒄 . 𝟒

2. Find the 𝟏𝟏th term of the sequence, 𝟏 + 𝒊, 𝟐, 𝟏+𝒊 , … 𝒙+𝒚

𝒙+𝒚

𝟐 𝟐 3. Which term of the sequence: 𝒙 − 𝒚 , 𝒙 + 𝒚, 𝒙−𝒚 , … is (𝒙−𝒚)𝟗 ?

4. Insert four real geometric means between 𝟑 and 𝟗𝟔. 𝟏

5. Sum the series 𝟐 + (𝟏 − 𝒊) + ( ) + ⋯ to 8 terms. 𝒊 𝟏 𝟏 𝟏

6. Find the 𝒏th and 𝟖th term of 𝟐 , 𝟓 , 𝟖 , … Q#3 (a)

(b)

LONG QUESTION

For what value 𝒏 , 𝟐

𝟒

If 𝒚 = 𝒙 + 𝒙𝟐 + 𝟑

𝟗

𝒂𝒏 +𝒃𝒏

𝒂𝒏−𝟏 +𝒃𝒏−𝟏

𝟖

𝟐𝟕

SECTION(III)

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is the positive geometric mean between 𝒂 and 𝒃? 𝟑

𝒙𝟑 + ⋯ and if 𝟎 < 𝑥 < , then 𝟐

show that 𝒙 =

𝟑𝒚

𝟐(𝟏+𝒚)

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TEST : MATHEMATICS ( UNIT # 7 )

TIME: 1:10 HOUR

TEST#:17

Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

1. 2.

𝟖! 𝟕!

=

(a) 8

(b) 7

𝒏 𝑷𝒓 =

𝟖

(c) 56

𝒏!

(a) 𝒏!

(b) 𝒓!

In how many ways three books can be arranged? 3. (𝒂)𝟐! ways (b) 𝟑! ways

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(d) 𝟕

𝒏!

(d) 𝒓!

(c) (𝒏−𝒓)!

(c) 𝟒! ways

(d) 𝟓! ways

How many arrangement of the word “MATHEMATICS” can be made 𝟏𝟏 𝟏𝟏 (a) 11! (b) ( ) (c) ( ) (d) 𝟏𝟎! 𝟑, 𝟐, 𝟏, 𝟏, 𝟏, 𝟏, 𝟏 𝟐, 𝟐, 𝟐, 𝟏, 𝟏, 𝟏, 𝟏, 𝟏 How many signals can be given by 5 flags of different colors , using 3 at a time 5. (a) 120 (b) 60 (c) 24 (d) 15 4.

For complementary combination 𝒏𝑪𝒓 = 6. (a) 𝒏 (b) 𝒏𝑪𝒏−𝒓 𝑪𝒏 𝒏𝑪 = 7. (a) 𝒏𝒏!

(b) 𝟎!

If 𝒏𝑪𝟖 = 𝒏𝑪𝟏𝟐 then 𝒏 = 8. (a) 10 (b) 20 Q#2

(c) 𝒏𝑪𝒓

(d) None of these

(c) 𝟏

(d) 0

(c) 30 SHORT QUESTION

(d) 40

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

SECTION(II)

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20.19.18.17 1. Write in factorial form: 2. Find the value of 𝟏𝟏𝒑𝒏 = 𝟏𝟏. 𝟏𝟎. 𝟗 3. How many necklace can be made from 6 beads of different colors ? 𝟏𝟐×𝟏𝟏

4. Find the value of 𝒏, when 𝒏𝑪𝟏𝟎 = 𝟐! 5. How many (a) diagonals and (b) triangles can be formed by joining the vertices of the polygon having 8 sides. 6. Make a Sample space for tossing 3 coin . Q#3 (a)

(b)

LONG QUESTION

SECTION(III)

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How many 5-digit multiplies of 5 can be formed from the digits 2,3,5,7,9, when no digit is repeated. Prove that 𝒏𝑷𝒓 = 𝒏 − 𝟏𝑷𝒓 + 𝒓. 𝒏 − 𝟏𝑷𝒓−𝟏

ILM GROUP OF COLLEGES KHANPUR NAME:____________________________ CLASS: 1ST YEAR ROLL# _______ MARKS: 30, TEST : MATHEMATICS ( UNIT # 8)

TIME: 1:10 HOUR

TEST#:18

Q#1 Each question has four options. Fill the right option from given MCQS SECTION(I)

The method of induction was given by Francesco who lived from: 1. (a) 1494-1575 (b) 1500-1575 (c) 1498-1575 𝒏 The statement 𝟑 < 𝑛! is true, when 2. (a) 𝑛 = 2 (b) 𝑛 = 4 (c) 𝑛 = 6

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(d) 1494-1570 (d) 𝑛 > 6

The statement 𝟒𝒏 + 𝟑𝒏 + 𝟒 =6 is true when : (a) 𝑛 = 0 (b) 𝑛 = 1 (c) 𝑛 ≥ 2 (d) 𝑛 is any +iv integer 𝒏 General term in the expansion of (𝒂 + 𝒃) is: 4. 𝑛 𝑛 (a) (𝑛+1 (b)(𝑟−1 (c) (𝑟+1 (d) (𝑛𝑟)𝑎𝑛−𝑟 𝑥 𝑟 )𝑎𝑛−𝑟 𝑥^𝑟 )𝑎𝑛−𝑟 𝑥 𝑟 )𝑎𝑛−𝑟 𝑥 𝑟 𝑟 3.

A

B

C

D

A

B

C

D

A

B

C

D

A

B

C

D

The number of terms in the expansion of (𝒂 + 𝒃)𝒏 are: 5. (a) 𝑛 (b) 𝑛 + 1 (c) 2𝑛

(d) 2𝑛−1

A

B

C

D

A

B

C

D

Middle term/s in the expansion of (𝒂 − 𝟑𝒙)𝟏𝟒 is/are : 7. (a) 𝑇7 (b) 𝑇8 (c) 𝑇6 &𝑇7

(d) 𝑇7 &𝑇8

A

B

C

D

A

B

C

D

The number of terms in the expansion of (𝒂 + 𝒃)𝟐𝟎 is: 6. (a) 18 (b) 20 (c) 21

8.

𝟏 + 𝒙 + 𝒙𝟐 + 𝒙𝟑 + ⋯ (a) (1 + 𝑥)−1

(b) (1 − 𝑥)−1

Q#2

(d) 19

(c) (1 + 𝑥)−2

SHORT QUESTION

𝟓 1. Using binomial theorem expand (𝒂 + 𝟐𝒃) 𝟒 2. Calculate (𝟐. 𝟎𝟐)

SECTION(II)

(d) (1 − 𝑥)−2

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𝟏

3. Use Binomial theorem find the value of (. 𝟗𝟖)𝟐 𝟓 4. Use Binomial theorem find the value of √𝟑𝟏 𝟏 𝟐𝒎+𝟏

5. Determine the middle term in (𝟐𝒙 − 𝟐𝒙)

6. State fundamental law of MATHEMATICAL INDUCTION Q#3

LONG QUESTION

(a)

Prove by mathematical induction that all positive integral values of 𝒏 𝟏 𝟐

(b)

𝟏

𝟏 + + + ⋯+ 𝟒

𝟏

𝟐𝒏−𝟏

= 𝟐 [𝟏 −

SECTION(III)

𝟏

𝟐𝒏

]

Prove by mathematical induction that all positive integral values of 𝒏 𝒓 + 𝒓𝟐 + 𝒓𝟑 + ⋯ + 𝒓𝒏 =

𝒓(𝟏−𝒓𝒏 ) 𝟏−𝒓

,𝒓 ≠ 𝟏

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