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KATTIGENAHALLI, YELHANKA,
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QUESTION NO 41 −1 2 1 −3
5. If 𝐀 = [ ], 𝐁 = [ ] verify that
1. Prove that the function 𝐟: 𝐑 → 𝐑 is given by 2 3 −3 4
𝐟(𝐱) = 𝟒𝐱 + 𝟓 is invertible and find the inverse of 𝐀𝐁 – 𝐁𝐀 is skew symmetric matrix and 𝐀𝐁 +
f. 𝐁𝐀 is symmetric matrix.
2. Prove that the function 𝐟: 𝐑 → 𝐑 is given by 𝟏 𝟐 𝟐 𝟎
6. If 𝐀 = [ ]𝐁=[ ]and
𝐟(𝐱) = 𝟏𝟎𝐱 + 𝟕 is invertible and find the inverse 𝟐 𝟏 𝟏 𝟑
𝟏 𝟏
of f. 𝐂=[ ]. Calculate 𝐀𝐂, 𝐁𝐂 and
𝟐 𝟑
3. Prove that the function 𝐟: 𝐍 → 𝐘 is given by (𝐀 + 𝐁). 𝐂. Also verify that (𝐀 + 𝐁). 𝐂 = 𝐀𝐂 + 𝐁𝐂
𝐟(𝐱) = 𝐱 𝟐 is invertible and find the inverse of f. (MARCH 2014)
4. Let R+ be the set of all non – negative real 𝟏 𝟐 𝟑
number. Show that the function 𝐟: 𝐑 + → [𝟒, ∞) 7. If 𝑨 = [𝟑 −𝟐 𝟏]then show that
is given by 𝐟(𝐱) = 𝐱 𝟐 + 𝟒 is invertible and write 𝟒 𝟐 𝟏
the inverse of f. A3 – 23 A – 40 I = 0.
(MARCH 2014)(JUNE 2016)(MARCH (MARCH 2015)(MARCH 2019)
2018)(JUNE 2018) 𝟏 𝟐 −𝟑 𝟑 −𝟏 𝟐
5. Prove that the function 𝐟: 𝐑 → 𝐑 is given by 8. If = [𝟓 𝟎 𝟐 ] , 𝐁 = [𝟒 𝟐 𝟓] and
𝐟(𝐱) = 𝟒𝐱 + 𝟑 is invertible and find the inverse of 𝟏 −𝟏 𝟏 𝟐 𝟎 𝟑
f. (JUNE 2014)(JUNE 2015)(JUNE 𝟒 𝟏 𝟐
𝐂 = [𝟎 𝟑 𝟐], then compute (A+B) and (B –
2017)(JUNE 2019)(MARCH 2022,2020,2023)
𝟏 −𝟐 𝟑
Prove that the function 𝐟: 𝐍 → 𝐘 defined by
C). Also, verify that 𝐀 + (𝐁 − 𝐂) = (𝐀 + 𝐁) − 𝐂
𝐟(𝐱) = 𝟒𝐱 + 𝟑, where 𝐘 = {𝐲: 𝐲 = 𝟒𝐱 + 𝟑, 𝐱 ∈ 𝐍}
(JUNE 2015)(MARCH 2020,2022)
is invertible. Also write inverse of 𝐟(𝐱).
−𝟐
(MARCH 2019)
9. If 𝐀 = [ 𝟒 ] and 𝐁 = [𝟏 𝟑 −𝟔], verify that
6. Verify whether the function 𝐟: 𝐑 ⟶ 𝐑 defined 𝟓
by 𝐟(𝐱) = 𝟏 + 𝐱 𝟐 is one-one, onto or bijective. (𝐀𝐁)′ = 𝐁 ′ 𝐀′ (MARCH 2016)
Justify your answer. 𝟎 𝟔 𝟕 𝟎 𝟏 𝟏
(MARCH 2022) 10. If 𝐀 = [−𝟔 𝟎 𝟖], 𝐁 = [𝟏 𝟎 𝟐] and
7. let 𝐀 = 𝐑 − {𝟑) and 𝐁 = 𝐑 − {𝟏). Consider the 𝟕 −𝟖 𝟎 𝟏 𝟐 𝟎
𝐱−𝟐 𝟐
function 𝐟: 𝐀 → 𝐁 defined by 𝐟(𝐱) = ( ). Is f
𝐱−𝟑 𝐂 = [−𝟐]. Calculate 𝐀𝐂, 𝐁𝐂 and
one-one and onto?
𝟑
QUESTION NO 42 (𝐀 + 𝐁)𝐂. Also verify that (𝐀 + 𝐁). 𝐂 = 𝐀𝐂 + 𝐁𝐂
𝟏
(JULY 2016)(JULY 2017)(MARCH 2018)(JULY
1. If 𝐀 = [−𝟒] and 𝐁 = [𝟏 𝟓 𝟕], verify that
𝟑 2019)
(𝐀𝐁)′ = 𝐁 ′ 𝐀′ 𝟏 𝟎 𝟐
𝟐 𝟎 𝟏 11. If 𝐀 = [𝟎 𝟐 𝟏], prove that 𝐀𝟑 − 𝟔𝐀𝟐 + 𝟕𝐀 +
2. Find 𝐀𝟐 – 𝟓𝐀 + 𝟔𝐈, 𝑨 = [𝟐 𝟏 𝟑]. 𝟐 𝟎 𝟑
𝟏 −𝟏 𝟎 𝟐𝐈 = 𝟎. (MARCH 2017)
𝟏 𝟏 −𝟏 𝟏 𝟑 𝟏
3. If 𝐀 = [𝟐 𝟎 𝟑 ], 𝐁 = [ 𝟎 𝟐] and 𝐂 = 12. If 𝐀 = [−𝟒] and 𝐁 = [−𝟏 𝟐 𝟏], verify that
𝟑 −𝟏 𝟐 −𝟏 𝟒 𝟑
𝟏 𝟐 𝟑 −𝟒 (𝐀𝐁)′ = 𝐁 ′ 𝐀′ (JULY 2018)
[ ], find 𝐀(𝐁𝐂), (𝐀𝐁)𝐂. Show that
𝟐 𝟎 −𝟐 𝟏 QUESTION NO 43
𝐀(𝐁𝐂) = (𝐀𝐁)𝐂. 1. 𝐒𝐨𝐥𝐯𝐞 𝐛𝐲 𝐦𝐚𝐭𝐫𝐢𝐱 𝐦𝐞𝐭𝐡𝐨𝐝:
𝛂
𝟎 −𝐭𝐚𝐧 ( ) 𝐱 + 𝟐𝐲 + 𝟑𝐳 = 𝟏𝟎 , 𝟐𝐱 − 𝟑𝐲 + 𝐳 = 𝟏,
𝟐
4. If 𝑨 = [ 𝛂 ] and I is the identity 𝟑𝐱 + 𝐲 − 𝟐𝐳 = 𝟗.
𝐭𝐚𝐧 ( ) 𝟎
𝟐
2. 𝐒𝐨𝐥𝐯𝐞 𝐛𝐲 𝐦𝐚𝐭𝐫𝐢𝐱 𝐦𝐞𝐭𝐡𝐨𝐝:
matrix of order 2. Show that
𝐜𝐨𝐬 𝛂 𝐬𝐢𝐧 𝛂 𝟑𝐱 − 𝐲 + 𝟐𝐳 = 𝟏𝟑 , 𝟐𝐱 + 𝐲 − 𝐳 = 𝟑,
𝐈 + 𝐀 = (𝐈 − 𝐀) [ ] 𝐱 + 𝟑𝐲 − 𝟓𝐳 = 𝟖.
𝐬𝐢𝐧 𝛂 𝐜𝐨𝐬 𝛂
3. 𝐒𝐨𝐥𝐯𝐞 𝐛𝐲 𝐦𝐚𝐭𝐫𝐢𝐱 𝐦𝐞𝐭𝐡𝐨𝐝.
𝐱 + 𝐲 − 𝐳 = 𝟏, 𝟑𝐱 + 𝐲 − 𝟐𝐳 = 𝟑,

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𝐱 − 𝐲 − 𝐳 = −𝟏 (MARCH 2017)(MARCH 2018)(MARCH
4. Solve by matrix method : 2020)(MARCH 2022)(MAY 2023)
𝐱 − 𝐲 + 𝐳 = 𝟒, 𝟐𝐱 + 𝐲 − 𝟑𝐳 = 𝟎 and 𝐝𝟐 𝐲 𝐝𝐲
6. If 𝐲 = 𝐬𝐢𝐧−𝟏 𝐱 show that (𝟏 − 𝐱 𝟐 ) −𝐱 =𝟎
𝐝𝐱𝟐 𝐝𝐱
𝐱 + 𝐲 + 𝐳 = 𝟐. (MARCH 2014)
(MARCH 2019)
5. Solve the system of linear equations by matrix
method: 𝟐𝐱 + 𝟑𝐲 + 𝟑𝐳 = 𝟓, 𝐱 − 𝟐𝐲 + 𝐳 = QUESTION NO 45
𝟏
−𝟒 𝐚𝐧𝐝 𝟑𝐱 − 𝐲 − 𝟐𝐳 = 𝟑 1. Find the integral of with respect to x and
𝐚𝟐 +𝐱𝟐
(JUNE 2015)(MARCH 2015)(MARCH 2020) 𝟑 𝐱𝟐 𝐝𝐱
6. Solve the system of linear equations by matrix hence evaluate ∫ .
𝐱𝟔 +𝟐
𝐝𝐱 𝐝𝐱
method: 𝟐𝐱 − 𝟑𝐲 + 𝟓𝐳 = 𝟏𝟏, 𝟑𝐱 + 𝟐𝐲 − 𝟒𝐳 = 2. Find ∫
𝐱𝟐 −𝐚𝟐
. Hence evaluate ∫
𝟑𝐱𝟐 +𝟏𝟑𝐱−𝟏𝟎
−𝟓, 𝐱 + 𝐲 − 𝟐𝐳 = −𝟑 (MARCH 2016) (MARCH 2014)
7. Solve the following system of equations by 3. Find the integral of
𝟏
with respect to x and
matrix method: √𝐱𝟐 +𝐚𝟐
𝟏
𝟑𝐱 − 𝟐𝐲 + 𝟑𝐳 = 𝟖, 𝟐𝐱 + 𝐲 − 𝐳 = 𝟏, hence evaluate ∫ 𝐝𝐱 (JUNE 2014)
√𝐱𝟐 +𝟕
𝟒𝐱 − 𝟑𝐲 + 𝟐𝐳 = 𝟒. 𝟏
4. Find the integral of with respect to x and
(JUNE 2016)(MARCH 2019)(JUNE √𝐱𝟐 −𝐚𝟐
𝐝𝐱
2019)(MARCH 2022) hence evaluate ∫ . (MARCH 2015)
√𝐱𝟐 +𝟔𝐱−𝟕
8. Solve the following system of linear equations by 𝟏
5. Find the integral of with respect to x and
matrix method. 𝐱𝟐 +𝐚𝟐
𝟏
𝐱 − 𝐲 + 𝟐𝐳 = 𝟏, 𝟐𝐲 − 𝟑𝐳 = 𝟏, hence evaluate ∫ 𝐝𝐱 (MARCH 2016)
𝟑+𝟐𝐱+𝐱𝟐
𝟑𝐱 − 𝟐𝐲 + 𝟒𝐳 = 𝟐. (MARCH 2017) 𝟏
6. Find the integral of with respect to x and
9. Solve the following system of equations by 𝐱𝟐 +𝐚𝟐
𝟏
matrix method. hence evaluate ∫ 𝐝𝐱 (MARCH 2018)
𝐱𝟐 −𝟔𝐱+𝟏𝟑
𝐱 + 𝐲 + 𝐳 = 𝟔, 𝐲 + 𝟑𝐳 = 𝟏𝟏 𝟏
7. Find the integral of with respect to x and
√𝐚𝟐 −𝐱𝟐
, 𝐱 − 𝟐𝐲 + 𝐳 = 𝟎. (JUNE 2017) 𝐝𝐱
10. Solve the following system of equations by hence evaluate ∫ (JUNE 2018)
√𝟕−𝟔𝐱−𝐱𝟐
matrix method. 8. Find the integral of √𝐱 𝟐 + 𝐚𝟐 w.r.t.x and hence
𝐱 − 𝐲 + 𝟐𝐳 = 𝟕, 𝟑𝐱 + 𝟒𝐲 − 𝟓𝐳 = −𝟓,
evaluate ∫ √𝐱 𝟐 + 𝟒𝐱 + 𝟓 𝐝𝐱 (JUNE 2019)
𝟐𝐱 − 𝐲 + 𝟑𝐳 = 𝟏𝟐. (MARCH 2018)
11. Solve the following system of equations by 9. Find the integral of √𝐚𝟐 − 𝐱𝟐 w.r.t.x and hence
matrix method. evaluate ∫ √𝟓 − 𝐱𝟐 + 𝟐𝐱 𝐝𝐱 (JULY 2016)
𝟒𝐱 + 𝟑𝐲 + 𝟐𝐳 = 𝟔𝟎, 𝟐𝐱 + 𝟒𝐲 + 𝟔𝐳 = 𝟗𝟎, 10. Find the integral of √𝐱 𝟐 − 𝐚𝟐 w.r.t.x and hence
𝟔𝐱 + 𝟐𝐲 + 𝟑𝐳 = 𝟕𝟎. (JULY 2018) evaluate ∫ √𝐱 𝟐 − 𝟖𝐱 + 𝟕 𝐝𝐱 (MARCH 2017)
QUESTION NO 44
11. Find the integral of √𝐚𝟐 + 𝐱𝟐 w.r.t.x and hence
𝐝𝟐 𝐲 𝐝𝐲
1. If 𝐲 = 𝟑 𝐞𝟐𝐱 + 𝟐 𝐞𝟑𝐱 show that 𝟐 −𝟓 + 𝟔𝐲 = evaluate ∫ √𝟏 + 𝐱𝟐 𝐝𝐱 (JUNE 2017)
𝐝𝐱 𝐝𝐱
𝟎. (MARCH 2014)(MARCH 2023) 12. Find the integral of √𝐱 𝟐 + 𝐚𝟐 w.r.t.x and hence
2. If 𝐲 = 𝐀 𝐞𝐦𝐱 + 𝐁 𝐞𝐧𝐱 show that
𝐝𝟐 𝐲 𝐝𝐲 evaluate ∫ √𝐱 𝟐 + 𝟒𝐱 + 𝟓 𝐝𝐱 (JUNE 2015)
− (𝐦 + 𝐧) + 𝐦𝐧𝐲 = 𝟎. 𝐝𝐱 𝐝𝐱
𝐝𝐱𝟐 𝐝𝐱 13. Find ∫ . Hence evaluate ∫
(MARCH 2015)(JULY 2018) 𝐱𝟐 −𝐚𝟐 𝐱𝟐 −𝟏𝟔

3. If 𝐲 = (𝐬𝐢𝐧−𝟏 𝐱)𝟐 , Show that (𝟏 − 𝐱 𝟐 )𝐲𝟐 − 𝐱 𝐲𝟏 = (MARCH 2019)(MARCH 2022)

𝟏
𝟐 (MARCH 2016) 14. Find the integral of
𝐱𝟐 +𝐚𝟐
with respect to x and
4. If 𝐲 = 𝟑 𝐜𝐨𝐬(𝐥𝐨𝐠 𝐱) + 𝟒 𝐬𝐢𝐧(𝐥𝐨𝐠 𝐱), Show that 𝟏
hence evaluate ∫ 𝐝𝐱
𝐱 𝟐 𝐲𝟐 + 𝐱𝐲𝟏 + 𝐲 = 𝟎. 𝐱𝟐 +𝟐𝒙+𝟐

(JULY 2016)(JULY 2017)(JUNE 2019) (MARCH 2020)(JUNE 2023)

𝟏
5. If 𝐲 = (𝐭𝐚𝐧−𝟏 𝐱)𝟐 , Show that 15. Find the integral of
√𝐱𝟐 +𝐚𝟐
with respect to x and
(𝐱 𝟐 + 𝟏)𝟐 𝐲𝟐 + 𝟐𝐱(𝐱 𝟐 + 𝟏) 𝐲𝟏 = 𝟐 𝟏
hence evaluate ∫ 𝐝𝐱 (MARCH 2023)
√𝐱𝟐 +𝟐𝐱+𝟐

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QUESTION NO 46 (JUNE 2017)
𝐱𝟐 𝐲𝟐 11. Find the general solution of the differential
1. 𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐚𝐫𝐞𝐚 𝐨𝐟 𝐭𝐡𝐞 𝐞𝐥𝐥𝐢𝐩𝐬𝐞 + =𝟏 𝐝𝐲
𝐚𝟐 𝐛𝟐
equation (𝐱 + 𝟑𝐲 𝟐 ) = 𝐲 (JUNE 2015)
𝐮𝐬𝐢𝐧𝐠 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧. 𝐝𝐱
(JUNE 2018)(MARCH 2022) QUESTION NO 48
𝐱𝟐 𝐲𝟐 1. Derive the equation of a line in a space through a
2. 𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐚𝐫𝐞𝐚 𝐨𝐟 𝐭𝐡𝐞 𝐞𝐥𝐥𝐢𝐩𝐬𝐞 + =𝟏
𝟏𝟔 𝟗 given point and parallel to a vector 𝑏⃗, both in
𝐮𝐬𝐢𝐧𝐠 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧. (JUNE 2016)
𝐱𝟐 𝐲𝟐
vector and Cartesian form.
3. 𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐚𝐫𝐞𝐚 𝐨𝐟 𝐭𝐡𝐞 𝐞𝐥𝐥𝐢𝐩𝐬𝐞 + = 𝟏 (JULY 2014) (MARCH 2015)(MARCH
𝟒 𝟗
𝐮𝐬𝐢𝐧𝐠 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧. 2019)(MARCH 2020)(MARCH 2022)
4. 𝐅𝐢𝐧𝐝 𝐭𝐡𝐞 𝐚𝐫𝐞𝐚 𝐨𝐟 𝐜𝐢𝐫𝐜𝐥𝐞 𝐱 𝟐 + 𝐲 𝟐 = 𝐚𝟐 2. Derive the equation for distance between two
𝐛𝐲 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧. skew lines.
5. Find area lying in the first quadrant and bounded
3. Derive the equation for distance between parallel
by the circle x2 + y2 = 4 and the
lines.
lines x = 0 and x = 2.
QUESTION NO 49 ( 6 MARKS)
6. Find Area of the region bounded by the curve
1. Solve the following problem graphically
y2 = 4x, y-axis and the line y = 3. minimize and maximize 𝒁 = 𝟓𝒙 + 𝟏𝟎𝒚 subject
(MARCH 2017) to the constraints 𝐱 + 𝟐𝐲 ≤ 𝟏𝟐𝟎, 𝐱 + 𝐲 ≥
7. Find the area bounded by the curve 𝟔𝟎, 𝐱 − 𝟐𝐲 ≥ 𝟎, 𝐱 ≥ 𝟎, 𝐲 ≥ 𝟎.
𝐲 = 𝐜𝐨𝐬 𝐱 between 𝐱 = 𝟎 and 𝐱 = 𝟐𝛑. 2. Solve the following problem graphically
(JULY 2017) minimize and maximize
8. Find the area bounded by the curve Z = 3x + 9y subject to the constraints
𝐲 = 𝐬𝐢𝐧 𝐱 between 𝐱 = 𝟎 and 𝐱 = 𝟐𝛑. 3. Minimize Z = 4x + y subject to the constraints
QUESTION NO 47 𝐱 + 𝐲 ≤ 𝟓𝟎, 𝟑𝐱 + 𝐲 ≤ 𝟗𝟎, 𝐱 ≥ 𝟎, 𝐲 ≥ 𝟎 by
1. Find the general solution of the differential graphical method.
equation (MARCH 2014) 4. Solve the following problem graphically
𝐱 𝐱 𝟐
𝐞 . 𝐭𝐚𝐧 𝐲 𝐝𝐱 + (𝟏 − 𝐞 ). 𝐬𝐞𝐜 𝐲 𝐝𝐲 = 𝟎. minimize and maximize 𝐙 = 𝐱 + 𝟐𝐲 subject to
𝐝𝐲 𝟐 the constraints 𝐱 + 𝟐𝐲 ≥ 𝟏𝟎𝟎, 𝟐𝐱 − 𝐲 ≤
2. Solve 𝐱 𝐥𝐨𝐠 𝐱 +𝐲= 𝐥𝐨𝐠 𝐱 (JULY 2014)
𝐝𝐱 𝐱 𝟎, 𝟐𝐱 + 𝐲 ≤ 𝟐𝟎𝟎, 𝐱 ≥ 𝟎, 𝐲 ≥ 𝟎.
3. Find the general solution of the differential 5. Solve the following problem graphically
𝐝𝐲
equation + 𝟑𝐲 = 𝐞−𝟐𝐱 maximize 𝐳 = 𝐱 + 𝐲, subject to constraints, 𝟐𝐱 +
𝐝𝐱
4. Find the general solution of the differential 𝐲 ≤ 𝟓𝟎, 𝐱 + 𝟐𝐲 ≤ 𝟒𝟎, 𝐱 ≥ 𝟎 , 𝐲 ≥ 𝟎.
𝐝𝐲 6. Minimize and maximize
equation 𝐱 + 𝟐𝐲 = 𝐱 𝟐 (𝐱 ≠ 𝟎) 𝒁 = 𝟔𝟎𝟎𝒙 + 𝟒𝟎𝟎𝒚 subject to the constraints
𝐝𝐱
(M D P 1)(JUNE 2016)(JUNE 2018) 𝟓
+ 𝟐𝐲 ≤ 𝟏𝟐, 𝟐𝐱 + 𝐲 ≤ 𝟏𝟐, 𝐱 + 𝐲 ≥ 𝟓 𝐱 ≥
𝐝𝐲 𝟒
5. Solve the differential equation 𝐜𝐨𝐬 𝟐 𝐱 +𝐲= 𝟎, 𝐲 ≥ 𝟎 by graphical method. (MARCH 2017)
𝐝𝐱
𝐭𝐚𝐧 𝐱 (MARCH 2017)(MARCH 2022) 7. Minimise Z = −3x + 4y subject to 𝐱 + 𝟐𝐲 ≤
6. Find the general solution of the differential 𝟖, 𝟑𝐱 + 𝟐𝐲 ≤ 𝟏𝟐, 𝐱 ≥ 𝟎, 𝐲 ≥ 𝟎 by graphical
𝐝𝐲 method. (JUNE 2017)
equation + 𝐲 𝐜𝐨𝐭 𝐱 = 𝟒𝐱 𝐜𝐨𝐬𝐞𝐜 𝐱
𝐝𝐱 8. Minimize and Maximise 𝐙 = 𝟏𝟎𝟓𝟎𝟎𝐱 + 𝟗𝟎𝟎𝟎𝐲
7. Find the general solution of the differential subject to the constraints,𝐱 + 𝐲 ≤
𝐝𝐲
equation 𝒙 + 𝟐𝐲 = 𝐱 𝟐 𝐥𝐨𝐠 𝐱 𝟓𝟎 , 𝟐𝟎, 𝐱 + 𝟏𝟎𝐲 ≤ 𝟖𝟎𝟎, 𝟐𝐱 + 𝐲 ≤ 𝟖𝟎 , 𝐱 ≥
𝐝𝐱
(MARCH 2018)(MARCH 2020) 𝟎, 𝐲 ≥ 𝟎 by graphical method.
(JULY 2018)
8. Find the general solution of the differential
𝐝𝐲
OR
equation + 𝐲 𝐬𝐞𝐜 𝐱 = 𝐭𝐚𝐧 𝐱 𝐚
1. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫𝟎 𝐟(𝐱)𝐝𝐱 = ∫𝟎 𝐟(𝐚 −
𝐚
𝐝𝐱
(MARCH 2015)(MARCH 2019) 𝐱)𝐝𝐱 𝐚𝐧𝐝 𝐡𝐞𝐧𝐜𝐞 𝐬𝐡𝐨𝐰 𝐭𝐡𝐚𝐭
9. Find the general solution of the differential ∞
𝐱 𝛑
𝐝𝐲 ∫ 𝐝𝐱 =
equation (𝐱 + 𝐲) = 𝟏 (JUNE 2019) (𝟏 + 𝐱)(𝟏 + 𝐱 𝟐 ) 𝟒
𝐝𝐱 𝟎
10. Find the general solution of the differential
equation 𝐲 𝐝𝐱 − (𝐱 + 𝟐𝐲 𝟐 )𝐝𝐲 = 𝟎

S. CHIRANJEEVI M.Sc., M.Phil. DCA

LECTURER IN MATHEMATICS Page 3
 REVA INDEPENDENT PU COLLEGE
KATTIGENAHALLI, YELHANKA,
BANGALORE-560064
𝐚 𝐚 𝐚
2. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫–𝐚 𝐟(𝐱)𝐝𝐱 = 12. Prove that ∫𝟎 𝐟(𝐱)𝐝𝐱 = ∫𝟎 𝐟(𝐚 − 𝐱) 𝐝𝐱 hence
𝐚 𝛑
𝟐 ∫ 𝐟(𝐱) 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐞𝐯𝐞𝐧 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐜𝐨𝐬𝟓 𝐱
{ 𝟎 evaluate ∫𝟎𝟐 𝐝𝐱
𝐜𝐨𝐬𝟓 𝐱+𝐬𝐢𝐧𝟓 𝐱
𝟎 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐨𝐝𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 (MARCH 2019)(MARCH 2023)
𝛑
𝐇𝐞𝐧𝐜𝐞 𝐞𝐯𝐚𝐥𝐮𝐚𝐭𝐞 ∫𝟐𝛑 𝐬𝐢𝐧𝟐 𝐱𝐝𝐱 QUESTION NO 50 ( 4 MARKS)
−
𝟐 𝟑 𝟏
𝐚
3. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫𝟎 𝐟(𝐱)𝐝𝐱 = ∫𝟎 𝐟(𝐚 − 𝐱) 𝐝𝐱 hence
𝐚 1. If 𝐀 = [ ], satisfies the equation 𝐀𝟐 −
−𝟏 𝟐
𝛑
𝛑 𝟓𝐀 + 𝟕𝐈 = 𝟎 , then find the inverse of A using
evaluate ∫𝟎𝟒 𝐥𝐨𝐠 𝐞 (𝟏 + 𝐭𝐚𝐧 𝐱)𝐝𝐱 = 𝐥𝐨𝐠 𝐞 𝟐
𝟖 this equation, where I is the identity matrix of
(MARCH 2014) order 2.
𝐛 𝐛
4. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫𝐚 𝐟(𝐱)𝐝𝐱 = ∫𝐚 𝐟(𝐚 + 𝐛 − 𝐱) 𝐝𝐱 𝟐 𝟑
2. Show that the matrix 𝑨 = [ ] satisfies the
𝛑
𝟏
𝟏 𝟐
𝟑 𝟐
hence evaluate ∫ 𝛑 𝐝𝐱 equation 𝑨 − 𝟒𝑨 + 𝑰 = 𝟎. Where I is 2X2
𝟏+√𝐭𝐚𝐧 𝐱
𝟔
identity matrix and O is 2x2 zero matrix. Using
(JUNE 2014)(JUNE 2015)(JUNE 2019)
𝐚 this equation. Find 𝑨−𝟏 .
5. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫–𝐚 𝐟(𝐱)𝐝𝐱 =
𝐚
OR
𝟐 ∫ 𝐟(𝐱) 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐞𝐯𝐞𝐧 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐤 𝐜𝐨𝐬 𝐱 𝛑
{ 𝟎 ,𝐱 ≠
𝛑−𝟐𝐱 𝟐
𝟎 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐨𝐝𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 1. Find the value of k , if 𝐟(𝐱) = { 𝛑 is
𝟏 𝟓 𝟒
𝟑 ,𝐱 =
𝐇𝐞𝐧𝐜𝐞 𝐞𝐯𝐚𝐥𝐮𝐚𝐭𝐞 ∫−𝟏𝐬𝐢𝐧 𝐱. 𝐜𝐨𝐬 𝐱 𝐝𝐱 𝛑
𝟐

(MARCH 2015)(MARCH 2020) continuous at 𝐱 =

𝟐
𝟐𝐚 (MARCH 2014) (MARCH 2017)(JUNE 2019)
6. Prove that ∫𝟎 𝐟(𝐱)𝐝𝐱 =
𝐤𝐱 + 𝟏 , 𝐱 ≤ 𝟓
𝟐
𝐚
∫𝟎 𝐟(𝐱) 𝐢𝐟 𝐟(𝟐𝐚 − 𝐱) = 𝐟(𝐱) 2. Find the value of k , if 𝐟(𝐱) = { is
{ and hence evaluate 𝟑 𝐱 − 𝟓 ,𝐱 > 5
𝟎 𝐢𝐟 𝐟(𝟐𝐚 − 𝐱) = −𝐟(𝐱) continuous at 𝐱 = 𝟓
𝟐𝛑 (MARCH 2015)(MARCH 2019)
∫𝟎 𝐜𝐨𝐬 𝟓 𝐱 𝐝𝐱 (MARCH 2016)
𝐚 𝐚 3. Find the value of k , if
7. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫𝟎 𝐟(𝐱)𝐝𝐱 = ∫𝟎 𝐟(𝐚 − 𝐱) 𝐝𝐱 hence 𝟏−𝐜𝐨𝐬 𝟐𝐱
,𝐱 ≠ 𝟎
𝛑
√𝐬𝐢𝐧 𝐱 𝐟(𝐱) = { 𝟏−𝐜𝐨𝐬 𝐱 is continuous at 𝐱 = 𝟎
evaluate ∫𝟎 𝟐 𝐝𝐱 𝟑𝐤 ,𝐱 = 𝟎
√𝐜𝐨𝐬 𝐱+√𝐬𝐢𝐧 𝐱
(JULY 2016)(MARCH 2022) (MARCH 2016)
𝐚 𝐚 𝐤 𝐱𝟐 , 𝐱 ≤ 𝟐
8. Prove that ∫𝟎 𝐟(𝐱)𝐝𝐱 = ∫𝟎 𝐟(𝐚 − 𝐱) 𝐝𝐱 hence 4. Find the value of k , if 𝐟(𝐱) = { is
𝟑 ,𝐱 = 𝟐
𝛑
evaluate ∫𝟎 (𝟐 𝐥𝐨𝐠 𝐬𝐢𝐧 𝐱 − 𝐥𝐨𝐠 𝐬𝐢𝐧 𝟐𝐱)𝐝𝐱
𝟐 continuous at 𝐱 = 𝟐
(JULY 2016)(JULY 2018)(MARCH 2023)(MAY
(MARCH 2017)
𝐚 2023)
9. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫–𝐚 𝐟(𝐱)𝐝𝐱 = 5. For what value of 𝛌 is the function defined by
𝐚
𝟐 ∫𝟎 𝐟(𝐱) 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐞𝐯𝐞𝐧 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝛌(𝐱 𝟐 − 𝟐𝐱), 𝐢𝐟 𝐱 ≤ 𝟎
{ 𝐟(𝐱) = { continuous at x=0?
𝟎 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐨𝐝𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝟒𝐱 + 𝟏, 𝐢𝐟 𝐱 > 0
𝛑 (JULY 2017)
𝐇𝐞𝐧𝐜𝐞 𝐞𝐯𝐚𝐥𝐮𝐚𝐭𝐞 ∫𝟐𝛑 (𝐱 𝟑 + 𝐱 𝐜𝐨𝐬 𝐱) 𝐝𝐱 6. Find the relationship between a and b so that
–
𝟐
the function f defined by
(JUNE 2017)
𝐚 𝐚 𝐚𝐱 + 𝟏, 𝐢𝐟 𝐱 ≤ 𝟑
10. Prove that ∫𝟎 𝐟(𝐱)𝐝𝐱 = ∫𝟎 𝐟(𝐚 − 𝐱) 𝐝𝐱 hence 𝐟(𝐱) = { is continous at x=3.
𝐛𝐱 + 𝟑, 𝐢𝐟 𝐱 > 3
𝐚 √𝐱 (MARCH 2018)
evaluate ∫𝟎 𝐝𝐱 (MARCH 2018)
√𝐱+√𝐚−𝐱
𝐚 7. Find the value of k , if
11. 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭 ∫–𝐚 𝐟(𝐱)𝐝𝐱 = 𝐤𝐱 + 𝟏 , 𝐱 ≤ 𝛑
𝐚 𝐟(𝐱) = { is continuous at 𝐱 = 𝛑
𝟐 ∫ 𝐟(𝐱) 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐞𝐯𝐞𝐧 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝐜𝐨𝐬 𝐱 , 𝐱 > 𝛑
{ 𝟎 (MARCH 2020)(MARCH 2022)
𝟎 𝐢𝐟 𝐟(𝐱)𝐢𝐬 𝐚𝐧 𝐨𝐝𝐝 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝛑
𝐇𝐞𝐧𝐜𝐞 𝐞𝐯𝐚𝐥𝐮𝐚𝐭𝐞 ∫𝟐𝛑 𝐬𝐢𝐧𝟕 𝐱𝐝𝐱
−
𝟐
(JUNE 2018)(JUNE 2023)

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