Buds Public School , Dubai
Grade 12 Science
Worksheet – 1 Mathematics
Matrices and Determinants
1
1. Find the value of 𝑥 𝑎𝑛𝑑 𝑦 if : 2 [
0
𝑎 + 𝑖𝑏 𝑐 + 𝑖𝑑
2. Evaluate |
|
−𝑐 + 𝑖𝑑 𝑎 − 𝑖𝑏
𝑦
3
]+[
𝑥
1
0
5
]=[
2
1
6
]
8
2 −3 5
3. Find the cofactor of 𝑎12 in the following : |6 0 4|
1 5 7
3 2 5
4. Let 𝐴 = |4 1 3|Express 𝐴 as sum of two matrices such that one is symmetric and the
0 6 7
other is skew symmetric .
1 2 2
5. If 𝐴 = |2 1 2| verify that 𝐴2 − 4𝐴 − 5𝐼 = 0
2 2 1
6. Using properties of determinants, prove the following :
𝛼
𝛽
𝛾
2
2
| 𝛼
𝛽
𝛾 2 | = (𝛼 − 𝛽)(𝛽 − 𝛾)(𝛼 + 𝛽 + 𝛾)
𝛽+𝛾 𝛾+𝛼 𝛼+𝛽
𝛼
𝛽
𝛾
𝛼 𝛽 𝛾
2
2
| 𝛼
𝛽
𝛾 2 | = (𝛼 + 𝛽 + 𝛾) |𝛼 2 𝛽 2 𝛾 2 |
𝛽+𝛾 𝛾+𝛼 𝛼+𝛽
1
1
1
7. Using Properties of determinants, prove the following
1 + 𝑎2 − 𝑏 2
2𝑎𝑏
−2𝑏
|
| = (1 + 𝑎2 + 𝑏 2 )3
2𝑎𝑏
1 − 𝑎2 + 𝑏 2
2𝑎
2
2
2𝑏
−2𝑎
1−𝑎 −𝑏
8. Using Properties of determinants, prove the following :
𝑎 + 𝑏 + 2𝑐
|
𝑐
𝑐
𝑥
9. If [
7
3𝑦
−𝑥
𝑎
𝑏 + 𝑐 + 2𝑎
𝑎
𝑦
4
]=[
0
4
𝑏
| = 2(𝑎 + 𝑏 + 𝑐)3
𝑏
𝑐 + 𝑎 + 2𝑏
−1
] find the values of 𝑥 𝑎𝑛𝑑 𝑦.
4
𝑥+2 3
10. If |
| = 3, find the value of 𝑥.
𝑥+5 4
1
2 3
1
7
11
11. (
)(
)= (
) find K .
3
4 2
5
𝑘
23
12. Express 𝐴 as sum of two matrices such that one is symmetric and the other is skew
3
−2
−4
symmetric . Verify your result : ( 3
− 2 − 5)
−1
1
2
13. Using matrices, solve the following system of linear equations.
𝑎)
𝑏)
2𝑥 − 𝑦 + 𝑧 = 3 , −𝑥 + 2𝑦 − 𝑧 = −4 𝑥 − 𝑦 + 2𝑧 = 1
3𝑥 − 2𝑦 + 3𝑧 = 8 , 2𝑥 + 𝑦 − 𝑧 = 1 , 4𝑥 − 3𝑦 + 2𝑧 = 4
c)
2𝑥 − 3𝑦 + 5𝑧 = 11 , 3𝑥 + 2𝑦 − 4𝑧 = −5 , 𝑥 + 𝑦 − 2𝑧 = −3
d)
𝑥 + 𝑦 + 𝑧 = 6 , x+2z = 7 , 3𝑥 + 𝑦 + 𝑧 = 12
12. Using elementary transformations, find the inverse of the following matrix:
2 −1 4
[4
0 2]
3 −2 7
13. Using properties of determinants, prove the following:
𝑎
𝑎 + 𝑏 𝑎 + 2𝑏
|𝑎 + 2𝑏
𝑎
𝑎 + 𝑏 | = 9𝑎2 (𝑎 + 𝑏)
𝑎 + 𝑏 𝑎 + 2𝑏
𝑎
14. Using elementary transformations, find the inverse of the following matrix:
2 5 3
3
0
−1
a)
b) | 2
[ 3 4 1]
3
0|
0
4
1
1 6 2
𝑎 𝑏 𝑐
15. If a ,b, and c are all positive and distinct .show that ∆ = |𝑏 𝑐 𝑎| has a negative value .
𝑐 𝑎 𝑏
16 . By using Properties of determinants prove the following :
𝑥−4
2𝑥
2𝑥
a) | 2𝑥
𝑥−4
2𝑥 | = (5𝑥 + 4)(4 − 𝑥)4
2𝑥
2𝑥
𝑥−4
𝑎
b) | 𝑎 − 𝑏
𝑏+𝑐
1
c) | 2
3
𝑏
𝑏−𝑐
𝑐+𝑎
1+𝑝
3 + 2𝑝
6 + 3𝑝
𝑥+𝑦
𝑥
d) | 5𝑥 + 4𝑦 4𝑥
10𝑥 + 8𝑦 8𝑥
𝑐
𝑐 − 𝑎 | = 𝑎3 +𝑏 3 + 𝑐 3 − 3𝑎𝑏𝑐
𝑎+𝑏
1+𝑝+𝑞
1 + 3𝑝 + 2𝑞 | = 1
1 + 6𝑝 + 3𝑞
𝑥
2𝑥 | = 𝑥 3
3𝑥
Buds Public School , Dubai
Grade 12 Science
Worksheet – 2 Mathematics
Continuity and Differentiability , Applications of Derivatives :
𝜋
1. Find the equation of tangent to the curve 𝑥 = sin 3𝑡, 𝑦 = cos 2𝑡 𝑎𝑡 𝑡 4
2. Show that the rectangle of maximum area that can be inscribed in a circle is square.
3. Show that the height of the cylinder of maximum volume that can be inscribed in a cone of
1
height h is 3 ℎ.
1
1
𝑑𝑦
4. If 𝑦 = √𝑥 2 + 1 − 𝑙𝑜𝑔 (𝑥 + √1 + 𝑥 2 ) find 𝑑𝑥
√1+sin 𝑥 + √1−sin 𝑥
5, If 𝑦 = cot −1 (
√1+sin 𝑥 − √1−sin 𝑥
𝑑𝑦
) find 𝑑𝑥
6. Discuss the continuity of the following function at 𝑥 = 0
𝑥 4 +2𝑥 3 +𝑥 2
,
tan−1 𝑥
𝑓(𝑥) =
𝑥 ≠ 0,
=
0
𝑥=0
7. Verify Lagrange’s mean value theorem for the following function :
𝑓(𝑥) = 𝑥 2 + 2𝑥 + 3, for [4, 6]
sec 𝑥−1
𝜋
8. If 𝑓(𝑥) = √sec 𝑥+1 , find 𝑓′(𝑥). Also find f ‘( 2 )
𝑑𝑦
9. If 𝑥√1 + 𝑦 + 𝑦√1 + 𝑥 = 0, find 𝑑𝑥
10. Prove that the curves 𝑥 = 𝑦 2 and 𝑥𝑦 = 𝑘 intersect at right angles if 8𝑘 2 = 1.
𝑑𝑦
11 . Find 𝑑𝑥 for the following functions :
a) (cos 𝑥) 𝑦 = (sin 𝑦) 𝑥
d)
b) 𝑥 sin 𝑥 + (sin 𝑥)cos 𝑥
c) y = (sin 𝑥) 𝑥 +sin−1 √𝑥
𝑥 sin 𝑥 + (sin 𝑥)tan 𝑥
1
12. Find the intervals in which the function f given by 𝑓(𝑥) = 𝑥 3 + 𝑥 3 , 𝑥 ≠ 0 is
a) increasing
b) decreasing .
13. Find the equation of the tangent to the curve y = √3𝑥 − 2 which is parallel to the line 4𝑥 −
2𝑦 + 5 = 0
Buds Public School , Dubai
Grade 12 Science
Worksheet – 3 Mathematics
Functions and Relations , Inverse Trigonometric Functions:
1. If 𝑓(𝑥) = 𝑥 + 7 𝑎𝑛𝑑 𝑔(𝑥) = 𝑥 − 7, 𝑥 ∈ 𝑅, 𝑓𝑖𝑛𝑑 (𝑓𝑜𝑔)(7)
𝜋
1
2. Evaluate : sin [3 − 𝑠𝑖𝑛−1
(− 2)]
3. (I) Is the binary operation defined on set N, given by 𝑎 ∗ 𝑏 =
𝑎+𝑏
2
for all 𝑎, 𝑏 ∈ 𝑁,
commulative? (II) Is the above binary operation associative?
4. Prove the following :
1
1
1
1
𝜋
a) 𝑡𝑎𝑛−1 3 + 𝑡𝑎𝑛−1 5 + 𝑡𝑎𝑛−1 7 + 𝑡𝑎𝑛−1 8 + 4
b) cot −1 (
√1+sin 𝑥 + √1−sin 𝑥
√1+sin 𝑥 − √1−sin
4
𝑥
)=
𝑥
𝜋
, x𝜖 (0, 4 )
2
5
16
c) sin−1 (5) + sin−1 (13) + sin−1(65) =
𝜋
2
5. Solve for x 2 tan−1 (cos 𝑥) = tan−1(2𝑐𝑜𝑠𝑒𝑐 𝑥)
6. Solve for 𝑥 ∶ 𝑡𝑎𝑛−1 (2𝑥) + 𝑡𝑎𝑛−1 (3𝑥) =
7. Solve for
𝑥−1
𝑥: tan−1 (𝑥−2)
+ tan
−1
𝜋
4
𝑥+1
(tan−1 𝑥+2)
𝜋
=4
8. If 𝑓(𝑥) is an invertible function, find the inverse of 𝑓(𝑥) =
9. Solve for
1−𝑥
𝑥: tan−1 1+𝑥
1
= 2 tan
−1
3𝑥−2
5
𝑥; 𝑥 > 0
10. Let T be the set of all triangles in plane with R as relation in T given by
𝑅 = {(𝑇1 , 𝑇2 ) ∶ 𝑇1 ≅ 𝑇2 } show that 𝑅 is an equivalence relation.
𝜋
1
𝑎
+ 1) + tan
−1 (𝑥
𝜋
1
𝑎
11. Prove that tan (4 + 2 cos −1 𝑏) + tan (4 + 2 cos−1 𝑏) =
12. Solve tan
−1 (𝑥
− 1) =
2b
a
8
tan−1 31
13. If the binary operation * on the set of integers Z , is defined by 𝑎 ∗ 𝑏 = 𝑎 + 3𝑏 2 ,
then find the value of 2*4
14. Show that the relation R in the set of real numbers , defined as R = { (𝑎, 𝑏): 𝑎 ≤ 𝑏 2 } is neither
reflexive , nor symmetric , nor transitive .
15. Let Z be the set of all integers and R be the relation on Z ,defined as R = { (𝑎, 𝑏): 𝑎, 𝑏 ∈
𝑍, 𝑎𝑛𝑑 (𝑎 − 𝑏)𝑖𝑠 𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5 } . 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑅 𝑖𝑠 𝑎𝑛 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑐𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 .
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