mathematics important questions based on latest format

1. MOST IMPORTANT QUESTIONS (FOR 2018)
Relations & Functions (7 Marks)
1 Mark Questions:
1. What is the range of the function f(x)=
𝑥−1
𝑥−1
2. F: R→R f(x)= (3-x3
)1/3
, find (fof)(x)
3. Let * be a binary operation defined by a * b = 2a + b – 3. Find 3 * 4.
4. If f(x) = x2
+ 2 and g(x) =
x
x+1
, find gof (5).
5. F : R→R is defined by f(x)= 3x+2. Find f(f(x))
6. A= {1,2,3} and B={4,5,6,7} and f={(1,4),(2,5),(3,6)} be a function from A to B. state whether f is one one or onto.
7. Let f : R - {
−3
5
} → R be a function defined as f (x) =
2x
5x+3
, find f-1
.
8. If *is defined on integers as a * b = a + 3b2
, find 2 * 4.
9. If f(x) is an invertible function , find the inverse of f(x) =
( 3x−2)
5
10. If f(x) = 2x + 5 , g(x) = 2x – 5 , x∈ R find (fog) (9).
11. If R = {(x , y) : x2
+ y2
≤ 4 ; x , y∈Z } is a relation in Z , write the domain of R.
12. Let f : R→ R be defined by f(x) = x2
. Find f-1
(x).
13. Let * be a binary operation defined by a * b = 3a + 4b – 2. Find 4 * 5.
14. If f(x) = x2
+ 2 and g(x) = 1 -
1
x+1
, find (fog) (3).
15. If * is a binary operator on Z defined by a*b= a+b-5, then write the identity element for * in Z.
6 Marks Questions:
1) Let N be the set of all natural Nos. & R be a relation on NxN defined by (a,b)R (c,d) such that
ad = bc. Prove that R is an equivalence relation
2) Let R be a relation on NxN defined by (a,b)R (c,d) : a+d = b+c. Prove that R is an equivalence relation.
3) Show that the relation R on Z defined by R = {(a, b) : a – b is divisible by 5} is an equivalence relation.
4) Show that the relation R in the set : A = { x : x ∈ Z, 0≤x ≤12} given by.
R = {( a, b): a − b is divisible by 4} is an equivalencerelation.
5) Prove that the relation R in the set A = {5, 6, 7, 8, 9} given by R = {(a, b) : 𝑎 − 𝑏 is divisible by 2}, is
an equivalence relation. Find all elements related to the element 6.
6) A=R-{3}and B= R-{1}. Consider the function F: A →B defined by f(x)=
𝑥−2
𝑥−3
is f is one one and onto? Justify
7) Consider f: R+
→ [-5,∞) given by f(x)= 9x2
+6x-5. Show that f is invertible and find inverse of f.
8) F: R→R, g: R→R such that f(x)= 3x+1 and g(x)= 4x-2. Find fog and show that fog is one one and onto.

2. 9) A binary operation * on Q-{-1} such that a*b= a+b+ab. Find the identity element on Q. Also find the inverse of an
element in Q-{-1}
10) Let A= N×N and * be the binary operator on A defined by (a,b)*(c,d)=(a+c, b+d). show that * is commutative and
associative. Find the identity element for * on A if any.
11) Prove that f: N→N defined by f(x)= x2
+x+1 is one- one but not onto.
12) Show that the relation R in the set Z of integers given by R = {( a, b) : 7 divides a – b} is an equivalence relation.
13) i) State whether the function is one – one, onto f : R → R : f (x) = 1 + x2
. Justify your answer.
ii) Show that f : [ -1 , 1 ] →R, given by f(x) =
x
x+2
is one – one. Also find its inverse
12. Let A = R – {3} and B = R -
2
3
. If f : A → B: f(x) =
2x−4
3x−9
, then prove that f is a bijective function.
13. Let ∗ be a binary on QxQ defined by (a,b) ∗(c,d) = (ac,b+ad).Determine, whether ∗ is commutative and
associative .Find the identity element for ∗ and the invertible elements of QxQ.
14. Show that the relation R in the set A ={x∈ 𝑍 : 0 ≤ 𝑥 ≤ 10}, 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦
𝑅 = {(𝑎, 𝑏): 𝑎 − 𝑏 𝑖s a multiple of 3} is an equivalence relation. Write all its equivalence classes.
INVERSE TRIGONOMETRY (3 Marks)
1 Mark Questions:
1. Find the principal value of following:
(i) sin−1 1
2
⁡ (ii)cos −1
−1 ⁡ (iii)sin−1
sin
2π
3
(iv) sec- 1
(2)
(v)cos−1
−
3
2
⁡ (vi)sin−1
sin
3π
5
⁡ (vii)cos−1
cos
5π
6
(viii)cos−1
cos
7π
6
2. Evaluate the following:
(i) tan−1
2 cos⁡( 2 sin−1 1
2
) (ii) sin
π
3
− sin−1
−
1
2
(iii) tan sin−1 3
5
+ cot−1 3
2
(iv) tan 2 tan−1 1
5
−
π
4
(v) cos-1 1
2
+ 2 sin−1 1
2
(vi) cos
π
6
+ cos−1 1
2
(vii)sin−1
sin
5π
6
+ cos−1
cos
π
6
(viii)tan−1
x + tan−1 1
x
(ix)sin−1
sin
3π
4
+ cos−1
cos
π
3
(x)cos−1
cos
3π
6
+ sin−1
sin
2π
3
3. Write the range of one branch of sin -1
x, other than principal branch.
4. Solve for x : sin -1
x – cos -1
x =
π
6
.
2 Marks Questions:
1. Show that cos -1 1−x2
1+ x2 = 2 tan−1
x .
2. Write the value of tan−1 1
5
+ tan−1 1
8
.

3. 3. If sin sin−1 1
5
+ cos−1
x = 1, find x.
4. Find the value of cos (sec-1
x + cosec -1
x).
5. Find the value of cot (tan -1
a + cot -1
a).
6. If tan-1
x + tan-1
y =
𝜋
4
, then write the value of x + y + xy.
7. Simplify: tan−1 1+x2−1
x
8. Simplify: sin−1 sin x+cos x
2
9. Prove that: tan-1
2 + tan -1
3 =
3π
4
10. Solve for x: tan−1 1 −x
1+x
=
1
2
tan−1
x
11. Solve: sin -1
(1 – x) – 2 sin -1
x =
π
2
12. Solve: tan−1
𝑥(𝑥 + 1) + sin−1
𝑥2 + 𝑥 + 1 =
𝜋
2
.
13. Simplify: cos−1
2𝑥 1 − 𝑥2 : -
1
2
≤ 𝑥 ≤
1
2
MATRICES AND DETERMINANTS (1+2+4+6=13 Marks)
1 MARK QUESTIONS.
1. If matrix
0 6 − 5𝑥
𝑥2
𝑥 + 3
is symmetric find x
2.
3𝑥 − 2𝑦 5
𝑥 −2
=
3 5
−3 −2
find y .
3. Find x if
𝑥 1
3 𝑥
=
1 0
2 1
4. For what value of x, matrix
3 − 2𝑥 𝑥 + 1
2 4
is singular?
5. ‘A’ is a square matrix of order 4 : 𝐴 = 1 find (i) 2𝐴 (ii) 𝑎𝑑𝑗 𝐴 (iii) −𝐴
6. Find cofactor of a12 in
2 −3 5
6 0 4
1 5 −7
7.
𝑥 + 1 𝑥 − 1
𝑥 − 3 𝑥 + 2
=
4 −1
1 3
find x.
8. A matrix A is of order 2 x 2 has determinant 4. What is the value of 2𝐴 ?
9. A is a square matrix of order 3 : |A|= -1, |B|=3 find |3AB|
10. If A is a skew symmetric matrix of order 3,what will be the value of det.(A).
11. Find x if
2 4
5 −1
=
2𝑥 4
6 𝑥
.
12. If A is a square matrix such that A2
=I, then write 𝐴−1
,
13. If A and B are square matrices of order 3 such that 𝐴 =-1 and 𝐵 =3 the find the value of 2𝐴𝐵

4. 2 MARKS QUESTIONS.
1. In the matrix eqn.
1 2
3 4
4 3
2 1
=
8 5
20 13
apply𝑅2 → 𝑅2 − 𝑅1 and then apply𝐶2 → 𝐶2 − 𝐶1.
2. If A=
6 5
7 6
,compute (AdjA) and very that A(AdjA)= 𝐴 I.
3. For the matrix A=
3 1
−1 2
,A2
-5A+7I=O,then find 𝐴−1
.
4. Find the matrix X for which
1 −4
3 −2
X=
−16 −6
7 2
5. Prove that the diagonal elements of a skew symmetric matrix are zero.
6. A and B are symmetric matrices of same order,then show that AB is symmetric iff A and B commute.
7. A is a square matrix: A2
=A, then write the value of 7A-(I+A)3
.
8. If A=
𝑐𝑜𝑠𝐴 𝑆𝑖𝑛𝐴
−𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐴
find A-1
.
9. If A is a skew symmetric matrix of order 3, then prove that |A|=0.
10. Solve:
5 4
1 1
𝐴 =
1 −2
1 3
4 MARKS QUESTIONS
1. Express the matrix
6 2 −5
−2 −5 3
−3 3 −1
as sum of symmetric and skew symmetric matrix.
2. Find x if [x 4 -1]
2 1 −1
1 0 0
2 2 4
𝑥
4
−1
= 0
3. Using Elementary Row operations & column operations find A-1
whose
a. A =
2 0 −1
5 1 0
0 1 3
b. A =
−1 1 2
1 2 3
3 1 1
4. Using properties of determinants, prove that
i)
𝑥 + 4 2𝑥 2𝑥
2𝑥 𝑥 + 4 𝑥
2𝑥 2𝑥 𝑥 + 4
= (5x + 4) (4 – x)2
ii)
𝑎 − 𝑏 − 𝑐 2𝑎 2𝑎
2𝑏 𝑏 − 𝑐 − 𝑎 2𝑏
2𝑐 2𝑐 𝑐 − 𝑎 − 𝑏
= (a + b + c )3
iii)
 2
 + 
 2
 + 
 2
 + 
= (  - ) ( - ) ( - ) ( +  + )
iv)
1 1 + 𝑝 1 + 𝑝 + 𝑞
2 3 + 2𝑝 4 + 3𝑝 + 2𝑞
3 6 + 3𝑝 10 + 6𝑝 + 3𝑞
= 1

5. v)
𝑎2
𝑏𝑐 𝑎𝑐 + 𝑐2
𝑎2
+ 𝑎𝑏 𝑏2
𝑎𝑐
𝑎𝑏 𝑏2
+ 𝑏𝑐 𝑐2
= 4a2
b2
c2
vi)
𝑎2
+ 1 𝑎𝑏 𝑎𝑐
𝑎𝑏 𝑏2
+ 1 𝑏𝑐
𝑎𝑐 𝑏𝑐 𝑐2
+ 1
= (1 + a2
+ b2
+ c2
)
vii)
1 𝑎 𝑎3
1 𝑏 𝑏3
1 𝑐 𝑐3
= (a – b) (b – c) (c – a) (a + b + c)
viii)
3𝑎 −𝑎 + 𝑏 −𝑎 + 𝑐
−𝑏 + 𝑎 3𝑏 −𝑏 + 𝑐
−𝑐 + 𝑎 −𝑐 + 𝑏 3𝑐
= 3(a + b + c) ( ab + bc + ca)
ix)
1 + 𝑎2
− 𝑏2
2𝑎𝑏 −2𝑏
2𝑎𝑏 1 − 𝑎2
+ 𝑏2
2𝑎
2𝑏 −2𝑎 1 − 𝑎2
− 𝑏2
= (1+a2
+b2
)3
.
x)
𝑎 𝑏 𝑐
𝑎 − 𝑏 𝑏 − 𝑐 𝑐 − 𝑎
𝑏 + 𝑐 𝑐 + 𝑎 𝑎 + 𝑏
= a3
+ b3
+ c3
– 3abc .
5. Find the matrix A satisfying the matrix equation
2 1
3 2
A
−3 2
5 −3
=
1 0
0 1
.
6. Solve the equation :
𝑥 + 𝑎 𝑥 𝑥
𝑥 𝑥 + 𝑎 𝑥
𝑥 𝑥 𝑥 + 𝑎
= 0, a≠0
7. Using properties of determinant prove :-
𝑥 𝑥2
1 + 𝑝𝑥3
𝑦 𝑦2
1 + 𝑝𝑦3
𝑧 𝑧2
1 + 𝑝𝑧3
= (1 + pxyz)(x – y) (y – z) ( z – x)
8. A trust fund has Rs. 30,000 to invest in two bonds. First pays 5% interest per year & second pays 7%. Using matrix
multiplication determine how to divide Rs. 30,000 among two types of bonds if trust find must obtain annual
interest of Rs. 2000.
9. Find A if
4
1
3
A=
−4 8 4
−1 2 1
−3 6 3
10. Find A if
2 −1
1 0
−3 4
A=
−1 −8 −10
1 −2 −5
9 22 15
11. Using elementary transformations find inverse of
2 0 −1
5 1 0
0 1 3
12. Using properties of determinants show that if
3 −2 𝑠𝑖𝑛3𝜃
−7 8 𝑐𝑜𝑠2𝜃
−11 14 2
=0 then sin 𝜃=0 or ½.
6 Marks Questions:
13. A =
1 2 −3
2 3 2
3 −3 −4
, find A-1
, solve the equation x + 2y – 3z = - 4, 2x + 3y + 2z = 2, 3x – 3y – 4z = 11.

6. 14. Find A-1
, where A =
1 −2 0
2 1 3
0 −2 1
. Hence solve the equations x – 2y = 10, 2x + y + 3z = 8, -2y + z = 7.
15. If A =
1 −1 1
2 1 −3
1 1 1
, find A-1
and use it to solve x + 2y + z = 4, -x + y + z = 0, x – 3y + z = 2
16. If A =
1 1 1
1 2 −3
2 −1 3
, find A-1
and use it to solve x + y +2z = 0, x +2 y - z = 9, x – 3y +3z = -14.
17. If A =
−4 4 4
−7 1 3
5 −3 −1
and B =
1 −1 1
1 −2 −2
2 1 3
, find AB and use it to solve the system of equations
x – y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.
18. Let the two matrices A and B be given by A =
1 −1 0
2 3 4
0 1 2
and B =
2 2 −4
−4 2 −4
2 −11 5
.
Verify AB = BA = 6I and hence solve the equations x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7.
19. If A =
1 1 1
1 2 −3
2 −1 3
show that A3
– 6A2
+ 5A + 11I = 0. Hence find A-1
.
CONTINUITY AND DIFFERENTIABILITY (2+4+4=10 Marks)
2 MARKS QUESTIONS.
1. Write the value of k for which 𝑓 𝑥 =
𝑘𝑥2
, 𝑥 < 2
3, 𝑥 ≥ 2
is continuous at x=2
2. Write the value of k for which 𝑓 𝑥 =
3𝑠𝑖𝑛𝑥
2𝑥
+ 𝑐𝑜𝑠𝑥, 𝑥 ≠ 0
𝑘, 𝑥 = 2
is continuous at x=0
3. Write two points at which 𝑓 𝑥 =
1
𝑥− 𝑥
is not continuous.
4. Write one point where f(x) = 𝑥 − 𝑥 + 1 is not differentiable
5. If y = 𝑒2𝑥3
, write
𝑑𝑦
𝑑𝑥
.
6. Find the value of k so that 𝑓 𝑥 =
1−𝑐𝑜𝑠4𝑥
8𝑥2 , 𝑥 ≠ 0
𝑘, 𝑥 = 0
is continuous at x=0.
7. If x= cos𝜃 − 𝑐𝑜𝑠2𝜃, y = sin 𝜃- sin2 𝜃 , find
𝑑𝑦
𝑑𝑥
.
8. If 𝑠𝑖𝑛2
𝑦 + cos 𝑥𝑦 = 𝜋,find
𝑑𝑦
𝑑𝑥
.
9. If y = tan−1 5𝑥
1−6𝑥2 ,-
1
6
< x <
1
6
,then show that
𝑑𝑦
𝑑𝑥
=
2
1+4𝑥2 +
3
1+9𝑥2 .
10. If it is given that for the function f(x)=x3
-5x2
-3x,Mean value theorem is applicable in [1,3],find all values of c.
11. f(x) =
𝐾𝑥 + 1 𝑖𝑓 𝑥 ≤ 𝜋
𝑐𝑜𝑠 𝑥 𝑖𝑓 𝑥 > 𝜋
𝑎𝑡 𝑥 = 𝜋, find k if function is continuous at π,
12. If x = 𝑎sin −1 𝑡 , 𝑦 = 𝑎cos −1 𝑡 , 𝑠𝑕𝑜𝑤 𝑡𝑕𝑎𝑡
𝑑𝑦
𝑑𝑥
= −
𝑦
𝑥
.
13. If (x2
+y2
)3
=3x2
y , find
𝑑𝑦
𝑑𝑥
.
4 MARKS QUESTIONS

7. 14. Differentiate log ( xsin x
+ cot2
x) with respect to x.
15. If y = log [ x + 𝑥2 + 𝑎2 ], show that ( x2
+ a2
)
𝑑2 𝑦
𝑑𝑥2 + x
𝑑𝑦
𝑑𝑥
= 0.
16. . If 1 − 𝑥2 + 1 − 𝑦2 = a (x – y), Prove
𝑑𝑦
𝑑𝑥
=
1−𝑦2
1−𝑥2
17. If x = a sin t and y = a ( cos t + log tan
𝑡
2
), find
𝑑2 𝑦
𝑑𝑥2 .
18. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that
𝑑𝑦 2
𝑑𝑥 2 =
𝑏 𝑠𝑒𝑐 3 𝑡
𝑎2 𝑡
.
19. Differentiate the following functions w.r.t.x :
(i) sin−1 2𝑥
1+𝑥2 (ii) tan−1 1−𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 𝑥
(iii) tan−1 𝑐𝑜𝑠 𝑥
1+𝑠𝑖𝑛 𝑥
(iv) tan−1 5 𝑥
1−6𝑥2
(v) tan-1 1+𝑥2
1+𝑥2
+ 1−𝑥2
− 1−𝑥2
(vi) tan-1 1+𝑠𝑖𝑛 𝑥
1+𝑠𝑖𝑛 𝑥
+ 1−𝑠𝑖𝑛 𝑥
− 1−𝑠𝑖𝑛 𝑥
(vii) cot-1 1−𝑥
1+𝑥
20. Differentiate tan-1 1+ 𝑥2− 1− 𝑥2
1+ 𝑥2+ 1− 𝑥2
with respect to cos-1
x2
.
21. Verify lagrange’s Mean value theorem: (i) f(x) = x(x – 1) (x – 2) [ 0, ½] (ii) f(x) = x3
-5x2
– 3x [1, 3]
22. Verify Rolle’s theorem for the following functions :
(i) f(x) = x2
+ x – 6[-3, 2] (ii) f(x) = x (x – 1) (x – 2) [0, 2] (iii) f(x) = sin x + cos x [0,
𝜋
2
]
23. Differentiate sin-1 2𝑥
1+𝑥2 w.r.t. tan-1
x .
24. If y =
sin −1 𝑥
1−𝑥2
, show that (1-𝑥2
)
𝑑2 𝑦
𝑑𝑥2 − 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦 = 0 .
25. Differentiate cos-1 1− 𝑥2
1+ 𝑥2 with respect of tan-1 3𝑥 − 𝑥3
1−3 𝑥2 .
26. If x = a sin 2t(1 + cos 2t), y = b cos 2t( 1 – cos 2t) Show that
𝑑𝑦
𝑑𝑥 𝑡=
𝜋
4
=
𝑏
𝑎
27. Find
𝑑𝑦
𝑑𝑥
, if y = sin-1
[x 1 − 𝑥 − 𝑥 1 − 𝑥2] .
28. If y = log [x + 𝑥2 + 1], prove that (𝑥2
+ 1)
𝑑2 𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
= 0 .
29. If y =
sin −1 𝑥
1−𝑥2
, show that (1-𝑥2
)
𝑑2 𝑦
𝑑𝑥2 − 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦=0
APPLICATION OF DERIVATIVES (2+4+4=10 Marks)
2 Marks Questions:
1. The volume of a cube is increasing at a rate of 9 cm3
/s, how fast is the surface area increasing when length of an
edge is 10 cm.
2. Show that the function f(x)= x3
-3x2
+4x is strictly increasing on R.
3. If the radius of a sphere is measured as 9m with an error of 0.03m,find the approximate error in calculating the
surface area.
4. Show that the function f(x)= x3
+x2
+x+1 do not have maxima or minima.
5. The side of an equilateral triangle is increasing at the rate of 2cm/s. at what rate is its area increasing when the side
of triangle is 20cm?
6. Using differentials, find approximate value of 25.2

8. 7. The volume of a spherical balloon is increasing at the rate of change of its surface area at the instant when radius is
6cm .
8. The total cost C(x) in rupees associated with the production of x units of an item is given by
C(x) = 0.007 x3
– 0.003 x2
+ 15x + 4000. Find the marginal cost when 17 units are produced.
9. The radius of a spherical diamond is measured as 7 cm with an error of 0.04 cm. Find the approximate error in
calculating its volume. If the cost of 1 cm3
diamond is Rs. 1000, what is the loss to the buyer of the diamond? What
lesson you get?
10. Using derivative, find the approximate percentage increase in the area of the circle if its radius is increased by 2%.
4 Marks Questions:
1. Separate the interval 0,
𝜋
2
into sub – intervals in which f(x) = sin4
x + cos4
x is increasing or decreasing.
2. Show that the curves 4x = y2
and 4xy = K cut at right angles if K2
= 512 .
3. Find the intervals in which the function f given be f(x) – sinx – cosx, 0  x  2 is strictly increasing or strictly
decreasing.
4. Find all points on the curve y = 4x3
– 2x5
at which the tangents passes through the origin.
5. Find the equation of Normal to the curve y = x3
+ 2x + 6 which are parallel to line x+14y+4=0.
6. Show that the curves y = aex
and y = be –x
cut at right angles if ab = 1.
7. Find the intervals in which the function f f(x) = x3
+
1
𝑥3 , x ≠ 0 is increasing or decreasing.
8. Prove that y =
4 𝑠𝑖𝑛 𝜃
(2+𝑐𝑜𝑠 𝜃)
– 𝜃 is an increasing function of 𝜃 in [0,
𝜋
2
]
9. Find the area of the greatest rectangle that can be inscribed in an ellipse
𝑥2
𝑎2 +
𝑦2
𝑏2= 1.
10. Find the equation of tangent to the curve y =
𝑋−7
𝑋−2 (𝑋−3)
at the point where it cuts x-axis. [x-20y=7]
11. A helicopter if flying along the curve y = x2
+ 2. A soldier is placed at the point (3, 2) .find the nearest distance
between the solider and the helicopter.
12. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to
diameter of base.
13. Show that the semi – vertical angle of cone of maximum volume and of given slant height is tan-1
2 .
14. The sum of perimeter of circle and square is K. Prove that the sum of their areas is least when side of square is
double the radius of circle.
15. Find the value of x for which f(x) = [x(x – 2)]2
is an increasing function. Also, find the points on the curve, where the
tangent is parallel to x- axis.
16. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth in 2 m and
volume is 8 m3
. If building of tank costs Rs. 70 per sq. meter for the base and
Rs. 45 per sq. Meter for sides, what is the cost of least expensive tank?

9. 17. Show that the function f defined f(x) = tan-1
(sin x + cos x) is strictly increasing in (0,
𝜋
4
) .
18. Find the intervals in which f(x)=
3
2
x4
-4x3
-12x2
+5 is strictly increasing or decreasing.
19. Find the intervals in which f(x)=
3
10
x4
-
4
5
x3
-3x2
+
36
5
𝑥 + 11 is strictly increasing or decreasing.
20. Find the minimum value of (ax+by), where xy=c2
.
21. Find the equations of tangents to the curve 3x2
-y2
=8, which passes through the point (4/3, 0).
22. A manufacturer can sell x items at a price of Rs. (5 -
𝑥
100
)each. The cost price of x items is Rs. (
𝑥
5
+ 500). Find the
number of items he should sell to earn maximum profit.
23. Find the point on the curve x2
=4ywhich is nearest to the point (-1,2).
INTEGRALS (2+4+6=12 Marks)
2 MARKS QUESTIONS.
1. (i) ∫ 𝑡𝑎𝑛8
𝑥𝑠𝑒𝑐4
𝑥 𝑑𝑥 (ii)∫
1
4+9𝑥2 𝑑𝑥 (iii) ∫
1
𝑥2 −
1
𝑥
𝑒 𝑥
𝑑𝑥 (iv)∫ 𝑠𝑖𝑛7
𝑥
𝜋
0
𝑑𝑥
(v) ∫
1
𝑥2+2𝑥+2
𝑑𝑥 (vi)∫
1
𝑠𝑖𝑛2 𝑥𝑐𝑜𝑠2 𝑥
𝑑𝑥 (vii). ∫
𝑐𝑜𝑠2𝑥
𝑠𝑖𝑛𝑥 +𝑐𝑜𝑠𝑥 2 𝑑𝑥
2. (i). ∫
1
𝑥2−4𝑥+9
𝑑𝑥 (ii). ∫ 10 + 4𝑥 − 2𝑥2 𝑑𝑥 (iii). ∫
1
1+𝑐𝑜𝑡 7 𝑥
𝜋
2
0
𝑑𝑥 (iv). ∫
1−𝑥
1+𝑥2
2
𝑒 𝑥
𝑑𝑥
3. Evaluate: ∫
𝑐𝑜𝑠2𝑥
𝑐𝑜 𝑠2 𝑥 𝑠𝑖𝑛2 𝑥
dx
4. Evaluate: ∫
𝑑𝑥
7−6𝑥−𝑥2
5. Evaluate: ∫
2+𝑠𝑖𝑛 2𝑥
1+𝑐𝑜𝑠 2𝑥
𝑒 𝑥
dx
6. Evaluate: ∫
𝑑𝑥
𝑥[6 𝑙𝑜𝑔 𝑥 2+7 𝑙𝑜𝑔 𝑥+2]
dx
7. Evaluate: ∫
𝑥
1−𝑥3
𝑑𝑥
8. Evaluate: ∫
𝑠𝑖𝑛 𝑥
𝑠𝑖𝑛 (𝑥+𝑎)
𝑑𝑥
9. Evaluate: ∫
(2𝑥−5)𝑒2𝑥
(2𝑥−3)3 dx
4 MARKS QUESTIONS.
10. Evaluate :-∫
𝑥2+ 1
( 𝑥−1)2 𝑥+3
dx.
11. Evaluate :-∫
sin ( 𝑥−𝑎)
sin ( 𝑥+𝑎)
dx.
12. Evaluate :-∫
5𝑥 2
1+2𝑥+3𝑥2 dx.
13. Evaluate :-∫( 2 sin 2𝑥 − cos 𝑥) 6 − 𝑐𝑜𝑠2 𝑥 − 4 sin 𝑥 dx.
14. Evaluate :- ∫
2
1−𝑥 ( 1+ 𝑥2 )
dx
15. Evaluate :- ∫
𝑑𝑥
𝑠𝑖𝑛𝑥 − sin 2𝑥
dx

10. 16. Evaluate :- ∫
𝑥2
𝑥2+ 3𝑥−3
𝑑𝑥
17. Evaluate :- ∫ 𝑒 𝑥 sin 4𝑥−4
1−cos 4𝑥
dx
18. Evaluate :- ∫
𝑥2
𝑥−1 3 (𝑥+1)
dx
19. Evaluate :- ∫
1
𝑠𝑖𝑛𝑥 ( 5−4 cos 𝑥)
dx
20. Evaluate: ∫
𝑐𝑜𝑠𝑥 𝑑𝑥
(𝑠𝑖𝑛2 𝑥+1)(𝑠𝑖𝑛2 𝑥+4)
21. Evaluate: ∫
𝑠𝑖𝑛4 𝑥 𝑐𝑜𝑠𝑥 𝑑𝑥
𝑠𝑖𝑛𝑥 +1 (𝑠𝑖𝑛𝑥 +4)2
22. Evaluate: ∫ 𝑠𝑖𝑛3𝑥 𝑒5𝑥
𝑑𝑥
23. Evaluate: ∫
2𝑥−1
𝑥−1 𝑥+2 𝑥−3)
dx
24. Evaluate: ∫
1−𝑥2
𝑥(1−2𝑥)
dx
25. Evaluate: ∫
𝑥4
𝑥+1 (𝑥+2)4 𝑑𝑥
26. Evaluate: ∫
𝑥4
𝑥−1 (𝑥2+1)
𝑑𝑥
27. Evaluate: ∫
𝑥3+𝑥+1
(𝑥2−1)
dx
6 MARKS QUESTIONS.
28. Evaluate :-∫
𝑥 sin 𝑥 cos 𝑥
𝑠𝑖𝑛4 𝑥+ 𝑐𝑜𝑠4 𝑥
𝜋
2
0
dx.
29. Show that ∫ 𝑡𝑎𝑛 𝑥 + 𝑐𝑜𝑡 𝑥 𝑑𝑥 = 2 𝜋
𝜋
2
0
30. Prove that :- ∫ sin−1 𝑥
𝑎+𝑥
𝑎
0
𝑑𝑥 = −
𝑎
2
𝜋 − 2 .
31. Evaluate :- ∫ 𝑥3
− 𝑥
2
1
dx
32. Prove that : ∫ 𝑓(𝑥)𝑑𝑥
𝑎
0
= ∫ 𝑓(𝑎 − 𝑥)
𝑎
0
𝑑𝑥 and hence Evaluate :∫
𝑥 𝑑𝑥
25 𝑠𝑖𝑛2 𝑥+16 𝑐𝑜𝑠2 𝑥
𝜋
0
.
33. Prove that : ∫ 𝑓(𝑥)𝑑𝑥
𝑏
𝑎
= ∫ 𝑓(𝑎 + 𝑏 − 𝑥)
𝑏
𝑎
𝑑𝑥 and hence Evaluate: ∫
10−𝑥
𝑥+ 10−𝑥
8
2
dx
34. Evaluate: ∫
𝑥 𝑠𝑖𝑛 𝑥
1+ 𝑐𝑜𝑠2 𝑥
𝜋
2
0
𝑑𝑥
35. Evaluate: ∫
𝑥 𝑑𝑥
4−𝑐𝑜𝑠2 𝑥
𝜋
0
36. Evaluate: ∫
𝑥 𝑑𝑥
1+𝑠𝑖𝑛𝑥
𝜋
0
37. Evaluate: ∫ 𝑥 − 1 + 𝑥 − 2 + 𝑥 − 4
4
1
dx
38. Evaluate: ∫
𝑠𝑖𝑛2 𝑥
𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥
𝜋
2
0
dx
39. Evaluate: ∫
𝑒 𝑐𝑜𝑠 𝑥
𝑒 𝑐𝑜𝑠 𝑥 +𝑒−𝑐𝑜𝑠 𝑥
𝜋
0
dx
40. Evaluate as the limit of sums :

11. (i)∫ 2𝑥2
− 5 𝑑𝑥
3
0
(ii) ∫ 𝑥2
+ 5𝑥 𝑑𝑥
3
1
(iii) ∫ 𝑥2
+ 𝑥 + 1 𝑑𝑥
2
0
(iv) ∫ 3𝑥2
+ 2𝑥 𝑑𝑥
3
1
APPLICATION OF INTEGRATION (6 MARKS)
6 MARKS QUESTIONS.
1. Using integration, find the area of the region enclosed between the two circles x2
+ y2
= 4 and
(x – 2)2
+ y2
= 4.
2. Find the area of the region { (x, y) : y2
 6ax and x2
+ y2
 16a2
} using method of integration.
3. Prove that the area between two parabolas y2
4ax and x2
= 4ay is 16 a2
/ 3 sq units.
4. Using integration, find the area of the following region. 𝑥, 𝑦 :
𝑥2
9
+
𝑦2
4
≤ 1 ≤
𝑥
3
+
𝑦
2
.
5. Find the area of the region {(x, y) : x2
+ y2
≤ 4, 𝑥 + 𝑦 ≥ 2}.
6. Find the area lying above x – axis and included between the circle x2
+ y2
= 8x and the parabola y2
= 4x.
7. Find the area of the region included between the curve 4y = 3x2
and line 2y = 3x + 12
8. Sketch the graph of y = 𝑥 + 3 and Evaluate ∫ 𝑥 + 3
0
−6
dx .
9. Using the method of integration, find the area of the region bounded by the lines 3x – 2y + 1 = 0,
2x + 3y – 21 = 0 and x – 5y + 9 = 0.
10. Using integration , find the area of ∆ ABC where A (2,3), B (4,7), C (6,2).
11. Using integration, find the area of the triangle formed by positive x-axis and tangent and normal to the circle
x2
+y2
=4 at (1, 3)
12. Find the area of the region included between the parabola y =
3 𝑥2
4
and the line 3x – 2y + 12 = 0.
13. Find the area bounded by the circle x2
+ y2
= 16 and the line y = x in the first Quadrant. [2𝜋]
DIFFERENTIAL EQUATIONS (6MARKS)
2 MARKS QUESTIONS.
1. What is the degree and order of following differential equation?
(i) y
𝑑2 𝑦
𝑑𝑥2 +
𝑑𝑦
𝑑𝑥
3
= 𝑥
𝑑3 𝑦
𝑑𝑥3
2
. (ii)
𝑑𝑦
𝑑𝑥
4
+ 3y
𝑑2 𝑦
𝑑𝑥2 = 0. (iii)
𝑑3 𝑦
𝑑𝑥3 +y2
+ 𝑒
𝑑𝑦
𝑑𝑥 = 0
2. Write the integrating factor of
𝑑𝑦
𝑑𝑥
+ 2y tan x = sin x
3. Form the differential equation of family of straight lines passing through origin.
4. Form the differential equation of family of parabolas axis along x-axis.
5. Verify that y-cosy=x is a solution of the diff.eqn.(ysiny+cosy+x)𝑦/
=y.
6. Form the diff. eqn. representing the family of curves y=asin(x+b).
7. Find the general solution of the differential equation:
ylogydx – xdy = 0 Or 𝑒 𝑥
tany dx + (1- 𝑒 𝑥
)𝑠𝑒𝑐2
y dy = 0

12. 8. Find the differential equation representing the curve y = e-x
+ax+b.
9. Form the differential equation of family of curves; (x-a)2
+2y2
=a2
.
10. Form the differential equation of family of curves; y2
=a(b-x2
)
4 MARKS QUESTIONS.
1. Show that the differential equation xdy – ydx = 𝑥2 + 𝑦2 dx is homogeneous and solve it.
2. Find the particular solution of the differential equation :- cos x dy = sin x ( cos x – 2y) dx, given that
y = 0, when x =
𝜋
3
.
3. Show that the differential equation 𝑥 𝑠𝑖𝑛2 𝑦
𝑥
− 𝑦 dx + x dy = 0 is homogeneous. Find the particular solution of
this differential equation, given that y =
𝜋
4
when x = 1.
4. Show that the differential equation x
𝑑𝑦
𝑑𝑥
sin
𝑦
𝑥
+ 𝑥 − 𝑦 sin
𝑦
𝑥
= 0 is homogeneous. Find the particular solution
of this differential equation, given that x = 1 when y =
𝜋
2
.
5. Solve the following differential equation :- (1 + y + x2
y) dx + (x + x3
)dy = 0, where y = 0 when x = 1.
6. Solve the following differential equation : 1 + 𝑥2 + 𝑦2 + 𝑥2 𝑦2 + xy
𝑑𝑦
𝑑𝑥
= 0 .
7. Find the particular solution of the differential equation :( xdy – ydx) y sin
𝑦
𝑥
= 𝑦𝑑𝑥 + 𝑥𝑑𝑦 𝑥 cos
𝑦
𝑥
,
given that y =  when x = 3.
8. 𝑆𝑜𝑙𝑣𝑒 ∶ 𝑥𝑒
𝑦
𝑥 − 𝑦 sin
𝑦
𝑥
+ 𝑥
𝑑𝑦
𝑑𝑥
sin
𝑦
𝑥
= 0 . given that y = 0 where x = 1, i.e., y(1) = 0
9. Solve the initial value problem : (x2
+ 1)
𝑑𝑦
𝑑𝑥
- 2xy = ( x4
+ 2x2
+ 1) cos x, y (0) = 0.
10. Solve : (x2
+ xy) dy = (x2
+ y2
) dx.
11. Show that the differential equation : 2y ex/y
dx + (y – 2x ex/y
) dy = 0 is homogenous and find its
particular solution, given x = 0 when y = 1.
12. Solve the equation : 𝑥 𝑐𝑜𝑠
𝑦
𝑥
+ 𝑦 𝑠𝑖𝑛
𝑦
𝑥
𝑦 𝑑𝑥 = 𝑦 𝑠𝑖𝑛
𝑦
𝑥
− 𝑥 𝑐𝑜𝑠
𝑦
𝑥
x dy.
13. Find the particular solution of the differential equation (1 + x3
)
𝑑𝑦
𝑑𝑥
+ 6x2
y = (1 + x2
), given that x = y = 1.
14. Solve : 1 + 𝑒
𝑥
𝑦 𝑑𝑥 + 𝑒
𝑥
𝑦 1 −
𝑥
𝑦
𝑑𝑦 = 0 .
15. Solve the differential equation : (tan-1
y – x) dy = (1 + y2
) dx .
16. Find the particular solution of the differential equation :
17. Solve: (1 + e2x
) dy+ (1 + y2
) ex
dx = 0, given that y = 1 when x = 0.
18. Find the general solution of (x + 2y3
)
𝑑𝑦
𝑑𝑥
= y.
19. Find the equation of a curve passing through origin and satisfying the differential equation
(1 + x2
)
𝑑𝑦
𝑑𝑥
+ 2xy = 4x2
.
20. Find the general solution of
𝑑𝑦
𝑑𝑥
- 3y = sin 2x.

13. VECTORS AND THREE DIMENSIONAL GEOMETRY(1+2+4+4+6=17 MARKS)
1 MARK QUESTIONS.
1. If 𝑝 is a unit vector and (𝑥 − 𝑝) 𝑥 + 𝑝 = 80 , then find 𝑥 .
2. Find the value of 𝜆 so that the vector 𝑎 = 2 𝑖 + 𝜆𝑗 + 𝑘 and 𝑏 = 𝑖 − 2𝑗 +3 𝑘 are perpendicular to each
other.
3. If two vectors 𝑎 𝑎𝑛𝑑 𝑏are : 𝑎 = 2, 𝑏 = 3 and 𝑎. 𝑏 = 4, find 𝑎 − 𝑏 .
4. Find the angle between 𝑖 − 2𝑗 + 3𝑘 𝑎𝑛𝑑 3𝑖 − 2𝑗 + 𝑘 .
5. 𝑎= 𝑖 + 2𝑗 − 𝑘, 𝑏 = 3𝑖 + 𝑗 − 5𝑘, find a unit vector in the direction 𝑎 − 𝑏 .
6. Write Direction ratios of
𝑥−2
2
=
2𝑦−5
−3
= 𝑧 − 1 .
7. Write the vectors representing the diagonals of a parallelogram whose sides are the vectors 𝑎 𝑎𝑛𝑑 𝑏.
8. Find the angle which the vector 2𝑖+𝑗 − 2𝑘 makes with y –axis.
9. Find a unit vector in the direction of 𝑎 = 3𝑖 - 2 𝑗 + 6𝑘
10. Find 𝑥 ,if for a unit vector 𝑐 , 𝑥 − 𝑐 𝑥 + 𝑐 = 15.
11. Find the distance of the plane 3x-2y+6z-14=0 from origin.
12. Find p if the line
𝑥−1
2
=
𝑦+2
𝑝
=
𝑧
3
is parallel to the plane 5x+2y-6z+9=0.
13. Write the eqn. of plane 2x-3y+4z+6=0 in intercept form.
14. Find the distance between the planes 2x-y+2z +5=0 and 4x-2y +4z+16=0.
15. Find a vector of magnitude 7 units which is perpendicular to both 𝑎 = 2 𝑖 + 𝜆𝑗 + 𝑘 and 𝑏 = 𝑖 − 2𝑗 +3 𝑘
2 MARKS QUESTIONS.
16. Find the volume of the parallelopiped whose adjacent sides are represented by 𝑎 , 𝑏 and 𝑐 where
𝑎= 3𝑖 − 2𝑗 + 5𝑘 , 𝑏 = 2𝑖 + 2𝑗 − 𝑘 , 𝑐 = -4𝑖 + 3𝑗 + 2𝑘 .
17. If two vectors 𝑎 𝑎𝑛𝑑 𝑏are : 𝑎 = 2, 𝑏 = 3 and 𝑎. 𝑏 = 4, find 𝑎 − 𝑏 .
18. Find ‘𝛌’ when the projection of 𝑎 = 𝛌𝑖 + 𝑗 + 4 𝑘 and 𝑏 = 2𝑖 + 6𝑗 + 3 𝑘 is 4 units.
19. Find the value of 𝜆 so that the vectors 𝑖 + 𝑗 + 𝑘 , 2𝑖 + 3𝑗 − 𝑘 , -𝑖 + 𝜆𝑗 + 2𝑘 are coplanar.
20. A line makes angles 900
𝑎𝑛𝑑 1350
with x-axis and y-axis respectively , find the angle which it makes with z-axis.
21. Write the vector eqn. of a line passing through the point (2,-1,3) and perpendicular to the plane 3x-5y+2z=6.
22. Find the angle between the lines:
3+𝑥
3
=
1−𝑦
−5
=
𝑧+2
4
𝑎𝑛𝑑
𝑥+1
1
=
𝑦−4
1
=
5−3𝑧
−6
23. Write the eqn. of plane on which the foot of perpendicular from origin is (5,2,1).
24. The line 𝑟 = 2𝑖 −3𝑗 + 𝑝𝑘 + 𝑡(𝑖 − 𝑗 + 𝑞𝑘) lies in the plane 𝑟.(3𝑖 + 𝑗 − 𝑘) − 4 = 0,then find the values of p and q.
25. Prove that: (𝑎 × 𝑏)2
= |𝑎| |𝑏|- (𝑎. 𝑏)2
.
26. If 𝑎 and𝑏 are unit vectors and 𝜃 is the angle between them, show that sin 𝜃/2 = ½ 𝑎 − 𝑏 .
27. If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is 3 .

14. 28. If 𝑎 , 𝑏 𝑎𝑛𝑑 𝑐 are three unit vectors such that 𝑎 = 5, 𝑏 = 12 and 𝑐 = 13 , and 𝑎 + 𝑏 + 𝑐 = 0 , find the value of
𝑎 . 𝑏 + 𝑏 . 𝑐 + 𝑐 . 𝑎 .
29. Find the sine of the angle between the vectors 𝑎 = 3𝑖 + 𝑗 + 2𝑘 and 𝑏 = 2𝑖 − 2𝑗 + 4𝑘
30. If 𝑎 = 5𝑖 − 𝑗 − 3𝑘 and 𝑏 = 𝑖 + 3𝑗 + 5𝑘 then show that the vectors (𝑎 + 𝑏) and ( 𝑎 − 𝑏 ) perpendicular
31. Find the value of 𝜆 so that the lines
1−𝑥
3
=
7𝑦−14
2𝜆
=
5𝑧−10
11
and
7−7𝑥
3𝜆
=
𝑦−5
1
=
6−𝑧
5
are perpendicular to
each other.
32. Find the angle between the line
𝑥+1
2
=
3𝑦+5
9
=
3−𝑧
−6
and the plane 10x + 2y – 11z = 3.
4 MARKS QUESTIONS.
1. Find a unit vector perpendicular to the plane of triangle ABC where the vertices are A (3, -1, 2), B ( 1, -1, -3)
and C ( 4, -3, 1).
2. Let 𝑎 = 4𝑖 + 5𝑗 + 𝑘, 𝑏 = 𝑖 − 4𝑗 + 5𝑘 and 𝑐= 3𝑖 + 𝑗 − 𝑘. Find a vector 𝑑 which is perpendicular to
Both 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑛𝑑 𝑑. 𝑐 = 21.
3. Let 𝑎, 𝑏 𝑎𝑛𝑑𝑐 be three vectors : 𝑎 = 3, 𝑏 = 4, 𝑐 = 5 and each one of them being perpendicular to
the sum of other two, find 𝑎 + 𝑏 + 𝑐 .
4. Using vectors, find the area of the triangle ABC with vertices A (1, 2, 3), B ( 2, -1, 4) and C ( 4, 5, -1) .
5. If 𝑎 = 𝑖 + 𝑗 + 𝑘 , 𝑏 = 4 𝑖 − 2𝑗 + 3𝑘 𝑎𝑛𝑑 𝑐 = 𝑖 − 2𝑗 + 𝑘, find a vector of magnitude 6 units which is parallel to
the vector 2 𝑎 - 𝑏 + 3 𝑐 .
6. The two adjacent side of a parallelogram are 2𝑖 − 4𝑗 − 5𝑘and 𝑖 − 2𝑗 + 3𝑘 . Find the unit vector
parallel to its diagonal. Also, find its area.
7. The scalar product of the vector 𝑖 + 𝑗 + 𝑘 with a unit vector along the sum of vectors 2𝑖 + 4𝑗 − 5𝑘 and
𝛾𝑖 + 2𝑗 + 3𝑘is equal to one, find the value of 𝛾 .
8. If 𝑖 + 𝑗 + 𝑘 , 2𝑖 + 5𝑗 , 3𝑖 + 2𝑗 − 3𝑘 and 𝑖 − 6𝑗 − 𝑘 are the position vectors of the points A, B, C and D, find the
angle between 𝐴𝐵 and 𝐶𝐷 . Deduce that 𝐴𝐵 and 𝐶𝐷 are collinear.
9. Find the value of 𝜆 so that the four points A, B, C, D with position vectors 4𝑖 + 5𝑗 + 𝑘 , −𝑗 − 𝑘 , 3𝑖 + 9𝑗 + 4𝑘 and
4(-𝑖 + 𝑗 + 𝑘) respectively are coplanar.
10. If 𝑎 , 𝑏 𝑎𝑛𝑑 𝑐 are three unit vectors such that 𝑎 . 𝑏 = 𝑎 𝑐= 0 and angle between 𝑏 𝑎𝑛𝑑 𝑐 is
𝜋
6
, prove that
𝑎 =  2( 𝑏 𝑐 ) .
11. Find the equation of plane passing through the point (1, 2, 1) and perpendicular to the line joining the points
(1, 4, 2) and ( 2, 3, 5) . Also, find the perpendicular distance of the plane from the origin.
12. Find the shortest distance between the lines :
𝑟 = 6𝑖 + 2𝑗 + 2𝑘 +  𝑖 − 2𝑗 + 2𝑘 𝑎𝑛𝑑𝑟 = −4𝑖 − 𝑘 +  3 𝑖 − 2𝑗 − 2𝑘 .
13. If any three vectors 𝑎 , 𝑏 and 𝑐 are coplanar, prove that the vectors 𝑎 + 𝑏 , 𝑏 + 𝑐 and 𝑐 + 𝑎 are also coplanar.

15. 14. If 𝑎 , 𝑏 and 𝑐 are mutually perpendicular vectors of equal magnitudes, show that the vector 𝑎 + 𝑏 + 𝑐 is equally
inclined to 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 .
15. Find the position vector of a point R which divides the line joining two points P and Q whose position
vectors are (2𝑎 + 𝑏 ) & (𝑎 − 3𝑏) respectively, externally in the ration 1 : 2. Also, show P is the mid
point of RQ .
16. If 𝛼 = 3𝑖 + 4𝑗 + 5𝑘 and 𝛽 = 2𝑖 + 𝑗 − 4𝑘 , then express 𝛽 in the form 𝛽 = 𝛽1 + 𝛽2, where 𝛽1 is parallel
to 𝛼 and 𝛽2 is perpendicular to 𝛼 .
17. Find whether the lines 𝑟 = ( 𝑖 − 𝑗 + 𝑘) + 𝜆(2𝑖 + 𝑗) and 𝑟 = (2𝑖 − 𝑗) + 𝜇( 𝑖 + 𝑗 − 𝑘) intersect or not. If
intersecting, find their point of intersection.
18. Find a vector of magnitude 5 units and perpendicular to each of the vectors (𝑎 + 𝑏) and (𝑎 − 𝑏), where
𝑎 = 𝑖 + 𝑗 + 𝑘, 𝑏 = 𝑖 + 2𝑗 + 3𝑘.
19. Show that the lines :
𝑥−1
2
=
𝑦−2
3
=
𝑧−3
4
and
𝑥−4
5
=
𝑦−1
2
= z intersect. Also find their point of intersection.
20. Prove that , for any three vectors 𝑎 , 𝑏 , 𝑐 : [𝑎 − 𝑏, 𝑏 − 𝑐, 𝑐 − 𝑎] =0
6 MARKS QUESTIONS.
1. Show that the lines
𝑥+3
−3
=
𝑦−1
1
=
𝑧−5
5
,
𝑥+1
−1
=
𝑦−2
2
=
𝑧−5
5
are coplanar. Also find the equation of the plane
containing the lines.
2. Find the co-ordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane determined by
points A(1, 2, 3), B (2, 2, 1) and C (-1, 3, 6) .
3. Find the equation of the line passing through the point (-1, 3, -2) and perpendicular to the lines
𝑥
1
=
𝑦
2
=
𝑧
3
and
𝑥+2
−3
=
𝑦−1
2
=
𝑧+1
5
4. Find the equation of the plane containing the lines :
𝑟 = 𝑖 + 𝑗 +  𝑖 + 2 𝑗 − 𝑘 and𝑟 = 𝑖 + 𝑗 +  − 𝑖 + 𝑗 − 2 𝑘
Find the distance of this plane from origin and also from the point (1, 1, 1).
5. Find the coordinates of the foot of the perpendicular drawn from the point A (1, 8, 4) to the line joining the point
B (0, -1, 3) and C ( 2, -3, -1).
6. Find the distance of the point (-1, -5, -10) from the point of intersection of the line
𝑟 = 2 𝑖 − 𝑗 + 2 𝑘 +  3 𝑖 + 4 𝑗 + 2 𝑘 and the plane 𝑟. 𝑖 − 𝑗 + 𝑘 = 5.
7. Show that the lines 𝑟 = 𝑖 + 𝑗 − 𝑘 +  3 𝑖 − 𝑗 and 𝑟 = 4𝑖 − 𝑘 +  2 𝑖 + 3 𝑘 are coplanar. Also, find the
equation of the plane containing both these lines.
8. Find the equation of the plane passing through the line of intersection of the planes 𝑟 = 𝑖 + 3𝑗 - 6 = 0
and 𝑟 = 3 𝑖 − 𝑗 − 4 𝑘 = 0, whose perpendicular distance from origin is unity.
9. Find the image of point (1, 6, 3) in the line
𝑥
1
=
𝑦−1
2
=
𝑧−2
3
.
10. Find the vector equation of the line passing through the point (2, 3, 2) and parallel to the line
𝑟 = −2 𝑖 + 3𝑗 +  2 𝑖 − 3 𝑗 + 6 𝑘 . Also find the distance between the lines.
11. Find whether the lines 𝑟 = 𝑖 − 𝑗 − 𝑘 +  𝑖 + 𝑗 and 𝑟 = 2𝑖 − 𝑗 +  𝑖 + 𝑗 − 𝑘 intersect or not. If
intersecting, find their point of intersection.

16. 12. Find the equation of plane which contains two parallel lines:
𝑥−3
3
=
𝑦+4
2
=
𝑧−1
1
and
𝑥+1
3
=
𝑦−2
2
=
𝑧
1
.
13. Find the distance between the point P (6, 5, 9) and the plane determined by the points A(3, -1, 2), B(5, 2, 4)
and C( -1, -1, 6).
14. Find the equation of plane through the points (1, 2, 3) & (0, -1, 0) and parallel to the line𝑟.(2 𝑖 + 3𝑗 + 4𝑘) + 5 = 0 .
15. Prove that the image of the point (3, -2, 1) in the plane 3x-y+4z = 2 lie on plane x+y+z+4 = 0.
16. Find the vector equation of the plane through the points ( 2, 1, -1) and ( -1, 3, 4) and perpendicular to the
plane x – 2y + 4z = 10.
17. Find the coordinates of the point, where the line
𝑥−2
3
=
𝑦+1
4
=
𝑧−2
2
intersects the plane x – y + z – 5 = 0. Also, find
the angle between the line and the plane.
18. Find the distance of the point (2,3,4) from the line
𝑥+3
3
=
𝑦−2
6
=
𝑧
2
measured parallel to the line 3x+2y+2z=5
19. Find the equation of plane passing through the points (3,4,1) and (0,1,2) and parallel to the line
𝑥+3
2
=
𝑦−3
7
=
𝑧−2
5
20. Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1) crosses the plane, passing through
the points (2,2,1), (3,0,1) and (4,-1,0)
21. Find the image of the point (1,3,4) in the plane 2x-y+z+3=0.
22. Find the vector equation of the plane passing through three points with position vectors 𝑖 + 𝑗 − 2𝑘,
2𝑖 − 𝑗 + 𝑘 and 𝑖 + 2𝑗 + 𝑘. Also find the coordinates of the point of intersection of this plane and the
line 𝑟 = 3𝑖 − 𝑗 − 𝑘 + 𝛾(2𝑖 − 2𝑗 + 𝑘) .
23. Find the vector equation of the plane which contains the line of intersection of the planes
𝑟 . 𝑖 + 2𝑗 + 3 𝑘 − 4 = 0 and𝑟 . 2𝑖 + 𝑗 − 𝑘 + 5 = 0 and which is perpendicular to the plane
𝑟 . 5𝑖 + 3𝑗 − 6 𝑘 + 8 = 0 .
24. Find the equation of plane which contains the line of intersection of planes x+2y+3z=4 and 2x+y-z+5=0 and whose
x- intercept is twice its z- coordinate. (7x+11y+14z+5=0)
25. Find equation of plane through (1,1,1) and containing the line 𝑟 = −3 𝑖 + 𝑗 + 5𝑘 +  3 𝑖 − 𝑗 − 5 𝑘 . Also show
that the plane containing the line 𝑟 = − 𝑖 + 2𝑗 + 5𝑘 +  𝑖 − 2 𝑗 − 5 𝑘 {x-2y+z=0}
26. Find the distance of the point (-1, -5, -10) from the point of intersection of the line
𝑟 = 2𝑖 − 𝑗 + 2𝑘 + ( 3𝑖 + 4𝑗 + 2𝑘 ) and the plane 𝑟 .(𝑖 − 𝑗 + 𝑘 ) = 5
PROBABILITY(2+4+4=10MARKS)
2 marks questions:
1. Mother , father and son line up at random for a family picture.If E: son on one end F: father in the middle,
determine P(E/F).
2. Given that the two numbers appearing on throwing two dice are different, find the probability of the event `the
sum of numbers on the dice is 4`.

17. 3. P(A)= 6/11, P(B) = 5/11 and P(𝐴 ∪ 𝐵) = 7/11 𝑓𝑖𝑛𝑑 𝑃(𝐵/𝐴)
4. An unbiased die is thrown twice.Let the event A be `odd number on the first throw` and B the event ` odd
number on the second throw`. Check the independence of the events A and B.
5. If A and B are two independent events, then show that the probability of occurrence of atleast one of A and B is
given by 1-P 𝐴/
𝑃 𝐵/
.
6. A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. Given that there is
atleast one girl on the committee, calculate the probability that there are exactly 2 girls on the committee.
7. If A and B are independent events,P(A)=p and P(B)=2p and P(exactly one of A,B)=
5
9
, then find p.
8. If P(A)= 6/11 , P(B)=5/11 , P(A∪B)=7/12, find P(A/B)
9. For 6 trials of an experiment, let X be a binomial variate: 9P(X=4) = P(X=2), find the probability of success.
10. Two dice are rolled once. Find the probability that the number on two dice is atleast 4.
11. If E and F are independent events , prove that E’ and F’ are also independent.
12. The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that
exactly one of A, B occurs is q, then prove that P(A) + P(B) = 2 – 2p + q.
13. A speaks truth in 60% of the cases and B in 90% of the cases. In what percentage of cases are they likely to
contradict each other in stating the same fact ?
14. A and B are two events such that P(A) = ½, P(B) =
1
3
and P(A ∩ B) =
1
4
. find
P(A I B) (ii) P(B I A) (iii) P(A I B) (iv) P(A I B)
4 MARKS QUESTIONS.
1. Probability of solving specific problem by X & Y are
1
2
and
1
3
. If both try to solve the problem, find the probability
that: (i) Problem is solved. (ii) Exactly one of them solves the problem.
2. Three balls are drawn without replacement from a bag containing 5 white and 4 green balls. Find the probability
distribution of number of green balls.
3. A die is thrown again and again until three sixes are obtained. Find the probability of obtaining 3rd
six in 6th
throw of
die.
4. Find the mean number of heads in three tosses of a fair coin.
5. Bag I contains 3 red and 4 black balls and Bags II contains 4 red and 5 black balls. One ball is transferred from
Bag I to bag II and then two balls are drawn at random ( without replacement) from Bag II. The balls so
drawn are found to be both red in colour. Find the probability that the transferred ball is red.
6. In a hockey match, both team A and B scored same number of goals up to the end of the game, so to decide the
winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain
gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respectively
probabilities of winning the match and state whether the decision of the referee was fair or not.

18. 7. In answering a question on a MCQ test with 4 choices per question, a student knows the answer, guesses or copies
the answer. Let ½ be the probability that he knows the answer, ¼ be the probability that he guesses and ¼ that he
copies it. Assuming that a student, who copies the answer, will be correct with the probability ¾ , what is the
probability that the student knows the answer, given that he answered it correctly?
8. The probability that a student entering a university will graduate is 0.4. find the probability that out of 3 students
of the university :
a)None will graduate, b)Only one will graduate, c) All will graduate.
9. How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80% ?
10. On a multiple choice examination with three possible answer (out of which only one is correct) for each of the five
questions, what is the probability that a candidate would get four or more correct answer just by guessing?
11. A family has 2 children. Find the probability that both are boys, if it is known that
(i) At least one of the children is a boy (ii) the elder child is a boy.
12. An experiment succeed twice often as it fails. Find the probability that in the next six trails there will be at least 4
successes.
13. From a lot of 10 bulbs, which include 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability
distribution of the number of defective bulbs.
14. Find the probability distribution of number of doublets in three throws of a pair of dice. Also find mean & variance.
15. A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at
random ( without replacement) and are found to be all spades. Find the probability of the lost card being spade.
16. Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course
reduces the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a
patient can choose any one of the two options with equal probabilities. It is given that after going through one of
the two options, the patient selected at random suffers a heart attack. Find the probability that the patient
followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more
beneficial for the patient.
17. An insurance company insured 2000 cyclist, 4000 scooter drivers and 6000 motorbike drivers. The probability of an
accident involving a cyclist, scooter driver and a motorbike driver are 0.01, 0.03 and 0.15 respectively. One of the
insured persons meets with an accident. What is the probability that he is a scooter driver? Which mode of
transport would you suggest to a student and why?
18. Two bags A and B contain 4 white and 3 black balls and 2 white and 2 black balls respectively. From bag A, two balls
are drawn at random and then transferred to bag B. A ball is then drawn from bag B and is found to be a black ball.
What is the probability that the transferred balls were 1 white and 1 black?
19. A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive
letter TA are visible. What is the probability that the letter came from TATANAGAR.

19. 20. There are three coins. One is two headed coin (having head on both faces), another is a biased coin that comes up tail
25% of the times and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads,
what is the probability that it was the two headed coin?
21. Three bags contain balls as shown in the table below :
Bag Number of White Balls Number of Black
balls
Number of Red
balls
I 1 2 3
II 2 1 1
III 4 3 2
A bag is chosen at random and two balls are drawn from it. They happen to be white and red.
What is the probability that they come from the III Bag?
22. A and B throw alternatively a pair of dice, A wins if he throws 6 before B throws 7 & B wins if he throws 7 beforeA
throw 6. Find their respective chances of winning , if A begins. [30/61, 31/61]
23. A girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2,
3 or 4, she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads,
What is the probability that she threw 1, 2, 3 or 4 with the die?
24. From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find
the Probability distribution of the number of defective bulbs.
25. A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the variance of number of success.
26. A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at
random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade
LINEAR PROGRAMMING (6MARKS)
6 MARKS QUESTIONS.
1. If a young man rides his motorcycle at 25 km/h, he has to spend Rs. 2/km on petrol. If he rides at a faster speed
of 40 km/h, the petrol cost increase at Rs. 5/km. He has Rs. 100 to spend on petrol and wishes to find what is the
maximum distance he can travel within one hour? Solve graphically.
2. Two tailors A and B earn Rs. 15 and Rs. 20 per day resp. A can stitch 6 shirts and 4 pants while B can stitch 10
shirts and 4 pants per day. How many days shall each work if it is desired to produce at least 60 shirts and 32
pants at minimum labour cost?
3. A dealer wishes to purchase a no. Of fans and sewing machines. He has only Rs. 5760 to invest and has space
for at most 20 items. A fan cost him Rs. 360 and a sewing machine Rs. 240. He expects to sell a fan at a profit of
Rs. 22 and a sewing machine for a profit of Rs. 18. Assuming that he can sell all the items that he buys, how
should he invest his money to maximize his Profit?

20. 4. A shopkeeper sells only tables and chairs. He has only Rs. 6000 invest and has space for at most 20 items. A
table cost him Rs. 400 and achair Rs. 250. He can sell a table at a profit of Rs. 40 and chain at Rs. 25. Solve it
graphically for maximum profit.
5. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made on each executives class
ticket and profit of Rs. 600 is made on each economy class ticket. The airlines reserves at least 20 seats for
executive class. However, at least 4 times as many passengers prefer to travel by economy class than by
executive class. Determine how many of each type of tickets must be sold in order to maximize the profit for
airline. What is the maximum profit?
6. Anil wants to invest at most Rs. 12000 in bonds A and B. According to the rules, he has to invest at least Rs.
2000 in Bond A and at least Rs. 4000 in bond B.If the rate of interest on bond A is 8% p.a. and on bond B is 10%
p.a., how should he invest his money for maximum interest? Solve the problem graphically.
7. A small firm manufactures necklace and bracelets. The combined no. Of necklace and bracelets that it can
handle for day is at most 24. The bracelets takes one hour to make and necklace takes half an hour. The
maximum no. Of hours available per day is 16. If the profit on bracelet is Rs. 2 and on necklace is Re. 1, how
many each product should be produced daily to maximize profit?
8. A manufacture produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to
produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of
bolts. He earns a profit of Rs. 17.50 per package on nuts and Rs. 7 per package on bolts. How many packages of
each should be produced each day so as to maximize his profits if he operates his machines for at the most 12
hours a day? Form the linear programming problem and solve it graphically.
9. A factory makes tennis rackets and crickets bats. A tennis racket takes 1.5 hours of machine time and 3 hours of
craftman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftman’s time.
In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftman’s
time. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets
and crickets bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve
graphically.
10. A manufacturer produces pizza and cakes. It takes 1 hour of work on machine. A and 3 hours on machine B to
produce a packet of pizza. It takes 3 hours on machine A and 1 hour on machine B to produce a packet of cakes.
He earns a profit of Rs. 17.50 per packet on pizza and Rs. 7 per packet of cake. How many packets of each
should be produced each day so as to maximize his profits if he operates his machines for at the most 12 hours
a day? Solve graphically.