1. This document contains mathematics questions on matrices and determinants ranging from 1 to 6 marks. Questions involve finding determinants, inverses, solving systems of equations using matrices, properties of determinants, and continuity and differentiability. 2. The document is divided into sections on 1, 2, 4 and 6 mark questions related to matrices and determinants as well as questions on continuity and differentiability ranging from 2 to 4 marks. 3. The questions cover a wide range of matrix and calculus concepts and techniques including finding determinants, inverses, solving systems of equations, properties of determinants, continuity, differentiation, mean value theorem, Rolle's theorem and related calculus topics.

1. MATRICES AND DETERMINANTS
1 MARK QUESTIONS.
1.
3𝑥 − 2𝑦 5
𝑥 −2
=
3 5
−3 −2
find y .
2. Find x if
𝑥 1
3 𝑥
=
1 0
2 1
3. For what value of x, matrix
3 − 2𝑥 𝑥 + 1
2 4
is singular?
4. ‘A’ is a square matrix of order 4 : 𝐴 = 1 find (i) 2𝐴 (ii) 𝑎𝑑𝑗 𝐴 (iii) −𝐴
5. Find cofactor of a12 in
2 −3 5
6 0 4
1 5 −7
6.
𝑥 + 1 𝑥 − 1
𝑥 − 3 𝑥 + 2
=
4 −1
1 3
find x.
7. A matrix A is of order 2 x 2 has determinant 4. What is the value of 2𝐴 ?
8. A is a square matrix of order 3 : |A|= -1, |B|=3 find |3AB|
9. If A is a skew symmetric matrix of order 3,what will be the value of det.(A).
10. Find x if
2 4
5 −1
=
2𝑥 4
6 𝑥
.
11. If A is a square matrix such that A2
=I, then write 𝐴−1
,
12. If A and B are square matrices of order 3 such that 𝐴 =-1 and 𝐵 =3 the find the value of 2𝐴𝐵
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. For the matrix A=
3 1
−1 2
,A2
-5A+7I=O,then find 𝐴−1
.
3. Find the matrix X for which
1 −4
3 −2
X=
−16 −6
7 2
4. Prove that the diagonal elements of a skew symmetric matrix are zero.
5. A and B are symmetric matrices of same order ,then show that AB is symmetric iff A and B commute.
6. If A=
𝑐𝑜𝑠𝐴 𝑆𝑖𝑛𝐴
−𝑆𝑖𝑛𝐴 𝐶𝑜𝑠𝐴
find A-1
.
7. If A is a skew symmetric matrix of order 3, then prove that |A|=0.
8. 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. 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
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. Find A if
4
1
3
A=
−4 8 4
−1 2 1
−3 6 3

3. 6 Marks Questions:
9. 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.
10. 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.
11. 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.
12. 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.
13. 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 −1 5
.
Verify AB = BA = 6I and hence solve the equations x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7.
14. 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 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 = 0
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. If x = 𝑎sin −1 𝑡 , 𝑦 = 𝑎cos −1 𝑡 , 𝑠𝑕𝑜𝑤 𝑡𝑕𝑎𝑡
𝑑𝑦
𝑑𝑥
= −
𝑦
𝑥
.
4 MARKS QUESTIONS
12. Differentiate log ( xsin x
+ cot2
x) with respect to x.
13. If y = log [ x + 𝑥2 + 𝑎2 ], show that ( x2
+ a2
)
𝑑2 𝑦
𝑑𝑥2 + x
𝑑𝑦
𝑑𝑥
= 0.

4. 14. . If 1 − 𝑥2 + 1 − 𝑦2 = a (x – y), Prove
𝑑𝑦
𝑑𝑥
=
1−𝑦2
1−𝑥2
15. If x = a sin t and y = a ( cos t + log tan
𝑡
2
), find
𝑑2 𝑦
𝑑𝑥2 .
16. 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
.
17. 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
18. Differentiate tan-1 1+ 𝑥2− 1− 𝑥2
1+ 𝑥2+ 1− 𝑥2
with respect to cos-1
x2
.
19. Verify lagrange’s Mean value theorem: (i) f(x) = x(x – 1) (x – 2) [ 0, ½] (ii) f(x) = x3
-5x2
– 3x [1, 3]
20. 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
]
21. Differentiate sin-1 2𝑥
1+𝑥2 w.r.t. tan-1
x .
22. If y =
sin −1 𝑥
1−𝑥2
, show that (1-𝑥2
)
𝑑2 𝑦
𝑑𝑥2 − 3𝑥
𝑑𝑦
𝑑𝑥
− 𝑦 = 0 .
23. Differentiate cos-1 1− 𝑥2
1+ 𝑥2 with respect of tan-1 3𝑥 − 𝑥3
1−3 𝑥2 .
24. If x = a sin 2t(1 + cos 2t), y = b cos 2t( 1 – cos 2t) Show that
𝑑𝑦
𝑑𝑥 𝑡=
𝜋
4
=
𝑏
𝑎
25. Find
𝑑𝑦
𝑑𝑥
, if y = sin-1
[x 1 − 𝑥 − 𝑥 1 − 𝑥2] .
26. If y = log [x + 𝑥2 + 1], prove that (𝑥2
+ 1)
𝑑2 𝑦
𝑑𝑥2 + 𝑥
𝑑𝑦
𝑑𝑥
= 0 .
APPLICATION OF DERIVATIVES
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
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?

5. 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 equation of tangent to the curve y =
𝑋−7
𝑋−2 (𝑋−3)
at the point where it cuts x-axis. [x-20y=7]
10. 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.
11. Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to
diameter of base.
12. Show that the semi – vertical angle of cone of maximum volume and of given slant height is tan-1
2 .
13. 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.
14. 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.
15. 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?
16. Show that the function f defined f(x) = tan-1
(sin x + cos x) is strictly increasing in (0,
𝜋
4
) .
17. Find the intervals in which f(x)=
3
2
x4
-4x3
-12x2
+5 is strictly increasing or decreasing.
18. Find the intervals in which f(x)=
3
10
x4
-
4
5
x3
-3x2
+
36
5
𝑥 + 11 is strictly increasing or decreasing.
19. Find the minimum value of (ax+by), where xy=c2
.
20. Find the equations of tangents to the curve 3x2
-y2
=8, which passes through the point (4/3, 0).
21. 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.
22. Find the point on the curve x2
=4ywhich is nearest to the point (-1,2).

6. INTEGRALS
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
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