Karnataka 2nd PUC Mid Term Question Paper 2025 Mathematics

Government of Karnataka
Karnataka Secondary Education Examination Board

Question Papers
Mid Term

|- OCTOBER 2025
MID TERM EXAMINATION
Time:3-00 Hrs.
PUC-Mathematics (35)
I Max. Marks :80
Instructions : A, B, C,D
namely and Answer allthe parts.
question paper has five parts
E.

1. The and 5 Fill in the blanks
2 Part A has 15 mnultiple choice questions of 1 mark each.
Use the graph sheet for question on linear programming
in Par
3

PART A
Answer all the Multiple-choice questions
15x1=15
1, Iff: R Rand g:RR are givenby f(x)=cosx and g(x) =x3 thenfog is

A) cosx B) cos(x3) C) x'coSx D) cos(x2)
2. Which of the following relation is not transitive on A= (1, 2, 3 }
A) ((1, 1), (1, 2)} do B) {(1, 1).(2,2)}
C) {(1, 1), (1,2), (2,1)} D) {(2, 2),(2, 1), (1, 2), (1, 1) }

3. The value of tan-1 sin(-)is

A) B) C) D)- 3

4. Match the list I with list II

List I List II

a) Domain of sinl x i) (0, T)eSa
b) Range of cos1 x ii) (-o, o)
c) Domain of tanlx iii) [-1, 1]

Choose the correctoptions given below
A) a-ii, b-i, c- iii B) a-iii, b-i, c-ii C) a-ii, b-iii, c-i D) a-ii, b-ii , c-i
5. IfA is a skew symmetric matrix of order3then A+A is
0 0 11

0 1
lo 0 o!
N
6. Square matrix A is the inverse of anothersquare matrix B if and only if

A) AB=BA B) AB=BA=0 C) AB=0, BA=I D) AB=BA=I

7. If the matrix*is singular then then x take the value
A) 2 B) 4 C) 8 D) #2

8.The rght hand derivative of f(x) atx0 is

A) 0 B) 1 C) -1 D)does not exist

Mathematics

9. The domain of the derivative of sinx w.rt.x is

A) [0, 1] B) [-1, 1] C) (0, 1)
D) (-1,1)
dy is
10. Ify= log, (logx) then dx
1

D)og7
1
B) Log7
A) xlog7log xlogx C) xlogx

11. The critical point of thefunction f(x)=|x 5| is -
A)o B) 5 D) 1
C) -5

12.The maximum value of thefunction f(x)=x, xE (0,1) is

A) 0 B) 1 C) -1 D) does not exist
13.

Statement 1: The function f(x) e is always increasing

Statement 2: The function logpx is always increasing in its domain (0, co)

Choose the correct answer

A) Statement 1 is false but statement 2 is true
B) Statement 1 is true but statement 2 is false

C) Statement 1 is true but statement 2 is true

D) Statement 1 is false but statement 2 is false
14.1f is a nonzero vector of magnitude 'a' and Ais ,
a non-zero scalar then Aå is a unit vector if

A) =1 B) =-1 C) a| D) a=
1

15. In the triangle ABC, which one of the following not true

A) AB +B +C=
C
B) AB +BC -AC=ö
C) AB +BC – C=

D) AB -CB +CÁ=

Ilin the blanks by choosingthe appropriateanswer from those given in the brackets 5x1=5

[5, 4, 25, 16, 6, 2]

16. The number of onto functions from{ 1,2, 3 }to itself is
17. The number of all 2x2 matriceswith each entry 0 and 1 is

18. If A=| then |adj A| is

19.The number of points at which the functionf RRis defined by f(x) |+ x- 2 is not
differentiable is

20.Themaximum value of the functionf(x)= sin4x +3|, xER is

PART B

Answer any SIX questions 6x2=12

21. Showthat the relation R in the set R of real numbers defined by R={ (a, b):as b'} is neither
reflexive nor transitive

22.Show that sin-'(2xV1-x) 2cos-1x SxS1
23.Prove that 2sin-1()tan-1()
; 1

24.Find the equation of the line passing through the point (3,1) and (9,3 )using determinants

25.Pind X and Y if X+Y 1, and X- Y=|PI

Z6. Find all the points discontinuity ofthefunction
+
f(v)~Z* 3 if xS2
of
(2x-3 if x >2
27.Differentiate sec(tanyx w. ,t x)
28.The radius ofa circle is increasing at the rate of 0.7cm/sec. what is the rate of increase of its

circumference

29. Find the unit vector in the direction of thesum of the vectors å =2i +2-5k and b =21 +j+3k

PART C

6x3=18
Answer any SIXanswers

30. Show that the relation R in the set A= {x: xE Z, 0sxs 12 }given by R={ (a,b) : -b| is la a

relation
multiple of 4} is an equivalence

sin-113 +Cos -1
31.Prove that tan-1
16

32. ExpressA= |?as the sum ofa symmetric anda skew
symmetricmatrices

show that AB is symmetricif and only
if
of the same order,then
33. If A and B are symmetricmatrices

AB=BA

34. Differentiate (x)sinx +(cosx) Stnx w.rtx

35.Differentiate sin²x w. r.t ecosx
of the function fgiven
by f(x)= 2x+
36. Find the absolute maximum and absolute minimum

-15x3+36x+1 on the interval (1, 5]

37. If the vertices A,

38.1f
B, C of a triangle

b and dbe three vectors
(1,

such
2,

that
3)
|
,(-1, 0, 0 and (0,

|b 4, 1 3,
)
1, 2) respectively
then find LABC

5 and d is perpendicular
to (
find+b+
d,

to (i+b)then
to (d +a) and 2is perpendicular
b+),bis perpendicular

DART D
Answer any FOUR questions
oe 5x4=202
39. Provethat the function f: N Y defined

findthe inverse of f(x)
(x) 4+3 where Y={ ye N: y=4x+3,xe N} is invertible. Also

40.

AC +BC
41.Solve the
-:3. find AC, BC and (A+B)C.Also verify that (A+B)C=

system of linear
equations by using
matrix method
X-y+2z=1; 2y-3z=1;3x-2y +4z=2
42. If y=eacos 'x
,-1<x<1, showthat (1-x2)
^dx -a²y=0
43. If y=Aemx +Benx
show that a'y
dx2 -(m+n)
dx tmny=0
44. Sand is pouring from a pipe at
the rate of 1 2cm3 /s The falling sand
forms a cone on the ground in such a
way that the height of the cone is always
one-sixth of the radius ofthe base. How
fast is the height of the
sand cone is increasingwhen the
height is 4cm
45. a) Ifd = + +3k ; b= -+21+k and =31 +i are such that å+ Ab
2i 2j
is perpendicular
to ,then find the value of A
b) Find the projection of thevector î + +7k on the vector 7i- +8k
3

PART E

Answer the following questions

46. Minimize and Maximize z= 5x+10y subjectto the constraints
X+2y < 120 ; x+y> 60;x-2y>0; x,y >0 using graphical method
OR
Maximizeand minimize z=200x+-500y subject to the constraints

x +2y > 10, 3x +4y < 24, x,y 0,using graphical method 6 Marks
3
– 54 + 7I =0.Hencefind A-1
1

47. If A= show that A2

OR

Find the value of k so that thefunction f definedby f(x) =k* +1,ifxs5
(3x-5,if > s x 1S Continuous at X=5

4 Marks

(
PART F For visually challenge students only)

15. The projection of the vector å =2i + 3)+ 3kon z- axis is

A) 0 B) 1 C) 3 D) 2

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