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DEPARTMENT OF MATHEMATICS, DEPARTMENT OF MATHEMATICS, DEPARTMENT OF MATHEMATICS, DEPARTMENT OF MATHEMATICS, SAPUC, JAVALLI.SAPUC, JAVALLI.SAPUC, JAVALLI.SAPUC, JAVALLI.

** CHAPTER WISE MOST LIKELY QUESTIONS **

Chapter-1

SETS RELATIONS AND FUNCTIONS

One mark problems

1. Define an empty set.

2. Define power set of a set.

3. If A has 4 elements. How many subsets does A has?

4. If A={1,2}, B ={x:x∊ N and x2-9=0 }. Find A×B.

5. Write the set { x : x ϵ R and -4 < x ≤6 } as an interval.

Two mark problems

1. In a school, there are 20 teachers who teach mathematics or physics . Of

these 12 teach mathematics and 4 teach both physics and mathematics.

How many teach physics.

2. Let A and B be two sets such that n(A)=3 and n(B)=2. If (5,a),(6,b),(7,a) are

in A×B then find the sets A and B, where a,b are distinct elements.

3. If A={1, 2, 3, 4} B={2, 3, 5} and C={3, 5, 6} ,find AU(B∩C).

4. If the universal set U = { 1, 2, 3, 4, 5, 6, 7 }, A = { 1, 2, 5, 7 } ,

B = { 3, 4, 5, 6} Verify ( ) .|||BABA ∩=∪

5. If U={1,2,3,4,5,6,7,8,9}, A={2,4,6,8} and B={2,3,5,7} verify (A∩B)| =A|∪B| .

6. If X and Y are two sets such that X∪Y has 50 elements , X has 28 elements

and Y has 32 elements. How many elements does X∩Y have?

7. If A={1,2} and B={3,4} write A×B. How many subsets will A×B have?

8. If A = {1 , 2} , form the set A×A× A .

9. Taking the set of natural numbers as the universal set.

If A={ x:xϵN and 2x+1 >10} B={ x:xϵN and 3x-1 >8} find A| and B| .

10. If A={3,5,7,9,11}, B={7,9,11,13} and C={11,13,15},then find A∩ (BUC).

11. If X and Y are two sets such that XUY has 18 elements, X has 8 elements

and Y has 15 elements, then how many elements does X∩Y have?

12. If X and Yare the two sets such that n(X)=17, n(Y)=23, n(XUY)= 38. Find

n(X∩Y) .

13. Find the range and domain of the real function f(x)= 29 x− .

14. In a class of 35 students, 24 likes to play cricket, 5 likes to play both

cricket and football. Find how many students like to play football?

15. If A = { 1, 2, 3 }, B = { 3, 4 } , C ={ 4, 5, 6 }. Find A× (BUC).

Three mark problems

1. Let A = {1,2,3,........14} Define a relation R from A to A by

R = {(x,y): 3x-y=0, x,y∊A } write its domain and range.

2. In a survey of 400 students in a school, 100 were listed as taking apple

Juice, 150 as taking orange juice and 75 were listed as taking both apple

and orange juices. Find how many students were taking neither apple juice

nor orange juice.


3. If f(x)=x2 and g(x)=2x+1 be two real functions find i)(f+g)(x) ii)(f-g)(x) iii)(fg)(x) .

4. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis.

How many like tennis? How many like tennis only and not cricket?

5. Write the relation R defined as R={(x, x+5): x∈ {0,1,2,3,43}} in roster system.

Write down its range and domain.

6. In a survey, it was found that 21 people liked product A, 26 liked product B

and 29 liked product C. If 14 liked products A and B, 12 liked products C

and A, 14 people liked product B and C and 8 liked all the three products.

Find how many liked product C only.

7. If A={1,2,3,4} , B={5,6}. Define a relation R from A to B by

R = {(x, y): xϵ A ,y ϵB, x-y is odd}. Write R in the roaster form. Write down its

domain and range.

8. There are 200 individuals with a skin disorder. 120 has been exposed to

the chemical A, 50 to chemical B and 30 to both chemical A and B. Find

the number of individuals exposed to i) Chemical A but not to chemical B

ii)Chemical A or chemical B

5. Let A={1,2} , B={1,2,3,4} , and C={5,6} . Verify that A×(B∩C)=(A×B)∩(A×C).

Five mark problems

1. Define Signum function. Draw the graph of it and write down it Domain

and Range.

2. Define modulus function. Draw the graph of modulus function, write down

its domain and range .

3. Define an identity function. Draw the graph of the identity function and

write down its range and domain.

4. Define greatest integer function. Draw the graph of greatest integer

function, Write the domain and range of the function.

5. Define a polynomial function. If the function from f:R→R is defined as

f(x)=x2 then draw the graph of and find the domain and range.

Chapter-2

TRIGONOMETRIC FUNCTIONS

One mark problems

1. Convert c

6

7π into degrees .

2. Convert 40o20|| into radian measure. 3. Convert 5200 into radian measure.

4. Convert 3

2π radians into degree measure

5. If cos x = -5

3 , x lies in the III quadrant then find the value of sin x.

Two mark problems

1. The minute hand of a clock is 1.5cm long. How far does its tip move in 40

Minute? (Use π=3.142).

2. Prove that sin3x =3sinx - 4sin3x.

3. A wheel makes 360 revolutions in one minute, through how many radians

does it turn in one second ?


4. Find the value of sin (150)

5. The minute hand of a clock is 2.1 cm long. How far does its tip move in 20

minute?

6. Prove that .tantan1

tantan)tan(

yx

yxyx

+−

=−

7. Find the value of 3

31sin

π.

8. Prove that : xxx cos3cos43cos 3 −= .

9. Prove that x

xx

2tan1

tan22sin

+= .

10. Find the value of cos (-17100).

Three mark problems

1. Prove that in any triangle ABC, .sinsinsin

c

C

b

B

a

A==

2. If 5

3sin =x ,

13

12cos −=x where x and y both lie in the second quadrant,

find the value of sin(x+ y).

3. Prove that (cos x+ cos y)2+(sin x - sin y)2 = .2

cos4 2

+ yx

4. Find the general solution of 2 cos2 x + 3sinx = 0 .

5. Find the general solution of sec22x=1-tan2x.

Five mark problems

1. Prove that .2

5sin5sin

2

9cos3cos

2cos2cos

xx

xx

xx =−

2. Prove that .tancos5cos

sin3sin25sinx

xx

xxx=

−+−

3. Prove that .3cot2sin3sin4sin

2cos3cos4cosx

xxx

xxx=

4. Prove that cos2 2x –cos2 6x =sin4x.sin8x .

Six mark problems

1. Prove geometrically that cos (A+B) = cos A cos B - sin A sin B.

2. Prove geometrically that cos(A+B) = cosA.cos B - sinA.sinB and

hence prove that cos2A = cos2A – sin2A.

3. Prove geometrically that cos(x + y) = cos x cos y - sin x siny using unit

circle method and hence find the value of cos xx sin2

−=

−π

4. Prove geometrically that cos(x + y) = cos x cos y – sin x sin y and hence

prove that cos(x-y)=cos x cos y + sin x sin y


Chapter-3

PRINCIPLE OF MATHEMATICAL INDUCTION

Five mark problems

1. Prove by mathematical induction that

.,46)23).(13(

1.......

11.8

1

8.5

1

5.2

1Nn

n

n

nn∈∀

+=

+−++++

2. Prove that Nnn

n

nn∈∀

+=

+⋅++

⋅+

⋅+

⋅,

1)1(

1....

43

1

32

1

21

1by the principle

of mathematical induction.

3. Prove by mathematical induction that

.)13()13)(23(

1........

107

1

74

1

41

1Nn

n

n

nn∈∀

+=

+−+

⋅+

⋅+

⋅

4. Prove by mathematical induction that 13+23+33+……n3= .2

)1(2

+nn

5. Prove that 102n-1+1 is divisible by 11,∀nϵN by the principle of

mathematical Induction.

6. Prove by mathematical induction that 12+22+32+……n2= .6

)12)(1( ++ nnn

Chapter-4

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

One mark problems

1. Find the real number x if (x-2i)(1+i) is purely imaginary.

2. Write the multiplicative inverse of .2

1

2

3i−

3. Find the conjugate of .13 −i

4. Express i

i

2

25 +in the form of x+ i y.

5. Evaluate: ( ) .1

26

24

+i

i

Two mark problems

1. Find the least positive integer m such that

m

i

i4

1

1

−+

= 1.

2. Express ( )( )( ) ( )ii

ii

2525

2323

−−+−+

in the form of a+ib.

3. Represent the complex number Z=-1+i in polar form.

4. Find the value of x and y, if (x+2y)+i(2x-3y)is the conjugate of 5+4i.

5. If iqp

iqpiyx

−+

=+ then prove that x2+y2=1.


Three mark problems

1. Express 2

1 i+−in the polar form.

2. Solve the equation x2 +2

x+2=0.

3. Convert 2

31 i+ into polar form.

4. Solve: 27x2-10x+1=0.

5. Find the real such that θθ

sin21

sin23

i

i

−+

is purely real.

6. Solve the equation x2 + 3x + 9=0.

7. Express i+3 in the polar form.

8. Solve: .03

2043 2 =+− xx

9. Express i

i

+−

1

1 into polar form.

10. Solve: 2x2 + 3 x - 1=0.

Chapter-5

LINEAR INEQUALITIES

Two mark problems

1. Solve the inequality (2x-5) > (1-5x) and represent the solution graphically

on the number line.

2. Solve 7x+1≤ 4x+5 and represent the solution graphically on the number

line.

3. Solve [3(2x–5)-7] ≥ 9(x–5).

4. The marks obtained by a student of class XI in first and second terminal

5. Solve 3x+2y>6 graphically.

Five mark problems

1. Solve the following system of linear inequalities graphically:x+y ≥ 5, x-y ≤3

2. Solve graphically 2x+y ≥ 4, x + y ≤ 3,2x-3y ≤ 6.

3. Solve the inequalities : 2x+3y< 12, x≥2 , y≥2 graphically.

4. Solve the following system of inequalities graphically : 5x+4y≤40, x≥2,y≥3.

5. A manufacture has 600 litters of a 12% solution of acid.. How many liters

of a 30% acid solution must be added to it so that the acid content in the

resulting mixture be more that 15% but less than 18%.


Chapter-6

PERMUTATION AND COMBINATION

One mark problems

1. Show that the middle term

2. If !8!7

1

!6

1 x=+ then find the value of `x`.

3. Find n if 9:1: 43

)1( =− PP nn .

4. Find the number of 4 digits that can be formed using the digits 1,2,3,4,5.

If no digit is repeated.

Three mark problems

1. The marks obtained by a student of class XI in first and second terminal

examinations are 62 and 48, respectively. Find the minimum marks he

should get in the annual examination to have an average of at least 60

marks.

2. Find the number of arrangement of the letters of the word

PERMUTATIONS. In how many of these arrangements

(i) word start with P and end with S

(ii) vowels are all together.

3. In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted

for both NCC and NSS. If one of these students is selected at random, find

the probability that

(i) the student has opted for NCC or NSS

(ii) The student has opted for NCC but not NSS

4. How many numbers greater than 10,00,000 can be formed by using the

digits 1,2,0,2,4,2,4.

5. If 1

54 .6.5 −= rr pp then find r.

Five mark problems

1. A committee of seven has to be formed from 9 boys and 4 girls. In how

many ways this can be done when the committee consists of

(i) exactly 3 girls,

(ii) at least 3 girls and

(iii) at most 3 girls.

2. How many words with or without meaning each of 3 vowels and 2

consonants can be formed from the letters of the word INVOLUTE?

3. A group consists of 4 girls and 7 boys, In how many ways can a team of 5

members be selected if the team has

(i) No girl

(ii) At least one boy and one girl

(iii) At least three girls

4. Find the number of ways of selecting 9 balls from 6 red balls, 5 white

balls and 5 blue balls. If each selection consists of 3 balls of each colour.


Chapter-7

BINOMIAL THEOREM

Three mark problems

1. Find the coefficient of x5 of (x+3)8.

2. Using Binomial theorem, which number is among (1.1)10000 and 1000 .

3. Find the coefficient of x6y3 in the expansion of (x+2y)6.

4. Expand 0,5

35

2 ≠

+ xx

5. Find the middle term in the expansion of .93

10

+ yx

Five mark problems

1. Prove Binomial Theorem for positive integers with real numbers. Hence

prove that ................ 731420 +++=+++ CCCCCC nnnnnn

2. State and prove Binomial theorem for all natural numbers.

3. Find “a” if 17th and 18th terms of the expansion (2+a)50 are equal.

4. Show that the middle term in the expansion of (1+x)2n is !

)12....(5.3.1

n

n −

5. For all real numbers a, b and positive integer ‘n’ prove that,

( ) n

n

nnnnnnnnbCbaCbaCaCba ++++=+ −− ..........22

2

1

10

Chapter-8

SEQUENCE AND SERIES

One mark problems

1. For what value of x the numbers2

7− , x,

7

2− are in G. P.

2. What is the 20th term of the sequence defined by an=(n-1)(2-n)(3+n)?

3. Find the 20th term of the G.P : .,.........8

5,

4

5,

2

5

4. Find the tenth term of G.P: 5, 25, 125, .......

5. Which term of 2, 2 2 ,4,………..is 128.

Three mark problems

1.The number of bacteria in a certain time double every hour. If there are 30

bacteria present in the culture originally. How many bacteria will be

present at the end of 2nd hour, 4th hour, and nth hour.

2. How many terms of the G.P 3, ,........4

3,

2

3 are needed to give the sum 512

3069 .

3. If the pth qth rth terms of a G.P. are a,b,c respectively then Prove that

1=⋅⋅ −−− qpprrq aaa

4. Find the sum of the sequence : 7,77,777,7777,…………

5. In an A.P, if mth term is n and nth term is m. Then find pth term (m≠n).

6. Find the sum of n terms of an A.P whose kth term is (5k+1).


7. The difference between any two consecutive interior angles of a polygon is

8. If the smallest angle is 1200. Find the number of sides of the polygon.

9. If 11 −− ++

nn

nn

ba

bais the A.M. between a and b then find the value of n.

10. Insert five number between 8 and 26 such that resulting sequence is an A.P.

11. The sum of first three terms of a G.P. is 12

13 and their product if -1. Find

the common ratio and the terms. 12. Two series A and B with equal means have standard deviations 9 and 10 respectively. Which series is more consistent?

Four mark problems

1.Find the sum to n terms of the series 5+11+19+29+. . .

2. Find the sum to ‘n’ terms of the series 12+(12+22)+(12+22+32)+……….

3.Find the sum to terms of the series : 1.2.3 + 2.3.4 + 3.4.5 + ……….

4. Find the sum of first n terms of the series 12+22+32+…….n2

5. Find the sum to n terms of series: ............32

1

21

1+

⋅+

⋅

Chapter-9

STRAIGHT LINES

One mark problems

1. Find the slope of the line .123

=+yx

2. Find the slope of a line 3x-4y+10=0.

3. Find the slope of the line making inclination of 600 with positive direction

of x-axis.

4. Find the slope of the line joining the points (3,-2) and (-1,4).

5. Reduce 6x+3y-5=0 into slope-intercept form.

Two mark problems

1. Find the angle between the lines y – 3 x – 5 = 0 and 3 y - x + 6 = 0.

2. Find the distance of a point (3 ,-5) from the line 3x – 4y – 5 = 0 .

3. Find the equation of line which make intercepts -3 and 2 on x and y axes

respectively.

4. Find the equation of the line passing through (-3,5) and perpendicular to

the line through the points (2,5) and (-3,6).

5. Find the equation of the line perpendicular to the line x+y+2=0 and passing

through the point (-1,0).

6. Find the distance between the parallel lines 3x-4y+7=0 and 3x-4y+5=0.

7. Find the equation of the line passing through (-1,1)and (2,-4).

8. Find the equation of the line passing through (-4,3) with slope 2

1 .

9. By using the concept of equation of the line prove that the three points

(3,0),(-2,-2) and (8,2) are collinear.

10. Find the equation of the line parallel to the line 3x-4y+2=0 and passing

through the point (-2,3).


Five mark problems

1. Derive a formula for the perpendicular distance of a point (x1,y1) from the

line Ax + By + C = 0.

2. Derive the equation of a straight line passing through the point (x1,y1)

having the slope m. Hence deduce the equation of a line which passes

through (2,1) which makes an angle 450 with positive direction of x-axis.

3. Derive the equation of the line with slope m and y-intercept c . Also find

the equation of the line for which tan θ=½ and y-intercept is -3/2.

4. Derive the formula for the angle between two straight lines with slopes m1

and m2 hence find the slope of the line which makes an angle 4

πwith the

line x = 2y+5.

5. P (a, b) is the midpoint of the line segment between axes. Show that the

equation of the line is .2=+b

y

a

x

Chapter-10

CONIC SECTIONS

One mark problems

1. Find the equation of parabola with vertex at the origin, axis along x-axis

and passing through the point (2,3) also find its focus.

2. Find the co-ordinates of the foci and latus rectum of the hyperbola 3x2-

y2=3.

3. Find the centre and radius of the circle x2+y2+8x+10y-8=0 .

4. Find the equation of circle which passes through (1,0) and (0,-1) and

whose centre lies on the line x-y+2=0.

5. Find the equation of the ellipse whose center at origin, major axis on the

X-axis and passes through the point (4, 3) and (6,2).

Six mark problems

1. Defined ellipse as a set of all points in the plane and derive its equation in

the standard form as 12

2

2

2

=+b

y

a

x , a>b.

2. Define ellipse and derive the equation of the ellipse in standard form as

12

2

2

2

=+b

y

a

x (a>b)

3. Define Hyperbola as a set of points. Derive its equation in the form

12

2

2

2

=−b

y

a

x

4. Define hyperbola as a set of all points in the plane and derive

its equation as 12

2

2

2

=−b

y

a

x


5. Define parabola as a set of all points in the plane and derive its equation

in the form y2=4ax ,a>0 and hence also find the focus and vertex.

Chapter-11

INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

One mark problems

1. Show that the points P( -2 ,3 ,5 ) ,Q( 1, 2,3) and R ( 7 ,0 ,-1) are collinear.

2. Show that the points A(1,2,3), B(-1,-2,-3), C(2,3,2) and D(4,7,6) are the

vertices of a parallelogram.

3. Find the ratio in which the line segment joining the points (4,8,10)and

(6,10,-8) is divided by YZ-Plane.

4. Find the ratio in which the YZ-plane divides the line segment formed by

joining the points (-2,4,7) and (3,-5,8).

Five mark problems

1. Derive the section formula in 3-D for internal division. Also find the

Co-ordinates of the midpoint of the line joining the points . A(1,-2,3)

and B(3,4,8).

2. Derive the section formula in 3-dimensions for the internal division. Also,

find the co-ordinates of the mid-point of the line joining the points

P(2,3,-4) and Q(4,-7,2).

3. Derive an expression for the coordinates of a point that divides the line

joining the points A(x1,y1,z1) and B(x2,y2,z2) internally in the ratio m:n .

Hence, find the coordinates of the midpoint of AB where A(2,3,-3 ) and

B(-1,2,1 ).

4. Derive the formula for the distance between two points. And hence find

the distance between (2,-1,3) and (-2,1,3).

5. Derive the formulae for distance between two points (x1,y1,z1) and

(x2,y2,z2) and hence find the distance between P(1,-3,4) and Q(-4,1,2).

Chapter-12

LIMITS AND DERIVATIVES

One mark problems

1. Find the derivative of x2 –2 at x = 0.

2. Find the derivative of x at x=1.

3. Find the derivative of 2x - 4

3.

4. Evaluate :

−→ x

x

x πcos

lim0

5. Find 5lim5

−→

xx

.


Two mark problems

1. Evaluate .2

2

12

lim2 +

+

−→ x

xx

2. Evaluate: .sin

coslim

0 xb

xxax

x

+→

3. Evaluate: [ ]102

1...........1lim xxx

x++++

−→.

4. Evaluate :

−→ x

x

x

cos1lim

0

5. Compute the derivative of sin2x.

Three mark problems

1. Verify by the method of contradiction that 2 is irrational.

2. Differentiate cos x with respect to x by using first principles.

3. Find the derivative of the function ‘-x’ with respect to x from first principle.

4. Find the derivative of `sin x` from first principle.

5. Differentiate: x

x1

+ from first principle.

Four mark problems

1. Find the derivative of f(x)=2x2 +3x-5 , also prove that f|(0)+3f|(-1)=0.

2. Find the derivative of x

xx

tan

cos+ .

3. Suppose )1()(lim

1

14

1

)(1

fxfand

xaxb

x

xbxa

xfx

=

>−

=

<+

=→

What are the possible

values of a and b.

4. Find the derivation of x

xx

sin

cos5 −with respect to x.

5. Find the derivate of .131

2 2

−−

+ x

x

x

Five mark problems

** Different ways

1. Prove geometrically: 1sin

lim0

=→ θ

θθ

where θ is in radian and hence deduce

that 1tan

lim0

=→ θ

θθ

.

2. Prove that 1sin

lim0

=→ x

x

x,where x is in radian and hence evaluate: .

2sin

4sinlim

0 x

x

x→

3. Prove that 1sin

lim0

=→ θ

θθ

( being in radians) and hence show that bx

ax

x sin

sinlim

0→.


4. Prove that 1sin

lim0

=→ x

x

x

, where x is in radian and hence evaluate 1tan

lim0

=→ x

x

x

.

5. Prove that 1lim −

→=

−− n

nn

axna

ax

ax where n is an integer hence evaluate

x

x

x

11lim

0

−+→

Chapter-13

MATHEMATICAL REASONING

One mark problems

1. Write the negation of ‘For all ’. a, b є I, a-b є I.

2. Write the negation of “Intersection of two disjoint sets is not an empty set”.

3. Write the negation of “For all a, b ∈ I, a-b ∈ I”.

4. Write the contrapositive of

“ if a number is divisible by 9 then it is divisible by 3”.

1. Write the negation of “Every natural number is greater than zero”.

Two mark problems

1. Write the converse and contrapositive of ‘If a parallelogram is a square,

then it is a rhombus’.

2. Write the contra positive and converse of “ If a parallelogram is a square,

then it is a rhombus”.

3. Write the component statement of the following compound statement and

check whether the compound statement is true or false ; “ Zero is less than

every positive integer and every negative integer”.

4. By giving a counter example, show that the following statements is false:

“If n is an odd integer then n is a prime”.

5. Write the converse and contrapositive of “If a parallelogram is a square,

then it is a rhombus‟.

Three mark problems


2. By the method of contradiction , check the validity of the statement:

“If a,b ϵ Z such that ab is odd, then both “a and b are odd”.


4. Given p:25 is a multiple of 5;q:25 is a multiple of 8. Write the compound

statement Connecting these two statements with “and ”, “or”. In 60th

cases check the validity of the statement.



Chapter-14

STATISTICS

Two mark problems

1. Write the mean of the given data 6 , 7 , 10 , 12 , 13 , 4 , 6 , 12.

2. Two series A and B with equal means have standard deviations 9 and 10

respectively. Which series is more consistent?

3. The co-efficient of variation and standard deviation are 60 and 21

respectively. What is the arithmetic mean of the distribution.

4. The mean and variance of heights of XI students are 162.6cm and

127.69cm2 respectively. Find the C.V.

Five mark problems

1. The mean and standard deviation of 20 observations are found to be 10

and 2 respectively. On rechecking it was found that on observation 8

was incorrect. Calculate the correct mean and the standard deviation in

each of the following cases i) if wrong item is omitted ii)if it is replaced

by 12.

2. Find the mean deviation about the mean for the following data:

Marks obtained 0-10 10-20 20-30 30-40 40-50 50-60

Number of girls 6 8 14 16 4 2

3. Find the mean deviation about the median age for the age distribution of

100 persons given below

4. Find the mean deviation about the mean for the following data

Marks

Obtained

10-20

20-30

30-40

40-50

50-60

60-70

70-80

Number of

Students

2

3

8

14

8

3

2

5. The mean and standard deviation of 100 observations were evaluated as

40 and 5.1 respectively. By a student who took by mistake, 50 instead of

40 for one observation. What are correct mean and standard deviation?

Chapter-15

PROBABILITY

One mark problems

1. Define sure event.

2. A die is rolled. Describe the event “a number less than 7” occurs.

3. If for some non-empty sets A and B containing 3 elements

A×B = {(3, 4), (5,-3), (6, 1)} Find the set A.

Age 16-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55

Number 5 6 12 14 26 12 16 9


4. Write the sample space for the experiment

“ a coin is tossed repeatedly three times”.

5. If 11

2is the probability of an event A then what is the probability of the event “not

A‟?

Two mark problems

1. Given P(A) =5

3 and P (B) =

5

1 find P (A or B).

2. Two series A and B with equal means have standard deviations 9 and 10

respectively. Which series is more consistent?

3. A coin is tossed 3 times. Events A and B are,

A: No head appears

B: No tail appears. Show that A and B are mutually exclusive.

4. A die is thrown. Write the sample space. Also find the probability of the

event “A number greater than or equal to 3 will appear”.

5. One card is drawn from a well-shuffled deck of 52 cards. Calculate the

probability that the card will be “not an ace”.

6. A card is selected from a pack of 52 cards calculate the probability that the

card is i)an Ace ii)a black card.

Three mark problems

1. One card is drawn from a well shuffled deck of 52 cards. If each outcome

is equally likely, calculate the probability that the card will be

i) diamond ii)not an ace iii)a black card.

2. A fair coin 1 marked on one face and 6 on the other and a fair die are

both tossed. Find the probability that the sum of numbers that turn up is

i)3 ii)12.

3. A and B are events such that P(A)= 0.42, P(B)= 0.48 and P(A and B)= 0.16.

Determine (i) P(notA), (ii) P(notB) and (iii) P(Aor B).

4. A committee of two persons is selected from two men and two women.

What is the probability that the committee will have

(i) no man? (ii) two men?

5. Find the probability that when a hand of 7 cards is drawn from a well

shuffled deck of 52 cards, it contains (i) 3 kings (ii) At least 3 kings.

6. A letter is chosen at random from the ward „ASSASSINATION‟, Find the

probability that the letter is an vowel consonant.

7. In a survey of 600 students in a school, 150 students were found to be

taking tea and 250 taking coffee, 100 were taking both tea and coffee.

Find how many student were taking neither tea nor coffee.

8. A bag contains 9 discs of which 4 are red 3 are blue and 2 are yellow. The

discs are similar in shape and size. The disc is drawn at random from

the bag. Calculate the probability that will be

(i)red (ii) no blue (iii) either red or blue.

9. A and B are events such that P(A)=1/4, P(B)=1/2 and P(A and B)=1/8.

Find (i)P(A or B) (ii) P(not A and not B).


SAMPLE BLUEPRINT

I PUC: MATHEMATICS(35)

CONTENT

PART

A

PART

B

PART

C

PART

D

PART

E TOTAL

MARKS 1M 2M

3M

5M

6M

4M

1 SETS 1 2 1 8

2 RELATIONS AND

FUNCTIONS 1 1 1 1

11

3 TRIGONOMETRIC

FUNCTIONS 1 2 1 1 1

19

4 PRINCIPLE OF

MATHEMATICAL

INDUCTION

1

5

5 COMPLEX NUMBERS

AND QUADRATIC

EQUATIONS

1

1

2

9

6 LINEAR INEQUALITIES 1 1 7

7 PERMUTATION AND

COMBINATION 1

1 1

9

8 BINOMIAL THEOREM 1 1 8

9 SEQUENCE AND SERIES 1 2 1 11

10 STRAIGHT LINES 1 2 1 10

11 CONIC SECTIONS 1 1 9

12 INTRODUCTION TO 3D

GEOMETRY

1

1

7

13 LIMITS AND

DERIVATIVES 1 1 1 1

1 15

14 MATHEMATICAL

REASONING 1 1 1

6

15 STATISTICS 1 1 7

16 PROBABILITY 1 1 2 9

TOTAL 10 14 14 10 2 2 150


KARNATAKA PU BOARD MODEL PAPER-01 I PUC MATHEMATICS

PART-A

I Answer ALL the questions 10 ×1=10

1. Given that the number of subsets of a set A is 16. Find the number of elements in A. 2. Let � = {2, 3, 4} and be a relation on A defined as

},,|),{( ydividesxAyxyxR ∈= find .

3. If tan � = �� and x lies in the third quadrant, findsin �.

4. Find the modulus of ��.

5. Find ‘n’ if .67CCnn ====

6. Find the 20th term of the G.P., �� , �� , �� , …

7. Find the distance between the lines 3� + 4! + 5 = 0#$%6� + 8! + 2 = 0. 8. Given )�* = +

,|,| ,� ≠ 02 ,� = 0/ , find )(lim

0xf

x +→.

9. Write the negation of "For all a,b∈I, a–b∈I". 10. A letter is chosen at random from the word “ASSASINATION”. Find the

probability that letter is vowel.

PART – B

II Answer any TEN questions 10 ×2 = 20

11. If A and B are two disjoint sets and n(A) = 15 and n(B) = 10 find $)� ∪0*#$%$)� ∩ 0*. 12. If 2 = {�: � ≤ 10, � ∈ 5}, � = {� ∶ � ∈ 5, �isprime} and 0 = {� ∶ � ∈ 5, �iseven}

write � ∩ 0� in roster form, where A and B are subsets of U.

13. If <: = → = is a linear function, defined by < = {)1,1*, )0, −1*, )2,3*}, find <)�*. 14. The minute hand of a clock is 2.1 cm long. How far does its tip move in 20

minute? 22

use7

π =

.

15. Find the general solution of cos� � − 3 sin � = 0. 16. Evaluate: .

)65(

)3(lim

23 +−

−→ xx

x

x

17. Coefficient of variation of distribution is 30 and the variance is 144, what is

the arithmetic mean of the distribution? 18. Write the converse and contra positive of ‘If a parallelogram is a square,

then it is a rhombus’. 19. In a certain lottery 10,000 tickets are sold and 10 equal prizes are awarded.

What is the probability of not getting a prize if you buy one ticket? 20. In a triangle ABC with vertices A (2, 3), B (4, −1) and C (1, 2). Find the

length of altitude from the vertex A.

21. Represent the complex number B = 1 + C in polar form. 22. Obtain all pairs of consecutive odd natural numbers such that in each pair

both are more than 50 and their sum is less than 120.


23. A line cuts off equal intercepts on the coordinate axes. Find the angle made by the line with the positive x-axis.

24. If the origin is the centroid of the triangle PQR with vertices P (2a, 4, 6), Q (-4, 3b,-10) and R (8, 14,2c) then find the values of a, b, c.

PART – C

III Answer any TEN questions 10 ×3=30

25. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis? How many like tennis only and not cricket?

26. Let R : Z Z→ be a relation defined by ( ){ }ZbaZbabaR ∈−∈= ,,|, Show

that i) RaaZa ∈∈∀ ),(,

ii) RabRba ∈⇒∈ ),(),(

iii) RcaRcbRba ∈⇒∈∈ ),(),(,),( .

27. Prove that

+=−++

2cos4)sin(sin)cos(cos 222 yx

yxyx .

28. Solve the equation .012

2 =++x

x

29. In how many ways can the letters of the word PERMUTATIONS be arranged if the

i) Words start with P and end with S ii) Vowels are all together iii) There are always four letters between P and S?

30. If � + C! = �� then prove that �� + !� = 1.

31. Find the middle term in the expansion of

10

93

+ yx

.

32. Insert 3 arithmetic means between 8 and 24. 33. A committee of two persons is selected from 2 men and 2 women. What is

the probability that the committee will have (i) At least one man, (ii) at most one man.

34. Find the derivative of the function ‘tanx’ with respect to ‘x’ from first principle.

35. A parabola with vertex at origin has its focus at the centre of

.091022 ====++++−−−−++++ xyx Find its directrix and length of latus rectum.

36. In an A.P. if mth term is ‘n’ and the nth term is ‘m’, where nm ≠ , find the pth

term.


38. Two students Anil and Sunil appear in an examination. The probability that Anil will qualify in the examination is 0.05 and that Sunil will qualify is 0.10. The probability that both will qualify the examination is 0.02. Find the probability that Anil and Sunil will not qualify in the examination.


PART D Answer any SIX questions 6 ×5=30

39. Define greatest integer function. Draw the graph of greatest integer function, Write the domain and range of the function.

40. Prove that1lim −=

−−

→

nnn

naax

ax

ax, where ‘n’ is a rational number and ‘a’ is a

real number. Hence evaluate .11

lim0 x

x

x

−+→

41. Prove by mathematical induction that 13 + 23 +………+ n3 4

)1( 22 +=

nn ∀$ ∈ 5.

42. A group consists of 7 boys and 5 girls. Find the number of ways in which a team of 5 members can be selected so as to have at least one boy and one girl.

43. For all real numbers a, b and positive integer ‘n’ prove that,

( ) ....22

2

1

10

n

n

nnnnnnnnbCbaCbaCaCba ++++=+ −−

Hence Prove that .2...210

n

nCCCC =++++

44. Derive an expression for the coordinates of a point that divides the line joining the points A (x1, y1, z1) and B (x2, y2, z2) internally in the ratio m: n. Hence, find the coordinates of the midpoint of AB where

).7,6,5()3,2,1( BandA

45. Derive a formula for the angle between two lines with slopes m1 and m2. Hence find the slopes of the lines which make an angle π/4 with the line x-2y+5=0.

46. Prove that .tancos5cos

sin3sin25sinx

xx

xxx=

−+−

47. Solve the following system of inequalities graphically, 2� + ! ≥ 4,� + ! ≤ 3,2� − 3! ≤ 6. 48. Find the mean deviation about the mean for the following data

Marks obtained

10-20 20-30 30-40 40-50 50-60 60-70 70-80

No. of students 2 3 8 14 8 3 2

PART–E

Answer any ONE question 1 ×10=10

49. a) Prove geometrically that sin(A - B)=sin A cos B-cos A sin B.

Hence find sin 15°.

(b) Find the sum to n terms of 13 + 33 + 53 + . . . . 50. (a) Define ellipse as a set of points. Derive its equation in the form

.12

2

2

2

=+b

y

a

x

(b) Find the derivative of x

xx

sin

cos5 − using rules of differentiation.


Model paper-02 Subject: MATHEMATICS

Time : 3 hours 15 minute Max. Mark: 100 Instructions: i) The question paper has five parts namely A, B, C, D and E. Answer all the parts. ii) Use the graph sheet for the question on Linear inequalities in PART D.

PART-A Answer ALL the questions 10×1=10

1. Write the power set of the set A={1,2}. 2. If G= {7, 8} and H= {5, 4, 2}, find H×G.

3. Convert 3

5π radians into degree measure.

4. Find the multiplicative inverse of 2-3i.

5. Compute .!5

!7

6. Write the first three terms of the sequence #E = EE��.

7. Find the x-intercept of the line 3x+2y-12=0.

8. Evaluate .)(

)sin(lim

x

x

x −−

→ πππ

π

9. Write the negation of the statement “All triangles are not equilateral triangles”.

10. Define simple event.

PART - B Answer any TEN questions 10×2 = 20

11. Let U={1,2,3,4,5,6,7,8,9}, B={2,4,6,8} and C={3,4,5,6}, find (B-C)1.

12. If X and Y are two sets such that n(X)=17, n(Y)=23 and n(X∪Y)=38 find n(X∩Y).

13. Find the domain and range of the real function f)�* = √9 − ��. 14. Find the radius of the circle in which a central angle of 600 intercepts an

arc of length 37.4 cm IJKLM = ��N O.

15. Prove that sin� IPQO + cos� IP�O − tan� IP�O = − ��.

16. If � + C! = R��SR��T, prove that �� + !� = 1.

17. Solve 7x+3 ≤ 5x+9. Show the graph of the solutions on number line. 18. Find the distance of the point (-1,1) from the line 12(x+6)=5(y-2). 19. Find the value x for which the points (x,-1),(2,1) and (4,5) are collinear. 20. Find the equation of the set of points which are equidistant from the

points (1,2,3) and (3,2,-1).

21. Evaluatelim,→� I,VW��,VX��O. 22. Write the contra positive and converse of the statement “If x is a prime

number, then x is odd”. 23. Co-efficient of variation of a distribution is 60 and its standard deviation

is 21. What is the arithmetic mean? 24. One card is drawn from a well shuffled deck of 52 cards. If each outcome

is equally likely, calculate the probability that the card will be i) a diamond ii) not an ace.


PART - C Answer any TEN questions 10×3=30

25. In a survey of 400 students in a school, 100 were listed as taking apple juice, 150 as taking orange juice and 75 were listed as taking both apple and orange juices. Find how many students were taking neither apple juice nor orange juice.

26. Let A={1,2,3,4,6}. Let R be a relation on set A defined by R= {(a,b): a, b ϵ A, b is exactly divisible by a} i) Write R in roster form ii)

Find the domain of R iii) Find the range of R. 27. Find the general solution of the equation sin 2� − sin 4� + sin 6� = 0. 28. Express

��√��√�� in the form of a+ib.

29. Convert i++++3 in the polar form.

30. How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter repeated, if i) 4 letters are used at a time ii) All letters are used at a time iii) All letters are used but first letter is a vowel?

31. Find the term independent of x in the expansion of I�� − ��,O

Q.

32. If the sum of certain number of terms of the arithmetic progression 25,22,19… is 116. Find the last term.

33. The sum of the first three terms of a G.P is �� and their product is -1. Find

the common ratio and the terms. 34. Find the equation of a circle with centre (2,2) and passes through the

point (4,5). 35. Find the derivative of cos x from first principle.

36. Verify by the method of contradiction that 7 is irrational. 37. A committee of two persons is selected from two men and two women.

What is the probability that the committee will have i) no man ii) one man iii) two men?

38. E and F are events such that P(E)=1/4, P(F)=1/2 and P(E and F)=1/8. Find (i)P(E or F) (ii) P(not E and not F).

PART D Answer any SIX questions 6×5=30

39. Define Signum function. Draw the graph of it and write down its Domain and Range.

40. Prove that )YZ[ N,�YZ[�,*�)YZ[ \,�YZ[ �,*)]^Y N,�]^Y�,*�)]^Y \,�]^Y�,* = tan 6�.

41. Prove by mathematical induction that 12+22+32+……n2=

( )Nn

nnn∈∀

++,

6

12)1(

42. Solve the inequalities : 5x+4y< 40, x≥2 , y≥3 graphically. 43. A group consists of 4 girls and 7 boys, In how many ways can a team of 5

members be selected if the team has (i) No girl (ii) At least one boy and one girl (iii) At least three girls?

44. State and prove Binomial theorem for any positive integer.


45. Derive the equation of a straight line passing through the point (x0,y0) having the slope m. hence deduce the equation of line passing through ther points (-4,3) with slope ½.

46. Derive an expression for the coordinates of a point that divides the line joining the points A(x1,y1,z1) and B(x2,y2,z2) internally in the ratio m:n.

47. Prove that 1sin

lim0

=→ θ

θx

, where x is in radian and hence evaluate 1tan

lim0

=→ θ

θx

.

48. Find the mean deviation about the mean for the following data.

xi 2 5 6 8 10 12

fi 2 8 10 7 8 5

PART-E

Answer any ONE question 1×10=10 49. a) Prove geometrically that cos(x+y)=cosx cosy-sinx siny and hence

show that cos2x=cos2x-sin2x. 6 b) find the sum to n-terms of the series 12+(12+22)+( 12+22+32)+… 4

50. a) Define hyperbola as a set of all points in the plane and derive

its equation as 12

2

2

2

=−b

y

a

x. 6

b) Find the derivative of YZ[,�]^Y,YZ[,�]^Y,. 4

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