Bhavitha 24.02.2011 EM Maths

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WWW.SAKSHI.COM/VIDYA/BHAVITHA çÜμÆý‡®Ä¶æ* Ð]lÆý‡®™ól ѧýlÅ {糆 VýS$Æý‡$-Ðé-Æý‡… Ýë„ìS-™ø E_-™èl… 24-&2&2011 Connectives: 1. and 2. or 3. if... then 4. if and only if Compound Statements: 1.Disjunction 2. Conjunction 3. Conditional 4. Biconditional Function: f:AB, 1. if for every aA there is b B such that (a/b)f One-one function (injection): f(x 1 ) = f(x 2 ) x 1 = x 2 Arithmetic progression(A.P) difference (d) is equal General form of A.P a, a + d, a + 2d, .... Modulus of 'x', |x| |x| = x if x > 0 or - x if x < 0 or 0 if x = 0 |x| = a solution: x=a or x= - a Quadratic equation ax 2 + bx+c=0 Discriminent Δ = b 2 –4ac Δ > 0 Roots are real, unequal Sexagesimal system Degree Centesimal system grade Circular measure Radian Convex Set: X is convex if the line segment joining any two points P, Q in x is contained in x Linear programming problem L.P.P consists of Minimising/maximising a function f = ax+by, a, bR subject to certain constraints Circum center Concurrence point of perpendicular bisector of the sides of the Triangle. In center concurrence point of angle bisector of the Triangle. Equation of X-axis y = K Equation of Y-axis x = K Slope of X-axis 0 10 th MATHEMATICS BITBANK SPECIAL Practice Bits Important Questions Question Trends Preparation Tips Quick Review 10 th MATHEMATICS BITBANK SPECIAL QUICK RE'VIEW'

Transcript of Bhavitha 24.02.2011 EM Maths

Page 1: Bhavitha 24.02.2011 EM Maths

&&&&&&&&&……………………………………

W W W . S A K S H I . C O M / V I D Y A / B H A V I T H A

çÜμÆý‡®Ä¶æ* Ð]lÆý‡®™ól ѧýlÅ

{糆 VýS$Æý‡$-Ðé-Æý‡… Ýë„ìS-™ø E_-™èl… 24--&2&2011

☞ Connectives: 1. and 2. or 3. if... then 4. if and only if

☞ Compound Statements: 1.Disjunction 2. Conjunction 3. Conditional4. Biconditional

☞ Function: f:A→B, 1. if for every a∈A there is b ∈ B such that (a/b)f

☞ One-one function (injection): f(x1) = f(x2) ⇒ x1 = x2

☞ Arithmetic progression(A.P) difference (d) is equal

☞ General form of A.P a, a + d, a + 2d, ....

☞ Modulus of 'x', |x| |x| = x if x > 0 or - xif x < 0 or 0 if x = 0

☞ |x| = a solution: x=a or x= - a

☞ Quadratic equation ax2 + bx+c=0

☞ Discriminent Δ = b2–4ac

☞ ΔΔ > 0 Roots are real, unequal

☞ Sexagesimal system Degree

☞ Centesimal system grade

☞ Circular measure Radian

☞ Convex Set: X is convex if the line segment joining any two points P, Q in x is contained in x

☞ Linear programming problem L.P.P consists of Minimising/maximising a function f = ax+by, a, b∈R subject to certain constraints

☞ Circum center Concurrence point of perpendicular bisector of the sides of the Triangle.

☞ In center concurrence point of angle bisector of the Triangle.

☞ Equation of X-axis y = K

☞ Equation of Y-axis x = K

☞ Slope of X-axis 0

10th

MMAATTHHEEMMAATTIICCSS

BITBANK SPECIAL

● Practice Bits

● Important Questions

● Question Trends

● Preparation Tips

● Quick Review

10th

MMAATTHHEEMMAATTIICCSS

BITBANK SPECIAL

QUICK RE'VIEW'

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p or not p is example forMATHEMATICS BIT BANK STATEMENTS, SETS

STATEMENTS

1. The terms which connect two statements are

called ________

2. If the switch ‘P’ is ‘OFF’ we represent it by

________

3. The complement law using ‘∧’ is ________

4. The truth value of (3≠2) ∨ (2=3) is _______

5. The statement of the form “ If...... then.......”

is called an ________

6. A combination of one or more simple state-

ments with a connective is called a _______

7. The symbol for existential quantifier

________ (June 2009), (June 2008)

8. ~(p ⇔ q) = ________

9. The contrapositive of “If a polygen is a

square then it is a rectangle” is _________

10. p, q, r are threee statements then p ∧ (q ∨ r)

= (p ∧ q) ∨ (q ∧ r) is ________ law

11. “For all” or “For every” is called ________

quantifier.

12. If p and q are switches. The combination of

p ∨ q is called _____________________

13. p and q are two statements. The symbolic

form of “Converse of a conditional is equiv-

alent to its inverse” is ____________

14. The statement which uses the connective

“OR” is called a __________________

15. The truth value of (4 × 7 = 20) ⇔ (4÷7=1)

is ___________________

16. P is the statement then ~(~(~p)) is _______

17. The symbolic form of “If x is not odd then

x2 is odd” ___________________

18. p: It is raining, q: The sun is shining .

Connect p,q using conjuction is ________

19. Denial of a statement is called its ________

20. p and q are two statements then example for

tautology is ________________

21. p∧(~p) is very simple example of a ______

(June 2009)

22. ~(p∨q) ≡ __________ (June 2009)

23. P∨p = p. This is ________ law. (June 2010)

24. The symbol of Universal Quantifier is

________ (March 2009)

25. ~(p∨q) ≡ (~p) ∧ (~q) is _________ law.

(June 2008)

26. p∨(q∧r) ≡ (p∨r) ∧ (p∨r) is ______ law.

(March 2008)

27. The truth value of implication statement :

If 3 ÷ 2 = 5 then 1 × 0 = 0 is _________

(March 2008)

28. The last column of truth table contains only

F it is called _________________

29. p or not p is example for ___________

30. The inverse of “~p ⇒ ~q” is __________

SETS

1. If A and B are disjoint sets, then n(A ∪ B) =

____________ (June 2009)

2. If A⊂B then A∩B = _______

(June 2009)

3. The complement of μ is ____________

(March 2009)

4. n(φ) = _________ (March 2009)

5. If A ⊂ B then A ∪ B = ______

(June 2008)

6. If A⊆B and B⊆A then ______

(June 2008)

7. A ∪ A' = ________ (June 2008)

8. If A⊂B and n(A) = 5, n(B) = 6 then n(A∪B)

= _______ (March 2008)

9. The set builder form of B = {1,8,27,64,125}

is ________ (March 2008)

10. (A ∪ B)' = ________ (March 2010)

11. If A = {3,4}, B = {4,5} then n (A×B) =

_________

12. (A ∩ B) ∪ (A∩C) = _________

13. If A sand B are two sets then A Δ B = _____

14. If A⊂B, n(A) = 10 and n(B) = 15

then n(A-B) = _________

15. If A∩B = φ, n(A∪B) = 12 then n(AΔB) =

__________

16. If A, B, C are three sets A–(B∪C) = ______

17. n(A∪B) = 8, n(A∩B) = 2, n(B) = 3 then

n(A) = _________

18. If A = {x; x ≤ 5, x ∈ N}, B = {2,3,6,8} then

A∩B = ________

19. If A, B are disjoint sets n(A) = 4, n(A∪B) =

12 then n(B) = _________

20. (A ∪ B)' = A' ∩ B' is ________ law.

21. A, B are two sets then x ∉ (A – B) = _____

22. A ⊂ B and n(A) = 5, n(B) = 6 then n(A∪B)

= _______

23. The sets which are having same cardnial

numbers are called __________

24. If A has ‘n’ elements then the number of ele-

ments in proper sub set is ________

25. If A and B are disjoint sets then n(A∩B) =

______________

26. If n(A) = 7, n(B) = 5 then the maximum

number of elements in A∩B is _________

27. If A∩B = φ then B∩A = _____________

28. If any law of quality of sets, if we inter-

change ∩ and ∪ and μ and φ the resulting

law also true, this is known as ___________

29. A – B' = _________

30. A, B are subsets of μ then A ∩ B' = _____

FUNCTIONS

1. If f(A) = B then f : A→B is a/an _________

function (June 2009)

2. Let f : R→R be defined by f(x) = 3x+2, then

the element of the domain of ‘f ’which has

11 as image is _______________

3. Range of a constant function is a _____ set.

4. If f : N→N is defined by f(x) = x+1, then the

range of ‘f ’ is __________ (June 2009)

5. If f(x) = x∀x, then f is a/an ___________

function (June 2009), ( March 2008)

6. If f(x) = x2 – x + 6 then f(4) = ___________

(March 2008)

7. f(x) = x2 + 4x – 12,what are the zeros of f(x)

__________ (March 2008)

8. f(x) = x3, g(x) = x2–2 for x∈R then (gof)(x)

= ________ (March 2008)

9. f(x) = x2 + 2x – K and if f(2) = 8 then k

=_________ (June 2007)

10. f : A→B is an objective and if n(A) = 4 then

n (B) = _________ (June 2007)

11. If f(x) = x then the function f is _________

(June 2010)

12. A function is one - one and on-to then the

function is _________ (June 2010)

13. If f = {(1,2),(2,3),(3,1)} then f–1(2) = _____

14. If f is Identity function f(5) = ___________

15. If f(x1) = f(x2) ⇔ x1 = x2 then f is ________

function.

16. f : A→B and f (x) = c∀x∈A then f is ______

17. If f : A→B such that f (A) ⊂ B then f is

_________

18. f = {(1,2), (2,3), (3,4)}, g = {(2,5), (3,6),

(4,7)} then fog = __________

19. The domain of the function is

_______

20. f : A→B and f(x) = 2x +5 then the inverse

of f is ___________

21. If f(x) = then _______

22. The range of constant function is ________

23. If f = {(1,2), (2,3), (3,4), (4,1)} then fof =

_________

24. If f(x) = ax + b and f (2) = 6 then the rela-

tion between a and b is _______

25. f(x) = x + 2 and g (x) = 2x–1 then

f (1) -g(-1) = ________

[ ]fo(fof ) (x) =x

2

1

x 16−

K.Umamaheswara ReddySr. TeacherBeechupally

BITBANK Written by

KEY

1. Connectivities 2. P1 3. (p ∧ (~p)) ≡ f 4. True

5. conditional (or) implication 6. Compound

statement 7. ∃; 8. ~p ⇔ q (or) p ⇔ ~q 9. If a

polygon is not a rectangle then it is not a

square. 10. Distributive law. 11. Universal 12.

Parallel combination 13. (q ⇒ p) ≡ ~(p ⇒ q)

14. Disjuction 15. True 16. ~p 17. “x is not

odd ⇒ x2 is odd” 18. p ∧ q 19. Negation 20.

p∨(~q) 21. contradiction 22. ~p∧~q 23. idem-

potent law 24. ∀ 25. De morgan’s law 26. dis-

tributive law 27. True 28. contradiction 29.

Tautology 30. p ⇒ q

4 Marks1. Using element wise prove that A – (B ∩ C)

= (A – B) ∪ (A – C)

2. Prove that A ∪ (B ∩ C) = (A∪B)∩ (A ∪C)

3. Let A,B are two subsets of a Universal set

μ show that A ∩ B = A – B1 = B – A1

4. Prove that (A ∧ B)1 = A1 ∪ B1

2 Marks1. Define implication and write truth table?

2. Write the truth table (~P) ∨ (P ∧ q).

3. Write the converse, inverse and contrapa-

sitive of the conditional “If in a triangle

ABC, AB > AC then ∠C > ∠B.

4. If A ∩ B = φ then show that B ∩ A1 = B

5. Using element wise proof show that A – B

= A∩B1

6. If A,B are any two sets, prove that A1-B1 =

B–A

7. Show that A ∪ B = φ, implies A = φ and B

= φ.

1 Mark1. Define Tautology and contradiction?

2. Write Truth table for conjunction?

3. Prove that (A1)1 = A

4. Write contrapasitive of a conditional ‘If

two triangles are congruent then they are

similar”.

5. Show that P ∧ (~P) is contradiction.

6. If A = {1,2,3}, B={2,3,4} then find AΔB.

7. Write set-builder form of

8. Prove that A∧ B ⊂ A for any two sets A, B.

9. Prove that ~(~P) = P

1 1 1 1 1A 1, , , , ,

2 3 4 5 6⎧ ⎫= ⎨ ⎬⎩ ⎭

STATEMENTS AND SETS: Important Questions

KEY

1. n(A)+n(B) 2. A 3. φ 4. 0 5. B 6. A = B 7. μ8. 6 9. {x/x = n3, n ∈ N, n ≤ 5} 10. A' ∩ B' 11.

4 12. A ∩ (B ∪ C) 13. (A∪B)-(A∩B) (or) (A-

B) ∪ (B-A) 14. 0 15. 12 16. (A-B)∩(A-C) 17.

7 18. {2,3} 19. 8 20. De Morgan’s law 21. x

∉ A and x ∈ B 22. 6 23. equivalent sets 24. 2n-

2 25. 0 26. 5 27. B 28. Principle of duality 29.

A∩B 30. A-B

°°p

qA B

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f : A→→B and B ⊆⊆ R then f is.. MATHEMATICS BIT BANK

POLYNOMIALS OVERINTEGERS

26. If a function is both one-one and on-to then

the function is _________

27. f : A→B is a function then B is called _____

28. f : A→B such that f (A) = B then f is ______

29. f : A→B and B ⊆ R then f is ___________

30. A constant function f : N→N is defined by

f (x) = 5 then f (15) = _______

31. _____

32. The range of the function f = {(a,x), (b,y),

(c,z)} is ________

33. The inverse of a function will be a function

again if it is _________

34. If f : x → log2 x then f (16) = __________

35. The set builder form of

R = {(1,3), (2,4), (3,5)} is ________

36. f–1 (x) = x–3, g–1 (x) = x–1 then (fog)–1 =

_________

37. What is the zeros of the adjacent function is_________

38. Number of elements in {3,5,7,9} × {4,6,8}

is __________

39. A function f : A→B is said to be ________

function, if for all y ∈ B there exists x ∈ A

such that f (x) = y.

40. If f(x) = 2–x, g(x) = 3x + 2 then (fog) (2) =

________

41. f(x) = x+1, then 3f(2)–2f(3) = ________

42. f = {(x,1004)/x ∈ N} then f is ________

43. The condition to define gof is ________

44. Let f : R→R, f(x) = 6x+5 then f–1(x) = ____

45. If f(x) = 2x – 3 the value of

is ________

POLYNOMIALS OVER INTEGERS

1. Product of the roots of equation

x2–(a+b)x = c is ________

2. If α, β are the roots of the equation

2x2–9x+8 = 0 then α + β = ________

3. The line y = mx+c cuts the y-axis at ______

4. The curve x = my2(m>0) lies in ________

quardrants (March 2008, Jun 2010)

5. If the co-efficient of x2 in the expansion of

(1+x)n is 28 then n = ________

6. If f(x) = a0x+a1x+a2x2+a3x

3+_____. If a0 +

a2 + a4 + ----- = a1 + a3 + a5 + ----- then the

factor of f(x) is ________

7. If 15Cr–1 = 15Cr+2 then r = ________

8. If |3x – 2|=10 then the positive value of

‘x’ is ________

9. a2 + bx + c = 0 is quadratic equation if

b2 – 4ac < 0 then the roots are ________

10. The no.of terms of the expansion (1+x)n+1 is

6 then n = ________

11. The sum of the roots of 2x2 – Kx + 4 = 0 is

-1 then K = ________

12. (x-1) is a factor of 2x3 – 5x2 + Kx + 7 then

K = ________

13. The last term in the expansion of

is ________

14. The quadratic equation in ‘x’ where roots

are 2, –3 is ________

15. If x2 – 3x + 2 > 0 then x is ________

16. The solution set which satisfies the inequa-

tion x2 – 4x + 3 < 0 is ________

17. The ineaquation with solution set 1 < x < 3

is ________(June 2008)

18. Product of the roots of 2x2 + 3x – 2 = 0 is

________

19. The condition for xy + yn is exactly divisible

by (x+y) then n = ________

20. If (2,K) lies on y = 2x2 - 3 then K = ______

21. The two factors of x3 + 3x2 – x – 3 are

(x – 1) (x+1) then the other factor is ______

22. The rationalising factor of a1/3-b1/3 is _____

23. Sum of the binomial co-efficients of the

expansion (x + y)4 is ________

24. If (x – y) is a factor of xn – yn then n is

________ (June 2007)

25. Y= mx2(m>0) is symmetric about ___axis.

26. The roots of 2x2 + Kx + 2 = 0 are equal then

K = ________

27. The standard form of second degree homog-

enous equation in two variables x and y is

________

28. x3 – 2x2 + 4x – 5 is divided by x–2 then the

remainder is ________

29. If f(x) is divided by ax+b then the remain-

der is ________ (March 2010)

30. Second term in the expansion of

is ________

31. If the roots of the equation Px2 + qx + r = 0

equal then the condition is ________

32. To solve graphically the roots of x2 + 2x –

15 = 0 we draw y = x2 and ________

33. The other name of pascal triangle is ______

34. If (x + y, 1) = (3, y – x) then (x,y) = ______

35. The descrimenent of 4x2 – 5x + 4 = 0 is

________

36. If then factor of f(x) = ________

37. The sum of the co-efficients of the quadrat-

ic expression is zero then ________ is a fac-

tor to it (June 2010)

38. The graph of y = x2 is a ________

39. If 2 is a root of the equation x2 – px + q = 0

and p2 = 4q then the other root is _______

40. The roots of ax2 + bx + c = 0 are ________

41. If x3 – 3x2 + 4x – 2 is divided by x–1, then

the quotient is ________ (June 2009)

42. The nature of the roots of 4x2 – 5x + 4 = 0

is ________

43. The product of the roots of

is ________44. (March 2009)

45. (–2, 3) ∈ ________ quadrant (March 2009)

46. The sum of the roots of x2 - 3x + 7 = 0 is

________

47. The discriminant of the quadratic equation

2x2 – 7x + 3 = 0 is ________ (June 2008)

48. If then x = ________June 2008)

49. The product of the roots of px2 + qx + r = 0

________ (June 2008)

x 1 x+ =

C0n __________=

23x 9x 6 3 0+ + =

bf 0

a⎛ ⎞ =⎜ ⎟⎝ ⎠

41

xx

⎛ ⎞−⎜ ⎟⎝ ⎠

81

1x

⎛ ⎞−⎜ ⎟⎝ ⎠

f (x h) f (x)

h

+ −

x 1 1f (x) (x 1) then f (x) f

x 1 x

− ⎛ ⎞= ≠ + ⎜ ⎟− ⎝ ⎠

KEY

1. Onto 2. 3 3. Singleton set 4. {2,3,4,5-----}

5. Identity 6. 18 7. -6 (or) 2 8. (x6-2) 9. 0 10. 4

11. Identity function 12. bijective 13. 1 14. 5

15. one-one 16. constant function 17. Into

function 18. does not find 19. x > 4

20. 21. 22. Singleton set 23.

{(1,3) (2,4)(3,1) (4,2)} 24. 2a+b = 6 25. 6 26.

bijective 27. co-domain 28. onto function 29.

real valued 30. 5 31. 0 32. {x,y,z} 33. bijective

34. 4 35. {(x,y)/ y=x+2, x ∈ N, x ≤ 3} 36. (x-

4) 37. {-3,-1,1,3} 38. 12 39. Onto 40. -6 41. 1

42. Constant function 43. The range of f is

equal to the domain of g. 44. 45. 2x 5

6

1/88 x or xx 5

2

4 Marks1. Let f : R → R be defined by f (x) = 2x + 3.

find f–1(4),

2. Let f,g,h be functions , f(x) = x+2, g (x) =

3x-1 and h(x)=2x show that

ho(gof)=(hog)of ?

3. If a function f : R → R is defined by f(x) =

3x-5, then find a formula that defines the

inverse function f–1?

4. Let f be given by f(x) = x+2 and f has the

domain {x : 2 ≤ x ≤ 5} find f-1and its domain

and Range?

2 Marks1. Let f : R -{2} → R be defined by

show that ?

2. Define one-one function show that f(x) = 3x

– 2; x ∈ N is one -to-one.?

3. If f(x) = x2 + 2x + 3, x ∈ R find the volue of

when h ≠ 0.?

4. f : R → R be defined by f(x) = 6x + 5, find

f–1 (x).?

5. f(x) = x + 2, g(x) = x2 – 3 find

1) (gof) (-2) 2) (fog) (-2).?

1 Mark1. Define on-to function?

2. Let f : A → B and let f have an inverse func-

tion f–1 : B → A. state the properties of f for

which its inverse exists.

3. Define equal functions?

4. Let f = {(1,2), (2,3), (3,4)} and g = {(2, 5),

(3, 6), (4, 7)} find gof?

5. Define a bijection?

6. Let f : R-{1} → R be defined by f(x) = 1 +

2x, g(x) = 3 – 2x, find (fog) (3)?

( ) ( )f x h f x

h

+ −

2x 1f x.

x 2

+⎛ ⎞ =⎜ ⎟−⎝ ⎠

2x 1f (x)

x 2

+=−

{ }{ }1 1f (x) : 2 x 3 , f (x) : x 5 .− −≤ ≤ ≤

FUNCTIONS: Important Questions

5 Marks1. Using graph of y = x2, solve x2 – 4x+3 = 0

2. Draw the graph of y = x2 + 5x + 6 and find

the solution of x2 + 5x + 6 = 0?

4 Marks1. If ax2 + bx + c is exactly divisible by (x-1),

(x-2) and leaves remainder 6 when divided

by (x+1). find a,b and c?

2. Resolve in to factors of the polynomial

3x4 – 10x3 + 5x2 + 10x – 8?

3. Find the independent term of ‘x’ in the

expansion of?

4. Find a quadratic function is in ‘x’ such that

when it is divided by (x-1),(x-2) and (x-3)

leaves remainders 1,2 and 4 respectively.

2 Marks1. Find the value of ‘m’ in order that x4 –2x3 +

3x2–mx+5 may be exactly divisible by(x-3)?

2. Find the roots of x2+x (c-b)+(c-a) (a-b) = 0.

3. Find the middle term of the expansion of

?

4. Solve the inequation x2 – 6x + 8 > 0?

5. The difference of two numbers is 5 and their

product is 84 find them?

6. Find the 5th term in the expansion

1 Mark1. Define mathematical induction?

2. Comment up on the roots of a quadratic

equation 3x2 – 7x + 2 = 0 ?

3. Find the quadratic equation having roots

?

4. Find the value of K so that x3 – 3x2 + 4x +

K is exactly divisible by x-2?

5. Find the sum and product of the roots of

the equation ?

6. Define Remainder theorem?

7. The product of two consecutive numbers

is 72. Find the number?

8. Write factor theorem?

9. Expand ?

10. Write General term of expression (x+y)n?

( )a a b c+ −∑

23x 9x 6 3 0+ + =

1 2 and 1 2+ −

81

2x3y

⎛ ⎞+⎜ ⎟

⎝ ⎠

71

3x2x

⎛ ⎞−⎜ ⎟⎝ ⎠

82

2

56x

x

⎛ ⎞−⎜ ⎟⎝ ⎠

POLYNOMIALS: Important Questions

(1,0) x

y

(–1,0)(–3,0) (3,0)

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4

The point on Y axis is... MATHEMATICS BIT BANK LINEAR PROGRAMMING

50. Middle term in the expansion of

is ________ ( March 2008)

51. If (a + b, 1) = (5, a – b) then 2a + 3b =

________ ( March 2006)

52. (x+1) is a factor to ax4 + bx3 + cx2 + dx + e

then the condition is ________

53. If |x| ≤ a then the solution set is ________

54. The middle term of expansion is

________ (March 2010)

55. Sum of the number and its reciprocal is 17/4

then the number is ________

56. Expand

LINEAR PROGRAMMING

1. Any line belonging to the system of parallel

lines of objective function is called _______

2. If none of the feasible solutions maximise

or minimise the objective function, then the

problem has ________

3. A line divides the plane in to ________ sets.

4. If x>0, y<0 then the point (x,y) lies in

________ quadrant (June 2008)

5. If then the value of P at the

point (4,9) is ________

6. The parallel lines that are determined by the

objective function are called ________

7. If the isoprofit line moves away from the

origin then the value of the objective func-

tion ‘f’ is ________

8. Polygon represented by the inequalities

x ≥ 1, y ≥ 1, x ≤ 3, y ≤ 3 is ________

9. The solution set of constraints of linear pro-

gramming is called ________

10. A line segment joining the points P and Q,

where P, Q ∈ x such that then X is

called ________ set

11. If the values of the expression f = ax+by is

attained maximum or minimum at one of

the vertices that is called _______

12. The expression ax+by which is to mini-

mized or maximized is called ________

13. The isoprofit line coincides with the sides of

polygon then it has ________ solution.

14. Any point (x,y) in the feasible region gives

a solution to LPP is called ________

15. If the point (-3,2) lies on 3x-5y+k<0 then

the maximum value of K is ________

16. f = A X + BY is called ________

17. The shaded region represents the inequation

is ________

18. The solution set of x ≥ y and x ≤ y is

________

19. The point on Y axis is ________

20. The value of x+y should not be less than8’ can be written as ________

21. The slope of Y-axis is ________22. Isoprofit lines are ________23. The profit of a chair is Rs 10 and table is

Rs 25. A man purchased x chairs and ytables. Then the total profit is ________

24. The c>0 then ax+by+c <0 represents theregion ________

25. If a<0 then the point (4,-a) lies in________ quadrant.

26. The knowledge of Linear Programming

helps to solve the problem in ________

27. The maximum (or minimum) value of foccurs on atleast one of the vertices of thefeasible region. This is the statement ofthe ________ theorem of LinearProgramming.

28. If Q1 and Q2 are first and second quad-rants then Q1 ∩ Q2 ________

29. If f = ax+by is objective function, thenthe line ax+by = c is called ________ line

30. If x = 0 then (x,y) is a point on ________axis.

31. The value of f = 2x+3y at (1,2) is________

32. Intersection of x ≥ 0, y ≥ 0 is ________33. The value of an objective function

at (0,9) is ________

REAL NUMBERS

1. If 2x+3 = 8x+3 then x = ______ (March 2009)

2. (16)1.25 =________ (March 2009)

3. ________ (March 2009)

4. (16)0.5 = ________ (March 2008)

5. (64)x = then x = _______ (March 08)

6. The limiting position of secant of a circle is

________ (March 2008)

7. If then a = ______ (March 2008)

8. If ax = b; by = c, cz = a then the value of xyz

= ________ (March 2008), (March 2009)

9. (March 2009)

10. If (x2/3)p=x2then the value of p is ____

11. (June 2009, 10)

12. a≠0 and if p + q + r = 0 then a3p + 3q + 3r =

________ (June 2007)

13. If (March'10)

14. If x = –3 then |x2 – 20| = ______ (March'10)

15. (June 2010)

16. If then 3x = ________

17. If then x = ________

18. If then a = ________

19. The rationalising factor of is _____

20.

21.

22.

23.

24.

25. If 35x + 2 = (27)4 then x = ________

1 1 1 1 1 14 4 2 4 4 2x y x x y y ________

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟+ − + =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

3

2x 2

x 2x 2Lt ________

2x 3x 5→−

− + =+ +

a b b a

1 1________

1 x 1 x− −+ =+ +

2

x 0

x 5xLt _______

x→

+ =

3 3 3 3

4n

1 2 3 nLt ______

n→∞

+ + + − − − + =

1 13 3a b+

(a a ) aa (a a )=

32x 0.027=

xy

1(64)

(256)=

3 i

i 04 ________

==∑

1 1x 4 then x _______

x x+ = − =

x

2x 3Lt _________

3x 5→∞

+ =+

3 2

n 1(n 1) _______

=+ =∑

2a 3x x=

2 2

x

1Lt

x→∞=

1 2f x y

3 3= +

PQ X⊆

1 2P x y

4 3= +

2a (b c) ______________− =∑

4x y

y x

⎛ ⎞+⎜ ⎟

⎝ ⎠

8x y

y x

⎛ ⎞+⎜ ⎟

⎝ ⎠

KEY

1. -C 2. 9/2 3. (0,c) 4. I & III quadrants 5. 8 6.

(x+1) 7. 7 8. 4 9. not real & complex 10. 4 11.

-2 12. -4 13. 1/x414. x2+x-6 = 0 15. doesnot

lies between 1 and 2 16. 1<x<3 17. x2-

4x+3<0 18. -1 19. any odd natural number 20.

5 21. (x+3) 22. (a2/3 + a1/3b1/3 + b2/3) 23. 16 24.

any natural number 25. positive y-axis 26. ±4

27. ax2+2hxy+by2 28. 3

29. 30. -4x2 31. q2 = 4pr 32. y = -

2x+15 33. Arithmetic triangle 34. (1,2) 35. -39

36. ax-b 37. (x-1) 38. parabola 39. 2

40. 41. x2-2x+2 42. not real

and complex43. 6 44. 1 45. II 46. 3 47. 25 48.

49. r/p 50. 5th term (70) 51. 12 52. a+c+e =

b+d 53. –a ≤ x ≤ a 54. 3rd term (6) 55. 4

56. a2(b–c) + b2(c–a) + c2(a–b)

1 5

2

±

2b b 4ac

2a

− ± −

bf

a

−⎛ ⎞⎜ ⎟⎝ ⎠

5 Marks1. Maximize f = 5x+7y, subject to the condi-

tions 2x + 3y ≤ 12, 3x + y ≤ 12, x ≥ 0, y ≥0?

2. Minimise f = x+y, subject to the conditions

x + y ≥ 6 and 2x + y ≥ 8, c ≥ 0, y ≥ 0?

3. Maximize f = 3x + y, subject to the con-

straints 8x + 5y ≤ 40, 4x + 3y ≥12, x ≥ 0,y

≥ 0?

4 Marks1. A sweet shop makes gift packets of sweets,

combines two special types of sweets A

and B which weight 7kg, atleast 3kg of A

and no more that 5kg of B should be used.

The shop makes a profit of 15 on A and 20

on B per kg, Determine the product mix so

as to obtain maximum profit. (graph not

necessary)?

2. A shop keeper sells not more than 30 shirts

of each colour. Atleast twice as many

white ones are sold as green ones. If the

profit on each of the white be Rs. 20 and

that of green be Rs. 25. How many of each

kind be sold to give him a maximum prof-

it?

2 Marks1. State the polygonal region represented by

the systems of inequations x ≥ 0; y ≥ 0; x

+ y ≤ 1.

2. Define convex set and iso-profit Line.

3. Draw the graph of the following inequa-

tion 2x + 3y ≤ 6.

4. Shade the region represented by the

inequation 4x + 3y ≥ 12.

5. Define objective function and Feasible

region.

6. From the given vertices (0,0),(2,3),(3,0)

and (0,5) at which point the objective

function 2x+3y will have maximum value.

1 Mark1. Define L.P.P.?

2. Define isoprofit line?

3. Define convex set?

4. What is an objective function?

5. Define feasible region?

6. At which of the point A (2,4),B (0,8) the

function f(x) = 4x-y is minimum?

7. What Kind of polygonal region do you get

from the system of inequations x ≥ 0, y ≥0, x + y ≤ 8? Write the vertices?

LINEAR PROGRAMMING : Important Questions

KEY

1. Isoprofit lines 2. no solution 3. three dis-

joint sets 4. IV (or) Q4 5. 7 6. Isoprofit lines 7.

Maximum 8. Square 9. Feasible region 10.

Convex set 11. Fundamental theorem of L.PP

12. Objective function 13. Infinite 14. feasible

solution 15. K = 18 16. objective function 17.

2x+3y≤6 18. x = y 19. (0,a) 20. x+y≤8 21.

undefined 22. Parallel lines 23. 10x+25y 24.

closed half plane containing zero 25. Q1 26.

Business, Industry and Transportation 27.

Fundamental 28. φ (or) positive y-axis 29. iso-

profit line 30. y-axis 31. 8 32. Q1(or) I quad-

rant. 33. 6

y

A(0,2)

B(3,0)x

Page 5: Bhavitha 24.02.2011 EM Maths

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5

If x,y,z are in A.P then 2y =.... MATHEMATICS BIT BANK PROGRESSIONS

26. If then (x + y + z)3 =

________

27. The limit of the sum is

________

28. The value of

29.

30.

31. If x = –8 then |x – 1| = ________

32.

33. If |2x – 3| = 7 then x = ________34. The solution of |2x – 3| ≤ 7 is ________

35.

36. If ax–1 = bc, by–1 = ac, ez–1 = ab then xy + yz + zx = ________

37. If then x = ________

38. If Σn = 10 then Σn3 = ________

39.

40. Σn3 = ________41. If Σn = 66 then n = ________42. a2/3[a1/3(a1/4)4]= ________

43. If then x = ________

44.

45. If then x2 = ________

PROGRESSIONS

1. If A,G,H denote the A.M. G.M. and H.M of

two positive numbers then their descending

order is ________

2. If there are ‘n’ Arithmetic means between

‘a’ and ‘b’ then the common difference d =

______

3. Sum to infinity terms of the G.P.

is ________ (June 2009)

4. 13+23+33+--------------+103 = ________

5. If there are ‘n’ G.M’s inserted between a and

b then the common ratio ‘r’ is ________

6. The nth term of the series 1.2+2.3+3.4+-----

--- is ________ ( March 2008)

7. If a,b,c are in G.P then a/b = ________

8. If then t2008 = ________

9. The G.M of 3 and 27 is ________

10. If 3,4,6 are in H.P then the fourth term is

________ (March 2008)

11. If the sum of first ‘n’ natural numbers is 66

then ‘n’ = ________

12. 1+2+3+--------+ 100 = ________

13. If x,y,z are in A.P then 2y = ________

14. The sum of ‘n’ terms of the series (a+1) +

(a+2) + (a+3) +------is _______

(June 2010)

15. nth term of A.P. is (2n2+2n+3) then the sec-

ond term is ________ (March 2010)

16. The arithmetic mean of (a-b)2 and (a+b)2 =

________17. In a H.P.

is

________ term.

18. If a1, a2, a3, -------- and b1, b2, b3, .......... are

in A.P then a1 – b1, a2 – b2, a3 – b3 are in

________ progression.

19. Sum of the first ‘n’ odd natural numbers is

________

20. The number of multiples of 9 between 1 and

1000 is ________

21. If are in H.P then c = ________

22. The 10th term of the series

is ________

23. K+2, 4K-6 and 3K-2 are in A.P. then K =

________

24. The nth term of A.P is 3n+1 and the sum of

‘n’ terms is ________

25. If are in A.P. then x = _______

26. The first term of a G.P is 3 and 6th term is 96

then its common ratio is ________

27. If a,b,c are in A.P then b+c, c+a, a+b are in

________

28. The arithmetic mean of is ________

29. The two geometric means inserted between

2, 16 are ________

30. g1, g2, g3 are G.M’s between a and b then

g1g3 = ________

31. In an A.P Sn = 2n2 + 5n then t4 = ________

32. The ‘n’th term of G.P is 2(0.2)n–1 its third

term is ________

33. The first term of an A.P is –1 and common

difference is –3 then 12th term is ________

34. 1 + 8 + 27 +---------- + n3 = ________

35. If A.M = 2, G.M = 8, then H.M = ________

36. If TanA, TanB, TanC are in A.P. then CotA,

CotB, CotC are in ________ progression.

37. form of is ________

38. If a,b,c are 3 consecutive terms of an A.P

then Ka,Kb,Kc are 3 consecutive terms of

_______

39. The relation between Σn & Σn3 is ________

40. The nth term of 13,8,3,-2, ------ is ______

41. If a,b are positive numbers then A.M, G.M,

H.M, are in ________ progression.

42. Sum of the squares of the first ‘n’ natural

numbers is ________

43. In an A.P, if 4 times of 4th term is equal to

5 times of 5th term then _____ term is zero.

44. The nth term of the series a,ar,ar2,ar3 -------

is ________

45. The sum of ‘n’ terms of the G.P 3,32,33, ----

-- is 120 then n = ________

46. 7th term of the series is ______

47. Sum of the 5 terms in the series 1.2 + 2.3 +

3.4 + ------- is ________

1 11, , ,

2 4

− − −

1.56p

q

1 1,

a b

2 16x

7 7

4x 5xx

3 3+ + + − − − − −

1 1 1, ,

a b c

1

x 21−1 1 1

, , , thenx 3 x x 3

− − − − − −+ −

nn

tn 1

=+

1 11, , ,

3 9− − − − −∞

5x 5 15 5− = −

( )( )

4

3x 0

1 x 1Lt ___________

1 x 1→

+ −=

+ −

2x3 1

9 3

⎛ ⎞=⎜ ⎟⎜ ⎟⎝ ⎠

3 4 25 5 5x x x __________

=

x3 81=

2

x 4

x 16Lt ______

x 4→

− =−

a b______

a b

− =−

5 5

3 3x a

x aLt ______

x a

− −

− =−

p q q r r pp q r

q r p

a a a. . _____

a a a

+ + +⎛ ⎞ ⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( )4

532 _____

=

2

1 11 .......

3 3+ + + α

3 33x y z 0,+ + =

KEY

1. x = -3 2. 32 3. zero 4. 4 5. 1/4 6. Tangent 7.

4/3 8. 1 9. 17 10. P = 3 11. 2/3 12. 1 13.

14. 11 15. 85 16. -4Y 17. 0.09 18. 9/4

19. 20. 1/4 21. 5 22. 1

23. -2/7 24. (x3/4 + y3/4) 25. 2 26. 27xyz 27. 3/2

28. 1/16 29. 1 30. 31. 9

32. 33. 5 (or) -2 34. –2 ≤ x ≤ 5 35. 8

36. xyz 37. 16 38. 100 39. x 40.(Σn)2 41. 11

42. a2 43. 7/4 44. 4/3 45. 9

a b+

88

5 5a (or)

3 3a−− −

2/3 1/3 1/3 2/3(a a b b )− +

2 3

REAL NUMBERS: Important Questions

4 Marks

1. If lmn = 1 show that ?

2. If a+b+c = 0 show that

?

3. If show that 3y3-9y = 10?

4. If ax–1= bc, by–1 = ca, cz–1 = ab, Prove thatxy+yz+zx = xyz?

5. Evaluate ?

2 Marks1. Solve |4x + 1| ≤ 7?

2. If then show that

3x+4y = 0?

3. Evaluate ?

4. Evaluate ?

5. If then show that

?

6. If ax = b, by = c, cz = a show that xyz = 1?

1 Mark

1. If find ‘P’?

2. Evaluate ?

3. Simplify ax(y–z) . ay(z–x) . az(x–y) ?

4. Evaluate ?

5. Evalute ?

6. Solve 2x+3 = 4x+1?7. If ax = b, by = c, cz = a show that xyz = 1?

8. Simplify ?

9. Simplify ?b c c a a bbc ca abx .x .x .− − −

1 1 2 1 1 23 3 3 3 3 3a b a a b b

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟− + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

x

8x 4Lt

2x 6→∞

++

2

x 0

x 5xLt

x→

+

3 32 2x a

x a

−−

P22

x x3

⎛ ⎞ =⎜ ⎟⎝ ⎠

11x (a a )

2= −

2a x x 1= + +

2

x 0

1 x x 1Lt

x→

+ + −

p p

q qx m

x mLt

x m→

−−

( )x 2

y

1(64) 2

256= =

x a

x a 2aLt

x a→

+ −−

33

1y 3

3= +

2 1 1 1 2 1 1 1 2 3a b c a b c a b c. . xx x x− − − − − −

=

1

11

1 l m− =∑+ +

4 Marks1. If the sum of the first ‘n’ natural numbers

is s1, and that of their squares s2 and cubes

s3, show that ?

2. Find the sum of ‘n’ terms of the series

0.5+0.55+0.555+----- n terms?

3. Insert 6 H.M’s between 2/3 and 2/31.

4. The A.M,G.M andH.M of two numbers are

A,G,H respectively show that A ≥ G ≥ H?

5. Find the sum to ‘n’ terms of the series

1.3+3.5+5.7+-----?

6. If 7 times the 7th term of an A.P is equal to

11 times the 11th term, show that the 18th

term of it is zero?

2 Marks1. Insert 4 arithmetic means between 3 and 33

2. The 8th term of an A.P is 17 and the 19th

term is 39 Find 25th term?

3. If g1, g2, g3 are three geometric meansbetween mand n. Show that g1g3= = mn

4. Determine the 12th term of a G.P where 8th

term is 192 and common ratio is 2?

5. Which term of the A.P.10,8,6....... is -28?

6. Find the sum to ‘n’ terms of the series

51+49+47+................?

7. Find the 15th term of the A.P (x+y),

(x-y),(x-3y), ...................?

1 Mark1. Find the sum to infinity of the G.P.?

2. Find the nth term of G.P 100, –110, 121, ....?

3. If K+2, 4K–6 and 3K–2 are in A.P find K?

4. In Arithmetic progression a = –3030,

l = –1530 and n = 51 find 5n ?

5. Find the 17th term in a series if

?

6. Find the 12th term of the progression

10,17,24 .................?

7. First term in A.P is ‘a’ and common differ-

ence is ‘d’ write general term of A.P.?

8. In G.P a = 2 , r = find s12?

9. Find the Harmonic mean of 6 and 12.?

10. Write the fractional form of ?0.423

2

( )( )n

n n 3t

n 2

+=

+

3 3 3, , ,.............

4 16 64

− − ∞

22g

( )22 3 19S S 1 8S= +

PROGRESSIONS: Important Questions

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6

Angle in a semi circle is...MATHEMATICS BIT BANK GEOMETRY

48. If |r| < 1, then the sum to infinite terms of

the series a+ar+ar2+------+ ∞ = ________

49. The nth term of an A.P is 2n+5 then the com-

mon difference is ________

50. If ‘a’ is the first term and ‘d’ is the common

difference of an A.P then 15th term of corre-

sponding H.P is ________.

51. In a G.P then tn = ______

GEOMETRY

1. ΔABC ~ΔPQR, If ∠A= 60°, ∠B = 70° then

∠R =________ (June 2009)

2. If P and Q are the mid points of AB and AC

of ΔABC, then ________ (June 2009)

3. Two circles of radii x and y touch each other

externally, the distnace between their cen-

tres is ________ (June 2009)

4. The number of common tangents for two

externally touching circles is ______(June

2009)

5. If C = 90° in ΔABC and a = 3, b = 4 then c

= ________(June 2009)

6. If a parallelogram is cyclic, then it is

________ (March 2008)

7. If ΔABC ~ΔPQR then = _______

8. The distance between the centres of two cir-

cles is ‘d’. If their radii are r1 and r2 then the

length of transverse common tangent is

________

9. Number of common tangent that can be

drawn to two circles touching externally are

______

10. Two circles touch externally at ‘O’ AB is

their direct common tangent, then ∠AOB=

______ (March 2008)

11. PT is a tangent and PAB is a secant of the

circle meeting the cirlce at A and B. If PA =

4cm, PB = 5cm then PT = ________

12. A ________ to a circle is perpendicular to

the radius through the point of contact.

13. ΔABC is an obtuse triangle, obtuse angle at

B. If AD ⊥ CB then AC2 = ________

14. If ΔABC ~ΔDEF then ΔDEF ~ΔABC. This

is ________ property.

15. In ΔABC, b2 = a2+c2 then ________ is a

right angle.

16. AD is the angle bisector of ∠A in ΔABC. If

BD:DC = 6:7 and AC = 3.5 then AB =

_______

17. The side and one diagonal of a rhombus are

5cm and 8cms. respectively. the length of

the other diagonal is ________

18. If two circles of radii 3cm and 5cm touch

internally, then the distance between their

centres is ________cm

19. The angle subtended by major arc at the

centre is ________

20. Altitude of two similar triangles are in

ratio, then the ratio of their areas is

________

21. The perimeter of two similar triangles are in

the ratio 1:3. The ratio of their correspon-

ding sides is ________

22. Angle in a semi circle is ________

23. The angle between the tangent and the

radius at the point of contanct is ________

24. The length of the tangent to a circle with

radius ‘r’ from a point, ‘p’ which is ‘d’ cm

away from the centre is ________

25. The point of intersection of the perpendicu-

lar bisectors of any two sides of a triangle is

its ________

26. The area of a rectangle is 24sq.cm. If its

length is 6cm then its peri meter is

________

27. In triangle ABC: if a circle drawn on BC as

diameter passes through A, the triangle

ABC is ________

28. If two circles touch externally, then the

number of direct common tangents are

________

29. P is a point outside a circle and PT is a tan-

gent to the circle PAB a secant, cuts the cir-

cle at A and B then PA.PB = ________

30. Angle in a semicircle at the centre is ______

31. Tangents drawn to a circle from an external

point are ________

32. In a square, the diagonal is ________ times

to its side.

33. In a ΔABC, the sides are 6,10,8 then it is a

________ triangle.

34. In a ΔABC, AD is the median drawn to BC

then AB2 + AC2 = ________

35. For two concentric circles., no.of tangents is

________

36. The number of circles that can be drawn

passing through three points which are not

collinear is ________

37. If two chords are subtending equal angles at

the centre of a circle, they are ________

38. In ΔABC ‘B’ is right angle triangle and BD

⊥ AC then BD2 = ________

39. The point which is equidistance from the

vertices of a triangle is ________

40. If a line divides any two sides of a triangle

in the same ratio then the line is ________

to the thrid side.

41. ΔABC, ∠B < 90° and AD ⊥ BC then

AC2 = AB2 + BC2– ________

42. If ABCD is a cyclic quadrilateral then

∠A+∠C = ________

43. Angles in the same segment of a circle are

________

44. The height of an equilateral triangle with

side is ________

45. ‘O’ is the centre of a circle, If ∠BOA = 140°

and ∠COA = 100° then ∠BAC = _______

46. Basic proportionality theorem is known as

________ theorem.

47. In the below circle the chords AB and CD

intersects at ‘O’ and AO = 8, OB = 6, CO =

4 then OD = ________

48. ‘O’ is the centre of the circle.

If ∠AOC = 130° then ∠B = ________

49. The point of concurrence of the medians of

a triangle is ________

2 3

1: 2

AB : AC

BC

PQ

( )n

n

1 ( 2)S

3

− −=

KEY

1. A ≥ G ≥ H 2. 3. 3/2 4. 3025

5. 6. n(n+1) 7. b/c 8. 2008/2009 9. 9

10. 12 11. 11 12. 5050 13. (x+z)

14. 15. 15 16. a2+b2 17. 9th

term 18. Arithmetic 19. n2 20. 111 21. 2b-a

22. 4x 23. k = 3 24. 25. x = 1 26. 2

27. Arithmetic progression 28. 29. 4&8

30. 31. 19 32. 0.08 33. –34

34. 35. 32 36. Harmonic

37. 155/99 38. Geometric 39. Σn3 = (Σn)2

40. 18-5n 41. Geometric 42.

43. 9th term 44. a.rn–1 45. 4 46. 1/64 47. 70

48. a/1–r 49. 2 50. 51. (-2)n–11

a 14d+

n(n 1)(2n 1)

6

+ +

2 2n (n 1)

4

+

22g

a b

2ab

+

23n 5n

2

+

n(2a (n 1))

2+ +

1

n 1b

a

+⎛ ⎞⎜ ⎟⎝ ⎠

b a

n 1

−+

5 Marks1. Construct a cyclic quadrilateral ABCD

where AC = 4 Cm ∠ABC = 57°, AB = 1.5

Cm and AD = 2 cm?

2. Construct a triangle ABC in which. AB = 4.4

cm ∠c = 65° and median through c =

2.7cm.?

3. Construct a triangle ABC in which Bc =

7cm, ∠A = 70° and foot of the perpendicular

D on BC from A is 4.5cm away from B?

4 Marks1. State and prove Basic proportionality theo-

rem?

2. State and prove pythagorean theorem?

3. State and prove Alternate segment theorem?

4. State and prove vertical angle Bisector theo-

rem?

5. State and prove converse of Alternate seg-

ment theorem?

6. State and prove converse of Basic proporti-

nality theorem?

2 Marks1. ∠B of ΔABC is an acute angle and AD ⊥

BC. Prove that AC2 = AB2 + BC2 – 2BC.BD?

2. ABCD is rhombus, prove that AB2 + BC2 +

CD2 + DA2 = AC2 + BD2?

3. Prove that the line Joining the mid - points of

two sides of a triangle is parallel to 3rd side?

4. Prove that the area of an equilateral triangle

at side ‘a’ is ?

5. Write two properties when two polygons are

said to be similar to each other?

6. In ΔABC, AD is drawn perpendicular to BC

, then prove that AB2–BD2 = AC2 – CD2?

7. If PAB is a secant to a circle intersecting the

circle at A and B and PT is tangent segment

then PA.PB = PT2?

1 Mark

1. If the radii 5 cms and 6cms of two circles

touch externally Find their direct common

tangents?

2. Define converse of the pythagorean theo-

rem?

3. Define Appolonius theorem?

4. A ladder 25 cm. long reches a window of a

building 24 cm. above the ground.

Determine the distance of the fort of ladder

from the building?

5. State two properties, when two triangles are

said to be similar?

6. State the converse of Alternate segment the-

orem?

7. There is a circle of radius 3cm. From a point

‘P’ which is at a distance of 5 cm. from the

centre of a circle, a tangent is drawn to the

circle. Find the length of the tanqent?

8. Two circles radii Terms 9cms touch internal-

ly .Find the distance between their centres?

23a .

4

GEOMETRY: Important Questions

A

CB

o

DA

BC

p

BCA

0

130°

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7

Equation of a line intercept form is...MATHEMATICS BIT BANK

ANALYTICAL GEOMETRY

1. The centroid of the triangle whose sides are

x = 0, y = 0 and x+y = 6 is ________

2. The inclination of the line x+y + 10 = 0 is

________

3. If D,E,F are the mid points of the sides BC,

CA, AB respectively and area of ΔABC is

64 sq.units then area of ΔDEF = ________

4. The area of the triangle formed by the line

with co-ordinate axis is _______

5. The equation of a line passing through

(-5,7) and having a slope 4 is ________

6. The point of intersection of the lines

y = 2x+1 and y = 3x-2 is ________

7. If ax+by+c = 0 represents a straight line

then the condition is ________

8. Slope of the line perpendicular to 3x+4y-10

= 0 is ________

9. The distance between the points (0,1) and

(8, K) is 10 then K = ________

10. Equation of the line making equal intercepts

on the axis and passing through (-2,3) is

_____

11. If 2x-3y + 5 = 0 and 4x-ky+3 = 0 are paral-

lel then K = ________

12. Equation of a line with intercepts 3 and 2

untis on x-axis and y-axis is ________

13. The center of the circle is (0,0) if one end of

a diameter is (1,2) then the other end is

______

14. If is the slope of the line then its incli-

nation is ________ degrees.

15. Equation of y-axis is ________

16. If (2,-4) and (6,-2) are the two vertices of

diameter of circle then its centre is

________

17. The perimeter of the triangle whose vertices

are A(0,0), B(1,0) and C(0,1) is ________

18. The equation of the line having slope 2 and

Y - intercept - 2 is ________

19. The equation of the line joining the points

( 1,2) and (3,4) is ________

20. The X and Y intercepts made by the line

4x+6y-9 = 0 are ________

21. If the line passes through the

point (2,–3) then a = ________

22. If the angle between two lines is 90° then

the product of their slopes is ________

23. The Area of the triangle formed by the point

A(0,0), B(a,0) and C(0,a) is ________

24. Slope of X-axis is ________

25. Equation of a line intercept form is

________

26. The lines y=2x+5, y=2x-5 are ________ to

each other.

27. Slope of the line joining the points (-a,a)

and is ________

28. Equation of a straight line passing through

(–1,–1) with 60° is ________

29. The lines y = 3x + 4 and x = -3y are ______

to each other.

30. Slope of is ________

31. The equation of a straight line parallel to

3x+4y = 10 and passing through origin is

________

32. In ________ratio is the segment joining the

points (4,-3) and (5,2) divides

by the x-axis

33. The centroid divides the median in the ratio

________

34. The line x-y+50 = 0 makes an angle of

x-axis is ________

35. The mid point of (Sin2α, Sec2α) and

(Cos2α, –Tan2α) is ________

36. The angle between x – 2 = 0 and y+3 = 0 is

________

37. The line x = 3y + 1 cuts x-axis at ________

38. Distance between the points (a cosθ, 0) and

(0, a sinθ) is ________

39. The line y = mx passes through ________

40. If the line joining the points (x1,y1) and

(x2,y2) is divded by a point R internally in

the ratio m:n then x-coordinate is ________

41. The line y = mx+c intersects the x-axis at

the point ________ (June 2009)

42. The line parallel to x-axis through (h,k) is

________ (June 2009)

43. If (1,3), (2,5) and (3,k) are collinear then K

= ________ (June 2009)

44. The slope of a line parallel to the line

3x – 2y + 1 = 0 is ________ (March 2009)

45. (4,7), (1,4), (3,2), (6,5) are the vertices of a

parallelogram, then the intersect point of its

diagonal is ________ (March 2009)

46. The slope of x = 2y is ________ (March

2009)

47. Analytical geometry was introduced by

________ (March 2009)

48. Slope-intercept form of an equation is

________ (March 2009)

49. Slope of ax+by+c = 0 is ________

(March 2009)

50. If two straight lines are parallel, their

slopes are ________ (March 2008)

51. Sum of the intercepts made by 3x+4y=12

on the axis is ________ (March 2007)

52. Slope of the line y = 5 is ________ (March

2007)

53. The point - slope form of an equation of a

straight line is ________ (June 2007)

54. Y-intercept made by line 3x+4y = 0 is

________ (June 2007)

55. The distance between origin to the given

point (a,b) is ________ (June 2005)

56. If two straight lines are parallel their equa-

tions differ only by a ________

3

x y1

a b+ =

(0,a a 3)+

x y1

a 2a+ =

3

x y1

a b+ =

KEY

1. 500 2. 2:1 3. x+y 4. 3 5. 5 6. Rectangle

7. PQ:PR 8. 9. 3 10. 90°

11. 12. Tangent 13. AB2+BC2+2BC.BD

14. Symmetric 15. ∠B 16. 3 17. 6 cm 18. 2 cm19. >180° 20. 1:2 21. 1:3 22. right angle (90°)

23. 90° 24. 25. Circum center 26. 20cm 27. right angle triangle 28. 2 29. PT2

30. 180° 31. equal 32.

33. right angle triangle34. 2(AD2+BD2)(or)2(AD2+DC2)35. Zero 36. One 37. Equal 38.AD.DC39. Circum center 40. Parallel 41. 2BC.BD 42.180° 43. Equal 44. 3cm 45. 60° 46. Thales the-orem 47. 12 48. 115° 49. centroid

2

2 2d r−

2 5

22 1 2d (r r )− +

KEY

1. (2,2) 2. 135° 3. 16 sq.units. 4.

5. 4x – y + 27 = 0 6. (3,7) 7. |a| + |b| ≠ 0 8. 4/39. K = 7 (or) -5 10. x + y – 1 = 0 11. k = 612. 2x + 3y – 6 = 0 13. (–1,–2) 14. 60°

15. x = K 16. (4,–3) 17. mt

18. y = 2x–2 19. x – y + 1 = 0 20. (9/4, 3/2 )

21. 1/2 22. –1 23. sq.units 24. zero

25. 26. Parallel 27.

28. 29. Perpendicular30. –b/a 31. 3x + 4y = 0 32. 3:2 33. 2:1 34. 60°35. (1/2, 1/2) 36. 90° 37. (1,0) 38. a

39. origin 40.

41.

42. y = k 43. k = 7 44. 3/2 45. (7/2, 9/2) 46. 1/2 47. Rene descortes 48. y = mx + c 49. –a/b50. equal 51. 7 52. zero 53. (y – y1) = m(x –

x1) 54. 0 55. 56. constant2 2a b+

c,0

m

−⎛ ⎞⎜ ⎟⎝ ⎠

2 1mx nx

m n

++

3x y ( 3 1) 0− + − =

3x y

1a b

+ =

2a

2

(2 2)+

1ab

2

4 Marks1. Find the equation of the line perpendicular to

the line joining (-1,3), (4,6) and passing

through the point (2,-5)?

2. Find the equation of a line whose slope is 4/5

and which bisects the line joining the points

P (1,2) and Q (4,-3)?

3. Find the equation of a line passing through

the point (5,-3) and whose sum of the inter-

cepts on the coordinate axis is 5/6?

4. Find the coordinates of the points of trisec-

tion of a segment joining A (-3,2)and B

(9,5)?

5. Show that the points A ( 1,2) , B (-3,4) and

c (7,-1) are collinear and find the ratio in

which A divides BC?

2 Marks1. If the three points A (P,2) B (-3,4) , C (7,-1)

are collinear find the volue of ‘p’.

2. In what ratio is the segment Joining the

points (4,6) and (-7,-1) divided by the X-

axis?3. Find the third vertice of the triangle, if two of

its vertices are (-1,4) and (5,2) and the medi-

ans intersect at (0,-3).?4. Find the equation of the line passing through

the point (3,4) and is parallel to 4x+7y=8?5. Find the area of the triangle formed by the

points (-2,3), (-7,5) and (3,-5)?6. One end of the diameter of a circle is (3,2)

and the centre is (0,0). Find the co-ordinates

of the other end of the diameter?7. Find the point on X-axis which is equidistant

from (2,3) and (4,-2)?8. Find the area of the triangle formed by the

line 2x-4x-7 = 0 with the co-ordinate axis?

1 Mark1. Find the equation of the line passing through

(4-7) and (1,5)?

2. A straight line makes intercepts 4 and -7 on

the X and Y -axis what is the equation of

that line?3. Find the equation of the line making an angle

150° with X-axis and having Y-intercept -1?4. Find the co-ordinates of the centroid of the

triangle whose vertices are (-4,4),(-2,2) and

(6,12) ?5. Find the intercepts of the equation

2x+3y-5=0 ?6. Find the slope of the line perpendicular to

the line 3x-2y+1=0?7. Find the area of a triangle with vertices at

(3,0) , (0,4) and (0,0) ?8. Find the slope of the line whose equation is

2x-7y = 12?9. Find the slope of the line joining (4,6) and

(2,-5)?10.Find the equation of the line passing through

the point (3,-5) and whose slope is 7/3?

ANALYTICAL GEOMETRY: Important Questions

ANALYTICAL GEOMETRY

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8

If Tan θθ = a/b then Sin θθ =...MATHEMATICS BIT BANK TRIGONOMETRY

TRIGONOMETRY

1. If Sin θ = 5/13 then Cos (900 – θ) = ______

2. Cos2 40° + Cos2 50°= ________ (June 09)

3. 270° = ________ radians. (June 2009)

4. Sec π/3 = ________ (June 2009)

5. =______(June 09)

6. Tan (90+θ) = ________ (March 2008)

7. If Tan θ = 3/4 then Sin θ = ________

(0 < 90°) (March 2008)

8. Cos π/3 = ________ (March 2008)

9. Cos 360° = ________ (March 2007)

10. If Sin θ = Cos 2θ then Cot 3θ = ________

(March 2007)

11. If x = a Cosec θ, y = a Cot θ then x2 – y2 =

________

12. From a ship most head 150 feet heigh, the

angle of depression of a boat is observed as

45°. Its distance from ship is ________

13. If SecA + TanA = p then SinA = ________

14. Maximum and minimum values of Sin θ is

________

15. Radians is the unit of measure in ________

system

16. (Sec θ + Tanθ) (1-Sin θ).Sec θ = ________

17. If SecA + Tan A = p then SecA - TanA =

________

18. Eliminate θ from x = Cos θ + Sin θ, y = Cos

θ . Sin θ then the equation is ________

19. = ________

20. Sin θ. Cosec θ + Cos θ . Sec θ + Tan θ . Cot

θ = ________

21. = ________ grades.

22. 135° = ________grades.

23. = ________ degrees.

24. A wheel makes 360° revolution in one

minute through ______ radians does it turn

in a one second.

25. The angles of a triangle are in A.P and the

greatest angle is three times the least. The

angles in circular measure are ________

26. The value of Cos 0°+Sin 90°+ Sin 45°

is ________

27. Sin θ in terms of Sec θ = ________

28. Sin 420° = ________

29. If Tan θ = 1 then θ = ________

30. = ________

31. 1 radian = ________ degrees.

32. Cos(–60°) = ________

33. Tanθ+Cotθ = 2 then Tan4θ + Cot4θ = _____

34. If Tan (15°+B) = then B = ________

35. If Tan θ = a/b then Sin θ = ________

36. Sec θ (1-Sin θ) (Sec θ + Tan θ) = _______

37. Cos1°.Cos2°.Cos3°.................Cos179° =

________

38. If Sinx + Sin2x = 1 then Cos2x + Cos4x =

________

39. If Sin θ = Cos θ then θ = ________

40. Sin29° + Sin281° = ________

41.________

42. Sin230°, Sin245°, Sin260° are in_______progressions.

43. If Tan(A+B) = , TanA = 1 then ∠B =

________

44.

45. A minute hand of a clock is 3cm long, thedistance moved in 20 minutes is ________

46. The value of (Sinθ+Cosθ)2 +(Sinθ –Cos θ)2

= ________47. The values of Tan 30°, Tan 45°, Tan 60° are

in ________ progressions.

48. Sec (270°– θ) = ________

49. A straight angle contains ________ degrees.

50. The side about which a rotation is made is

called ________

51. Find the length of side of a regular hexagon

inscribed in a circle of a radius 2mt is _____

STATISTICS

1. The mean of the first ‘n’ natural numbersis ________ (June 2009)

2. Range of first 20 natural numbers is________ ( March 2009 )

3. The formula for the arithmetic mean bythe deviation method is ________ (June2008)

4. The class internal of the frequency distri-bution having the classes 1-8, 9-16, 16-24......... is ________

5. The arithmetic mean 39 and mode 34.5then the median is ________ (June 2008)

6. The mid value of the class is used to cal-culate for ________ (March 2007)

7. For 20,30,20,30,40,10,50 Mode of thescore is ________ (June 2006)

8. The Median of scores x1,x2,2x1 is 6 andx1 < 2x1 < x2 , then x1 = ________(March 2006)

9. The arithemetic mean of a–2 , a and a+2is ________ (June 2005)

10. The value of Δ1While calculating themode in delta method is ________

11. 1-8, 9-16, 17-24, ................ are ________classes.

12. Formula for grouped data of Median is________

13. In a histogram, the breadths of the rectan-gles represent the ________

14. For the construction of a frequency poly-gon ________ and frequencies are takeninto considaration.

15. In the frequency distribution with classes1-10,11-20,........ the upper boundary ofclass 1-10 is ________

16. The median of is ________

17. If the mean of the data 12,15,x,19,25,44 is

25 then x = ________

18. The relation among mean, median and mode

is ________

19. The upper boundary of a class is 30. Class

interval is 10. Lower boundary of the class

is ____

20. Cumilative frequencies are used to measure

in ________

3 1 2 1 7, , , ,

4 2 3 6 12

Sin18

Cos72

° =°

3

2 2

2

Sin 81 Sin 9

Tan 45

+ =

1

3

2Cosec 1

Cosec

θ −θ

3

2

c3

5

π

c5

2

π

4 4

2 2

Sin A Cos A

Sin A Cos A

−−

2 2 2Sin Cos Tanθ + θ + θ

5 Marks1. There are two temples, one on each bank of

a river , just opposite to each other. one of

the temples A is 40 mts high. AB observed

from the top of this temple A , the angle of

depression of the top and foot of the other

temple B are 12°30' and 21°48' respective-

ly. Find the width of the river and the

height of the temple B?

2. From the ground and first floor of a build-

ing , the angle of elevation of the top of the

spire of a church was found to be 60° and

45° respectively. The first floor is 5 mts

high. Find the height of the spire?

3. A glider is flying at an altitude of 5000

mts. The angle of depression of the cotrol

tower of the air port from the glider is 18°.

What is the horizontal distance between the

glider and control tower?

4. An aeroplane at an altitude of 2500 mts

observe the angles of depression of oppa-

site points on the two banks of river to be

41°20' and 52°10'. Find in meters, the

width of the river?

4 Marks1. If cosecθ + cot = P then prove that (P2+1)

cos θ = p2 –1(p ≠ 0)?

2. Show that 3 (sin x - cosx)4 + 6 (sinx+cosx)2

+4 (sin6 x+cos6x)=13?

3. Eliminate θ from the equations x cos θ + y

sin θ = a and x sin θ – y cos θ = b?

4. Prove that ?

5. Find the value of 32 cot2 -8sec2 +

8cot3 ?

2 Marks

1. Show that = cosecθ + cotθ?

2. If cos θ = and θ is acute find 4 sin2 +

Tan2θ.?

3. Show that = Tan2θ?

4. If Tan (A+B) = and Tan A=1 What is

the measure of B?

5. If Tanθ + cotθ = 2 find the value of

Tan2θ+cot2θ?

6. Prove that sec2θ +cosec2θ = sec2θ. cosec2θ.

7. Prove that 1–(sin6θ + cos6θ) = 3 sin2θ.

cos2θ?

8. Show that sin2 A+cos2 A = 1?

1 Mark

1. Find the value of cos 0° + sin90° + sin45°?

2. If cos θ = then find values of sin?

3. Eliminate ‘θ’ from x = a sin θ, y = acos θ?4. Write Tan θ value interms of cos θ?5. Define Radian?

6. Show that = cos 60°?

7. Express in sexagesimal measure?

8. Convert 200° in to circular measure ? 9. Find the value of cot 240°?10. If sec θ + Tan θ = p then Find sec θ – Tan

θ Value?

c5

6

π

2

2

1 Tan 30

1 Tan 30

−+

3

2

2

3

2

2

1 Tan

cot 1

− θθ −

3

2

1 cos

1 cos

+ θ− θ

6

π 3

π4

π

Tan sec 1 1 sin

Tan sec 1 cos

θ + θ − + θ=θ − θ + θ

TRIGONOMETRY: Important Questions

KEY

1. 5/13 2. 1 3. 4. 2 5. Secθ 6. –Cotθ 7. 3/5

8. 1/2 9. 1 10. 0 11. a2 12. 150mt 13. 14. [+1,-1] 15. Circular 16. 1 17. 1/p18. x2–2y = 1 19. 1 20. 3 21. 500g 22. 150g 23.108°

24. 12π 25. 26. 3 27.

28. 29. 30° 30. Cos θ 31.

57°.16' 32. 1/2 33. 2 34. 15° 35.

36. 1 37. 0 38. 1 39. 45° (or) 40. 1 41. 1

42. A.P 43. B = 15° 44. 1 45. 44/7 cm 46. 247. Geometric Progression 48. –Cosecθ49. 180° 50. initial side 51. 2mt

c

4

π

2 2

a

a b+

3 / 2

2Sec 1

Sec

θ −θ

c c c

and6 3 2

π π π

2

2

p 1

p 1

−+

c3

2

π

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9

Number of rows in a Row matrix...MATHEMATICS BIT BANK MATRICES

21. The most common and widely used measure

is ________

22. Father of statistics is ________23. Given data, frequency of modal class

f = 36, f2 = 24 then Δ2 = ________24. The average which is not affected by the

extraction value is ________25. The median of 7,5,7.5,5.5,6,6.5 is ______26. The mean of 10 observations is 7 and the

mean of 15 observation is 12 then themean of all observations is ________

27. Mid value of the class 1-10 is ________28. In a frequency distribution, the mid value

of a class is 35 and the lower boundary is30 then upper boundary is ________

29. 0-10,10-20,20-30 are ________ type ofclasses.

30. Unlike mean , median is not affected bythe ________ observations

31. A.M = where A is called

________32. In a data having two modes, then it is

called ________33. Sum of 20 observations is 420 then the

mean is ________34. The difference between two consecutive

lower limits of the class is ________35. Circular diagram consists of ________36. The mode of 4,8,9,p,2,6,4,9 is 9 then p =

________37. The Arithmetic mean of sum of the even

natural numbers is ________38. The median of natural numbers from 1 to

9 is ________39. A Histogram Consists of ________40. In a distribution

Δ1 = 6, Δ2 = 4, c = 10 and L=25 thenmode = ________

MATRICES

1. If then |A|=_____ (March09)

2. If then t = ________

3. If then the value of ‘x’

is ________

4. = ________

5. If |A| = 0 then the matrix has ________6. The mathematician who introduced

matrices is ________ (June 2006)7. A,B are two matrices (AB)T = ________8. The condition to multiply two matrices

A,B is ________

9. then order of M =____

10. If has no multiplicative

inverse then x = ________11. If the transpose of a given matrix is equal

to its additive inverse, then the matrix iscalled ________

12. Matrix obtained by interchanging rowsand columns is called ________ (March2009)

13. If the rows and columns of a matrix aresame, then it is called ________ (March09)

14. If then a and b

are ________

15. If then x = ________

16. If then d = ________

17.

then AB = ________

18. If is to be scalar matrix then λ

= ________

19. If A and B are two matrices then (AB)–1 =________

20. If and ad = bc then A is

________ matrix

21. If and AD = A then D is

________ Matrix 22. If A2×3, B3×2 then the order of A×B is

________23. If AB = KI, where K ∈ R, then A–1 =

________24. If A is a matrix then (AT)T =________

25. If then a+b+c+d =

________ (June 2005)26. The order of A is 3 × 2 then the order of

AT is ________

27. is example of ________

28. ________

29. If A is matrix then A.A–1=A–1. A= ______30. Number of rows in a Row matrix

________31. The order of A and B are 3×4 and 5×3

then the order of BA is ________32. If A is 2 × 2 matrix such that A=A–1 then

A2 = ________ (June 2009)

33. A is any 2 × 2 matrix. if then

AB = ________ (June 2009)

1 0B

0 1

⎛ ⎞= ⎜ ⎟

⎝ ⎠

1 3

3 1

2

3 (1 2 3)

×

⎛ ⎞⎜ ⎟ =⎜ ⎟⎜ ⎟⎝ ⎠

4 0 0

0 4 0

0 0 4

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

a b 1 2

c d 3 1

⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

1 2A

3 4

−⎛ ⎞= ⎜ ⎟−⎝ ⎠

a bA

c d

⎛ ⎞= ⎜ ⎟

⎝ ⎠

3 0P

0

⎛ ⎞= ⎜ ⎟λ⎝ ⎠

2 3 2 2

1 2 3 1 0A ;B

3 0 1 0 1× ×

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

2 414

d 5

−=

1 3 2 x

0 1 1 1

⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠

a 5 4 6 2 1

8 b 7 2 1 5

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

x 3A

3 x

⎛ ⎞= ⎜ ⎟

⎝ ⎠

2 3M (6 10)

0 1

⎛ ⎞× =⎜ ⎟

⎝ ⎠

Tan sec

sec Tan

θ θθ θ

x 3 2 5

1 2 1 0

⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠

t

4 34 3

2 32 2 2

−⎛ ⎞−⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

4 3A

2 1

⎛ ⎞= ⎜ ⎟−⎝ ⎠

fxA c

N∑+ ×

KEY

1. 2. 19 3. 4. 7 5. 37.56. Arithmetic mean 7. 20,30 8. 3 9. a 10. f – f1

11. inclusive 12. 13. class inter-

vals 14. Midvalues of the classes 15. 10.5 16.

17. 35 18. Mode = 3Median-2A.M 19. 20

20. Median 21. Arithmetic mean 22. SirRonald A. Fisher 23. 12 24. Median 25. 6.2526. 10 27. 5.5 28. 40 29. Exclusive 30.Extreme 31. Assumed mean 32. Bi modal 33.21 34. Class interval 35. Sectors 36. 9 37.(n+1) 38. 5 39. Rectangles 40. 31

7

12

NF

2L cf

−+ ×

i i1

A f cN

+ Σ μ ×(n 1)

2

+

4 Marks1. Calculate the A,M for the following data

by deviation method?

2. Find the median for the following data ?

2 Marks1. The mean of 20 observation is 135. By an

error, one observation is registered as-25

instead of 25 . Find the correct mean?

2. Write four merits of the Arithmetic mean ?

3. The mean and median of Uni-modal

grouped data are 72.5 and 73.9 respective-

ly. Find the mode of the data?

4. Observations of some data are

where x>0. If the median

of the data is 8. Find the value of ‘x’?

5. The observations of an ungrouped data are

x1, x2 and 2x1 and x1 < x2 < 2x. If the mean

and median of the data are each equal to 6.

Find the observations of the data?

1 Mark1. The mean of 9,11,13,P,18,19, is P. Find the

value of ‘P’?

2. Find the mode of the data 12, 11, 15, 12,

11, 15, 12, 9, 12?

3. Write two properties of mode?

4. A.M= x, Median= y find mode of the data?

5. Find the median of the observations 1.8,

4.0, 2.7, 1.2, 4.5, 2.3 and 3.7?

6. The observation of an ungrouped data in

the assending order is 12, 15, x, 19, 25. If

the median of the data is 18 find the value

of ‘x’?

x x x x,x, , and

5 4 2 3

STATISTICS: Important Questions

Marks 0-10 10-20 20-30 30-40 40-50 50-60No.of

5 7 15 8 3 2students

Class 60-64 65-69 70-74 75-79 80-84 85-89

Frequency 13 28 35 12 9 3

4 Marks

1. If ?

find 1) A–1 2) B–1 3) (AB)–1 4) B–1A–1 ?2. Solve the following linear system of equa-

tions using cramers method 4x–y=16 and

?

3. Solve the following equations by using

Matrix inversion method and y

= 13 – 6x?

4. If show that

A2 – (a+d) A= (bc-ad) I.?

5. If

Show that A(B+C) = AB+AC?

2 Marks

1. If find the order ofM

and determine the Matrix ‘M’ ?

2. If find ‘m’ if

AB=BA.?

3. If find the

Matrix B+A–1?

4. If findx,y?

5. If

find A2 + BC?

1 Mark

1. If find the value of A+AT?

2. If find 3A-2B?

3. If find A+A–1 = 4I?

4. =0 find ‘d’?

5. If find A–1?

6. Define Non-singular Matrix

7. If and then Find

AB?

0 0B

0 1

⎛ ⎞= ⎜ ⎟

⎝ ⎠

1 0A

0 0

⎛ ⎞= ⎜ ⎟

⎝ ⎠

2 3A

1 5

−⎛ ⎞= ⎜ ⎟

⎝ ⎠

d 2 5

4 2

−−

1 2A

1 3

⎛ ⎞= ⎜ ⎟

⎝ ⎠

2 4 4 3A , B

6 5 5 7

−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

1 3A

5 6

⎛ ⎞= ⎜ ⎟

⎝ ⎠

1 4 3 2 1 0A ;B ;C

2 1 4 0 0 2

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

3x 2y 6 5 6

2 2x 3y 2 1

+⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

1 2 2 0A ;B

1 3 5 3

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

2 m1 4

A ;B 10 1 0

2

⎛ ⎞⎛ ⎞ ⎜ ⎟= = −⎜ ⎟ ⎜ ⎟− ⎜ ⎟⎝ ⎠ ⎝ ⎠

( )1 2M 2 3

0 5

⎛ ⎞× =⎜ ⎟

⎝ ⎠

2 4 2 5 1 2A , B , C .

3 6 6 1 3 0

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

a b 1 0A and I

c d 0 1

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

7 3yx

2

−=

3x 7y

2

− =

2 1 2 0A ,B

3 1 5 3

−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

MATRICES: Important Questions

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10

The unit that gains results from C.P.U.. MATHEMATICS BIT BANK COMPUTING

34. The Inverse of an identity matrix is________ (March 2009)

35. If then A–1= ____ (March08)

36. If then x =

________ (March 2008)

37. In a Matrix the element in

2nd row and 3rd column is_____(June 07)

38. , then AB =

________ (June 2007)

39. While solving the equations 3x+4y = 8and x – 6y = 10 by Cramer’s method thenthe matrix B1 =________

40. The determinant of a singular matrix is________

41. If and A+B = A then B is

________ matrix

42. If and P+R=I then R=______

43. If and A–B+X=0

then the Matrix X is ________

44. In a Matrix the number of rows are notequal to number of columns then thematrix is ________

45. A square matrix in which each of theprincipal diagonal elements are equal toone and all other elements are zero iscalled a ________ matrix

46. If the transpose of a given matrix is equalto its additive inverse that matrix is called_______

COMPUTING

1. Small Transistors are used in _______generation of computers. (March 06,June 09)

2. All parts of computer are controlled by________ (2006, 2007, 2009)

3. Input, Output, CPU are ________ of thecomputer. (June 2006)

4. An example for output is ________ (June2006)

5. Vacuum tubes are used in ________ gen-eration of computers

(March 2007)6. The language known to the computers is

called ________ (June 2009)7. ________ is used to make a diagrammat-

ic representation of an algorithm (March2008)

8. The father of computer is ________(March 2008)

9. To express the algorithm in a languageunderstandable by a computer is called________

10. The number of major parts in a computeris ________ (June 2009)

11. C.P.U means ________12. large amount of information is stored in

________ unit of computers.13. The method of solving a problem is

called ________14. ________ are used in fourth generation

of computers.15. All the mathematical operations are car-

ried out in ________ units.16. The input unit, C.P.U and output unit all

together is called ________17. The unit that gains results from C.P.U is

________18. Example for computer language is

________19. The present day computers are made as

________ generation computers.20. In the preparation of flow charts, we use

Rhombus shaped box for ________21. A computer is an ________ device.22. Pictorial representation of algorithm is

called ________23. Printer is example for ________ unit24. COBOL means ________25. The computers built in between 1950-

1960 are called as ________ generationof computers.

26. ________ is example for Input unit27. An algorithm means ________28. The Rhombus shaped box is used in a

flow chart for ________29. Each computer consists of three essential

units, namely Input unit, output unit andthe ________ unit.

30. BASIC is ________ language.31. Father of modern computers is

________

32. ________ are used in third generation ofcomputers.

33. A.L.U means ________

1 2 2 4A ,B

3 4 3 5

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

4 5P

7 6

−⎛ ⎞= ⎜ ⎟−⎝ ⎠

5 7A

0 8

⎛ ⎞= ⎜ ⎟

⎝ ⎠

1 22 1

xA , B (5 2)

y ××

⎛ ⎞= =⎜ ⎟

⎝ ⎠

1 8 4

2 3 0

5 7 4

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠

x y x y 2 0

2x 3y 2x 3y 5 1

+ −⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟+ − −⎝ ⎠ ⎝ ⎠

1 4A

0 1

⎛ ⎞= ⎜ ⎟−⎝ ⎠

1. Negation2. And3. Or4. Implie5. If and only if6. For all7. For some8. Belongs9. Not belongs10. Subset11. Superset12. Union13. Intersection14. Powerset15. Null set16. Complement of A17. Cartesian product of

A, B is18. Identity function19. Discriminant20. Transpose of A21. Inverse of A22. Fistle function A to B23. Composite function of f

and g24. Sum of first 'n' natural

numbers25. nth term26. Sum of 'n' terms27. Arithmetic mean28. Sum of frequencies

KEY

1. 10 2. 5 3. 4 4. –1 5. has no multiplicative

inverse 6. Author Cayley 7. BT.AT 8. No.of

Columns in A = Rows in B 9.(1×2) 10. ±3 11.

Skew symmetric 12. Transpose of matrix 13.

Square matrix 14. 6,7 15. –1 16. 1 17. is not

defined 18. 3 19. B–1.A–1 20. Singular matrix

21. Identity matrix 22. 2 × 2 23.

24. A 25. 5 26. 2 × 3 27. 3 × 3 scalar matrix

28. 29. I 30. l 31. 5 × 4 32. I

33. A 34. also identity matrix 35.

(or) A 36. 1 37. 0 38. 39.

40. zero 41. null 42. 43.

44. Rectangle matrix 45. Identity matrix

46. Skew symmetric matrix

1 2

0 1

⎛ ⎞⎜ ⎟⎝ ⎠

3 5

7 7

−⎛ ⎞⎜ ⎟−⎝ ⎠

8 4

10 6

⎛ ⎞⎜ ⎟−⎝ ⎠

5x 2x

5y 2y

⎛ ⎞⎜ ⎟⎝ ⎠

1 4

0 1

⎛ ⎞⎜ ⎟−⎝ ⎠

2 4 6

3 6 9

4 8 12

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

1B.

K

KEY

1. Second 2. C.P.U 3. Hardware 4. print-er 5. First 6. Higher language (or) soft-ware programming language 7. Flowchart 8. Charles Babbage 9.Programming language 10. 3 11. CentralProcessing Unit 12. Memory 13.Programme 14. Very large scale integrat-ed circuites 15. Arithmetic and logicalunit 16. Hardware 17. Out put 18.COBOL (or) PASCAL 19. IVth genera-tion 20. Decision box 21. Eelectronic 22. Flowchart 23.Output 24. Common business orientedlanguage 25. Ist generation 26. Key board27. Plan of obtaining a solution to a prob-lem 28. Decision making 29. Central Processing Unit (C.P.U.) 30.Computer 31. Von Newmann 32. Verysmall electronic circuits33. Arithmetic and Logic unit

4 Marks1. Give the principal amount and the rate

of interest write an algorithm to obtain atable of compound interest at the end ofeach year for 1 to 5 years and draw aflow chart?

2. Gopal purchased a radio set for 500 andsold it for 600. Execute a flow chart todetermine loss or gain percentages?

3. Draw the flow chart to find the value ofproduct of the first ‘n’ natural numbers?

2 Marks1. What are the different boxes used in a

flow chart? Write for what functionsthey are used?

2. Write the characteristics of a computer?3. Draw a structure diagram of computer?4. What is Flow chart and define

Algorithm?5. What are the essential components of a

computer?6. What is meant by step-wise refinement

in computer?7. What are the types of operations that a

computer performs?

1 Mark1. What is meant by Computer Hardware?2. Expand C.P.U.?3. Write 4 computer languages?4. What are the essential components of

C.P.U.?5. What are the shapes of terminal and

decision boxes in the flow chart?6. Define a computer?7. What is the difference between Hard

ware and software?

COMPUTING: Important Questions

∼∼∧∧∨∨⇒⇒⇔⇔∀∀∃∃∈∈∉∉⊂⊂⊃⊃∪∪∩∩μμφφ

A1 / Ac

A × BI (A)

ΔΔ or DA T

A–1

f:A→→B

gofΣΣ n

tnsnx

ΣΣf or N

Important symbols

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11

IMPORTANT POINTS PAPER- 1 AND 2

STATEMENTS and SETS

1. Function: f:A→B, 1. if for every a∈A there is b ∈ B such that(a/b)f

2. One-one function (injection: f(x1) = f(x2) ⇒ x1 = x2

3. Onto function (surjection): f:A→Β, f(A)=b

4. BiJection: Both one-one and onto

5. Constant function: f:A→B, The range of f(A) = singleton set6. Identity function: I:A→A, I(x)=x ∀ x ∈ A7. Inverse function: If fi is Bijection then f-1is also a function8. Equal function: Two functions domains are equal.9. In to function: f: A→Β, f(Α) ⊂ Β 10. Real function: f: A→Β, A,B are subsets of R.11. Composite function (gof): A,B,C are three sets, f:A→Β, g:B→C then gof:

A→C.

FUNCTIONS

Item Explanation1. Arithmetic progression(A.P) difference (d) is equal

2. General form of A.P a, a + d, a + 2d, ....

3. 'n'th term in A.P (tn) a+ (n–1)d

4. Sum of n terms in A.P. (sn)

5. Geometric progression (G.P) Ratio (r) is equal

6. General form of G.P a, ar, ar2, .....

7. nth term in G.P (tn) a⋅rn – 1

8. Sum of n terms in G.P. (sn)

9. Harmonic progression (H.P) Reciprocal of the terms form an A.P.

10. Arithmetic mean of a, b

11. Geometric mean of a , b

12. Harmonic mean of a, b

13. Σ n = 1 + 2 + 3 + . . . . + n

14. Σ n2 = 12 + 22 + 32 + . . . . + n2

15. Σ n3 = 13 + 23 + 33 + . . . . + n3

2 2n (n 1)

4

+

n(n 1)(2n 1)

6

+ +

n(n 1)

2

+

2ab

a b+

ab

a b

2

+

( ) ( )n na r 1 a 1 rif r 0 or if r 0

r 1 1 r

− −> <

− −

( ) ( )n n2a n 1 d or a l

2 2+ − +⎡ ⎤⎣ ⎦

PROGRESSIONS

Statement Explanation1. Modulus of 'x', |x| |x| = x if x > 0 or - x

if x < 0 or 0 if x = 0

2. |x| = a solution: x=a or x= - a

3. |x| ≤ a solution: – a ≤ x ≤ a

4. |x| ≥ a solution: x≥a or x≤– a

5. n⋅xn – 1

6.

m nma

n−

m m

n nx a

x aLt

x a→

−−

n n

x a

x aLt

x a→

−−

REAL NUMBERS

Item Explanation1. Convex Set: X is convex if the line segment joining any

two points P, Q in x is contained in x

2. Linear programming problem L.P.P consists of Minimising/maximising a

function f = ax+by, a, b∈R subject to certain

constraints.

3. Objective function: f = ax + by, a, b∈R which is to be minimised

or maximised

4. Feasible Region: Solution set of constraints of LPP is convex

set is called

5. Feasible solutions: Anypoint (x, y) in the feasible region gives a

solution to LPP.

LINEAR PROGRAMMING

Item Explanation1. Quadratic equation ax2 + bx+c=0

2. Discriminent Δ = b2–4ac

3. Δ > 0 Roots are real, unequal

4. Δ = 0 Roots are equal and real

5. Δ < 0 Roots are imaginary

6. Sum of the roots –b/a

7. Product of the roots c/a

8. Quadratic equation whose roots α, β x2–(α+β)x + αβ= 0

9. (x–α) (x–β) <0, (α<β) Solution: α < x < β10. (x–α) (x–β) >0, (α<β) Solution: x < α ∪ x > β11. y = mx2 (m > 0) graph I, II Quadrants, Symmetric about Y-axis

12. y = mx2 (m < 0) III, IV Quadrants, Symmetric about Y-axis

13. x = my2 (m > 0) I, IV Quadrants, Symmetric about X-axis

14. x = my2 (m < 0) II, III Quadrants, Symmetric about X-axis

15. Remainder theorem f(x) is divided by (x-a) then Remainder is f(a)

16. Sum of the co-efficents to the polynomial ( x – 1) is a factor

is zero

17. Sum of the co-efficents of even power of 'x' = ( x + 1) is a factor

Sum of the co-efficents of odd powers of 'x'

18. The general term of (x+y)n Tr+1 = nCr xn – r yr

19. Roots of ax2 + bx + c = 0

20. y=mx2 graph Parabola.

2b b 4ac

2a

− ± −

POLYNOMIALS

1. Sexagesimal system - Degree

2. Centesimal system - grade

3. Circular measure - Radian

4. 90° = 1009 =

5. Sin2 θ + cos2 θ = 1

6. Sec2 θ – Tan2 θ = 1

7. Cosec2 θ – Cot2 θ = 1

8. Range of Sin x = [–1, 1]

9. Range of Cos x = [–1, 1]

c

2

π

TRIGONOMETRY

Chapterwise Quick Review1. Connectives: 2. Compound Statements:3. Quantifiers: 4. Tautology: 5. Contradiction:6. Proofs:

7. Switching networks:

1. and 2. or 3. if...then 4. if and only if

1. Disjunction 2. Conjunction 3. Conditional 4. Biconditional

1. Universal (∀) 2. Existential (∃)

Compound Statement always true

Compound statement always False.

1.Direct method 2.Indirect Method 3. disproof by counter

example.

1. Series combination 2. Parallel combination.

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12

IMPORTANT POINTS Paper 1, 2

GEOMETRY

STATISTICS

ANALYTICAL GEOMETRY

Preparation Tips and Blue Print

Paper - IChapter 5 Marks 4 Marks 2 Marks 1 Mark 1/2 MarkStatements & Sets & 1 2 1 5Functions (Mappings) & 2 1 1 5Polynomials 1 1 1 1 6Real Numbers & 1 2 1 3Linear Programming 1 1 1 1 6Progressions & 2 1 1 5

Paper - IIChapter 5 Marks 4 Marks 2 Marks 1 Mark 1/2 MarkGeometry 1 1 1 1 5Analytical Geometry & 2 2 1 5Trignometry 1 1 1 1 5Statistics & 1 1 1 5Matrices & 2 1 1 5Computing & 1 2 1 5

Item Formula

1. The slope of (x1, y1)and (x1, y2)

2. Distance between (x1, y1) and (x2, y2)

3. Equation of X-axis y = K

4. Equation of Y-axis x = K

5. Slope of X-axis 0

6. Slope of y-axis Not defined

7. General form of a straight line ax + by +c = 0

8. Slope of ax + by +c = 0

9. Mid point of (x1, y1) and (x2, y2)

10. Given points is divided by internally

in the ratio m : n is

11. Given points is divided by externally

in the ratio m : n is

12. Gradien form of equation y = mx

13. Slope - intercept form y = mx + c

14. Point - slope form y – y1 = m(x – x1)

15. Two intercepts form

16. Two - points form (y – y1) (x2 – x1) = (x – x1) (y2 – y1)

17. Centroid of the Triangle

18. Area of the Triangle

19. Area of the Triangle = 0 given points are collinear.

20. Product of the slopes = –1 Lines are perpendicular

21. Slopes are equal Lines are parallel.

( ) ( ) ( )1 2 3 2 3 1 3 1 21

x y y x y y x y y2

− + − + −

1 2 3 1 2 3x x x y y y,

3 3

+ + + +⎛ ⎞⎜ ⎟⎝ ⎠

x y1

a b+ =

1 1 2 1mx nx my ny,

m n m n

− +⎛ ⎞⎜ ⎟− −⎝ ⎠

2 1 2 1mx nx my ny,

m n m n

+ +⎛ ⎞⎜ ⎟+ +⎝ ⎠

1 2 1 2x x y y,

2 2

+ +⎛ ⎞⎜ ⎟⎝ ⎠

a

b

( ) ( )2 22 1 2 1x x y y− + −

2 1

2 1

y y

x x

−−

� 'Mathematics is a hard nut to crack' it isgeneral perception of students. Actuallyregular practice and good command onbasics will make mathematics easy andscoring subject.

� Out of 12 chapters, if you prepare 8chapters thoroughly you can get 90marks easily.

� Before solving the problems you shouldbe well aware of the definitions andlaws of that particular chapter.

� Please do not by-heart steps in theproblem. Understand the problem andmake steps accordingly to arrivesolution.

� Avoid tension and be cool in exam hall.� First of all read all the questions in the

given question paper, then only answerthe question first which you feel wellprepared.

� Avoid illegible writing and striking. Useonly the allotted place for rough work.

� Before answering the question, checktwice whether the number of thequestion written correctly or not in theleft side of the margin.

� In every section answer according tochoice. If you have time after answeringall you may try for some otherproblems.

� Highlight the answers and laws bymaking boxes if necessary.

� You got more scope to get good score inPart B only. So prepare accordingly.

� The allotted time for Part B is only 30minutes. Utilize the allotted time fullyto this part only. Start writing knownanswers first, then only go for otherbits.

� Students must aware of basic factors inexamination hall like quoting correctnumber of question concerned, andindentify the key factor of the problemetc.

PREPARATION TIPS

Statement Explanation1. Circum center Concurrence point of perpendicular bisector of the

sides of the Triangle.

2. In center concurrence point of angle bisector of the Triangle.

3. Centroid Concurrence point of the medians

4. Ortho center Concurrence point of the heights

5. Basic proportionality theorem In ΔABC, DE //BC then

6. Vertical angle bisector theorem In ΔABC, the bisector of A intersects BC in D then

7. Pythagorean theorem ΔABC, =90° then AC2 = AB2 + BC2

8. Appollonius theorem R,

r are radii of two circles

d is distance between centers

ΔABC, AD is median then AB2 + AC2 = 2(BD2 + AD2)

9. d > R + r Do not intersect, Total Tangents 4

10. d = R + r Intersect at one points Total Tangents 3

11. d < R + r Intersect at two points Total Tangents 2

12. d = R – r Intersect at one points internally Total Tangents 1

13. Concentric circles No Tangent lines.

B

AB BD

AC DC=

AD AE

DB EC=

Item Formula

1. Ungrouped data mean

2. Grouped data mean

3. Mean (By short cut method)

4. Grouped data median =

5. Grouped data mode =

6. The relation between mean,median and mode Mode =3 Median – 2 Mean.

( )( )

1 1

1 2 1 2

f f c CL or L

2f f f

− Δ+ +− + Δ + Δ

N2 F

L cf

−+ ×

fdA c

N

∑+ ×

fx

n

x

n

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Mathematics Chapter wise marks weightage analysis chart