Bhavitha 24.02.2011 EM Maths
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Transcript of 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'
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
2
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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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°
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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×
×
⎛ ⎞⎜ ⎟ =⎜ ⎟⎜ ⎟⎝ ⎠
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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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
Ýë„ìS ¿¶æÑ™èl VýS$Æý‡$ÐéÆý‡… 24 íœ{ºÐ]lÇ, 2011
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