https://youtu.be/h3Wy4RAs8Ls

THE NAMAKKAL TEACHERS VIDHYAASHARAM MAT HR SEC SCHOOL

1. If A and B are two sets so that n(B – A) = 2n (A – B) = 4n(A ∩ B) and if n(A ∪ B) = 14, then find n(P(A)).
1. SETS, RELATIONS AND FUNCTIONS

2. In the set Z of integers, define mRn if m-n is divisible by 7. Prove that R is an equivalence relation.

  
3. Find the range of the function .
4. If ∶ ℝ → ℝ is defined by  = 2 − 3 prove that is a bijection and find its inverse.
5. If : ℝ → ℝ is defined by  = 3 − 5, prove that is a bijection and find its inverse.
6. Let ,  ∶ ℝ → ℝ be defined as  = 2 − || and  = 2 + ||. Find
.
7. From the curve
= ||, draw (i)
= | − 1| + 1 (ii)
= | + 1| − 1 (iii)
= | + 2| − 3.

=  , !, ", !, !, "#;  ∶ % → %. (ii) If X = {x, y, z} and
8. State whether the following relation are function or not. If it is a function check for one-to-oneness and ontoness. If it is

= ,
, , ', ', #;  ∶ ( → (.
not a function, state why?(i) If A = {a, b, c} and

9. From the curve
= )*+ , graph the functions (i)
= )*+ − (ii)
= −)*+ − (iii)
= )*+ , + . which is cos x
-

(iv)
= )*+ , − . which is also cos x (refer trigonometry)
-

CHAP-2 BASIC ALGEBRA
= = , then prove that 
' = 1.
/0 /0 1 /0 2
9. Prove that √3 is an irrational number.
1 2 2 1
1. If
2. If one root of 4 − 1 = 5 − 7 is double the other root, show that 4 = 2 or – 25.
3. Find the square root of 7 − 4√3. − + − +
    
7 √8 √8 √9 √9 √: √: √; √; 
12. Simplify : .
4. Use the method of undetermined coefficients to find the sum of 1 + 2 + 3 +……….++ − 1 + +, + ∈ ℕ.
 > @ @  > @;  9@
> ?> > ; @: > @ 8. @ @ > 
7
5. 6. (7) 7.
9. Solve the linear inequalities and exhibit the solution set graphically  +
≥ 3, 2 −
≤ 5, − + 2
≤ 3.
10. 2 + 3
≤ 6,  + 4
≤ 4,  ≥ 0,
≥ 0 11.  − 2
≥ 0,2 −
≤ −2,  ≥ 0,
≥ 0
12. 2 +
≥ 8,  + 2
≥ 8,  +
≤ 6. 13. !
) , + . , when )*+ H = .
- G 8
F  I


1. Prove that cos (J + K= cos J !
) K-sin J sin K
CHAP-3 TRIGONOMETRY

3. Solve !
) H + !
)L! H = √3 8. Solve √3 M + H + N√3 − 1O M + H − 1 = 0.
2. If A+B=45°, show that (1 + tan A) (1 + tan B) = 2.

= =
? T 
PQR S PQR U PQR V
4. State and prove Sine formula. (OR) prove that
.= !
M
S U ? T V
 ?@T 
5. State and prove Napier’s formula. (OR) Prove that tan,
T > @ > ?>
6. State and prove Cosine formula. (OR) Prove that cos % = 7. In a ∆ABC, prove that )*+ , .= !
) .
U V T  S
T  ? 


2. Prove that nXY +nXY  =n+1XY
CHAP-4 COMBINATORICS AND MATHEMATICAL INTRODUCTION
1. If10 Pr – 1 = 2 x 6 Pr , find r.
3. By the principle of mathematical induction, prove that for all integers + ≥ 1, 1 + 2 + 3 + ⋯ + + =
[[@[@
:
+ + +…….+ =
    [
. .7 7.F [[@ [@
4. Using the Mathematical induction, show that for any natural number n, .
5. Prove that 3[@ − 8+ − 9 is divisible by 8 for all + ≥ 1 by using the principles of mathematical induction.
6. By the principle of Mathematical induction, prove that, for + ≥ 1, 1.2 + 2.3 + 3.4 +……++. + + 1 =
[[@[@
7
.
7. Use induction to prove that +7 − 7+ + 3, is divisible by 3, for all natural numbers n.
8. Prove that for any natural number +, [ − " [ is divisible by − ", where a > b.

NTVMHSS
9. By the principle of mathematical induction, prove that, for + ≥ 1 1 + 3 + 5 +……..+2+ − 1 =
10. Using the induction, show that for any natural number n,
11. By the principle of mathematical induction, prove that 1 + 2 + 3 + ⋯ + + =


..7
+

.7.F
+

7.F.;
, for all integers + ≥ 1.
+….. +
[[ 

[.[@.[@
=
[[ [@

[[@7
F[@[@
7
.
.




1. The 2[] , 3Y] and 4^_ terms in the binomial expansion of  + [ are 240, 720 and 1080 for a suitable value
CHAP-5 BINOMIAL THEOREM, SEQUENCE AND SERIES

of x. Find x, a and n.
a ;
2. Find the coefficient of  ; in ,  + . . 3. Find the constant term of ,2 7 − >. .
 
` 7
4. If the binomial coefficients of three consecutive terms in the expansion of  + [ are in the ratio 1 : 7 : 42, then find n.
5. Prove that √ 7 + 7 − √ 7 + 4 is approximately equal to > when x is large.
` ` 


6. Prove that √ 7 + 6 − √ 7 + 3 is approximately equal to
` ` 
> when x is sufficiently large.
>
7. Prove that b is approximately equal to 1 −  +

@ 
when x is very small.

1. If the equation c − 10
+ 12
 + 5 − 16
− 3 = 0 represent a pair of straight lines, find (i) The value of λ and the
CHAP-6 TWO DIMENSIONAL ANALYTICAL GEOMETRY




2. Show that the equation 2  − 
− 3
 − 6 + 19
− 20 = 0. represent a pair of straight lines. Show further that the
separate equations of the lines. (ii) Point of intersection of the lines (iii) Angle between the lines.




MGK
3. The slope of one of the straight lines   + 2ℎ
+ "
 = 0 is twice that of the other, show that 8ℎ = 9 ".
angle between them is tan-1(5)

4. Find e and f, if the following equations represents a pair of perpendicular lines 6  + 5
− e
 + 7 + f
− 5 = 0.