mathematics practice sheets for olympiads

1. WORK SHEET – 1 ( FOR MANASTHALI) CLASS XI
1. Find the derivative of 𝑒 𝑥
by first principle.
2. Find sum to n terms of series 0.7 + 0.77 + 0.777 + .......
3. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on a second
day, 4 more workers dropped out on 3rd
day & so on it took 8 more days to finish the work. Find the number
of days in which the work was completed.
4. If the coefficient of ar – 1
, ar
& ar + 1
in exp of (1 + a)n
are in A.P., prove that n2
– n (4r + 1) + 4r2
– 2 = 0.
5. Using P.M.T. prove that :- (1 + x)n
 1 + nx for all x > -1.
6. Find the value of cos 20o
cos 40o
cos 60o
cos 80o
.
7. Draw the graph of f(x) = [x] & find its domain & range
8. f (x) =
𝑘 cos 𝑥
𝜋−2𝑥
𝑥 <
𝜋
2
3 𝑥 >
𝜋
2
, find k if lim 𝑥 →
𝜋
2
𝑓 𝑥 exists
9. Find the rank of the word “: MOTHER”.
10. Prove that the product of lengths of the perpendiculars drawn from ( 𝑎2 − 𝑏2 , 0) & (− 𝑎2 − 𝑏2 , 0) to
line
𝑥
𝑎
cos 𝜃 +
𝑦
𝑎
sin 𝜃 = 1
WORKSHEET – 2( FOR MANASTHALI) CLASS XI
1. If 4 digit numbers greater than 5000 are formed from the digit 0, 1, 3, 5 & 7. What is the probability of family
a number divisible by 5 when digits can be repeated.
2. Solve :- x2
– (7 – i) x + (18 – i) = 0
3. Find the coefficient of x40
in (1 + 2x + x2
)27
.
4. The first, second & last terms of an A.P. are a, b, c respectively. Prove that the sum of n terms is
𝑎+𝑐 (𝑏+𝑐−2𝑎)
2 (𝑏−𝑎)
.
5. If x = a +
𝑎
𝑟
+
𝑎
𝑟2 +, , , , , , , , , ∞, y = b -
𝑏
𝑟
+
𝑏
𝑟2 + .............. ∞ , z = c -
𝑐
𝑟2 +
𝑐
𝑟4 + .............. ∞, prove
𝑥𝑦
𝑧
=
𝑎𝑏
𝑐
6. If the image of the point (2, 1) with respect to line mirror is (5, 2), find the equation of the mirror.
7. Find the centre & radius of a circle passing through (5, -8), (2, -9) & (2, 1).
8. Evaluate :- lim 𝑥→0
1−cos 𝑥 cos 2𝑥
𝑥2 .
9. Find the derivative of sin 𝑥 by first principle.
10. Prove :-
cos 8𝐴 cos 5𝐴−cos 12𝐴 cos 9𝐴
sin 8𝐴 cos 5𝐴+cos 12𝐴 sin 9𝐴
= tan 4A
11. Find domain & range of f(x) =
𝑥−2
3−𝑥

2. WORK SHEET – 3 ( FOR MANASTHALI) CLASS XI
1. Let R1 be a relation on R : (a, b)  R1 1 + ab > 0 a, b  R Show that
a. (a, 0)  R1 for all a  R
b. (a, b)  R1 ⇒ (b, a)  R1 for all a, b  R
2. Find domain & range of f(x) =
𝑥2
𝑥−4
3. If 10 sin4
 + 15 cos 4
 = 6, find the value of 27 cosec 6
 + 8 sec6
 .
4. Evaluate :- lim 𝑥 →2 𝑓(𝑥)if f(x) =
𝑥 − [𝑥] 𝑥 < 2
4 𝑥 = 2
𝑥 − 5 𝑥 > 2
5. Prove that
𝑑
𝑑𝑥
𝑖𝑛 𝑥−𝑥 cos 𝑥
𝑥 sin 𝑥+cos 𝑥
=
𝑥2
(𝑥 sin 𝑥+cos 𝑥)2
6. The letters of word SOCIETY are placed at random in a row. What is the probability that three vowels come
together?
7. Find the probability of getting an even numbers on first die or a total of in a single throw of two dice.
8. An arc is in form of semi ellipse. It is sin wide & 2m high at centre. Find the height of arch at a point 1.5 m
from one end.
9. If a is the A.M. of b & c and two G.M. are G1 & G2, prove G1
3
+ G3
2 = 2abc
10. Find sum of 24 terms of AP if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
WORK SHEET – 4( FOR MANASTHALI) CLASS XI
1. If the first term of an A.P. is z & sum of first 5 terms is equal to ¼ of sum of rent 5 terms. Find S30 .
2. Prove that :-
tan 3𝑥
tan 𝑥
never lies between
1
3
& 3.
3. Prove that :- sin2
72 – sin2
60 =
5− 1
8
.
4. Find real value of x & y:- (x4
+ 2xi) – ( 3x2
+ iy) = (3 – 5i) + (1 + 2iy) .
5. Solve :-
5𝑥
4
+
3𝑥
8
>
39
8
,
2𝑥−1
12
−
𝑥−1
3
<
3𝑥+1
4
6. The letter of word “RANDOM” are written in all possible orders in dictionary. Find the rank of word “
RANDOM”.
7. Prove by PMI that :-
𝑥5
5
+
𝑥3
3
+
7𝑛
15
is a natural number for all n  N .
8. Using concept of slope, prove that the line joining the mid points of the two sides of a triangle is 11 to third
side.
9. Evaluate :- lim 𝑥 →2
𝑥2− 4
3𝑥−2− + 2
10. Prove that :- lim 𝑥 →
𝜋
4
𝑡𝑎𝑛 3 𝑥−tan 𝑥
cos (𝑥+
𝜋
4
)
= -4
11. Find the derivative of cos3
x by first principle.
12. Three dice are thrown together. Find the probability of getting a total of at least 6.

3. WORKSHEET – 5( FOR MANASTHALI) CLASS XI
1. f(x) =
5𝑥
𝑥 − 2𝑥2 𝑥 ≠ 0
0 = 0
does lim 𝑥 →0 𝑓(𝑥) exist ?
2. if a1, a2, a3 ....... an are in A.P. where ai > 0. Show that
1
𝑎1+ 𝑎2
+
1
𝑎2+ 𝑎3
+ … . +
1
𝑎 𝑛−1+ 𝑎 𝑛
=
𝑥− 1
𝑎1+ 𝑎 𝑛
.
3. the natural number are grouped as follows : (1), (2, 3), (4, 5, 6), (7, 8, 9, 10) ....... find the first term of nth
group.
4. If z1 , z2 are complex numbers :-
𝑧1− 3𝑧2
3− 𝑧1 𝑧2
= 1 & 𝑧2 ≠ 1, then find 𝑧1 .
5. Find all non zero complex numbers z satisfying 𝑧 = iz2
.
6. If 𝑛 𝑐𝑟 : 𝑛 𝑐𝑟+2 = 1 : 2 : 3 find n & r.
7. Find the derivative of tan 𝑥 by first principle.
8. Solve :- sin 3x + cos 2 x = 0
9. In a triangle ABC. If a cos A = b cos B, then prove that either triangle is isosceles or right angled.
10. Prove :- 3 cosec 20o
– sec 20o
= 4.
11. Find the least value of n for which 1 + 3 + 32
+ ............. to n terms is greater than 7000.
12. Find the derivative of
a. sin
𝑥2
3
− 1 b. log (sec x + tan x) c.
1
𝑎2− 𝑥2
.