GRADES 11 & 12
1. Find 𝑓 ′ (1), 𝑖𝑓 𝑓(𝑥 ) =
A) 1/e
ln (𝑥)
𝑒𝑥
B) e-1
C) 2e
D) √2
2. What is the average number of pairs of consecutive integers in a randomly
selected subset of 5 distinct integers chosen from the set { 1, 2, 3, …, 30}.
(For example the set {1, 17, 18, 19, 30} has 2 pairs of consecutive integers.)
A) 2/3
B) 5/6
C) 29/30
D) 1
2
log2 √3+√2
3
3. Simplify: 8
A) 15
B) 7
· 27
1
log3 (3−√2)
3
C) 12
D) 6
4. How many squares whose sides are parallel to the axes and whose vertices
have coordinates that are integers lie entirely within the region bounded by
the line
, the line
and the line
A) 30
B) 45
C) 50
D) 64
5. Find 𝑓’(3) + 𝑓’(8) , if 𝑓(𝑥) = 6𝑥√1 + 𝑥
A) 27.5
B) 23
6. Find the limit
A) 0
B) -1
C) 65
lim
D) 42.5
√𝑥2 +2𝑥+5+√𝑥2 −2𝑥+5
𝑥→+∞
C) 1
2𝑥+1
D) 2
7. If the equation 2 ∙ 32𝑥 − (𝑘 − 3) ∙ 3𝑥 − 2 ∙ (𝑘 + 1) = 0 has no solution.
Find all The set of values of such 𝒌 parameter.
A) [−1; 3]
B) (−∞; −1]
C) (-1;1)
D) R
8. The tangent line passing through the point A on the graph of the
function 𝑦 = 𝑥 2 − 4𝑥 − 7 is parallel to the line given by the equation
5𝑥 + 3𝑦 = 0 Find the x coordinate of the point A.
A) 7/6
B) -4/3
C) 1/2
D) 17/10
9. It is known that the minimum value of a 𝑥 2 + 𝑝𝑥 + 𝑞 is equal to 10. Find
the minimum possible value of the 𝑝 + 𝑞.
A) 5
B) 6
C) 8
D) 9
10.Each of the 20 balls is tossed independently and at random into one of the
5 bins. Let p be the probability that some bin ends up with 3 balls,
another with 5 balls, and the other three with 4 balls each. Let q be the
probability that every bin ends up with 4 balls. What is p/q?
A) 2
B) 4
C) 8
D) 16
11. What is the sum of the digits of the largest prime factor of 16383 ?
A) 7
B) 10
C) 16
D) 18
12.A person appears for an examination in which there are four papers with
maximum of ‘m’ marks from each paper. Find the number of ways in
which one can get 2m marks.
1
B) (𝑚 + 1)(2𝑚2 + 4𝑚 + 1)
A) 2𝑚 + 3
1
2
C) (𝑚 + 1)(2𝑚 + 4𝑚 + 3)
3
3
1
D) (𝑚 + 1)(2𝑚2 + 4𝑚 + 5)
3
13.Let S(n) equal the sum of the digits of positive integer . For example,
S(1507)=13. For a particular positive integer , S(n)=1274. Which of the
following could be the value of S(n+1)?
A) 12
B) 1239
C) 1265
D) 1333
14. For any integer 𝑛 ≥ 9 , the value of
following?
A) a multiply of 4
C) a perfect square
(𝑛+2) !−(𝑛+1)!
𝑛!
is always which of the
B) a prime number
D) a multiply of 10
15. Triangle ABC has side lengths AB = 11, BC=24, and CA = 20. The bisector of
the angle <BAC intersects BC at point D, and intersects the circumcircle of
the triangle ABC at point E≠A. The circumcircle of the triangle BED
intersects the line AB at points B and F≠B. What is CF?
A) 28
B) 30
C) 32
D) 20√2
16. Let a,b,c be positive integers such 𝑎 + 𝑏 + 𝑐 = 23 and
𝑔𝑐𝑑(𝑎, 𝑏) + 𝑔𝑐𝑑(𝑏, 𝑐) + 𝑔𝑐𝑑(𝑐, 𝑎) = 9 . What is the sum of all possible
distinct values of 𝑎2 + 𝑏2 + 𝑐 2 ?
A) 259
B) 438
C) 516
D) 625
17.The product of the lengths of the two congruent sides of an obtuse
isosceles triangle is equal to the product of the base and twice the
triangle’s height to the base. What is the measure, in degrees, of the vertex
angle of this triangle?
A) 120
B) 135
C) 135
D) 150
18. Let 𝑐 =
A) -1
2𝜋
11
. what is the value of :
𝑠𝑖𝑛3𝑐 ∗ 𝑠𝑖𝑛6𝑐 ∗ 𝑠𝑖𝑛9𝑐 ∗ 𝑠𝑖𝑛12𝑐 ∗ 𝑠𝑖𝑛15𝑐
𝑠𝑖𝑛𝑐 ∗ 𝑠𝑖𝑛2𝑐 ∗ 𝑠𝑖𝑛3𝑐 ∗ 𝑠𝑖𝑛4𝑐 ∗ 𝑠𝑖𝑛5𝑐
B) 10/11
C)
√11
5
D) 1
19. Jimmy chooses a real number uniformly at random from the interval
[0,2022]. Independently, Jack chooses a real number uniformly at random
from the interval [0,4044]. What is the probability that Jack's number is
greater than Jimmy's number?
A) 1/2
B) 2/3
C) 3/4
D) 4/5
20. All the roots of the polynomial 𝑥 5 − 10𝑥5 + 𝐴𝑥 4 + 𝐵𝑥 3 + 𝐶𝑥 2 + 𝐷𝑥 + 16
are positive integers. What is the value of B?
A) -88
B) -80
C) -64
D) -41
21. For some positive integer , the number
has
positive integer
divisors, including and the number
. How many positive integer
divisors does the number
have?
A) 110
B) 191
C) 261
D) 325
22. A function f is defined recursively by 𝑓(1) = 𝑓(2) = 1 and
𝑓(𝑛) = 𝑓(𝑛 − 1) − 𝑓(𝑛 − 2) + 𝑛 for all integers 𝑛 ≥ 3. What is 𝑓(2022) ?
A) 2022
B) 2024
C) 4044
D) 2022*2022
23. Right triangle ABC has side lengths BC=6, AC=8, AB=10. A circle centered at
O is tangent to the line BC at B and passes through A. A circle centered at P
is tangent to line AC at A and passes through B. What is OP ?
A) 23/8
B) 29/10
C) 35/12
D) 73/25
24. What is the maximum value of
A) 1/9
B) 1/10
( 2𝑥 −3𝑥)𝑥
C) 1/11
4𝑥
for real values of x?
D) 1/12
25. Let ABCD be a convex quadrilateral with BC=2 and CD=6. Suppose that the
centroids of ABC, BCD and ACD from the vertices of an equilateral triangle.
What is the maximum possible value of the area of ABCD?
A) 27
B) 16√3
C) 12 + 10√3
D) 30