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- Find the sum of all the prime factors of 𝑁 if: $$\log_3 \log_5 (\frac{N}{2})=6$$
A. 6
B. 5
C. 7
D. 3
E. 2 - If 𝑁 = 5! 8! 9! then find the number of factors of N which are square integers.
A. 144
B. 24
C. 72
D. 128
E. 180 - Find the value of: $$X = \cos (57) \cos (27) + \sin (57) \sin (27)$$ If $\cos (57) = 0.5446, \cos (27) = 0.8910$ and all values in side sin and cos are in degrees.
A. 0.1045
B. 0.5
C. $\frac{\sqrt{3}}{2}$
D. $\frac{1}{\sqrt{2}}$
E. 1 - If $x+\frac{1}{x}=\sqrt{3}$ then find the value of $x^3+\frac{1}{x^3}$.
A. 1
B. $3\sqrt{3}$
C. $-\sqrt{3}$
D. 0
E. 3 - Find the following sum if 𝑖 = $\sqrt{−1}$: $$𝑆 = 𝑖 + 2𝑖^2 + 3𝑖^3 + ⋯ + 2020𝑖^{2020} + 2021𝑖^{2021}$$
A. 1011 + 1010𝑖
B. 1010 + 1011𝑖
C. 1010 − 1010𝑖
D. 2020 + 2021𝑖
E. −1011 − 1011𝑖 - If $a>b>1$ and $\frac{1}{\log_a b}+\frac{1}{\log_b a}=\sqrt{293}$, then find the value of: $$\frac{1}{\log_{ab} b}-\frac{1}{\log_{ab} a}$$
A. 18
B. 17
C. 1
D. 29
E. 19 - A tower 40m tall, has an elevation of 45 degrees from a point B on the ground, and 53 degrees from a point C on the ground. If point A lies on the base of the tower 40m tall, then what is the distance between point B and C on the ground if ∠𝐵𝐴𝐶 = 90° (Given sin(37) = $\frac{3}{4}$).
A. 50m
B. 40m
C. 40$\sqrt{2}$m
D. 25$\sqrt{3}$m
E. 37m - In the adjoining figure, 4 semi-circles are constructed and the area of overlap between any 2 semi-circles is shaded purple. Find the area of the shaded region if the side of the square is 10.
A. 50𝜋 − 50
B. 100 − 25𝜋
C. 25𝜋 − 50
D. 50𝜋 − 100
E. 100𝜋 − 100 - A quadratic equation of the form $𝑓(𝑥) = 𝑥^2 + 𝑏𝑥 + 𝑐$ opens upwards and has its vertex 5 units above the x-axis and 7 units to the right of the $y$-axis. If a linear equation 𝑔(𝑥) = 𝑚𝑥 + 10 were added to the original equation to give ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥). Find the x coordinate of the vertex if the graph of h(x) just touches the positive $x$ axis.
A. 7
B. 5
C. 6
D. 10
E. 8 - In a class of 40 students, 20 boys and 20 girls, when given a choice to play either football or chess 5 girls and 12 boys preferred football with the rest choosing chess. What is the probability that the winner of the chess tournament held between the students who choose chess is a girl if all students are equally likely to win?
A. $\frac{3}{8}$
B. $\frac{1}{2}$
C. $\frac{15}{23}$
D. $\frac{12}{17}$
E. $\frac{4}{5}$ - If $\frac{1}{x+1}+\frac{12}{y+12}+\frac{144}{z+144}=1$, then find the value of $\frac{x^2}{x^2+x}+\frac{y^2}{y^2+12y}+\frac{z^2}{z^2+144z}$.
A. 3
B. 12
C. 157
D. 2
E. 11 - Find the product of all integers 𝑛 ≥ 1 such that $\frac{n^3+9}{n^2+13}$ is an integer:
A. 2
B. 13
C. 65
D. 22
E. 26 - In the adjoining figure, if value of the larger 2 circles is 72 and 18 then find the radius of the smallest circle. It is known that all circles are tangent to each other and the common ray B.
A. 4.5
B. 14.4
C. 8
D. 6
E. 7.2 - A person invests $\$$100 every month in a Recurring deposit which compounds monthly promising a 10% increase, in the principal invested, every year. Find the total value to the recurring deposit after 4 years or after the 48th instalment is paid. (Given $1.1^{\frac{1}{12}}$ = 1.00797)
A. $\$$582.3
B. $\$$766
C. $\$$7660
D. $\$$5823
E. $\$$4800 - If 𝑎, 𝑏, 𝑐 ∈ 𝑅, 𝑎 + 𝑏 + 𝑐 = 36, $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=39$ find: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a}{c}+\frac{c}{b}+\frac{b}{a}$$
- Find $abc$ if $a, b, c$ are distinct prime numbers, and the following is given: $$𝑎 + 𝑏 + 𝑐 = 92 $\text{ and } 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 = 2201$$
- If $a$ and $b$ are relatively prime positive integers, then find the value of $a + b$ if: $$\frac{a^3-b^3}{(a-b)^3}=\frac{133}{3}$$
- In the adjoining figure, line segments DE is constructed perpendicular to the diameter BC of the circle intersecting BC at point P. Find the length of the line segment if the radius of the circle is 10 and PB = 2.
- 10 lines are drawn on a plane. Find the maximum number of unique line segments that can be identified from these 10 lines on the plane. Note: the line segment identified must be between 2 points that already lies on one of the 10 lines and AB is equivalent to BA.
- In the adjoining figure area of the regions are mentioned in the figure (not to scale). Find the value of $m + n$, if $\frac{𝑚}{𝑛}$ is the unknown area $x$, of the quadrilateral, in the lowest terms.
- Find the maximum value of the following expression. $$\sqrt{781-X}+\sqrt{X-59}$$
- Find the ordered set of 4-tuples, of positive integers, [$w, x, y, z$] such that: $$𝑤 + 𝑥 + 𝑦 + 𝑧 = 21$$
- If the following is true: $$\frac{3+\cot 78\times\cot 18}{\cot 78+\cot 18}=\frac{\tan n+\sin m}{\cos m}$$ Where $n$ and $m$ are in degrees and are acute angles, find the product $mn$.
- Find the value of: $$X=\sum_{n=1}^{7}\tan^2 \frac{n\pi}{16}$$
This topic was modified 3 minggu ago by Admin dot
Posted : 19/06/2026 9:14 am
