# 2015 Grade 10 MTAP Sectoral Level

This is the 2015 Metrobank-MTAP-NCR Math Challenge Sectoral Level for  Grade 10. Please inform me if you see any errors.

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Easy

1.) What is the next term in the arithmetic sequence whose first two terms are −18 and then 19?

2.) Express $-3 + \log_2 x + 3 \log_2 y$ as a single logarithm with base $2$.

3.) What should be the value of $c$ if the point where $f(x) = -x^2 + 2x + c$ attains its maximum, is also a zero of $f$?

4.) What is the reciprocal of $\sqrt[3]{9}- 2$ in simplest terms?

5.) If $\log_x25 = 4$, what is $\log_25x$

6.) What is the smallest positive integer n which satisfies $n^{20} \geq 5^{30}$?

7.) Find $k$ such that the circle $x^2 + y^2 = 4$ intersects the parabola $y = x^2 + k$ in exactly three points.

8.) Find the value of $\theta$ in $[0^\circ, 180^\circ]$ assuming $\frac {\sin20^\circ}{\cos70^\circ} = \tan \theta$

9.) The cube of a number $x$ equals $2015^{15}$. What is the product of $x$ and the square of $2015^{14}$?

10.) Find the smallest positive value of $\theta$ such that $\sin(x - \theta) = \cos \theta$.

11.) Which real number is not attained by the function $f(x) = \frac {2x + 4}{1 - 6x}$?

Average

1.) Find the domain of $f(x) = \dfrac {2 - \log_3x}{3-\log_2x}$

2.) Find the 12th term of the harmonic sequence 3/2, 3/7, 4, …

3.) What is the numerical value of $\log_8 (\cos(\frac {7 \pi}{4}))$?

4.) An isosceles right triangle, with area $n$ sq. cm, has a hypotenuse which also measures $n$ cm. Find $n$.

5.) A linear function $f$ is such that $f(2015) -f(2005) = 100$. What is $f(2051) - f(2015)$?

6.) If $f(1+\ln x) = 2x$, what is $f(x)$?

Difficult

1.) The numerical value of $\log_32015$ is in between which consecutive integers?

2.) Evaluate the sum $\sin 30^\circ + \sin60^\circ +\sin90^\circ + \cdots + \sin 510^\circ + \sin540^\circ$.

3.) The roots of $x^3 +ax^2 +b x - 18$ are prime numbers. Find $a + b$.

4.) Solve for $x$ in the equation $\sqrt{2x + 3} + \sqrt{-x} = 2$

5.) How many positive integers n are there such that 1n and n/3 are three-digit integers?

6.) For n ≥ 2, the nth term of the sequence equals the sum of all the terms before it. If the eight therm is 320, what the first term of the sequence?

Clincher Questions

1.) Solve for x: $(\ln x)^2 = 3 + \ln x^2$.

2.) For which real x can we sum the geometric series x, −1, 1/x, …to infinity?

3.) If $p$, $q$, $r$ are the prime factors of 2015, find $\log_7(p + q + r)$

Do or Die Question

1.) Find all possible values of $x$ so that the sequence $x + 6$, $6$, $x + 1$ becomes geometric.

Answers

Easy
1.) 56
2.) $\log_2 (\frac {xy^3}{8}$)
3.) -1
4.) $3 \sqrt[3]{3}+ 2 ^3\sqrt[3]{9} + 4$
5.) 1/4
6.) 12
7.) -4
8.) 45°
9.) $2015^{33}$
10.) $\frac {3 \pi}{2}$
11.) $-\frac {1}{3}$

Average
1.) (0, 8) ∪ (8, inf)
2.) 1/19
3.) -1/6
4.) 4
5.) 360
6.) $2e^{x-1}$ or equivalent

Difficult
1.) 6 and 7
2.) $2 + \sqrt{3}$
3.) 13
4.) -1, -1/9
5.) 67
6.) 5

Clincher
1.) $e^3, \frac{1}{e}$
2.) x > 1 or x < −1
3.) 2

Do or Die
1.) -10, 3

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