Metrobank-MTAP-DepEd Math Challenge 2013
NCR Sectoral Orals • Third Year • Category A 15-Second Questions [2 points each]
1. If n n is a positive integer, what is the value of ( −1)4
+3
n
2. Factor completely: completely: x6 − 1.
+ (−1)2
+4
n
?
[0]
[ (x − 1)(x 1)(x + 1)(x 1)(x2 − x + x + 1)(x 1)(x2 + x + x + 1) ]
3. The 1st 1st and 5th 5th terms terms of an an arithme arithmetic tic seque sequence nce are are 3 and 23, 23, respect respectiv ively ely.. Find Find the 3rd 3rd term. term.
[ 13 ]
4. A circle circle is inscri inscribed bed in a square square.. If the circle circle has area 36π 36π cm2 , what is the area of the square? [ 144 cm2 ] 5. As two chords chords of a circle interse intersect, ct, one chord is divided into into segments segments of lengths 8 and 15, and the other chord into segments of lengths 10 and x and x + + 2. Find Find x. [ 10 ] 6. What is the center center of the circle circle x x 2 − 6x 6x + y + y 2 + 12y 12y + 38 = 0? 7. If 100
2
x
[ (3, −6) ]
= 104 , find x.
[ 0, 2 ]
x
8. The ratio ratio of the vertex vertex angle angle to a base base angle of an isoscele isosceless triangle triangle is 5 : 2. Find Find the base angle. angle. [ 40 ] ◦
9. What are the coordinates coordinates of the vertex vertex of the parabola y parabola y = −2(x 2(x + 2)2 − 6 ?
[ (−2, −6) ]
10. What is the area of the largest triangle triangle that can be inscribed inscribed in a semicircle semicircle with radius radius 14 cm? [ 196 cm2 ] 11. For what values values of the constant constant k will x2 − kx + kx + 3 = 0 have two distinct real roots?
√
√
[ k > 2 3 or k < −2 3 ]
30-Second Questions [3 points each]
5 3 27 + + + ·· [ 50 50//3 ] · · · . 3 2 20 2. An equilateral equilateral triangle triangle is inscribed in a circle having having radius 6 cm. Find the area of the triangle. triangle. 1. Find the sum of the infinite geometric geometric series series
√
[ 27 3 cm2 ] 3. A line and the coordinate axes form a triangle in the first quadrant, quadrant, having area 24 square units. units. If the y-intercept is thrice the x the x-in -interce tercept, pt, give the equation equation of the line in slope-interce slope-intercept pt form. [ y = −3x +12 ] 4. A point outside a circle is 11 units away from from its center. If a tangent segment segment from the point√ to the circle is 9 units, what is the circumference of the circle in terms of π? [ 4 10 10π π units ] 3x+2
5. For what values values of x x is
1 4
> 8 4
+7
x
?
[ x < −25 25//18 ]
6. A rectangle rectangle has dimensions dimensions 2x 2x cm and 12 − x cm. cm. If x If x is free to vary, what is the largest possible area the rectangle can have? [ 72 cm2 ] 1-Minute Questions [5 points each]
· · · + f 1. If f ( f (x) = 2x + 1912, find f find f (1) (1) + f + f (2) (2) + f + f (3) (3) + ·· + f ((100).
[ 201300 ]
2. Find the constan constantt k if the circle x circle x 2 + y 2 − kx + kx + 4ky 4ky − 4k 4k2 + 7 = 0 passes through (4, (4, 3).
[ −2, 4 ]
3. Two Two towers towers have have heigh heights ts 35 ft and 15 ft. Wires Wires are attached attached from from the top of each each towe towerr to the base of the the othe otherr tow tower. er. H Hoow high high from from the the grou ground nd is the the inte inters rsec ecti tion on of thes thesee two wire wires? s? [ 10 10..5 ft ] 4. Three numbers numbers in arithmetic arithmetic sequence sequence have sum 9 and product − 48. What What is the largest largest of the three three numbers? [8] 5. Three cylindri cylindrical cal cans, each with radius radius 5 cm, are tied together with a rope. How long is the rope around the cans? [ 10π + 30 cm ] 6. If m If m and a nd n are the roots of x2 − 6x 6x + 6 = 0, find the quadratic equation whose roots are and . The 2 2 coefficients should be integers and the coefficient of x x positive. [ x − 4x 4x + 1 = 0 ] C.1. Find the area of a rhombus rhombus whose diagonals diagonals are 8/ 8/3 cm and 9 cm. C.2. The area area of a right right triang triangle le is 24 ft2 . If its hypotenuse is 10 ft, find its perimeter. C.3. The graphs graphs of y = x3 + 3x 3 x2 points?
− 10x 10x + 6 and y
m
n
n
m
[ 12 cm2 ] [ 24 ft ]
= x2 + 5x 5 x + 6 inter intersec sectt at three three points. points. What What are these these [ (0, 6), 6), (3, (3, 30), 30), (−5, 6) ]
DoD. In a geometric sequen sequence, ce, the 3rd term is 20 and the 7th term is 320. If the sum of the first 9 terms is 855, what is the sum of the first 10 terms? [ −1705 ]
3.A.1.
If f (x) = 2x + 1912, find f (1) + f (2) + f (3) + · · · + f (100).
the constant k if the circle x2 + y 2 − kx + 4ky − 4k 2 +7 = 0 passes through (4, 3).
3.A.2. Find
3.A.3. Two
towers have heights 35 ft and 15 ft. Wires are attached from the top of each tower to the base of the other tower. How high from the ground is the intersection of these two wires?
3.A.4. Three
numbers in arithmetic sequence have sum 9 and product − 48. What is the largest of the three numbers?
3.A.5. Three
cylindrical cans, each with radius 5 cm, are tied together with a rope. How long is the rope around the cans?
3.A.6.
If m and n are the roots of x2 − 6x + 6 = 0, find the quadratic equation whose n x2 roots are m n and m . The coefficients should be integers and the coefficient of positive.
graphs of y = x3 + 3x2 − 10x + 6 and y = x2 + 5 x + 6 intersect at three points. What are these points?
3.A.C.3. The
3.A.DoD. In
a geometric sequence, the 3rd term is 20 and the 7th term is 320. If the sum of the first 9 terms is 855, what is the sum of the first 10 terms?
