NSMQ Past Questions and Answers – Mathematics

The National Science and Math Quiz, NSMQ is an annual Science and Mathematics based national level quiz competition for high schools in Ghana.

The NSMQ competition encourages quick thinking, problem solving, and ultimately, a healthy academic rivalry among students across the country.

NSMQ Competition Structure

The NSMQ competition consist of five rounds:

NSMQ Round 1

Fundamentals: This includes questions across Biology, Chemistry, Physics, and Mathematics.

All the contesting schools usually receive 4 Biology, 4 Chemistry, 4 Physics, and 4 Mathematics questions, making 16 questions to answer.

NSMQ Round 2

Speed Race: This consist of quick-response applied questions. The Speed Race questions are thrown to all the contesting schools, and the first school to answer gets full three (3) points.

NSMQ Round 3

Problem of the Day: Each contesting school gets one high stake question worth 10 points. These contestants must solve a single question within 3 minutes.

NSMQ Round 4

True or False: Turn based statements are given to contestants with penalties for wrong answers.

NSMQ Round 5

Riddles: These are clue based, progressively scored riddles. Clues are given to the contesting schools. The schools are to compete against each other to find the answers to these riddles.

NSMQ Math Past Questions

Here are some frequently asked National Science And Math Quiz questions with respect to Mathematics. Answers to these NSMQ questions have been provided as well.

A. Find the values of the constants a and b if the straight lines

1. ax + 5y = 3, and 4x + by = 1 intersect at the point (2, -1)

ANSWER: a = 4, b = 7

[2a – 5 = 3, 2a = 8, a = 4, 8 – b = 1, b = 7]

2. 3x + ay = 4, and bx – 5y = 6 intersect at the point (2, -2)

ANSWER: a = 1, b = -2

[6 – 2a = 4, 2a = 2, a = 1, 2b + 10 = 6, 2b = -4, b = –2] 

3.  ax + 2y = -3, and 2x + by = 6   meet at the point (1, 2)

ANSWER: a = -7, b = 2

[a + 4 = -3, a = -7, 2 + 2b = 6, 2b = 4, b = 2]

B.

1. Factorize completely a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

ANSWER: (a + b)⁴

2. In how many ways can 5 persons be seated in a row of 5 seats.

ANSWER: 120

[5! = 5 x 4 x 3 x 2 x 1 = 20 x 6 = 120]

3. Find the coordinates of the point of inflexion of the curve y = x³ – 6x² + 16x. 

ANSWER: (2, 16)

[dy/dx = 3x² – 12x + 16, d²y/dx² = 6x – 12 = 0, x = 2, y = 8 – 24 + 32 = 16, (2, 16)]

C. Find the degree measures of the interior angles of a triangle if 

1. the exterior angles are in the ratio 3: 4: 5            

ANSWER: 90°, 60°, 30°

[3x + 4x + 5x = 12x = 360, x = 30, exterior angles are 90, 120, 150, interior angles are 180 – 90 = 90, 180 – 120 = 60, 180 – 150 = 30]

2. the exterior angles are in the ratio 11: 12: 13   

ANSWER: 70°, 60°, 50°

[11x + 12x + 13x = 36x = 360, x = 10, exterior angles are 110, 120, 130, interior angles are 180 – 110 = 70, 180 – 120 = 60, 180 – 130 = 50]

3. the exterior angles are in the ratio 5: 6: 7   

ANSWER: 80°, 60°, 40°

[5x + 6x + 7x = 18x = 360, x = 20, exterior angles are 100, 120, 140, interior angles are 180 – 100 = 80, 180 – 120 = 60, 180 – 140 = 40]

D. Find the values of A, B, C such that

1. 9x² + 12x + A = (3x + B)²

ANSWER: A = 4, B = 2

[ 9x² + 12x + A = 9x² + 6Bx + B², 12 = 6B, B = 2, A = B² = 2² = 4]

2. 4x² + 16x + 25 = A(x + B)² + C

ANSWER: A = 4, B = 2, C = 9

[4x² + 16x + 25 = 4(x² + 4x) + 25 = 4(x + 2)² + 25 – 16, A = 4, B = 2, C = 9]

3. 5x² – 30x – 6 = A(x + B)² + C

ANSWER: A = 5, B = -3, C = -51

[5(x² – 6x) – 6 = 5(x – 3)² – 45 – 6 = A(x + B)² + C, A = 5, B = -3, C = – 51]

E.

1. Find the sum to infinity of the series 5 – 5/3 + 5/9 – 5/27 + . . .

ANSWER: 15/4, or 3.75

[exponential series a = 5, r = -1/3, S_∞ = a/(1 – r) = 5/(1 + 1/3) = 15/4]

2. Find the solution set of the inequality 2x² – 3x – 5 > 0.

ANSWER: {x: x > 5/2, or x < – 1}

[2x² – 3x – 5 = (2x – 5)(x + 1) > 0, x > 5/2 or x <- 1]

3. Find the sum in radians of the interior angles of a polygon of 17 sides.

ANSWER: 15π radians

[(n – 2)π = (17 – 2)π = 15π radians]

F. Solve the equation for x from the logarithmic equation

1. log₆x + log₆x² = 3

ANSWER: x = 6

[ log₆x + 2log₆x = 3log₆x = 3, log₆x = 1, x = 6]

2. log₃x – log₃(x – 1) = 2

ANSWER: x = 9/8

[ log₃(x/(x – 1)) = 2, x/(x – 1) = 9, x = 9(x – 1), 8x = 9, x = 9/8]

3. log₂ x = log₂ (x + 3) – 1            

ANSWER: x = 3       

[log₂ (x/(x + 3)) = log₂(1/2), x/(x + 3) = ½, 2x = x + 3, x = 3]

G. Find the equation of the locus of the point P (x, y) moving in the coordinate plane such that AP = BP given

1. A(-4, 2) and B(2, – 4)             

ANSWER:  y = x

[(x + 4)² + (y – 2)² = (x – 2)² + (y + 4)², 8x – 4y = -4x + 8y, y = x  ]

2.  A(3, -2) and B(2, – 3)               

ANSWER: y = -x

[(x – 3)² + (y + 2)² = (x – 2)² + (y + 3)², -6x + 4y = -4x + 6y, -2x = 2y, y = -x]

3.  A(2, 4) and B (-2, -4)                

ANSWER: y = – x/2

[(x – 2)² + (y – 4)² = (x + 2)² + (y + 4)², -4x – 8y = 4x + 8y, 16y = -8x,  y = -x/2]

H.

1. Find the equation of the tangent to the curve y² = 4x at the point A(1, -2)

ANSWER: y = -x – 1, or x + y + 1 = 0

[ 2ydy/dx = 4, dy/dx = 2/y, m  = 2/-2 = -1, y + 2 = -1(x – 1), y = – x – 1]

2. Solve for x given (1/25)^{x + 2} = 125^{x – 2}

ANSWER:  x = 2/5

[ 5^{-2(x + 2)} = 5^{3(x – 2)}, -2x – 4 = 3x – 6, 5x = 2, x = 2/5]

3. If (2x + 3)/(x² – x – 6) = A/(x – 3) + B(x + 2), find the value of (A + B)

ANSWER: 2

[2x + 3 = A(x + 2) + B(x – 3), for x = 3, 9 = 5A, A = 9/5, for x = -2, -1 = -5B, B = 1/5 A + B = 9/5 + 1/5 = 2]

I. Find the coordinates of the vertices of a triangle whose sides are along the lines

1.  x + y = 3, x = 4, y = 5         

ANSWER: (4, 5), (4, -1), (-2, 5)

[x + y = 3 and x = 4, y = – 1, (4, -1), x + y = 3 and y = 5, x = -2, (-2, 5), x = 4 and y = 5 gives (4, 5)]

2.  x – y = 5, x = -2, y = 3   

ANSWER: (8, 3), (-2, 3), (-2, -7)

[x – y = 5 and x = -2, y = -7, (-2, -7), x – y = 5 and y = 3, x = 8, (8, 3),  x = -2 and y = 3 gives (-2, 3)]

3. 2x + y = 8, x = 3, y = 4        

ANSWER: (3, 2), (2, 4), (3, 4)  

[2x + y = 8 and x = 3, y = 2, (3, 2), 2x + y = 8 and y = 4, gives x = 2, (2, 4),  x = 3, y = 4 gives (3, 4)]

This is part 1 of the NSMQ Past Questions and Answers with respect to Mathematics. Part 2 will follow soon.