2010 JAMB Mathematics Past Questions

Questions

1. 1.

   If y = (2x + 1)$^3$ find $\frac{\text{d}}{\text{dx}}$

   1. A.3(2x+1)2
   2. B.6(2x+1)
   3. C.3(2x+1)
   4. D.6(2x+1)2

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   Answer: D. 6(2x+1)2

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2. 2.

   If y varies directly as the square root of x and y = 3 when x = 16. Calculate y when x = 64

   1. A.12
   2. B.6
   3. C.3
   4. D.5

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   Answer: B. 6

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3. 3.

   The interior angles of a quadrilateral are (x + 15)°, (2x - 45)°, ( x - 30)° and (x + 10)°. Find the value of the least interior angle.

   1. A.112o
   2. B.102o
   3. C.82o
   4. D.52o

   Show answer

   Answer: D. 52o

   AI explanation

   1. Analyze the question

   We are given the four interior angles of a quadrilateral in terms of $x$. We need to find the value of $x$ first, then substitute it back into each expression to identify the smallest angle.

   2. Explain the concept

   The sum of the interior angles of any quadrilateral is always 360°. The formula for the sum of interior angles of a polygon with $n$ sides is $(n - 2) \times 180^\circ$. For a quadrilateral ($n=4$): $(4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ$.

   3. Show the solution

   Step 1: Set up the equation Sum the given angles and equate them to 360°: $(x + 15) + (2x - 45) + (x - 30) + (x + 10) = 360$

   Step 2: Simplify the equation Group the $x$ terms and the constant numbers: $(x + 2x + x + x) + (15 - 45 - 30 + 10) = 360$ $5x - 50 = 360$

   Step 3: Solve for $x$ $5x = 360 + 50$ $5x = 410$ $x = \frac{410}{5}$ $x = 82$

   Step 4: Calculate each angle Now, substitute $x = 82$ into each expression:

   1. $(x + 15) = 82 + 15 = 97^\circ$
   2. $(2x - 45) = 2(82) - 45 = 164 - 45 = 119^\circ$
   3. $(x - 30) = 82 - 30 = 52^\circ$
   4. $(x + 10) = 82 + 10 = 92^\circ$

   Step 5: Identify the least angle Comparing $97^\circ, 119^\circ, 52^\circ,$ and $92^\circ$, the smallest value is 52°.

   4. Identify the answer

   The calculated value for the least interior angle is 52°, which corresponds to option D.

   5. Tip

   When dealing with geometry problems involving variables, always check your final answers by adding them up. If they don't sum to the required total (360° for a quadrilateral), you know you made a calculation error in your algebra!

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4. 4.

   If x * y = x + y$^2$, find then value of (2*3)*5

   1. A.36
   2. B.25
   3. C.11
   4. D.55

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   Answer: A. 36

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5. 5.

   From the cyclic quadrilateral TUVW above, find the value of x

   1. A.26o
   2. B.24o
   3. C.20o
   4. D.23o

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   Answer: C. 20o

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6. 6.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

7. 7.

   Find the distance between the points ($\frac{1}{2}$, $\frac{1}{2}$) and (-$\frac{1}{2}$, -$\frac{1}{2}$).

   1. A.1
   2. B.o
   3. C.√3
   4. D.√2

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   Answer: D. √2

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8. 8.

   Make Q the subject of formula if p = $\frac{M}{5}$(X + Q) + 1

   1. A.$\frac{5P - MX + 5}{M}$
   2. B.$\frac{5P - MX - 5}{M}$
   3. C.$\frac{5P + MX + 5}{M}$
   4. D.$\frac{5P + MX - 5}{M}$

   Show answer

   Answer: B. $\frac{5P - MX - 5}{M}$

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9. 9.

   Find ∫(sin x + 2)dx

   1. A.cos x + x2 + K
   2. B.cos x 2x + K
   3. C.-cos x + x2 + K
   4. D.-cos x + 2x + K

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   Answer: D. -cos x + 2x + K

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10. 10.

    The 3rd term of an arithmetic progression is -9 and the 7th term is -29. Find the 10th term of the progression

    1. A.-44
    2. B.-165
    3. C.165
    4. D.44

    Show answer

    Answer: A. -44

    AI explanation

    Analysis

    To solve this problem, we need to use the formula for the $n^{th}$ term of an Arithmetic Progression (AP):

    $T_n = a + (n - 1)d$

    Where:

    • $a$ is the first term
    • $d$ is the common difference
    • $n$ is the position of the term

    Concept

    An arithmetic progression is a sequence where the difference between consecutive terms is constant. We are given the 3rd term ($T_3$) and the 7th term ($T_7$). We will use these to create a system of linear equations to find $a$ and $d$, then use those values to find the 10th term ($T_{10}$).

    Step-by-Step Solution

    1. Set up the equations based on the given information:

    • For the 3rd term ($n=3$): $T_3 = a + (3 - 1)d = -9$ $a + 2d = -9$ --- (Equation 1)

    • For the 7th term ($n=7$): $T_7 = a + (7 - 1)d = -29$ $a + 6d = -29$ --- (Equation 2)

    2. Solve the system of equations: Subtract Equation 1 from Equation 2 to eliminate $a$: $(a + 6d) - (a + 2d) = -29 - (-9)$ $4d = -29 + 9$ $4d = -20$ $d = -5$

    3. Find the first term ($a$): Substitute $d = -5$ into Equation 1: $a + 2(-5) = -9$ $a - 10 = -9$ $a = -9 + 10$ $a = 1$

    4. Calculate the 10th term ($T_{10}$): Using $a = 1$, $d = -5$, and $n = 10$: $T_{10} = a + (10 - 1)d$ $T_{10} = 1 + 9(-5)$ $T_{10} = 1 - 45$ $T_{10} = -44$

    Conclusion

    Comparing our result to the options provided, the 10th term is -44, which matches option A.

    Exam Tip

    When dealing with AP questions, always remember that any term $T_n$ can be expressed as $T_m + (n-m)d$. For example, you could have jumped straight to the common difference by saying $T_7 = T_3 + 4d$. This saves time during the exam!

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11. 11.

    Find r, if 6r7$_8$ = 511$_9$

    1. A.3
    2. B.2
    3. C.6
    4. D.5

    Show answer

    Answer: A. 3

    AI explanation

    1. Analyze the question

    We are given an equation involving numbers in different bases: $6r7_8 = 511_9$. We need to find the value of the digit $r$.

    2. Explain the concept

    To solve problems involving different bases, we convert both sides of the equation to Base 10 (decimal system). The formula to convert a number $d_n...d_2d_1d_0$ in base $x$ to base 10 is: $(d_n \times x^n) + ... + (d_2 \times x^2) + (d_1 \times x^1) + (d_0 \times x^0)$

    3. Show the solution

    Step 1: Convert the left side ($6r7_8$) to base 10 $6r7_8 = (6 \times 8^2) + (r \times 8^1) + (7 \times 8^0)$ $= (6 \times 64) + (8r) + (7 \times 1)$ $= 384 + 8r + 7$ $= 391 + 8r$

    Step 2: Convert the right side ($511_9$) to base 10 $511_9 = (5 \times 9^2) + (1 \times 9^1) + (1 \times 9^0)$ $= (5 \times 81) + (1 \times 9) + (1 \times 1)$ $= 405 + 9 + 1$ $= 415$

    Step 3: Equate both sides and solve for $r$ $391 + 8r = 415$ Subtract 391 from both sides: $8r = 415 - 391$ $8r = 24$ Divide by 8: $r = \frac{24}{8}$ $r = 3$

    4. Identify the answer

    Comparing our result with the given options, $r = 3$ corresponds to option A.

    5. Tip

    When dealing with base conversion, always remember that the digit in the base must be smaller than the base itself. Since the number is in base 8, the digit $r$ must be less than 8. Our answer, 3, satisfies this condition perfectly!

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12. 12.

    For what range of values of x is $\frac{1}{2}$x + $\frac{1}{4}$ > $\frac{1}{3}$x + $\frac{1}{2}$?

    1. A.x < $\frac{3}{2}$
    2. B.x > $\frac{3}{2}$
    3. C.x < -$\frac{3}{2}$
    4. D.x > -$\frac{3}{2}$

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    Answer: B. x > $\frac{3}{2}$

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13. 13.

    If p and q are two non zero numbers and 18(p+q) = (18+p)q, which of the following must be true?

    1. A.q = 18
    2. B.p <1
    3. C.p = 18
    4. D.q < 1

    Show answer

    Answer: A. q = 18

    AI explanation

    Analysis of the Question

    We are given the equation $18(p + q) = (18 + p)q$ where $p$ and $q$ are non-zero numbers. We need to simplify this expression to determine the relationship between $p$ and $q$ and identify which of the given options must be true.

    Concept Explanation

    This is an algebraic manipulation problem. The goal is to expand the brackets on both sides of the equation and rearrange the terms to isolate one variable or find a constant relationship.

    Step-by-Step Solution

    Step 1: Expand both sides of the equation. Given: $18(p + q) = (18 + p)q$

    Expand the left side: $18p + 18q$

    Expand the right side: $18q + pq$

    Step 2: Set the expanded sides equal to each other. $18p + 18q = 18q + pq$

    Step 3: Simplify the equation. Subtract $18q$ from both sides: $18p = pq$

    Step 4: Solve for the variables. Since the problem states that $p$ and $q$ are non-zero numbers, we can safely divide both sides by $p$: $\frac{18p}{p} = \frac{pq}{p}$ $18 = q$

    This shows that regardless of the value of $p$ (as long as it is not zero), $q$ must be equal to 18 for the equation to hold true.

    Conclusion

    Comparing our result ($q = 18$) with the given options, we can see that option A is the correct statement.

    Exam Tip

    When you see an equation with variables on both sides, always expand the brackets first. If you see the same term on both sides (like the $18q$ we had here), you can cancel them out immediately to save time and reduce the risk of calculation errors!

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14. 14.

    In how many ways can a committee of 2 women and 3 men be chosen from 6 men and 5 women?

    1. A.100
    2. B.200
    3. C.30
    4. D.50

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    Answer: B. 200

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15. 15.

    Evaluate $\int^{2}_{0}(x^3 + x^2)$dx.

    1. A.4$\frac{5}{6}$
    2. B.6$\frac{2}{3}$
    3. C.1$\frac{5}{6}$
    4. D.2$\frac{5}{6}$

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    Answer: B. 6$\frac{2}{3}$

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16. 16.

    If p : q = $\frac{2}{3}$ : $\frac{5}{6}$ and q : r = $\frac{3}{4}$ : $\frac{1}{2}$, find p : q : r

    1. A.12 : 15 : 10
    2. B.12 : 15 : 16
    3. C.10 : 15 : 24
    4. D.9 : 10 : 15

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    Answer: A. 12 : 15 : 10

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17. 17.

    Find the standard deviation of 2, 3, 5 and 6

    1. A.√6
    2. B.√10
    3. C.√$\frac{2}{5}$
    4. D.√$\frac{5}{2}$

    Show answer

    Answer: D. √$\frac{5}{2}$

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18. 18.

    Find the sum to infinity of the following series. 0.5 + 0.05 + 0.005 + 0.0005 + .....

    1. A.$\frac{5}{8}$
    2. B.$\frac{5}{7}$
    3. C.$\frac{5}{11}$
    4. D.$\frac{5}{9}$

    Show answer

    Answer: D. $\frac{5}{9}$

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19. 19.

    A student measures a piece of rope and found that it was 1.26m long. If the actual length of the rope was 1.25m, what was the percentage error in the measurement?

    1. A.0.25%
    2. B.0.01%
    3. C.0.80%
    4. D.0.40%

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    Answer: C. 0.80%

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20. 20.

    If P = $\begin{pmatrix} 2 & -3 \\ 1 & 1 \end{pmatrix}$ , what is P$^-1$

    1. A.$\begin{pmatrix} -{\frac{1}{5}} & -{\frac{3}{5}} \\ -{\frac{1}{5}} & -{\frac{2}{5}} \end{pmatrix}$
    2. B.$\begin{pmatrix} {\frac{1}{5}} & {\frac{3}{5}} \\ {\frac{1}{5}} & {\frac{2}{5}} \end{pmatrix}$
    3. C.$\begin{pmatrix} -{\frac{1}{5}} & {\frac{3}{5}} \\ -{\frac{1}{5}} & {\frac{2}{5}} \end{pmatrix}$
    4. D.$\begin{pmatrix} {\frac{1}{5}} & {\frac{3}{5}} \\ -{\frac{1}{5}} & {\frac{2}{5}} \end{pmatrix}$

    Show answer

    Answer: D. $\begin{pmatrix} {\frac{1}{5}} & {\frac{3}{5}} \\ -{\frac{1}{5}} & {\frac{2}{5}} \end{pmatrix}$

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21. 21.

    If two smaller sides of a right angled triangle are 4cm and 5cm, find its area

    1. A.10 cm2
    2. B.6 cm2
    3. C.24 cm2
    4. D.8 cm2

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    Answer: A. 10 cm2

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22. 22.

    Determine the value of x for which (x$^2$ - 1)>0

    1. A.x < -1 or x > 1
    2. B.-1 < x < 1
    3. C.x > 0
    4. D.x < -1

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    Answer: A. x < -1 or x > 1

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23. 23.

    If three unbiased coins are tossed, find the probability that they are all heads

    1. A.$\frac{1}{2}$
    2. B.$\frac{1}{3}$
    3. C.$\frac{1}{9}$
    4. D.$\frac{1}{8}$

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    Answer: D. $\frac{1}{8}$

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24. 24.

    Evaluate $\begin{vmatrix} 2 & 0 & 5 \\ 4 & 6 & 3 \\ 8 & 9 & 1 \end{vmatrix}$

    1. A.5y - 2x -18 = 0
    2. B.102
    3. C.-102
    4. D.-42

    Show answer

    Answer: C. -102

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25. 25.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

26. 26.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

27. 27.

    An arc subtends an angle of 50º at the center of circle of radius 6cm. Calculate the area of the sector formed

    1. A.100/7
    2. B.80/7
    3. C.110/7
    4. D.90/7

    Show answer

    Answer: C. 110/7

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28. 28.

    At what rate will the interest on N400 increases to N24 in 3 years reckoning in simple interest?

    1. A.3%
    2. B.2%
    3. C.5%
    4. D.4%

    Show answer

    Answer: B. 2%

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29. 29.

    If 9x$^2$ + 6xy + 4y$^2$ is a factor of 27x$^3$ - 8y$^3$, find the other factor.

    1. A.2y + 3x
    2. B.2y - 3x
    3. C.3x + 2y
    4. D.3x - 2y

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    Answer: D. 3x - 2y

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30. 30.

    Evaluate $\left(\frac{81}{16}\right)^{\frac{-1}{4}}\times 2^{-1}$

    1. A.1/3
    2. B.3
    3. C.6
    4. D.1/6

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    Answer: A. 1/3

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31. 31.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

32. 32.

    W is directly proportional to U. If W = 5 when U = 3, find U when W = $\frac{2}{7}$

    1. A.$\frac{6}{35}$
    2. B.$\frac{10}{21}$
    3. C.$\frac{21}{10}$
    4. D.$\frac{35}{6}$

    Show answer

    Answer: A. $\frac{6}{35}$

    AI explanation

    Analysis

    The question involves Direct Variation. When a variable $W$ is directly proportional to $U$, it means that as $U$ increases, $W$ increases at a constant rate. Mathematically, this is expressed as $W = kU$, where $k$ is the constant of proportionality.

    Concept

    1. Find the constant ($k$): Use the initial values given ($W=5, U=3$) to determine the value of $k$.
    2. Form the equation: Substitute $k$ back into the general formula $W = kU$.
    3. Solve for the unknown: Use the new value of $W$ to find the corresponding value of $U$.

    Step-by-Step Solution

    Step 1: Find $k$ Given $W = kU$: $5 = k \times 3$ $k = \frac{5}{3}$

    Step 2: Form the equation Now that we know $k = \frac{5}{3}$, our specific equation is: $W = \frac{5}{3}U$

    Step 3: Solve for $U$ when $W = \frac{2}{7}$ Substitute $W = \frac{2}{7}$ into the equation: $\frac{2}{7} = \frac{5}{3}U$

    To isolate $U$, multiply both sides by the reciprocal of $\frac{5}{3}$, which is $\frac{3}{5}$: $U = \frac{2}{7} \times \frac{3}{5}$ $U = \frac{2 \times 3}{7 \times 5}$ $U = \frac{6}{35}$

    Conclusion

    Comparing our result to the options provided, we can see that our calculated value matches option A.

    Exam Tip: In variation problems, always ensure you find the constant ($k$) first. A common mistake students make is trying to use ratios incorrectly; writing out $W = kU$ clearly at the start of your working prevents this error.

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33. 33.

    Find the gradient of the line passing through the points P(1, 1) and Q(2, 5).

    1. A.3
    2. B.2
    3. C.5
    4. D.4

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    Answer: D. 4

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34. 34.

    If $\begin{vmatrix} x & 3 \\ 2 & 7 \end{vmatrix} = 15$, find the value of x

    1. A.3
    2. B.4
    3. C.5
    4. D.2

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    Answer: A. 3

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35. 35.

    Factorize completely $\frac{x^{3} + 3x^{2} - 10x}{2x^{2} - 8}$

    1. A.x(x-5)2(x+2)
    2. B.x(x-5)2(x-2)
    3. C.x(x+5)2(x+2)
    4. D.x2+52x+4

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    Answer: C. x(x+5)2(x+2)

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36. 36.

    At what value of X does the function y = -3 - 2x + X2 attain a minimum value?

    1. A.-1
    2. B.14
    3. C.4
    4. D.1

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    Answer: D. 1

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37. 37.

    If the area of ΔPQR above is 12$\sqrt{3}$ cm$^2$, find the value of q?

    1. A.6 cm
    2. B.7 cm
    3. C.8 cm
    4. D.5 cm

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    Answer: C. 8 cm

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38. 38.

    Solve the inequalities -6 $\leq$ 4 - 2x < 5 - x

    1. A.-1 < x < 5
    2. B.-1 < x $\leq$ 5
    3. C.-1 $\leq$ x $\leq$ 6
    4. D.-1 $\leq$ x < 6

    Show answer

    Answer: B. -1 < x $\leq$ 5

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39. 39.

    Express the product of 0.21 and 0.34 in standard form

    1. A.7.14 x 10-3
    2. B.7.14 x 10-2
    3. C.7.14 x 10-1
    4. D.7.14 x 10-4

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    Answer: B. 7.14 x 10-2

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40. 40.

    If x is inversely proportional to y and x = 2$\frac{1}{2}$ when y = 2, find x if y = 4

    1. A.4
    2. B.5
    3. C.1$\frac{1}{4}$
    4. D.2$\frac{1}{4}$

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    Answer: C. 1$\frac{1}{4}$

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41. 41.

    Find the equation of a line parallel to y = -4x + 2 passing through (2,3)

    1. A.y + 4x + 11 = 0
    2. B.y - 4x - 11 = 0
    3. C.y + 4x - 11 = 0
    4. D.y - 4x + 11 = 0

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    Answer: C. y + 4x - 11 = 0

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42. 42.

    Solve for x and y if x - y = 2 and x2 - y2 = 8

    1. A.(-1, 3)
    2. B.(3, 1)
    3. C.(-3, 1)
    4. D.(1, 3)

    Show answer

    Answer: B. (3, 1)

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43. 43.

    In a survey of 50 newspaper readers, 40 read Champion and 30 read Guardian, how many read both papers?

    1. A.10
    2. B.5
    3. C.15
    4. D.20

    Show answer

    Answer: D. 20

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44. 44.

    If y = x sinx, find $\frac{dy}{dx}$

    1. A.sin x - x cosx
    2. B.sinx + x cosx
    3. C.sinx - cosx
    4. D.sinx + cosx

    Show answer

    Answer: B. sinx + x cosx

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45. 45.

    A cylindrical pipe 50m long with radius 7m has one end open. What is the total surface area of the pipe?

    1. A.749$\pi$m2
    2. B.700$\pi$m2
    3. C.350$\pi$m2
    4. D.98$\pi$m2

    Show answer

    Answer: A. 749$\pi$m2

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46. 46.

    Simplify ($\frac{3}{4}$ of $\frac{4}{9}$ $\div$ 9$\frac{1}{2}$) $\div$ 1$\frac{5}{19}$

    1. A.$\frac{1}{5}$
    2. B.$\frac{1}{4}$
    3. C.$\frac{1}{36}$
    4. D.$\frac{1}{25}$

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    Answer: C. $\frac{1}{36}$

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47. 47.

    What is the locus of point that is equidistant from points P(1,3) and Q(3,5)?

    1. A.y = -χ + 6
    2. B.y = -χ - 6
    3. C.y = χ + 6
    4. D.y = χ - 6

    Show answer

    Answer: A. y = -χ + 6

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48. 48.

    Rationalize $\frac{2\sqrt{3} + \sqrt{5}}{\sqrt{5} - \sqrt{3}}$

    1. A.(3 √ 15 +11) / 2
    2. B.3 √ 15 -11
    3. C.(3 √ 15 -11) / 2
    4. D.3 √ 15 +11

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    Answer: A. (3 √ 15 +11) / 2

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49. 49.

    If cot$\theta$ = $\frac{8}{15}$, where $\theta$ is acute, find sin$\theta$

    1. A.$\frac{8}{17}$
    2. B.$\frac{15}{17}$
    3. C.$\frac{16}{17}$
    4. D.$\frac{13}{17}$

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    Answer: B. $\frac{15}{17}$

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