JAMB UTME - Mathematics - 1978

A hollow right prism of equilateral triangular base of side 4cm is filled with water up to a certain height. If a sphere of radius \(\frac{1}{2}\)cm is immersed in the water, then the rise of water is

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The quantity (x + y) is a factor of

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If sec² \(\theta\) + tan² \(\theta\) = 3, then the angle \(\theta\) is equal to

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A man runs a distance of 9km at a constant speed for the first 4 km and then 2 km\h faster for the rest of the distance. The whole run takes him one hour. His average speed for the first 4 km is

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A rectangular picture 6cm by 8cm is enclosed by a frame \(\frac{1}{2}\)cm wide. Calculate the area of the frame

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Simplify \(\frac{(a^2 - \frac{1}{a}) (a^{\frac{4}{3}} + a^{\frac{2}{3}})}{a^2 - \frac{1}{a}^2}\)

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Given that \(a*b = ab + a + b\) and that \(a ♦ b = a + b = 1\). Find an expression (not involving * or ♦) for (a*b) ♦ (a*c) if a, b, c, are real numbers and the operations on the right are ordinary addition and multiplication of numbers

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Add the same number to the numerator and denominator of \(\frac{3}{18}\). If the resulting fraction is \(\frac{1}{2}\), then the number added is

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The smallest number such that when it is divided by 8 has a remainder of 6 and when it is divided by 9, has a remainder of 7 is

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Five years ago, a father was 3 times as old as his son, now their combined ages amount to 110years. thus, the present age of the father is

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If x⁴ - kx³ + 10x² + 1x - 3 is divisible by (x - 1), and if when it is divided by (x + 2) the remainder is 27, find the constants k and 1

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Evaluate without using tables sin(-1290^o)

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If \(3x - \frac{1}{4})^{\frac{1}{2}} > \frac{1}{4} - x \), then the interval of values of x is

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A force of 5 units acts on a particle in the direction to the east and another force of 4 units acts on the particle in the direction north-east. The resultants of the two forces is

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The vectors a and b are given in terms of two perpendicular units vectors i and j on a plane by a = 2i - 3j, b = -i + 2j. Find the magnitude of the vector a + 3b

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The locus of all points having a distance of 1 unit from each of the two fixed points a and b is

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The sum of \(3\frac{7}{8}\) and \(1\frac{1}{3}\) is less than the difference between \(\frac{3}{8}\) and \(1\frac{2}{3}\) by:

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A pyramid is constructed on a cuboid. The figure has

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If (25)^{x - 1} = 64(\(\frac{5}{2}\))⁶, then x has the value

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An arithmetic progression has first term 11 and fourth term 32. The sum of the first nine terms is

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If the four interior angles of a quadrilateral are (p + 10)o, (p - 30)o, (2p - 450o, and (p + 15)o, then p is

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In a geometric progression, the first term is 153 and the sixth term is \(\frac{17}{27}\). The sum of the first four terms is

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A regular hexagon is constructed inside a circle of diameter 12cm. The area of the hexagon is

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The number of telephone calls N between two cities A and B varies directly as the population P\(_{A}\), P\(_B\) respectively and inversely as the square of the distance D between A and B. Which of the following equations represents this relation?

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In a soccer competition in one season, a club had scored the following goals: 2, 0, 3, 3, 2, 1, 4, 0, 0, 5, 1, 0, 2, 2, 1, 3, 1, 4, 1 and 1. The mean, median and mode are respectively

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Father reduced the quantity of food bought for the family by 10% when he found that the cost of living had increased 15%. Thus the fractional increase in the family food bill is now

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In the figure, 0 is the centre of the circle ABC, < CED = 30^o, < EDA = 40^o. What is the size of < ABC?

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Arrange \(\frac{3}{5}\),\(\frac{9}{16}\), \(\frac{34}{59}\) and \(\frac{71}{97}\) in ascending order of magnitude.

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If x2 + 4 = 0, then x ?

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In the figure, DE//BC: DB//FE: DE = 2cm, FC = 3cm, AE = 4cm. Determine the length of EC.

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In the figure, AB is parallel to CD then x + y + z is

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Find the square root of 170 - 20\(\sqrt{30}\)

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In the Figure FD\\AC, the area of AEF = 6sq.cm. AE = 3cm, BC = 3cm, CD = 5cm, < BCD is an obtuse angle. Find the length of BD.

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A canal has rectangular cross section of width10cm and breadth 1m. If water of uniform density 1 gm cm-3 flows through it at a constant speed of1000mm per minute, the adjacent sea is

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Assuming log_e 4.4 = 1.4816 and log_e 7.7 = 2.0142, then the value of log_e \(\frac{7}{4}\) is

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If a circular paper disc is trimmed in such a way that its circumference is reduced in the ratio 2:5, In what ratio is the surface area reduced?

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If y = 2x² + 9x - 35. Find the range of values for which y < 0.

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What is the number whose logarithm to base 10 is 2.3482?

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A solid sphere of radius 3cm, a solid right cone of radius 3cm and height 12cm and a solid right circular cycular of radius 3cm and height 4cm.Which of the following statements is true?

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Multiply (x + 3y + 5) by (2x2 + 5y + 2)

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The set of value of x and y which satisfies the equations x² - y - 1 = 0 and y - 2x + 2 = 0 is

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Without using tables, simplify \(\frac{1n \sqrt{216} - 1n \sqrt{125} - 1n\sqrt{8}}{2(1n3 - 1n5)}\)

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The minimum point on the curve y = x2 - 6x + 5 is at

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Simplify \(\frac{a - b}{a + b}\) - \(\frac{a + b}{a - b}\)

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Two triangles have the same areas if

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The sum of the progression is 1 + x + x2 + x3 + ......

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The angle of elevation of the top of a vertical tower from a point A on the ground is 60^o. From a point B, 2 units of distance further away from the foot of the tower, the angle of elevation of the tower is 45^o. Find the distance of A from the foot of the tower

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A triangle has angles 30o, 15o and 135o. The side opposite to the angle 30o is length 6cm. The side opposite to the angle 135o is equal to

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