Mathematics (Core), 1985, JAMB Exam, Exam, Paper 1 | Objectives

Paper Info

Year :

1985

Title :

Mathematics (Core)

Exam :

JAMB Exam

Paper 1 | Objectives

11 - 20 of 47 Questions

#	Question	Ans
11.	Find the values of p for which the equation x² - (p - 2)x + 2p + 1 = 0 A. (21, 0) B. (0, 12) C. (1, 2) D. (3, 4) E. (4, 5) Show Content Detailed Solution Equal roots implies b² - 4ac = 0 a = 1b = - (p - 2), c = 2p + 1 [-(p - 2)]² - 4 x 1 x (2p + 1) = 0 p² - 4p + 4 - 4(2p + 1) = 0 p² - 4p = 4 - 8p - 4 = 0 p² - 12p = 0 p(p - 12) = 0 p = 0 or 12	B
12.	If \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ... \), find \(\frac{1}{e^{\frac{1}{2}}}\) A. 1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + ......... B. 1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{2.4.3}\) + .. C. 1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2}\) + \(\frac{x^3}{1.2.3}\) + \(\frac{x^4}{1.23.4}\) + ......... D. 1 - x + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + ......... E. 1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{1.2.6}\) + ......... Show Content Detailed Solution \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ...\) \(\frac{1}{e^{\frac{1}{2}}} = e^{-\frac{1}{2}}\) \(e^{-\frac{1}{2}} = 1 - \frac{x}{2} + \frac{x^{2}}{1.2^{3}} - \frac{x^{3}}{1.2^{4}.3} + ... \)	C
13.	\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\) is equal to A. 1 B. (\(\sqrt{2} + 4\sqrt{2}\)) C. (6\(\sqrt{2}\) D. 8 Show Content Detailed Solution \((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\) \((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2}) = \sqrt{3} - \sqrt[4]{6} + \sqrt[4]{6} - \sqrt{2}\) = \(\sqrt{3} - \sqrt{2}\) \((\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 + \sqrt{6} - \sqrt{6} - 2\) = \(3 - 2 = 1\)	A
14.	In a restaurant, the cost of providing a particular type of food is partly constant and partially inversely proportional to the number of people. If cost per head for 100 people is 30k and the cost for 40 people is 60k, Find the cost for 50 people? A. 15k B. 20k C. 50k D. 40k E. 45k Show Content Detailed Solution C = a + k \(\frac{1}{N}\) = c = \(\frac{aN + k}{N}\) CN = aN + K 30(100) = a(100) + k 3000 = 100a + k.......(i) 60(40) = a(40) + k 2400 = 40a + k.......(ii) eqn (i) - eqn (ii) 600 = 60a a = 10 subt. for a in eqn (i) 3000 = 100(10) + K 3000 - 1000 = k k = 2000 CN = 10N + 2000. when N = 50, 50C = 10(50) + 2000 50C = 500 + 2000 C = \(\frac{2500}{50}\) = 50k	C
15.	The factors of 9 - (x² - 3x - 1)² are A. -(x - 4)(x + 1) (x - 1)(x - 2) B. (x - 4)(x - 2) (x - 1)(x + 1) C. -(x - 2)(x + 1) (x - 2) (x - 1) D. (x - 2)(x + 2) (x - 1)(x + 1) Show Content Detailed Solution 9 - (x² - 3x - 1)² = [3 - (x² - 3x - 1)] [3 + (x² - 3x - 1)] = (3 - x² + 3x + 1)(3 + x² - 3x - 1) = (4 + 3x - x²)(x² - 3x + 2) = (4 - x)(1 + x)(x - 1)(x - 2) = -(x - 4)(x + 1) (x - 1)(x - 2)	A
16.	If 3^2y + 6(3^y) = 27. Find y A. 3 B. -1 C. 2 D. -3 E. 1 Show Content Detailed Solution 3^2y + 6(3^y) = 27 This can be rewritten as (3^y)² + 6(3^y) = 27 Let 3^y = x x² + 6x - 27 = 0 (x + 9)(x - 3) = 0 when x - 3 = 0, x = 3 sub. for x in 3y = x 3^y = 3 log₃3 = y y = 1	E
17.	Factorize abx² + 8y - 4bx - 2axy A. (ax - 4)(bx - 2y) B. (ax + b)(x - 8y) C. (ax - 2y)(bx - 4) D. (bx - 4)(ax - 2y) E. (abx - 4)(x - 2y) Show Content Detailed Solution abx² + 8y - 4bx - 2axy = (abx² - 4bx) + (8y - 2axy) = bx(ax - 4) 2y(ax - 4) 2y(ax - 4) = (bx - 2y)(ax - 4)	A
18.	At what real value of x do the curves whose equations are y = x³ + x and y = x² + 1 intersect? A. -2 B. 2 C. -1 D. 9 E. 1 Show Content Detailed Solution y = x³ + x and y = x² + 1 \(\begin{array}{c\|c} x & -2 & -1 & 0 & 1 & 2 \\ \hline Y = x^3 + x & -10 & -2 & 0 & 2 & 10 \\ \hline y = x^2 + 1 & 5 & 2 & 1 & 2 & 5\end{array}\) The curves intersect at x = 1	E
19.	If the quadratic function 3x² - 7x + R is a perfect square, find R A. \(\frac{49}{24}\) B. \(\frac{49}{12}\) C. \(\frac{49}{13}\) D. \(\frac{49}{3}\) E. \(\frac{49}{36}\) Show Content Detailed Solution 3x² - 7x + R. Computing the square, we have x² - \(\frac{7}{3}\) = -\(\frac{R}{3}\) (\(\frac{x}{1} - \frac{7}{6}\))² = -\(\frac{R}{3}\) + \(\frac{49}{36}\) \(\frac{-R}{3}\) + \(\frac{49}{36}\) = 0 R = \(\frac{49}{36}\) x \(\frac{3}{1}\) = \(\frac{49}{12}\)	B
20.	Solve for (x, y) in the equation 2x + y = 4: x^2 + xy = -12 A. (6, -8): (-2, 8) B. (3, -4): (-1, 4) C. (8, -4): (-1, 4) D. (-8, 6): (8, -2) E. (-4, 3): (4, -1) Show Content Detailed Solution 2x + y = 4......(i) x^2 + xy = -12........(ii) from eqn (i), y = 4 - 2x = x2 + x(4 - 2x) = -12 x2 + 4x - 2x2 = -12 4x - x2 = -12 x2 - 4x - 12 = 0 (x - 6)(x + 2) = 0 sub. for x = 6, in eqn (i) y = -8, 8 =(6,-8); (-2, 8)	A

11.	Find the values of p for which the equation x² - (p - 2)x + 2p + 1 = 0 A. (21, 0) B. (0, 12) C. (1, 2) D. (3, 4) E. (4, 5) Show Content Detailed Solution Equal roots implies b² - 4ac = 0 a = 1b = - (p - 2), c = 2p + 1 [-(p - 2)]² - 4 x 1 x (2p + 1) = 0 p² - 4p + 4 - 4(2p + 1) = 0 p² - 4p = 4 - 8p - 4 = 0 p² - 12p = 0 p(p - 12) = 0 p = 0 or 12	B
12.	If \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ... \), find \(\frac{1}{e^{\frac{1}{2}}}\) A. 1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + ......... B. 1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{2.4.3}\) + .. C. 1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2}\) + \(\frac{x^3}{1.2.3}\) + \(\frac{x^4}{1.23.4}\) + ......... D. 1 - x + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + ......... E. 1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{1.2.6}\) + ......... Show Content Detailed Solution \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ...\) \(\frac{1}{e^{\frac{1}{2}}} = e^{-\frac{1}{2}}\) \(e^{-\frac{1}{2}} = 1 - \frac{x}{2} + \frac{x^{2}}{1.2^{3}} - \frac{x^{3}}{1.2^{4}.3} + ... \)	C
13.	\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\) is equal to A. 1 B. (\(\sqrt{2} + 4\sqrt{2}\)) C. (6\(\sqrt{2}\) D. 8 Show Content Detailed Solution \((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\) \((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2}) = \sqrt{3} - \sqrt[4]{6} + \sqrt[4]{6} - \sqrt{2}\) = \(\sqrt{3} - \sqrt{2}\) \((\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 + \sqrt{6} - \sqrt{6} - 2\) = \(3 - 2 = 1\)	A
14.	In a restaurant, the cost of providing a particular type of food is partly constant and partially inversely proportional to the number of people. If cost per head for 100 people is 30k and the cost for 40 people is 60k, Find the cost for 50 people? A. 15k B. 20k C. 50k D. 40k E. 45k Show Content Detailed Solution C = a + k \(\frac{1}{N}\) = c = \(\frac{aN + k}{N}\) CN = aN + K 30(100) = a(100) + k 3000 = 100a + k.......(i) 60(40) = a(40) + k 2400 = 40a + k.......(ii) eqn (i) - eqn (ii) 600 = 60a a = 10 subt. for a in eqn (i) 3000 = 100(10) + K 3000 - 1000 = k k = 2000 CN = 10N + 2000. when N = 50, 50C = 10(50) + 2000 50C = 500 + 2000 C = \(\frac{2500}{50}\) = 50k	C
15.	The factors of 9 - (x² - 3x - 1)² are A. -(x - 4)(x + 1) (x - 1)(x - 2) B. (x - 4)(x - 2) (x - 1)(x + 1) C. -(x - 2)(x + 1) (x - 2) (x - 1) D. (x - 2)(x + 2) (x - 1)(x + 1) Show Content Detailed Solution 9 - (x² - 3x - 1)² = [3 - (x² - 3x - 1)] [3 + (x² - 3x - 1)] = (3 - x² + 3x + 1)(3 + x² - 3x - 1) = (4 + 3x - x²)(x² - 3x + 2) = (4 - x)(1 + x)(x - 1)(x - 2) = -(x - 4)(x + 1) (x - 1)(x - 2)	A

16.	If 3^2y + 6(3^y) = 27. Find y A. 3 B. -1 C. 2 D. -3 E. 1 Show Content Detailed Solution 3^2y + 6(3^y) = 27 This can be rewritten as (3^y)² + 6(3^y) = 27 Let 3^y = x x² + 6x - 27 = 0 (x + 9)(x - 3) = 0 when x - 3 = 0, x = 3 sub. for x in 3y = x 3^y = 3 log₃3 = y y = 1	E
17.	Factorize abx² + 8y - 4bx - 2axy A. (ax - 4)(bx - 2y) B. (ax + b)(x - 8y) C. (ax - 2y)(bx - 4) D. (bx - 4)(ax - 2y) E. (abx - 4)(x - 2y) Show Content Detailed Solution abx² + 8y - 4bx - 2axy = (abx² - 4bx) + (8y - 2axy) = bx(ax - 4) 2y(ax - 4) 2y(ax - 4) = (bx - 2y)(ax - 4)	A
18.	At what real value of x do the curves whose equations are y = x³ + x and y = x² + 1 intersect? A. -2 B. 2 C. -1 D. 9 E. 1 Show Content Detailed Solution y = x³ + x and y = x² + 1 \(\begin{array}{c\|c} x & -2 & -1 & 0 & 1 & 2 \\ \hline Y = x^3 + x & -10 & -2 & 0 & 2 & 10 \\ \hline y = x^2 + 1 & 5 & 2 & 1 & 2 & 5\end{array}\) The curves intersect at x = 1	E
19.	If the quadratic function 3x² - 7x + R is a perfect square, find R A. \(\frac{49}{24}\) B. \(\frac{49}{12}\) C. \(\frac{49}{13}\) D. \(\frac{49}{3}\) E. \(\frac{49}{36}\) Show Content Detailed Solution 3x² - 7x + R. Computing the square, we have x² - \(\frac{7}{3}\) = -\(\frac{R}{3}\) (\(\frac{x}{1} - \frac{7}{6}\))² = -\(\frac{R}{3}\) + \(\frac{49}{36}\) \(\frac{-R}{3}\) + \(\frac{49}{36}\) = 0 R = \(\frac{49}{36}\) x \(\frac{3}{1}\) = \(\frac{49}{12}\)	B
20.	Solve for (x, y) in the equation 2x + y = 4: x^2 + xy = -12 A. (6, -8): (-2, 8) B. (3, -4): (-1, 4) C. (8, -4): (-1, 4) D. (-8, 6): (8, -2) E. (-4, 3): (4, -1) Show Content Detailed Solution 2x + y = 4......(i) x^2 + xy = -12........(ii) from eqn (i), y = 4 - 2x = x2 + x(4 - 2x) = -12 x2 + 4x - 2x2 = -12 4x - x2 = -12 x2 - 4x - 12 = 0 (x - 6)(x + 2) = 0 sub. for x = 6, in eqn (i) y = -8, 8 =(6,-8); (-2, 8)	A