NATIONAL SCIENCE AND MATHS QUIZ PAST QUESTIONS-1
NATIONAL SCIENCE AND MATHS QUIZ PAST QUESTIONS-1
Find the slope m of the tangent to the given curve at the given point on the curve.
1. y = 3x2 – 5x + 2 at x = 1
ANSWER: m = 1
[dy/dx = 6x – 5, at x = 1, m = 6 – 5 = 1]
2. y = x3 – x2 – 3x at x = 2
ANSWER: m = 5
[dy/dx = 3x2 – 2x – 3, at x = 2, m = 3(4) – 2(2) – 3 = 12 – 7 = 5]
3. y = x4 – 3x2 + 2x at x = 1
ANSWER: m = 0
[dy/dx = 4x3 – 6x + 2, at x = 1, m = 4 – 6 + 2 = 0]
Solve for x in the interval 0 < x < π/2. Give answer in radians.
1. sinx = √3/2
ANSWER: x = π/3 radians
2. cosx = √3/2
ANSWER: x = π/6 radians
3. tanx = √3
ANSWER: x = π/3 radians
1. Find the number of sides of a regular polygon if an exterior angle has measure π/12 radians
ANSWER: 24
[2π/(π/12) = 2(12) = 24]
2. Evaluate 10C7 (10 combination 7)
ANSWER: 120
[ (10 x 9 x 8)/(3 x 2 x 1) = 10 x 3 x 4 = 120]
3. simplify (tan3x + tanx)/(1 – tan3xtanx)
ANSWER: tan4x
[tan(3x + x) = tan4x]
Find each interior angle of a quadrilateral if its interior angles are represented by
1. (x – 5)°, (x + 20)°, (2x – 45)° and (2x – 30)֯
ANSWER: 65°, 90°, 95°, 110֯
[ 6x – 60 = 360, 6x = 420, x = 70, hence 65, 90, 95, 110]
2. (2x – 10)֯, (x + 30)֯, (2x + 50)֯, and (x + 20)֯
ANSWER: 80°, 75°, 140°, 65°
[ 6x + 90 = 360, 6x = 270, x = 45, hence 80, 75, 140, 65]
3. (2x – 30)֯, (x + 50)֯, (x + 30)֯, (2x + 10)֯
ANSWER: 70°, 100°, 80°, 110°
[ 6x+ 60 = 360, 6x = 300, x = 50, hence 70, 100, 80, 110]
1. Identify the polygon if in its regular form an interior angle has measure
5π/6 radians.
ANSWER: duo-decagon [2π/n = π/6, n = 12 hence a duo-decagon]
2. Evaluate the expression (tan87° – tan42°)/(1 + tan87°tan42°)
ANSWER: 1
[ tan(87 – 42) = tan45 = 1]
3. Find the general term Un of the linear sequence, -5, – 2, 1, 4, 7, . . .
ANSWER: Un = 3n – 8
[a = -5, d = 3, Un = a + (n – 1)d = – 5 + 3(n – 1) = 3n – 8
Factorize completely
1. x3 + 3x2 – x – 3
ANSWER: (x – 1)(x + 1)(x + 3)
[x2(x + 3) – 1(x + 3) = (x2 – 1)(x + 3) = (x – 1)(x + 1)(x + 3)
2. x3 – 2x2 – 4x + 8
ANSWER: (x + 2)(x – 2)2
[ x2(x – 2) – 4(x – 2)= (x2 – 4)(x – 2) = (x + 2)(x – 2)2
3. 3x3 – 4x2 – 27x + 36
ANSWER: (3x – 4)(x – 3)(x + 3)
[ 3x(x2 – 9) – 4(x2 – 9) = (3x – 4)(x2 – 9) = (3x – 4)(x – 3)(x + 3)]
Find the slope of the line passing through the given points
1. (a, b) and (3a, – 2b)
[ (-2b – b)/(3a – a) = – 3b/2a
ANSWER: -3b/2a
2. (a, b + c) and (a + c, b – c)
[(b – c – b – c)/(a + c – a) = -2c/c = -2
ANSWER: -2
3. (a, a2) and (b, b2)
[(b2 – a2)/(b – a) = (b – a)(b + a)/(b – a) = (b + a)]
ANSWER: (b + a)
1. Find a natural number n such that the number nnn3 = 2610
ANSWER: n = 2
[9n + 3n + n = 13n = 26, n = 2]
2 A fair coin is tossed three times. Find the probability of obtaining an even number of heads
ANSWER: ½
[A= {TTT, THH, HHT, HTH}, P(A) = 4/8 = ½ ]
3 Evaluate the trigonometric expression cos33°cos27° – sin33°sin27°
ANSWER: ½
[cos(33 + 27) = cos60° = ½ ]
Find the value of x if the vectors
1. a = 2xi + 4j and b = 8i + xj are parallel,
ANSWER: x = ± 4
[4/2x = x/8, x2 = 16, x = ± 4]
2. a = 2xi + 4j and b = 8i + xj are perpendicular,
ANSWER: x = 0
[a.b = 16x + 4x = 0, x = 0]
3. a = 5i + xj and b = xi + 5j are parallel.
ANSWER: x = ± 5
[x/5 = 5/x, x2 = 25, x = ± 5]
Find an expression for f(x), if
1. f(x + 1) = x2 + 5x + 7
ANSWER: f(x) = x2 + 3x + 3
[x2 + 5x + 7 = (x + 1)2 + 3x + 6 = (x + 1)2 + 3(x + 1) + 3, f(x) = x2 + 3x + 3 ]
2. f(x – 1) = x2 + x + 2
ANSWER: f(x) = x2 + 3x + 4
[ x2 + x + 2 = (x – 1)2 + 3x + 1 = (x – 1)2 + 3(x – 1) + 4, f(x) = x2 + 3x + 4]
3. f(x + 2) = x2 + 5x + 8
ANSWER: f(x) = x2 + x + 2
[ x2 + 5x + 8 = (x + 2)2 + x + 4 = (x + 2)2 + (x + 2) + 2, hence f(x) = x2 + x + 2]
1. Find the quadratic equation the sum of whose root is 10 and the product is -7
ANSWER: x2 – 10x – 7 = 0
2. Find the term without x in the expansion of (x + 1/x)8
ANSWER: 56
[8Crxr(1/x)8 – r = 8Crx2r – 8, 2r – 8 = 0, r = 4, 8C4 = (8 x 7 x 6 x 5)/(4 x 3 x 2 x 1)= 56]
3. Find the value of (tan87° + tan33°)/(1 – tan87° tan33°)
ANSWER: -√3
[tan(87 + 33) = tan120 = -tan60 = -√3]
Find the length of a side of a rhombus if the diagonals are of lengths
1. 12cm and 16cm
ANSWER: 10cm
[ 62 + 82 = x2 = 100, x = 10]
2. 10cm and 24cm
ANSWER: 13cm
[ 52 + 122 = 25 + 144 = 169 = x2, x = 13]
3. 16cm and 30cm
ANSWER: 17cm
[ 82 + 152 = 64 + 225 = 289 = x2, x = 17]
Find the binomial expansion in ascending powers of x of
1. (1 + 2x)4
ANSWER: 1 + 8x + 24×2 + 32×3 + 16×4
[1 + 4(2x) + 6(2x)2 + 4(2x)3 + (2x)4 = 1 + 8x + 24×2 + 32×3 + 16×4
2. (1 – 3x)3
ANSWER: 1 – 9x + 27×2 – 27×3
[ 1 + 3(-3x) + 3(-3x)2 + (-3x)3 = 1 – 9x + 27×2 – 27×3
3. (1 + 2x)5
ANSWER: 1 + 10x + 40×2 + 80×3 + 80×4 + 32×5
[ 1 + 5(2x) + 10(2x)2 + 10(2x)3 + 5(2x)4 + 1(2x)5]
1. Solve for x from the equation 125x = 625
ANSWER: x = 4/3
[53x = 54, 3x = 4,x = 4/3]
2 Express the decimal number 19510 as a number in base 7.
ANSWER: 3667
(195 = 3 x 49 + 48 = 3 x 72 + 6 x 7 + 6 = 3667)
Factorize the cubic expression
1. x3 – 27
ANSWER: (x – 3)(x2 + 3x + 9)
[ x3 – 33 = (x – 3)(x2 + 3x + 9) ]
2. x3 + 125
ANSWER: (x + 5)(x2 – 5x + 25)
[ x3 + 53 = (x + 5)(x 2 – 5x + 25)]
3. (x3 – 64)
ANSWER: (x – 4)(x2 + 4x + 16)
[ x3 – 43 = (x – 4)(x2 + 4x + 16)]
Solve the logarithmic equation
1. log27x = 1/3
ANSWER: x = 3
[ x = 271/3 = 3 ]
2. log25x = 3/2
ANSWER: x = 125
[ x = 253/2 = (251/2)3 = 53 = 125]
3. log81x = 3/4
ANSWER: x = 27
[x = 813/4 = (811/4)3 = 33 = 27]
1. Find the number of digits when the decimal number 70 is expressed as a binary number.
ANSWER: 7
[Highest power of 2 in 70 is 64 = 26, number of digits is 6 + 1 = 7]
2. The general term of a sequence is Un = 3 – 2n. Find the sum of the first 20 terms.
ANSWER: -360
[Linear sequence a = 1, d = -2, S20 = (20/2)(2(1) + 19(-2)) = 10(2 – 38) = -360]
3. Given that sinx = a/b and x is acute, find cos(π/2 – x).
ANSWER: a/b
[cos(π/2 – x) = sinx = a/b]
1. Which of the following fractions is greater than 1/3?
A) 3/10, B) 6/17, C) 5/18, D) 10/33
ANSWER: B) 6/17
2. Which of the following fractions is between 1/3 and 5/14?
A) 2/3, B) ¾, C) 29/84, D) 3/7
ANSWER: C) 29/84
3. Which of the following fractions is less than 25%?
A) ¼, B) 2/5, C) 3/11, D) 5/21
ANSWER: D) 5/21
Differentiate the rational function with respect to x.
1. y = x/(x + 1)
ANSWER: dy/dx = 1/(x + 1)2
[dy/dx = ((x + 1) – x)/(x + 1)2 = 1/(x + 1)2]
2. y = (x – 2)/(x + 2)
ANSWER: dy/dx = 4/(x + 2)2
[ dy/dx = ((x + 2) – (x – 2))/(x + 2)2 = 4/(x + 2)2
3. y = (2x – 1)/(2x + 1)
ANSWER: dy/dx = 4/(2x + 1)2
[ dy/dx = (2(2x + 1) – 2(2x – 1))/(2x + 1)2 = 4/(2x + 1)2]
1. Find the next term in the sequence 1, 27, 125, 343, . . .
ANSWER: 729
[ 13, 33 , 53, 73, . . ., next term is 93 = 81 x 9 = 729]
2. A bicycle wheel has 18 spokes. Find the measure of angle between adjacent spokes.
ANSWER: 20°
[ 360/18 = 20]
3. Find the total surface area of a solid cube if a side has length 20 cm.
ANSWER: 2400 cm2
[ Area = 6 x 202 = 6 x 400 = 2400 cm2]
Given that i2 = -1, simplify
1. i6
ANSWER: -1
[ i6 = i4i2 = 1(-1) = -1]
2. i15
ANSWER: -i
[i15 = i12i3 = (i4)3i3 = i2i = (-1)i = -i
3. i12
ANSWER: 1
[ i12 = i4i4i4 = 1 x 1 x 1 = 1]
Find a polynomial function f(x) of degree 3 with integer coefficients, with the given numbers as its zeros:
1. 1, -2, 3
ANSWER: f(x) = (x – 1)(x + 2)(x – 3)
2. 2, -3, 5
ANSWER: f(x) = (x – 2)(x + 3)(x – 5)
3. -5, 3, -1
ANSWER: f(x) = (x + 5)(x – 3)(x + 1)
Solve the logarithmic equation for x
1. 2logx = log5 + log(x + 10)
ANSWER: x = 10
[ x2 = 5(x + 10), x2 – 5x – 50 = 0, (x – 10)(x + 5) = 0, x = 10]
2. logx + log(x – 3) = log18
ANSWER: x = 6
[ x(x – 3) = 18, x2 – 3x – 18 = (x – 6)(x + 3) = 0, x = 6]
3. logx + log(x + 5) = log36
ANSWER: x = 4
[ x(x + 5) = 36, x2 + 5x – 36 = 0, (x – 4)(x + 9) = 0, x = 4]
Evaluate and simplify
1. sin60° + cos60° + tan60°
ANSWER: (1 + 3√3)/2
[√3/2 + ½ + √3 = ½ (1 + 3√3) = (1 + 3√3)/2
2. cos45° + sin45° + tan45°
ANSWER: 1 + √2
[ 1/√2 + 1/√2 + 1 = 1 + 2/√2 = 1 + √2
3. sin30° + cos30° + tan30°
ANSWER: (3 + 5√3)/6
[1/2 + √3/2 + 1/√3 = ½ (1 + √3) + √3/3 = ½ + (3√3 + 2√3)/6 = (3 + 5√3)/6
1. Find the coordinates of the intercepts of the curve x2/9 – y2/16 = 1 on the axes.
ANSWER: (3, 0), (-3, 0)
[y = 0, x2/9 = 1, x2 = 9, x = ± 3, intercepts are (3, 0) and (-3, 0,), no y-intercepts]
2. Find a natural number n such that 123n = 6610
ANSWER: n = 7
[n2 + 2n + 3 = 66, n2 + 2n = 63, n2 + 2n – 63 = 0, (n – 7)(n + 9) = 0, n = 7]
3. Find the sum of the first 20 even natural numbers
ANSWER: 420
[S20 = (20/2)[2(2) + 2(19)] = 10(42) = 420
A fair coin and a fair die are tossed. Find the probability of obtaining
1. a Head and a number greater than 3,
ANSWER: ¼
[ A = {(H, 4), (H, 5), (H, 6)}, P(A) = 3/12 = ¼ ]
2. a Head or a Tail and a prime number,
ANSWER: 1/2
[ B = {(H,2), (T, 2), ( H, 3), (T, 3), (H,5), (T, 5)}, P(B) = 6/12 = ½ ]
3. a Tail and an odd prime or a number greater than 4.
ANSWER: ¼
[ C = {T, 3) (T, 5), (T, 6)}, P(C ) = 3/12 = ¼ ]
Find the equation of the line y = x – 3 after reflection in the
1. line y = x
ANSWER: y = x + 3
[(x, y) →(y, x), hence x = y – 3, y = x + 3]
2. x-axis
ANSWER: y = 3 – x
[(x, y)→(x, -y), -y = x – 3, y = 3 – x]
3. y-axis
ANSWER: y = -x – 3
[(x, y) →(-x, y), y = -x – 3]
1. If a, b, c are the degree measures of angles in an equilateral triangle, find a relation between a, b, c.
ANSWER: a = b = c = 60֯
2. If a, b, c are the degree measures of the interior angles of a triangle, find a relation between a, b, c.
ANSWER: a + b + c = 180°
3. if a, b are the degree measures of the non-congruent angles of an isosceles trapezium, find a relation between a and b.
ANSWER: a + b = 180°
Solve for n given that nCr represents ‘n combination r’.
1. nC2 = 10C8
ANSWER: n = 10
[n = 2 + 8 = 10]
2. 20Cn = 20C12 with n ≠ 12
ANSWER: n = 8
[ n + 12 = 20, n = 8
3 15C5 = nC10
ANSWER: n = 15
[ n = 5 + 10 = 15]
Assume x is the measure of an acute angle.
1. If sinx = 3/5, find secx
ANSWER: 5/4
[cosx = 4/5, hence secx = 5/4]
2. If tanx = 5/12 find cosecx
ANSWER: 13/5
[ sinx = 5/13, hence cosecx = 13/5]
3. If cosx = 12/13, find cotx
ANSWER: 12/5
[ tanx = 5/12, hence cotx = 12/5]
Given f(x) = 1 – 2x and g(x) = x2 + 1, find an expression for
1. f(g(x))
ANSWER: -2x2 – 1
[ f(g(x)) = 1 – 2g(x) = 1 – 2(x2+ 1) = -2x2 – 1]
2. g(f(x))
ANSWER: (1 – 2x)2 + 1, or 4x2 – 4x + 2
[ g(f(x)) = f(x)2 + 1 = (1 – 2x)2 + 1 = 4x2 – 4x + 2]
3. g(g(x))
ANSWER: (x2 + 1)2 + 1, or (x4 + 2x2 + 2)
[ g(g(x)) = g(x)2 + 1 = (x2 + 1)2 + 1 = x4 + 2x2 + 2]
Evaluate
1. (13.75)2 – (3.75)2
ANSWER: 175.00
[ (13.75 + 3.75)(13.75 – 3.75) = 17.50(10.00) = 175.00
2. (77.77)2 – (22.23)2
ANSWER: 5,554.00
[ (77.77 + 22.23)(77.77 – 22.23) = 100.00(55.54) = 5,554.00]
3. (17.85)2 – (7.85)2
ANSWER: 257.00
[ (17.85 + 7.85)(17.85 – 7.85) = 25.70(10.00) = 257.00]
1. Solve the equation tanx = √3 in the interval 90° < x < 180°.
ANSWER: NO SOLUTION
[ in the interval 90° < x < 180°, tanx is negative]
2. Evaluate 10C4
ANSWER:210
[ 10 x 9 x 8 x 7/(4 x 3 x 2 x 1) = 210
3. Find the number of sides of a regular polygon if an interior angle measures 174°.
ANSWER: 60 sides
[ n = 360/(180 – 174) = 360/6 = 60]
Find the number of sides of a regular polygon if an interior angle has measure
1. 171°
ANSWER: 40
[ n = 360/9 = 40]
2. 177°
ANSWER: 120
[ n = 360/3 = 120]
3. 160°
ANSWER: 18
[ n = 360/20 = 18]
If f(x) = 2x – 1, g(x) = x + 2 and h(x) = x2, evaluate
1. f(g(h(2)))
ANSWER: 11
[ h(2) = 4, g(4) = 4 + 2 = 6, f(6) = 12 – 1 = 11]
2. g(f(h(-1)))
ANSWER: 3
[ h(-1) = 1, f(1) = 1, g(1) = 1 + 2 = 3]
3. h(f(g(-2)))
ANSWER: 1
[ g(-2) = -2 + 2 = 0, , f(0) = 0 – 1 = -1, h(-1) = 1]
1. Solve the quadratic equation 4x2 + x – 5 = 0
ANSWER: x = 1, -5/4
[4x2 + 5x – 4x – 5 = 0, x(4x + 5) – 1(4x + 5) = (x – 1)(4x + 5) = 0, x = 1, -5/4]
2. Find the first term a and the common difference d, of a linear sequence given third term is 7 and seventh term is 15.
ANSWER: a = 3, d = 2
[a + 2d = 7, a + 6d = 15, 4d = 8, d = 2 and a = 7 – 4 = 3]
Find the coordinates of the image of the point A(3, 5) after reflection
1. in the origin,
ANSWER: (-3, -5)
2. in the line y = 3,
ANSWER: (3, 1)
3. in the line x = 2,
ANSWER: (1, 5)
Find dy/dx if
1. x2 + xy – y2 = 9
ANSWER: dy/dx = (2x + y)/(2y – x)
[2x + y + xdy/dx -2ydy/dx = 0, dy/dx(2y – x) = (2x + y), dy/dx = (2x + y)/(2y – x)]
2. 2x2 – xy + y2 = 6
ANSWER: dy/dx = (4x – y)/(x – 2y)
[ 4x – y – xdy/dx + 2ydy/dx = 0, dy/dx(x – 2y) = (4x – y), dy/dx = (4x – y)/(x – 2y)]
3. x2 + xy – 3y2 = 10
ANSWER: dy/dx = (2x + y)/(6y – x)
[2x + y + xdy/dx – 6ydy/dx = 0, dy/dx(6y – x) = (2x + y), dy/dx = (2x + y)/(6y – x)]
1. Find n if 123n = 3810
ANSWER: n = 5
[ n2 + 2n + 3 = 38, n2 + 2n – 35 = 0, (n + 7)(n – 5) = 0, n = 5]
2. Find the greatest value of the expression 3sinx + 4cosx
ANSWER: 5
[R = √(32 + 42) =√25 = 5]
3. Rationalize the denominator 3/(√5 – √2)
ANSWER: √5 + √2
[3(√5 + √2)/(5 – 2) = √5 + √2]
Find the coordinates of the vertex of the quadratic curve given by
1. y = x2 + 6x + 10
ANSWER: (-3, 1)
[ x2 + 6x + 9 + 1 = (x + 3)2 + 1, hence vertex is (-3, 1)]
2. y = x2 – 8x + 12
ANSWER: (4, – 4)
[ x2 – 8x + 16 – 4 = (x – 4)2 – 4, hence (4, -4)]
3. y = x2 + 6x – 1
ANSWER: (-3, -10)
[ x2 + 6x + 9 – 10 = (x + 3)2 – 10, hence (-3, -10)]
Evaluate the trigonometric expression
1. sin17°cos13° + sin13°cos17°
ANSWER: ½ or 0.5
[ sin(17 + 13) = sin30° = ½ ]
2. sin25֯sin35֯ – cos25֯cos35֯
ANSWER: -1/2 or -0.5
[ -(cos25cos35 – sin25sin35) = – cos(25 + 35) = -cos60 = – ½]
3. (tan32° + tan28°)/(1 – tan32°tan28°)
ANSWER: √3
[ tan(32° + 28°) = tan60° = √3]
1. Find the prime numbers between 80 and 90
ANSWER: 83, 89
[ 81, 82, (83), 84, 85, 86, 87, 88, (89)]
2. Find the equation of the image of the line y = 2x after reflection in the y-axis.
ANSWER: y = -2x
[(x, y)→(-x, y), y = 2(-x) = -2x]
3. A fair die is thrown. Find the probability of getting a multiple of 2 or a multiple of 3.
ANSWER: 2/3
[ A = {2, 3, 4, 6}, P(A) = 4/6 = 2/3]
Determine the nature of the roots of the given equation without solving.
1. x2 + 5x + 7 = 0
ANSWER: NO REAL ROOTS
[ b2 – 4ac = 25 – 28 = -3 < 0, hence no real roots]
2. x2 – 2x – 3 = 0
ANSWER: 2 REAL RATIONAL ROOTS
[ b2 – 4ac = 4 + 12 = 16, hence 2 real rational roots]
3. x2 + 5x – 7 = 0
ANSWER: 2 REAL IRRATIONAL ROOTS
[ b2 – 4ac = 25 + 28 = 53 > 0, hence 2 real irrational roots]
Find the distance between the pair of points
1. A(2, 4) and B(6, 7)
ANSWER: 5
[d2 = 32 + 42 = 25, d = 5]
2. C(-2, -3) and D(2, 3)
ANSWER: 2√13
[d2 = 42 + 62 = 16 + 36 = 52 = 4(13), d = 2√13]
3. E(3, -1) and F(-2, 6)
ANSWER: √74
[ d2 = 52 + 72 = 25 + 49 = 74, d = √74]
1. Multiply and simplify (x – 3)(x2 + 3x + 9)
ANSWER: x3 – 27 (difference of two cubes)
2. Find an equation of a line having x- intercept of 4 and a y-intercept of -4.
ANSWER: x – y = 4
[ x/4 + y/-4 = 1, x – y = 4]
3. If i2 = -1, evaluate (5 – 3i)(5 + 3i)
ANSWER: 34
[ 25 – 9i2 = 25 + 9 = 34]
Find the quadratic equation whose roots are
1. 3 ± √5
ANSWER: x2 – 6x + 4 = 0
[sum of roots = 6, product = 9 – 5 = 4, hence x2 – 6x + 4 = 0]
2. 5 ± √7
ANSWER: x2 – 10x + 18 = 0
[ sum of roots = 10, product =25 – 7 = 18, hence x2 – 10x + 18 = 0]
3. -4 ± 3√2
ANSWER: x2 + 8x – 2 = 0
[ sum of roots = -8, product = 16 – 18 = -2, hence x2 + 8x – 2 = 0]
1. Express a relation between x and y if x and y are degree measures of adjacent angles of a parallelogram.
ANSWER: x + y = 180 (Accept x + y = 180°)
2. Two fair coins are tossed once. Find the probability of obtaining an even number of heads.
ANSWER: ½ [A = {HH, TT}, P(A) = 2/4 = ½]
3. Differentiate the function y = (2x3 – 3x2)4
ANSWER: dy/dx = 24(x2 – x)(2x3 – 3x2)3
[dy/dx = 4(6x2 – 6x)(2x3 – 3x2)3 = 24(x2 – x)(2x3 – 3x2)3]
Suppose for some function f(x + 2) = 2x + 3. Evaluate
1. f(-2)
ANSWER: -5
[x + 2 = -2, x = – 4, f(-2) = 2(-4) + 3 = -5]
2. f(2)
ANSWER: 3
[ x + 2 = 2, x = 0, f(2) = 2(0) + 3 = 3]
3. f(5)
ANSWER: 9
[ x + 2 = 5, x = 3, f(5) = 2(3) + 3 = 9],
Rationalize the denominator
1. 3/(√5 – √2)
ANSWER: √5 + √2
[ 3(√5 + √2)/(5 – 2) = √5 + √2
2. 5/(√15 + √10)
ANSWER: √15 – √10
[ 5(√15 – √10)/(15 – 10) = √15 – √10]
3. 6/(√13 – √7)
ANSWER: √13 + √7
[6(√13 + √7)/(13 – 7) = √13 + √7
1. Solve the equation sinx = √3/2 in the interval 90° < x < 180°
ANSWER: x = 120°
2. Find the coordinates of the image of the point A(2, -3) after reflection in the x-axis.
ANSWER: (2, 3)
3. For what values of k will the equation kx2 + 4x – k = 0 have real roots?
ANSWER: All real values of k
[ 16 + 4k2 > 0, hence all real values of k]
Find the greatest value of
1. (sec2x – tan2x)
ANSWER: 1
[ sec2x = tan2x + 1, hence sec2x – tan2x = 1]
2. (sinxcos30° + cosxsin30°)
ANSWER: 1
[ sin(x + 30), greatest value = 1]
3. 2sinxcosx + 5
ANSWER: 6
[ sin2x + 5, greatest value is 1 + 5 = 6]
Differentiate the given function with respect to x.
1. y = (2x + 1)6
ANSWER: dy/dx = 12(2x + 1)5
2. y = (3 – 2x)-6
ANSWER: dy/dx = 12(3 – 2x)-7
[ dy/dx = -6(-2)(3 – 2x)-7 = 12(3 – 2x)-7]
3. y = (x2 – 2x)8
ANSWER: dy/dx = 16(x – 1)(x2 – 2x)7
[ dy/dx = 8(2x – 2)(x2 – 2x)7 = 16(x – 1)(x2 – 2x)7 Use chain rule]
1. If A is acute and cosA = 12/13, find tanA
ANSWER: 5/12
[sinA = 5/13, tanA = sinA/cosA = (5/13)/(12/13) = 5/12]
2. Solve the exponential equation 62x – 1 = 216
ANSWER: x = 2
[ 216 = 63, 2x – 1 = 3, x = 2]
3. Find the coefficient of x3 in the expansion of (1 + 2x)6
ANSWER: 160
[ 6C3(2x)3, 20(23) = 160]
Express the variation as an equation
1. y varies partly as x and partly as the inverse of x2
ANSWER: y = ax + b/x2 (a and b are constants)
2. y is partly a constant and partly varies as x3
ANSWER: y = a + bx3
3. y varies jointly as x and z2 and inversely as √w.
ANSWER: y = kxz2/√w
Factorise as an exact cube.
1. x3 – 6x2 + 12x – 8
ANSWER: (x – 2)3
[ x3 – 3(x)2(-2) + 3(x)(-2)2 + (-2)3 = (x – 2)3]
2. x3 + 9x2 + 27x + 27
ANSWER: (x + 3)3
[ x3 + 3x2(3) + 3(x)(3)2 + 33 = (x + 3)3]
3. x3 + 12x2 + 48x + 64
ANSWER: ( x + 4)3
[ x3 + 3(x)24 + 3(x)(4)2 + 43 = (x + 4)3]
1. Find x if the vectors a = xi – 3j and b = 5i + 10j are perpendicular.
ANSWER: x = 6
[ a.b = 5x – 30 = 0, x = 6]
2. Find the stationary point of the function y = (x – 2)2 + 1 and state whether a maximum or a minimum.
ANSWER: (2, 1) minimum
3. Simplify (1 + √3)/(1 – √3)
ANSWER: -(2 + √3)
[(1 + √3)2/(1 – 3) = (4 + 2√3)/-2 = -2 – √3 = -(2 + √3)
Given the relation R = {(1, 2), (2, -3), (3, 5), (4, 7)}
1. find the domain of the relation,
ANSWER: {1, 2, 3, 4}
2. find the range of the relation,
ANSWER: {2, -3, 5, 7}
3. determine if R is a function.
ANSWER: R is a function
Suppose for some function f, f(x – 1) = 5x. Evaluate
1. f(2)
ANSWER: 15
[x – 1 = 2, x = 3, f(2) = 5(3) = 15]
2. f(-2)
ANSWER: -5
[ x – 1 = -2, x = -1, f(-2) = 5(-1) = -5]
3. f(-5)
ANSWER: -20
[ x – 1 = -5, x = -4, f(-5) = 5(-4) = -20]
1. Find the midpoint of the line segment connecting the points A(1/2, 1/3) and B(1/3, ½)
ANSWER: (5/12, 5/12)
[(1/2 + 1/3)/2, (1/3 + ½)/2) = ((5/6)/2, (5/6)/2) = (5/12, 5/12)
2. Solve the linear equation 2(7x – 4) – 4(2x – 6) = 3x + 31
ANSWER: x = 5
[14x – 8x – 8 + 24 = 3x + 31, 3x = 39 – 24 = 15, x = 5]
3. Multiply and simplify (x + 2)(x2 – 2x + 4)
ANSWER: (x3 + 8)
[sum of two cubes formula]
Simplify the trigonometric expression
1. (tanx + tan3x)/(1 – tanxtan3x)
ANSWER: tan4x
[ tan(x + 3x) = tan4x]
2. (tan5x – tan2x)/(1 + tan5xtan2x)
ANSWER: tan3x
[ tan(5x – 2x) = tan3x]
3. (1 – tan2x)/(2tanx)
ANSWER: cot2x or 1/tan2x
[ 1/tan(x + x) = 1/tan(2x) = cot2x]
Find the sum of the interior angles of the given polygon in radians
1. duo-decagon
ANSWER: 10π radians
[ (12 – 2)π = 10π]
2. nonagon
ANSWER: 7π radians
[ (9 – 2)π = 7π]
3. icosagon
ANSWER:18π radians
[ (20 – 2)π = 18π]
1. Find the value of x if (x – 2), (2x + 2), (x + 12) are consecutive terms of a linear sequence.
ANSWER: x = 3
[ 2(2x + 2) = x – 2 + x + 12 = 2x + 10, 2x = 10 – 4 = 6, x = 3]
2. Find the coordinates of the image of the point P(2, 5) after reflection in the line y = 6.
ANSWER: (2, 7)
[5 + 1 + 1 = 7, hence (2, 7)
3. Find the term without x in the expansion of (x2 + 1/x)9
ANSWER: 84
[ 9Crx2r(1/x)9 – r, x2r + r – 9, 3r – 9 = 0, r = 3, 9C3 = 9 x 8 x 7 /3 x 2 x 1 = 84]
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