# 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 = 3x^{2} – 5x + 2 at x = 1

ANSWER:** m = 1**

[dy/dx = 6x – 5, at x = 1, m = 6 – 5 = 1]

#### 2. y = x^{3} – x^{2} – 3x at x = 2

ANSWER:** m = 5**

[dy/dx = 3x^{2} – 2x – 3, at x = 2, m = 3(4) – 2(2) – 3 = 12 – 7 = 5]

#### 3. y = x^{4} – 3x^{2} + 2x at x = 1

ANSWER:** m = 0**

[dy/dx = 4x^{3} – 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 10C_{7} (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 U_{n} of the linear sequence, -5, – 2, 1, 4, 7, . . .

ANSWER: U_{n} = 3n – 8

[a = -5, d = 3, U_{n} = a + (n – 1)d = – 5 + 3(n – 1) = 3n – 8

Factorize completely

#### 1. x^{3} + 3x^{2} – x – 3

ANSWER: (x – 1)(x + 1)(x + 3)

[x^{2}(x + 3) – 1(x + 3) = (x^{2} – 1)(x + 3) = (x – 1)(x + 1)(x + 3)

#### 2. x^{3} – 2x^{2} – 4x + 8

ANSWER: (x + 2)(x – 2)^{2}

[ x^{2}(x – 2) – 4(x – 2)= (x^{2} – 4)(x – 2) = (x + 2)(x – 2)^{2}

#### 3. 3x^{3} – 4x^{2} – 27x + 36

ANSWER: (3x – 4)(x – 3)(x + 3)

[ 3x(x^{2} – 9) – 4(x^{2} – 9) = (3x – 4)(x^{2} – 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, a^{2}) and (b, b^{2})

[(b^{2} – a^{2})/(b – a) = (b – a)(b + a)/(b – a) = (b + a)]

ANSWER: (b + a)

#### 1. Find a natural number n such that the number nnn_{3} = 26_{10}

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, x^{2} = 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, x^{2} = 25, x = ± 5]

Find an expression for f(x), if

#### 1. f(x + 1) = x^{2} + 5x + 7

ANSWER: f(x) = x^{2} + 3x + 3

[x^{2} + 5x + 7 = (x + 1)^{2} + 3x + 6 = (x + 1)^{2} + 3(x + 1) + 3, f(x) = x^{2} + 3x + 3 ]

#### 2. f(x – 1) = x^{2} + x + 2

ANSWER: f(x) = x^{2} + 3x + 4

[ x^{2} + x + 2 = (x – 1)^{2} + 3x + 1 = (x – 1)^{2} + 3(x – 1) + 4, f(x) = x^{2} + 3x + 4]

#### 3. f(x + 2) = x^{2} + 5x + 8

ANSWER: f(x) = x^{2} + x + 2

[ x^{2} + 5x + 8 = (x + 2)^{2} + x + 4 = (x + 2)^{2} + (x + 2) + 2, hence f(x) = x^{2} + x + 2]

#### 1. Find the quadratic equation the sum of whose root is 10 and the product is -7

ANSWER: x^{2} – 10x – 7 = 0

#### 2. Find the term without x in the expansion of (x + 1/x)^{8}

ANSWER: 56

[8C_{r}x^{r}(1/x)^{8 – r} = 8C_{r}x^{2r – 8}, 2r – 8 = 0, r = 4, 8C_{4} = (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

[ 6^{2} + 8^{2} = x^{2} = 100, x = 10]

#### 2. 10cm and 24cm

ANSWER: 13cm

[ 5^{2} + 12^{2} = 25 + 144 = 169 = x^{2}, x = 13]

#### 3. 16cm and 30cm

ANSWER: 17cm

[ 8^{2} + 15^{2} = 64 + 225 = 289 = x^{2}, 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 125^{x} = 625

ANSWER: x = 4/3

[5^{3x} = 5^{4}, 3x = 4,x = 4/3]

2 Express the decimal number 195_{10} as a number in base 7.

ANSWER: 366_{7}

(195 = 3 x 49 + 48 = 3 x 72 + 6 x 7 + 6 = 3667)

Factorize the cubic expression

#### 1. x^{3} – 27

ANSWER: (x – 3)(x^{2} + 3x + 9)

[ x^{3} – 3^{3} = (x – 3)(x^{2 }+ 3x + 9) ]

#### 2. x^{3} + 125

ANSWER: (x + 5)(x^{2} – 5x + 25)

[ x^{3} + 5^{3} = (x + 5)(x ^{2} – 5x + 25)]

#### 3. (x^{3} – 64)

ANSWER: (x – 4)(x^{2} + 4x + 16)

[ x^{3} – 4^{3} = (x – 4)(x^{2} + 4x + 16)]

Solve the logarithmic equation

#### 1. log_{27}x = 1/3

ANSWER: x = 3

[ x = 27^{1/3 }= 3 ]

#### 2. log_{25}x = 3/2

ANSWER: x = 125

[ x = 25^{3/2} = (25^{1/2})^{3} = 5^{3} = 125]

#### 3. log_{81}x = 3/4

ANSWER: x = 27

[x = 81^{3/4} = (81^{1/4})^{3} = 3^{3} = 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 = 2^{6}, number of digits is 6 + 1 = 7]

#### 2. The general term of a sequence is U_{n} = 3 – 2n. Find the sum of the first 20 terms.

ANSWER: -360

[Linear sequence a = 1, d = -2, S_{20} = (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

[ 1^{3}, 3^{3} , 5^{3}, 7^{3}, . . ., next term is 9^{3} = 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 cm^{2}

[ Area = 6 x 20^{2} = 6 x 400 = 2400 cm^{2}]

Given that i^{2} = -1, simplify

#### 1. i^{6}

ANSWER: -1

[ i^{6} = i^{4}i^{2} = 1(-1) = -1]

#### 2. i^{15}

ANSWER: -i

[i^{15} = i^{12}i^{3} = (i^{4})^{3}i^{3} = i^{2}i = (-1)i = -i

#### 3. i^{12}

ANSWER: 1

[ i^{12} = i^{4}i^{4}i^{4} = 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**

[ x^{2} = 5(x + 10), x^{2} – 5x – 50 = 0, (x – 10)(x + 5) = 0, x = 10]

#### 2. logx + log(x – 3) = log18

ANSWER:** x = 6**

[ x(x – 3) = 18, x^{2} – 3x – 18 = (x – 6)(x + 3) = 0, x = 6]

#### 3. logx + log(x + 5) = log36

ANSWER:** x = 4**

[ x(x + 5) = 36, x^{2} + 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 x^{2}/9 – y^{2}/16 = 1 on the axes.

ANSWER: (3, 0), (-3, 0)

[y = 0, x^{2}/9 = 1, x^{2} = 9, x = ± 3, intercepts are (3, 0) and (-3, 0,), no y-intercepts]

#### 2. Find a natural number n such that 123_{n} = 66_{10}

ANSWER: n = 7

[n^{2} + 2n + 3 = 66, n^{2} + 2n = 63, n^{2} + 2n – 63 = 0, (n – 7)(n + 9) = 0, n = 7]

#### 3. Find the sum of the first 20 even natural numbers

ANSWER: 420

[S_{20} = (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 nC_{r} represents ‘n combination r’.

#### 1. nC_{2} = 10C_{8}

ANSWER: n = 10

[n = 2 + 8 = 10]

#### 2. 20C_{n} = 20C_{12 } with n ≠ 12

ANSWER: n = 8

[ n + 12 = 20, n = 8

#### 3 15C_{5} = nC_{10}

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) = x^{2} + 1, find an expression for

#### 1. f(g(x))

ANSWER: -2x^{2} – 1

[ f(g(x)) = 1 – 2g(x) = 1 – 2(x^{2}+ 1) = -2x^{2} – 1]

#### 2. g(f(x))

ANSWER: (1 – 2x)^{2} + 1, or 4x^{2} – 4x + 2

[ g(f(x)) = f(x)^{2} + 1 = (1 – 2x)^{2} + 1 = 4x^{2} – 4x + 2]

#### 3. g(g(x))

ANSWER: (x^{2} + 1)^{2} + 1, or (x^{4} + 2x^{2} + 2)

[ g(g(x)) = g(x)^{2} + 1 = (x^{2} + 1)^{2} + 1 = x^{4} + 2x^{2} + 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 10C_{4}

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) = x^{2}, 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 4x^{2} + x – 5 = 0

ANSWER:** x = 1, -5/4**

[4x^{2} + 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. x^{2} + xy – y^{2} = 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. 2x^{2} – xy + y^{2} = 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. x^{2} + xy – 3y^{2} = 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 123_{n} = 38_{10}

ANSWER: n = 5

[ n^{2} + 2n + 3 = 38, n^{2} + 2n – 35 = 0, (n + 7)(n – 5) = 0, n = 5]

#### 2. Find the greatest value of the expression 3sinx + 4cosx

ANSWER: 5

[R = √(3^{2} + 4^{2}) =√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 = x^{2} + 6x + 10

ANSWER: (-3, 1)

[ x^{2} + 6x + 9 + 1 = (x + 3)^{2} + 1, hence vertex is (-3, 1)]

#### 2. y = x^{2} – 8x + 12

ANSWER: (4, – 4)

[ x^{2} – 8x + 16 – 4 = (x – 4)^{2} – 4, hence (4, -4)]

#### 3. y = x^{2} + 6x – 1

ANSWER: (-3, -10)

[ x^{2} + 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. x^{2} + 5x + 7 = 0

ANSWER: NO REAL ROOTS

[ b^{2} – 4ac = 25 – 28 = -3 < 0, hence no real roots]

#### 2. x^{2} – 2x – 3 = 0

ANSWER: 2 REAL RATIONAL ROOTS

[ b^{2} – 4ac = 4 + 12 = 16, hence 2 real rational roots]

#### 3. x^{2} + 5x – 7 = 0

ANSWER: 2 REAL IRRATIONAL ROOTS

[ b^{2} – 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

[d^{2} = 3^{2} + 4^{2} = 25, d = 5]

#### 2. C(-2, -3) and D(2, 3)

ANSWER: 2√13

[d^{2} = 4^{2} + 6^{2} = 16 + 36 = 52 = 4(13), d = 2√13]

#### 3. E(3, -1) and F(-2, 6)

ANSWER: √74

[ d^{2} = 5^{2} + 7^{2} = 25 + 49 = 74, d = √74]

#### 1. Multiply and simplify (x – 3)(x^{2} + 3x + 9)

ANSWER: x^{3} – 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 i^{2} = -1, evaluate (5 – 3i)(5 + 3i)

ANSWER: 34

[ 25 – 9i^{2} = 25 + 9 = 34]

Find the quadratic equation whose roots are

#### 1. 3 ± √5

ANSWER: x^{2} – 6x + 4 = 0

[sum of roots = 6, product = 9 – 5 = 4, hence x^{2} – 6x + 4 = 0]

#### 2. 5 ± √7

ANSWER: x^{2} – 10x + 18 = 0

[ sum of roots = 10, product =25 – 7 = 18, hence x^{2} – 10x + 18 = 0]

#### 3. -4 ± 3√2

ANSWER: x^{2} + 8x – 2 = 0

[ sum of roots = -8, product = 16 – 18 = -2, hence x^{2} + 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 = (2x^{3} – 3x^{2})^{4}

ANSWER: dy/dx = 24(x^{2} – x)(2x^{3} – 3x^{2})^{3}

[dy/dx = 4(6x^{2} – 6x)(2x^{3} – 3x^{2})^{3} = 24(x^{2} – x)(2x^{3} – 3x^{2})^{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 kx^{2} + 4x – k = 0 have real roots?

ANSWER: All real values of k

[ 16 + 4k^{2} > 0, hence all real values of k]

Find the greatest value of

#### 1. (sec^{2}x – tan^{2}x)

ANSWER: 1

[ sec^{2}x = tan^{2}x + 1, hence sec^{2}x – tan^{2}x = 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 = (x^{2} – 2x)^{8}

ANSWER: dy/dx = 16(x – 1)(x^{2} – 2x)^{7}

[ dy/dx = 8(2x – 2)(x^{2} – 2x)^{7} = 16(x – 1)(x^{2} – 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 6^{2x – 1} = 216

ANSWER: x = 2

[ 216 = 6^{3}, 2x – 1 = 3, x = 2]

#### 3. Find the coefficient of x^{3} in the expansion of (1 + 2x)^{6}

ANSWER: 160

[ 6C_{3}(2x)^{3}, 20(2^{3}) = 160]

Express the variation as an equation

#### 1. y varies partly as x and partly as the inverse of x^{2}

ANSWER:** y = ax + b/x ^{2}** (a and b are constants)

#### 2. y is partly a constant and partly varies as x^{3}

ANSWER:** y = a + bx ^{3}**

#### 3. y varies jointly as x and z^{2} and inversely as √w.

ANSWER:** y = kxz ^{2}/**

**√w**

Factorise as an exact cube.

#### 1. x^{3} – 6x^{2} + 12x – 8

ANSWER: (x – 2)^{3}

[ x^{3} – 3(x)^{2}(-2) + 3(x)(-2)^{2} + (-2)^{3} = (x – 2)^{3}]

#### 2. x^{3} + 9x^{2} + 27x + 27

ANSWER: (x + 3)^{3}

[ x^{3} + 3x^{2}(3) + 3(x)(3)^{2} + 3^{3} = (x + 3)^{3}]

#### 3. x^{3} + 12x^{2} + 48x + 64

ANSWER: ( x + 4)^{3}

[ x^{3} + 3(x)^{2}4 + 3(x)(4)^{2} + 4^{3} = (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)(x^{2} – 2x + 4)

ANSWER: (x^{3} + 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 – tan^{2}x)/(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 (x^{2} + 1/x)^{9}

ANSWER: 84

[ 9C_{r}x^{2r}(1/x)^{9 – r}, x^{2r + r – 9}, 3r – 9 = 0, r = 3, 9C_{3} = 9 x 8 x 7 /3 x 2 x 1 = 84]