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PROBLEM SHEET 1 1.1 Find the radius and centre of the circle described by the equation x2 + y2 − 2x − 4y + 1 = 0 by writing it in the form (x − a)2 + (y − b)2 = c2 for suitable a, b and c. 1.2 Find the equation of the line perpendicular to y = 3x passing through the point (3, 9). 1.3 Given sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B, show that cos A sin B = 1 2 [sin(A + B) − sin(A − B)] and sin2 A = 1 2 [1 − cos 2A]. 1.4 Show that 4 cos(αt) + 3 sin(αt) = 5 cos(αt + φ) where φ =arc tan(−3/4). 1.5 Show that, for −1 ≤ x ≤ 1, cos ¡ sin−1 x ¢ = ± √ 1 − x2. 1.6 Given sinh(A ± B) = sinh A cosh B ± cosh A sinh B and cosh(A ± B) = cosh A cosh B ± sinh A sinh B, show that cosh A cosh B = 1 2 [cosh(A + B) + cosh(A − B)] and sinh2 A = 1 2 [cosh 2A − 1]. 1.7 Given that sinh x = 1 2 [ex − e−x ], show that sinh−1 x = ln h x + √ 1 + x2 i . 1.8 Express x (x − 1)(x − 2) in partial fractions. 1.9 If an = 1 n , find P5 i=1 an as a fraction. 1.10 If S = PN i=0 xi , show that xS = PN+1 i=1 xi . Hence show that S − xS = 1 − xN+1 and therefore that S = 1 − xN+1 1 − x . .
PROBLEM SHEET 2 2.1 Given that sinh x = 1 2 [ex − e−x ] show that dy dx = cosh x. 2.2 Given that cosh x = 1 2 [ex + e−x ], show that dy dx = sinh x. 2.3 Let n be a positive integer. Show that dn (xn ) dxn = n! 2.4 If y = ln x, show that dy dx = 1 x ; d2 y dx2 = −1 x2 ; d100 y dx100 = −99! x100 . 2.5 Find the equation of the tangent to the curve y = x2 at (1, 1). 2.6 Find the slope of the curve y = 4x + ex at (0, 1). 2.7 Find the angle of inclination of the tangent to the curve y = x2 + x + 1 at the point (0, 1). 2.8 The displacement y(t) metres of a body at time t seconds (t ≥ 0) is given by y(t) = t−sint. At what times is the body at rest? 2.9 A particle has displacement y(t) metres at time t seconds given by y(t) = 3t3 + 4t + 1. Find its acceleration at time t = 4 seconds. 2.10 If y = N X n=0 anxn show that dy dx = N X n=1 nanxn−1 .
PROBLEM SHEET 3 3.1 If y = ln(1 + x2 ), find dy/dx. 3.2 If y = x 1 + x2 find dy/dx. 3.3 If y = cosh(x4 ), find dy/dx. 3.4 If y = x2 ln x, find d2 y/dx2 . 3.5 Find dy/dx for y = (1 + x2 ) −1/2 . 3.6 Show that for y = sinh−1 x, dy dx = 1 √ 1 + x2 . 3.7 Show that for y = ln[x + √ 1 + x2], dy dx = 1 √ 1 + x2 . 3.8 Find dy/dx for y = cos−1 (sinx). 3.9 A curve is given in polar coordinates by r = 1+sin2 θ. Find dy/dx at θ = π/4. 3.10 Show that if y = 1 2a ln ¯ ¯ ¯ ¯ x − a x + a ¯ ¯ ¯ ¯ , then dy dx = 1 x2 − a2 .
PROBLEM SHEET 4 4.1 Given f(x − ct), where x and c are constant, show that d2 dt2 f(x − ct) = c2 f 00 (x − ct), and calculate this expression when f(u) = sinu. 4.2 Classify the stationary point of y = x−2 lnx, where x > 0. 4.3 Classify the stationary points of y(x) = x2 − 3x + 2. 4.4 The numbers x and y are subject to the constraint x + y = π. Find the values of x and y for which cos(x) sin(y) takes its minimum value. 4.5 Sketch the graph of y = x 1 + x2 . 4.6 Sketch the graph of y(x) = tan(2x) for − 3π 4 ≤ x ≤ 3π 4 . 4.7 Sketch the graph of y = xlnx for x > 0. 4.8 Sketch the graph of y = x3 2x − 1 showing clearly on your sketch any asymptotes. 4.9 Sketch the graph of y = x cos(3x) for 0 ≤ x ≤ 2π.
PROBLEM SHEET 5 5.1 Verify the following Taylor expansions (taking the ranges of validity for granted). (a) ex = 1 + 1 1! x + 1 2! x2 + 1 3! x3 + ... + 1 n! xn + ...valid for any x. (b) sin x = x − x3 3! + x5 5! − ... + (−1)n x2n+1 (2n + 1)! + ...valid for any x. (c) cos x = 1 − x2 2! + x4 4! − ... + (−1)n x2n (2n)! + ...valid for any x. (d) Let α be a constant. (1 + x)α = 1 + αx + α(α − 1) 2! x2 + α(α − 1)(α − 2) 3! x3 + ...valid for − 1 < x < 1 (e) ln(1 + x) = x − x2 2 + x3 3 − ... + (−1)n−1 xn n + ...valid for − 1 < x ≤ 1. 5.2 Obtain a four-term Taylor polynomial approximation valid near x = 0 for each of the following. (a) (1 + x)1/2 , (b) sin(2x), (c) ln(1 + 3x).
PROBLEM SHEET 6 6.1 Reduce to standard form (a) 3 + i 4 − i , and (b) (1 + i)5 . 6.2 Prove (a) |z1z2| = |z1| |z2| , and (b) ¯ ¯ ¯ ¯ z1 z2 ¯ ¯ ¯ ¯ = |z1| |z2| when z2 6= 0. 6.3 Given that eiθ = cos θ + i sin θ, prove that cos(A + B) = cos A cos B − sin A sin B. 6.4 Let z = 1+i. Find the following complex numbers in standard form and plot their corresponding points in the Argand diagram:- (a) z̄2 , and (b) z z̄ . 6.5 Find the modulus and principal arguments of (a) −2 + 2i, (b) 3 + 4i. 6.6 Find all the complex roots of (a) cosh z = 1; (b) sinh z = 1; (c) ez = −1; (d) cos z = √ 2. 6.7 Show that the mapping w = z + c z , where z = x + iy, w = u + iv and c is a real number, maps the circle |z| = 1 in the z plane into an ellipse in the w plane and find its equation. 6.8 Show that cos6 θ = 1 32 (cos 6θ + 6 cos 4θ + 15 cos 2θ + 10).
PROBLEM SHEET 7 7.1 The matrix A = (aij) is given by A = ⎛ ⎜ ⎜ ⎝ 1 2 3 −1 0 1 2 −2 4 1 5 −3 ⎞ ⎟ ⎟ ⎠ Identify the elements a13 and a31. 7.2 Given that A = µ 1 3 0 2 1 1 ¶ , B = ⎛ ⎝ 1 0 2 1 −1 −1 ⎞ ⎠ , C = ⎛ ⎝ 2 1 −1 1 −0 1 ⎞ ⎠ , verify the distributive law A(B + C) = AB + AC for the three matrices. 7.3 Let A = µ 4 2 2 1 ¶ , B = µ −2 −1 4 2 ¶ . Show that AB = 0, but that BA 6= 0. 7.4 A general n ×n matrix is given by A = (aij). Show that A +AT is a symmetric matrix, and that A − AT is skew-symmetric. Express the matrix A = ⎛ ⎝ 2 1 3 −2 0 1 3 1 2 ⎞ ⎠ . as the sum of a symmetric matrix and a skew-symmetric matrix. 7.5 Let the matrix A = ⎛ ⎝ 1 0 0 a −1 0 b c 1 ⎞ ⎠ . Find A2 . For what relation between a, b, and c is A2 = I (the unit matrix)? In this case, what is the inverse matrix of A? What is the inverse matrix of A2n−1 (n a positive integer)? 7.6 Using the rule for inverses of 2 × 2 matrices, write down the inverse of µ 1 1 2 −1 ¶ 7.7 If A and B are both n × n matrices with A non-singular, show that (A−1 BA)2 = A−1 B2 A.
PROBLEM SHEET 8 8.1 Obtain the components of the vectors below where L is the magnitude and θ the angle made with the positive direction of the x axis (−180◦ < θ ≤ 180◦ ). (a) L = 3, θ = 60◦ ; (b) L = 3, θ = −150◦ . 8.2 Two ships, S1 and S2 set off from the same point Q. Each follows a route given by successive displacement vectors. In axes pointing east and north, S1 follows the path to B via − → QA = (2, 4), and − → AB = (4, 1). S2 goes to E via − → QC = (3, 3) − − → CD = (1, 1) and − − → DE = (2, −3). Find the displacement vector − − → BE in component form. 8.3 Sketch a diagram to show that if A, B, C are any three points, then − → AB + − − → BC + − → CA = 0. Formulate a similar result for any number of points. 8.4 Sketch a diagram to show that if A, B, C, D are any four points, then − − → CD = − − → CB + − → BA + − − → AD. Formulate a similar result for any number of points. 8.5 Two points A and B have position vectors a and b respectively. In terms of a and b find the position vectors of the following points on the straight line passing through A and B. (a) The mid-point C of AB; (b) a point U between A and B for which AU/UB = 1/3. 8.6 Suppose that C has position vector r and r = λa +(1 − λ)b where λ is a parameter, and A, B are points with a, b as position vectors. Show that C describes a straight line. Indicate on a diagram the relative positions of A, B, C, when λ < 0, 0 < λ < 1, and λ > 1. 8.7 Find the shortest distance from the origin of the line given in vector parametric form by r = a+tb, where a = (1, 2, 3) and b = (1, 1, 1) , and t is the parameter (Hint: use a calculus method, with t as the independent variable.) 8.8 ABCD is any quadrilateral in three dimensions. Prove that if P, Q, R, S are the mid-points of AB, BC, CD, DA respectively, then PQRS is a parallelogram. 8.9 ABC is a triangle, and P, Q, R are the mid-points of the respective sides BC, CA, AB. Prove that the medians AP, BQ, CR meet at a single point G (called the centroid of ABC; it is the centre of mass of a uniform triangular plate.) 8.10 Show that the vectors 0A = (1, 1, 2), 0B = (1, 1, 1), and 0C = (5, 5, 7) all lie in one plane.
PROBLEM SHEET 9 9.1 The figure ABCD has vertices at (0, 0), (2, 0), (3, 1) and (1, 1). Find the vectors − → AC and − − → BD. Find − → AC · − − → BD. Hence show that the angles between the diagonals of ABCD have cosine −1/ √ 5. 9.2 Show that the vectors a = i + 3j + 4k and b = −2i + 6j − 4k are perpendicular. Obtain any vector c = c1i + c2j + c3k which is perpendicular to both a and b. 9.3 Find the value of λ such that the vectors (λ, 2, −1) and (1, 1, −3λ) are perpendicular. 9.4 Find a constant vector parallel to the line given parametrically by x = 1 − λ, y = 2 + 3λ, z = 1 + λ. 9.5 A circular cone has its vertex at the origin and its axis in the direction of the unit vector â. The half-angle at the vertex is α. Show that the position vector r of a general point on its surface satisfies the equation â · r = |r| cos α. Obtain the cartesian equation when â = (2/7, −3/7, −6/7) and α = 60◦ .
PROBLEM SHEET 10 10.1 For vectors a and b, show |a + b|2 + |a − b| = 2(|a|2 + |b|2 ) and a · b = 1 4 (|a + b|2 − |a − b|2 ) 10.2 In component form, let a = (1, −2, 2), b = (3, −1, −1), and c = (−1, 0, −1). Evaluate a × b, a· (b × c) , c· (a × b) . 10.3 What is the geometrical significance of a × b = 0? 10.4 Show that the vectors a = 2i +3j +6k and b = 6i +2j − 3k are perpendicular. Find a vector which is perpendicular to a and b. 10.5 Let a, b, c be three non-coplanar vectors, and v be any vector. Show that v can be expressed as v = Xa + Y b + Zc where X, Y, Z, are constants given by X = v· (b × c) a· (b × c) , Y = v· (c × a) a· (b × c) , Z = v· (a × b) a· (b × c) . (Hint: start by forming, say, v· (b × c)).
PROBLEM SHEET 11 11.1 Integrate cos(3x + 4). 11.2 Integrate (1 − 2x)10 . 11.3 Integrate e4x−1 . 11.4 Integrate (4x + 3)−1 . 11.5 Find the equation of the curve passing through the point (1, 2) satisfying dy/dx = 2x. 11.6 A particle has acceleration (3t2 + 4) ms−2 at time t seconds. If its initial speed is 5ms−1 , what is its speed at time t = 2 seconds? 11.7 Find the area between the graph of y = sinx and the x−axis from x = 0 to x = π/2. 11.8 Find the area between the graph y = 1 x − 1 and the x−axis between x = 2 and x = 3. 11.9 Find the signed area between the graph y = 2x + 1 and the x-axis between x = −1 and x = 3. 11.10 Find y, given that d2 y dx2 = sin x − 4 x3 .

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allsheets.pdfnnmmmmmmlklppppppplklkkkkkkkj

  • 1. PROBLEM SHEET 1 1.1 Find the radius and centre of the circle described by the equation x2 + y2 − 2x − 4y + 1 = 0 by writing it in the form (x − a)2 + (y − b)2 = c2 for suitable a, b and c. 1.2 Find the equation of the line perpendicular to y = 3x passing through the point (3, 9). 1.3 Given sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B, show that cos A sin B = 1 2 [sin(A + B) − sin(A − B)] and sin2 A = 1 2 [1 − cos 2A]. 1.4 Show that 4 cos(αt) + 3 sin(αt) = 5 cos(αt + φ) where φ =arc tan(−3/4). 1.5 Show that, for −1 ≤ x ≤ 1, cos ¡ sin−1 x ¢ = ± √ 1 − x2. 1.6 Given sinh(A ± B) = sinh A cosh B ± cosh A sinh B and cosh(A ± B) = cosh A cosh B ± sinh A sinh B, show that cosh A cosh B = 1 2 [cosh(A + B) + cosh(A − B)] and sinh2 A = 1 2 [cosh 2A − 1]. 1.7 Given that sinh x = 1 2 [ex − e−x ], show that sinh−1 x = ln h x + √ 1 + x2 i . 1.8 Express x (x − 1)(x − 2) in partial fractions. 1.9 If an = 1 n , find P5 i=1 an as a fraction. 1.10 If S = PN i=0 xi , show that xS = PN+1 i=1 xi . Hence show that S − xS = 1 − xN+1 and therefore that S = 1 − xN+1 1 − x . .
  • 2. PROBLEM SHEET 2 2.1 Given that sinh x = 1 2 [ex − e−x ] show that dy dx = cosh x. 2.2 Given that cosh x = 1 2 [ex + e−x ], show that dy dx = sinh x. 2.3 Let n be a positive integer. Show that dn (xn ) dxn = n! 2.4 If y = ln x, show that dy dx = 1 x ; d2 y dx2 = −1 x2 ; d100 y dx100 = −99! x100 . 2.5 Find the equation of the tangent to the curve y = x2 at (1, 1). 2.6 Find the slope of the curve y = 4x + ex at (0, 1). 2.7 Find the angle of inclination of the tangent to the curve y = x2 + x + 1 at the point (0, 1). 2.8 The displacement y(t) metres of a body at time t seconds (t ≥ 0) is given by y(t) = t−sint. At what times is the body at rest? 2.9 A particle has displacement y(t) metres at time t seconds given by y(t) = 3t3 + 4t + 1. Find its acceleration at time t = 4 seconds. 2.10 If y = N X n=0 anxn show that dy dx = N X n=1 nanxn−1 .
  • 3. PROBLEM SHEET 3 3.1 If y = ln(1 + x2 ), find dy/dx. 3.2 If y = x 1 + x2 find dy/dx. 3.3 If y = cosh(x4 ), find dy/dx. 3.4 If y = x2 ln x, find d2 y/dx2 . 3.5 Find dy/dx for y = (1 + x2 ) −1/2 . 3.6 Show that for y = sinh−1 x, dy dx = 1 √ 1 + x2 . 3.7 Show that for y = ln[x + √ 1 + x2], dy dx = 1 √ 1 + x2 . 3.8 Find dy/dx for y = cos−1 (sinx). 3.9 A curve is given in polar coordinates by r = 1+sin2 θ. Find dy/dx at θ = π/4. 3.10 Show that if y = 1 2a ln ¯ ¯ ¯ ¯ x − a x + a ¯ ¯ ¯ ¯ , then dy dx = 1 x2 − a2 .
  • 4. PROBLEM SHEET 4 4.1 Given f(x − ct), where x and c are constant, show that d2 dt2 f(x − ct) = c2 f 00 (x − ct), and calculate this expression when f(u) = sinu. 4.2 Classify the stationary point of y = x−2 lnx, where x > 0. 4.3 Classify the stationary points of y(x) = x2 − 3x + 2. 4.4 The numbers x and y are subject to the constraint x + y = π. Find the values of x and y for which cos(x) sin(y) takes its minimum value. 4.5 Sketch the graph of y = x 1 + x2 . 4.6 Sketch the graph of y(x) = tan(2x) for − 3π 4 ≤ x ≤ 3π 4 . 4.7 Sketch the graph of y = xlnx for x > 0. 4.8 Sketch the graph of y = x3 2x − 1 showing clearly on your sketch any asymptotes. 4.9 Sketch the graph of y = x cos(3x) for 0 ≤ x ≤ 2π.
  • 5. PROBLEM SHEET 5 5.1 Verify the following Taylor expansions (taking the ranges of validity for granted). (a) ex = 1 + 1 1! x + 1 2! x2 + 1 3! x3 + ... + 1 n! xn + ...valid for any x. (b) sin x = x − x3 3! + x5 5! − ... + (−1)n x2n+1 (2n + 1)! + ...valid for any x. (c) cos x = 1 − x2 2! + x4 4! − ... + (−1)n x2n (2n)! + ...valid for any x. (d) Let α be a constant. (1 + x)α = 1 + αx + α(α − 1) 2! x2 + α(α − 1)(α − 2) 3! x3 + ...valid for − 1 < x < 1 (e) ln(1 + x) = x − x2 2 + x3 3 − ... + (−1)n−1 xn n + ...valid for − 1 < x ≤ 1. 5.2 Obtain a four-term Taylor polynomial approximation valid near x = 0 for each of the following. (a) (1 + x)1/2 , (b) sin(2x), (c) ln(1 + 3x).
  • 6. PROBLEM SHEET 6 6.1 Reduce to standard form (a) 3 + i 4 − i , and (b) (1 + i)5 . 6.2 Prove (a) |z1z2| = |z1| |z2| , and (b) ¯ ¯ ¯ ¯ z1 z2 ¯ ¯ ¯ ¯ = |z1| |z2| when z2 6= 0. 6.3 Given that eiθ = cos θ + i sin θ, prove that cos(A + B) = cos A cos B − sin A sin B. 6.4 Let z = 1+i. Find the following complex numbers in standard form and plot their corresponding points in the Argand diagram:- (a) z̄2 , and (b) z z̄ . 6.5 Find the modulus and principal arguments of (a) −2 + 2i, (b) 3 + 4i. 6.6 Find all the complex roots of (a) cosh z = 1; (b) sinh z = 1; (c) ez = −1; (d) cos z = √ 2. 6.7 Show that the mapping w = z + c z , where z = x + iy, w = u + iv and c is a real number, maps the circle |z| = 1 in the z plane into an ellipse in the w plane and find its equation. 6.8 Show that cos6 θ = 1 32 (cos 6θ + 6 cos 4θ + 15 cos 2θ + 10).
  • 7. PROBLEM SHEET 7 7.1 The matrix A = (aij) is given by A = ⎛ ⎜ ⎜ ⎝ 1 2 3 −1 0 1 2 −2 4 1 5 −3 ⎞ ⎟ ⎟ ⎠ Identify the elements a13 and a31. 7.2 Given that A = µ 1 3 0 2 1 1 ¶ , B = ⎛ ⎝ 1 0 2 1 −1 −1 ⎞ ⎠ , C = ⎛ ⎝ 2 1 −1 1 −0 1 ⎞ ⎠ , verify the distributive law A(B + C) = AB + AC for the three matrices. 7.3 Let A = µ 4 2 2 1 ¶ , B = µ −2 −1 4 2 ¶ . Show that AB = 0, but that BA 6= 0. 7.4 A general n ×n matrix is given by A = (aij). Show that A +AT is a symmetric matrix, and that A − AT is skew-symmetric. Express the matrix A = ⎛ ⎝ 2 1 3 −2 0 1 3 1 2 ⎞ ⎠ . as the sum of a symmetric matrix and a skew-symmetric matrix. 7.5 Let the matrix A = ⎛ ⎝ 1 0 0 a −1 0 b c 1 ⎞ ⎠ . Find A2 . For what relation between a, b, and c is A2 = I (the unit matrix)? In this case, what is the inverse matrix of A? What is the inverse matrix of A2n−1 (n a positive integer)? 7.6 Using the rule for inverses of 2 × 2 matrices, write down the inverse of µ 1 1 2 −1 ¶ 7.7 If A and B are both n × n matrices with A non-singular, show that (A−1 BA)2 = A−1 B2 A.
  • 8. PROBLEM SHEET 8 8.1 Obtain the components of the vectors below where L is the magnitude and θ the angle made with the positive direction of the x axis (−180◦ < θ ≤ 180◦ ). (a) L = 3, θ = 60◦ ; (b) L = 3, θ = −150◦ . 8.2 Two ships, S1 and S2 set off from the same point Q. Each follows a route given by successive displacement vectors. In axes pointing east and north, S1 follows the path to B via − → QA = (2, 4), and − → AB = (4, 1). S2 goes to E via − → QC = (3, 3) − − → CD = (1, 1) and − − → DE = (2, −3). Find the displacement vector − − → BE in component form. 8.3 Sketch a diagram to show that if A, B, C are any three points, then − → AB + − − → BC + − → CA = 0. Formulate a similar result for any number of points. 8.4 Sketch a diagram to show that if A, B, C, D are any four points, then − − → CD = − − → CB + − → BA + − − → AD. Formulate a similar result for any number of points. 8.5 Two points A and B have position vectors a and b respectively. In terms of a and b find the position vectors of the following points on the straight line passing through A and B. (a) The mid-point C of AB; (b) a point U between A and B for which AU/UB = 1/3. 8.6 Suppose that C has position vector r and r = λa +(1 − λ)b where λ is a parameter, and A, B are points with a, b as position vectors. Show that C describes a straight line. Indicate on a diagram the relative positions of A, B, C, when λ < 0, 0 < λ < 1, and λ > 1. 8.7 Find the shortest distance from the origin of the line given in vector parametric form by r = a+tb, where a = (1, 2, 3) and b = (1, 1, 1) , and t is the parameter (Hint: use a calculus method, with t as the independent variable.) 8.8 ABCD is any quadrilateral in three dimensions. Prove that if P, Q, R, S are the mid-points of AB, BC, CD, DA respectively, then PQRS is a parallelogram. 8.9 ABC is a triangle, and P, Q, R are the mid-points of the respective sides BC, CA, AB. Prove that the medians AP, BQ, CR meet at a single point G (called the centroid of ABC; it is the centre of mass of a uniform triangular plate.) 8.10 Show that the vectors 0A = (1, 1, 2), 0B = (1, 1, 1), and 0C = (5, 5, 7) all lie in one plane.
  • 9. PROBLEM SHEET 9 9.1 The figure ABCD has vertices at (0, 0), (2, 0), (3, 1) and (1, 1). Find the vectors − → AC and − − → BD. Find − → AC · − − → BD. Hence show that the angles between the diagonals of ABCD have cosine −1/ √ 5. 9.2 Show that the vectors a = i + 3j + 4k and b = −2i + 6j − 4k are perpendicular. Obtain any vector c = c1i + c2j + c3k which is perpendicular to both a and b. 9.3 Find the value of λ such that the vectors (λ, 2, −1) and (1, 1, −3λ) are perpendicular. 9.4 Find a constant vector parallel to the line given parametrically by x = 1 − λ, y = 2 + 3λ, z = 1 + λ. 9.5 A circular cone has its vertex at the origin and its axis in the direction of the unit vector â. The half-angle at the vertex is α. Show that the position vector r of a general point on its surface satisfies the equation â · r = |r| cos α. Obtain the cartesian equation when â = (2/7, −3/7, −6/7) and α = 60◦ .
  • 10. PROBLEM SHEET 10 10.1 For vectors a and b, show |a + b|2 + |a − b| = 2(|a|2 + |b|2 ) and a · b = 1 4 (|a + b|2 − |a − b|2 ) 10.2 In component form, let a = (1, −2, 2), b = (3, −1, −1), and c = (−1, 0, −1). Evaluate a × b, a· (b × c) , c· (a × b) . 10.3 What is the geometrical significance of a × b = 0? 10.4 Show that the vectors a = 2i +3j +6k and b = 6i +2j − 3k are perpendicular. Find a vector which is perpendicular to a and b. 10.5 Let a, b, c be three non-coplanar vectors, and v be any vector. Show that v can be expressed as v = Xa + Y b + Zc where X, Y, Z, are constants given by X = v· (b × c) a· (b × c) , Y = v· (c × a) a· (b × c) , Z = v· (a × b) a· (b × c) . (Hint: start by forming, say, v· (b × c)).
  • 11. PROBLEM SHEET 11 11.1 Integrate cos(3x + 4). 11.2 Integrate (1 − 2x)10 . 11.3 Integrate e4x−1 . 11.4 Integrate (4x + 3)−1 . 11.5 Find the equation of the curve passing through the point (1, 2) satisfying dy/dx = 2x. 11.6 A particle has acceleration (3t2 + 4) ms−2 at time t seconds. If its initial speed is 5ms−1 , what is its speed at time t = 2 seconds? 11.7 Find the area between the graph of y = sinx and the x−axis from x = 0 to x = π/2. 11.8 Find the area between the graph y = 1 x − 1 and the x−axis between x = 2 and x = 3. 11.9 Find the signed area between the graph y = 2x + 1 and the x-axis between x = −1 and x = 3. 11.10 Find y, given that d2 y dx2 = sin x − 4 x3 .