Solve your IGNOU Doubts

Question:

a) Find the distance of the point of intersection of the line

$\frac{x-2}{1}=\frac{y+3}{-1}=\frac{z}{3}$

and the plane 2x−3y+4z+4 = 0 from the origin.

b) Find the equation of the tangent plane to the conicoid x² +y² = kz at the point (k, k,2k), where k is a constant. Represent the plane geometrically. Now take different values of k, including both positive and negative, and see how the shape of the conicoid changes.

c) Find the equation of the plane which passes through the line of intersection of the planes x+y−2z = 1 and 2x+y−4z = 3 and which is perpendicular to the plane x+y+z = 1.

d) Find the equation of the cylinder with base x² +y² +z²+3x+3y−z = 1, x+y+2z = 2.

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Question:

a) Obtain the equation of the conic, a focus of which lies at (2,1), the directrix of which is x+y = 0 and which passes through (1,4). Also identify the conic.

b) Find the equations of the spheres which pass through the circle x²+y² +z² = 9,2x+2y−7 = 0 and the touch the plane x−y+z+3 = 0

c) Find the transformation of the equation 12x²−2y² +z² = 2xy if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1,−3,0; 3,1,0; 0,0,1.

d) Identify and trace the conicoid y² +z² = x. Describe its sections by the planes x = 0, y = 0 and z = 0.

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Question:

a) Find the equation of the normal to the parabola y² +4x = 0 at the point where the line y = x+c touches it.

b) Identify and trace the conic x² −2xy+y²−3x+2y+3 = 0.

c) Prove that the plane 2x−3y+6z = 6 touches the conicoid 4x²−9y² +36z² = 36. Find the point of contact.

d) Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

i) x² +y²+z²+x+y+z = 1

ii) 2x² +4xy+xz−x−3y+5z+3 = 0

iii) x² −y² −z² +xy+4yz+x = 0

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Question:

Which of the following statements are true and which are false? Give reasons for your answer.

i) The equation r = $a \cos \theta + b \sin \theta$ represents a circle.

ii) If 1,1/2,0 are direction ratios of a line, then the line makes an angle of 90^◦with the x-axis, an angle of 60^◦with the y-axis, and is parallel to the z-axis.

iii) The intersection of a plane and a cone can be a pair of lines.

iv) If a cone has three mutually perpendicular generators then its reciprocal cone has three mutually perpendicular tangent planes.

v) The equations $2x^{2}+y^{2}+3z^{2}+4x+4y+18z+34=0,2x^{2}-y^{2}=4y-4y-4x$

represent a real conic.

vi) $4x^{2}-9y^{2}+z^{2}+36=0$ represents a hyperboloid of one sheet.

vii) The intersection of any plane with an ellipsoid is an ellipse.

viii) No plane passes through the points (1,2,3),(1,−1,0) and (1,1,2).

ix) The circle with centre (a,0) and radius a, where a > 0, touches all the sides of the square x = 0,x = a,y= $\pm a$

x) If the projection of a line segment AB on a line L is 0, then AB lies in L.

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Question:

Image ignouassignments-ignouacademy-com--p-ignou-65237

Image ignouassignments-ignouacademy-com--p-solve-61589

c) Write a 3 × 3 square matrix A whose determinant is 75.

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Question:

a) A lady bought a plot of land for ` Rs.30 lakhs. She wanted to landscape it. So she bought 15 bushes and 18 trees from a nursery for ` Rs.975/-. A month later she bought 7 bushes and 5 trees from the same nursery for ` Rs.470/-. She paid a gardener `Rs. 5000/- to plant them. How much did each bush and tree cost her?

b) A collection of 58 coins, consisting of 25p, 50p, `Rs. 1 and ` Rs.2 coins are in a bag. The ` Rs.1 coins number 5 times that of the 25p coins. The ` Rs.2 coins are double the number of `Rs. 1 coins, and thrice the number of the 50p coins. If the total value of the coins is ` Rs.80/75, how many coins of each kind are there?

c) Create a meaningful problem related to your life that can be represented by the equations x – 4 = y, 2x + y = 5.

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Question:

Solve the following linear systems by the method given alongside each. Verify your solution also, in each case.

i) 2x – 3y + z = 1, x + y + z = 2, 3x – 4z – 17 = 0

(by Elimination Method).

ii) 2x – 3y = 1, 5 – 2y = z

(by Substitution Method)

iii) $2a+\gamma = 3\beta ,a+\beta =5+3\gamma , a-\beta =\pi$

(by Cramer’s Rule)

iv) 4 3x − 5 = y, y = (geometrically)

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Question:

Obtain the resolvent cubic of $4x^{4}+16x^{3}-17x^{2}-10x-45=0$

using i) Ferrari’s method; ii) Descartes’ method. Are the cubics you get from

(i) and

(ii) above the same?

Further, use either method to obtain the roots of the equation.

[Hint: To obtain a solution of the resolvent cubic, you can apply Theorem 5 of Unit 6.]

b) Solve the equation $x^{5}-5x^{4}-5x^{3}+25x^{2}+4x-20=0$ , given that its roots are of the form $a,-a,b,-b,c.$

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Question:

a) Show the geometric representation of the set

Image ignouassignments-ignouacademy-com--p-doubts-33480

b) Prove, by contradiction, that if w,z $\epsilon$ $\mathbb{C}$ such that $| w |\leq 1$ and $W^{n}z+W^{n-1}z^{2}+...........=1$ ,then $| z|> \frac{1}{2}.$

c) If z = a + 2i is a root of $x^{2}+6x+k=0,$ where a,k $\epsilon$ $\mathbb{R}$ , find a, as well as the modulus and principal argument of z. Which quadrant does z lie in?

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Question:

a) Give an example related to the life of a schoolchild, of each of the following. Justify your choice of example also.

i) Two non-empty sets, whose intersection is the null set.

ii) A set with exactly 4 subsets.

b) Give three sets A,B,C such that $A\cap B\neq \varnothing ,$ $B\cap C\neq \varnothing ,$ $A\cap B\cap C=\varnothing$ Show them in a Venn Diagram also, clearly stating what U is.

c) Prove that if A and B are sets such that $A\times B\neq \varnothing$ , then $A\cup C\neq \varnothing$ for any set C.

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Question:

a) Prove that DO, DB and DE are the AM,GM and HM of a and b, as shown in Fig. 1, Unit 6.

b) Give an example, with justification, to show why 0 a 1 < i < in Theorem 6, Unit 6.

c) Prove that $\frac{[1.2+2.3+...+n(n+1)]}{n(n+3)}\geq \frac{n+1}{4}$ for $n\geq 1$

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Question:

Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that ‘{1, padma, blue} is a set’ is true, you need to say that this is true because it is a well-defined collection of 3 objects.)

i) The contrapositive of ‘ not A⇒not B’ is ‘ A ⇒ B’, where A and B are two statements.

ii) Any set can be represented by the listing method.

iii) For any three sets A,B,C in a universal set U, (A \ B) $\times$ C=A \ ( B $\times$ C)

iv) The operation of conjugation is closed on $\mathbb{C}$ .

v) If f(x) x and g(x ) are polynomials for which f (0) = g (0) and f = (1) f(1) g (1), then f (x)= g(x).

vi) The solution set of 3 2x =1, y + 5 = x, x = y + is a singleton.

vii) ( ) 1 2 3 ...... n is a matrix.

viii) Bunyakovskii and Weirstrass were contemporaries.

ix) The Substitution Method for solving a linear system should be employed when the Elimination Method fails.

x) If a monic polynomial of degree n has n roots in $\mathbb{Z}$ , then all its coefficients are in $\mathbb{Z}$

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Question:

(a) Measurements of a sample of 6 weights were found to be 14⋅ 16,3 ⋅ 15,6 ⋅ 14,7 ⋅ 16,8 ⋅ 2 and 15 ⋅ 4 kilogram respectively. (i) Determine an unbiased estimate of the population mean. (ii) Compare the sample standard deviation with the estimated population standard deviation.

(b) Evaluate: $\lim_{x\rightarrow 3}\frac{\sqrt{x-3}+\sqrt{x}-\sqrt{3}}{\sqrt{x^{2}-9}}$

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Question:

a)The number of accidents in a highway as recorded every month over a 9-month period are 15, 18, 9, 11, 14, 10, 8, 13, 19. Test at 5% frequencies are in agreement with the belief that the number of accidents was the same during the 9 months. It is given that the table values of $\chi ^{2}$ at 5% level for 8 d.o.f. and 9 d.o.f. are 15⋅5 and 16 ⋅9 respectively.

(b) Find the equation of the straight line passing through the intersection of the lines x + y =1 and 2x − 3y + 2 = 0 and perpendicular to the line 3x + y + 9 = .0

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Question:

(a) Find the equation of the sphere whose radius is 5 and centre is the point of intersection of the plane x+y+2z=2 and the straight line $\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-0}{-1}$ .

(b) Find the equation of the normal to the curve y(x − 2) (x −3 ) − x + 7 = ,0 at the point where it cuts the x -axis.

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Question:

(a) Find the sine of the angle between the vectors α = 2i − j + k and β = 3i + 4 j − k.

(b) Given $\frac{dy}{dx}=-\frac{y^{2}}{25}$ and y = 5 when x = 0. Find x when y = .2

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Question:

(a) If the roots of the equation 0 2 x − lx + m = differ by 1, then prove that

$l^{2}=4m+1$ .

(b) In a Binomial distribution consisting of 5 independent trials, probabilities of 1 and 2 successes are 0 ⋅ 4096 and 0 ⋅ 2048 respectively. Find the probability of success. Also, find the mean and variance of the distribution.

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Question:

Find dz/dt where z = x² + 3xy + 5y² and x = cos t, y = 2 sint.

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Question:

For 5 observations of pairs (x, y) of variables x and y, the following results are obtained:

$\sum x=15,\sum y=25,\sum x^{2}=55,\sum y^{2}=135,\sum xy=83$ .