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S.No QUESTIONS ANSWERS Q.1

If A= , find satisfying 0< when A + AT= √2I 2; where AT is transpose of A.

Sol.1 Consider the given matrix

Q.2 If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k. Sol.2

Q.3 For what values of k, the system of linear equations x + y + z = 2 2x + y – z = 3 3x + 2y + kz = 4 has a unique solution?

Sol.3

Q.4 Write the sum of intercepts cut off by the plane

on the three axes.

Sol.4

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Q.5 Find λ and μ if

Sol.5

Q.6 If , then find a unit vector parallel to

the vector

Sol.6

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Q.7 Solve for x:

OR

Prove that

Sol.7

OR

Consider the left hand side

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Q.8 A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

Sol.8 Let charges for typing one English page be Rs. x. Let charges for typing one Hindi page be Rs.y. Thus from the given statements, we have, 10x+3y=145 3x+10y=180 Thus the above system can be written as,

AX = B, where, A= Multiply A-1 on both the sides, we have, A-1 X AX =A-1B IX = A-1B X = A-1B Thus, we need to find the inverse of the matrix A.

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We know that, if P= Thus,

Therefore,X=

Amount taken from Shyam = 2 × 5 = Rs.10 Actual rate = 15 × 5 =75 Difference amount = Rs.75 – Rs.10 = Rs.65 Rs. 65 less was charged from the poor boy Shyam. Humanity and sympathy are reflected in this problem.

Q.9

is continuous at x = 0, then find the values of a and b.

Sol.9 Given that f is continuous at x=0

Since f x is continuous at x=0,

Thus,R.H.L

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Q.10

If x cos(a+y)= cosy then prove that

Hence show that sina OR

Sol.10 Given that

Differentiating both sides of the equation (1), we have,

OR Given that

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Q.11 Find the equation of tangents to the curve y= x3 + 2x – 4, which are perpendicular to line x + 14y + 3 = 0.

Sol.11 Consider the given equation, Y=x3 + 2x – 4 Differentiating the above function with respect to x, we have,

Given that the tangents to the given curve are perpendicular to the line x+14y+3=0 Slope of this line,m2 = Since the given line and the tangents to the given curve are perpendicular, we have, m1x m2 = -1

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Equation of the tangent having slope m at the point (x1,y1) is (y-y1) =m(x-x1) Equation of the tangent at P(2,8) with slope 14 (y-8)=14(x-2) y-8 =14x -28 14x-y=20 Equation of the tangent at P(-2,-16) with slope 14 (y+16) =14(x+2) Y+16=14x + 28 14x –y=-12

Q.12

Find: OR

Find :

Sol.12 Consider the given integral

Rewriting the above integral as

Let us consider,2x-3=t dx = dt

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OR

Consider the given function

Thus equating the coeffcients, we have, A+B=1….(1) 2B+C=1…..(2) 2C+A=1….(3) Solve the above three equations ,we have,

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\

Q.13

Evaluate :

Sol.13 Consider the given integral

Let us use the property,

Adding equations (1) and (2), we have,

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Q.14 Find : Sol.14

Q.15 Find the particular solution of differential equation:

Sol.15

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Q.16 Find the particular solution of the differential equation:

Sol.16

Given differential equation is a homogeneous differential equation.

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Integrating on both the sides

Q.17 Show that the four points A(4,5,1), B(0,1,1), C(3,9,4) and D(4,4,4) are coplanar.

Sol.17 Given points are A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4).

Q.18 Find the coordinates of the foot of perpendicular drawn from the point A (1,8,4) to the line joining the points B(0,1,3) and C(2,3,1).

Sol.18 Let P be the foot of the perpendicular drawn from point A on the line joining points B and C.

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Hence find the image of the point A in the line BC.

Let P’ (a, b, c) be the coordinates of image of point A. Equation of line BC is given by,

General Coordinates of P is (

Direction ratios of AP (

Coordinates of foot of perpendicular is (-2,1,7) Coordinates of image of A is P’(a,b,c)is

Q.19 A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y. OR A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

Sol.19 Bag X = 4 white, 2 black. Bag Y = 3 white, 3 black Let A be the event of selecting one white and one black ball. E1=first bag selected E2=second bag selected

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OR

Q.20 Three numbers are selected at random (without replacement) from

first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X.Also, find the mean and variance of the distribution.

Sol.20 X=Larger of three numbers X= 3,4,5,6

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Q.21 Let A= R × R and * be a binary operation on A defined by (a, b) * (c, d) = (a+c, b+d) Show that * is commutative and associative. Find the identity element for *on A. Also find the inverse of every element (a, b) A.

Sol.21 (a, b) * (c, d) = (a + c, b + d) (i) Commutative (a, b) * (c, d) = (a+c, b+d) (c, d) * (a, b) = (c+a, d+b) for all, a, b, c, d R * is commulative on A (ii) Associative : ______ (a, b), (c, d), (e, f) A { (a, b) * (c, d) } * (e, f) = (a + c, b+d) * (e, f) = ((a + c) + e, (b + d) + f) = (a + (c + e), b + (d + f)) = (a*b) * ( c+d, d+f) = (a*b) {(c, d) * (e, f)} is associative on A Let (x, y) be the identity element in A. then, (a, b) * (x, y) = (a, b) for all (a,b) A (a + x, b+y) = (a, b) for all (a, b) A a + x = a, b + y = b for all (a, b) A x = 0, y = 0

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(0, 0) A (0, 0) is the identity element in A. Let (a, b) be an invertible element of A. (a, b) * (c, d) = (0, 0) = (c, d) * (a, b) (a+c, b+d) = (0, 0) = (c+a, d+b) a + c = 0 b + d = 0 a = - c b = - d c = - a d = - b (a, b) is an invertible element of A, in such a case the inverse of (a, b) is (-a, -b)

Q.22

Prove that y= is an increasing function of on OR

Show that semi-vertical angle of a cone of maximum volume and

given slant height is

Sol.22

OR

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Q.23 Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).

Sol.23

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Q.24 Find the equation of the plane which contains the line of intersection of the Planes

and whose intercept on x-axis is equal to that of on y-axis.

Sol.24

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Equation of the required plane - x - y + 4z = -1 x + y - 4z - 1 = 0 Vector eqnof the required Plane r.(ˆi + ˆj - 4kˆ)-1= 0

Q.25 A retired person wants to invest an amount of Rs. 50, 000. His broker recommends investing in two type of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond ‘A’ and at least Rs. 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximise his returns.

Sol.25 Maximize Z = 0.1x + 0.09 y x + y 50000 x 20000 y 10000 y x

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When A invest 40000 & B invest 10000 in his return is maximum.

Q.26 Using properties of determinants, prove that

OR

Sol.26

Multiplying R1,R2 and R3 by z,x,y respectively

take common z,x,y from C1,C2 & C3

C1C1 - C3 and C2 C2 - C3 taking common x+y+z from C1 & C

Taking z and x common from R1 & R2

expansion along R3 = (x+y+z)2 × 2xz ((x + y) (z + y) – xz)

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= (x+y+z)2 × 2xz (xz + xy + yz + y2 - xz) = (x+y+z)2 × 2xz (xy + yz + y2) = 2xyz (x + y + z)3

OR

k=2