Inverse trigo
1) If sin⁻¹(sin⁻¹(1/5) + cos⁻¹x)= 1 then find the value of x. 1/5
2) Solve: tan⁻¹{(1- x)/+1+ x)}= (1/2) tan⁻¹x, x> 0. 1/√3
3) Prove: cot⁻¹7 + cot⁻¹8 + cot⁻¹18= cot⁻¹3.
4) Prove: sin⁻¹(8/17) + sin⁻¹(3/5) = cos⁻¹(36/85).
5) Show that tan⁻¹√x = (1/2) cos⁻¹{(1- x)/(1+ x)}, x ∈ [0,1].
6) Show that cos[tan⁻¹{sin(cot⁻¹x)}]= √{(1+ x²)/(2+ x²)}.
7) Find the value of cot{π/2 - 2 cot⁻¹√3}. √3
8) Find the value of tan⁻¹[2 sin(2 cos⁻¹(√3/2))]. π/3
9) If sin[cot⁻¹(x +1)]= cos(tan⁻¹x) then find x. -1/2
10) If tan⁻¹{(x -2)/(x -4)} + tan⁻¹{(x +2)/(x +4)}= π/4 then find the value of x. ±√2
11) Solve for x : cos⁻¹x + sin⁻¹+x/2)=π/6. 1
12) Show: tan⁻¹(3/4)+ tan⁻¹(3/5) - tan⁻¹(8/19)=π/4.
13) Show: sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36).
14) Find the principal value of tan⁻¹√3 - sec⁻¹(-2).
15) Show: tan⁻¹(1/4) + tan⁻¹(2/9) = (1/2) sin⁻¹(4/5).
16) Evaluate: tan[2 tan⁻¹(1/2) - cot⁻¹3]. 9/13
17) If cos⁻¹x + cos⁻¹y + cos⁻¹z =π then show that x²+ y²+ z²+ 2xyz = 1.
1) Choose the correct answer from the given alternatives: 1 x 10=10
i) The domain for which the function f(x)= 3x²- 2x and g(x)= 3(3x -2) are equal, will be
a) {1,2/3} b) {1,3} c) {2/3,3} d) {2/3,0}
ii) The value of tan {π/2 - tan⁻¹(1/3)} is equals to
a) 1/3 b) 3 c) 2/3 d) 3/2
iii) If two rows or two columns of a determinant are identical then value of the determinant is
a) 0 b) 2 c) - 1 d) 1
iv) if f(x)= - f(x), then the value of ᵃ₋ₐ∫ f(x) dx is equals to
a) 2a b) a c) a/2 d) 0
v) If y= tan⁻¹{5 - x).(1+ 5x)}, then value of dy/dx is
a) -1/(1+ x²) b) 1/(1+ x²) c) 5 d) 5/(1+ x²)
vi) If P(A)= 3/7, P(B)= 4/7 and P(A∩B) =2/9, then the value of P(A/B) is equals to
a) 7/18 b) 14/27 c) 5/18 d) 4/9
vii) A coin is tossed 10 times . The probability of getting had six times is
a) ¹⁰C₅. 1/2¹⁰ b) ¹⁰C₃. 1/2¹⁰ c) ¹⁰C₄. 1/2¹⁰ d) ¹⁰C₈. 1/2¹⁰
viii) If xy =1 then the value of d²y/dx² is
a) 0 b) 3 c) -1 d) 2
ix) The integrating factor of the differential equation (x + y+ z) dy/dx =1 is
a) e⁻ʸ b) eˣ c) e⁻ˣ d) eʸ
x) Form the differential equation of the family of curves y= A cos(x + B), where A and B are arbitrary constants.
2)a) Find the maximum value of
1 1 1
1 1+sinx 1
1 1 1 + cosx (1)
b) Solve the equation for x:
cos(tan⁻¹x)= sin(cot⁻¹(3/4)). (1)
c) Find the intervals in which the function f(x)= 3x⁴- 4x³- 12x²+ 5 is strictly decreasing. (1)
4) Given that event A and B are such that P(A)= 1/2, P(B)= p, P(A U B)= 3/5. Find p if A and B are
a) mutually exclusive.
b) independent. (2)
5) Evaluate ∫ (x sinx)/(1+ sinx) dx at (π,0). (2)
6) For what value of x is the given matrix
2x+4 4
x+ 5 3 is a singular matrix ? (2)
7) If y = xʸ, prove that x dy/dx = y²/(1- y logx). (2)
8) The probabilities of A, B, and C solving a problem are 1/2, 1/3 and 1/4 respectively . Find the probability that the problem will be solved. (3)
9) Prove: 1+ a 1 1
1 1+b 1 = abc(1/a+1/b+1/c). (3)
1 1 1+c
10) If the following function is differentiable at x=2, then find the value of a and b.
f(x)= x², if x ≤ 2
ax + b, if x > 2. (3)
OR
Find the value of k. So that the function f defined by
f(x)= kx +1, if x ≤π
cosx, if x > π
is continuous at x =π
11) if cos⁻¹(x/a)+ cos⁻¹(y/b)= K, Show that x²/a² - (2xy cosK)/ab + y²/b² = sin²K. (4)
12) If y= {x + √(1+ x²)}ⁿ, then show that (1+ x²) d²y/dx² + x dy/dx = n²y. (4)
13) Two balls are drawn one after another (without replacement) from a bag containing 2 white, 3 red and 5 blue balls. What is the probability that atleast one ball is red ? (4)
14) Evaluate ∫ (3x +1)/√(5 - 2x - x²) dx. (4)
OR
Prove ³₁∫ dx/{x²(x +1} = 2/3 + log(2/3).
14) Find the equations of the tangent to the curve y= x²- 2x +7 which is
a) parallel to the line 2x - y +9=0.
b) perpendicular to the line 5y - 15x =13. (4)
OR
Find the intervals in which the function f given by f(x)= sinx - cosx, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing.
15) a) Solve : dy/dx + y/x = x². (2)
b) Solve : dy/dx + 1 = eˣ⁺ʸ. (2)
16) Let f: N --> N be a function defined as f(x)= 4x²+ 12x +15. Show that f: N --> S is invertible (where S is the range of f). Find the inverse of f and hence f⁻¹(31) and f⁻¹(87). (4)
17) Using matrices , solve the following equations: 5x+ 3y + z =16, 2x+ y + 3z =19, x+ 2y + 4z =16pp25. (6)
OR
If A= 1 -1 0
2 5 3
0 2 1 Find A⁻¹.
18) Prove that the area of a right angle triangle of given hypotenuse is maximum, when the triangle is isosceles. (6)
OR
Show that of all the rectangle inscribed in a given fixed circle, the square has the maximum area.
19) Evaluate
a) ∫ x² sin⁻¹x dx. (3)
b) ∫ x/(x²+ 4x +3) dx. (3)
20) A pair of dice is thrown 4 times. If getting a double is considered a success, find the probability distribution of the number of successes and show that its mean is 2/3. (3)
21) a) One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls . A ball is transferred from the first bag to the second and then a ball is drawn from the second. Find the probability that the ball drawn is white. (3)
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22) i) If a= i+ 3j - k and b = 2i + 6j + μk. If a and b vectors are parallel, then the value of μ is
a) 3 b) -6 c) -3 d) -2 (1)
ii) The angle between the two planes x - y +2z =9 and 2x + y +z =7 is
a) 30° b) 45° c) 60° d) 90° (1)
iii) If a= 2i + j + 3k and b = 3i + 5j - 2k represent two adjacent sides of a triangle, then find the area of the triangle. (1)
iv) If A(8,2,0), B(4,6,-7), C(-3,1,2) and D(-9,-2,4) are four given points , then find the angle between AB and CD. (1)
23) If a and b are two vectors, then show that
(a x b)²= a.a a.b
a. b b.b (4)
24) If a= 7i - 2j + 3k, b= i - j +2k, c= 2i + 8j are three vectors then find a.(b x c) and (a x b).c. (4)
25) Find the equation of plane passing through the line of intersection number planes x + 2y + 3z -5=0 and 3x - 2y - z +1=0 and cutting off equal intercepts on the OX and OZ axes. (4)
OR
Find the coordinates of the points in the line (x -1)/2 = (y +2)/3= (z -3)/6 which are at a distance of 3 units from the point (1,-2,3).
26) Find the area of the region included between the parabola y= 3x²/4 and the line 3x - 2y +12=0. (4)
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27)a) The fixed cost of a product is Rs18000 and the variable cost per unit is Rs550. If the demand function is p(x)= 4000 - 150x, find the break-even values. (2)
b) Given x + 4y =4 and 3x + y =16/3 are regression lines. Find the line of regression of x and y. (2)
c) The cost function for a commodity is C(x)= Rs(200+20x - x²/2)
i) Find the marginal cost (MC).
ii) Calculate the marginal cost when x= 4 and interprete it. (2)
28) Two regression lines are represented by 2x + 3y -10=0 and 4x + y -5=0. Find the lines of regression of y on x. (4)
OR
Fit a straight line to the following data, treating y as the dependent variable.
X: 1 2 3 4 5
Y: 7 6 5 4 3
Hence, estimate the value of y when x=3.5.
29) The marginal cost function of a firm is MC= 33 logx. Find the total cost function when the cost producing one unit is Rs11. (4)
30) A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours and both machines and each unit product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at profit of Rs7 and that of B at a profit of Rs4. Find graphically the production level per day for maximum profit.
(6)
REVISION - 1
1) If x= a sin³t and y= a cos³t, find dy/dx.
2) If y= ₑ(m cos⁻¹x), show that (1- x²) d²y/dx² - x dy/dx = m²y.
3) If y= eˣ log(tan2x), find dy/dx.
4) If xʸ = eˣ⁻ʸ, show that dy/dx = (logₑx)/(1+ logₑx)².
5) If y= (cosx)ᶜᵒˢˣ, find dy/dx.
6) If y= log √{(1- cosx)/(1+ cosx)}, find dy/dx.
7) If y= ₑsin²x, find dy/dx.
8) If x= a(cosθ+ θ sinθ), y= a(sinθ - θ cosθ). Then find dy/x at θ= π/4.
9) If y= logₑ{x + √(x²+ k²), find dy/dx.
10) If xʸyˣ = 5, then show that dy/dx = |- (log y + (y/x))/(log x + (x/y)).
11) If y= (sin⁻¹x)/√(1- x²), then show that (1- x²) dy/dx - xy = 1.
12) If xᵖ yᑫ = (x + y)ᵖ⁺ᑫ, then show that dy/dx = y/x.
13) If y= xʸ, show that x. dy/dx = y²/(1- y log x).
14) If √(1- x⁴)+ √(1- y⁴)= λ(x² - y²), then show that y √(1- x⁴) dy/dx = x √(1- y⁴).
15) If sin(xy)+ cos(xy)= 1 and tan(xy)≠ 1, then show that dy/dx = -y/x.
Test paper - XII maths
Part- A
1) Choose the correct option (1x 10=10)
a) If P is a linear on the set A={1,2,3,5,6} defined by P={(a, b) belongs to A x A, a - b=15} then P is
i) reflexive ii) symmetric iii) transitive iv) void relation
b) If A and B are arbitrary square matrix
i) (AB)'= A' B' ii) (A')'(B')'= B'A' iii) (A+ B)' = A'B'. d) (AB)'= B'A'
c) The number of points of discontinuous of f(x)= (x²+4)/(4x - x³) is
i) 1 ii) 2 iii) 3 iv) 4
d) If a and b are two vectors such that
a, b< 0 and|a. b|= |a x b| then the angle between a and b is
i) π ii) 7π/4 iii) 3π/4 d) π/2
e) i. (j x k)+ j. (k x i)+ k. (i x j)
i) 1 b) 2 c) 3 d) -3
f) f(x)= logx , then d/dx(f(logx))
i) (1/x) logx ii) x/logx iii) 1/(x logx) iv) x log x
g) Two integers are selected at random from 1 to 15. If the sum is even, then the probability that both the number will be odd is
i) 3/7 ii) 5/7 iii) 4/7 iv) 6/7
h) if f(x)={3x -4, 0≤ x < 2
2x + k, 2 < x ≤ 3
is continuous at x= 2. Find the value of k.
i) 0 ii) 2 iii) -2 iv) -1
i) Two vectors are α = 2i - 5j - 3k and β = -4i + 10j + 6k then α and β are
i) like ii) collinear iii) equal iv) none
j) The tangent to the curve y= x³ at (0,0) is
i) x axis ii) y-axis iii) x= y iv) none
Type- B
2) answer any one question (2 x2= 2)
a) i) Show that, the function f: IR --> IR, f(x) = |x| is neither oneone nor onto
ii) Differentiate log(logx).
b) i)Answer any one question. 2x 2= 2
Without expanding prove:
a a² bc = 1 a² a³
b. b² ca 1 b² b³
c c² ab 1 c² c³
ii) Find real values of k for which the following system of operations has non trivial solution
x- ky - z=0; Kx - y - z= 0, x+ y - z= 0.
c) Answer any three questions. (2cpx3=6)
i) Show that f(x)= |x|/x, x≠ 0
0, x= 0
is discontinuous at x= 0
ii) Evaluate ∫ xeˣ dx.
iii) Find order and degree of differential equation d³y/dx³+ y = ³√(1+ dy/dx).
iv) Show that f(x)= logx has neither a maximum value nor a minimum value.
v) Find dy/dx when eˣʸ - 4xy = 4.
d) Answer any one (2x2= 2)
i) Find direction cosine of the straight line in YZ plane which makes π/6 with zaxis.
ii) If the Cartesian equation of a line AB is (3- x)/1= (y+2)/-2= (z -5)/4. Find the direction ratio of a line parallel to AB.
e) Answer any one (2x2=2)
i) P(A' UB')= 5/6, P(A)= 1/2, P(B')= 2/3, show A and B are two independent event.
ii) If a random variable x has the following probability distribution
X: 4 5 6 8
P(x). 0.1 0.3 0.4 0.2
Find expectation of random variable of x.
Type- 3
3) a) Answer any one. (4x 1= 4
i) Show that, the relation R is reflexive, transitive not symmetric on W where
R={(x,y): x belongs to IN, y belongs to IN, x is multiple of y}
ii) If two function f: IR ---> IR g: IR--> IR Defined by
f(x)= 2x+ 3 and g(x)= 4x -1. Show that (fog)⁻¹= g⁻¹ o f⁻¹.
b) Answer the following: (4x2= 8
i) If matrix
A= 1 -2 2 & B= 1 2 0
0 1 -1 2 3 -1
0 0 1 0 -1 -2 show that (A'B)A is diagonal matrix.
OR
Prove that, the inverse of a given square matrix if it exists, is unique.
ii) Prove the determinants
x y z
x² y² z²
yz zx xy
= (x - y)(y- x)= (z - x)(xy + yz + zx)
OR
Solve by Cramer's Rule
2/x + 3/y - 4/z= -3,
1/x + 2/y - 6/z= 2
3/x - 1/y - 2/z= 5
c) Answer the following: (4x 3= 12)
i) If y= eᵘ and u= f(x) show that d²y/dx²= eᵘ [d²u/dx²+ (du/dx)²]
OR
i) If y= 2 sin⁻¹{(x -2)/√6} - √(2+ 4x - x²), then show that dy/dx |ₓ₌₂ = 2/√6
ii) Evaluate , ∫ (x -1)dx/{(x -3)(x+2)}
OR
Find , ∫ eˣ(x log x +1)/x dx.
iii) Solve, (1+ e²ˣ) dy + eˣ(1+ y²) dx = 0, given y= 1, x= 0
OR
Solve, cos⁻¹(dy/dx)= x + y.
d) Answer any one. 4x1= 4
i) Let a= i + 4j + 2k, b = 3i - 2j + 7k and c= 2i - j + 4k. Find a vector d whris perpendicular to both a and b and c.d= 18
ii) ABCD is a quadrilateral. M and N are midpoint of AB and CD. O is the midpoint of MN, show that OA+ OB+ OC + OD= 0.
e) Answer any one. 4x4= 4
i) Evaluate ¹₀∫ (x +2) dx from defination of definite integral.
ii) ²₁∫ dx/{√(x +2). + √(x -2)}
f) Answer any one. 4x 1= 4
i) What is probability of getting 53 Sundays in a leap year?
ii) There are 3 black balls and 5 white balls in a bag A and 5 black balls and 4 white balls in bag B. One ball is chosen at random from any bag. What is the getting the ball from bag B when the ball is white?
Type- 4
4) a) Answer any one. 5 x 1= 5
i) A furniture manufacturing rplana to make two products, chairs and tables from its available resources of 400 board feet of mahogany lumber and 10 hours. A chair requires 5 board feet of lumber and 10 manhours and gives a profit of Rs 45 while each table uses 20 board feet of lumber and 15 man hours and a profit of Rs 80. Formulate the LPP to maximize the profit.
ii) Solve graphically the LPP given
Min: z= 3x + 2y
2x + y≥ 14; 2x + 3y≥ 22; x+ y≥ 5, x≥ 0, y≥ 0
b) Answer any two (5x.2=10)
i) Prove that the sum of the x and y intercepts made by the tangents to the curve √x + √y at any point on its a constant.
ii) Find the area between two curves y= x² and y²= x.
iii) Show that, a conical tent of given capacity will require the least amount of canvas, when the height is √2 times of its radius.
iv) If the straight line lx + my = n is a normal to the hyperbola a²/l² - b²/m²= (a²+ b²)²/n².
c) Answer any one. (5x 1=5)
i) Find the vector equation of a line which passes through the point with position vector i- 2j + 4k and is in the direction of i + 2j - k. Also reduce it to Cartesian form.
ii) Show that, the lines (x -1)/3= (y+1)/2= (z -1)/5 and (x +2)/4 = (y -1)/3= (z +1)/-2 do not intersect.
MATRIX (1)
1) if A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A+ I
2) The multiplicative inverse of matrix
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
3) Assuming that the sums and products given below are defined, which of the following is not a true for matrices?
a) AB= AC doesn't imply B= C
b) A+ B = B+ A
c) (AB)'= B'A'
d) AB= 0 implies A= O or B= O
4) If A= 1 0 2 & Adj A= 5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2 b
Then the values of a and b are
a) -4,1 b) -4, -1 c) 4, 1 d) 4, -1
5) If A= -1 0
0 2 then the value of A³- A² is equal to
a) I b) A c) 2A d) 2I
6) If A= - x - y
z t then the transpose of adj A is
a) t z b) t y c) t -z d) none
-y - x -z -x y -x
Hy
7) If A=3 5 & B= 1 17
2 0 0 -10 then|AB| is equal to
a) 80 b) 100 c) -110 d) 92
8) If A= 5. -2
3 1 find the inverse of A
9) If A is singular matrix of order n then A. (Adj A) is equal to
a) a null matrix
b) a row matrix
c) a column matrix
d) none
10) If A= 3 -5
-4 2 then find the value of A²- 5A is equal to
a) I b) 14I c) O d) none
11) If A= 1 2 & B= 1 2
2 3 2 1
3 4
Then
a) both AB and BA exist
b) neither AB nor BA exist
c) AB exists but BA does not exist
d) AB does not exist but BA exist
12) If A= 2 -1 & B= 1 0
0 1 -1 -1 then+A+ B)² is not equal to
a) A²+ AB+ BA+ B²
b) A²+ AB+ BA+ B²I
c) A²I + AB+ BA+ B²
d) A²+ 2AB+ B²
13) If A be an n × n matrix and k any scalar then det. kA is equal to
a) k detA b) nᵏdetA c) kⁿ detA d) kn detA
14) If A= 1 2
3 -5 find inverse of A
15) If A= -1 2 & B= 5
2 -1 7 and AX= B, then X is equal to
a) 19 17 b) 19/3 c)19/3 17/3 d) 19
17/3 17
16) If A≠ O and B≠ O are two 2 x 2 matrices such that AB= O, then which of the following is correct?
a) detA= 0 or detB= 0
b) detA= 0 and detB= 0
c) detA= 0 = detB≠ 0 d) none
17) If A is a square matrix 3x3 and k is a scalar, then adj(kA) is equal to which of the following?
a) k adj A B) k² adj A c) k³ adj A d) 2k adj A
18) If A= 0 1 2
1 2 3
3 1 1 and its inverse B= [bᵢⱼ] , then the element b₂₃ of matrix B is
a) -1 b) 1 c) -2 d) 2
19) If A=a₁₁ a₁₂ a₁₃ & B=1 2 3
a₂₁ a₂₂ a₂₃ 2. 3. 4
a₃₁ a₃₂ a₃₃ 3 4 5
C= -1 -2 & D= -4 -5 -6
3 0 0 0 1
0 -4
With the relation A= BCD, then the value of a₂₂ is
a) 40 b) -40 c) -20 d) 20
20) If A= 1 2 & B= 3 8
3 4 7 2 with the relation 2X+ A= B, then the matrix X is equal to
a) 2 6 b) 1 -3 c) 1 3 d) 2 -6
4 -2 2 -1 2 -1 4 -2
21) If A= a 2
2 a and |A³|= 125, then the value of a is
a) ±2 b) ±2 c) ±5 d) 0
22) If A= |aᵢⱼ| and Aᵢⱼ denotes the Cofactor of aᵢⱼ, then which of the following is not equal to zero ?
a) a₃₁A₁₁ + a₃₂A₁₂+ a₃₃A₁₃
b) a₁₁A₃₁+ a₁₂A₃₂+ a₁₃A₃₃
c) a₂₁A₂₁+ a₂₂A₂₂+ a₂₃A₂₃
d) a₃₁A₂₁+ a₃₂A₂₂+ a₃₃A₂₃
23) A= 1 0 0
a 1 0
b c 1 find inverse of A
24) The minors of (-4) and 9 and the Cofactor of (-4) and 9 in matrix
-1 -2 3
-4 -5 -6
-7 8 9 are respectively
a) 42,3; -42,3 b) -42,-3; 42,-3 c) 42,3; -42,-3 d) 42,3; 42,3
25) If A= 0 3 & kA= 0 4a
4 5 3b 60
Then the values of k, a and b are respectively
a) 12,9,16 b) 9,12,16 c) 12,9,12 d) 16,12,9
26) B= 1 3 & C= 1 1
0 1 0 -1
If the matrix A satisfies the equation BA= C, then which one of the following represents A?
a) 1 4 b) 1 4 c) 1 -4 d) 1 -2
-1 0 0 -1 1 0 0 -1
27) For any matrix A, if A⁻¹ exists then which of the following is not true?
a) (A⁻¹)⁻¹= A
b) (Aᵀ)⁻¹= (A⁻¹)ᵀ
c) (A²)⁻¹ = (A⁻¹)²
d) |A⁻¹|= |A|⁻¹
28) If A and B are two square matrices of the same order, then (A - B)² is equal to
a) A²- 2AB+ B²
b) A²- AB - BA + B²
c) A²- 2BA+ B²
d) A²+ 2AB+ B²
29) For how many values of x in the closed interval [-4,-1], the matrix
3 -1+ x 2
3 -1 x+2
x+3 -1 2
is singular
a) 0 b) 1 c) 2 d) 3
30) A= 7 1 2 & B= 3 & C= 4
9 2 1 4 2
5
find the relation AB+ 2C
a) 43 b) 43 c) 45 d) 44
44 45 44 45
31) If A= 3 4
5 7
Then the value of A(adj A) is equal to
a) I b) |A| c) |A| I d) none
32) If x+ y 2x + z = 4 7
x -y 2z+ w 0 10 then the values of x, y, z and w are
a) 2,3,1,2 b) 2,2,3,4 c) 3,3,0,1 d) 2,2,4,3
33) The matrix
2 k. -4
-1 3 4
1 -2 -3 is non-singular if
a) k≠ 2 b) k≠ 3 c) k≠ -3 d) k≠ -2
34) If matrix A= 3 2
4 5 and
AC= 19 24
37 46
Then the matrix C is equal to
a) 3 4 b) 3 5 c) 5 4 d) 3 2
5 6 4 6 2 6 6 4
35) Let A= [5] be a matrix of order 1x1. Then adj A is equal to
a) [1] b) [5] c) [0] d) 1
5
36) If A²- A + I= 0 then the inverse of matrix A is
a) A- I b) A+ I c) A d) I - A
37) Let A and B are two square matrices such that AB= A and BA= B. Then A² is equal to
a) O b) I c) A d) B
38) If A= 4 2
-1 1 then the value of (A - 2I)(A- 3I) is
a) A b) I c) O d) 4I
39) If A= 1 -1 & B= a 1
2 -1 b -1 and (A+ B)²= A²+ B², then the values of a and b are
a) a=4, b= 1 b) a=1, b= 4 c) a=9, b= 4 d) a=2, b= 4
40) Let A= a 0 & B= 1 0
1 1 5 1 if A²= B, then the value of a is
41) The matrix 0 7 4
-7 0 -5
-4 5 0 is
a) symmetric b) skew symmetric c) nonsingular d) orthogonal
42) Let A= 1 -1 1 & B= 4 2 2
2 1 -3 -5 0 a
1 1 1 1 -2 3
If B is the inverse of the matrix A, then the value of a is
a) 2 b) 1 c) -2 d) 5
43) If A= 0 0 -1
0 -1 0
-1 0 0
Then the only correct statement about the matrix A is
a) inverse of A does not exist
b) A= (-1) I
c) A is a zero matrix
d) A²= I
SECTION : A (80 Marks)
Question 1) (10x2= 20 Marks)
i) If A= 2 3
4 5 , find Inverse of A.
ii) Show that the function f: R-> R, given by f(x) = | x | is neither one one or onto.
iii) Show sin⁻¹cos sin⁻¹x+ cos⁻¹sin cos⁻¹ x = π/2
iv) If y= tan⁻¹(secx+ tanx), find d²y/dx²
v) ∫ x eˣ dx
vi)
vii) Prove without expanding:
a - b 1 a a 1 b
b - c 1 b = b 1 c
c - a 1 c c 1 a
viii) If x > 1/2, show that the function f(x)= x(4x²-3) is strictly increasing.
ix) Solve 2ˣ⁻ʸ dx + 2ʸ ⁻ˣ dy = 0
x) A and B are two Independent events with P(A)= 2/5 and P(B)= 1/3, Evaluate P(AUB).
Question 2). (4)
Prove: 1+a²-b² 2ab - 2b
2ab 1 - a²+b² 2a
2b - 2a 1 - a² - b²
= (1+a²+b²)³.
Question 3). (4)
If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) = π/2 then Prove that, x² + y² + z² = r².
Question 4). (4)
A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement. Find the probability that none is red.
Question 5). (4)
If y= (tan⁻¹x)², show that (1+x²) d²y/dx² + 2x(1+x²) dy/dx - 2= 0.
Question 6). (4)
Evaluate ∫ 2⁴ˣ sin 3x dx.
OR
Evaluate:
∫ (cosx + x sinx)/{x(x+ cosx) dx
Question 7) (4)
Find the equations of the tangent to the curve y= x² - 2x +7 which is:
a) Parallel to the line 2x - y+9= 0
b) Perpendicular to 5y-15x= 13
OR
Show that the maximum value of 2x + 1/2x is less than its minimum value.
Question 8). (4)
Solve by matrix inversion method
x+2y+z= 7; x + 3z= 11; 2x - 3y=1
OR
Show that
a + b+ 2c a b
c b+c + 2a b
c a. c+a+2b = 2(a+b+c)³.
Question 9). (6)
Given x+y= 3, find the maximum and minimum values of 9/x + 36/y
OR
A closed right circular cylinder is has a volume of 2156cm³. What will be the radius of the base so that total surface area is minimum.
Question 10). (4)
Show that the function f in A= R - {2/3} defined as f(x)=(4x-3)/(6x-4) is one-one and onto .
Question 11). 2+2=4
Solve:
a) dy/dx + y secx = tan x.
b) tan x dy/dx= 1+y² where x= π/2 and y= 1.
Question 12). 3+3= 6
a) Evaluate ∫ |sin x| dx at (π/2,-π/2).
b) Prove ∫ {log(1+x)}/(1+x²) dx at (1,0) = π/8 . Log 2.
Question 13). (3x2= 6)
a) If x= sint and y= cos pt, p is constant, then find the value of (1-x²) d²y/dx² - x dy/dx.
b) If m² = p² cos² t + q²sin²t, then show that m+ d²m/dt² = p²q²/m²
Question 14). 3+3=6
A) It is known that 5 men out of 100 and 25 women out of 1000 are colour blind. A colour blind person is chosen at random. Assuming that males and females are in equal proportion, find the probability of the person to be male.
B) Rajiv and Robin play 12 games of chess. Rajiv wins 6 games, Robin wins 4 games and and 2 games end in a draw. They agree to play 3 more games. Calculate the probability that out of these 3 games, two games end in a draw.
Or
Evaluate:
A) ∫ x² sin⁻¹x dx
B) ∫x² eᵃˣ dx at (a,0)
SECTION C. (20 Marks)
Question 15). 2+2+2
A) Given demand function x= 50- 0.5 P and cost function C=50+40x, find price for break-even price.
B) 4x+y-10= 0, 2x + 5y -14= 0 are two regression lines. Find the correlation coefficient between variables x and y.
C) The total cost C(x) of a firm is C(x)= 0.0005x³ - 0.7x² - 30x + 3000 where x is the output. Determine:
a) average cost (AC)
b) Marginal cost (MC)
Question 16). (4)
The two lines of Regression for a distribution (x,y) are 3x+2y= 7 and x+4y= 9. Find the regression coefficient X on Y and Y on X.
OR
Treating x as an independent variable. Find the line of best fit for the following date:
X: 15 12 11 14 13
Y: 25 28 24 22 30
Hence, predict the value of y when x= 10.
Question 17) (4)
The marginal cost function of manufacturing x units of a commodity is 6+10x - 6x². The total cost of producing one unit of the commodity is ₹ 12. Find the total and average cost functions.
OR
If c= 2x{(x+4)/(x+1)} + 6 is the total cost of production of x units of a commodity, show that marginal cost falls continuously as a x increases.
Question 18). (6)
A small firm manufacturers gold rings and chains. The combined number of rings and chains manufactured per day is almost 24. It takes one hour to make a ring and half an hour for a chain. The maximum number of hours available per day is 16. If the profit on a ring is ₹300 and on a chain is ₹190, how many of each should be manufactured daily so as to maximize the profit?
MATRIX (2)
1) If A= 1 0 0
0 1 0
a b -1
Then A² is equal to
a) a null matrix b) a unit matrix c) - A d) A
2) If A and B are symmetric matrices of the same order, then ABᵀ - BAᵀ is a
a) skew symmetric matrix
b) null matrix
c) symmetric matrix d) none
3) If A and B are symmetric matrices of the same order, then AB= A and -BA = B, then B² is equal to
a) B b) A c) 1 d) 0
4) If AB= A and BA= B, where A and B are square matrices, then
a) B²= B b) B²≠ B and A²= A c) A²≠ A, B²= B d) A²≠ A, B²≠ B
5) If A and B are two matrices such that AB= B and BA= A, then A²+ B² is equal to
a) 2AB b) 2BA c) A + B d) AB
6) A= 1 0 0
0 2 0
0 0 4 is
a) Identity matrix
b) symmetric matrix
c) Skew symmetric matrix
d) diagonal matrix
7) If the matrix AB is zero, then
a) it is not necessary that either A= O or B= O
b) A= O or B= O
c) A= O and B= O
d) all the above statements are wrong.
8) A= 0 -5 8
5 0 12
-8 -12 0 is a
a) diagonal matrix
b) symmetric matrix
c) skew symmetric matrix
d) scalar matrix
9) If A, B are square matrices of order 3, A is a non singular and AB= O, then B is a
a) null matrix
b) singular matrix
c) unit matrix
d) non singular matrix
10) A= n 0 0 & B= a₁ a₂ a₃
0 n 0 b₁ b₂ b₃
0 0 n c₁ c₂ c₃
Then AB is equal to
a) B b) nB c) Bⁿ d) A+ B
11) If A=2 -1 3 & B= 2. 3
-4 5 1 4 2
1 5
Then
a) only AB is defined
b) only BA is defined
c) AB and BA both are defined
d) AB and BA both are not defined
12) If A=1 2 x & B= 1 -2 y
0 1 0 0 1 0
0 0 1 0. 0 1 and AB= I₃, then x+ y is equal to
a) 0 b) -1 c) 2 d) none
13) If A= 1 -1 & B= a 1
2 -1 b -1 and (A+ B)²= A²+ B², then values of a and b are
a) 4,1 b) 1,4 c) 0,4 d) 2,4
14) If A= a b
c d is such that A²= I, then
a) 1+ a²+ bc= 0
b) 1- a²+ bc= 0
c) 1- a²- bc= 0
d) 1+ a²- bc= 0
15) If S= [sᵢⱼ] is a scalar matrix such that sᵢⱼ= k and A is a square matrix of the same order, then AS= SA= ?
a) Aᵏ b) k+ A c) kA d) kS
16) If A is a square matrix such that A²= A, then (I+ A)³- 7A is equal to
a) A b) I - A c) I d) 3A
17) If a matrix A is both symmetric and skew symmetric, then
a) A is a diagonal matrix
b) A is a zero matrix
c) A is a scalar matrix
d) A is a square matrix
18) The matrix
0 5 -7
-5 0 11
7 -11 0
is
a) a skew symmetric matrix
b) a symmetric matrix
c) a diagonal matrix
d) an upper triangular matrix
19) If A is a square matrix, then AA is a
a) skew symmetric matrix
b) symmetric matrix
c) diagonal matrix
d) none of these
20) If A and B are symmetric matrices, then ABA is
a) symmetric matrix
b) skew symmetric matrix
c) diagonal matrix
d) scalar matrix
21) If A= 5 x
y 0 and A= Aᵀ, then
a) x=0, y= 5
b) x + y= 5
c) x=y d) none
22) If A is 3x4 matrix and B is a matrix such that AᵀB and BAᵀ are both defined. Then, B is of the type
a) 3x4 b) 3x3 c) 4x4 d) 4x3
23) If A= [aᵢⱼ] is a square matrix of even order that aᵢⱼ= i²- j², then
a) A is a skew symmetric matrix and |A|= 0
b) A is symmetric matrix and |A| is a square
c) A is symmetric matrix and |A|= 0 d) none
24) If A and B are square matrices of the same order, then (A+ B)(A- B) is equal to
a) A²- B² b) A²- BA - AB - B² c) A²- B²+ BA - AB d) A²- BA+ B²+ AB
25) If A= 2 0 -3
4 3 1
-5 7 2
is expressed as the sum of a symmetric and skew symmetric matrix, then the symmetric matrix is
a) 2 2 -4 b) 2 4 -5
2 3 4 0 3 7
-4 4 2 -3 1 2
c) 4 4 -8 d) 1 0 0
4 6 8 0 1 0
-8 8 4 0 0 1
26) Out of the following matrices, choose that matrix which is scalar matrix
a) 0 0 b) 0 0 0 c) 0 0 d) 0
0 0 0 0 0 0 0 0
0 0 0
27) The number of all possible matrices of order 3 x 3 with each entry 0 or 1 is
a) 27 b) 18 c) 81 d) 512
28) Which of the given values of x and y make the following pairs of matrices equal?
a) x=-1/3, y = 7
b) x=-2/3, y = 7
c) x=-1/3, y = -2/5
d) not possible to find
29) If A=0 2 & kA= 0 3a
3 -4 2b 24
Then the values of k,a,b are respectively
a) -6,-12,-18 b) -6, -4,9 c) -6,-4,-9 dx) -6,12,18
30) If matrix A= [aᵢⱼ] 2x2,
where aᵢⱼ={ 1, if i≠ j
0, if i+ j
Then A² is equal to
a) I b) A c) O d) -I
31) The trace of the matrix
1 -5 7
0 7 9
11 8 9 is
a) 17 b) 25 x) 3 d) 12
32) If A= [aᵢⱼ] is a scalar matrix of order n x n such that aᵢⱼ= k for all i, then trace of A is equal to
a) nk b) n+ k c) n/k d) none
33) A= 0 0 4
0 4 0
4 0 0 is a
a) square matrix
b) diagonal matrix
c) unit matrix
d) none
34) The number of possible matrices of order 3x3 with each entry 2 or 0 is
a) 9 b) 27 c) 81 d) none
35) If 2x+ y 4x = 7 7y-13
5x-7 4x y x+ 6
Then the value of x, y is
a) 3,1 b) 2,3 c) 2,4 d) 3,3
36) If A is a square matrix such that A²= I, then (A- I)³+ (A+ I)³- 7A is equal to
a) A b) I - A c) I+ A d) 3A
37) If A and B are two matrices of order 3x m and 3x n respectively and m= n, then the order of 5A - 2B is
a) mx3 b) 3x3 c) m x n d) 3 x n
38) If A is a matrix of order m x n and B is matrix such that ABᵀ and BᵀA are both defined, then the order of matrix B is
a) mx n b) nx n c) nx n d) mx n
39) If A and B are matrices of the same order, then ABᵀ - BᵀA is a
a) skew symmetric matrix
b) null matrix
c) unit matrix
d) symmetric matrix
MATRIX (1)
1) if A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A+ I
2) The multiplicative inverse of matrix
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
3) Assuming that the sums and products given below are defined, which of the following is not a true for matrices?
a) AB= AC doesn't imply B= C
b) A+ B = B+ A
c) (AB)'= B'A'
d) AB= 0 implies A= O or B= O
4) If A= 1 0 2 & Adj A= 5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2 b
Then the values of a and b are
a) -4,1 b) -4, -1 c) 4, 1 d) 4, -1
5) If A= -1 0
0 2 then the value of A³- A² is equal to
a) I b) A c) 2A d) 2I
6) If A= - x - y
z t then the transpose of adj A is
a) t z b) t y c) t -z d) none
-y - x -z -x y -x
7) If A=3 5 & B= 1 17
2 0 0 -10 then|AB| is equal to
a) 80 b) 100 c) -110 d) 92
8) If A= 5. -2
3 1 find the inverse of A
9) If A is singular matrix of order n then A. (Adj A) is equal to
a) a null matrix
b) a row matrix
c) a column matrix
d) none
10) If A= 3 -5
-4 2 then find the value of A²- 5A is equal to
a) I b) 14I c) O d) none
11) If A= 1 2 & B= 1 2
2 3 2 1
3 4
Then
a) both AB and BA exist
b) neither AB nor BA exist
c) AB exists but BA does not exist
d) AB does not exist but BA exist
12) If A= 2 -1 & B= 1 0
0 1 -1 -1 then+A+ B)² is not equal to
a) A²+ AB+ BA+ B²
b) A²+ AB+ BA+ B²I
c) A²I + AB+ BA+ B²
d) A²+ 2AB+ B²
13) If A be an n × n matrix and k any scalar then det. kA is equal to
a) k detA b) nᵏdetA c) kⁿ detA d) kn detA
14) If A= 1 2
3 -5 find inverse of A
15) If A= -1 2 & B= 5
2 -1 7 and AX= B, then X is equal to
a) 19 17 b) 19/3 c)19/3 17/3 d) 19
17/3 17
16) If A≠ O and B≠ O are two 2 x 2 matrices such that AB= O, then which of the following is correct?
a) detA= 0 or detB= 0
b) detA= 0 and detB= 0
c) detA= 0 = detB≠ 0 d) none
17) If A is a square matrix 3x3 and k is a scalar, then adj(kA) is equal to which of the following?
a) k adj A B) k² adj A c) k³ adj A d) 2k adj A
18) If A= 0 1 2
1 2 3
3 1 1 and its inverse B= [bᵢⱼ] , then the element b₂₃ of matrix B is
a) -1 b) 1 c) -2 d) 2
19) If A=a₁₁ a₁₂ a₁₃ & B=1 2 3
a₂₁ a₂₂ a₂₃ 2. 3. 4
a₃₁ a₃₂ a₃₃ 3 4 5
C= -1 -2 & D= -4 -5 -6
3 0 0 0 1
0 -4
With the relation A= BCD, then the value of a₂₂ is
a) 40 b) -40 c) -20 d) 20
20) If A= 1 2 & B= 3 8
3 4 7 2 with the relation 2X+ A= B, then the matrix X is equal to
a) 2 6 b) 1 -3 c) 1 3 d) 2 -6
4 -2 2 -1 2 -1 4 -2
21) If A= a 2
2 a and |A³|= 125, then the value of a is
a) ±2 b) ±2 c) ±5 d) 0
22) If A= |aᵢⱼ| and Aᵢⱼ denotes the Cofactor of aᵢⱼ, then which of the following is not equal to zero ?
a) a₃₁A₁₁ + a₃₂A₁₂+ a₃₃A₁₃
b) a₁₁A₃₁+ a₁₂A₃₂+ a₁₃A₃₃
c) a₂₁A₂₁+ a₂₂A₂₂+ a₂₃A₂₃
d) a₃₁A₂₁+ a₃₂A₂₂+ a₃₃A₂₃
23) A= 1 0 0
a 1 0
b c 1 find inverse of A
24) The minors of (-4) and 9 and the Cofactor of (-4) and 9 in matrix
-1 -2 3
-4 -5 -6
-7 8 9 are respectively
a) 42,3; -42,3 b) -42,-3; 42,-3 c) 42,3; -42,-3 d) 42,3; 42,3
25) If A= 0 3 & kA= 0 4a
4 5 3b 60
Then the values of k, a and b are respectively
a) 12,9,16 b) 9,12,16 c) 12,9,12 d) 16,12,9
26) B= 1 3 & C= 1 1
0 1 0 -1
If the matrix A satisfies the equation BA= C, then which one of the following represents A?
a) 1 4 b) 1 4 c) 1 -4 d) 1 -2
-1 0 0 -1 1 0 0 -1
27) For any matrix A, if A⁻¹ exists then which of the following is not true?
a) (A⁻¹)⁻¹= A
b) (Aᵀ)⁻¹= (A⁻¹)ᵀ
c) (A²)⁻¹ = (A⁻¹)²
d) |A⁻¹|= |A|⁻¹
28) If A and B are two square matrices of the same order, then (A - B)² is equal to
a) A²- 2AB+ B²
b) A²- AB - BA + B²
c) A²- 2BA+ B²
d) A²+ 2AB+ B²
29) For how many values of x in the closed interval [-4,-1], the matrix
3 -1+ x 2
3 -1 x+2
x+3 -1 2
is singular
a) 0 b) 1 c) 2 d) 3
30) A= 7 1 2 & B= 3 & C= 4
9 2 1 4 2
5
find the relation AB+ 2C
a) 43 b) 43 c) 45 d) 44
44 45 44 45
31) If A= 3 4
5 7
Then the value of A(adj A) is equal to
a) I b) |A| c) |A| I d) none
32) If x+ y 2x + z = 4 7
x -y 2z+ w 0 10 then the values of x, y, z and w are
a) 2,3,1,2 b) 2,2,3,4 c) 3,3,0,1 d) 2,2,4,3
33) The matrix
2 k. -4
-1 3 4
1 -2 -3 is non-singular if
a) k≠ 2 b) k≠ 3 c) k≠ -3 d) k≠ -2
34) If matrix A= 3 2
4 5 and
AC= 19 24
37 46
Then the matrix C is equal to
a) 3 4 b) 3 5 c) 5 4 d) 3 2
5 6 4 6 2 6 6 4
35) Let A= [5] be a matrix of order 1x1. Then adj A is equal to
a) [1] b) [5] c) [0] d) 1
5
36) If A²- A + I= 0 then the inverse of matrix A is
a) A- I b) A+ I c) A d) I - A
37) Let A and B are two square matrices such that AB= A and BA= B. Then A² is equal to
a) O b) I c) A d) B
38) If A= 4 2
-1 1 then the value of (A - 2I)(A- 3I) is
a) A b) I c) O d) 4I
39) If A= 1 -1 & B= a 1
2 -1 b -1 and (A+ B)²= A²+ B², then the values of a and b are
a) a=4, b= 1 b) a=1, b= 4 c) a=9, b= 4 d) a=2, b= 4
40) Let A= a 0 & B= 1 0
1 1 5 1 if A²= B, then the value of a is
41) The matrix 0 7 4
-7 0 -5
-4 5 0 is
a) symmetric b) skew symmetric c) nonsingular d) orthogonal
42) Let A= 1 -1 1 & B= 4 2 2
2 1 -3 -5 0 a
1 1 1 1 -2 3
If B is the inverse of the matrix A, then the value of a is
a) 2 b) 1 c) -2 d) 5
43) If A= 0 0 -1
0 -1 0
-1 0 0
Then the only correct statement about the matrix A is
a) inverse of A does not exist
b) A= (-1) I
c) A is a zero matrix
d) A²= I
TEST- 10/7/26
1) Prove: cot⁻¹(1/2) - (1/2) cot⁻¹(4/3) =π/4.
2) y= 2 sin{(x -2)/√6 - √(2+ 4x - x²)}, find dy/dx at x= 2
3) Let y= (sin⁻x)²+ (cosx)², find dy/dx
4) If (a+ bx)eʸ/ˣ= x, find the value of (x dy/dx - y)².
5) Three normals are drawn from the point (14,7) to the curve y² - 16x - 8y =0. Find the coordinates of the feet of the normal.
6) For what values of x the function y= 2 sinx + cos2x(0< x ≤2π) attains maximum and minimum values.
7) d/dx (logₑ(ax)ˣ), where a is a constant), is equal to
a) 1 b) logₑ(ax) c) 1/a d) logₑ(ax) +1
8) Find the value of a if f(x)= a |sin x|+ 2x is continuous at x=0.
H. W -1
SAP- 1
1) If A= 2 0 3 & B= 1 2 3
1 2 0 2 1 4 find A+ B
2) If A= 1 2 3 & B= 1 2
4 5 6 3 4
6 8 9 5 6 find A+ B
3) If A= 0 2 3 & B= 7 8 3
2 1 4 1 4 3 find
a) 2A + 3B
b) 3A - B
c) AB
4) If A= 1 3
3 4 and A²- kA - 5=0, then find k.
5) If A= x y z & B= a h g & C= x
h b f y
g f c z
Then show ABC= ax²+ by²+ cz²+ 2fyz + 2gzx+ 2hxy.
6) If A= 1 0 & B= 0 1
0 1 -1 0
Then show that (aA+ bB)(cA+ dB)= (ac - bd)A+ (ad + bc)B.
7) If A= 1 0 0 & B= x₁ y₁ z₁
0 1 0 x₂ y₂ z₂
0 0 1 x₃ y₃ z₃
Then show that AB= BA = B
Sap-2 (H. W)
1) If A= 1 -1 & B= a 1
2 -1 b -1 and (A+ B)¹= A²+ B², find a, b, using the value of a, b, verify whether AB= BA.
2) If A= 1 -1
1 1 show A/√2 is a orthogonal matrix. A. A'= I is orthogonal matrix
3) Find the inverse of A
A= 1 -3 2
2 5 -1
-3 1 4
4) A= 20 10
10 20 find inverse of A
5) Solve: x + 2y - z= 9, 2x - y - 4z= -7; 3x + 2y - 3z= 2.
SAP-1
1) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A + I
2) The multiplicative inverse of
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
3) Assuming that the sums and products given below are defined, which of the following is not true for matrices?
a) AB= AC does not imply B= C
b) A+ B= B+ A
c) (AB)'= B'A'
d) AB= O implies A= O or B= O.
4) If A=1. 0 2 & Adj A= 5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2 b
Then the values of a and b are
a) a= -4, b= 1 b) a= -4, b= -1 c) a= 4, b= 1 d) a= 4, b= -1
5) If A= -1 0
0 2 then the value of A³- A² is
a) I b) A c) 2A d) 2I.
6) If A= -x - y
z t, then transpose of adj A
a.) t z b) t y c) t -z d) none
-y -x -z -x y -x.
7) If A square metrix of order 3x3 and λ is a scalar, then adj(λA) is equal to
a) λ adj A B) λ² adj A c) λ³ adj A d) λ⁴ adj A.
8) The inverse of 5 -2
3 1
a) -2/13 5/13 b) 1 2 c) 1/11 2/11 d) 1 3
1/13 3/13 -3 5 -3/11 5/11 -2 5
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