Part-A (Marks: 10)
Group - A
1. Choose the correct alternative [MCQ]. 1x10= 10
I) let Z be the be the set of integers and the mapping f: Z --> Z be defined by, f(x)= x², State which of the following is equal f⁻¹(-4)
A) {2}. B) {- 2}. C) {2,- 2} D) ∅
ii) If sec⁻¹x= cosec⁻¹y state which of the following is the value of (cos⁻¹1/x + cos⁻¹1/y) ?
A) π. B) 2π/3. C) 5π/6. D) π/2
iii) The degree of the differential equation d³y/dx³ + x(dy/dx)⁴= 4 log (d⁴y/dx⁴)
A) 1. B) 3. C) 4. D) undefined
iv) If f(x) is defined as f(x)= x⁵ +3x +1 and x belongs to R. Then f(x) is..
A) one one and onto
B) oneone but but on to
C) onto but not oneone D) none
v) If d/dx{(1+x²+x⁴)/(1+x+x²)}= ax + b, then the value of a and b are
A) -2 B) 1, -2 C) 2,-1 D) -1,2
vi) 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
vii) Angle between the straight lines (x-5)/7= (y+2)/-5= z/1 and x/1= y/2=z/3 is ---
A) π/4. B)π/3. C) π/2. D) π
viii) Integrating factor of the differential equation (x+y+1) dy/dx =1 is
A) eˣ B) e⁻ʸ C) e⁻ˣ D) eʸ
ix) Without expanding prove that
0 99 98
-99 0 97 = 0
-98 -97 0
x) The slope of the normal to the circle x²+ y²= a² at the point (m,n)
A) m/n. B) -m/n. C) -n/m. D) n/m
PART -A (Marks : 70)
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Part A: 1.a). Any one. 2x1= 2
i) Prove tan⁻¹+ tan⁻¹2+ tan⁻¹= π
ii) f: R--> R and g: R --> R and f(x)= 5|x| - x² and g(x)= 2x-3 then find the value of (gof)(5)
b) any one. 2x1= 2
i) If A = 2x x And A⁻¹= 1 -1
0 x 0 2 then Prove x= 1/2
ii) Prove a+b a+2b a+3b
a+2b a+3b a+4b = 0
a+4b a+5b a+6b
c) Any three 2x3= 6
i) f(x)= (sinx/x) +k, x= 0
2 , x= 0
continuous at x=0 , find k
ii) ∫ dx/{x(x²+1)
iii) Solve dy/dx= cos²y/(1+x²) at y(0)1
iv) If y= sin⁻¹x then prove (1-x²)y" - x y'= 0
v) Show that maximum value is less than minimum value of the function y= x²+ 1/x²
d) Any one. 2x1= 2
i) If a, b, c are the vector as a+b+c= 0, |a|= 3, |b|= 4 and |c|= 5 then prove a.b + b.c+ c.a= -25
ii) Find the value of M for which (x-2)/1 =(y-1)/1 = (z-4)/-M and (x-1)/M = (y-3)/1 = (z-4)/-M are coplanar.
e) Any one. 2x1= 2
i) If for two events A and B, 2P(A)= P(B)= 5/13 and P(A/B)= 2/3, find the value of P(AUB)
ii) If A and B independent event then prove A' and B' also independent.
2a) Answer any one. 4x1= 4
i) If cos⁻¹x+ cos⁻¹y+ cos⁻¹z= π then show x²+ y²+ z²+ 2xyz= 1
ii) f: R --> R as f(x)= x³ +x. Verify the function f is bijective or not.
b) Answer the following. 4x2= 8
i) If A= 4 2 -1
3 5 7
1 -2 1 show as the sum of symmetric and skew-symmetric matrix.
ii) Prove with the help of determinants. a²+1 ab ac
ab b²+1 bc
ca cb c²+1 = 1 + a² + b² + c². OR
Solve with the help of Cramer's rule 2x - y+ 3z= 4; x - 3y +5z = 3; x+y+z= 3
c) Answer the following. 4x3=12
i) If (1-x²)y" - xy' = m²y.
ii) Solve: dy/dx=(x+y)² given (0)=1
iii) ∫ x² dx/(2x⁴ - 7x²-4) OR
∫ x tanx sec ²x dx
d) Answer any one:. 4x1= 4
i) If M= 2i+j- 3k and N= i-2j+k, find a vector of magnitude if M and N are perpendicular to both.
ii) If Prove [a+b, b+ c, c+a] =2[a.b.c]
e) Answer any one:. 4x1= 4
i) Show that cot⁻¹(1 - x+x²) at (1,0) = π/2 - log 2
ii) ∫ (x sin x)/(1+cos²x) dx at (π,0)
f) Answer any one:. 4x1= 4
i) Bag A contain 22 white and 3 red balls, bag B contain 4 white and 5 red balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it was drawn from bag A.
ii) A die is rolled 6 times. If getting an odd number is a success, find the probability of getting atleast 3 success.
3a) Answer any one:. 5x1= 5
i) In one kg. Food I contain 6 unit of vitamin A and 7 unit of vitamin B. In one kg. of Food II contain 8 unit of vitamin A and 12 unit vitamin B. Cost of vitamin A and vitamin B are Rs 12, Rs 20 respectively. A man required daily 100 unit of vitamin A and 120 unit of vitamin B. Find minimum cost to get Food I and Food II, with the help of L.P.P.
ii) Solve Graphically with the help of L. P. P , z= 2x+ y;
x +y ≤2,
- x + x ≤1; x ≤ z, x≥ 0, y≥0
b) Answer any two: 5x2= 10
i) Solve: xy dy/dx - y²= (x+y)²eʸ⁾ˣ
ii) If the curve xᵐ yⁿ = zᵐ⁺ⁿ touches the straight line x cos a + y sin a = p then prove pᵐ⁺ⁿm ᵐ nⁿ = (m+n)ᵐ⁺ⁿ zᵐ⁺ⁿ sinⁿa cosᵐa.
Continue......
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Very short question type)
1) ∫ cos x e ˢⁱⁿ ˣ dx at (π/2, 0)
2) ∫ sin x/(3+ 4cos²x) dx
3) Find the solution of dy/dx= 2ʸ⁻ˣ
4) The differential equation representing the family of curves y= A sin x + B cos x dx
5) ∫dx/{cos²x(1+m²tan²x) at(π/2,0)
6) Calculate the area under the curve y= 2√(x) included between the lines x=0 and x=1.
7) Solve the differential equation (1+y²)tan⁻¹ x dx +2y(1+x²) dy =0
8) Evaluate ∫(1 - x)√(x) dx
9) ∫(e ⁶ˡᵒᵍ ˣ- e⁵ˡᵒᵍ ˣ)/(e⁴ˡᵒᵍ ˣ-e ³ˡᵒᵍ ˣ)
10) ∫ dx/(sin²x cos²x)
11) The area enclosed by the circle x²+ y₂ =2
12) dy/dx = y eˣ and x=0, y=e. Find the value of y when x=1
SHORT ANSWER TYPE:
13) Compute the area bounded by the lines x+2y=2, y-x=1 and 2x+y=7.
14) Evaluate ∫ sin (log x) dx
15) Solve the following initial value problem
2x² dy/dx - 2xy +y² =0, y(e)=e
LONG ANSWER TYPE
16) Solve the following initial value problems.
x- sin y)+(tan y) dy =0, y(0)=0
17) Find the area bounded by the curve y= 2x -x² and the straight line y= - x
18) ∫{2x(1+sin x)/(1+cos²x)} dx
19) ∫ x²/{(x-1)³(x+1) dx
20) ∫(1- cos x)/{cos x(1+ cos x) dx
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