Saturday, 3 February 2024

Quick Test - XII-1

1) Fill in the gap: The value of the determinant
3        1975       1978
4        1982       1986
5        1995       2000 is _____



3) If y= logₑlogₑx, x>1
a) xdy/dx = 1    b) (x logₑx)dy/dx = 1
c) (logₑx) dy/dx = 1 d) (logₑx) dy/dx= x

4) If x= a(k - sink) and y= a(1+ cosk) then dy/dx is
a) -cot(k/2)  b) -cotk c) - tan(k/2) d) cot(k/2)

5) If y= cos²x then d²y/dx² is
a) - 2 cos2x b) 2 cos2x c) - 2 sin2x d) cos2x 

6) ∫ tan²x dx = _____

7) The value ²₀∫ dx/(x²+ 4) is π/16.   T/F

7) ˣ⁾²∫₋ₓ/₂ sin⁵x dx= ____

8) The gradient of the tangent at the point (8,-4) to the parabola y²= 8(x - 6) is -1.          T/F

9) f(x)= (x -1)(3- x) has an extreme value at the point x= 2.       T/F



11) If y= (Secx)ᵗᵃⁿˣ, find dy/dx.

12) If x³+ y³= 2xy then find the value of dy/dx at the point (1,1).

13) ∫ 2 sin2x/(5+ 3cosx) dx

14) ∫ xe²ˣ dx.

15) What are the order and degree of (d²y/dx²)² + 5(dy/dx)³ + 2y =0.

16) Find the area bounded by the curve y= x², the x-axis and the straight line x= 2.

17) An integer is chosen at random from 100 integers 1,2,3,....100. What is the probability that the selected integer is divisible by 5 or 7 ?

18) Solve by Creamer's rule: 
x+ y=2; y+ z= 2; z+ x =2

19) ∫ dx/√(3- 2x - x²), 3< x <1.

20) ∫ sin⁻¹√{x/(1+ x)} dx 

21) ∫ sin²x/(sinx + cosx) dx at (π/2,0)

22) Solve: dy/dx = (3x + 2y)/(2x - 3y); given that y=0 when x=1.

23) Show that the minimum value of {(2x -1)(x -8)}/{(x -1)(x -4)} is greater than its maximum value.

24) Prove that, of all rectangles of a given area, the square has the smallest perimeter.

25) Find the area in the first quadrant bounded by the parabola y²= ax, a> 0 and the straight line y= x 

26) If A and B are two events connected to a random experiment E, then show that P(AUB)= P(A)+ P(B)- P(A∩B)

27) Prove by property of determinant
1+ a        1        1
   1       1+ b      1
   1          1     1+ c
= ABC(1+ 1/a + 1/b + 1/c)

28) If eˣʸ - 4xy = 4, find the value of dy/dx.

29) If y= tan⁻¹{t/√(1- t²)} and x= sec⁻¹{t/√(2t²- 1)}, then find the value of dy/dx 

30) If f(x)= {(1+x)/(2+ x)}³⁺²ˣ, find the value of f(0).

31) If y= sin(2sin⁻¹x), then show (1- x²) d²y/dx² = x dy/dx - 4y.

32) ∫ xeˣ/(x+1)² dx 

33) ∫ (x +1)/(x²+ 4x +5) dx.

34) ∫ dx/(5+ 4 sinx) at (π/2,0)

35) ∫ (4eˣ + 6e⁻ˣ)/(9eˣ - 4e⁻ˣ) dx.

36) ∫ dx/(a² sin²x + b² cos²x), (a,b > 0)







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