R-3
1) Find dy/dx when y= ₓx²+ ₐx².
2) if y= xⁿ{a cos(logx)+ b sin(logx)}, show that x² d²y/dx²+ (1- 2n) x dy/dx + (1+ n²)y= 0.
3) find the derivative of x² cosx.
4) Find derivative of tan(√x).
5) If cosy = x cos(a+ y), show that dy/dx = (cos²(a+ y)/sina.
6) evaluate: tan⁻¹(a/b) - tan⁻¹{(a-b)/(a+ b)}.
7) If y= c₁x⁻¹ + c₂x², the value of x² d²y/dx² is:
a) x b) y c) 2x d) 2y
R-2
1) 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 zero matrix
d) A²= I
2) The value of determinant
a² - ab - ac
- ab b² - bc
ca bc - c² is
a) 4a²b² b) 4b²c² c) 4c²a² d) 4a²b²c²
3) If y= sin(πeˣʸ)/6, then the value of dy/dx at x= 0 is
a) √3/24 b) √3 π/24 c) √3/12 d) √3π/12
4) If f(x)= √(1+ cos(x²)) then the value of f'(√π/2) is
a) -√π/√6 b) √π/√6 c) π/2√2 d) π/√6
5) The slope of the tangent to the curve x= 3t²+ 1, y= t³- 1 at t=1 is
a) 1/2 b) 0 c) -2 d) undefined
6) If f(x)= x³- 6x²+ 9x +3 be a decreasing function, then x lies in
a) (1,3) b) (-∞,1) U(3,∞) c) (3,∞) d) none
7) The equation of the normal to the curve x³+ y³= 8xy at the point where it meets the parabola y²= 4x is
a) x+ y= 0 b) x- y= 0 c) x- y= 4 d) x+ y= -4
8) The difference between the maximum and minimum values of the function f(x)= x³/3 - 2x² + 3x +1 is
a) 4 b) 2 c) 1 d) 4/3
9) If 2x+ 3y = 4, the maximum value of xy is
a) 3 b) 2/3 c) 1 d) 1/3
R-1
a₁ a₂
a₃ a₄ is _____
2) State whether the following statement is true or false: if f(x) is a polynomial of degree n(≥1), then the degree of f'(x) is (n+1).
3) If y= (x -1)eˣ, then the value of dy/dx at the point x= 1 is
a) e b) 2e c) 1 d) 0
4) If y= 1+ cos2x then d²y/dx² + 4y= __
5) ∫ 3ᵅˣ dx ?
a) 3ᵅˣ⁺¹ b) 3ᵅˣ c) 3ᵅˣ logₑ3ᵅ d) 3ᵅˣ/(alogₑ3)
6) Without expanding at any stage, find the value of the determinant:
2 x y+z
∆= 2 y z+ x
2 z x+y
a) x+ y b) xy c) xyz d) 1 e) none
7) Solve: sin⁻¹ cos(sin⁻¹x)=π/3
a) 1 b) 0 c) ±1/2 d) 2
8) Find the value of k if
M= 1 2
2 3 and M²- kM - I₂= 0.
a) ±2 b) 1 c) 3 d) 4
9) Find the intervals in which the function f(x) is strictly increasing where f(x)= 10 - 6x - 2x².
a) 3/2 b) -3/2 c) (-∞,-3/2) d) (3/2,∞)
10) Find d²y/dx², if x= at² and y= 2at
a) 1/t b) 2at c) 1/2at d) 1/2at²
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