1) Fill the gap: The value of the determinant 3 1975 1978
4 1982 1986
5 1995 2000 is_____
OR
State whether the following statement is true or false :
" the product of two non zero matrices must be a non-zero metrix."
2) If y= log log x, x> 1 which one of the following answer is true?
A) x dy/dx= 1 B)(x logx)dy/dx= 1
C) (logx)dy/dx= 1
D) (logx)dy/dx= x
OR
* If x= a(t - sint) and y= a(1+cos t) then which one of the following is the value of dy/dx.
A) - cot(t/2) B) cot t
C) - tan(t/2) D) cot(t/2)
3) If y= cos²x, then Which one of the following is the value of d²y/dx² ?
A) - 2 cos 2x B) 2 cos 2x
C) -2 sin 2x D) cos 2x
4) State whether the following statement is true or false:
the gradient of the tangent at the point (8,-4) to the parabola y²= 8(x-6) is - 1.
OR
* State whether the following is true or false:
f(x)= (x-1)(3-x) had an extreme value at the point x= 2.
5) The equation of the normal to the ellipse x²+ 4y²= 4 which is parallel to the line 8x+ 3y= 0 be..
A) 8x+ 3y= ±12 B) 16x+ 6y= ±15
C) 40x+ 15y= ±36 D) 24x+ 9y= ±15
6) 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.
7) The minimum value of 1/2 (7 - cos 2x) is...
A) 7/2 B) 4 C) 5/2 D) 3.
8) The length of the rectangle of maximum area that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter, is..
A) √2. B) 2 C) √2/3 D) √3 unit
9) If the tangent at the point P on the circle x²+ y²+ 6x + 6y= 2 meets the straight line 5x - 2y+6= 0 at the point Q on the y-axis, then the length of PQ is...
A) 4 B) 2√5 C) 5. D) 3√5
10) Equations of the tangent and normal drawn at the point (6.0) on the ellipse x²/36+ y²/9= 0 respectively are...
A) x= 6, y= 0. B) x+y= 6, y-x+6= 0
C) x= 0, y= 3 D) x= -6, y= 0
11) If the function f(x)= x²(x-2)² is an increasing function of 3, then
A) 1< x<2 B) x<0
C)0< x<1 or x>2. D)1< x<2 or 0< x
12) If f(x)=kx³ - 9x²+ 9x+3 is an increasing function then...
A) k< 3 B) k≤3 C) k>3 D) k is indeterminate.
13) The function f(x)= 1- x³ - x⁵ is decreasing for.....
A) 1≤ x≤5 B) all real values of x
C) x≤ 3. D) x≥ 5
14) If v= 4πr³/3, then the rate (in cubic unit) at which v is increasing when r= 10 and dr/dt= 0.01, is ..
A) 4π. B) π C) 40π. D) 4π/3
15) If the time rate of change of the radius of a sphere is 1/2π, then the rate of change of its surface area (in sq cm), when the radius is 5cm is..
A) 20 B) 10. C) 4 D) 5
16) If f(x)= x when 0≤ x ≤ 1
2x -1 when x> 1 then
A) f(x) is discontinues at x= 1
B) f(x) is discontinues but not differentiable at x= 1
C) f(x) is differentiable at x= 1
D) none of these
17) Which of the following statements is not true?
A) a polynomial function is always continuous
B) a differentiable function is always continuous
C) a continuous function is always differentiable
D) log x is continuous for all x> 0
18) If the function
f(x)= (x²-9)/(x-3) when x≠ 3
2x+a when x= 3 is continuous at x=3, then the value of a is...
A) 3 B) 6 C) 0 D) 4
19) If y= √(x+1) - √(x-1), then the value of (x²-1)d²y/dx² + x dy/dx is
A) 2y B) -2y C) y/4 D) y/2
20) In the mean value theorem f(b) - f(a) = (b - a) f'(c), (a<c<b), if f(x)= x³ - 3x -1, a= -11/7, b= 13/7 then the value of c is...
A) 0 B) 1 C) -1 D) ±1
21) The derivative of the function
tan⁻¹[{2x√(1-x²)}/(1-2x²)] w.r.t. the function tan⁻¹[{√(1+x²) - 1}/x] at x= 0 is...
A) 1 B) 2 C) 4 D) 8
22) If f(x)g(x)= k(a constant) and g"(x)/g'(x)= f"(x)/f'(x) + a.f'(x)/f(x) , then the value of a is...
A) 4 B) -4 C) 2 D) -2
23) If 2x= y¹⁾⁵ + y⁻¹⁾⁵ and (x²-1)y₂ + xy₁= ky, then the value of k is..
A) 5 B) -5 C) 25 D) -25
24) lim ₓ→∞{1 - 4/(x-1)}³ˣ⁻¹=
A) e⁴ B) e³ C) e¹² D)1/e¹²
25) If the function f(x)= 4x³ + ax² + bx -1 satisfies all the conditions of Rolle's theorem in -1/4 ≤x ≤ 1 and f'(1/2)=0, then the values of a and bare
A) a=2, b =-3 B) a=1, b =-4
C) a=-1, b =4 D) a=-4, b =-1
26) lim ₓ→∞ {(n-3)/(n+2)ⁿ is..
A) 1/e⁵ B) 1/e⁴ C) 1/e² D) 1/e
27) The value of c in Rolle's theorem f(x)= 2x³ - 5x² - 4x +3, x belongs [1/2, 3] is...
A) -1/3 B) 2/3 C) 2 D) -2
28) If sin⁻¹(x/5) + cosec⁻¹(5/4)= π/2, then the value of x is..
A) 1 B) 2 C) 3 D) 4
29) The equation sin⁻¹x - cos⁻¹x = cos⁻¹(√3/2) has
A) unique solution
B) two solution
C) no solution
D) infinite number of solutions
30) If x+y+z= xyz, then the value of (tan⁻¹x + tan⁻¹y + tan⁻¹z) is equal to
A) 3π/2 B) π C) π/2 D) 2π
31) The value of cos⁻¹{(3+ 5 cosx)/(5+ 3cosx)} is...
A) 1/2 tan⁻¹(2 tan(x/2))
B) 2 tan⁻¹(1/2 tan(x/2))
C) tan⁻¹(1/2 tan x)
D) 2 tan⁻¹(2 tan(x/2))
32) If the determinant of the matrix a₁ b₁ c₁
a₂ b₂ c₂
a₃ b₃ c₃ is denoted by D, then the determinant of the matrix a₁+3b₁- 4c₁ b₁ 4c₁
a₂ +3₂- 4c₂ b₂ 4c₂
a₃ +3b₃-4c₃ b₃ 4c₃ will be
A) D B) 2D C) 3D D) 4D
33) If A= 3 - 5
-4 2 , then the value of A² - 5A is equal to
A) I B) 14I C) O D) none
34) If A= 1 2 and 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
35) The value of the determinant b²c² bc b+ c
c²a² ca c+ a
b²a² ab a+ b
A) abc(a²+b²+c²) B) 0
C) abc(bc+ca+ab)
D) (a+b+c)(a²+b²+c²) (ab+bc+ca)
36) The maximum value of the function 3cosx - 4 sinx -2 is..
A) 0 B)1 C) 4 D) 3
37) If y= cot⁻¹{(b- ax)/(a+ bx)}, then the value of dy/dx is...
A) 1 B) 1/(1+x²) C) -1 D)-1/(1+x²)
38) If y= cot⁻¹{8x⁴ - 8x²+1}, then the value of dy/dx.
A) 4/√(1-x²) B) -4/√(1-x²)
C) 4/(1+x²) D) - 4/(1+x²)
39) State whether sin(sin⁻¹(√2))= √2 is true or false
OR
Find y in terms of x where tan⁻¹{x/√(1- x²)}= sin⁻¹y.
40) tan⁻¹x + tan⁻¹y=π/4, then...
A) x+y+z+1=0 B) x+y+xy-1=0
C) x+y- xy+1=0 D) x+y- xy-1=0
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