INTERMEDIATE
PART-II (12th CLASS)
MATHEMATICS
TIME ALLOWED: 30 Minutes
OBJECTIVE
MAXIMUM MARKS: 20
Note: You have four choices for each objective type question as A, B, C and D. The choice which you think is correct, fill that circle in front of that question number. Use marker or pen to fill the circles. Cutting or filling two or more circles will result in zero mark in that question. Attempt as many questions as given in objective type question paper leave others blank. No credit will be awarded in case BUBBLES are not filled. Do not solve question on this sheet of OBJECTIVE PAPER

Q.No.1

equals:-

(A) 0 
(B) -1  
(C) 1
(D) -2

• ∫Sec x dx equals:-
  

(A) ℓn sec x + c
(B) ℓn |sec x + tanx| +c
(C) Sec x tan x + c
(D) Sec x + tan x + c

• Solution of differential equation dy/dx = cos x is:-
  

(A) y = cosx + c
(B) y = tan x + c
(C) y = cotx + c
(D) y = cotx + c

• If A = (- 3, 6 ) and B= (3,2) then slope of AB is:-
  

(A) 3 /2
(B) -2/3
(C) 1/3
(D) -3/2

• Scope intercept form of straight line is:-
  

(A) y = mx + c
(B) xcosα + ysinα = p
(C) x/a + y/b = 1
(D) x= 0

• (1, 0) is. not solution of inequality:-
  

(A) 7x + 2y < 8
(B) x - 3y < 0
(C) 3x + 5y < 7 
(D) 3x + 5y ≤ 3

• The directrix of parabola x² = -16y is
  

(A) y-1=0
(B) y +1 = 0
(C) y - 4 = 0
(D) y + 4 = 0

• If a > b then foci of ellipse x²/a² +y²/b² = 1 are
  

(A) (± a, 0)
(B) (0, ± a)
(C) (0 ± ae)
(D) ab cosɵ

• If a and b are oppositely directed then a. b equals:
  

(A) ab
(B) –ab
(C) ab sinɵ
(D) abcosɵ

• The magnitude of dot and cross produc of two vectors are 1 and 1 respectively. Then angle between vectors is:-
  

(A) 90°
(B) 60°
(C) 45°
(D) 30°

• Domain of f (x) = 2+
  

(A) (0, 1)
(B) (2, ∞)
(C) (1, ∞)
(D) (0, ∞)

• equals:-
  

(A) 0
(B) 1  
(C) – 1
(D) - ∞

• If y = x^1/2 then dy/dx equals:
  

(A) -3/2- x1/2
(B) -3/2. x1/2
(C) -3 x1/2
(D) -3/2. x1/2

• If y = ℓn (tan hx) then dy/dx is
  

(A) Sech²xcothx
(B) 2Sechx
(C) SechxCothx
(D) -2Sechx Cothx

• d/dx (2^√n) equals:
  

(A) 2√x
(B) 2√x ℓn²
(C)
(D)

• d/dx (Sec^-1x) is
  

(A)
(B)
(C)
(D) Cot^-1x

• f(x) increases if
  

(A) f(x) < 0
(B) f’(x) > 0
(C) f’(x) = 0
(D) f’(x) ≥ 0

• ∫ex (Cosx + Sinx) dx equals:-
  

(A) exCos x + C
(B) ex Sinx + c
(C) –ex Sinx + c
(D) xesinx + c

•  equals:-
  

(A) √tanx + c
(B) 2√tanx + c
(C) √cosx + c
(D) 2√cotx + c

• equals:

(A)
(B)
(C)
(D)

INTERMEDIATE
PART-II (12th CLASS)
MATHEMATICS
TIME ALLOWED: 2.30 Hours
SUBJECTIVE
MAXIMUM MARKS: 80
NOTE: Write same question number and its part number on answer book, as given in the question paper.

SECTION-I

2. Attempt any eight parts.  (8 x 2=16)

• Find f^-1(x) if f(x) = -2x + 8
  
• Evaluate
  
• Find the derivative of f(x)=c by definition.
  
• Differentiate w.r.t. 'x'
  
• Differentiate x² – 1/x²  w.r.t. 'x'
  
• Differentiate Cos^-1(x/a) w.r.t. 'x'
  
• Find f’ (x) if f (x) = ℓn (ex + e-x)
  
• Find y² if x³ – y³ = a
  
• Prove that e^x = 1 + x + x²/2 + x²/3 + ------------
  
• Determine the intervals in which f is increasing or decreasing for the domain f(x) = 4-x² when x € ( -2, 2)
  
• Find dy/dx if y=cosh(2x)
  
• Find the derivative of

3. Attempt any eight parts. (8x2=16)

• Evaluate  ∫x/x+2 dx
  
• Evaluate ∫
  
• Integrate 
  
• Find ∫secx dx
  
• Find ∫x.ex dx
  
• Evaluate 2∫1
  
• Evaluate ∫tanxx.dx dr
  
• Evaluate ∫ 3x+1/ x²-x-6 dx
  
• Integrate ∫x²ℓnx dx
  
• Integrate
  
• Define feasible region and feasible solution.
  
• Define optimal solution. 

4. Attempt any nine parts. (9x2=18)

• Show that points A(3,1), B(-2,-3) and C(2, 2) are the vertices of an isosceles triangle.
  
• Find the distance between a(-√5, -1/3) and B(-3√2, 5)
  
• The xy-coordinate axes are rotated about the origin through the indicated angle. The new axes are OX and OY. Find the XY-coordinates of the point P with the given xy-coordinates:  P( 5, 3 ) and ɵ = 45"
  
• Find the equation of the line bisecting first and third quadrants.
  
• Convert the equation 2x - 4y + 11 = 0 into two intercepts form.
  
• Find the centre and radius of the circle x2 + y2 + 12x - 10) = 0
  
• Find the length of the tangent from the point P(- 5, 10 ) to the circle 5x2+5y2 + 14x + 12y - 10 = 0
  
• Write an equation of the parabola with focus (-3,1) and directrix: x= 3
  
• Find the foci and vertices of the ellipse 9x2 + y2 = 18    
  
• Find α so that |α i + (α + 1) 1 + 2k|  = 3    
  
• Find a real number ά so that the vectors u = 2αi + j – k and v = I + αj +4k are perpendicular
  
• Find the area of the triangle determined by the points P(0, 0, 0), Q(2, 3, 2) and R(-1, 1. 4)
  
• Find a vector of length 5 in the direction opposite to that of v = i - 2 j + 3k,

SECTION-II

Attempt any three questions. (3x10=30)

5. (a) Evaluate
(b) If y = tan(2Tan-1x/2), show that dy/dx = 4(1+y2) / 4+x2

6. (a) Evaluate

(b) Find an equation of line through the intersection of the lines x-y-4=0 and  7x + y + 20 = 0 and parallel to the line 6x + y – 14 = 0

7. (a) Evaluate 1/4∫1/6 cos2ɵ cot2ɵ dɵ
(b) Graph the feasible region of the system of linear inequalities.2x+y≤10, x+4y≤12, x+2y≤10, and x≥0, y≥0

8. (a) Find an equation of the circle that passes through the points A(4, 5), B(- 4, -3), C(8, -3)
(b) Prove that the line segment joining the mid points of the sides of a quadrilateral taken in order form a parallelogram.

9. (a) Prove that the latusrectum of the ellipse
(b) By using vectors prove that Cos(α + β) = Cosα.Cosβ-Sinα.Sinβ