Inter (Part-II) Gujrawala Board 2015
Mathematics
Part I (Objective Type)
Time Allowed: 30 Minutes 
Max. 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.

Question #1
Circle the correct option i.e. A/B/C/D. Each part carries one mark.

• .

(a)                        
(b)                            
(c)                
(d) π

• .

(a) ex cosx          
(b) ex sinx                         
(c)-ex sinx       
(d)-ex cosx

• .
  

(a)cosec2x       
(b)           
(c) 
(d)

• dx

(a) lnsin2x      
(b)(lnsinx)2 
(c) (lnsinx)2      
(d) none

• ____________________

(a)    -cosec2x          
(b) –cotx cosecx        
(c)ln cos x          
(d) ln sin x

• If f’ (c) = 0, then f has relative maxima at x = c if:

(a) f" (c)=0     
(b) f" (c) < 0   
(c) f" (c) 0   
(d) f" (c) > 0

• is Maclaurin expansion of:

(a) cos x        
(b) sin x          
(c) n(l+x)      
(d)

• (cot-1x)=

    
(a)   
(b)        
(c)            
(d)

• If x=a sin y=a cos=

(a) cot x        
(b) tan x          
(c) -cot θ       
(d) -tan θ

(a) -sin 2 π                
(b)-2 sin 2 π               
(c) 0                
(d) 1

• .

(a) 1                            
(b) 0                             
(c)        
(d)

• If f(x)=

(a) R      
(b) (-        
(c)  (0,     
(d) [-1, 1]

• The angle which the vector   +        makes with x - axis is:

(a)                
(b)                
(c)                 
(d)

• Projection of a on b is:

(a) .
(b) .      
(c).     
(d) .      

• Latus-rectum of the parabola x2 = -8y is:

(a)x=2 
(b)x = -4         
(c)y=2 
(d) y = -2

• Covertices of the ellipse x2 + 4y = 16 are:

(a) (± 4, 0)      
(b) (0, ± 4)      
(c) (0, ± 2)      
(d)(± 2, 0)

• Solution of inequality x + 2y < 6 is:

(a) (1, 1)         
(b) (1, 3)         
(c) (1, 4)         
(d) (1, 5)

• If the Slopes of two lines areand 3, then the measure of the angle from 1st line to 2nd line is:

(a)π                
(b)         
(c)         
(d)

• The line ax + by + c 0 is parallel to y - axis if:

(a) c = 0          
(b) a = 0         
(c) a=b          
(d) b =0

• The order of the differential equation 4+x=0 is

(a) 4                
(b) 3               
(c) 2                
(d) 1

Inter (Part-II) Gujrawala Board 2015
Mathematics
Part I (Subjective)
Time Allowed: 2.30 Hours 
Max. Marks: 80 

Section I

Q2.write short answers to any eight questions:                         (2*7=16)

• Find f-1(x), if f(x) = 2+. Also write domain and range of f-1(x).
• Evaluate
• Prove that -1=
• Find  if y2-xy-x2+4=0.
• Prove that  
• Find y4 if y=
• Find y4 if y = cos3x.
• Expand ax in Maclaurin series.
• Prove that derivative of a constant is zero.
• Define point of inflexion.
• Determine the interval in which f is increasing if f(x) = x3-6x2+9x.
• Differentiate cos with respect to x.

Q3. Write short answers to any EIGHT questions:      (2x8=16)

• Use differentials to approximate the value of (31)1/5
• Evaluate:  dx
• Evaluate: dx.
• Evaluate:  dx.
• Evaluate: 
• Evaluate: 
• Find area between x-axis and the curve y=sin2x from x=0 x=
• Solve. +=x
• Evaluate:
• Solve:
• Graph the inequalities 2x + y  2 ; x + 2y 10, x  0, y  ()
• Indicate solution set of linear inequality 2x + 3y  12 by shading.

Q4. Write short answers to any NINE questions:  (2x9=18)

• Transform the equation 5x - 12y + 39 = 0 into slope – intercept form.  
• Find the slope and angle of inclination of the line joining the points (4, 6) and (4, 8).
• Find the point of intersection of the lines x - 2y + 1 = 0 and 2x- y + 2 = 0
• Find the interior angles of the triangle whose vetices are A(2 -5), B(-4, -3), C(-1, 5).
• Find the equation of the lines represented by 20x2 + 17xy -24y2= 0
• Find the centre and radius of the circle x2+ y2- 6 x + 4y + 13 =0.
• Write the equations of tangent and normal to the circle x2 + y2 =25 at (4, 3).
• Write an equation of parabola with focus (-3, 1) and directrix x= 3.
• Find the equation of hyperbola with centre (0, 0), focus (6, 0) and vertex (4, 0).
• Prove that sin (α+β)=sin αcos β+cos αsin β
• Prove that +)+()+
• Find the volume of tetrahedron whose vetices are A(2, 1, 8), B(3, 2, 9), C(2, 1, 4) and D(3, 3, 10).
• Find the value of [

(Section-II)

Question #5
(a) Evaluate the limit                                    
(b) If y = x4+ 2x2+ 2, prove that                            

Question #6
(a) Evaluate   
(b)Find the equation of line through the intersection of two lines x+ 2y + 3 = 0 and 3x+4y+ 7 = 0 and making equal intercept on the axes.

Question #7
(a) Evaluate tdt
(b) Maximize z = x + 3y Subject to 2 x + 5y 30
5x + 4y  20
x 0, y 0

Question #8
(a) Find the equations of tangents to the circle x2 + y2 = 2 parallel to the line x - 2y + 1 = 0
(b) Prove that the line segment joining the mid - points of two sides of a triangle is parallel to third side and half as long.

Question #9
(a) Find focus, vertex and directrix of parabola x2- 4x - 8y + 4= 0.
(b) Use vector method to prove sin (