Inter (Part-II) Gujrawala Board 2016
Mathematics
Part II (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.

• 2 i x (2 k x i) =:

(A) 4 k
(B) -4
(C) 0
(D) 4 i

• Focus of the parabola x2 = -4ay:

(A) (a,o)
(B) (o,-a)
(C) (o,a)
(D) (-a,o)

• x=-3 is the solution of the inequality:

(A) 2x-1>0
(B) 2x+1>0
(C) x+4<0
(D) 2x-1<0

• The lines represented by ax2+2 h xy + by2=0 are orthogonal if:

(A) a-b=0
(B) a+b=0
(C) a+b>0
(D) a+b0

• Solution of differential equation = cosec x cot x is y=

(A) cosec x + c
(B) – cosec x + c
(C) cot x + c
(D) – cot x +  c

• sec22 x dx=

(A)  tan 2x
(B) tan 2x
(C)  tanx
(D) 2 tan 2x

• dx

(A) x
(B)  
(C)
(D) ℓn(ℓnx)

• The minimum value of the function f(x) = x2 + 2x - 3 is at=x

(A) -3 
(B) 1
(C) 0   
(D) -1

• If y = x-, then =

(A) 1+ 
(B) 1- 
(C) 1+           
(D) 1-           

(A) 1  
(B) 0
(C)
(D)

• Projection of the vector r = ai + bj + ck on x-axis is:

(A) a   
(B) b   
(C) c   
(D)

• The equation Ax2 + By2 + Gx + Fy + C = 0 represents a circle if:

(A) A = 0, B0          
(B) AB
(C) A=B 0   
(D) C = 0

• The distance of the line 3x + 4y + 10 = 0 from origin is:

(A) 0   
(B) 1
(C) 2   
(D) 10

• esinx .cos dx =

(A) ecosx         
(B) esinx
(C) -sin x esin x
(D) - sin x ec°sx

• If v = x3, then differential of v is:

(A) 3x3
(B) 3x2dv
(C) X3dv        
(D) 3x2dx

• Slope of the line bisecting 2nd and 4th quadrant is:

(A) π 
(B) 0
(C) -1  
(D) 1

• [ℓn(ℓnx)]=

(A)   
(B) 
(C)
(D)           

•  (cosh-1 x) =

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

• If f(X)=tan x, then f’=

(A) 1   
(B)
(C) 2
(D)

(A) sin x
(B) cos x
(C) cos hx
(D) sin hx

Inter (Part-I) Gujrawala Board 2016
Mathematics
Part I (Subjective)
Time Allowed: 2.30 Hours 
Max. Marks: 80 

Section I

2. Attempt any EIGHT parts.                                                                      (2x8=16)

• What is linear function? What is its graph?
• Evaluate:
• Discuss continuity of f(x) =
• Differentiate
• If x = at2, y = 2at. Find .
• Differentiate x2 -  w.r.t.x2.
• Prove that  (tan-1 x)=
• Prove that (sinh-1x)=
• If y=x2ℓn. Find .
• If x= sin θ, y = sin (m θ). Find .
• Apply Maclaurin series to expand cos x =1-
• Differentiate cos2 x w.r.t. sin2 x.

3. Attempt any EIGHT parts.                                                                 (2x8=16)

• Using differential find  ,when x2 + 2y2 = 16.
• Evaluate
• Find
• Evaluate dx
• Find xℓnxdx
• Evaluate e2x (-sin x + 2 cos x) dx
• Compute  (x2 +1) dx
• Calculate  sec x(sec x + tan x) dx
• Find the area between x-axis and the curve y = 4x - x2.
• Solve the differential equation sin y cosec x  = 1
• Define objective function.
• Graph the solution region of 2x + y > 2.

4.  Attempt any NINE parts.                                                                (2x9=18)

• Find h if A(-1, h), B(3, 2), C(7, 3) are collinear.
• Find the point three fifth of the way along the line segment from A(-5, 8) to B(5, 3).
•  Find equation of straight line if its slope is 2 and y-intercept is 5.
•  Transform 5x - 12y + 39 = 0 in two intercept form.
• Find angle between the lines represented by 2x2 + 3xy-5y2 =0.
• Find the centre and radius of the circle given by the equation 4x2 + 4y2 - 8x + 12y - 25 = 0.
• Find equation of tangent to the circle x2 + y2 = 2 parallel to the line x - 2y + 1 =0.
• Find focus and directrix of parabole x2 = - 16y.
• Find equation of ellipse with vertices (0,± 5) ,eccentricity .
• Find position vector of a point which divide the pin of P and Q with position vectors 2i – 3j and 3i + 2j in ratio 4:3.
• Find a and b so that the vectors 3i - j+ 4k and ai + bj + 2k are parallel.
• Find cos of the angle betwee u.v If u = 3i +j-k,v = 2i -j-k.
• Find a unit vector perpendicular to the plane of vectors a = 2i -j + 4k, b = -i+j-2k.

SECTION-II

NOTE: - Attempt any three questions. 10 x3=30

Question #5.   
(a) Evaluate the                                                                      (5)
(b) Show that                                                      (5)

Question #6.  
(a) Evaluate dx   (5)
(b) Find an equation of the 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           (5)

Question #7
(a) Evaluate (x3+3x2) dx                                                                                                       (5)
(b) Graph the feasible region subject to the following constraint:                  (5)
2x-3y<6
2x+3<12
x> 0,                y>0

Question #8.  
(a) Prove vectorically that in any triangle ABC, a2 = b2 + c2 - 2bc cos A            (5)
(b) Find equation of circle passing through A(4, 5), B( -4, -3), C(8, -3)               (5)

Question #9.   
(a) Show that the equation 9x2 _ 18x + 4y2 + 8y - 23 = 0 represents an ellipse.  (5)
(b) Using vectors prove that the points A(-3, 5, -4), B(-1, 1, 1), C(-1, 2, 2) and D(-3, 4,    -5) are coplaner. (5