VIBRANT ACADEMY MATHS

Practice Problems IRP

(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-1
Single Choice Questions :
1. If a, b, c are 3 consecutive positive multiples of 5, then the variable line ax – by + c = 0 passes through
the fixed point
(A) (1, – 2) (B) (1, 2) (C) (2, 1) (D) (2, –1)

1
2. If sin x + cos x = , 0 < x < , then tan x =
5

4 4 4 4 3
(A) only (B) – only (C) (D) – or –
3 3 5 3 4

3. Maximum integral value of n so that sin x (sin x + cos x) = n has at least one solution is
(A) 2 (B) 1 (C) 3 (D) 0

1 
4. The value of tan –1  tan 2A  + tan–1(cot A) + tan–1 (cot3A), for 0 < A < is
2  4

(A) tan–1 2 (B) tan–1(cot A) (C) 4 tan-1 (1) (D) 2 tan–1 2

5. The greatest value of (sin–1x)3 + (cos–1x)3 is

133 73 3 53

(A) (B) (C) (D)
32 8 8 16

4
6. In a triangle ABC, a = 3 and b = 6 and cos (B – A) = . The area of the  is
5

9
(A) 9 (B) 18 (C) 9 2 (D)
2

7. In a ABC, the measures of angles A and B (A > B) satisfy the equation 3 sin x – 4 sin3 x – k = 0, 0 < k <

1. Then the measure of C 

  2 5
(A) (B) (C) (D)
3 2 3 6

Multiple Choice Questions :

8. A (1, 3) and C (5, 1) are two opposite vertices of a rectangle ABCD. If the slope of BD is 2, then the
coordinates of B can be
(A) (4, 4) (B) (5, 4) (C) (2, 0) (D) (1, 0)

9. If (x, y) be a variable point on the line y = 2x lying between the lines L1 : 2x + y + 2 = 0 and
L2 : x + 3y – 3 = 0 then

 1 3  1 6  3  6
(A) x   – ,  (B) x   – ,  (C) y   – 1,  (D) y   – 1, 
 2 7  2 7  7  7

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10. Two adjacent sides AB and BC of a cyclic quadrilateral are 5 and 2 and the angle between them is 60º.

The area of the quadrilateral is 4 3 . If x and y are the lengths of AD and CD, then

4 3 –5 19
(A) xy = 6 (B) xy = (C) x + y = 5 (D) area of circumcircle = 
2 3

 3
11. If  and satisfies the equation 2cos2 + sin 2, then
2 2

  5   3   2 3    
(A)    ,  (B)   ,  (C)    ,  (D)    , 
2 6   2  3 2 2 

Subjective :
12. In a ABC, the minimum value of 4 (cos2 A + cos2 B + cos2C) is

2 
13. The value of 7 

  
sin –1 cos cos –1(cos x )  sin –1(sin x ) where
2
< x <  is

14. In a ABC, the median to the side BC is perpendicular to AB. Then tan ( – A) cot B =

7–n
15. If A, B, C are acute angles, A + B + C =  and the maximum value of cot A cot B cot C = 27  2 , then n =


16. If the coordinate axes are turned through an angle about oriign, 0 <  < and the equation of the curve
2
x2 + 4xy + y2 = 0 becomes AX2 + BY2 = 0 then A =

17. The vertices of a triangle are origin, (4, –2) and (1, – 3). If (, ) is its orthocenter, then  –  =

Match the Column :

18. Consider the lines L1 : x + 4y – 5 = 0 ; L2 : 3x + 8y – 11 = 0 ; L3 : x + ky – 2 = 0.
Column I Column II
(A) L1, L2, L3 are concurrent if k = (P) 4
(B) One of L1, L2, L3 is parallel to (Q) 8/3
at least one of the other two if k =
(C) L1, L2, L3 form a triangle if k = (R) –1
(D) L1, L2, L3 do not form a triangle if k = (S) 1
(T) 3

19. Column I Column II

Equation Number of solution in the interval [0, 2]
(A) 4 sin2 x – 4 |cos x| – 1 = 0 (P) 1
(B) sin 2x = cos 3x (Q) 4
(C) tan2x + cot2 x = 2 (R) 6
(D) cot x + tan x = 2 cosec x (S) 5
(T) 2

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-2
Single Choice Questions :
 
1. If x, y  0,  and sin4 2x + cos4 2y + 2 = 4 sin 2x cos 2y, then the value of sin 2x + cos 2y is
 4
3 7
(A) 1 (B) (C) (D) 2
2 4

2. The solution of the equation 3 tan (x – 15°) = tan (x + 15º) is

  
(A) (8n + 1) (B) (4n + 1) (C) (8n – 1) (D) None of these
4 4 4
(where, n  )

3. The sum to infinity of the series cot–1 (3) + cot–1 (7) + cot–1 (13) + ...... belongs to

     
(A) 0,  (B)  3 , 2  (C) 0, 4  (D) None of these
 6    

4. If the distances of the sides BC, CA, AB of ABC from its circumcentre are d1, d2, d3 respectively then
a b c
the value of   is
d1 d2 d3

abc 1 abc 1 abc 1 abc

(A) d d d (B) 2 d d d (C) 3 d d d (D) 4 d d d
1 2 3 1 2 3 1 2 3 1 2 3

5. With the usual notation in ABC, it is given that b = c and the circumradius of the triangle is equal to b,
a
then the value of is
b
1
(A) 2 (B) 2 (C) 3 (D)
2
6. The slope of the straight line joining the orthocenter, circumcentre, centroid and the incentre of the triangle
formed by the lines whose equation is x + y = 2a and x and y axes where a is positive is

1 2 a
(A) (B) (C) (D) None of these
2 3 3
7. In an isosceles triangle ABC, the points B and C of the base BC are (3, 2) and (2, 3) respectively. If the
equation of the line AB is 2x – 3y = 0 then A is

3 5  5  2

(A) sin–1   (B) sin –1   (C) sin–1  13  (D) cos–1  3 
4 7    

 4x – 4x3   2x 
 
8. If  = tan–1  1 – 6x 2  x 4  and = 2 sin–1  1  x 2  then  – =  if
 

(A) – < x <  (B) < x < 1 (C) 0 < x < 1 (D) none of these


(where,  = tan )
8

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Multiple Choice Questions :
9. A straight line through the point P(3, 4) making an angle , 0 <  < /2, with the positive direction of the x-
axis meets the lines x = 12 and y = 8 at Q and R respectively then
4 PQ 9 PQ
(A) the slope of the line is (B) slope of the line is
9 PR 4 PR
2(9 sin   4 cos  ) 2(9 cos  – 4 sin  )
(C) PQ + PR = (D) PQ – PR =
sin 2 sin 2

10. With the usual notation in ABC, it is given that r1, r2, r3 are in H.P., the perimeter of the triangle is 24
units and its area is 24 sq. units then
(A) r2 + r = 8
(B) a and c are the roots of the equation t2 – 16t + 60 = 0
(C) a and c are the roots of the equation x2 – 16x + 63 = 0
(D) r2 + r = 10
sin 6x
11. A solution of the equation 8 cos x cos 2x cos 4x = in (0, ) is
sin x
 3 2 9
(A) (B) (C) (D)
14 14 7 14

12. A solution of the equation cot–12 = cot–1 x + cot–1 (10 – x) where 1 < x < 9 is
(A) 7 (B) 3 (C) 2 (D) 5
Comprehension (Q.13 to Q.15)
Given the curve whose equation is x2 + y2 – 4x – 4y + 3 = 0 ....(1)
and the straight line whose equation is x + y = k ....(2)
13. The sum of the possible values of k such that the lines joining the origin to the points of intersection of (1) and
(2) subtends a right angle at the origin is
(A) 0 (B) 3 (C) 4 (D) None of these
14. If k = 2 then the acute angle between the lines joining the origin to the points of intersection of (1) and (2) is

(A) tan –1  21 (B) tan –1 2 6  (C) tan –1  7 (D) tan –1  3

15. If k = 1, then the length of the intercept made by (2) on (1) is
(A) 2 (B) 3 (C) 2 (D) 3
Subjective :
16. The number of solutions of the equation sin3 + sin2 + cos3  + cos2 = 0 in [0, 3] is
B C
17. If in  ABC, A, B, C are in A.P. and a, b, c are in G.P. then the value of tan A tan + tan B tan + tan C
2 2
tan A/2 is

18. The number of solutions of cos 3x tan 5x = sin 7x in 0  x  is
2
19. If the image of the point A(k, k –1) on the line whose equation is 3x + y = 6k is the point B(k2 + 1, k), then the
abscissa of the point B is
20. A straight line through the point (4, 3) meets the x and y axes in A and B respectively. If the locus of the point
of intersection of the line through A parallel to y axis and the line through B parallel to x-axis is
x + my = xy then the value of  + m +  is _______
(where , m, are coprime natural numbers)

  b a  c b
21. If in  ABC, B  then the value of 1    1  –  is _______
3  c c  a a
22. A is the point which is equidistant from each of the points (0, 0) (2, 0) and (0, 4). B is the point through which
all the members of the family of lines represented by (4x – y – 18) +  (2x + y – 12) = 0 pass. Then the length
of AB is _______ .
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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-3
Single Choice Question :

1. If the coordinates of two consecutive vertices of a regular hexagon are (2, 0) and (4, 2 3 ) , then the equation
of the circumcircle of the hexagon which contains the origin is

(A) x 2  y 2 – 4 3 y – 4  0 (B) x 2  y 2  4 3 y – 4  0

(C) x 2  y 2  4 3 x – 4  0 (D) x 2  y 2 – 4 3 x – 4  0

3 8
2. The radius of the circle which touches the parabola 15y2 = 64x internally at  ,  and also touches the X-
5 5
axis is
(A) 2 (B) 4 (C) 1 (D) 3

3. P1Q1 and P2Q2 are two focal chords of the parabola x2 – 4y = 0. Then the chords P1P2 and Q1Q2 intersect on
the
(A) Y-axis (B) X-axis (C) line y = –1 (D) line y = –2

4. A circle has its centre at one end B of the minor axis and passes through S1 and S2 the foci of the ellipse
9x2 + 16y2 = 144. The area of the sector S1BS2 of the circle is


–1 3 7
  
–1 3 7 
3 7  
–1 3 7

   
(A) 8 sin  8  (B) 16 sin  8  (C) sin–1  2  (D) 8 sin  16 
       

x2 y2 x2 y2
5. The angle between the common tangents to   1 and   1 is
a2  b2 b2 a2 a2  b 2
(a > 0, b > 0)

–1 a –1 b –1 a –1 b
(A) 2 tan (B) 2 tan (C) tan (D) tan
b a b a

6. The distance between the foci of a hyperbola is 16 and its eccentricity is 2 . The length of the latus rectum
is

(A) 2 (B) 1 (C) 4 2 (D) 8 2

One or more than one correct :

7. If a circle C1 : x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord
3
is of maximum length and has a slope , then the centre of C2 is
4

9 12   9 12   12 9   12 9 
(A)  , –  (B)  – ,  (C)  ,–  (D)  – , 
5 5   5 5   5 5  5 5

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8. PSQ is a focal chord of the parabola y2 = 8x such that SP = 6. SP is inclined at an acute angle with the X-
axis. Then

(A) SQ = 3 (B) SQ = 4 (C) P  (4, 4 2 ) (D) Q  (1, – 2 2 )

 x2 y2
9. A tangent is drawn at a point ,  (0, ) of the ellipse   1 . The least value of the sum of the
2 27 1
intercepts on the coordinate axes by this tangent is attained at  =

   
(A) (B) (C) (D)
6 8 4 3

Comprehension (Q.10 to Q.11)

The normals at (xr, yr), r = 1, 2, 3, 4 on the rectangular hyperbola. xy + 144 = 0 meet at Q(h, k).
10. The sum of the ordinates of the four points (xr, yr) is equal to

1
(A) (B) k (C) –k (D) 2k
k

11. The product of the ordinates of the four points (xr, yr) is
(A) –122 (B) 124 (C) –124 (D) 122

Comprehension (Q.12 to Q.14)

Consider the parabola defined parametrically as x = t2 + t + 1, y = t2 – t + 1.
12. The focus of the parabola is

 2 2   1 1 

(A) 1  2, 1  2  
(B) 1  4 , 1  4 
 
(C) 1 
 2
, 1
2
 (D) None of these

13. The equation of the directrix is

1  1 
(A) x  y – 0 (B) x  y –  2 –   0
2  2

1 3
(C) x  – (D) x + y =
2 2

14. The tangent at the vertex meets the axes of coordinates at A and B. The area of the OAB where O is the
origin is
(A) 1 sq. unit (B) 4 sq. unit (C) 2 sq. unit (D) None of these

Subjective :
15. If p and q be the greatest and shortest distance respectively of P(–7, 2) from any point (, ) on the curve
x2 + y2 – 10x – 14y – 51 = 0, the integer nearest to pq is ________.

16. From a point A(1, 1) on the circle x2 + y2 – 4x – 4y + 6 = 0, two equal chords AB and AC of length 2 units are
drawn. The sum of the intercepts made by BC on the X and Y axis is _________.

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17. The locus of the centre of the circle (x cos  + y sin  – 3)2 + (x sin  – y cos  – 4)2 = 7,
(where  is a parameter) is a circle. Its radius is ________.

18. If the normals at the ends of the latus rectum of the parabola y2 = x meet the parabola again at A and B, then
AB = ________.

x2 y2
19. The locus of the midpoints of the portion of the tangent to   1 included between the X and Y-axes is
25 16
25 16
2
   , where  = ––––––.
x y2

2 2
20. If 2x2 + 3y2 – 8x – 18y + 35 – k = 0 represents an ellipse of whose latus rectum is of length , then k =
3
______.

3 
21. The point P , 2 5  is joined to the foci S1 and S2 of the hyperbola 16x2 – 9y2 + 144 = 0. (y coordinate of
2 
S1 > y coordinate of S2). Then PS2 – PS1 = _______

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-4
Single Choice Questions :
1. If  is the angle subtended between the tangents to circle x2 + y2 + 8x + 4y + 4 = 0 from the point (1, 1) then
the value of sec is :
(A) 17 (B) 18 (C) 33 (D) none of these

2. From any point on the tangent to y2 = 4x at (4, –4), tangents are drawn to the circle x2 + y2 = 1. Then the
chords of contact pass through a fixed point

 1 1  1 – 1
(A) (0, 0) (B) (1, 0) (C)  – , –  (D)  – , 
 2 4  4 2 

3. If the tangent from a point P to the circle x2 + y2 = 1 is perpendicular to the tangent from the same point P to
the circle x2 + y2 = 4 then the locus of P is :
(A) a straight line (B) a circle (C) an ellipse (D) a hyperbola

4. A parabola is drawn with the vertex at (0, –3), the axis of symmetry is along the conjugate axis of the

x2 y2
hyperbola –  1 and to pass through the foci of the above hyperbola, then the focus of the parabola is
49 9
:

 11   11   11   11 
(A)  0,  (B)  0,–  (C)  0,  (D)  0,– 
 6  6  12   12 

B C
5. In ABC, B is (3, 0) and C is (9, 0). The vertex A moves in such a way that cot . cot = 4 is satisfied,
2 2
then the locus of A is the conic whose eccentricity is :

2 3 4 5
(A) (B) (C) (D)
5 5 5 4

6. If a and b are the lengths of the intercepts made by two perpendicular tangents to the ellipse 4x2 + 9y2 = 36
on its auxiliary circle, then the value of a2 + b2 is :
(A) 15 (B) 25 (C) 20 (D) 30

7. A tangent to the ellipse x2 + 2y2 = 1 meets its director circle at the points A and B, then the product of the
slopes of OA and OB, where O is the origin is :

1 1 1
(A) (B) – (C) – (D) 1
4 4 2


8. If the normals at “” and “” where + = to 16x2 – 9y2 = 144 meet at the point (, ) then is :
2

9 2 4 2 5 2 3 2
(A) – e (B) e (C) – e (D) – e
4 9 4 4
where e is the eccentricity of the above conic.

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Multiple Correct Answer(s) Type :
9. The centre of the circle passing through the points (0, 0) and (1, 0) and touching the circle x2 + y2 = 4 is :

1 3 1 – 3 1 3 1 – 3
       
(A)  2 , 2  (B)  2 , 2  (C)  2 , 4  (D)  2 , 4 
       

10. Three distinct normals are drawn to the parabola 2y2 – 16y – x + 35 = 0 through the point (, 4) then can be

9 3
(A) 2 (B) 4 (C) (D)
2 2

( x  y – 2)2
11. The equation to the tangent at an end of the major axis of the ellipse whose equation is +
9

( x – y )2
= 1 is :
4
(A) x + y – 5 = 0 (B) x + y – 3 = 0 (C) x + y + 1 = 0 (D) x + y + 2 = 0

12. A chord of the hyperbola whose equation is x2 – 2y2 = 1 is bisected at the point (–1, 1). Then

1
(A) the slope of the chord is –
3

1
(B) the area of the triangle formed by the chord with the coordinate axes is sq. units
4
(C) the length of the intercept made by this chord on the given hyperbola is 4 units

 1
(D)  2,–  is a point on this chord
 2

Subjective :
13. From a point A outside to a circle of radius k, tangents AP and AQ are drawn touching them at P and Q such

1 1 1
that 2 + 2 = , then the length of the chord PQ is ............ .
k AP 9

14. The number of possible integral values of k such that the two circles x2 + y2 = 4 and x2 + y2 – 6x – 8y + k2 =
0 have exactly two common tangents is ...................

15. The normal at a point P of the parabola y2 = x meets the curve again at Q, then the least distance of Q from
the tangent at the vertex is .................

16. If the area of the quadrilateral formed by the tangents at the ends of the latera recta of the ellipse

x2 y2 3A
 = 1 is A, then the value of is .................
16 7 16

x2 y 2
17. P is any point on the ellipse   1 . P is the corresponding point on its auxiliary circle. If the locus
9 4
of a point which divides PP internally in the ratio 1 : 2 is an ellipse with the lengths of semi major and semi
a  3b
minor axis as a and b respectively, then the value of is ..............
2

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 x2 y2
18. The tangent at any point P on the hyperbola –  1 meets the lines 3 x – 2y = 0 and 3 x + 2y = 0
4 3
at Q and R, and C is the centre of the hyperbola, then the value of CQ · CR is .................... .

19. Column-I Column-II

(A) The angle subtended by the chord x + y + 1 = 0 on the major segment


of the circle x2 + y2 – 4x – 2y – 11= 0 is (P)
6

(B) AB is a chord of the parabola y2 = 4x of length 8 3 units. The chord

is parallel to the tangent at the vertex. Then angle subtended by AB at


the vertex is (Q)
2
(C) If the angle between the tangents at (a cos, bsin ) and

x2 y2 
(–asin, bcos) to 2
 2
 1 is minimum, then the value of  is (R)
a b 3
(D) Tangents are drawn from a point on the circle x2 + y2 = 8, to the hyperbola

x2 y2 
–  1 , then the angle between the two tangents is (S)
17 9 12


(T)
4

20. Column-I Column-II

(A) P(1, –2) and Q(4, 4) are points on the parabola (P) 6x – 8y + 25 = 0
y2 = 4x whose focus is S. If PS and QS meet
the parabola again at P and Q, then the equation
of PQ is
(B) The equation to the directrix of the ellipse (Q) 2x – y + 1 = 0
2 2
9x + 5y – 30y = 0 which is closer to (0, 6) is
(C) The extremities of a diagonal of rectangle are (0, 0) and (4, 3). (R) 2x + y + 1 = 0
The equation of the tangent to the circum circle of the
rectangle which is parallel to the above diagonal is (S) 4x – y – 2 = 0
(D) The equation of the common tangent to the curves y2 = 8x and
3x2 – y2 = 3 is (T) 2y – 15 = 0

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-5
Single Choice Questions :
1. If f(x) = cos |x| – 2ax + b be a non-decreasing function for all x  R, then

1 1 3
(A) a  – (B) a = b (C) a = b (D) a > –
2 2 2

x
2. The function f(x) = has
1  x tan x
   
(A) one point of minimum in  0,  (B) one point of maximum in  0, 
 2  2

   
(C) no point of maximum or minimum in  0,  (D) 2 critical points in  0, 
 2  2

2 – x , – 3  x  0
3. Let f(x) = 
x – 2 , 0  x  4
Then f–1 (x) is discontinuous at x =
(A) 0 (B) 2 (C) 3 (D) 1

x
4
4. If (x) =  sin
0
 d , then (x + ) is equal to

(A) (x) + () (B) (x) · () (C) (x) (D) (x) – ()

x 3 x5
 x
 ........ dx – dy
5. The solution of the differential equation 3! 5! = is :
x2 x4 dx  dy
1   ........
2! 4!

(A) 2ye2x = Ce2x + 1 (B) 2ye2x = Ce2x – 1 (C) ye2x = Ce2x + 2 (D) ye2x = Ce2x – 2

Multiple correct Answers Type :

6. The point on the curve y2 – 2y – 4x + 1 = 0 which is nearest to (3, 1) is :

1 
(A) (1, 3) (B) (1, – 1) (C)  ,2  (D) (0, 1)
4 

x (cos x  sin x )
7. If e dx  Ae x sec Bx + C, then
1  cos 2x
1
(A) B = 2 (B) B = 1 (C) A = (D) A = 1
2

x2
dy
8. If f(x) =  (log y ) 3
, x  1, then f(x) is :
x
(A) monotonically increasing in (4, ) (B) monotonically increasing in (2, )
(C) monotonically decreasing in (1, 3) (D) monotonically decreasing in (0, 4)
9. A focus of the curve which satisfies the differential equation (1 + y2) dx – xy dy = 0 and which passes through
(1, 0)
(A) (1, 0) (B) (0, 2 ) (C) ( 2 , 0) (D) (– 2 , 0)
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Comprehension : Q.10 to Q.11
Let f(x) = [x] + {x}2 where [.] and {.} are respectively the greatest integer function and the fractional part
function.
10. The area between y = f(x) and y = f–1(x) in the interval [0, 1) is :

2 4 1
(A) (B) (C) (D) 1
3 3 3
11. The area between y = f(x) and y = f–1 (x) in the interval [1, 2] is :

4 1 1 2
(A) (B) (C) (D)
9 9 3 3

Comprehension : Q.12 to Q.14

Let f : R  R be a continuous and differentiable function such that f(x + y) = f(x) · f(y)  x, y, f(x) 0 and
f(0) = 1 and f(0) = 2.
Let g(xy) = g(x) · g(y)  x, y and g(1) = 2 ; g(1) 0
12. Then
(A) f(2) = e4 (B) f(2) = 2e2 (C) f(1) < 4 (D) f(3) > 729
13. Then
(A) g(2) = 2 (B) g(3) = 3 (C) g(3) = 9 (D) g(3) = 6
14. The area between the tangent to the curve y = f(x) at the point whose abscissa is zero and the curve y = g(x)
is :

28 2 8 2
(A) 8 2 (B) (C) 6 2 (D)
3 3

Subjective :

n
x  
–1  x 
15. If  8 – x3
dx  k sin
 2  + C, then n =
 

  1 
16. The area bounded by curves y = 6  x    , y2 – 18x + 18 = 0 and 6x – 5y – 6 = 0, (where [.] denotes the
  x 

greatest integer function) is A, then (A – 4) is equal to :

17. Area of the region between the curves y + 2x2 – 6x = 0 and y – 2x2 + 6x = 0 is just less than the area of the
smallest rectangle enclosing the region by .......... .
18. The maximum value of the function f(x) = 2x3 – 9x2 + 12x – 25 on the set A = {x/x2 + 12  7x} is ............ .

19. If f(x + y + 1) =  f (x)  f(y)  2

 x, y  R and f(0) = 1, then f(2) = .............. .

 2x 
20. If the domain of f(x) = 12 – 3 x – 3 3 – x + sin–1  3  is [a, b], then a = ..........
 

21. The tangent other than the X-axis from (2, 0) to the curve y = x3 touches the curve at P. The abscissae of P
is ................ .

 3 2 1
 2 3  d2
22. If matrix A = 6x 2x x 4  , then (|AadjA|1/3) at x = a is :
2 dx 2
 1 a a 

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-6
Single Choice Questions :

(140 ) x – ( 35) x – ( 28) x – ( 20 ) x  7 x  5 x  4 x – 1

1. The value of lim is
x 0 x sin 2 x

(A) n 140 (B) 0 (C) n 2 n 5 n 7 (D) 2 n 2 n 5 n 7

2. If f(x) is continuous in [a, b] and differentiable in (a, b), then there exists at least one c in (a, b) such that

f (b) – f (a)
is equal to
b2 – a2

f ' (c )
(A) (B) 2c f(c) (C) f(c) f(c) (D) None of these
2c

x
3. 
If f(x) = sin x – ( x – y ) f ( y ) dy , then the value of f(x) + f(x) is
0

(A) 0 (B) – sin x (C) sin x (D) cos x

4. If g(x) > 0 and g(1) = 0 such that f(x) = g (cot2x + 2 cot x + 2) where 0 < x < , then the interval in which f(x)
is decreasing is

 3   3   
(A)  0,  (B)  ,   (C)  ,   (D) (0, )
 4  4  2 

1
x 1/ 4
5.  1
0
x
dx is equal to

 8 8 8 8
(A)  (B)   (C) 4  (D)  –
4 3 3 3 3

( x 2 – 2)
6. If x dx is equal to – 1 log u  u 2  1  C , then u is
8 4 2 2
x  x – 2x  1

1 1  1 1   1 1 
(A) 2
– 4 (B) 2  4 – 2  (C) 2  3 – 2  (D) None of these
x x x x  x x 

2x
7. If f : [0, ) [0, ) and f(x) = , then f is :
3  2x
(A) one-one and onto (B) one-one but not onto
(C) neither one-one nor onto (D) onto but not one-one

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8. The tangent at any point P on the curve y = f(x) meets the x and y axes at L and M respectively and the
normal at P meets these axes at Q and R respectively. If the centre of the circle through O, Q, P and M,
where O is the origin lies on the line whose equation is y = 2x then the differential equation of the curve is

dy x  2y dy y – 2x dy 2x  y dy x  y
(A) dx  y – 2x (B) dx  x  2y (C) dx  2x – y (D) dx  x – y

One or more than one correct :

a(1 – x sin x)  b cos x  7

9. Given f(x) = if x  0
x2
5 if x  0
1
  cx  dx  3  x
 1    if x  0
  x 
2


such that f(x) is continuous at x = 0 then,

(A) a = –3 (B) b = –4 (C) c = 1 (D) d = loge5

x
10. If f(x) =  | x  3 | dx , then
–4

(A) The right hand derivative at x = – 4 is 2 (B) The right hand derivative at x = –3 is zero
(C) f(x) is continuous at x = –3 (D) f(x) is not continuous at x = –3


4
n
11. If un =
 tan
0
x d{ x – [ x]} where [] denotes the greatest integer function then

4 3
(A) u0 + 2u2 +u4 = (B) u1 + 2u3 + u5 =
3 4

2 3
(C) u2 – u6 = (D) u0 + u1 + 2u2 + u3 + u4 =
15 2

12. If f(x + y) = f(x) + f(y) – 2xy  x, y  R and f(0) = 4 then

(A) A turning point of the curve y = f(x) is (2, 4)
(B) A turning point of the curve y = f(x) is (1, 3)

3
(C) The length of the sub tangent of the curve y = f(x) at x = 1 is
2

5
(D) The area of the curve y = f(x) bounded by x-axis between x = 0 and x = 1 is
3
Subjective :
13. The number of elements in the range of the function

 2 5  2 4
y = sin–1  x   + cos–1  x – 9  where [ ] denotes the greatest integer function is
 9  

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14. If f(x) + 2f(1 – x) = x2 + 2  x  R and f(x) is a differentiable function, then the value of f(8) is

15. Let f(x) = signum (x) and g(x) = x(x2 – 10x + 21), then the number of points of discontinuity of f[g(x)] is

d2  sin 4 x  sin 2 x  1 
16. If = a sin2x + b sin x + c then the value of b + c – a is
dx 2  sin2 x  sin x  1 


4
dx
17. If I = , then the integral part of I is
 sin
0
1/ 2
x  cos7 / 2 x

18. The degree of the differential equation of the family of circles touching the lines y2 – x2 = 0 and lying in the first
and second quadrant is

Match the column :

19. Match Column I and Column II
Column-I Column-II
 2 tan x 
(A) If f   (P) [5, )
 1  tan 2 x 

(sec 2 x  2 tan x )
 (cos 2x  1) ,
2
then the range of y = f(x) is

x 4 – x 2 – 2( x – 1)
(B) If f(x) = , then (Q) [6, )
x 4 – x 2 – 2( x – 2)
range of f(x) is
(C) If f(x) = 3 – x if x < 0 and (R) (1, 2]
= 3 + x if x  0, then the range of f{f(x)} is
(D) If f(x) = 4x + 93x + 2–2x + 3–6x + 1, (S) [0, 1)
then the range of f(x) is
(T) [0, 2]

20. Column-I Column-II

Cartesian Equation Differential Equation

e 3t d2 x dx
(A) x = Aet + Be–t + (P) –3  2x  e 3 t
8 dt 2 dt

e 3t d2 x dx
(B) x= Aet + Be2t + (Q) 2  x  e3t
2 dt 2 dt

e 3t d2 x dx e3t
(C) x = A cos t + B sin t + (R) 2
2 x
10 dt dt 2

e 3t d2 x
(D) x = (A + Bt) e–t + (S) – x  e3t
16 dt 2

d2 x
(T)  x  e3t
dt 2

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-7
Single Choice Question :
1 1  2i – 5i
1. If D  1 – 2i – 3 5  3i then
5i 5 – 3i 7
(A) D is purely real (B) D is purely imaginary

(C) D  D  0 (D) None of these

2. If  1 is a cube root of unity, then the sum of the series S = 1 + 2 + 32 + ...... + (3n) 3n – 1 is

3n 3nn 3n
(A) (B) (C) (D) 0
1–  1–  –1

3. The number of 5 digit numbers that can be formed with 0, 1, 2, 3, 7, 8 (with no repetition of digits) which are
divisible by 3 and not ending in 0 is
(A) 216 (B) 192 (C) 120 (D) 96

4. The number of triplets x, y, z  N, x < y < z such that x + y + z = 100 is

(A) 784 (B) 1617 (C) 4851 (D) 4704

5. An examiner wishes to assign 30 marks to 8 questions, giving not less than 2 marks to each question. The
number of ways this can be done is
(A) 15000 (B) 116280 (C) 123750 (D) 116200

9
 x 2 
6. In the expansion of  2x – , the sum of the two middle terms is
 4 

63 14 63 14 63 13 63 13
(A) x (x  8) (B) x ( x – 8) (C) x ( x – 8) (D) x (8 – x )
32 32 32 32

 sin  – cos  0
 
7. If A  cos  sin  0 then A is
 0 0 1
(A) singular (B) orthogonal (C) skew-symmetric (D) unitary

b 2 – ab b – c bc – ac
8. The value of the determinant ab – a 2 a – b b 2 – ab 
bc – ac c – a ab – a 2

(A) abc (B) 0 (C) a + b + c (D) a2 + b2 + c2

One or more than one correct :
1
9. z1 is any point on the curve |z| = 2 and z  z1  . Then locus of P(z) is
z1
5
(A) An ellipse with major axis of length (B) an ellipse with major axis of length 5
2
4 9
(C) an ellipse with eccentricity (D) an ellipse with length of latus rectum =
5 5

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10. Let a, b, c be the number of ways of distributing n rupee coins among 5, 6, 7 children respectively, with each
one getting at least one. If a, b, c, are in A.P., then n =
(A) 8 (B) 14 (C) 7 (D) 15

11. In the expansion of (x + y + z)25

(A) every term is of the form 25Cr rCs x25 – r yr – s zs (B) number of terms is 351
(C) number of terms is 325 (D) coefficient of x8 y9 z9 is 0

e 2iA e –iC e – iB
– iC
12. If A + B + C =  and z  e e 2iB e – iA , then
e –iB e –iA e 2iC

(A) Re z = 4 (B) Im z = 0 (C) Re z = –4 (D) Im z = –1

Subjective :
13. Number of divisors of 2456 × (143)2 of the form 4n + 2 is

14. Let A = {x / x is a prime number and x < 30}. The number of rational numbers whose numerator and
denominator belong to A is of the form 10a + 1, then a =

1024
 1 1
15. In the expansion of  5 2  7 8  , if the number of integral terms is N3, then the integer just greater than N is
 
 

2 sin2 x 2 cos 2x 2 cos 4x

16. When the determinant 2 cos 2x 2 sin 2 x 2 cos 2 x is expanded as a polynomial in sin x, the constant
2 cos 2 x 2 cos 4 x 2 cos 2x

term is
 | z |2 – | z | 1 
17. The minimum value of |z| for which " log    2" is true is
3 
 2  | z | 

 z – 1
18. If sin –1  can be the angle of a triangle, then the maximum value of Im(z) is
 i 
Match the Column :
19. Column I (Equations) Column II (Number of solutions)
(A) |z – 1| = |z + 1| = |z – i| (P) 0
(B) z 2  z  0 (Q) 2
(C) z2 + |z| = 0 (R) 1
(D) |z – 25i| = 15 and z  z  0 (S) 4
(T) 3

20. Column I Column II

10
 3
k
(A) If the independent term in  x  2  is 405, then k = (P) 48
 x 

(B) Last two digits in the expanded form of 231 (Q) 55
n
 x
(C) If coefficient of x7 and x8 in  2   are equal, then n = (R) 38
 3
(S) 27
(T) 56

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-8
Single Choice Question :

1. If z is a complex number such that iz2  z 2  z, then |z| is

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

1 13
2. If z1 and z2 are two complex numbers which satisfy | z1  2  2i |  and | z 2 – 1 – 2i |  , then the minimum
3 6

possible value of |z1 – z2| is

5 13 17 49
(A) (B) (C) (D)
2 3 6 6

3. Words with meaning or without meaning are formed with the letters of the word ERROR in the order opposite
to that found in the dictionary. Then the rank of the word ERROR is
(A) 24 (B) 22 (C) 18 (D) 19

4. The number of terms with integral coefficients in the expansion of (71/3 – 51/4 x2)500 is
(A) 41 (B) 42 (C) 83 (D) 84

3n
5. If (8  4 x  2x 2  x 3 )n  r
r
 a x , then the value of a
0 + (24)a4 + (28)a8 + (212)a12 + ...... is
r 0

(A) 22(2n – 1) (B) 25n – 2 (C) 22n – 2 (D) None of these

6. If A and B are two matrices such that AB = B and BA = A, then A4 + B4 is equal to

(A) AB (B) 4 AB (C) 4(A + B) (D) A + B

One or more than one correct :

7. ABCD is a quadrilateral inscribed in |z| = 1 where z1, z2, z3 and z4 are the complex numbers corresponding
to A, B, C, D respectively and AB is parallel to CD. Then the distance between these parallel lines is
1
(A) | z 2 – z 3 || z1 – z 3 | (B) |z2 – z3||z1 – z3|
2
1
(C) | z 2 – z 4 || z1 – z 4 | (D) |z2 – z4||z1 – z4|
2

8. Let N = 3n + 8n + 10n where n is a natural number. If n is odd, then the remainder when N is divided by 13 can
be
(A) 5 (B) 8 (C) 0 (D) 3

9. If the number of seven digit numbrs divisible by 9 formed by using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 without
repetition is expressed in the form2a · 3b · 5c · 7d, then
(A) a + b + c + d = 10 (B) b + c + d = 5 (C) b – c = 1 (D) c + d = 3

x y z
 
A  y z x
10. If is an orthogonal matrix, then the value of x3 + y3 + z3 – 3xyz is
z x y
 

(A) 2 (B) 1 (C) –1 (D) –2

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Comprehension : (Q.11 to Q.12)
 – 4 – 3 – 3
 
Let A be the matrix =  1 0 1
 4 4 3 
11. Adj A is
(A) 2A (B) –3A (C) –2A (D) None of these

12. If  satisfies |A – I| = 0, then the sum of the roots of the equation is
(A) 0 (B) –1 (C) 2 (D) 3

Comprehension : (Q.13 to Q.15)

z1, z2, z3 are three complex numbers which represent the vertices of an equilateral triangle.
( z1 – z 2 )3 – ( z1 – z 3 ) 3
13. is
( z 3 – z 2 )3
(A) 0 (B) –2 (C) 2 (D) None of these

z12  z 22  z 23
14. If z0 is the complex number representing the circumcentre of the triangle, then is
z 02
(A) 1 (B) 2 (C) 3 (D) 4

15. If further |z1| = |z2| = |z3| = 3, then the equation of the circumcircle is
(A) |3z – (z1 + z2 + z3)| = 27 (B) |3z – (z1 + z2 + z3)| = 6
(C) |z – (z1 + z2 + z3)| = 3 (D) None of these

Subjective :
16. Let f(x) = x2 – ax + b, where a is an odd positive integer and the roots of equation f(x) = 0 are two distinct

( f (1)  f (2)  f (3)  .....  f (10))f (10)

prime numbers. If a + b = 35, then the value of =
440

17. If z1, z2 are two complex numbers such that z13 – 3z1z22 = 2 and 3z12z2 – z32 = 11, then the value of
|z12 + z22| is _______.

1  2x 1  x 1  x
18. If x  0, y  0, z  0 and 1  2y 1  3 y 1  y  0 , then the value of x–1 + y–1 + z–1 + 9 is
1  2z 1  2z 1  4z

19. If N be the number of triangles that can be formed with the sides of lengths a, b, c such that a  b  c and
c = 13, then the unit digit in N is _______.

n
 1 
20. If in the expansion of   3  the ratio of the 4th term from the beginning to the 4th term from the end is
 2 

1
, then the value of n is ______.
6

21. If the 7th term is the middle term in the expansion of (1 + 4x + 6x2 + 4x3 + x4)n , then the value of n is _____.

2  cos2 x sin 2 x sin 2x

2
22. If m and  are the maximum and minimum values of the determinant cos x 2  sin2 x sin 2x , then
2 2
cos x sin x 2  sin 2x
m
the value of is _____.


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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-9
Single Choice Question :
  
1. If ˆ and ˆ be two perpendicular unit vectors such that x  ˆ – (ˆ  x), then x is equal to

1
(A) 1 (B) 2 (C) (D) None of these
2

2
2. The minimum value of (x2 – x1)2 +  1 – x12 – 4 – x 22  is equal to
 

(A) 13 (B) 6 (C) 4 (D) 1

( x 3  x  1)
3. The value of  ex dx is
(1  x 2 )3 / 2
xe x x 2e x ex
(A) C (B) C (C) C (D) None of these
(1  x 2 )1/ 2 (1  x 2 )1/ 2 (1  x 2 )1/ 2

4. The area bounded by y = max{|x – 2| + 2, 3 – |x – 2|} and y = min {|x – 2| +2, 3 – |x – 2|} is

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


5. If f(x) > 0,  x  R, f(3) = 0 and g(x) = f(tan2 x – 2 tan x + 4), 0 < x < , then g(x) is increasing in
2

     
(A)  0,  (B)  ,  (C)  0 ,  (D) None of these
 4 6 3  3

6. If foci of hyperbola lie on y = x and one of the asymptote is y = 2x, then equation of the hyperbola, given that
it passes through (3, 4), is

5
(A) x2 – y2 – xy + 5 = 0 (B) 2x2 – 2y2 + 5xy + 5 = 0
2
(C) 2x2 + 2y2 – 5xy + 10 = 0 (D) None of these

Comprehension (Q.7 to Q.9)

A sphere is the locus of a point in space such that its distance from a fixed point in space is constant. The
fixed point is called the centre and the constant distance is called the radius of the sphere. In Cartesian
system, the equation of the sphere, with centre at (–g, –f, –h) is

x2 + y2 + z2 + 2gx + 2fy + 2hz + c = 0 and its radius is f 2  g2  h 2 – c

In vector form the equation r – 0  a represents a sphere with centre 0 and radius a.

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 
7. Equation of the sphere having centre at (3, 6, –4) and touching the plane r . (2 î – 2 ĵ – k̂ )  10 , is (x – 3)2 +
(y – 6)2 + (z + 4)2 = k2, where k is equal to
(A) 3 (B) 4 (C) 6 (D) 17

 
8. Radius of the circular section of the sphere r  8 cut off by the plane r . ( î  2 ĵ  2k̂ ) = 15 is

(A) 39 (B) 29 (C) 59 (D) 7

9. Radius of the circular section of sphere x2 + y2 + z2 = 25 cut off by the sphere x2 + y2 + z2 –4x–4y–4z–13 = 0
is

(A) 5 (B) 22 (C) 24 (D) None of theses

Comprehension (Q.10 to Q.11)

f : R  R, f(x) is a differentiable function such that all its successive derivatives exist. f (x) can be zero at
discrete points only and f(x) f (x)  0  x  R.
Now answer the following questions based on the above comprehension :
10. If f(a) = 0, then which of the following is correct ?
(A) f(a+ h) f (a – h) < 0 (B) f(a + h) f (a – h) > 0
(C) f (a + h) f (a – h) < 0 (D) Nothing can be said

11. If f (x)  0, then maximum number of real roots of f(x) = 0 is/are

(A) No real root (B) One (C) Two (D) Three

Match the Column :

12. Column I Column II
–1
(A) lim cos ec sin x (P) –/2
x  / 2

k
(B) lim tan –1 , k  R may be equal to (Q) 0
x 0 x2

tan(  cos 2 x )
(C) lim (R) /2
x 0 2x 2

 2 x 2  2x  3
(D) lim (S) Does not exist
x –  2x  3
(T) 

13. Column I Column II

(A) If the point (a, a) lies between the lines |x + y| = 6 then [|a|] (P) 1
can be equal to, ([] represents the integral part)
(B) Number of circle touching both the axes and the line x + y = 4. (Q) 2
(C) A point can be located in the x-y plane in such a way that two tangents (R) 3
drawn from it to the parabola y2 = 4ax are normals to the parabola x2 = 4y,
then possible values of a can be
(D) If normals are drawn at four points with eccentric angles 1, 2, 3 and 4 (S) 4

x2 y2
on the ellipse 2
  1 , passes through a same point and value of
a b2
1 + 2 + 3 + 4 = n then n may be
(T) 5

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Subjective :

p
14. If four vectors are equally inclined to each other at an angle  and cos  = – (where p, q are coprime
q

number) then p + q =

15. A line L1 with direction ratios (–3, 2, 4) passes through the point A(7, 6, 2) and a line L2 with direction ratios
(2, 1, 3) passes through the point B(5, 3, 4). A line L3 with direction ratios (2, –2, –1) intersects L1 and L2 at
C and D. Find CD.


 n2  2n  3 
16. Let '' denotes the sum of the infinite series  

n1  2 n
.


Compute the value of [(13 + 23 + 33 + ....... + 3)1/4] (where [.] denotes greatest integer function).

17. The lengths of two perpendicular focal chords of the parabola y2 = 4(x + 1), are p and q. Then the value of

pq
is
pq

18. A box has 2 white, 4 black and 6 green balls. Person A, draws a ball from it. Then from the remaining balls

p
person B draws two balls which are found to be green. The probability that A has drawn a black ball is
q

(p and q are co-prime), p, q  I+ then p + q =

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-10
Single correct choice Type :

x2 y2
1. An ellipse E :   1 and hyperbola ‘H’ whose length of transverse axis is equal to length of semi
25 16

minor axis of ellipse E ; are confocal. Point ‘P’ is the common point for E and H. If M1 and M2 are feet of
perpendiculars from point ‘P’ on the corresponding directrices of E and H respectively then (M1M2) is :

11
(A) 7 (B) (C) 5 (D) none of these
15

2. A triangle ABC of area is inscribed in the parabola y2 = 4ax (a > 0) such that A is the vertex and BC is a focal
chord of the parabola. The difference of the ordinates of B and C is equal to :

2  2a 3 22
(A) (B) (C) (D)
a a  a3

x2 y2
3. Tangents are drawn to the ellipse   1 (a > b > 0) and the circle x2 + y2 = a2 at the points where a
a2 b2

common ordinate cuts them (on the same side of x-axis) then the greatest acute angle between these
tangents is given by

a–b ab 2ab 2ab

(A) tan–1 (B) tan–1 (C) tan–1 (D) tan–1
2 ab 2 ab a–b ab

4. If mn distinct coins have been distributed in ‘m’ purses of different colours, n coins into each then the
probability that two specified coins will be found in same purse :

n 1 n–1 m 1 m –1
(A) (B) (C) (D)
mn  1 mn – 1 mn  1 mn  1

5. If the median AM, angle bisector AD and altitude AH drawn from vertex A of a ABC divide angle A into four
equal parts and D lies in between H and M, then

 AC AC 1
(A) A = (B) A = 90º (C)  2 –1 (D) AB 
3 AB 22

6. If z1 and z2 satisfy the condition |z – 3| = 4 and |z – 1| + |z + 1| = 3 respectively then A = |z1 – z2| satisfies

15 15 17 17
(A) 0  A  (B) 0 < A  (C) 0  A  (D) 0  A <
2 2 2 2

 –1 
7. lim  1  5( x – 2 )–1   [ x – 1]  equal to (consider [.] as greatest integer function)
x 2  
  

(A) 1 (B) 0 (C) does not exist (D) none of these

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8. Which of the following is true? (consider [.] as greatest integer function)
(A) ([2x] – 2x + 2) is a non periodic function
(B) x [x] is discontinuous at x = 0
(C) x f(x) will be differentiable whenever f(x) is continuous
(D) none of these

Paragraph for Question Nos. 9 to 10

The shortest or longest distance between any two curves can be measured along the common normals for
the two given curve or by considering the distance between two parallel tangents to them. Considering this,
answer the following questions.
9. The point on the straight line 2x + 2y + 7 = 0 which is nearest to the curve having parametric equation
x = t2 – 2t + 3 and y = t2 + 2t + 3 ; is :

 7 7  7  7   7
(A)  – , –  (B)  0, –  (C)  – , 0  (D)  – 7, 
 4 4  2  2   2

10. The point on the curve y = |x2 – 4x + 3| which is nearest to the circle x2 + y2 – 4x – 4y + 7 = 0 is :
(A) (1, 0) (B) (0, 2) (C) (2, 1) (D) (2, 0)

Paragraph for Question Nos. 11 to 13

Consider a family of curves having the property that the ordinate is proportional to the cube of the abscissa
and let A be a fixed point in the plane, which has coordinates (a, b), with reference to the rectangular
coordinates axes.
11. If tangents be drawn through A to the members of the family of curves, then the locus of the points of contact
is
(A) xy + bx – 3ay = 0 (B) xy – 4bx + 3ay = 0
(C) 2xy + bx – 3ay = 0 (D) 2xy – 4bx + 3ay + 2 = 0

12. If normals be drawn through A to the member of the family of curves, then the feet of these normals on the
curves also lie on the curve
(A) xy + bx – 3ay = 0 (B) xy – 4bx + 3ay = 0
2 2
(C) x – 3y = ax – 3by (D) x2 + 3y2 = ax + 3by

13. If the tangent through A to a curve cuts the curve again at a point B, then the locus of B is
(A) xy – 4bx + 3ay = 0 (B) 2xy + bx – 3ay = 0
2 2
(C) x – 3y = ax – 3by (D) xy – by + ay = 0

Matrix Type :
14. Column-I Column-II

 –1 1  –1 1   3 

(A) The value of sin    sin    sin –1   (P) 0
 3  3 11   11 


(B) If 2sec 2= tan+ cotthen (2+ 2) may be equal to (Q)
2
(C) (R) 
2
| sin x |
(D) 2  dx  (S) 2
x
0

(T) 3

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15. Column-I Column-II
(A) The last digit of (1! + 2! + ........+ (2010)!)102 is (P) 3

1 2 2 
 
2 1 – 2
(B) If 3A =  is an orthogonal matrix then (|a| + |b|) equals to (Q) 9
a 2 b 

(C) If (1 + x)a (1 – x)b = a0 + a1x + a2x2 + ............ ; (R) 4 2

a0 = 1, a1 = a2 = 10, then (a + b) equal to a, b  N

 2 2 
(D) Twice of the minimum value of  ( x  4)  2  is
 (S) 16
 ( x  4 ) 

(T) 80

Integer Type :

2
2 –1
16. If f(x) = x + sin x, then find
2
.  (f

( x )  sin x ) dx .

17. Find the numbers of points of intersection of curve sin x = cos y and circle x2 + y2 = 1.

18. f(x) = cos–1 (cos x) and g(x) = sin–1 [x + 1] + cos–1[x] (where [.] denotes greatest integer function) then find
number of solutions of equation f(x) + g(x) = 3.

1 –1  1 
19. If f(x)= max.  cos (cos x ), {x }  and g(x) = min  cos –1(cos x ), {x }  (where {.} represents fractional
     

 f (x ) dx
1
part of x). Then find the value of 2 + n, where n is the number of points where y = f(x) + g(x) is non

 g(x ) dx
1

1 
differentiable  x   , 2 .
2 

 
k
20. The value of the sum   2n k is equal to
k 1 n 1

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 VIBRANT ACADEMY MATHS
Practice Problems IRP
(India) Private Limited Target IIT ADVANCE-2019
(IRP-BATCH) TIME : 60 MIN. IRP DPP. NO.-11
Single Choice Question :

 1
1. If '' be the only real root of the equation x3 + bx2 + cx + 1 = 0 (b < c); then the vlaue of tan–1 + tan–1   is

equal to

 
(A) (B) – (C) 0 (D) None of these
2 2
2. If the functions f : {1, 2, ......., n}  {2011, 2012} satisfying f(1) + f(2) + ....... + f(n) = odd integer, are formed;
then the number of such functions are
(A) 2n (B) 2n/2 (C) n2 (D) 2n – 1

x
3. Let f(x) be a polynomial with positive leading co-efficient satisfying f(0) = 0 and f f x   x f ( x ) dx  x  R

0

then 3 f (2) is
(A) 2 (B) 3 (C) 4 (D) 1


4. The area between the curve y = 2x4 – x2, the x-axis and the ordinates of two minima of the curve is , then
120
the value of '' is
(A) 7 (B) 11 (C) 13 (D) 17

x
t dt
5. If f(x) = eg(x) and g( x )   1 t 4
then f'(2) is
2

17 2 4 17
(A) (B) (C) (D)
2 17 17 4

6. If 2x3 – 6x + a = 0 has three real and distinct roots then number of distinct integral values of 'a' is
(A) 5 (B) 6 (C) 7 (D) 

3
7. If P(x) be a polynomial satisfying P(x2) = x2P(x) and P(0) = –2, P'    0 and P(1) = 0. Then maximum value
2
of P(x) is

1 1 1 1
(A) – (B) – (C) (D) –
4 3 4 2

8. The shortest distance between the point (0, –3) and the curve y = 1 + a1x2 + a2x4 + ...... + anx2n where all
ai > 0 (i = 1, 2, 3, ......, n) is
(A) 2 (B) 1 (C) 3 (D) 4

9. A polynomial f(x) of degree 6 satisfies f(x) = f(2 – x)  x  R. If f(x) = 0 has 4 distinct and two equal roots; then
the sum of the roots of f(x) = 0 is
(A) 0 (B) 4 (C) 6 (D) 8

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Comprehension (Q.10 to Q.12)
1
ex
Let P  dx then answer the following.

0
x 1

1
x 2e x
10. 
0
1 x
dx equals to

(A) P – e (B) e – 2 + P (C) 2 + P (D) 2 – e + P

1 2
 x  x
11. 
0
  e dx equals to
 x  1

e e e e
(A) P – (B)  1– P (C) –P (D) P – 1 
2 2 2 2
1
x2  1
12.  (x  1) 3
e x dx equals to
0

e 3e e e
(A) –2P (B) 3P – (C) P – 2 (D) P 
2 2 2 2

Comprehension (Q.13 to Q.15)

An acute angled triangle ABC is inscribed in a circle of radius unity. Let H be its orthocentre and x, y, z be the
circum radii of AHB, AHC and BHC respectively. If AH produced meets the circum circle at M and
intersects BC at L. Then
13. The value of xy + yz + zx
(A) 1 (B) 2 (C) 3 (D) 4

14. Area of AHB

(A) 2 cos A cos B cos C (B) cos A cos B cos C (C) cos A cos B sin C (D) 2 cosA cos B sin C

15. Ratio of area of AHB to BML is

(A) cos B : 2 cos A (B) 2 : 1 (C) 3 : 2 (D) cos A : cos B cos C

Comprehension (Q.16 to Q.18)

Let y = f(x) be a curve such that x = 1 – 3t2; y = t – 3t3 where t is a parameter. f(x) is not identically zero.
16. If the tangent at A(–2, 2) meets the curve again at B then co-ordinates of the point B is

 –9  1 2  1 2  1 2
(A)  – 3,  (B)  – , –  (C)  – ,  (D)  ,– 
 2   3 9  3 9 3 9

17. The acute angle between the tangents at x = 0 to the curve y = f(x) is

  
(A) (B) (C) 0 (D)
4 6 3

18. The number of points where curve crosses the x-axis

(A) 3 (B) 2 (C) 1 (D) None of these

Subjective :
19. Find the sum of all values of 'x' which satisfy x2 + 2x sin(xy) + 1 = 0.

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20. If f : R  R is a monotonic, differentiable real valued function, a, b are two real numbers and

b f (b )

 f ( x)  f (a) f (x)  f (a) dx  k  xb – f 

–1
( x ) dx then find the value k
a f (a )

21. In an acute angled triangle ABC, A = 20º, let DEF be the feet of altitudes through A, B, C respectively and

AH BH CH
H is the orthocentre of ABC. Find   .
AD BE CF

x3 x2 x4 11x 2
22. Let f ( x )    x  2; g( x)  – 2x 3  – 6 x then f(g(x)) has local minima at x 1 and x 2,
3 2 4 2
find the value of x1 + x2

 n n n n  2
23. If lim  tan –1 2 2  tan –1 2  tan –1 2  ......  tan –1 2  . Where k  N. Then find 'k'.
n  n 1 n  22 n  32 n  n2  k


 1 – cos 7 x  – x 1  100 
24. If  
0
x
e dx  ln
 2   
. Find ''.


xk
25. Let p(x) = 2x6 + 4x5 + 3x4 + 5x3 + 3x2 + 4x + 2. Let  k   dx ; where 0 < k < 5.
p( x )
0

Find the value of 'k' for which Ik is the smallest.

1 1 1 1 1 
26. Let 1 –  –  .......  –  ........  where a and b are relatively prime positive integers.
5 7 11 6n – 5 6n – 1 a b
Find the value of (a + b).


626 e – x sin25 x dx

0
27. If 
 . Find the sum of the digits of .
–x 23
e
0
sin x dx

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