VELAMMAL IIT ACADEMY

MATHEMATICS :: 10-05-2023 :: 12TH CLASS

AGT-26

SECTION – I INTEGER ANSWER TYPE

This section contains 8 questions. The answer is a single digit integer ranging from 0 to 9 (both
inclusive).
Marking scheme +4 for correct answer , 0 if not attempted and 0 in all other cases.
1. Number of equilateral triangles that can be formed with points lying on the curve 8 x3 + y 3 + 6 xy = 1
is/are
2. Each of the 11 letters A, H , I , M , O, T ,U ,V ,W , X and Y appears same when looked at in the mirror.
They are called symmetric letters. Other letters are asymmetric letters. Number of three letter computer
passwords which can be formed (without repetion) with at least one symmetric letter is x , then the sum
of the digits of the sum of digits of x is______.
area   ABC 
3. A point P lying inside ABC satisfies the condition PA + 2 PB + 3PC = 0, then is
area   APC 
equal to_______.
4. The least real value of a for which the equation x 4 − 2ax 2 + x + a 2 − a = 0, has all of its roots real is k,
then the value of 8k is
5. Die A has 4 red and 2 white faces whereas die B has 2 red and 4 white faces. A fair coin is flipped once.
If it shows a head, the game continues by throwing die A, if it shows tail, then die B is to be used. If the
64
probability that die A is used is where it is given that red turns up every time in first n throws, then
66
n is_____
x3 x 6 x4 x7 x 2 x 5 x8
6. Given a = 1 + + + ......, b = x + + + ......, c = + + + ....., If a 3 + b3 + c3 − zabc = 1.
3! 6! 4! 7! 2! 5! 8!
Find the value of z

 n2 + 3 
7. The sum of the infinite series,   n  is equal to
n =1  2 
dy
8. The solution of x 3 + 4x 2 tan y = e x sec y satisfying y(1) = 0 is sin y = e x (x − 1)x − k then k =
dx
Section – II MULTIPLE CORRECT ANSWER TYPE
This section contains 10 multiple choice questions. Each question has 4 options (A), (B), (C) and (D) for
its answer, out of which ONE OR MORE than ONE option can be correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases.
x x+ y 2x + 2 y + z
9. If f ( x, y, z ) = 2 x 3 x + 2 y 7 x + 5 y + 2 z & F ( x, 2 y, z ) = 2 F ( x, y, z ) and F ( x, y,3z ) = 3F ( x, y, z )
3 x 6 x + 3 y 16 x + 9 y + 3 z
then the value of F ( x, y, z ) equal to
A) F ( 3x, y, z ) B) F ( 2 x, y, z ) C) F ( 2 x, 2 y, 2 z ) D) F ( 4 x, y, z )
10. If x satisfies the equation
1 dt   3 t 2 sin 2t dt 
x  2  −x    − 2 = 0 ( 0     )  . Then the values of x is
2

 0 t + 2t cos  + 1   −3 t + 1 
4
 sin  sin  sin  sin 
A) 2 B) −2 C) 4 D) −4
   
x 3 − 3x 2 
11. Let f ( x ) = cos −1 x + cos −1  +  , then
2 2 
 
  1  −7  
A) f   = 
2
B) f   = cos −1   −
3 3 3  9  3
 −1  5  −7  2
C) f   = − cos −1   D) f   = 
 3  3  9  3 6
12. Two points A ( x1 , y1 ) , B ( x2 , y2 ) where x1  x2 and x1 , x2 , y1 , y2 are all non negative integers lie on a
circle passing through the origin and also lie on the curve y 3 = x3 + 8 x 2 − 6 x + 8 . Then which of the
following is/are true.
A) Point of intersection of normals at A, B to the circle can be (10,1)
B) Point of intersection of normals at A, B to the circle can be (9, 11)
 5 114 
C) point of intersection of tangents at A, B to the circle is  , 
 11 11 
 9 112 
D) Point of intersection of tangents at A, B to the circle is  , 
 11 11 
 2
sin ( 2n − 1) x  2 2
 sin nx 
13. If n  N and f ( n ) = 0 dx, g ( n ) =    dx then
sin x 0  sin x 

A) f ( n + 1) = f ( n ) B) g ( n + 1) = g ( n )
C) g ( n + 1) − f ( n + 1) = g ( n ) D) g ( n + 1) = f ( n ) + g ( n )
14. The circle C1 : x 2 + y 2 = 3, with centre at O, intersect the parabola x 2 = 2 y at the point P in the first
quadrant. Let the tangent to the circle C1 at P touches other two circles C2 and C3 at R2 and R3
respectively. Suppose C2 and C3 have equal radii 2 3 and centres Q and Q3 , respectively. If Q2 and
Q3 lie on the y-axis, then
A) Q2 Q3 = 12 B) R2 R3 = 4 6
C) Area of OR2 R3 is 6 2 D) Area of PQ2 Q3 is 4 2

(x + y2 )
2 p

15. If x = a cos  sin  , y = a sin  cos 

3 2 3 2
and ( p, q  N ) is independent of  , then which of
( xy )
q

the following is correct. ( 

. denotes fractional part)
pq pq
A) the digit at the tenth’s place of q is 0 B) the digit at the tenth’s place of q is 2
 2qc   2qc 
C)  q  = 1 D)  q  = 0
 p   p 
16. If a parabola touches the lines y = x and y = − x at A ( 3,3) and B (1, −1) respectively then
A) equation of directrix is 2 x + y = 0
B) equation of line through origin and focus is x + 2 y = 0
 −3 6 
C) focus is  , 
 5 5
D) directrix passes through (1, -2)
 SECTION – III PARAGRAPH TYPE
This section contains 2 groups of questions. Each group has 2 multiple choice questions based on a
paragraph. Each question has 4 choices A), B), C) and D) for its answer, out of which ONE OR MORE
is/are correct.
Marking scheme: +4 for correct answer, 0 if not attempted and -2 in all other cases.
Paragraph
For positive l , m and n, if the planes x = ny + mz , y = lz + nx, z = mx + ly intersect in a straight line, then
17. cos −1 l + cos −1 m + cos −1 n is equal to
A) 900 B) 500
C) 1800 D) None of these
18. l , m, n satisfy the equation.
A) l 2 + m 2 + n 2 = 2 B) l 2 + m 2 + n 2 + 2lmn = 1
C) l 2 + m 2 + n 2 = 1 D) None of these
Paragraph
Consider the curve C : x 2 + y 2 − 6 x + 5 = 0
19. From a point ‘A’ on the circumference of circle ‘C’, a line is drawn which is perpendicular to the
tangent at ‘B’. The point of intersection of the tangent at ‘B’ and the line drawn above
from ‘A’ is denoted ‘L’. The maximum area of ALB is
3 3 3 3 3 3
A) B) C) D) None of these
8 4 2
20. A point P is taken on the circumference of circle ‘C’. The chord ‘QR’ is drawn to the circle such that it
is parallel to tangent at P. The maximum area of PQR is
3 3
A) B) 3 3 C) 3 D) None of these
2

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