NAME : .....................................................................................................................................................
JEE (Main + Advanced) 2024
JEE (Main + Advanced) 2024
ENTHUSIAST COURSE
ENTHUSIAST COURSE
ASSIGNMENT # A (TANGENT & NORMAL) MATHEMATICS
One or more than one correct :
1. A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when the
depth of the liquid in it is x cm, the volume of the liquid in it is, x2 (15 - x) cu. cm. The length EF is:
(A) 7.5 cm (B) 8 cm (C) 10 cm (D) 12 cm
2. A horse runs along a circle with a speed of 20 km/hr . A lantern is at the centre of the circle. A fence is
along the tangent to the circle at the point at which the horse starts . The speed with which the shadow
of the horse moves along the fence at the moment when it covers 1/8 of the circle in km/hr is
(A) 20 (B)40 (C) 30 (D) 60
3. Which of the following pair(s) of curves touch each other ?
(A) y2 = 4x and x2 + y2 = 6x – 1 (B) xy = 4 and x2 + y2 = 8
x+2
(C) y = 2x – 3 and y = x3 – x + 1 (D) y = 6 + x – x2 and y =
x-2
x 2 y2
4. If the curves + = 1 and y2 = 16x intersects at right angles, then :-
a2 4
1
(A) a = ± 1 (B) a = ± 3 (C) a = ± (D) a = ± 2
3
5. Tangent of acute angle between the curves y = |x2 – 1| and y = |x2 – 3| at their points of intersection
is :-
4 2 4 2
(A) 0 (B) (C) (D) 2 2
3 7
6. At any two points of the curve represented parametrically by x = a (2 cos t - cos 2t) ;
y = a (2 sin t - sin 2t) the tangents are parallel to the axis of x corresponding to the values of the
parameter t differing from each other by :
(A) 2p/3 (B) 3p/4 (C) p/2 (D) p/3
Comprehension Type :
Paragraph for Q.7 to 9
da
Let a(t) is a function of t such that = 2 for all values of t and a = 0 when t = 0. Further
dt
y = m(t) x + c(t) is tangent to the curve y = x2 – 2ax + a2 + a at the point whose abscissa is 0. Then
7. If the rate of change of distance of vertex of y = x2 – 2ax + a2 + a from the origin with respect to
t is k, then k = :-
(A) 2 (B) 2 2 (C) 2 (D) 4 2
8. If the rate of change of c(t) with respect to t, when t = k, is l, then :-
(A) 16 2 – 2 (B) 8 2 + 2 (C) 10 2 + 2 (D) 16 2 + 2
9. The rate of change of m(t), with respect to t, at t = l is :-
(A) –2 (B) 2 (C) –4 (D) 4
Subjective :
10. A and B are points of the parabola y = x2. The tangents at A and B meet at C. The median of the triangle
ABC from C has length 'm' units. Find the area of the triangle in terms of 'm'.
11. A particle moves along the curve 6 y = x3 + 2. Find the points on the curve at which the y coordinate is
changing 8 times as fast as the x coordinate.
MATHEMATICS / ASSIGNMENT E-
, JEE (Main + Advanced) 2024
ENTHUSIAST COURSE
12. An air force plane is ascending vertically at the rate of 100 km/h. If the radius of the earth is R Km, how
fast the area of the earth, visible from the plane increasing at 3min after it started ascending. Take visible
2
area A = 2pR h Where h is the height of the plane in kms above the earth.
R+h
13. If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining elements are
changed slightly, show that da + db = 0.
cos A cos B
14. Find the equation of the normal to the curve y = (1 + x)y + sin-1 (sin2x) at x = 0.
15. Find all the lines that pass through the point (1, 1) and are tangent to the curve represented parametrically
as x = 2t – t2 and y = t + t2.
41x 3
16. A line is tangent to the curve f (x) = at the point P in the first quadrant, and has a slope of 2009.
3
This line intersects the y-axis at (0, b). Find the value of 'b'.
17. Find all the tangents to the curve y = cos (x + y), - 2p £ x £ 2p, that are parallel to the line x + 2y = 0.
18. The curve y = ax3 + bx2 + cx + 5 , touches the x - axis at P (- 2 , 0) & cuts the y-axis at a point Q where
its gradient is 3. Find a , b , c.
19. Find the gradient of the line passing through the point (2,8) and touching the curve y = x3.
20. Find the equations of the tangents drawn to the curve y2 – 2x3 – 4y + 8 = 0 from the point (1, 2).
21. Find the point of intersection of the tangents drawn to the curve x2y = 1 – y at the points where it is
intersected by the curve xy = 1 – y.
22. Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos2t ; y = - a cos3t contained
between the co-ordinate axes is equal to 2a.
23. Let P(x0, y0) be a point on the curve C : (x2 – 11) (y + 1) + 4 = 0 where x0, y0 Î N. If area of the
æaö
triangle formed by the normal drawn to the curve 'C' at P and the co-ordinate axes is ç ÷ , a, b Î
èbø
N then find the least value of (a – 6b).
24. A point P (x, y) moves on the curve x2/3 + y2/3 = a2/3, a > 0, For each position (x, y) of P, perpendiculars
are drawn from origin upon the tangent and normal at P, the length (absolute value) of them being
dp1 dp2
p1(x) and p2(x) respectively. Prove that . £ 0 :-
dx dx
25. If the sides and angles of a plane triangle vary in such a way that its circumradius remains constant,
da db dc
prove that + + = 0 where da, db and dc are small increments in the sides a, b, c
cos A cos B cos C
respectively
ANSWER KEY
1. C 2. B 3. A,B 4. D 5. C
m m
6. A 7. B 8. C 9. C 10.
2
11. (4 , 11) & (- 4, - 31/3) 12. 200 p r3 / (r + 5)² km² / h
14. x+y–1=0 15. x = 1 when t = 1, m ® ¥; 5x – 4y = 1 if t ¹ 1, t = 1/3
82 ·7 3
16. – 17. x + 2 y = p/2 & x + 2 y = - 3 p/2
3
18. a = - 1/2 ; b = - 3/4 ; c = 3 19. 3, 12
20. 2 3 x-y=2 ( )
3 -1 or 2 3 x + y = 2 ( 3 +1) 21. (0, 1) 23. 3
E- MATHEMATICS / ASSIGNMENT
JEE (Main + Advanced) 2024
JEE (Main + Advanced) 2024
ENTHUSIAST COURSE
ENTHUSIAST COURSE
ASSIGNMENT # A (TANGENT & NORMAL) MATHEMATICS
One or more than one correct :
1. A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when the
depth of the liquid in it is x cm, the volume of the liquid in it is, x2 (15 - x) cu. cm. The length EF is:
(A) 7.5 cm (B) 8 cm (C) 10 cm (D) 12 cm
2. A horse runs along a circle with a speed of 20 km/hr . A lantern is at the centre of the circle. A fence is
along the tangent to the circle at the point at which the horse starts . The speed with which the shadow
of the horse moves along the fence at the moment when it covers 1/8 of the circle in km/hr is
(A) 20 (B)40 (C) 30 (D) 60
3. Which of the following pair(s) of curves touch each other ?
(A) y2 = 4x and x2 + y2 = 6x – 1 (B) xy = 4 and x2 + y2 = 8
x+2
(C) y = 2x – 3 and y = x3 – x + 1 (D) y = 6 + x – x2 and y =
x-2
x 2 y2
4. If the curves + = 1 and y2 = 16x intersects at right angles, then :-
a2 4
1
(A) a = ± 1 (B) a = ± 3 (C) a = ± (D) a = ± 2
3
5. Tangent of acute angle between the curves y = |x2 – 1| and y = |x2 – 3| at their points of intersection
is :-
4 2 4 2
(A) 0 (B) (C) (D) 2 2
3 7
6. At any two points of the curve represented parametrically by x = a (2 cos t - cos 2t) ;
y = a (2 sin t - sin 2t) the tangents are parallel to the axis of x corresponding to the values of the
parameter t differing from each other by :
(A) 2p/3 (B) 3p/4 (C) p/2 (D) p/3
Comprehension Type :
Paragraph for Q.7 to 9
da
Let a(t) is a function of t such that = 2 for all values of t and a = 0 when t = 0. Further
dt
y = m(t) x + c(t) is tangent to the curve y = x2 – 2ax + a2 + a at the point whose abscissa is 0. Then
7. If the rate of change of distance of vertex of y = x2 – 2ax + a2 + a from the origin with respect to
t is k, then k = :-
(A) 2 (B) 2 2 (C) 2 (D) 4 2
8. If the rate of change of c(t) with respect to t, when t = k, is l, then :-
(A) 16 2 – 2 (B) 8 2 + 2 (C) 10 2 + 2 (D) 16 2 + 2
9. The rate of change of m(t), with respect to t, at t = l is :-
(A) –2 (B) 2 (C) –4 (D) 4
Subjective :
10. A and B are points of the parabola y = x2. The tangents at A and B meet at C. The median of the triangle
ABC from C has length 'm' units. Find the area of the triangle in terms of 'm'.
11. A particle moves along the curve 6 y = x3 + 2. Find the points on the curve at which the y coordinate is
changing 8 times as fast as the x coordinate.
MATHEMATICS / ASSIGNMENT E-
, JEE (Main + Advanced) 2024
ENTHUSIAST COURSE
12. An air force plane is ascending vertically at the rate of 100 km/h. If the radius of the earth is R Km, how
fast the area of the earth, visible from the plane increasing at 3min after it started ascending. Take visible
2
area A = 2pR h Where h is the height of the plane in kms above the earth.
R+h
13. If in a triangle ABC, the side 'c' and the angle 'C' remain constant, while the remaining elements are
changed slightly, show that da + db = 0.
cos A cos B
14. Find the equation of the normal to the curve y = (1 + x)y + sin-1 (sin2x) at x = 0.
15. Find all the lines that pass through the point (1, 1) and are tangent to the curve represented parametrically
as x = 2t – t2 and y = t + t2.
41x 3
16. A line is tangent to the curve f (x) = at the point P in the first quadrant, and has a slope of 2009.
3
This line intersects the y-axis at (0, b). Find the value of 'b'.
17. Find all the tangents to the curve y = cos (x + y), - 2p £ x £ 2p, that are parallel to the line x + 2y = 0.
18. The curve y = ax3 + bx2 + cx + 5 , touches the x - axis at P (- 2 , 0) & cuts the y-axis at a point Q where
its gradient is 3. Find a , b , c.
19. Find the gradient of the line passing through the point (2,8) and touching the curve y = x3.
20. Find the equations of the tangents drawn to the curve y2 – 2x3 – 4y + 8 = 0 from the point (1, 2).
21. Find the point of intersection of the tangents drawn to the curve x2y = 1 – y at the points where it is
intersected by the curve xy = 1 – y.
22. Prove that the segment of the normal to the curve x = 2a sin t + a sin t cos2t ; y = - a cos3t contained
between the co-ordinate axes is equal to 2a.
23. Let P(x0, y0) be a point on the curve C : (x2 – 11) (y + 1) + 4 = 0 where x0, y0 Î N. If area of the
æaö
triangle formed by the normal drawn to the curve 'C' at P and the co-ordinate axes is ç ÷ , a, b Î
èbø
N then find the least value of (a – 6b).
24. A point P (x, y) moves on the curve x2/3 + y2/3 = a2/3, a > 0, For each position (x, y) of P, perpendiculars
are drawn from origin upon the tangent and normal at P, the length (absolute value) of them being
dp1 dp2
p1(x) and p2(x) respectively. Prove that . £ 0 :-
dx dx
25. If the sides and angles of a plane triangle vary in such a way that its circumradius remains constant,
da db dc
prove that + + = 0 where da, db and dc are small increments in the sides a, b, c
cos A cos B cos C
respectively
ANSWER KEY
1. C 2. B 3. A,B 4. D 5. C
m m
6. A 7. B 8. C 9. C 10.
2
11. (4 , 11) & (- 4, - 31/3) 12. 200 p r3 / (r + 5)² km² / h
14. x+y–1=0 15. x = 1 when t = 1, m ® ¥; 5x – 4y = 1 if t ¹ 1, t = 1/3
82 ·7 3
16. – 17. x + 2 y = p/2 & x + 2 y = - 3 p/2
3
18. a = - 1/2 ; b = - 3/4 ; c = 3 19. 3, 12
20. 2 3 x-y=2 ( )
3 -1 or 2 3 x + y = 2 ( 3 +1) 21. (0, 1) 23. 3
E- MATHEMATICS / ASSIGNMENT
