.
Chapter 5 STD 12 Date : 16/12/25
Continuity and Differentiability Maths
//X Section A
• Write the answer of the following questions. [Each carries 2 Marks] [30]
dy
1. Find in the following : sin2y + cos xy = k
dx
dy
2. Find of the function : xy + yx = 1
dx
ì k cos x p
ïï p - 2x , x ¹ 2
If the function f ( x ) = í p
3.
ï3 p is continuous at x = 2 , then find the value of k.
,x =
ïî 2
4. If y = 3 cos(log x) + 4 sin(log x), show that x2y2 + xy1 + y = 0.
dy
5. Find in the following : sin2x + cos2y = 1
dx
dy
6. Find in the following : x2 + xy + y2 = 100
dx
7. Find the values of k so that the function f is continuous at the indicated point in :
ìïkx 2 , if x £ 2
f (x ) = í at x = 2
ïî 3 , if x > 2
dy log x
8. If xy = ex–y, then prove that =
dx (1 + log x )2
9. 1.5) Differentiate the following function with respect to x : sin x + sin y = tan (xy)
10. Find the values of k so that the function f is continuous at the indicated point in :
ìkx + 1, if x £ p
f (x ) = í at x = p
î cos x , if x > p
11. Find the values of k so that the function f is continuous at the indicated point in :
ì kx + 1, if x £ 5
f (x) = í at x = 5
î 3x - 5, if x > 5
d2 y
12. If y = 5 cos x – 3 sin x, prove that + y = 0
dx 2
13. Differentiate the following w.r.t. x : e x ,x >0
p 3p
14. Differentiate w.r.t. x : <x < , (sin x – cos x)(sin x – cos x)
4 4
dy
15. Find , if x = a(q + sin q), y = a(1 – cosq).
dx
//X Section B
• Write the answer of the following questions. [Each carries 3 Marks] [42]
1 -1
3.8) If 2x = y m + y
16. (n ³ 1) then prove that, (x2 – 1) y2 + xyy1 = m2y.
m
17. If y = (tan–1 x)2 show that (x2 + 1)2 y2 + 2x(x2 + 1) y1 = 2.
æ tö d2y
18. If x = a ç cos t + logtan ÷ , y = a sin t, then find .
è 2ø dx 2
2 2
19. Differentiate w.r.t. x : x x - 3 + ( x - 3)x , for x > 3
20. Differentiate the function w.r.t. x : (sin x )x + sin -1 x
, 2
x +1
21. Differentiate the function w.r.t. x : x x cos x +
2
x -1
dy
22. Find of the function : (cos x)y = (cos y)x
dx
23. Differentiate the function w.r.t. x : (log x)x + xlog x
24. æ
Differentiate the function w.r.t. x : ç x +
1ö
x
÷ø + x
( 1)
1+
x
è x
ì5 , If x £ 2
ï
25. Find the values of a and b such that the function defined by f ( x ) = íax + b , If 2 < x < 10 is a
ï 21 , If x ³ 10
î
continuous function.
2
d2 y æ dy ö
26. If e y (x + 1) = 1, show that =ç ÷
dx 2 è dx ø
2 d2y dy
27. If y = sin–1x, show that (1 - x ) 2
-x = 0.
dx dx
1
x
28. Differentiate the function w.r.t. x : ( x cos x ) + ( x sin x )x
dy -1
29. If x 1 + y + y 1 + x = 0 , for –1 < x < 1, prove that =- .
2
dx (1 + x )
//X Section C
• Write the answer of the following questions. [Each carries 4 Marks] [44]
3
é
( )
dy 2 ù 2
ê1+ dx ú
ë û
30. If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that 2 is a constant independent of a
d y
2
dx
and b.
d2y p
31. If x = a(cos t + t sin t) and y = a(sin t – t cos t), find 2
at x =
dx 4
-1 2
2 d y dy 2
32. If y = e a cos x
show that (1- x ) - x - a y = 0. Where –1 < x < 1.
2 dx
dx
d2 y
33. If y = 500 e7x + 600 e–7x, show that = 49 y
dx 2
d2y
34. If y = cos–1 x, find
2 in terms of y alone.
dx
35. Using the fact that sin(A + B) = sinA × cosB + cosA sinB and the differentiation, obtain the sum
formula for cosines.
a
dy t +1 æ 1ö
36. For a positive constant a find e y= a
, where t, and x = ç t + ÷ .
dx è tø
2
dy cos (a + y )
37. If cos y = x cos(a + y), with cos a ¹ ±1, prove that = .
dx sin a
-1 æ 3 x - x ö
3
dy 1 1
38. Find in the following : y = tan ç 2 ÷, - < x <
dx è 1 - 3x ø 3 3
dy -1
æ 2x ö
39. Find in the following : y = sin ç ÷
dx è 1 + x2 ø
40. If x and y are connected parametrically by the equations without eliminating the parameter, Find
dy sin 3 t cos 3 t
. : x= ,y =
dx cos 2t cos 2t
, .
Chapter 5 STD 12 Date : 16/12/25
Continuity and Differentiability Maths
Section [ A ] : 2 Marks Questions
No Ans Chap Sec Que Universal_QueId
1. - Chap 5 S3 7 QP25P11B1213_P1C5S3Q7
2. - Chap 5 S5 12 QP25P11B1213_P1C5S5Q12
3. - Chap 5 S15 7 QP25P11B1213_P1C5S15Q7
4. - Chap 5 S7 13 QP25P11B1213_P1C5S7Q13
5. - Chap 5 S3 8 QP25P11B1213_P1C5S3Q8
6. - Chap 5 S3 5 QP25P11B1213_P1C5S3Q5
7. - Chap 5 S1 27 QP25P11B1213_P1C5S1Q27
8. - Chap 5 S15 3 QP25P11B1213_P1C5S15Q3
9. - Chap 5 S14 6.1.5 QP25P11B1213_P1C5S14Q6.1.5
10. - Chap 5 S1 28 QP25P11B1213_P1C5S1Q28
11. - Chap 5 S1 29 QP25P11B1213_P1C5S1Q29
12. - Chap 5 S7 11 QP25P11B1213_P1C5S7Q11
13. - Chap 5 S4 7 QP25P11B1213_P1C5S4Q7
14. - Chap 5 S8 9 QP25P11B1213_P1C5S8Q9
15. - Chap 5 S9 33 QP25P11B1213_P1C5S9Q33
Section [ B ] : 3 Marks Questions
No Ans Chap Sec Que Universal_QueId
16. - Chap 5 S14 13.8 QP25P11B1213_P1C5S14Q13.8
17. - Chap 5 S7 17 QP25P11B1213_P1C5S7Q17
18. - Chap 5 S15 14 QP25P11B1213_P1C5S15Q14
19. - Chap 5 S8 11 QP25P11B1213_P1C5S8Q11
20. - Chap 5 S5 8 QP25P11B1213_P1C5S5Q8
21. - Chap 5 S5 10 QP25P11B1213_P1C5S5Q10
22. - Chap 5 S5 14 QP25P11B1213_P1C5S5Q14
23. - Chap 5 S5 7 QP25P11B1213_P1C5S5Q7
24. - Chap 5 S5 6 QP25P11B1213_P1C5S5Q6
25. - Chap 5 S1 30 QP25P11B1213_P1C5S1Q30
26. - Chap 5 S7 16 QP25P11B1213_P1C5S7Q16
27. - Chap 5 S9 38 QP25P11B1213_P1C5S9Q38
28. - Chap 5 S5 11 QP25P11B1213_P1C5S5Q11
29. - Chap 5 S8 14 QP25P11B1213_P1C5S8Q14
Welcome To Future - Quantum Paper
Chapter 5 STD 12 Date : 16/12/25
Continuity and Differentiability Maths
//X Section A
• Write the answer of the following questions. [Each carries 2 Marks] [30]
dy
1. Find in the following : sin2y + cos xy = k
dx
dy
2. Find of the function : xy + yx = 1
dx
ì k cos x p
ïï p - 2x , x ¹ 2
If the function f ( x ) = í p
3.
ï3 p is continuous at x = 2 , then find the value of k.
,x =
ïî 2
4. If y = 3 cos(log x) + 4 sin(log x), show that x2y2 + xy1 + y = 0.
dy
5. Find in the following : sin2x + cos2y = 1
dx
dy
6. Find in the following : x2 + xy + y2 = 100
dx
7. Find the values of k so that the function f is continuous at the indicated point in :
ìïkx 2 , if x £ 2
f (x ) = í at x = 2
ïî 3 , if x > 2
dy log x
8. If xy = ex–y, then prove that =
dx (1 + log x )2
9. 1.5) Differentiate the following function with respect to x : sin x + sin y = tan (xy)
10. Find the values of k so that the function f is continuous at the indicated point in :
ìkx + 1, if x £ p
f (x ) = í at x = p
î cos x , if x > p
11. Find the values of k so that the function f is continuous at the indicated point in :
ì kx + 1, if x £ 5
f (x) = í at x = 5
î 3x - 5, if x > 5
d2 y
12. If y = 5 cos x – 3 sin x, prove that + y = 0
dx 2
13. Differentiate the following w.r.t. x : e x ,x >0
p 3p
14. Differentiate w.r.t. x : <x < , (sin x – cos x)(sin x – cos x)
4 4
dy
15. Find , if x = a(q + sin q), y = a(1 – cosq).
dx
//X Section B
• Write the answer of the following questions. [Each carries 3 Marks] [42]
1 -1
3.8) If 2x = y m + y
16. (n ³ 1) then prove that, (x2 – 1) y2 + xyy1 = m2y.
m
17. If y = (tan–1 x)2 show that (x2 + 1)2 y2 + 2x(x2 + 1) y1 = 2.
æ tö d2y
18. If x = a ç cos t + logtan ÷ , y = a sin t, then find .
è 2ø dx 2
2 2
19. Differentiate w.r.t. x : x x - 3 + ( x - 3)x , for x > 3
20. Differentiate the function w.r.t. x : (sin x )x + sin -1 x
, 2
x +1
21. Differentiate the function w.r.t. x : x x cos x +
2
x -1
dy
22. Find of the function : (cos x)y = (cos y)x
dx
23. Differentiate the function w.r.t. x : (log x)x + xlog x
24. æ
Differentiate the function w.r.t. x : ç x +
1ö
x
÷ø + x
( 1)
1+
x
è x
ì5 , If x £ 2
ï
25. Find the values of a and b such that the function defined by f ( x ) = íax + b , If 2 < x < 10 is a
ï 21 , If x ³ 10
î
continuous function.
2
d2 y æ dy ö
26. If e y (x + 1) = 1, show that =ç ÷
dx 2 è dx ø
2 d2y dy
27. If y = sin–1x, show that (1 - x ) 2
-x = 0.
dx dx
1
x
28. Differentiate the function w.r.t. x : ( x cos x ) + ( x sin x )x
dy -1
29. If x 1 + y + y 1 + x = 0 , for –1 < x < 1, prove that =- .
2
dx (1 + x )
//X Section C
• Write the answer of the following questions. [Each carries 4 Marks] [44]
3
é
( )
dy 2 ù 2
ê1+ dx ú
ë û
30. If (x – a)2 + (y – b)2 = c2, for some c > 0, prove that 2 is a constant independent of a
d y
2
dx
and b.
d2y p
31. If x = a(cos t + t sin t) and y = a(sin t – t cos t), find 2
at x =
dx 4
-1 2
2 d y dy 2
32. If y = e a cos x
show that (1- x ) - x - a y = 0. Where –1 < x < 1.
2 dx
dx
d2 y
33. If y = 500 e7x + 600 e–7x, show that = 49 y
dx 2
d2y
34. If y = cos–1 x, find
2 in terms of y alone.
dx
35. Using the fact that sin(A + B) = sinA × cosB + cosA sinB and the differentiation, obtain the sum
formula for cosines.
a
dy t +1 æ 1ö
36. For a positive constant a find e y= a
, where t, and x = ç t + ÷ .
dx è tø
2
dy cos (a + y )
37. If cos y = x cos(a + y), with cos a ¹ ±1, prove that = .
dx sin a
-1 æ 3 x - x ö
3
dy 1 1
38. Find in the following : y = tan ç 2 ÷, - < x <
dx è 1 - 3x ø 3 3
dy -1
æ 2x ö
39. Find in the following : y = sin ç ÷
dx è 1 + x2 ø
40. If x and y are connected parametrically by the equations without eliminating the parameter, Find
dy sin 3 t cos 3 t
. : x= ,y =
dx cos 2t cos 2t
, .
Chapter 5 STD 12 Date : 16/12/25
Continuity and Differentiability Maths
Section [ A ] : 2 Marks Questions
No Ans Chap Sec Que Universal_QueId
1. - Chap 5 S3 7 QP25P11B1213_P1C5S3Q7
2. - Chap 5 S5 12 QP25P11B1213_P1C5S5Q12
3. - Chap 5 S15 7 QP25P11B1213_P1C5S15Q7
4. - Chap 5 S7 13 QP25P11B1213_P1C5S7Q13
5. - Chap 5 S3 8 QP25P11B1213_P1C5S3Q8
6. - Chap 5 S3 5 QP25P11B1213_P1C5S3Q5
7. - Chap 5 S1 27 QP25P11B1213_P1C5S1Q27
8. - Chap 5 S15 3 QP25P11B1213_P1C5S15Q3
9. - Chap 5 S14 6.1.5 QP25P11B1213_P1C5S14Q6.1.5
10. - Chap 5 S1 28 QP25P11B1213_P1C5S1Q28
11. - Chap 5 S1 29 QP25P11B1213_P1C5S1Q29
12. - Chap 5 S7 11 QP25P11B1213_P1C5S7Q11
13. - Chap 5 S4 7 QP25P11B1213_P1C5S4Q7
14. - Chap 5 S8 9 QP25P11B1213_P1C5S8Q9
15. - Chap 5 S9 33 QP25P11B1213_P1C5S9Q33
Section [ B ] : 3 Marks Questions
No Ans Chap Sec Que Universal_QueId
16. - Chap 5 S14 13.8 QP25P11B1213_P1C5S14Q13.8
17. - Chap 5 S7 17 QP25P11B1213_P1C5S7Q17
18. - Chap 5 S15 14 QP25P11B1213_P1C5S15Q14
19. - Chap 5 S8 11 QP25P11B1213_P1C5S8Q11
20. - Chap 5 S5 8 QP25P11B1213_P1C5S5Q8
21. - Chap 5 S5 10 QP25P11B1213_P1C5S5Q10
22. - Chap 5 S5 14 QP25P11B1213_P1C5S5Q14
23. - Chap 5 S5 7 QP25P11B1213_P1C5S5Q7
24. - Chap 5 S5 6 QP25P11B1213_P1C5S5Q6
25. - Chap 5 S1 30 QP25P11B1213_P1C5S1Q30
26. - Chap 5 S7 16 QP25P11B1213_P1C5S7Q16
27. - Chap 5 S9 38 QP25P11B1213_P1C5S9Q38
28. - Chap 5 S5 11 QP25P11B1213_P1C5S5Q11
29. - Chap 5 S8 14 QP25P11B1213_P1C5S8Q14
Welcome To Future - Quantum Paper
