Integrals: Class 12 Previous Year Questions

Integrals questions asked in CBSE board exams

Repeated questions first. Open the free app to see each question as it appeared in the paper, with its marking scheme.

1. Equation of a line passing through point (1, 1, 1) and parallel to z-axis is y-1 x-1 z-1 (A) (B) 1-1-1 z-1 x- 1 y-1 z-1 (C) (D) Page 7 (A) (B) (C) (D) tan x- 1 13. tan x+ 1 (A) sec2 (B) sec2 log (C) (D) + c sec - log sec 14. ABC (a, b), (c, d) (e, f) ABC 242 (…
   Board 20231 markMCQRepeated: 2023,2025
2. dy 10. Find the particular solution of the differential equation x dx given that when x = l, y =
   Board 20221 markvery shortRepeated: 2022,2026
3. Find the product of the order and the degree of the differential equation 0 y dx dy ) xy ( dx d 2 . .
   Board 20221 markvery shortRepeated: 2022,2026
4. Which of the following expressions will give the area of region bounded by the curve y = x2 and line y = 16 ? 1 (A)  4 0 2 x dx (B) 2 4 0 2 x dx (C)  16 0 y dy (D) 2  16 0 y dy d
   Board 20261 markMCQ
5. { } 7. The least value of f(x) = x3 – 12x, x  [0, 3] is 1 (A) –16 (B) –9 (C) 0 (D) 16 6 { } 8.  . . 3ax b2 + c2x2 dx = A log | b2 + c2x2 | + K ,  A    : 1 (A) 3a (B) 3a 2b2 (C) 3a b2c2 (D) 3a 2c2 9.  –1 1 . . x3 x2 + 2| x | + 1 dx   …
   Board 20261 markMCQ
6. { } 23. (a) Find the absolute maximum value of f(x) = cos x + sin2 x, x  [0, ] 2 OR (b) If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius. 2 14 { } 24.   – AB = ^j + ^k   – AC …
   Board 20262 marksshort answer
7. Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are ma…
   Board 20264 marksshort answer
8. { } 22. A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm3/min. Find the rate at which level of perfume is dropping at an instant w…
   Board 20262 marksshort answer
9. dy (a) If Vi-x2 + /i-y2 = a(x-y), then prove that 32. dx OR If x= a (b) cos 0 +log tan and y = sin ·, then find at = d2
   Board 20255 markslong answer
10. The line x = 1 + 5·, y = -5 + ·, z = -6 -3· passes through which of the following point ? (A) (1, -5, 6) (B) (1, 5, 6) (1, -5, -6) (C) (D) (-1, -5, 6) Page 9 of 24 14. A B (B) (C) (D) 15. =, 02: x=2 (B) 2 dx (A) Yy dy x2 dx (C) (D) /ydy Z=5+ y , , : (A) (B) (C…
    Board 20251 markMCQ
11. The area of the shaded region (figure) represented by the curves y = x2, 0 · x · 2 and y-axis is given by x=2 dx (B) (A) Vy dy x2 dx (C) (D) y dy
    Board 20251 markMCQ
12. (a) Evaluate : x 2 sin 16 9 x cos x sin 4 0 + + ò p dx OR (b) Evaluate : ) x (sin tan x 2 sin 1 – 2 0ò p dx 2
    Board 20245 markslong answer
13. Page 10 of 23 23. (H$) kmV H$s{OE : x 2 1 x + ò dx AWdm (I) _mZ kmV H$s{OE : x x sin 2 4 0ò p dx 666 19. Assertion (A) : For matrix A = ú ú ú û ù ê ê ê ë é q q q q 1 cos – 1 – cos 1 cos – 1 cos 1 , where q ∊ [0, 2p], |A| ∊ [2, 4]. Reason (R) : cos q ∊ [– 1, 1]…
    Board 20242 marksAssertion-Reason
14. JJJJ Page 15 31. (a) Evaluate : 2 0 ex sin x dx OR (b) Find : ) b x cos( ) a x cos( 1 dx JJJJ Page 16 IÊS> K (LA) 5 32. EH$ g§~§Y R, na Bg àH$ma n[a^m{fV h¡ {H$ R = {(x, y) : x . y EH$ An[a_o` g§»`m h¡} & Om±M R, ñdVwë`, g_{_V `m g§H«$m_H$ h¡ `m Zht & 33. (H$)…
    Board 20233 marksshort answer
15. IfA then A2023 is equal to 1. 2023 (A) (B) 2023 (C) (D) 2023 Page 3 2. · oll h ol hh d · + d = 5/2 -5/2 (A) (B) 5/2 5/2 5/2 -5/2 (C) (D) -5/2 5/2 3. (A) (B) (0) (C) (4) (D) R-(4) 4. (A) (B) -1 (C) (D) 5. [(x)] = ax + b f(0) = 0·, f(x) · : ax? a+b (A) (B) + bx …
    Board 20231 markMCQ
16. Evaluate log (1 + tan x) dx. 26. (a) OR dx Find (b) sin3x cos (x-·) Page 13 - x + x2 27. : dx 1 +x2 log1 28.· dx (ex + e-r) (er - e-) log V2 29. I h x = p (z - x) n (a) (b) 30. (a) 1 1 1 K qei : x + 2y · 120, x + y · 60, x- 2y · 0, x·0, y·0 3 Z = 5x + 10y 1 Pa…
    Board 20233 marksshort answer
17. A particle moves along the curve 3y = ax3 + 1 such that at a point with x-coordinate l, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.
    Board 20232 marksshort answer
18. Find the values of 2, for which the distance of point (2, 1, 2) from plane 3x + 5y + 4z = 11 is 2/2 units. Page 3 of 8 ··· =4(2) ,2,21 (b) 242+2+ (2+ 5+ )=9 (2, 1, 3)1 dx 9. (a) ·/2 cOS x (b) (1 + sin ·) (4 + sin x) dx. dy. 10. xcos dx 11. (a) , ((, ) : 4x2 + …
    Board 20221 markvery short
19. (a) Using integration, find the area of the region (x, y) : 4x2 + 9y2 · 36, 2x + 3y · 6). OR (b) Using integration, find the area of the region bounded by lines x-y + 1 = 0, x=-2, x= 3 and x-axis. Page 5 of 8 1 J P PIK 13. dx. 1 + sin x Grounded Overhead metal…
    Board 20221 markvery short
20. Study the equation given below. Find f(y). Show your steps. OR Integrate: Show your steps.
    CBSE practice 2021-222 marksshort answer

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