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CBSE / NCERT Class 12 Maths Practice Question Papers

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Brain Grain · braingrain.in
Maths — Practice Paper · Set 1
Class: 12CBSE / NCERTMax Marks: 43
Name: ____________________Reg No: ____________
Part I — Multiple Choice Questions 15 × 1 = 15

Choose the correct answer. (Answer all questions.)

1.If $\sin^{-1}x = y$i. $0 \le y \le \pi$ii. $-\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2}$iii. $0 \lt y \lt \pi$iv. $-\dfrac{\pi}{2} \lt y \lt \dfrac{\pi}{2}$[1]
2.The general solution of the differential equation $e^xdy+(ye^x+2x)dx=0$i. $xe^y+x^2=C$ii. $xe^y+y^2=C$iii. $ye^x+x^2=C$iv. $ye^y+x^2=C$[1]
3.A homogeneous differential equation of the form $\dfrac{dx}{dy}=h\left(\dfrac{x}{y}\right)$i. $y=vx$ii. $v=yx$iii. $x=vy$iv. $x=v$[1]
4.$\int_{1}^{\sqrt3}\dfrac{dx}{1+x^2}$i. $\dfrac{\pi}{3}$ii. $\dfrac{2\pi}{3}$iii. $\dfrac{\pi}{6}$iv. $\dfrac{\pi}{12}$[1]
5.If $A$i. $A$ii. $I-A$iii. $I$iv. $3A$[1]
6.$\int \dfrac{e^x(1+x)}{\cos^2(xe^x)}\,dx$i. $-\cot(e^x x)+C$ii. $\tan(xe^x)+C$iii. $\tan(e^x)+C$iv. $\cot(e^x)+C$[1]
7.If $\vec a$i. $\lambda=1$ii. $\lambda=-1$iii. $a=|\lambda|$iv. $a=1/|\lambda|$[1]
8.The maximum value of $[x(x-1)+1]^{1/3}$i. $\left(\dfrac13\right)^{1/3}$ii. $\dfrac12$iii. $1$iv. $0$[1]
9.Let $f : \mathbb{R} \to \mathbb{R}$i. $f$ is one-one ontoii. $f$ is many-one ontoiii. $f$ is one-one but not ontoiv. $f$ is neither one-one nor onto[1]
10.$\int\dfrac{\cos2x}{(\sin x+\cos x)^2}\,dx$i. $-\dfrac{1}{\sin x+\cos x}+C$ii. $\log|\sin x+\cos x|+C$iii. $\log|\sin x-\cos x|+C$iv. $\dfrac{1}{(\sin x+\cos x)^2}$[1]
11.The order of the differential equation $x^2\dfrac{d^2y}{dx^2}-3\dfrac{dy}{dx}+2y=0$i. $2$ii. $1$iii. $0$iv. not defined[1]
12.$\int\dfrac{dx}{e^x+e^{-x}}$i. $\tan^{-1}(e^x)+C$ii. $\tan^{-1}(e^{-x})+C$iii. $\log(e^x-e^{-x})+C$iv. $\log(e^x+e^{-x})+C$[1]
13.If $\dfrac{d}{dx}f(x)=4x^3-\dfrac3{x^4}$i. $x^4+\dfrac1{x^3}-\dfrac{129}{8}$ii. $x^3+\dfrac1{x^4}+\dfrac{129}{8}$iii. $x^4+\dfrac1{x^3}+\dfrac{129}{8}$iv. $x^3+\dfrac1{x^4}-\dfrac{129}{8}$[1]
14.The value of the integral $\int_{1/3}^{1}\dfrac{(x-x^3)^{1/3}}{x^4}\,dx$i. $6$ii. $0$iii. $3$iv. $4$[1]
15.On which of the following intervals is the function $f$i. $(0,1)$ii. $\left(\dfrac{\pi}{2},\pi\right)$iii. $\left(0,\dfrac{\pi}{2}\right)$iv. None of these[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Integrate $\dfrac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}$[2]
17.Find the maximum and minimum values, if any, of the following functions given by (i) $f(x)=(2x-1)^2+3$[2]
18.Find the equation of a curve passing through the point $(0,-2)$[2]
19.Is $f(x)=x^2-\sin x+5$[2]
20.Find a particular solution of the differential equation $\dfrac{dy}{dx}-3y\cot x=\sin 2x$[2]
21.Verify that $y=\sqrt{x^2+1}$[2]
22.Integrate the function $x\sqrt{x+2}$[2]
23.Prove that the function $f(x)=x^n$[2]
24.Let $A = \{1, 2, 3\}$[2]
25.Find the integral of $\dfrac{1-\cos x}{1+\cos x}$[2]
26.Integrate the function $x\log 2x$[2]
27.Show that $y=\log(1+x)-\dfrac{2x}{2+x}$[2]
28.Verify that $y=x\sin x$[2]
29.Find values of $x$[2]
🔑 Show Answer Key — Set 1
  1. 1. (ii) $-\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2}$
  2. 2. Option (iii), $ye^x+x^2=C$ .
  3. 3. Option (iii), $x=vy$ .
  4. 4. Option (iv), $\dfrac{\pi}{12}$ .
  5. 5. (iii) $I$
  6. 6. Option (ii), $\tan(xe^x)+C$ .
  7. 7. Option (iv), $a=1/|\lambda|$ .
  8. 8. $1$ , option (iii).
  9. 9. (i) $f$ is one-one onto.
  10. 10. Option (ii), $\log|\sin x+\cos x|+C$ .
  11. 11. Option (i), $2$ .
  12. 12. Option (i), $\tan^{-1}(e^x)+C$ .
  13. 13. Option (i), $x^4+\dfrac1{x^3}-\dfrac{129}{8}$ .
  14. 14. Option (i), $6$ .
  15. 15. None of these, option (iv).
  16. 16. $\dfrac{x^3}{3}+C$ .
  17. 17. (i) Minimum $3$ at $x=\dfrac12$ , no maximum. (ii) Minimum $-2$ at $x=-\dfrac23$ , no maximum. (iii) Maximum $10$ at $x=1$ , no minimum. (iv) No maximum and no minimum.
  18. 18. $y^2=x^2+4$ .
  19. 19. Yes.
  20. 20. $y=4\sin^3x-2\sin^2x$ .
  21. 21. $y=\sqrt{x^2+1}$ satisfies $y'=\dfrac{xy}{x^2+1}$ .
  22. 22. $\dfrac{2}{15}(3x-4)(x+2)^{3/2}+C$ .
  23. 23. Continuous at $x=n$ .
  24. 24. $f$ is one-one.
  25. 25. $2\tan\dfrac{x}{2}-x+C$ .
  26. 26. $\dfrac{x^2}{2}\log 2x-\dfrac{x^2}{4}+C$ .
  27. 27. $y$ is increasing for $x\gt -1$ .
  28. 28. $y=x\sin x$ satisfies $xy'=y+x\sqrt{x^2-y^2}$ under the stated condition.
  29. 29. (i) $x=\pm\sqrt{3}$ (ii) $x=2$ .
Brain Grain · braingrain.in
Maths — Practice Paper · Set 2
Class: 12CBSE / NCERTMax Marks: 43
Name: ____________________Reg No: ____________
Part I — Multiple Choice Questions 15 × 1 = 15

Choose the correct answer. (Answer all questions.)

1.$\int \sqrt{x^2-8x+7}\,dx$i. $\dfrac12(x-4)\sqrt{x^2-8x+7}+9\log|x-4+\sqrt{x^2-8x+7}|+C$ii. $\dfrac12(x+4)\sqrt{x^2-8x+7}+9\log|x+4+\sqrt{x^2-8x+7}|+C$iii. $\dfrac12(x-4)\sqrt{x^2-8x+7}-3\log|x-4+\sqrt{x^2-8x+7}|+C$iv. $\dfrac12(x-4)\sqrt{x^2-8x+7}-\dfrac92\log|x-4+\sqrt{x^2-8x+7}|+C$[1]
2.The number of arbitrary constants in the general solution of a differential equation of fourth order are:i. $0$ii. $2$iii. $3$iv. $4$[1]
3.If $A=\begin{bmatrix}1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1\end{bmatrix}$i. $\det(A)=0$ii. $\det(A)\in(2,\infty)$iii. $\det(A)\in(2,4)$iv. $\det(A)\in[2,4]$[1]
4.If $\theta$i. $0\lt\theta\lt\dfrac\pi2$ii. $0\le\theta\le\dfrac\pi2$iii. $0\lt\theta\lt\pi$iv. $0\le\theta\le\pi$[1]
5.Area of a rectangle having vertices A, B, C and D with position vectors $-\hat{i}+\dfrac12\hat{j}+4\hat{k}$i. $\dfrac12$ii. $1$iii. $2$iv. $4$[1]
6.If $A, B$i. Skew symmetric matrixii. Symmetric matrixiii. Zero matrixiv. Identity matrix[1]
7.The point on the curve $x^2=2y$i. $(2\sqrt2,4)$ii. $(2\sqrt2,0)$iii. $(0,0)$iv. $(2,2)$[1]
8.If $\vec a$i. $\vec b=\lambda\vec a$ , for some scalar $\lambda$ii. $\vec a=\pm\vec b$iii. the respective components of $\vec a$ and $\vec b$ are not proportionaliv. both the vectors $\vec a$ and $\vec b$ have same direction, but different magnitudes.[1]
9.The probability of obtaining an even prime number on each die, when a pair of dice is rolled isi. $0$ii. $\dfrac13$iii. $\dfrac1{12}$iv. $\dfrac1{36}$[1]
10.$\tan^{-1}\sqrt{3}-\cot^{-1}(-\sqrt{3})$i. $\pi$ii. $-\dfrac{\pi}{2}$iii. $0$iv. $2\sqrt{3}$[1]
11.$\int_{0}^{2/3}\dfrac{dx}{4+9x^2}$i. $\dfrac{\pi}{6}$ii. $\dfrac{\pi}{12}$iii. $\dfrac{\pi}{24}$iv. $\dfrac{\pi}{4}$[1]
12.If area of triangle is 35 square units with vertices $(2,-6)$i. $12$ii. $-2$iii. $-12,-2$iv. $12,-2$[1]
13.If the matrix $A$i. $A$ is a diagonal matrixii. $A$ is a zero matrixiii. $A$ is a square matrixiv. None of these[1]
14.The rate of change of the area of a circle with respect to its radius $r$i. $10\pi$ii. $12\pi$iii. $8\pi$iv. $11\pi$[1]
15.Area of the region bounded by the curve $y^2=4x$i. $2$ii. $\dfrac94$iii. $\dfrac93$iv. $\dfrac92$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Integrate the function $\sqrt{x^2+4x+1}$[2]
17.Integrate the function $\sec^2(7-4x)$[2]
18.If $x=a(\theta-\sin\theta)$[2]
19.Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\pm\left(\dfrac1{\sqrt3},\dfrac1{\sqrt3},\dfrac1{\sqrt3}\right)$[2]
20.Prove that $3\cos^{-1}x = \cos^{-1}(4x^3 - 3x)$[2]
21.Determine order and degree, if defined, of the differential equation $\dfrac{d^4y}{dx^4}+\sin(y''')=0$[2]
22.Integrate the rational function $\dfrac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$[2]
23.Solve the differential equation $\dfrac{dy}{dx}+y\sec x=\tan x$[2]
24.Write down a unit vector in XY-plane, making an angle of $30^\circ$[2]
25.Find $\dfrac{dy}{dx}$[2]
26.Simplify $\cos\theta\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}+\sin\theta\begin{bmatrix}\sin\theta & -\cos\theta \\ \cos\theta & \sin\theta\end{bmatrix}$[2]
27.Show that the points $A(1,2,7),B(2,6,3)$[2]
28.Differentiate $\dfrac{\cos^{-1}(x/2)}{\sqrt{2x+7}}$[2]
29.Let $A$[2]
🔑 Show Answer Key — Set 2
  1. 1. Option (iv), $\dfrac12(x-4)\sqrt{x^2-8x+7}-\dfrac92\log|x-4+\sqrt{x^2-8x+7}|+C$ .
  2. 2. Option (iv), $4$ .
  3. 3. (iv) $\det(A)\in[2,4]$ .
  4. 4. Option (ii), $0\le\theta\le\dfrac\pi2$ .
  5. 5. Option (iii), $2$ .
  6. 6. (i) Skew symmetric matrix
  7. 7. $(2\sqrt2,4)$ , option (i).
  8. 8. Options (ii), (iii) and (iv) are incorrect.
  9. 9. $\dfrac1{36}$ , option (iv).
  10. 10. (ii) $-\dfrac{\pi}{2}$
  11. 11. Option (iii), $\dfrac{\pi}{24}$ .
  12. 12. (iv) $12,-2$ .
  13. 13. (ii) $A$ is a zero matrix
  14. 14. $12\pi$ , option (ii).
  15. 15. $\dfrac94$ , option (ii).
  16. 16. $\dfrac{x+2}{2}\sqrt{x^2+4x+1}-\dfrac32\log|x+2+\sqrt{x^2+4x+1}|+C$ .
  17. 17. $-\dfrac14\tan(7-4x)+C$ .
  18. 18. $-\dfrac{\sin\theta}{1-\cos\theta}=-\cot\dfrac\theta2$ .
  19. 19. They are $\pm\left(\dfrac1{\sqrt3},\dfrac1{\sqrt3},\dfrac1{\sqrt3}\right)$ .
  20. 20. $3\cos^{-1}x = \cos^{-1}(4x^3 - 3x)$ for $x \in \left[\dfrac{1}{2},1\right]$ .
  21. 21. Order $4$ ; degree not defined.
  22. 22. $x+\dfrac{2}{\sqrt3}\tan^{-1}\left(\dfrac{x}{\sqrt3}\right)-3\tan^{-1}\left(\dfrac{x}{2}\right)+C$ .
  23. 23. $y(\sec x+\tan x)=\sec x+\tan x-x+C$ .
  24. 24. $\dfrac{\sqrt3}{2}\hat{i}+\dfrac12\hat{j}$ .
  25. 25. $-\dfrac{a}{2by+\sin y}$ .
  26. 26. $\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ .
  27. 27. They are collinear.
  28. 28. $-\dfrac{1}{2\sqrt{1-x^2/4}\sqrt{2x+7}}-\dfrac{\cos^{-1}(x/2)}{(2x+7)^{3/2}}$ .
  29. 29. $f$ is bijective.
Brain Grain · braingrain.in
Maths — Practice Paper · Set 3
Class: 12CBSE / NCERTMax Marks: 43
Name: ____________________Reg No: ____________
Part I — Multiple Choice Questions 15 × 1 = 15

Choose the correct answer. (Answer all questions.)

1.Let $A = \{1, 2, 3\}$i. $1$ii. $2$iii. $3$iv. $4$[1]
2.If $\theta$i. $0$ii. $\dfrac\pi4$iii. $\dfrac\pi2$iv. $\pi$[1]
3.If $A$i. $A \subset B$ii. $B \subset A$iii. $B=\emptyset$iv. $A=\emptyset$[1]
4.Which of the following functions are decreasing on $\left(0,\dfrac{\pi}{2}\right)$i. $\cos x$ii. $\cos 2x$iii. $\cos 3x$iv. $\tan x$[1]
5.The Integrating Factor of the differential equation $(1-y^2)\dfrac{dx}{dy}+yx=ay$i. $\dfrac{1}{y^2-1}$ii. $\dfrac{1}{\sqrt{y^2-1}}$iii. $\dfrac{1}{1-y^2}$iv. $\dfrac{1}{\sqrt{1-y^2}}$[1]
6.If $f(a+b-x)=f(x)$i. $\dfrac{a+b}{2}\int_a^b f(b-x)\,dx$ii. $\dfrac{a+b}{2}\int_a^b f(b+x)\,dx$iii. $\dfrac{b-a}{2}\int_a^b f(x)\,dx$iv. $\dfrac{a+b}{2}\int_a^b f(x)\,dx$[1]
7.The interval in which $y=x^2e^{-x}$i. $(-\infty,\infty)$ii. $(-2,0)$iii. $(2,\infty)$iv. $(0,2)$[1]
8.$\int e^x\sec x(1+\an x)\,dx$i. $e^x\cos x+C$ii. $e^x\sec x+C$iii. $e^x\sin x+C$iv. $e^x\tan x+C$[1]
9.If $A=\begin{bmatrix}\alpha & \beta \\ \gamma & -\alpha\end{bmatrix}$i. $1+\alpha^2+\beta\gamma=0$ii. $1-\alpha^2+\beta\gamma=0$iii. $1-\alpha^2-\beta\gamma=0$iv. $1+\alpha^2-\beta\gamma=0$[1]
10.$\int x^2e^{x^3}\,dx$i. $\dfrac13e^{x^3}+C$ii. $\dfrac13e^{x^2}+C$iii. $\dfrac12e^{x^3}+C$iv. $\dfrac12e^{x^2}+C$[1]
11.Area bounded by the curve $y=x^3$i. $-9$ii. $-\dfrac{15}{4}$iii. $\dfrac{15}{4}$iv. $\dfrac{17}{4}$[1]
12.If $A$i. $A\subset B$ but $A\ne B$ii. $A=B$iii. $A\cap B=\phi$iv. $P(A)=P(B)$[1]
13.$\int \dfrac{x\,dx}{(x-1)(x-2)}$i. $\log\left|\dfrac{(x-1)^2}{x-2}\right|+C$ii. $\log\left|\dfrac{(x-2)^2}{x-1}\right|+C$iii. $\log\left|\dfrac{x-1}{x-2}\right|^2+C$iv. $\log|(x-1)(x-2)|+C$[1]
14.The value of $\int_{0}^{\pi/2}\log\left(\dfrac{4+3\sin x}{4+3\cos x}\right)dx$i. $2$ii. $\dfrac34$iii. $0$iv. $-2$[1]
15.Let $A$i. $|A|$ii. $|A|^2$iii. $|A|^3$iv. $3|A|$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) $f : \mathbb{R} \to \mathbb{R}$[2]
17.By using the properties of definite integrals, evaluate $\int_{0}^{\pi}\dfrac{x\,dx}{1+\sin x}$[2]
18.Find both the maximum value and the minimum value of $3x^4-8x^3+12x^2-48x+25$[2]
19.If $F(x)=\begin{bmatrix}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{bmatrix}$[2]
20.Find $\dfrac{dy}{dx}$[2]
21.Determine order and degree, if defined, of the differential equation $y''+(y')^2+2y=0$[2]
22.Find $\dfrac{dy}{dx}$[2]
23.Integrate the function $\dfrac{3x^2}{x^6+1}$[2]
24.By using the properties of definite integrals, evaluate $\int_{0}^{1}x(1-x)^n\,dx$[2]
25.Integrate the function $\tan^{-1}x$[2]
26.In the matrix $A = \begin{bmatrix}2 & 5 & 19 & -7 \\ 35 & -2 & \dfrac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17\end{bmatrix}$[2]
27.Find the integral of $\sin^{-1}(\cos x)$[2]
28.Find the integral of $\sin^2(2x+5)$[2]
29.For the matrices $A$[2]
🔑 Show Answer Key — Set 3
  1. 1. (i) $1$
  2. 2. Option (ii), $\dfrac\pi4$ .
  3. 3. Option (i), $A \subset B$ .
  4. 4. Options (i) and (ii): $\cos x$ and $\cos 2x$ .
  5. 5. Option (iv), $\dfrac{1}{\sqrt{1-y^2}}$ .
  6. 6. Option (iv), $\dfrac{a+b}{2}\int_a^b f(x)\,dx$ .
  7. 7. $(0,2)$ , option (iv).
  8. 8. Option (ii), $e^x\sec x+C$ .
  9. 9. (iii) $1-\alpha^2-\beta\gamma=0$
  10. 10. Option (i), $\dfrac13e^{x^3}+C$ .
  11. 11. $\dfrac{17}{4}$ , option (iv).
  12. 12. $P(A)=P(B)$ , option (iv).
  13. 13. Option (ii), $\log\left|\dfrac{(x-2)^2}{x-1}\right|+C$ .
  14. 14. Option (iii), $0$ .
  15. 15. (ii) $|A|^2$ .
  16. 16. (i) $f$ is bijective (one-one and onto). (ii) $f$ is neither one-one nor onto.
  17. 17. $\pi$ .
  18. 18. Maximum $25$ at $x=0$ ; minimum $-39$ at $x=2$ .
  19. 19. $F(x)F(y)=F(x+y)$ .
  20. 20. $-\dfrac{2}{1+x^2}$ .
  21. 21. Order $2$ ; degree $1$ .
  22. 22. $-\dfrac{2x+y}{x+2y}$ .
  23. 23. $\tan^{-1}(x^3)+C$ .
  24. 24. $\dfrac{1}{(n+1)(n+2)}$ .
  25. 25. $x\tan^{-1}x-\dfrac12\log(1+x^2)+C$ .
  26. 26. (i) $3 \times 4$ (ii) $12$ (iii) $a_{13}=19$ , $a_{21}=35$ , $a_{33}=-5$ , $a_{24}=12$ , $a_{23}=\dfrac{5}{2}$ .
  27. 27. $\dfrac{\pi x}{2}-\dfrac{x^2}{2}+C$ .
  28. 28. $\dfrac{x}{2}-\dfrac{\sin(4x+10)}{8}+C$ .
  29. 29. (i) $(AB)'=B'A'=\begin{bmatrix}-1 & 4 & -3 \\ 2 & -8 & 6 \\ 1 & -4 & 3\end{bmatrix}$ . (ii) $(AB)'=B'A'=\begin{bmatrix}0 & 1 & 2 \\ 0 & 5 & 10 \\ 0 & 7 & 14\end{bmatrix}$ .

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