Practice Question Papers · with Answers

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.Assume $X, Y, Z, W$ and $P$ are matrices of order $2\times n$ , $3\times k$ , $2\times p$ , $n\times3$ and $p\times k$ , respectively. The restriction on $n, k$ and $p$ so that $PY+WY$ will be defined are:i. $k=3,\ p=n$ii. $k$ is arbitrary, $p=2$iii. $p$ is arbitrary, $k=3$iv. $k=2,\ p=3$[1]
2.If $\theta$ is the angle between two vectors $\vec a$ and $\vec b$ , then $\vec a\cdot\vec b\ge0$ only wheni. $0\lt\theta\lt\dfrac\pi2$ii. $0\le\theta\le\dfrac\pi2$iii. $0\lt\theta\lt\pi$iv. $0\le\theta\le\pi$[1]
3.If the matrix $A$ is both symmetric and skew symmetric, theni. $A$ is a diagonal matrixii. $A$ is a zero matrixiii. $A$ is a square matrixiv. None of these[1]
4.The area bounded by the curve $y=x|x|$ , $x$ -axis and the ordinates $x=-1$ and $x=1$ is given byi. $0$ii. $\dfrac13$iii. $\dfrac23$iv. $\dfrac43$[1]
5.If $\dfrac{d}{dx}f(x)=4x^3-\dfrac3{x^4}$ such that $f(2)=0$ . Then $f(x)$ isi. $x^4+\dfrac1{x^3}-\dfrac{129}{8}$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]
6.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]
7.The value of $\int_{-\pi/2}^{\pi/2}(x^3+x\cos x+\tan^5x+1)\,dx$ isi. $0$ii. $2$iii. $\pi$iv. $1$[1]
8.Which of the following is a homogeneous differential equation?i. $(4x+6y+5)dy-(3y+2x+4)dx=0$ii. $(xy)dx-(x^3+y^3)dy=0$iii. $(x^3+2y^2)dx+2xy\,dy=0$iv. $y^2dx+(x^2-xy-y^2)dy=0$[1]
9.If $A, B$ are symmetric matrices of same order, then $AB-BA$ is ai. Skew symmetric matrixii. Symmetric matrixiii. Zero matrixiv. Identity matrix[1]
10.The general solution of the differential equation $\dfrac{dy}{dx}=e^{x+y}$ isi. $e^x+e^{-y}=C$ii. $e^x+e^y=C$iii. $e^{-x}+e^y=C$iv. $e^{-x}+e^{-y}=C$[1]
11.Area lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and the lines $x=0$ and $x=2$ isi. $\pi$ii. $\dfrac{\pi}{2}$iii. $\dfrac{\pi}{3}$iv. $\dfrac{\pi}{4}$[1]
12.The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x)=3x^2+36x+5$ . The marginal revenue, when $x=15$ isi. $116$ii. $96$iii. $90$iv. $126$[1]
13.$\int \dfrac{dx}{\sqrt{9x-4x^2}}$ equalsi. $\dfrac19\sin^{-1}\left(\dfrac{9x-8}{8}\right)+C$ii. $\dfrac12\sin^{-1}\left(\dfrac{8x-9}{9}\right)+C$iii. $\dfrac13\sin^{-1}\left(\dfrac{9x-8}{8}\right)+C$iv. $\dfrac12\sin^{-1}\left(\dfrac{9x-8}{9}\right)+C$[1]
14.If $A$ and $B$ are two events such that $A\subset B$ and $P(B)\ne0$ , then which of the following is correct?i. $P(A|B)=\dfrac{P(B)}{P(A)}$ii. $P(A|B)\lt P(A)$iii. $P(A|B)\ge P(A)$iv. None of these[1]
15.The general solution of a differential equation of the type $\dfrac{dx}{dy}+P_1x=Q_1$ isi. $y e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C$ii. $y e^{\int P_1\,dx}=\int\left(Q_1e^{\int P_1\,dx}\right)dx+C$iii. $x e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C$iv. $x e^{\int P_1\,dx}=\int\left(Q_1e^{\int P_1\,dx}\right)dx+C$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Find the shortest distance between the lines whose vector equations are $\vec r=\hat{i}+2\hat{j}+3\hat{k}+\lambda(\hat{i}-3\hat{j}+2\hat{k})$ and $\vec r=4\hat{i}+5\hat{j}+6\hat{k}+\mu(2\hat{i}+3\hat{j}+\hat{k})$ .[2]
17.If $x=a(\cos\theta+\theta\sin\theta)$ , $y=a(\sin\theta-\theta\cos\theta)$ , find $\dfrac{dy}{dx}$ .[2]
18.Find the integral of $\sin 4x\sin 8x$ .[2]
19.If $x\sqrt{1+y}+y\sqrt{1+x}=0$ , $-1\lt x\lt1$ , prove that $\dfrac{dy}{dx}=-\dfrac1{(1+x)^2}$ .[2]
20.Given that $E$ and $F$ are events such that $P(E)=0.6$ , $P(F)=0.3$ and $P(E\cap F)=0.2$ , find $P(E|F)$ and $P(F|E)$ .[2]
21.If $y=5\cos x-3\sin x$ , prove that $\dfrac{d^2y}{dx^2}+y=0$ .[2]
22.Evaluate $\int_{2}^{3}\dfrac{dx}{x^2-1}$ .[2]
23.Prove that the logarithmic function is increasing on $(0,\infty)$ .[2]
24.Suppose a girl throws a die. If she gets a $5$ or $6$ , she tosses a coin three times and notes the number of heads. If she gets $1,2,3$ or $4$ , she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw $1,2,3$ or $4$ with the die?[2]
25.Using cofactors of elements of third column, evaluate $\Delta = \begin{vmatrix}1 & x & yz \\ 1 & y & zx \\ 1 & z & xy\end{vmatrix}$ .[2]
26.Consider the experiment of throwing a die; if a multiple of $3$ comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event the coin shows a tail, given that at least one die shows a $3$ .[2]
27.Find the value of $\tan^{-1}(1) + \cos^{-1}\left(-\dfrac{1}{2}\right) + \sin^{-1}\left(-\dfrac{1}{2}\right)$ .[2]
28.Of all the closed cylindrical cans of volume $100$ cm $^3$ , find the dimensions of the can which has the minimum surface area.[2]
29.Examine the consistency of the system of equations $2x-y=5$ , $x+y=4$ .[2]
🔑 Show Answer Key — Set 1
  1. 1. (i) $k=3,\ p=n$
  2. 2. Option (ii), $0\le\theta\le\dfrac\pi2$ .
  3. 3. (ii) $A$ is a zero matrix
  4. 4. $\dfrac23$ , option (iii).
  5. 5. Option (i), $x^4+\dfrac1{x^3}-\dfrac{129}{8}$ .
  6. 6. Options (i) and (ii): $\cos x$ and $\cos 2x$ .
  7. 7. Option (iii), $\pi$ .
  8. 8. Option (iv), $y^2dx+(x^2-xy-y^2)dy=0$ .
  9. 9. (i) Skew symmetric matrix
  10. 10. Option (i), $e^x+e^{-y}=C$ .
  11. 11. $\pi$ , option (i).
  12. 12. $126$ , option (iv).
  13. 13. Option (ii), $\dfrac12\sin^{-1}\left(\dfrac{8x-9}{9}\right)+C$ .
  14. 14. $P(A|B)\ge P(A)$ , option (iii).
  15. 15. Option (iii), $x e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C$ .
  16. 16. $\dfrac3{\sqrt{19}}$ .
  17. 17. $\tan\theta$ .
  18. 18. $\dfrac{\sin4x}{8}-\dfrac{\sin12x}{24}+C$ .
  19. 19. $\dfrac{dy}{dx}=-\dfrac1{(1+x)^2}$ .
  20. 20. $P(E|F)=\dfrac23$ , $P(F|E)=\dfrac13$ .
  21. 21. $y''+y=0$ .
  22. 22. $\dfrac12\log\dfrac32$ .
  23. 23. $\log x$ is increasing on $(0,\infty)$ .
  24. 24. $\dfrac8{11}$ .
  25. 25. $\Delta=(x-y)(y-z)(z-x)$ .
  26. 26. $0$ .
  27. 27. $\dfrac{3\pi}{4}$
  28. 28. $r=\left(\dfrac{50}{\pi}\right)^{1/3}$ cm and $h=2\left(\dfrac{50}{\pi}\right)^{1/3}$ cm.
  29. 29. The system is consistent with unique solution $(x,y)=(3,1)$ .
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.$\int_{1}^{\sqrt3}\dfrac{dx}{1+x^2}$ equalsi. $\dfrac{\pi}{3}$ii. $\dfrac{2\pi}{3}$iii. $\dfrac{\pi}{6}$iv. $\dfrac{\pi}{12}$[1]
2.If $\sin^{-1}x = y$ , theni. $0 \le y \le \pi$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]
3.The general solution of the differential equation $e^xdy+(ye^x+2x)dx=0$ isi. $xe^y+x^2=C$ii. $xe^y+y^2=C$iii. $ye^x+x^2=C$iv. $ye^y+x^2=C$[1]
4.Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = 3x$ . Choose the correct answer.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]
5.If $A$ and $B$ are any two events such that $P(A)+P(B)-P(A\cap B)=P(A)$ , then choose the correct answer.i. $P(B|A)=1$ii. $P(A|B)=1$iii. $P(B|A)=0$iv. $P(A|B)=0$[1]
6.Let $R$ be the relation in the set $\mathbb{N}$ given by $R = \{(a, b) : a = b - 2,\ b > 6\}$ . Choose the correct answer.i. $(2, 4) \in R$ii. $(3, 8) \in R$iii. $(6, 8) \in R$iv. $(8, 7) \in R$[1]
7.If $f(a+b-x)=f(x)$ , then $\int_a^b x f(x)\,dx$ is equal toi. $\dfrac{a+b}{2}\int_a^b f(b-x)\,dx$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]
8.$\int\dfrac{\cos2x}{(\sin x+\cos x)^2}\,dx$ is equal toi. $-\dfrac{1}{\sin x+\cos x}+C$ii. $\log|\sin x+\cos x|+C$iii. $\log|\sin x-\cos x|+C$iv. $\dfrac{1}{(\sin x+\cos x)^2}$[1]
9.$A=[a_{ij}]_{m\times n}$ is a square matrix, ifi. $m \lt n$ii. $m \gt n$iii. $m=n$iv. None of these[1]
10.$\int x^2e^{x^3}\,dx$ equalsi. $\dfrac13e^{x^3}+C$ii. $\dfrac13e^{x^2}+C$iii. $\dfrac12e^{x^3}+C$iv. $\dfrac12e^{x^2}+C$[1]
11.$\int \sqrt{x^2-8x+7}\,dx$ is equal toi. $\dfrac12(x-4)\sqrt{x^2-8x+7}+9\log|x-4+\sqrt{x^2-8x+7}|+C$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]
12.If $A$ is an invertible matrix of order 2, then $\det(A^{-1})$ is equal toi. $\det(A)$ii. $\dfrac{1}{\det(A)}$iii. $1$iv. $0$[1]
13.If $\begin{vmatrix}x & 2 \\ 18 & x\end{vmatrix}=\begin{vmatrix}6 & 2 \\ 18 & 6\end{vmatrix}$ , then $x$ isi. $6$ii. $\pm6$iii. $-6$iv. $0$[1]
14.The interval in which $y=x^2e^{-x}$ is increasing isi. $(-\infty,\infty)$ii. $(-2,0)$iii. $(2,\infty)$iv. $(0,2)$[1]
15.The order of the differential equation $x^2\dfrac{d^2y}{dx^2}-3\dfrac{dy}{dx}+2y=0$ isi. $2$ii. $1$iii. $0$iv. not defined[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Integrate the function $\dfrac{\sqrt{\tan x}}{\sin x\cos x}$ .[2]
17.Solve the differential equation $\dfrac{dy}{dx}+\dfrac{y}{x}=x^2$ .[2]
18.Find the second order derivative of $x^{20}$ .[2]
19.By using the properties of definite integrals, evaluate $\int_{0}^{\pi}\dfrac{x\,dx}{1+\sin x}$ .[2]
20.Solve system of linear equations, using matrix method: $2x-y=-2$ , $3x+4y=3$ .[2]
21.Evaluate $P(A\cup B)$ , if $2P(A)=P(B)=\dfrac{5}{13}$ and $P(A|B)=\dfrac25$ .[2]
22.Evaluate $\int_{0}^{\pi/2}\cos^2x\,dx$ .[2]
23.Integrate the function $\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)$ .[2]
24.Integrate the function $\sqrt{x^2+4x-5}$ .[2]
25.Differentiate $x^x-2^{\sin x}$ w.r.t. $x$ .[2]
26.Discuss the continuity of the cosine, cosecant, secant and cotangent functions.[2]
27.Integrate the function $\dfrac{x\cos^{-1}x}{\sqrt{1-x^2}}$ .[2]
28.By using the properties of definite integrals, evaluate $\int_{0}^{\pi/2}(2\log\sin x-\log\sin2x)\,dx$ .[2]
29.Integrate the function $(x^2+1)\log x$ .[2]
🔑 Show Answer Key — Set 2
  1. 1. Option (iv), $\dfrac{\pi}{12}$ .
  2. 2. (ii) $-\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2}$
  3. 3. Option (iii), $ye^x+x^2=C$ .
  4. 4. (i) $f$ is one-one onto.
  5. 5. Option (ii), $P(A|B)=1$ .
  6. 6. (iii) $(6, 8) \in R$
  7. 7. Option (iv), $\dfrac{a+b}{2}\int_a^b f(x)\,dx$ .
  8. 8. Option (ii), $\log|\sin x+\cos x|+C$ .
  9. 9. (iii) $m=n$
  10. 10. Option (i), $\dfrac13e^{x^3}+C$ .
  11. 11. Option (iv), $\dfrac12(x-4)\sqrt{x^2-8x+7}-\dfrac92\log|x-4+\sqrt{x^2-8x+7}|+C$ .
  12. 12. (ii) $\dfrac{1}{\det(A)}$ .
  13. 13. (ii) $\pm6$ .
  14. 14. $(0,2)$ , option (iv).
  15. 15. Option (i), $2$ .
  16. 16. $2\sqrt{\tan x}+C$ .
  17. 17. $y=\dfrac{x^3}{4}+\dfrac{C}{x}$ .
  18. 18. $380x^{18}$ .
  19. 19. $\pi$ .
  20. 20. $(x,y)=\left(-\dfrac5{11},\dfrac{12}{11}\right)$ .
  21. 21. $\dfrac{11}{26}$ .
  22. 22. $\dfrac{\pi}{4}$ .
  23. 23. $2x\tan^{-1}x-\og(1+x^2)+C$ .
  24. 24. $\dfrac{x+2}{2}\sqrt{x^2+4x-5}-\dfrac92\log|x+2+\sqrt{x^2+4x-5}|+C$ .
  25. 25. $x^x(1+\log x)-2^{\sin x}(\log2)\cos x$ .
  26. 26. $\cos x$ is continuous on $\mathbb{R}$ ; $\csc x$ and $\cot x$ for $x\ne n\pi$ ; $\sec x$ for $x\ne\dfrac\pi2+n\pi$ .
  27. 27. $-\sqrt{1-x^2}\cos^{-1}x-x+C$ .
  28. 28. $-\dfrac{\pi}{2}\log2$ .
  29. 29. $\left(\dfrac{x^3}{3}+x\right)\log x-\dfrac{x^3}{9}-x+C$ .
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.If $\Delta = \begin{vmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{vmatrix}$ and $A_{ij}$ is the cofactor of $a_{ij}$ , then $\Delta$ is equal toi. $a_{11}A_{31}+a_{12}A_{32}+a_{13}A_{33}$ii. $a_{11}A_{11}+a_{12}A_{21}+a_{13}A_{31}$iii. $a_{21}A_{11}+a_{22}A_{12}+a_{23}A_{13}$iv. $a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}$[1]
2.The general solution of the differential equation $\dfrac{y\,dx-x\,dy}{y}=0$ isi. $xy=C$ii. $x=Cy^2$iii. $y=Cx$iv. $y=Cx^2$[1]
3.For all real values of $x$ , the minimum value of $\dfrac{1-x+x^2}{1+x+x^2}$ isi. $0$ii. $1$iii. $3$iv. $\dfrac13$[1]
4.$\int \dfrac{dx}{x^2+2x+2}$ equalsi. $x\tan^{-1}(x+1)+C$ii. $\tan^{-1}(x+1)+C$iii. $(x+1)\tan^{-1}x+C$iv. $\tan^{-1}x+C$[1]
5.$\int \dfrac{x\,dx}{(x-1)(x-2)}$ equalsi. $\log\left|\dfrac{(x-1)^2}{x-2}\right|+C$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]
6.$\sin^{-1}(1-x)-2\sin^{-1}x=\dfrac{\pi}{2}$ , then $x$ is equal toi. $0,\ \dfrac{1}{2}$ii. $1,\ \dfrac{1}{2}$iii. $0$iv. $\dfrac{1}{2}$[1]
7.If $A=\begin{bmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha\end{bmatrix}$ , and $A+A'=I$ , then the value of $\alpha$ isi. $\dfrac{\pi}{6}$ii. $\dfrac{\pi}{3}$iii. $\pi$iv. $\dfrac{3\pi}{2}$[1]
8.If $\vec a$ is a nonzero vector of magnitude $a$ and $\lambda$ a nonzero scalar, then $\lambda\vec a$ is unit vector ifi. $\lambda=1$ii. $\lambda=-1$iii. $a=|\lambda|$iv. $a=1/|\lambda|$[1]
9.Matrices $A$ and $B$ will be inverse of each other only ifi. $AB=BA$ii. $AB=BA=0$iii. $AB=0,\ BA=I$iv. $AB=BA=I$[1]
10.If $A$ and $B$ are events such that $P(A|B)=P(B|A)$ , theni. $A\subset B$ but $A\ne B$ii. $A=B$iii. $A\cap B=\phi$iv. $P(A)=P(B)$[1]
11.The degree of the differential equation $\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^2+\sin\left(\dfrac{dy}{dx}\right)+1=0$ isi. $3$ii. $2$iii. $1$iv. not defined[1]
12.Two events $A$ and $B$ will be independent, ifi. $A$ and $B$ are mutually exclusiveii. $P(A^prime B^prime)=[1-P(A)][1-P(B)]$iii. $P(A)=P(B)$iv. $P(A)+P(B)=1$[1]
13.A homogeneous differential equation of the form $\dfrac{dx}{dy}=h\left(\dfrac{x}{y}\right)$ can be solved by making the substitution.i. $y=vx$ii. $v=yx$iii. $x=vy$iv. $x=v$[1]
14.Let $A = \{1, 2, 3\}$ . Then number of equivalence relations containing $(1, 2)$ isi. $1$ii. $2$iii. $3$iv. $4$[1]
15.The value of $\hat{i}\cdot(\hat{j}\times\hat{k})+\hat{j}\cdot(\hat{i}\times\hat{k})+\hat{k}\cdot(\hat{i}\times\hat{j})$ isi. $0$ii. $-1$iii. $1$iv. $3$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Minimise $Z=x+2y$ subject to $2x+y\ge3$ , $x+2y\ge6$ , $x\ge0$ , $y\ge0$ . Show that the minimum of $Z$ occurs at more than two points.[2]
17.Prove that $\int_0^{\pi/2}\sin^3x\,dx=\dfrac23$ .[2]
18.Find the equation of the line which passes through the point $(1,2,3)$ and is parallel to the vector $3\hat{i}+2\hat{j}-2\hat{k}$ .[2]
19.If a line has the direction ratios $-18,12,-4$ , then what are its direction cosines?[2]
20.In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3 - 4x$ (ii) $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 1 + x^2$ .[2]
21.Find $a$ and $b$ such that $f(x)=\begin{cases}5, & x\le2\\ax+b, & 2\lt x\lt10\\21, & x\ge10\end{cases}$ is continuous.[2]
22.Find the value of $\tan^{-1}\left(\tan\dfrac{3\pi}{4}\right)$ .[2]
23.Evaluate $\int_{0}^{1}\dfrac{x}{x^2+1}\,dx$ using substitution.[2]
24.Find the points at which the function $f$ given by $f(x)=(x-2)^4(x+1)^3$ has (i) local maxima (ii) local minima (iii) point of inflexion.[2]
25.Find values of $k$ if area of triangle is 4 square units and vertices are: (i) $(k,0)$ , $(4,0)$ , $(0,2)$ (ii) $(-2,0)$ , $(0,4)$ , $(0,k)$ .[2]
26.Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (i) $f(x)=x^2$ (ii) $g(x)=x^3-3x$ (iii) $h(x)=\sin x+\cos x$ , $0\lt x\lt\dfrac{\pi}{2}$ (iv) $f(x)=\sin x-\cos x$ , $0\lt x\lt2\pi$ (v) $f(x)=x^3-6x^2+9x+15$ (vi) $g(x)=\dfrac{x}{2}+\dfrac{2}{x}$ , $x\gt0$ (vii) $g(x)=\dfrac{1}{x^2+2}$ (viii) $f(x)=x\sqrt{1-x}$ , $0\lt x\lt1$ .[2]
27.Find $\dfrac{dy}{dx}$ if $x^3+x^2y+xy^2+y^3=81$ .[2]
28.Integrate the function $\cot x\log\sin x$ .[2]
29.Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\dfrac{4r}{3}$ .[2]
🔑 Show Answer Key — Set 3
  1. 1. (iv) $a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}$ .
  2. 2. Option (iii), $y=Cx$ .
  3. 3. $\dfrac13$ , option (iv).
  4. 4. Option (ii), $\tan^{-1}(x+1)+C$ .
  5. 5. Option (ii), $\log\left|\dfrac{(x-2)^2}{x-1}\right|+C$ .
  6. 6. (iii) $0$
  7. 7. (ii) $\dfrac{\pi}{3}$
  8. 8. Option (iv), $a=1/|\lambda|$ .
  9. 9. (iv) $AB=BA=I$
  10. 10. $P(A)=P(B)$ , option (iv).
  11. 11. Option (iv), not defined.
  12. 12. $P(A^prime B^prime)=[1-P(A)][1-P(B)]$ , option (ii).
  13. 13. Option (iii), $x=vy$ .
  14. 14. (ii) $2$
  15. 15. Option (iii), $1$ .
  16. 16. Minimum $Z=6$ at every point of the segment $x+2y=6$ , $0\le x\le6$ , $y=3-\dfrac{x}{2}$ .
  17. 17. $\int_0^{\pi/2}\sin^3x\,dx=\dfrac23$ .
  18. 18. $\vec r=\hat{i}+2\hat{j}+3\hat{k}+\lambda(3\hat{i}+2\hat{j}-2\hat{k})$ ; cartesian form $\dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z-3}{-2}$ .
  19. 19. $-\dfrac9{11},\dfrac6{11},-\dfrac2{11}$ .
  20. 20. (i) $f$ is bijective (one-one and onto). (ii) $f$ is neither one-one nor onto.
  21. 21. $a=2$ , $b=1$ .
  22. 22. $-\dfrac{\pi}{4}$
  23. 23. $\dfrac12\log2$ .
  24. 24. Local maximum at $x=\dfrac27$ ; local minimum at $x=2$ ; points of inflexion at $x=-1$ and $x=\dfrac{2\pm3\sqrt2}{7}$ .
  25. 25. In both cases, $k=0$ or $k=8$ .
  26. 26. (i) Local minimum $0$ at $x=0$ . (ii) Local maximum $2$ at $x=-1$ , local minimum $-2$ at $x=1$ . (iii) Local maximum $\sqrt2$ at $x=\pi/4$ . (iv) Local maximum $\sqrt2$ at $x=3\pi/4$ , local minimum $-\sqrt2$ at $x=7\pi/4$ . (v) Local maximum $19$ at $x=1$ , local minimum $15$ at $x=3$ . (vi) Local minimum $2$ at $x=2$ . (vii) Local maximum $1/2$ at $x=0$ . (viii) Local maximum $\dfrac{2}{3\sqrt3}$ at $x=2/3$ .
  27. 27. $-\dfrac{3x^2+2xy+y^2}{x^2+2xy+3y^2}$ .
  28. 28. $\dfrac12(\log\sin x)^2+C$ .
  29. 29. The altitude is $\dfrac{4r}{3}$ .

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