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Samacheer Kalvi Class 12 Maths Practice Question Papers

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Brain Grain · braingrain.in
Maths — Practice Paper · Set 1
Class: 12Samacheer KalviMax Marks: 93
Name: ____________________Reg No: ____________
Part I — Multiple Choice Questions 15 × 1 = 15

Choose the correct answer. (Answer all questions.)

1.If A, B and C are invertible matrices of some order, then which one of the following is not true? (A) adj A = |A|A -1 (B) adj (AB) = (adj A)(adj B) (C) det A -1 = (det A) -1 (D) (ABC) -1 = C -1 B -1 A -1[1]
2.By using Gaussian elimination method, balance the chemical reaction equation: C6H12O6 + O2 → CO2 + H2O.[1]
3.The tangent to the curve y² – xy + 9 = 0 is vertical when (A) y = 0 (B) y = ±√3 (C) –$\frac { 1 }{ 2 }$ (D) y = ±3[1]
4.If the distance of the point $(1,1,1)$ from the origin is half of its distance from the plane $x+y+z+k=0$, then the values of $k$ are (A) $\pm3$ (B) $\pm6$ (C) $-3,9$ (D) $3,-9$[1]
5.The change in surface area S(x)=6x^2 of a cube when edge changes from x_0 to x_0+dx is ?[1]
6.Find the parameter values for which the given system has the stated rank condition.[1]
7.Find $k$ from the given rotation-matrix condition. Options: (A) $0$ (B) $\sin\theta$ (C) $\cos\theta$ (D) $1$.[1]
8.$\sin^{-1}(\cos x)=\dfrac\pi2-x$ is valid for (A) $-\pi\le x\le0$ (B) $0\le x\le\pi$ (C) $-\dfrac\pi2\le x\le\dfrac\pi2$ (D) $-\dfrac\pi4\le x\le\dfrac{3\pi}4$[1]
9.sin -1 (tan$\frac {π}{4}$) – sin -1 ($\sqrt{\frac {3}{x}}$) = $\frac {π}{6}$. Then x is root of the equation (A) x² – x – 6 = 0 (B) x² – x – 12 = 0 (C) x² + x – 12 = 0 (D) x² + x – 6 = 0[1]
10.If A = $\begin{bmatrix} 7 & 3 \\ 4 & 2 \end{bmatrix}$ then 9I 2 – A = (A) A -1 (B) $\frac{A^{-1}}{2}$ (C) 3A -1 (D) 2A -1[1]
11.If the length of the perpendicular from the origin to the plane 2x + 3y + λz = 1, λ > 0 is $\frac { 1 }{ 5 }$, then the value of λ is (A) 2√3 (B) 3√2 (C) 0 (D) 1[1]
12.Find the number of the solutions of the equations tan -1 (x – 1) + tan -1 x + tan -1 (x + 1) = tan -1 3x[1]
13.If the line $\dfrac{x-2}{3}=\dfrac{y-1}{-5}=\dfrac{z+2}{2}$ lies in the plane $x+3y-\alpha z+\beta=0$, then $(\alpha,\beta)$ is (A) $(-5,5)$ (B) $(-6,7)$ (C) $(5,-5)$ (D) $(6,-7)$[1]
14.The equation tan -1 x – cot -1 x = tan -1 ($\frac {1}{√3}$) has (A) no solution (B) unique solution (C) two solutions (D) infinite number of solutions[1]
15.The degree of the differential equation $y=x\left(1+\dfrac{dy}{dx}+\dfrac1{2!}\left(\dfrac{dy}{dx}\right)^2+\dfrac1{3!}\left(\dfrac{dy}{dx}\right)^3+\cdots\right)$ is (A) $2$ (B) $3$ (C) $1$ (D) $4$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Find $\left(\dfrac{1+\sin(\pi/10)+i\cos(\pi/10)}{1+\sin(\pi/10)-i\cos(\pi/10)}\right)^{10}$.[2]
17.If his speed never exceeds 150 km/hr, what is the maximum kilometre stone he can reach in the next two hours?[2]
18.Show that p → q and q → p are not equivalent.[2]
19.The principal argument of $(\sin40^\circ+i\cos40^\circ)^5$ is: (1) $-110^\circ$ (2) $-70^\circ$ (3) $70^\circ$ (4) $110^\circ$.[2]
20.The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. Find (i) absolute error (ii) relative error (iii) percentage error in the radius.[2]
21.Find the inverse of each of the following by Gauss–Jordan method: (i) [[2,1],[5,2]] (ii) [[1,1,0],[1,0,1],[6,2,3]] (iii) [[1,2,3],[2,5,3],[1,0,8]].[2]
22.The relation between the number of words y a person learns in x hours is given by y = 52√x, 0 ≤ x ≤ 9. What is the approximate number of words learned when x changes from (i) 1 to 1.1 hours? (ii) 4 to 4.1 hours?[2]
23.Write the Maclaurin series expansion of the following functions: (i) e^x (ii) sin x (iii) cos x (iv) ln(1-x), -1 ≤ x <1 (v) arctan x, |x| ≤1 (vi) cos 2x.[2]
24.Find the centre and radius of the following circles. (i) x² + (y + 2)² = 0 (ii) x² + y² + 6x – 4y + 4 = 0 (iii) x² + y² – x + 2y – 3 = 0 (iv) 2x² + 2y² – 6x + 4y + 2 = 0[2]
25.Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle.[2]
26.Does there exist a differentiable function f such that f(0)=0, f(1)=2 and f'(x)≤2 for all x? Justify your answer.[2]
27.A particle moves along a straight line in such a way that after t seconds its distance from the origin is s(t)=t^3+2t^2 metres. (i) Find the average velocity between t=3 and t=6 seconds. (ii) Find the instantaneous velocities at t=3 and t=6 seconds.[2]
28.The values of m for which the line y = mx + 2√5 touches the hyperbola 16x² – 9y² = 144 are the roots of x² – (a + b)x – 4 = 0, then the value of (a + b) is (A) 2 (B) 4 (C) 0 (D) -2[2]
29.Evaluate $\lim _{(x, y) \rightarrow(0,0)}$ cos($\frac { x^3+y^2 }{ x+y+2 }$), if the limit exists.[2]
Part III — Long Answer Questions 10 × 5 = 50

Answer in detail. (Answer all questions.)

30.If $\overline { a }$ = $\hat { i }$ – 2$\hat { j }$ + 3$\hat { k }$, b = 2$\hat { i }$ + $\hat { j }$ – 2$\hat { k }$, c = 3$\hat { i }$ + 2$\hat { j }$ + $\hat { k }$ find $\overline { a }$.($\overline { b}$ × $\overline { c }$).[5]
31.Find parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, -2, 3) and parallel to the straight line passing through the points (2, 1, -3) and (-1, 5, -8)[5]
32.Let $\overline { a }$ = $\hat { i }$ + $\hat { j }$ + $\hat { k }$, $\overline { b }$ = $\hat { i }$ and $\overline { c }$ = c 1 $\hat { i }$ + c 2 $\hat { j }$ + c 3 $\hat { k }$. If c 1 = 1 and c 2 = 2, find c 3 such that $\overline { a }$, $\overline { b }$ and $\overline { c }$ are coplanar.[5]
33.The volume of a cylinder is given by the formula V = \pi r^2 h. For fixed volume V, show that the cylinder of minimum total surface area has height equal to the diameter (h = 2r).[5]
34.Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.[5]
35.For any vector a, prove that i×(a×i) + j×(a×j) + k×(a×k) = 2a.[5]
36.A family of 3 people went out for dinner. The cost of 2 dosai, 3 idlies and 2 vadais is ₹150. The cost of 2 dosai, 2 idlies and 4 vadais is ₹200. The cost of 5 dosai, 4 idlies and 2 vadais is ₹250. Find the cost of one dosai, one idli and one vadai. Will a family with ₹350 who ate 3 dosai, 6 idlies and 6 vadais be able to pay the bill?[5]
37.Show that the line x - y + 4 = 0 is a tangent to the ellipse x^2 + 3y^2 = 12. Also find the coordinates of the point of contact. (Hint: use parametric form)[5]
38.Find the square roots of (i) 4+3i (ii) 6+8i (iii) −5+12i.[5]
39.A boy is walking along the path y = ax^2 + bx + c through the points (−6, 8), (−2, 12) and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination.)[5]
🔑 Show Answer Key — Set 1
  1. 1. adj (AB) = (adj A)(adj B)
  2. 2. C6H12O6 + 6 O2 → 6 CO2 + 6 H2O
  3. 3. y = ±3
  4. 4. $k=3$ or $k=-9$
  5. 5. ΔS ≈ dS = 12 x_0 dx
  6. 6. Answer: The preserved key is $\lambda=7$, $\mu=-5$. The augmented matrix is missing from the text extract, so this item is not marked validated. Q.21 Let X = { 1, 2, 3, 4 }, Y = { a, b, c, d } and f = { (1, a), (4, b), (2, c), (3, d), (2, d) }. Then f is (1) a one-to-one function (2) an onto function (3) a function which is not one-to-one (4) not a function (1)) 2 (2)) 4 (3)) 3 (4)) 1 Answer: (4) not a function. Explanation: The set contains both (2,c) and (2,d), so the element 2 in the domain is assigned two different images. That violates the definition of a function (which must assign exactly one image to each domain element).
  7. 7. Answer: The preserved key is $k=1$, using $\cos^2\theta+\sin^2\theta=1$. The full condition involving $k$ is missing from the text extract, so this item is not marked validated. Q.16 If $A=\begin{bmatrix}2&3\\5&-2\end{bmatrix}$ is such that $\lambda A^{-1}=A$, then $\lambda$ is 1) $17$ 2) $14$ 3) $19$ 4) $21$ Answer: Option 3 From $\lambda A^{-1}=A$, multiplying both sides by $A$ gives $\lambda I=A^2$. Now $A^2=\begin{bmatrix}2&3\\5&-2\end{bmatrix}^2=\begin{bmatrix}4+15&6-6\\10-10&15+4\end{bmatrix}=\begin{bmatrix}19&0\\0&19\end{bmatrix}=19I$. Hence $\lambda I=19I$, so $\lambda=19$.
  8. 8. $0\le x\le\pi$
  9. 9. x² – x – 12 = 0
  10. 10. 2A -1
  11. 11. 2√3
  12. 12. x = 0, x² = 1 x = ±1 Number of solutions are three (0, 1 -1)
  13. 13. $(-6,7)$
  14. 14. unique solution
  15. 15. $1$
  16. 16. $1$.
  17. 17. 300 km farther (maximum displacement 300 km).
  18. 18. Not equivalent in general; give a counterexample: p = True, q = False.
  19. 19. Option 1: $-110^\circ$.
  20. 20. (i) 0.15 cm (ii) 0.012 (iii) 1.2%
  21. 21. s: (i) A = [[2,1],[5,2]], det(A)=2·2−1·5=−1. A^{-1} = (1/det)[[2,−1],[−5,2]] = [[−2,1],[5,−2]]. (ii) A = [[1,1,0],[1,0,1],[6,2,3]]. det(A)=1. Using cofactors one finds A^{-1} = [[−2,−3,1],[3,3,−1],[2,4,−1]]. (iii) A = [[1,2,3],[2,5,3],[1,0,8]], det(A)=−1. The adjugate is [[40,−16,−9],[−13,5,3],[−5,2,1]] so A^{-1} = (1/−1)·adj(A) = [[−40,16,9],[13,−5,−3],[5,−2,−1]].
  22. 22. $\frac { 26 }{ √4 }$ × 0.1 = 13 × 0.1 = 1.3 ≅ 1 word
  23. 23. (i) e^x=∑_{n=0}^∞ x^n/n!. (ii) sin x=∑_{n=0}^∞ (-1)^n x^{2n+1}/(2n+1)!. (iii) cos x=∑_{n=0}^∞ (-1)^n x^{2n}/(2n)!. (iv) ln(1-x)=-∑_{n=1}^∞ x^n/n, |x|<1. (v) arctan x=∑_{n=0}^∞ (-1)^n x^{2n+1}/(2n+1), |x|≤1 (endpoint x=±1 conditional). (vi) cos2x=∑_{n=0}^∞ (-1)^n (2x)^{2n}/(2n)! =∑_{n=0}^∞ (-1)^n 2^{2n} x^{2n}/(2n)!.
  24. 24. So centre = (-g, -f) = ($\frac{3}{2}$, -1) and radius = $\sqrt{g^{2}+f^{2}-c}=\sqrt{\frac{9}{4}+1-1}=\frac{3}{2}$ ∴ Centre = ($\frac{3}{2}$, -1) and radius = $\frac{3}{2}$
  25. 25. If diagonals equal in length then adjacent sides are perpendicular, so parallelogram is rectangle.
  26. 26. Yes. Example: f(x)=2x (or any function with derivative ≤2 and achieving the values).
  27. 27. (i) 81 m/s. (ii) v(3)=39 m/s, v(6)=132 m/s.
  28. 28. 0
  29. 29. = cos ($\frac { 0+0 }{ 0+0+2 }$) = cos 0 = $\lim _{(x, y) \rightarrow(0,0)}$ cos($\frac { x^3+y^2 }{ x+y+2 }$) = 1
  30. 30. \end{array}\right|\) = 1(1 + 4) + 2(2 + 6) + 3(4 – 3) = 5 + 16 + 3 = 24
  31. 31. -12x + 11y + 16z = 14 12x – 11y – 16z = -14 12x – 11y – 16z + 14 = 0
  32. 32. 1(0) – 1(c 3 ) + 1(c 2 ) = 0 -c 3 + c 2 = 0 c 3 = c 2 = 2
  33. 33. h = 2r
  34. 34. ρ(A) = 2, ρ(A | B) =2 ρ(A) = ρ(A | B) = 2 < n The system is consistent. It has infinitely many solution.
  35. 35. Use vector triple product identity: p×(q×r) = q(p·r) - r(p·q). For p = i, q = a, r = i we get i×(a×i) = a(i·i) - i(i·a) = a - i a_x, where a_x = i·a. Similarly j×(a×j) = a - j a_y and k×(a×k) = a - k a_z. Summing gives 3a - (a_x i + a_y j + a_z k) = 3a - a = 2a. Hence proved.
  36. 36. Price per item: dosai = ₹30, idli = ₹10, vadai = ₹30. Bill for 3 dosai,6 idli,6 vadai = ₹330 ≤ ₹350, so yes (₹20 change).
  37. 37. The line is a tangent, and the point of contact is (-3, 1).
  38. 38. (i) ±(3/√2 + (1/√2)i) (ii) ±(2√2 + √2 i) (iii) ±(2 + 3i)
  39. 39. No; the parabola determined by those three points does not pass through P(7,60).
Brain Grain · braingrain.in
Maths — Practice Paper · Set 2
Class: 12Samacheer KalviMax Marks: 93
Name: ____________________Reg No: ____________
Part I — Multiple Choice Questions 15 × 1 = 15

Choose the correct answer. (Answer all questions.)

1.sin -1 (2 cos²x – 1) + cos -1 (1 – 2 sin²x) = (A) $\frac {π}{2}$ (B) $\frac {π}{3}$ (C) $\frac {π}{4}$ (D) $\frac {π}{6}$[1]
2.If $\operatorname{adj}A$ and $\operatorname{adj}B$ are given, find $\operatorname{adj}(AB)$.[1]
3.If w (x, y) = x y, x > 0, then $\frac{\partial w}{\partial x}$ is equal to (A) x y log x (B) y log x (C) y x y-1 (D) x log y[1]
4.Find the point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is (A) (4, 11) (B) (4, -11) (C) (-4, 11) (D) (-4, -11)[1]
5.If $x^a y^b=e^m$ and $x^c y^d=e^n$, express $x$ and $y$ using Cramer determinants $\Delta_1$, $\Delta_2$ and $\Delta_3$.[1]
6.If cot -1 ($\sqrt {sinα}$) + tan -1 ($\sqrt {sinα}$) = u, then cos 2u is equal to (A) tan²α (B) 0 (C) -1 (D) tan 2α[1]
7.Find $B$ from the given trigonometric matrix equation.[1]
8.If u (x, y) = x² + 3xy + y – 2019, then $\frac{\partial u}{\partial x}$| (4, -5) is equal to (A) -4 (B) -3 (C) -7 (D) 13[1]
9.If $\vec a\cdot\vec b=\vec b\cdot\vec c=\vec c\cdot\vec a=0$, then the value of $[\vec a\ \vec b\ \vec c]$ is (A) $|\vec a||\vec b||\vec c|$ (B) $\tfrac13|\vec a||\vec b||\vec c|$ (C) $1$ (D) $-1$[1]
10.Two coins are flipped independently. The first lands heads with probability $0.6$ and the second with probability $0.5$. Let $X$ be the total number of heads. Find $E(X)$.[1]
11.If $\omega=\operatorname{cis}(2\pi/3)$, then the number of distinct roots of $\begin{vmatrix}z+1&\omega&\omega^2\\\omega&z+\omega^2&1\\\omega^2&1&z+\omega\end{vmatrix}=0$ is: (1) $1$ (2) $2$ (3) $3$ (4) $4$.[1]
12.If P = $\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] $ is the adjoint of 3 × 3 matrix A and |A| = 4, then x is (A) 15 (B) 12 (C) 14 (D) 11[1]
13.tan -1 ($\frac {1}{4}$) + tan -1 ($\frac {2}{9}$) is equal to (A) $\frac {1}{2}$cos -1 ($\frac {3}{5}$) (B) $\frac {1}{2}$sins -1 ($\frac {3}{5}$) (C) $\frac {1}{2}$tan -1 ($\frac {3}{5}$) (D) tan -1 ($\frac {1}{2}$)[1]
14.If $p$ and $q$ are the order and degree of the differential equation $y\dfrac{dy}{dx}+x^3\dfrac{d^2y}{dx^2}+xy=\cos x$, then (A) $p q$ (D) $p$ exists and $q$ does not exist[1]
15.The maximum slope of the tangent to the curve y = e x sin x, x ∈ [0, 2π] is at (A) x = $\frac { π }{ 4 }$ (B) x = $\frac { π }{ 2 }$ (C) x = π (D) x = $\frac { 3π }{ 2 }$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Find the values z_k = cos((2k+1)π/9) + i sin((2k+1)π/9).[2]
17.Evaluate $\lim _{(x, y) \rightarrow(0,0)}$ cos($\frac { x^3+y^2 }{ x+y+2 }$), if the limit exists.[2]
18.If the integrating factor of dy/dx + P(x) y = Q(x) is μ(x)=x, then P(x)= ?[2]
19.Let g(x, y) = $\frac { x^2y }{ x^4+y^2 }$ for (x, y) ≠ (0, 0) and f(0, 0) = 0 (i) Show that $\lim _{(x, y) \rightarrow(0,0)}$ g(x, y) = 0 along every line y = mx, m ∈ R (ii) Show that $\lim _{(x, y) \rightarrow(0,0)}$ g(x, y) = $\frac { k }{ 1+k^2 }$ along every parabola y = kx², k ∈ R\{0}[2]
20.Examine for the rational roots of (i) 2x³ – x² – 1 = 0[2]
21.Find the vertex, focus, equation of directrix and length of the latus rectum of the following: (i) y² = 16x (ii) x² = 24y (iii) y² = -8x (iv) x² – 2x + 8y + 17 = 0 (v) y² – 4y – 8x + 12 = 0[2]
22.Find the equation of the circle with centre (2,3) and passing through the intersection of the lines 3x-2y-1=0 and 4x+2y-7=0. (Assumed second line is 4x+2y-7=0.)[2]
23.If a vector α lies in the plane of β and γ, then which is true of [α β γ]? (options: 1) 1, 2) −1, 3) 0, 4) 2 )[2]
24.A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution? (Use Cramer's rule to solve the problem).[2]
25.Find the area of the region bounded by 3x – 2y + 6 = 0, x = -3, x = 1 and x axis.[2]
26.$\frac { dy }{ dx }$ + $\frac { 3y }{ x }$ = $\frac { 1 }{ x^2 }$, given that y = 2 when x = 1[2]
27.Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions: (i) f(x) = x² – x, x ∈ [0, 1] (ii) f(x) = $\frac { x^2-2x }{ x+2 }$, x ∈ [-1, 6] (iii) f(x) = √x – $\frac { x }{ 3 }$, x ∈ [0, 9][2]
28.A rectangular garden is fenced with 40 m of wire. What is the largest possible area?[2]
29.The value of $\int_{0}^{∞}$ e -3x x² dx (A) $\frac { 7 }{ 27 }$ (B) $\frac { 5 }{ 27 }$ (C) $\frac { 4 }{ 27 }$ (D) $\frac { 2 }{ 27 }$[2]
Part III — Long Answer Questions 10 × 5 = 50

Answer in detail. (Answer all questions.)

30.Find the domain of (i) f(x) = sin^{-1}(x+1/2) (ii) g(x) = sin^{-1}(2x−1/4).[5]
31.Find all values of x such that (i) −10π ≤ x ≤ 10π and sin x = 0. (ii) −3π ≤ x ≤ 3π and sin x = −1.[5]
32.Find the value of (i) tan(tan -1 ($\frac {7π}{4}$)) (ii) tan(tan -1 (1947)) (iii) tan(tan -1 (-0.2021))[5]
33.A ladder 17 m long is leaning against the wall. The base of the ladder is pulled away from the wall at 5 m/s. When the base is 8 m from the wall, (i) how fast is the top moving down? (ii) at what rate is the area of the triangle formed by ladder, wall and floor changing? Also: A police jeep approaching an intersection from the north is chasing a car moving east. When the jeep is 0.6 km north and the car 0.8 km east of the intersection the distance between them is increasing at 20 km/h. If the jeep's speed is 60 km/h at that instant, what is the speed of the car?[5]
34.Find the foot of the perpendicular drawn: from the point (5, 4, 2) to the line $\frac { x+1 }{ 2 }$ = $\frac { y-3 }{ 3 }$ = $\frac { z-1 }{ -1 }$. Also, find the equation of the perpendicular.[5]
35.Verify whether the following are tautologies, contradictions or contingencies: (i) (p ∧ q) ∧ ¬(p ∨ q). (ii) (p ∨ q) ∧ ¬(p → q). (iii) (p → q) ↔ ¬(p → q). (iv) (p → q) ∧ (q → r) → (p → r). (Reconstructed.)[5]
36.Find the least positive integer $n$ for which $(\sqrt3+i)^n$ is (i) real (ii) purely imaginary.[5]
37.If u(x,y)=x^2+4y+3, x=e^t, y=sin t. Find du/dt and evaluate at t=0.[5]
38.Using a truth table check whether the statements (¬p ∨ q) ∨ (¬p ∧ q) and ¬p are logically equivalent.[5]
39.Find Δf and df for the function f for the indicated values of x, Δx and compare: (i) f(x) = x³ – 2x², x = 2, Δx = dx = 0.5 (ii) f(x) = x² + 2x + 3, x = -0.5, Δx = dx = 0.1[5]
🔑 Show Answer Key — Set 2
  1. 1. $\frac {π}{2}$
  2. 2. Method: Use $\operatorname{adj}(AB)=\operatorname{adj}(B)\operatorname{adj}(A)$. The option key in the degraded card is not safely supported by the visible matrix entries, so this item is not marked validated. Q.18 The rank of the matrix [[1,2,3,4],[2,4,6,8],[1,2,3,4]] is: 1) 1 2) 2 3) 4 4) 3 Answer: 1 Rows: r1 = [1,2,3,4], r2 = 2·r1, r3 = r1. All rows are scalar multiples of r1, so only one independent row ⇒ rank = 1. Option (1). <div
  3. 3. y x y-1
  4. 4. (4, 11)
  5. 5. Answer: $x=e^{\Delta_1/\Delta_3}$ and $y=e^{\Delta_2/\Delta_3}$. Taking logarithms gives $a\log x+b\log y=m$ and $c\log x+d\log y=n$. Applying Cramer's rule to $\log x$ and $\log y$ gives the stated exponential form.
  6. 6. -1
  7. 7. Answer: The preserved key is $(\cos^2\frac{\theta}{2})A^T$. The exact matrix equation is missing from the text extract, so this item is not marked validated. <div
  8. 8. -7
  9. 9. $|\vec a||\vec b||\vec c|$
  10. 10. $E(X)=1.1$
  11. 11. Option 1: one distinct root.
  12. 12. 11
  13. 13. tan -1 ($\frac {1}{2}$)
  14. 14. $p>q$
  15. 15. x = $\frac { π }{ 2 }$
  16. 16. z_k = e^{i(2k+1)π/9}, k = 0,1,...,8 (the nine 9th-roots of −1).
  17. 17. = cos ($\frac { 0+0 }{ 0+0+2 }$) = cos 0 = $\lim _{(x, y) \rightarrow(0,0)}$ cos($\frac { x^3+y^2 }{ x+y+2 }$) = 1
  18. 18. Option (3): 1/x
  19. 19. Hence proved (ii) for parabola y = kx² Hence proved
  20. 20. a n = 1; a 0 = 1 If $\frac{p}{q}$ is a root of the polynomial. (p, q) = 1 By rational root theorem, it has no rational roots.
  21. 21. x + 1 = 0 (d) Length of latus rectum is 4a = 4 × 2 = 8 units.
  22. 22. (x-2)^2+(y-3)^2 = 769/196.
  23. 23. 3
  24. 24. 6 litres of 50% solution and 4 litres of 25% solution.
  25. 25. 3x – 2y + 6 = 0 2y = 3x + 6 y = $\frac { 1 }{ 2 }$(3x + 6)
  26. 26. 4 – 1 = c c = 3 ∴ 2yx³ = x² + 3 is a required solution.
  27. 27. Now f'(x) = $\frac { 1 }{ 2√x }$ – $\frac { 1 }{ 3 }$ Since, the tangent is parallel to x-axis. f'(x) = 0
  28. 28. 100 m^2 (square 10 m × 10 m)
  29. 29. $\frac { 2 }{ 27 }$
  30. 30. (i) −3/2 ≤ x ≤ 1/2. (ii) (1/4 −1)/2 ≤ x ≤ (1/4 +1)/2 i.e. −3/4 ≤ x ≤ 5/4.
  31. 31. (i) x = kπ, k = −10, −9, …, 0, …, 9, 10. (ii) x = 3π/2 + 2kπ; within the interval these are x = −5π/2, −π/2, 3π/2.
  32. 32. (i) $\tan \left(\tan ^{-1} \frac{7 \pi}{4}\right)=\frac{7 \pi}{4}$ (ii) tan(tan -1 (1947))= 1947 (iii) tan(tan -1 (-0.2021)) = -0.2021
  33. 33. (Ladder) (i) dy/dt=−8/3 m/s. (ii) dA/dt=161/6 m^2/s. (Police) speed of car = 70 km/h.
  34. 34. t = 1 ∴ F (2 – 1, 3 + 3, -1 + 1) = F (1, 6, 0) is foot point. Equation of the perpendicular. (x 1, y 1, z 1 ) = (5, 4, 2), (x 2, y 2, z 2 ) = (1, 6, 0).
  35. 35. (i) Contradiction. (ii) Contingency (can be true for some valuations). (iii) Contradiction (a statement equivalent to its negation can't be true). (iv) Tautology (transitivity of implication).
  36. 36. (i) $n=6$ (ii) $n=3$.
  37. 37. du/dt = 2e^{2t} + 4 cos t; at t=0, du/dt = 6.
  38. 38. Not equivalent.
  39. 39. f(x) = f(-0.5) = (-0.5) 2 + 2(-0.5) + 3 = 0.25 – 1 + 3 = 3.25 – 1 = 2.25 So ∆ f = f(x + ∆x) – f(x) = 2.36 – 2.25 = 0.11
Brain Grain · braingrain.in
Maths — Practice Paper · Set 3
Class: 12Samacheer KalviMax Marks: 93
Name: ____________________Reg No: ____________
Part I — Multiple Choice Questions 15 × 1 = 15

Choose the correct answer. (Answer all questions.)

1.Using truth table prove that p → (q → r) ≡ ¬p ∨ ¬q ∨ r.[1]
2.If sin -1 $\frac {x}{5}$ + cosec -1 $\frac {5}{4}$ = $\frac {π}{2}$, then the value of x is (A) 4 (B) 5 (C) 2 (D) 3[1]
3.If $[\vec a\ \vec b\ \vec c]=1$, then the value of $[\vec a\times\vec b,\ \vec b\times\vec c,\ \vec c\times\vec a]$ is (A) $1$ (B) $-1$ (C) $2$ (D) $3$[1]
4.If the normals of the parabola y² = 4x drawn at the end points of its latus rectum are tangents to the circle (x – 3)² + (y + 2)² = r², then the value of r² is (A) 2 (B) 3 (C) 1 (D) 4[1]
5.If $X$ is a binomial random variable with expected value $6$ and variance $2.4$, find $P(X=5)$.[1]
6.Points A and B are 10 km apart. From the sound heard at A and B it is determined the explosion was 6 km closer to A than to B. Determine the locus of possible explosion points and give its equation (place origin at midpoint of AB and AB on x-axis).[1]
7.The angle between the line $\vec r=(\hat i+2\hat j-3\hat k)+t(2\hat i+\hat j-2\hat k)$ and the plane $\vec r\cdot(\hat i+\hat j)+4=0$ is (A) $0^\circ$ (B) $30^\circ$ (C) $45^\circ$ (D) $90^\circ$[1]
8.If the coordinates at one end of a diameter of the circle x² + y² – 8x – 4y + c = 0 are (11, 2) the cordinates of the other end are (A) (-3, 2) (B) (2, -5) (C) (5, -2) (D) (-2, 5)[1]
9.The value of $int_{-4}^{4}$ $left[ an^{-1} rac { x^2 }{ x^4+1 }+ an^{-1} rac { x^4+1 }{ x^2 } ight]$ dx is (A) $pi$ (B) $2pi$ (C) $3pi$ (D) $4pi$[1]
10.According to the rational root theorem, which number is not possible rational root of 4x 7 + 2x 7 – 10x³ – 5? (A) -1 (B) $\frac{5}{4}$ (C) $\frac{4}{5}$ (D) 5[1]
11.If sin -1 x + cot -1 ($\frac {1}{2}$) = $\frac {π}{2}$, then x is equal to (A) $\frac {1}{2}$ (B) $\frac {1}{√5}$ (C) $\frac {2}{√5}$ (D) $\frac {√3}{2}$[1]
12.Tangents are drawn to the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{4}=1$ parallel to the line $2x-y=1$. One of the points of contact is (A) $\left(\tfrac{9}{2\sqrt2},-\tfrac{1}{\sqrt2}\right)$ (B) $\left(-\tfrac{9}{2\sqrt2},\tfrac{1}{\sqrt2}\right)$ (C) $\left(\tfrac{9}{2\sqrt2},\tfrac{1}{\sqrt2}\right)$ (D) $\left(3\sqrt3,-2\sqrt2\right)$[1]
13.sin -1 $\frac {3}{5}$ – cos -1 $\frac {12}{13}$ + sec -1 $\frac {5}{3}$ – cosec -1 $\frac {13}{12}$ is equal to (A) 2π (B) π (C) 0 (D) tan -1 $\frac {12}{65}$[1]
14.If $\int_{0}^{x}$ f(t) dt = x + $\int_{x}^{1}$ f(t) dt, then the value of f(1) is (A) $\frac { 1 }{ 2 }$ (B) 2 (C) 1 (D) $\frac { 3 }{ 4 }$[1]
15.The vector equation $\vec r=(\hat i-2\hat j-\hat k)+t(6\hat j-\hat k)$ represents a straight line passing through the points (A) $(0,6,-1)$ and $(1,-2,-1)$ (B) $(0,6,-1)$ and $(-1,-4,-2)$ (C) $(1,-2,-1)$ and $(1,4,-2)$ (D) $(1,-2,-1)$ and $(0,-6,1)$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Find the intervals of monotonicities and hence find the local extremum for the following functions: (i) f(x) = 2x³ + 3x² – 12x (ii) f(x) = $\frac { x }{ x-5 }$ (iii) f(x) = $\frac { e^x }{ 1-e^x }$ (iv) f(x) = $\frac { x^3 }{ 3 }$ – log x (v) f(x) = sin x cos x+ 5, x ∈ (0, 2π)[2]
17.Let f(x) = $\sqrt[3] { x }$. Find the linear approximation at x = 27. Use the linear approximation to approximate $\sqrt[3] { 27.2 }$[2]
18.Evaluate the following integrals as limits of sums: (i) ∫_4^5 (x+5) dx (ii) ∫_1^4 (2−x) dx[2]
19.If the straight line joining the points (2, 1, 4) and (a – 1, 4, -1) is parallel to the line joining the points (0, 2, b – 1) and (5, 3, -2) find the values of a and b.[2]
20.Let z (x, y) = x² y + 3xy 4, x, y ∈ R. Find the linear approximation for z at (2, -1).[2]
21.Expand sin x in ascending powers of (x-π/4) up to three non-zero terms.[2]
22.$\lim _{x \rightarrow 0^+}$ (cos x) $\frac { 1 }{ x^2 }$[2]
23.Show that y = ae -3x + b, where a and b are arbitrary constants, is a solution of the differential equation $\frac { d^2y }{ dx^2 }$ + 3 $\frac { dy }{ dx }$ = 0.[2]
24.The differential equation of the family of curves y = A e^x + B e^{-x} (A,B constants) is:[2]
25.The solution of the differential equation 2x$\frac { dy }{ dx }$ – y = 3 represents (A) straight lines (B) circles (C) parabola (D) ellipse[2]
26.If $A\begin{bmatrix}1&-2\\1&4\end{bmatrix}=\begin{bmatrix}6&0\\0&6\end{bmatrix}$, find $A$.[2]
27.Find the period and amplitude of (i) y = sin 7x (ii) y = − sin(x/3) (iii) y = −4 sin 2x.[2]
28.Solve ye $\frac { x }{ y }$ dx = (x $\frac { x }{ y }$ + y)dy[2]
29.The region enclosed between the graphs of y = x and y = x^2 is denoted by R. Find the volume generated when R is rotated through 360° about the x-axis.[2]
Part III — Long Answer Questions 10 × 5 = 50

Answer in detail. (Answer all questions.)

30.Let A = {1,2,3,4,5}. Check whether the usual multiplication is a binary operation on A.[5]
31.For each of the following differential equations, determine its order and degree (if exists): (i) dy/dx + x y = cot x. (ii) (d^3 y/dx^3)^2 − (d^2 y/dx^2)^3 + 5(dy/dx)^4 = 0. (iii) (contains sin of derivatives — non‑polynomial). (iv) (highest derivative order 7, appears linearly). (v) (highest derivative order 3, appears linearly). (vi) (equation linear in highest derivative of order 2). (vii) (highest derivative order 2 appears linearly). (viii) (contains cos of derivative — non‑polynomial). (ix) (contains an integral — non‑algebraic in derivatives). (x) x e^{x} y (dy/dx) = ... (derivative appears polynomially).[5]
32.Find all values of x such that (i) −6π ≤ x ≤ 6π and cos x = 0. (ii) −5π ≤ x ≤ 5π and cos x = −1.[5]
33.Find the volume of the parallelepiped whose coterminous edges are represented by the vectors -6$\hat { i }$ + 14$\hat { j }$ + 10$\hat { k }$, 14$\hat { i }$ – 10$\hat { j }$ – 6$\hat { k }$ and 2$\hat { i }$ + 4$\hat { j }$ – 2$\hat { k }$[5]
34.Consider the binary operation $*$ on $A=\{a,b,c,d\}$ defined by the table below. Is it commutative? Is it associative? $\begin{array}{c|cccc}*&a&b&c&d\\\hline a&a&c&b&d\\b&d&a&b&c\\c&c&d&a&a\\d&d&b&a&c\end{array}$[5]
35.Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work (use matrix inversion).[5]
36.(i) Let A = ℚ \ {1}. Define ∗ on A by x ∗ y = x + y − xy. Is ∗ binary on A ? Examine commutativity and associativity. (ii) For same A and ∗, examine identity and inverses.[5]
37.Find, by integration, the volume generated by revolving about the x-axis the region enclosed by y = e^x − 2, y = 0, x = 0 and x = 1.[5]
38.Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals: (i) f(x) = $\frac { 1 }{ 2√x }$, x ∈ [-1, 2] (ii) f(x) = |3x + 1|, x ∈ [-1, 3][5]
39.Find the angle between the rectangular hyperbola xy = 2 and the parabola x² + 4y = 0[5]
🔑 Show Answer Key — Set 3
  1. 1. Equivalent: p → (q → r) ≡ ¬p ∨ ¬q ∨ r.
  2. 2. 3
  3. 3. $1$
  4. 4. 2
  5. 5. $P(X=5)=\dbinom{10}{5}(0.6)^5(0.4)^5\approx0.2007$
  6. 6. Hyperbola: \displaystyle \frac{x^2}{9}-\frac{y^2}{16}=1
  7. 7. $45^\circ$
  8. 8. (-3, 2)
  9. 9. $4pi$
  10. 10. $\frac{4}{5}$
  11. 11. $\frac {1}{√5}$
  12. 12. $\left(\dfrac{9}{2\sqrt2},\dfrac{1}{\sqrt2}\right)$
  13. 13. 0
  14. 14. $\frac { 1 }{ 2 }$
  15. 15. $(1,-2,-1)$ and $(1,4,-2)$
  16. 16. ∴ f(x) attains local maximum at x = $\frac { 3π }{ 4 }$ and x = $\frac { 5π }{ 4 }$ Read More: KOTAKBANK Pivot Point Calculator
  17. 17. We know that f(x 0 + Δx) = f(x 0 ) + f'(x 0 ) Δx ∴ Approximate value of $\sqrt[3] { 27.2 }$ = 3.0074
  18. 18. (i) 19/2 (ii) −3/2
  19. 19. Cartesian equation (ii) Points are (0, 2, b – 1) (5, 3, -2) Cartesian equation
  20. 20. = 2 – x + 2 – 20y – 20 = -x – 20y – 16 = -(x + 20y + 16)
  21. 21. sin x = \frac{\sqrt2}{2} + \frac{\sqrt2}{2}(x-\frac{\pi}{4}) - \frac{\sqrt2}{4}(x-\frac{\pi}{4})^2 + … (first three non-zero terms include also the cubic term -\frac{\sqrt2}{12}(x-\frac{\pi}{4})^3 ).
  22. 22. $\lim _{x \rightarrow 0^+}$ (cos x) $\frac { 1 }{ x^2 }$ [1 ∞ indeterminate form let g(x) = (cos x) $\frac { 1 }{ x^2 }$ Taking log on both sides,
  23. 23. Given y = ae -3x + b …… (1) Differentiating equation (1) w.r.t ‘x’, we get Therefore, y = ae -3x + b is a solution of the given differential equation.
  24. 24. Differentiate twice: y'' = A e^x + B e^{-x} = y. Hence y'' − y = 0, which contains no arbitrary constants.
  25. 25. parabola
  26. 26. Answer: $A=\begin{bmatrix}4&2\\-1&1\end{bmatrix}$. Let $M=\begin{bmatrix}1&-2\\1&4\end{bmatrix}$. Since $\det M=6$, $M^{-1}=\frac16\begin{bmatrix}4&2\\-1&1\end{bmatrix}$. Hence $A=6I\,M^{-1}=\begin{bmatrix}4&2\\-1&1\end{bmatrix}$.
  27. 27. (i) Amplitude 1, period 2π/7. (ii) Amplitude 1, period 6π. (iii) Amplitude 4, period π.
  28. 28. The given equation can be written as
  29. 29. 2π/15
  30. 30. Yes, multiplication is not closed on A in general; (counterexample) so it is not a binary operation on A.
  31. 31. (i) order = 1, degree = 1. (ii) order = 3, degree = 2. (iii) order = 2, degree does not exist (non‑polynomial because of sin). (iv) order = 7, degree = 1 (highest derivative assumed 7 and appears linearly). (v) order = 3, degree = 1. (vi) order = 2, degree = 1. (vii) order = 2, degree = 1. (viii) order = 2, degree does not exist (contains cos — non‑polynomial in derivatives). (ix) order = 1, degree does not exist (contains an integral — not algebraic in derivatives). (x) order = 1, degree = 1 (derivative appears to first power).
  32. 32. (i) x = π/2 + kπ with k integer and −6π ≤ x ≤ 6π ⇒ k = −6, −5, …, 5, 6 giving x = π/2 + kπ. (ii) cos x = −1 ⇒ x = π + 2kπ; within interval x = π + 2kπ with k = −3, −2, −1, 0, 1 gives x = −5π, −3π, −π, π, 3π, 5π? Check: π+2(−3)π= −5π included, up to 5π.
  33. 33. = -6(44) -14(-16) + 10(76) = -264 + 224 + 760 = 720 cu. units.
  34. 34. Neither commutative ($a*b=c\ne d=b*a$) nor associative ($(a*b)*c=a\ne c=a*(b*c)$).
  35. 35. One man alone: 18 days. One woman alone: 36 days.
  36. 36. (i) Yes ∗ is binary on A, commutative and associative. (ii) Identity is 0; every element x ∈ A has inverse x' = −x/(1−x) ∈ A.
  37. 37. V = (π/2)(e^2 − 8e + 15)
  38. 38. (ii) f(x) =|3x + 1|, x ∈ [-1, 3] The function is not differentiable at x = $\frac{-1}{3}$. So Lagrange’s mean value theorem is not applicable in the given interval.
  39. 39. xy = 2 ⇒ y = $\frac { 2 }{ x }$ ……….. (1) Differentiating w.r.t. ‘x’ The angle between the curves

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