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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 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]
2.For a binomial distribution with $n=5$, $P(X=1)=0.4096$ and $P(X=2)=0.2048$. Find $p$, the mean and the variance.[1]
3.Sketch the graphs of the following functions (i) y = –$\frac { 1 }{ 3 }$ (x³ – 3x + 2) (ii) y = x $\sqrt { 4-x }$ (iii) y = $\frac { x^2+1 }{ x^2-4 }$ (iv) y = $\frac { 1 }{ 1+e^{-x} }$ (v) y = $\frac { x^3 }{ 24 }$ – log x[1]
4.Find $k$ from the given rotation-matrix condition. Options: (A) $0$ (B) $\sin\theta$ (C) $\cos\theta$ (D) $1$.[1]
5.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]
6.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]
7.The minimum value of the function $f(x)=|3-x|+9$ is (A) $0$ (B) $3$ (C) $6$ (D) $9$[1]
8.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]
9.If $\operatorname{adj}A$ and $\operatorname{adj}B$ are given, find $\operatorname{adj}(AB)$.[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.Use the given $A^{-1}$ and $(AB)^{-1}$ matrices to find $B^{-1}$.[1]
12.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]
13.At a water fountain, water attains a maximum height of 4 m at horizontal distance of 0.5 m from its origin. If the path of water is a parabola, find the height of water at a horizontal distance of 0.75 m from the point of origin.[1]
14.The equation of the normal to the circle x² + y² – 2x – 2y + 1 = 0 which is parallel to the line 2x + 4y = 3 is (A) x + 2y = 3 (B) x + 2y + 3 = 0 (C) 2x + 4y + 3 = 0 (D) x – 2y + 3 = 0[1]
15.If cot -1 x = $\frac {2π}{5}$ for some x ∈ R, the value of tan -1 x is (A) –$\frac {π}{10}$ (B) $\frac {π}{5}$ (C) $\frac {π}{10}$ (D) –$\frac {π}{5}$[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.Suppose that for a function f, f'(x) ≤ 1 for all 1 ≤ x ≤ 4. Show that f(4)-f(1) ≤ 3.[2]
17.A rectangular page must contain 24 cm^2 of print. Margins: top/bottom 1.5 cm, left/right 1 cm. Find page dimensions minimizing total paper area.[2]
18.Evaluate $\lim _{(x, y) \rightarrow(0,0)}$ cos($\frac { x^3+y^2 }{ x+y+2 }$), if the limit exists.[2]
19.$\lim _{x \rightarrow 0^+}$ (cos x) $\frac { 1 }{ x^2 }$[2]
20.The value of $\int_{-π/2}^{π/2}$ sin² x cos x dx (A) $\frac { 3 }{ 2 }$ (B) $\frac { 1 }{ 2 }$ (C) 0 (D) $\frac { 2 }{ 3 }$[2]
21.Find the adjoint of the matrices: (i) $-\begin{bmatrix}3&4\\6&2\end{bmatrix}$ (ii) $\begin{bmatrix}2&3&1\\3&4&1\\3&7&2\end{bmatrix}$. The third matrix in this textbook item needs source-text restoration before validation.[2]
22.The probability that Mr. Q hits a target at any trial is $\tfrac14$. He tries $10$ times. Find the probability that he hits the target (i) exactly $4$ times (ii) at least once.[2]
23.If $\cos\alpha+\cos\beta+\cos\gamma=\sin\alpha+\sin\beta+\sin\gamma=0$, show that (i) $\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta+\gamma)$ (ii) $\sin3\alpha+\sin3\beta+\sin3\gamma=3\sin(\alpha+\beta+\gamma)$.[2]
24.Find dimensions of rectangle of maximum area inscribed in a circle of radius 10 cm.[2]
25.Let W (x, y, z) = x² – xy + 3sinz, x, y, z ∈ R. Find the linear approximation at (2, -1, 0).[2]
26.Evaluate (i) lim_{x→∞} x e^{-x}. (ii) lim_{x→0} (sin x)/x.[2]
27.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]
28.Show that the differential equation representing the family of curves y² = 2a(x + a 2/3 ), where a is a positive parameter, is (y² – 2xy$\frac { dy }{ dx }$)³ = 8(y$\frac { dy }{ dx }$) 5[2]
29.Obtain the Cartesian form of the locus of $z=x+iy$: (i) $[\operatorname{Re}(iz)]^2=3$ (ii) $\operatorname{Im}[(1-i)z+1]=0$ (iii) $|z+i|=|z-1|$ (iv) $\overline z=z^{-1}$.[2]
Part III — Long Answer Questions 10 × 5 = 50

Answer in detail. (Answer all questions.)

30.Evaluate $\lim _{(x, y) \rightarrow(1,2)}$ g(x, y), if the limit exists where g(x, y) = $\frac { 3x^2-xy }{ x^2+y^2+3 }$[5]
31.Show that $y=ax+\dfrac{b}{x}$ ($x\ne0$) is a solution of the differential equation $x^2y''+xy'-y=0$.[5]
32.Find the value of (i) tan(cos -1 ($\frac {1}{2}$) – sin -1 (-$\frac {1}{2}$)) (ii) sin(tan -1 ($\frac {1}{2}$) – cos -1 ($\frac {4}{5}$)) (iii) cos(sin -1 ($\frac {4}{5}$) – tan -1 ($\frac {3}{4}$))[5]
33.If u(x,y)=x^2+4y+3, x=e^t, y=sin t. Find du/dt and evaluate at t=0.[5]
34.Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is 1/2 |AC × BD|.[5]
35.(i) Define ∗ on ℚ by a ∗ b = (a+b)/2. Examine closure, commutativity, associativity. (ii) For same ∗, examine existence of identity and inverses on ℚ.[5]
36.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]
37.Find the equations of the tangent and normal to hyperbola 12x² – 9y² = 108 at θ = $\frac {π}{3}$. (Hint: use parametric form)[5]
38.Solve the equation 3x³ – 26x² + 52x – 24 = 0 if its roots form a geometric progression.[5]
39.If the vectors a$\hat { i }$ + a$\hat { j }$ + c$\hat { k }$, $\hat { i }$ + $\hat { k }$ and c$\hat { i }$ + c$\hat { j }$ + b$\hat { k }$ are coplanar, prove that c is the geometric mean of a and b.[5]
🔑 Show Answer Key — Set 1
  1. 1. -7
  2. 2. $p=\dfrac15,\ q=\dfrac45$; mean $=1$, variance $=\dfrac45$.
  3. 3. No point of inflection exists. No Horizontal asymptotes are possible, but the vertical asymptote is x = 0 (y-axis). ESCORTS Pivot Calculator
  4. 4. 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 $\lambda=19$.
  5. 5. (-3, 2)
  6. 6. adj (AB) = (adj A)(adj B)
  7. 7. $9$
  8. 8. -1
  9. 9. 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
  10. 10. 2A -1
  11. 11. Method: From $(AB)^{-1}=B^{-1}A^{-1}$, multiply on the right by $A$ to get $B^{-1}=(AB)^{-1}A$. The preserved option key is $\begin{bmatrix}2&-5\\-3&8\end{bmatrix}$, but the source matrices are missing from the text extract, so the card is not marked validated. Q.11 If A^{-1}A^T is symmetric, then A^2 = ? (Options: A^{-1}, (A^T)^2, A^T, (A^{-1})^2 ) 1) A^{-1} 2) (A^T)^2 3) A^T 4) (A^{-1})^2 Answer: 3 Assume A^{-1}A^T is symmetric ⇒ A^{-1}A^T = (A^{-1}A^T)^T = A (A^{-1})^T = A A^{-T}. Multiplying on left by A and on right by A^{-1} leads to A^2 = A^T. Thus A^2 = A^T. Hence option (3). (This uses A invertible.)
  12. 12. $(-6,7)$
  13. 13. 3 m.
  14. 14. x + 2y = 3
  15. 15. $\frac {π}{10}$
  16. 16. f(4)-f(1) ≤ 3.
  17. 17. Page width = 6 cm, height = 9 cm
  18. 18. = 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
  19. 19. $\lim _{x \rightarrow 0^+}$ (cos x) $\frac { 1 }{ x^2 }$ [1 ∞ indeterminate form let g(x) = (cos x) $\frac { 1 }{ x^2 }$ Taking log on both sides,
  20. 20. $\frac { 2 }{ 3 }$
  21. 21. Answer: (i) $\operatorname{adj}(A)=\begin{bmatrix}-2&4\\6&-3\end{bmatrix}$. (ii) $\operatorname{adj}(A)=\begin{bmatrix}1&1&-1\\-3&1&1\\9&-5&-1\end{bmatrix}$. The missing third matrix is intentionally left unverified here; no answer is marked validated until the exact textbook entries are restored. <div
  22. 22. (i) $\approx0.146$ (ii) $1-\left(\tfrac34\right)^{10}\approx0.944$
  23. 23. Both identities follow by equating the real and imaginary parts of $e^{3i\alpha}+e^{3i\beta}+e^{3i\gamma}=3e^{i(\alpha+\beta+\gamma)}$.
  24. 24. Square of side \(10\sqrt2\) cm; max area = 200 cm^2
  25. 25. L (x, y, z) = 6 + 5 (x – 2)- 2 (y + 1) + 3 (z) = 6 + 5x – 10 – 2y – 2 + 3z = 5x – 2y + 3z – 6
  26. 26. (i) 0. (ii) 1.
  27. 27. 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.
  28. 28. Differentiating equation (1) w.r.t ‘x’ we get Hence y² = 2ox + 2 a³ is a solution of the differential equation (y² – 2xy$\frac { dy }{ dx }$)³ = 8(y$\frac { dy }{ dx }$) 5
  29. 29. (i) $y^2=3$ (ii) $y=x$ (iii) $x+y=0$ (iv) $x^2+y^2=1$.
  30. 30. $\lim _{(x, y) \rightarrow(1,2)}$ g(x, y) = $\lim _{(x, y) \rightarrow(1,2)}$ $\frac { 3x^2-xy }{ x^2+y^2+3 }$ = $\frac { 3(1)-1×2 }{ 1+4+3 }$ = $\frac { 1 }{ 8 }$
  31. 31. Substituting $y=ax+\dfrac bx$ makes $x^2y''+xy'-y=0$ identically, so it is a solution.
  32. 32. (i) Not defined (ii) -2/(5√5) (iii) 24/25
  33. 33. du/dt = 2e^{2t} + 4 cos t; at t=0, du/dt = 6.
  34. 34. Area(ABCD) = (1/2)|AC × BD|.
  35. 35. (i) Closed and commutative; not associative. (ii) No identity and so no (two-sided) inverses.
  36. 36. -12x + 11y + 16z = 14 12x – 11y – 16z = -14 12x – 11y – 16z + 14 = 0
  37. 37. ⇒ 4x – 3y – 6 = 0 (ii) Equation of the normal to hyperbola be ⇒ 3x + 4y – 42 = 0
  38. 38. 3λ² – 10λ + 3 = 0 (λ – 3) (3λ – 1) = 0 λ = 6 or λ = $\frac{1}{3}$
  39. 39. c² – ab = 0 c² = ab ⇒ c in the geometric mean of a and b.
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.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]
2.Find the parameter values for which the given system has the stated rank condition.[1]
3.Find the exact number of real roots and imaginary of the equation x 9 + 9x 7 + 7x 5 + 5x³ + 3x.[1]
4.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]
5.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]
6.Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centered at (0, y) passing through the origin and touching the circle C externally, then the radius of T is equal to (A) $\frac {√3}{√2}$ (B) $\frac {√3}{2}$ (C) $\frac {1}{2}$ (D) $\frac {1}{4}$[1]
7.Using truth table prove that p → (q → r) ≡ ¬p ∨ ¬q ∨ r.[1]
8.By using Gaussian elimination method, balance the chemical reaction equation: C6H12O6 + O2 → CO2 + H2O.[1]
9.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]
10.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]
11.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]
12.The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² – 2t – 8. The time at which the particle is at rest is (A) t = 0 (B) t = $\frac { 1 }{ 3 }$ (C) t = 1 (D) t = 3[1]
13.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]
14.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]
15.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]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.The value of the limit $\lim _{x \rightarrow 0}$ (cot x – $\frac { 1 }{ x }$) is (A) 0 (B) 1 (C) 2 (D) ∞[2]
17.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]
18.If $|z|=2$, show that $8\le |z+6+8i|\le12$.[2]
19.Find a linear approximation for the following functions at the indicated points (i) f(x) = x³ – 5x + 12, x 0 = 2 (ii) g(x) = $\sqrt { x^2+9 }$, x 0 = -4 (iii) h(x) = $\frac { x }{ x+1 }$, x 0 = 1[2]
20.Forces of magnitudes 5√2 and 10√2 units acting in the directions 3i + 4j + 5k and 10i + 6j - 8k, respectively, act on a particle which is displaced from the point with position vector 4i - 3j - 2k to the point with position vector 6i + j - 3k. Find the work done by the forces.[2]
21.A rectangular garden is fenced with 40 m of wire. What is the largest possible area?[2]
22.Find the differential equation of the family of circles passing through the origin and having their centres on the x‑axis.[2]
23.If $\overline { a }$ = $\hat { i }$ + 2$\hat { j }$ + 3$\hat { k }$, $\overline { b }$ = 2$\hat { i }$ – $\hat { j }$ + $\hat { k }$, $\overline { c }$ = 3$\hat { i }$ + 2$\hat { j }$ + $\hat { k }$ and $\overline { a }$ × ($\overline { b }$ × $\overline { c }$) = l$\overline { a }$ + m$\overline { b }$ + n$\overline { c }$, find the values of l, m, n.[2]
24.Solve the following system of linear equations by matrix inversion method. (i) 2x + 5y = -2, x + 2y = -3[2]
25.For what value of x does sin(sin^{-1} x) = −1 ?[2]
26.Expand sin x in ascending powers of (x-π/4) up to three non-zero terms.[2]
27.(1 + x + xy²) $\frac { dy }{ dx }$ + (y + y³) = 0[2]
28.The value of $\int_{0}^{2/3}$ $\frac { dx }{ \sqrt{4-9x^2} }$ (A) $\frac { π }{ 6 }$ (B) $\frac { π }{ 2 }$ (C) $\frac { π }{ 4 }$ (D) π[2]
29.Find the differential dy for each of the following functions. (i) y = $\frac { (1-2x)^3 }{ 3-4x }$ (ii) y = (3 + sin2x) 2/3 (iii) y = e x 2 – 5x +7 cos(x² – 1)[2]
Part III — Long Answer Questions 10 × 5 = 50

Answer in detail. (Answer all questions.)

30.Prove the identity: tan^{-1}x + tan^{-1}y + tan^{-1}z = tan^{-1}\left(\dfrac{x+y+z-xyz}{1-xy-yz-zx}\right) (under the usual conditions on principal values).[5]
31.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]
32.Using mean value theorem prove that for a,b>0, |e^a-e^b| = e^c|a-b| for some c between a and b; hence min(e^a,e^b)|a-b| ≤ |e^a-e^b| ≤ max(e^a,e^b)|a-b|.[5]
33.Determine k and solve the equation 2x³ – 6x² + 3x + k = 0 if one of its roots is twice the sum of the other two roots.[5]
34.An amount of ₹65,000 is invested in three bonds at the rates 6%, 8% and 9% p.a. The total annual income is ₹4,800. The income from the third bond is ₹600 more than that from the second bond. Determine the amounts invested in each bond (use Gaussian elimination).[5]
35.Prove that the point of intersection of the tangents at t1 and t2 on the parabola y^2 = 4ax is (a t1 t2, a(t1 + t2)).[5]
36.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]
37.A projectile (rocket cracker) reaches maximum height 4 m at horizontal distance 6 m from the point of projection. It finally hits the ground 12 m away from the start. Find the angle of projection.[5]
38.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]
39.Find the area of the region bounded by the line $y=2x+5$ and the parabola $y=x^2-2x$.[5]
🔑 Show Answer Key — Set 2
  1. 1. 2√3
  2. 2. 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).
  3. 3. P(x) = x 9 + 9x 7 + 7x 5 + 5x 3 + 3x. There is no change in the sign of P(x) and P(-x), P(x) has no positive and no negative real roots, but 0 is the root of the polynomial equation P(x).
  4. 4. $4pi$
  5. 5. y x y-1
  6. 6. $\frac {1}{4}$
  7. 7. Equivalent: p → (q → r) ≡ ¬p ∨ ¬q ∨ r.
  8. 8. C6H12O6 + 6 O2 → 6 CO2 + 6 H2O
  9. 9. $\left(\dfrac{9}{2\sqrt2},\dfrac{1}{\sqrt2}\right)$
  10. 10. 0
  11. 11. $\frac {π}{2}$
  12. 12. t = $\frac { 1 }{ 3 }$
  13. 13. $\frac{4}{5}$
  14. 14. Hyperbola: \displaystyle \frac{x^2}{9}-\frac{y^2}{16}=1
  15. 15. Option 1: one distinct root.
  16. 16. ∞
  17. 17. 6 litres of 50% solution and 4 litres of 25% solution.
  18. 18. $8\le |z+6+8i|\le12$.
  19. 19. (ii) g(x) = $\sqrt { x^2+9 }$, x 0 = -4 g(x 0 ) = g(14) = $\sqrt {16+9 }$ = 5 (iii) h(x) = $\frac { x }{ x+1 }$, x 0 = 1
  20. 20. 69 units
  21. 21. 100 m^2 (square 10 m × 10 m)
  22. 22. Differential equation: y^2 - x^2 = 2 x y y' .
  23. 23. m = 3 + 4 + 3; n = -(2 – 2 + 3) m = 10; n = -3 l = 0; m = 10; n = -3
  24. 24. ≠ 0 A -1 Exists ∴ x = 3, y = -2, z = 1
  25. 25. x = −1.
  26. 26. 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 ).
  27. 27. So, the solution of the equation is given by xy + tan -1 y = c Which is the required solution.
  28. 28. $\frac { π }{ 6 }$
  29. 29. (ii) y = (3 + sin2x) 2/3
  30. 30. Proof (sketch): Use the addition formula for tangent twice. Let A = tan^{-1}x, B = tan^{-1}y so tan(A+B) = (x+y)/(1−xy) provided 1−xy ≠ 0. Then tan(A+B+tan^{-1}z) = ( (x+y)/(1−xy) + z ) / ( 1 − z·(x+y)/(1−xy) ) . Simplify numerator and denominator: numerator = (x+y+z−xyz)/(1−xy), denominator = (1−xy−yz−zx)/(1−xy). Divide numerator by denominator to get (x+y+z−xyz)/(1−xy−yz−zx). Hence tan^{-1}x + tan^{-1}y + tan^{-1}z = tan^{-1}\left(\dfrac{x+y+z-xyz}{1-xy-yz-zx}\right), with the caveat that the principal-value branch of tan^{-1} and possible additions of π must be handled so the equality holds in the correct range; the algebraic identity for the tangent of the sum is as given.
  31. 31. V = (π/2)(e^2 − 8e + 15)
  32. 32. By MVT ∃c between a and b with e^a-e^b=e^c(a-b). Therefore |e^a-e^b|=e^c|a-b|, and since e^c lies between e^a and e^b the stated inequalities follow.
  33. 33. product of roots α β γ = –$\frac{k}{2}$ α = 2 ⇒ 2βγ = –$\frac{k}{2}$ βγ = –$\frac{k}{4}$ ………… (3)
  34. 34. Investments: at 6% = ₹20,000; at 8% = ₹15,000; at 9% = ₹30,000.
  35. 35. For parameter t the tangent to y^2 = 4ax is: t y = x + a t^2 . For t = t1 and t = t2 we have the two equations: t1 y = x + a t1^2 t2 y = x + a t2^2 Subtracting gives (t1 − t2) y = a(t1^2 − t2^2) = a(t1 − t2)(t1 + t2), so y = a(t1 + t2). Substitute y into one tangent equation, say t1 y = x + a t1^2: x = t1·a(t1 + t2) − a t1^2 = a t1 t2. Hence the intersection point is ( a t1 t2 , a(t1 + t2) ).
  36. 36. (Ladder) (i) dy/dt=−8/3 m/s. (ii) dA/dt=161/6 m^2/s. (Police) speed of car = 70 km/h.
  37. 37. \theta = \arctan\frac{4}{3}
  38. 38. h = 2r
  39. 39. Area $=36$ square units.
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.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]
2.The area of the quadrilateral formed with the foci of the hyperbolas $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ and $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=-1$ is (A) $4(a^2+b^2)$ (B) $2(a^2+b^2)$ (C) $a^2+b^2$ (D) $\tfrac12(a^2+b^2)$[1]
3.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]
4.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]
5.If $z=2-2i$, find its rotation through $\theta$ radians counterclockwise about the origin when (i) $\theta=\pi/3$ (ii) $\theta=2\pi/3$ (iii) $\theta=3\pi/2$.[1]
6.If x + y = k is a normal to the parabola y² = 12x, then the value of k is 14. (A) 3 (B) -1 (C) 1 (D) 9[1]
7.Find $B$ from the given trigonometric matrix equation.[1]
8.If the function f(x) = sin -1 (x² – 3), then x belongs to (A) [-1, 1] (B) [√2, 2] (C) [-2, -√2]∪[√2, 2] (D) [-2, -√2][1]
9.Determine the required entry of the matrix from the given adjoint relation. Options: (A) $0$ (B) $-2$ (C) $-3$ (D) $-1$.[1]
10.sin(tan -1 x), |x| < 1 is equal to (A) $\frac {x}{\sqrt{1-x^2}}$ (B) $\frac {1}{\sqrt{1-x^2}}$ (C) $\frac {1}{\sqrt{1+x^2}}$ (D) $\frac {x}{\sqrt{1+x^2}}$[1]
11.If A is a non-singular matrix such that A -1 = $\begin{bmatrix} 5 & 3 \\ -2 & -1 \end{bmatrix}$, then (AT) -1 =[1]
12.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]
13.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]
14.Find the number of the solutions of the equations tan -1 (x – 1) + tan -1 x + tan -1 (x + 1) = tan -1 3x[1]
15.The solution of the differential equation $\frac { dy }{ dx }$ + $\frac { 1 }{ \sqrt{1-x^2} }$ = 0 is (A) y + sin -1 x = c (B) x + sin -1 y = 0 (C) y² + 2sin -1 x = c (D) x² + 2sin -1 y = 0[1]
Part II — Short Answer Questions 14 × 2 = 28

Answer briefly. (Answer all questions.)

16.If $\omega\ne1$ is a cube root of unity and $(1+\omega)^7=A+B\omega$, then $(A,B)$ is: (1) $(1,0)$ (2) $(-1,1)$ (3) $(0,1)$ (4) $(1,1)$.[2]
17.Find intervals of concavity and points of inflection for the following functions: (i) f(x) = x(x – 4)³ (ii) f(x) = sin x + cos x, 0 < x < 2π (iii) f(x) = $\frac { 1 }{ 2 }$(e x – e -x )[2]
18.(x² + y²) dy = xy dx. It is given that y (1) = y(x 0 ) = e. Find the value of x 0.[2]
19.Find the rank of the listed matrices by the row reduction method.[2]
20.Show that the points (2,3,4), (−1,4,5) and (8,1,2) are collinear.[2]
21.Find the acute angle between the following lines. ] (i) $\overline { r }$ = (4$\hat { i }$ – $\hat { j }$) + t($\hat { i }$ + 2$\hat { j }$ – 2$\hat { k }$), $\overline { r }$ = ($\hat { i }$ – 2$\hat { j }$ + 4$\hat { k }$) + s(-$\hat { i }$ – 2$\hat { j }$ + 2$\hat { k }$) (ii) $\frac { x+4 }{ 3 }$ = $\frac { y-7 }{ 4 }$ = $\frac { z+5 }{ 5 }$, $\overline { r }$ = 4$\hat { k }$ + t(2$\hat { i }$ + $\hat { j }$ + $\hat { k }$) (iii) 2x = 3y = -z and 6x = -y = -4z.[2]
22.For a binomial distribution $B(n,p)$, compute $P(X=k)$ for (i) $n=6,\ p=\tfrac13,\ k=3$; (ii) $n=10,\ p=\tfrac15,\ k=4$; (iii) $n=9,\ p=\tfrac12,\ k=7$.[2]
23.Construct a cubic equation with roots (i) 1, 2, and 3 (ii) 1, 1, and −2 (iii) 2, 1/2, and 1.[2]
24.Prove that (i) $\tan^{-1}\dfrac{2}{11}+\tan^{-1}\dfrac{7}{24}=\tan^{-1}\dfrac12$ (ii) $\sin^{-1}\dfrac35+\cos^{-1}\dfrac{12}{13}=\sin^{-1}\dfrac{56}{65}$.[2]
25.Solve ye $\frac { x }{ y }$ dx = (x $\frac { x }{ y }$ + y)dy[2]
26.The order of the differential equation of all circles with centre at (h,k) and radius a is:[2]
27.Evaluate the following integrals as limits of sums: (i) ∫_4^5 (x+5) dx (ii) ∫_1^4 (2−x) dx[2]
28.$\lim _{x \rightarrow ∞}$ (1 + $\frac { 1 }{ x }$) x[2]
29.If a and b are parallel vectors, then [a,c,b] is equal to (1) 2 (2) −1 (3) 1 (4) 0[2]
Part III — Long Answer Questions 10 × 5 = 50

Answer in detail. (Answer all questions.)

30.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]
31.Write in rectangular form: (i) $\overline{(5+9i)+(2-4i)}$ (ii) $\dfrac{10-5i}{6+2i}$ (iii) $\overline{3i}+\dfrac{1}{2-i}$.[5]
32.Is cos^{-1}(cos(1−x)) = 1−x−π true? Justify.[5]
33.Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval: (i) f(x) = x³ – 3x + 2, x ∈ [-2, 2] (ii) f(x) = (x – 2) (x – 7), x ∈ [3, 11][5]
34.Find the point of intersection of the line with the plane (x – 1) = $\frac { y }{ 2 }$ = z + 1 with the plane 2x – y – 2z = 2. Also, the angle between the line and the plane.[5]
35.For any vector a, prove that i×(a×i) + j×(a×j) + k×(a×k) = 2a.[5]
36.Discuss the maximum possible number of positive and negative roots of the polynomial equations x² – 5x + 6 and x² – 5x + 16. Also, draw a rough sketch of the graphs.[5]
37.Find the area of the region common to the circle x^2 + y^2 = 16 and the parabola x = y^2/6 (equivalently y^2 = 6x).[5]
38.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]
39.Find the equations of tangents to the hyperbola x^2/16 - y^2/64 = 1 which are parallel to 10x-3y+9=0.[5]
🔑 Show Answer Key — Set 3
  1. 1. unique solution
  2. 2. $2(a^2+b^2)$
  3. 3. $45^\circ$
  4. 4. 11
  5. 5. (i) $(1+\sqrt3)+i(\sqrt3-1)$ (ii) $(\sqrt3-1)+i(\sqrt3+1)$ (iii) $-2-2i$.
  6. 6. 9
  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. [-2, -√2]∪[√2, 2]
  9. 9. Answer: The preserved key is $-1$. The defining matrix relation is missing from the text extract, so this item is not marked validated. <div
  10. 10. $\frac {x}{\sqrt{1+x^2}}$
  11. 11. D
  12. 12. $1$
  13. 13. $(1,-2,-1)$ and $(1,4,-2)$
  14. 14. x = 0, x² = 1 x = ±1 Number of solutions are three (0, 1 -1)
  15. 15. y + sin -1 x = c
  16. 16. Option 4: $(1,1)$.
  17. 17. f'(x) changes its sign when passing through x = 0 Now f(0) = – (e° – e°) = $\frac { 1 }{ 2 }$ (1 – 1) = 0 ∴ The point of inflection is (0, 0).
  18. 18. The given differential equation is of the form
  19. 19. Answer: The preserved row-echelon conclusions give ranks $2$, $3$ and $3$. Some starting matrices are absent from the text extract, so this combined item is not marked validated. <div
  20. 20. Collinear
  21. 21. See the worked solution above.
  22. 22. (i) $\dfrac{160}{729}\approx0.220$ (ii) $\dfrac{860160}{9765625}\approx0.088$ (iii) $\dfrac{9}{128}\approx0.070$
  23. 23. (i) x^3-6x^2+11x-6=0 (ii) x^3-3x+2=0 (iii) 2x^3-7x^2+7x-2=0
  24. 24. Both identities are verified: (i) $=\tan^{-1}\dfrac12$, (ii) $=\sin^{-1}\dfrac{56}{65}$.
  25. 25. The given equation can be written as
  26. 26. A general circle (x−h)^2+(y−k)^2=a^2 contains three arbitrary constants h, k and a. The differential equation representing this family therefore has order 3.
  27. 27. (i) 19/2 (ii) −3/2
  28. 28. Applying L’ Hôpital’s Rule Exponentiating we get, $\lim _{x \rightarrow ∞}$ g(x) = e 1 = e
  29. 29. 4
  30. 30. (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.
  31. 31. (i) $7-5i$ (ii) $\dfrac54-\dfrac54i$ (iii) $\dfrac25-\dfrac{14}{5}i$.
  32. 32. No in general. cos^{-1}(cos θ) = principal value in [0,π]; equality holds only when θ ∈ [0,π].
  33. 33. 2x – 9 = $\frac { 40 }{ 8 }$ = 5 2c = 14 ⇒ c = 7 ∈ (3, 11) ∴ x = 7.
  34. 34. 2λ + 2 – 2λ + 2λ – 2 = 2 λ = 1 ∴ The required point of intersection is (2, 2, 0)
  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. P(-x) = x 4 + 10x³ + 47x² + 110x + 96 It has no sign change, no negative real roots y = x 2 – 5x + 16
  37. 37. 16√3 / 3
  38. 38. Price per item: dosai = ₹30, idli = ₹10, vadai = ₹30. Bill for 3 dosai,6 idli,6 vadai = ₹330 ≤ ₹350, so yes (₹20 change).
  39. 39. The required tangents are 10x - 3y + 32 = 0 and 10x - 3y - 32 = 0.

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