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## TN State Board 12th Maths Model Question Paper 4 English Medium

Instructions:

- The question paper comprises of four parts.
- You are to attempt all the parts. An internal choice of questions is provided wherever applicable.
- questions of Part I, II. III and IV are to be attempted separately
- Question numbers 1 to 20 in Part I are objective type questions of one -mark each. These are to be answered by choosing the most suitable answer from the given four alternatives and writing the option code and the corresponding answer
- Question numbers 21 to 30 in Part II are two-marks questions. These are to be answered in about one or two sentences.
- Question numbers 31 to 40 in Parr III are three-marks questions, These are to be answered in about three to five short sentences.
- Question numbers 41 to 47 in Part IV are five-marks questions. These are to be answered) in detail. Draw diagrams wherever necessary.

Time: 3 Hours

Maximum Marks: 90

Part – I

I. Choose the correct answer. Answer all the questions. [20 × 1 = 20]

Question 1.

If A^{T} A^{-1} is symmetric, then A^{2} = _______

(a) A^{-1}

(b) (A^{T})^{2}

(c) A^{T}

(d) (A^{-1})^{2}

Answer:

(b) (AT)2

Question 2.

If p + iq = \(\frac{a+i b}{a-i b}\), then p^{2} + q^{2} = ________.

(a) 0

(b) 2

(c) 1

(d) -1

Answer:

(c) 1

Question 3.

If ω ≠ 1 is a cubic root of unity and \(\left|\begin{array}{ccc}

1 & 1 & 1 \\

1 & -\omega^{2}-1 & \omega^{2} \\

1 & \omega^{2} & \omega^{7}

\end{array}\right|\) = 3k, then k is equal to _______.

(a) 1

(b) -l

(c) \(\sqrt{3} i\)

(d) \(-\sqrt{3} i\)

Answer:

(d) \(-\sqrt{3} i\)

Question 4.

The value of sin^{-1} (cos x), 0 ≤ x ≤ π is _______.

(a) π – x

(b) x – \(\frac{\pi}{2}\)

(c) \(\frac{\pi}{2} – x\)

(d) π – x

Answer:

(c) \(\frac{\pi}{2} – x\)

Question 5.

The radius of the circle 3x^{2} + by^{2} + 4bx – 6by + b^{2} = 0 is ________.

(a) 1

(b) 3

(c) \(\sqrt{10}\)

(d) \(\sqrt{11}\)

Answer:

(c) \(\sqrt{10}\)

Question 6.

The equation of the directrix of the parabola y^{2} = -8x is ______.

(a) y + 2 = 0

(b) x – 2 = 0

(c) y – 2 = 0

(d) x + 2 = 0

Answer:

(b) x – 2 = 0

Question 7.

If \(\vec{a}\) and \(\vec{b}\) are parallel vector, then \([\vec{a}, \vec{c}, \vec{b}]\) is equal to _____

(a) 2

(b) -1

(c) 1

(d) 0

Answer:

(d) 0

Question 8.

The length of the perpendicular from the origin to the plane \(\vec{r} \cdot(3 \vec{i}+4 \vec{j}+12 \vec{k})=26\) is _______.

(a) 26

(b) \(\frac{26}{169}\)

(c) 2

(d) \(\frac{1}{2}\)

Answer:

(c) 2

Question 9.

The curve y = ax^{4} + bx^{2} with ab > 0

(a) has no horizontal tangent

(b) is concave up

(c) is concave down

(d) has no points of inflection

Answer:

(d) has no points of inflection

Question 10.

The asymptote to the curve y^{2 }(1 + x) = x^{2 }(1 – x) is _______.

(a) x = 1

(b) y = 1

(c) y = -1

(d) x = -1

Answer:

(d) x = -1

Question 11.

If f(x, y, z) = xy + yz + zx, then f_{x} – f_{z} is equal to ________.

(a) z – x

(b) y – z

(c) x – z

(d) y – x

Answer:

(a) z – x

Question 12.

If f(x, y) = e^{xy} , then \(\frac{\partial^{2} f}{\partial x \partial y}\) is equal to ________.

(a) xye^{xy}

(b) (1 + xy) e^{xy}

(c) (1 + y) e^{xy}

(d) (1 + x) e^{xy}

Answer:

(b) (1 + xy) e^{xy}

Question 13.

The value of \(\int_{0}^{\frac{\pi}{6}} \cos ^{3} 3 x d x\) is _______.

(a) \(\frac{2}{3}\)

(b) \(\frac{2}{9}\)

( c) \(\frac{1}{9}\)

(d) \(\frac{1}{3}\)

Answer:

(b) \(\frac{2}{9}\)

Question 14.

If f(x) is even then \(\int_{-a}^{a} f(x) d x \) _______.

Answer:

(b) \(2 \int_{0}^{a} f(x) d x\)

Question 15.

The order and degree of the differential equation \(\sqrt{\sin x}\)(dx + dy) = \(\sqrt{\sin x}\) (dx- dy) is ________.

(a) 1, 2

(b) 2, 2

(c) 1, 1

(d) 2, 1

Answer:

(c) 1, 1

Question 16.

The solution of the differential equation \(\frac{d y}{d x}\) = 2xy is _______.

(a) y = c e^{x2}

(b) y = 2x^{2} + c

(c) = ce^{-x2} + c

(d) y = x^{2} + c

Answer:

(a) y = c e^{x2}

Question 17.

If P{X = 0} = 1 – P{X = 1}. If E[X] = 3Var(X), then P{X = 0} ________.

(a) \(\frac{2}{3}\)

(b) \(\frac{2}{5}\)

(c) \(\frac{1}{5}\)

(d) \(\frac{1}{3}\)

Answer:

(d) \(\frac{1}{3}\)

Question 18.

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are _________.

(a) i + 2n, i = 0, 1, 2 … n

(b) 2i – n, i = 0, 1, 2 … n

(c) n – i, i = 0, 1, 2 … n

(d) 2i + 2n, i = 0, 1, 2 … n

Answer:

(b) 2i – n, i = 0, 1, 2 … n

Question 19.

In the set Q define a Θ b= a + b + ab. For what value of y, 3 Θ (y Θ 5) = 7 ?

Answer:

(b) y = \(\frac{-2}{3}\)

Question 20.

If X is a continuous random variable then P(X > a) =

(a) P (X < a)

(b) 1 – P (X > a)

(c) P (X > a)

(d) 1 – P (x ≥ a)

Answer:

(c) P (X > a)

Part – II

II. Answer any seven questions. Question No. 30 is compulsory. [7 × 2 = 14]

Question 21.

Reduce the matrix \(\left[\begin{array}{ccc}

3 & -1 & 2 \\

-6 & 2 & 4 \\

-3 & 1 & 2

\end{array}\right]\) to a row-echelon form.

Answer:

Question 22.

Find the least positive integer n such that \(\left(\frac{1+i}{1-i}\right)^{n}=1\)

Answer:

Question 23.

Find the value of \(\sin ^{-1}\left(\sin \left(\frac{5 \pi}{4}\right)\right)\)

Answer:

Question 24.

Identify the type of conic section for the equation 3x^{2} + 3y^{2} – 4x + 3y + 10 = 0

Answer:

Comparing this equation with the general equation of the conic

Ax^{2} + Bxy + cy^{2} + Dx + Ey +F = 0

We get A = C also B = 0

So the given conic is a circle.

Question 25.

If U(x, y, z) = log(x^{3} + y^{3} + z^{3}), find \(\frac{\partial \mathrm{U}}{\partial x}+\frac{\partial \mathrm{U}}{\partial y}\) and \(\frac{\partial U}{\partial z}\)

Answer:

Question 26.

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = 2x^{2}, y = 0 and x = 1.

Answer:

Question 27.

Solve the differential equation: \(\frac{d y}{d x}-x \sqrt{25-x^{2}}=0\)

Answer:

Question 28.

Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred.

Answer:

When three coins are tossed, the sample space is

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

‘X’ is the random variable denotes the number of heads.

∴ ‘X’ can take the values of 0, 1, 2 and 3

Hence, the probabilities

P(X = 0) = P (No heads) = \(\frac{1}{8}\);

P(X = 1) = P (1 head) = \(\frac{3}{8}\);

P(X = 2) = P (2 heads) = \(\frac{3}{8}\);

P(X = 3) = P (3 heads)= \(\frac{1}{8}\);

∴ The probability mass function is

\(f(x)=\left\{\begin{array}{lll}

1 / 8 & \text { for } & x=0,3 \\

3 / 8 & \text { for } & x=1,2

\end{array}\right.\)

Question 29.

Construct the truth table for the following statements. \(\neg p \wedge \neg q\)

Answer:

Truth table for \(\neg p \wedge \neg q\)

Question 30.

Write the Maclaurin series expansion of the function: ex

Answer:

f (x) = e^{x}; f (0) = e^{0} = 1

f’ (x) = e^{x}; f’ (0) = 1

f”(x) = e^{x}; f”(0) = 1

Part – III

III. Answer any seven questions. Question No. 40 is compulsory. [7 × 3 = 21]

Question 31.

If A = \(\left[\begin{array}{lll}

0 & 1 & 1 \\

1 & 0 & 1 \\

1 & 1 & 0

\end{array}\right]\)show that A^{-1} = \(\frac{1}{2}\) (A^{2} – 3I).

Question 32.

Find the values of the real numbers x and y, if the complex numbers.

(3 – i)x – (2 – i) y + 2i + 5 and 2x + (-1 + 2i) y + 3 + 2i are equal.

Question 33.

It is known that the roots of the equation x^{3} – 6x^{2} – 4x + 24 = 0 are in arithmetic progression. Find its roots.

Question 34.

Prove that: \(\cos \left[\tan ^{-1}\left\{\sin \left(\cot ^{-1} x\right)\right\}\right]=\sqrt{\frac{x^{2}+1}{x^{2}+2}}\)

Question 35.

Find the equation of a circle of radius 5 whose centre lies on x-axis and which passes through the point (2, 3).

Question 36.

Using the l’ Hopital Rule prove that, \(\lim _{x \rightarrow 0^{+}}(1+x)^{\frac{1}{x}}=e\)

Question 37.

If v(x, y) = x^{2} – xy + \(\frac{1}{4}\) y^{2} + 7, x, y ∈ R, find the differential dv.

Question 38.

Find the area of the region bounded by 2x – y + 1 =0, y = – 1, y = 3 and y-axis..

Question 39.

Solve: \(\frac{d y}{d x}\) + 2y cot x = 3x^{2} cosec^{2}x

Question 40.

If the straight lines \(\frac{x-5}{5 m+2}=\frac{2-y}{5}=\frac{1-z}{-1}\) and x = \(\frac{2 y+1}{4 m}=\frac{1-z}{-3}\) are perpendicular to each other, find the value of m.

Part – IV

IV. Answer all the questions. [7 × 5 = 35]

Question 41.

(a) Investigate the values of X and p 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.

[OR]

(b) If z(x, y) = x tan^{-1} (xy), x = t^{2}, y = s e^{t}, s, t ∈ R, Find \(\frac{\partial z}{\partial t}\) and \(\frac{\partial z}{\partial t}\) at s = t = 1.

Question 42.

(a) Form the equation whose roots are the squares of the roots of the cubic equation

x^{3} + ax^{2} + bx + c = 0.

[OR]

(b) Find the intervals of concavity and the points of inflection of the function.

f(θ) = sin 2θ in (0, π)

Question 43.

(a) If a = cos 2α + i sin 2α, b = cos 2β + i sin 2β and c = cos 2γ + i sin 2γ, prove that.

(b) A closed (cuboid) box with a square base is to have a volume of 2000 c.c. The material for the top and bottom of the box is to cost Rs. 3 per square cm and the material for the sides is to cost Rs. 1.50 per square cm. If the cost of the materials is to be the least, find the dimensions of the box.

Question 44.

(a) Prove that a straight line and parabola cannot intersect at more than two points.

[OR]

(b) Solve \(\left(y-e^{\sin ^{-1} x}\right) \frac{d x}{d y}+\sqrt{1-x^{2}}=0\)

Question 45.

(a) Solve \(\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}\)

[OR]

(b) Show that the lines \(\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}\) and \(\frac{x-4}{2}=\frac{y}{0}=\frac{z+1}{3}\) the point of intersection.

Question 46.

(a) A tank initially contains 50 liters of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 liters per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.

[OR]

(b) If X ~ B(n, p) such that 4P (X = 4) = P (x = 2) and n = 6 . Find the distribution, mean and standard deviation.

Question 47.

(a) Find the centre, foci, and eccentricity of the hyperbola 11x^{2} – 25y^{2} – 44x + 50y – 256 = 0

[OR]

(b) Verify (i) closure property (ii) commutative property (iii) associative property (iv) existence of identity and (v) existence of inverse for the operation +5 on Z5 using table corresponding to addition modulo 5.