Tamil Nadu 12th Maths Model Question Paper 2 English Medium

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TN State Board 12th Maths Model Question Paper 2 English Medium

Instructions:

1.  The question paper comprises of four parts.
2.  You are to attempt all the parts. An internal choice of questions is provided wherever applicable.
3. questions of Part I, II. III and IV are to be attempted separately
4. 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
5. Question numbers 21 to 30 in Part II are two-marks questions. These are to be answered in about one or two sentences.
6. Question numbers 31 to 40 in Parr III are three-marks questions, These are to be answered in about three to five short sentences.
7. 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 = \(\left[\begin{array}{cc}
2 & 3 \\
5 & -2
\end{array}\right]\) be such that λA^-1 = A, then λ is _______.
(a) 17
(b) 14
(c) 19
(d) 21
Answer:
(c) 19

Question 2.
If ω ≠ 1 is a cubic root of unity and (1+ ω)⁷ = A+ B ω, then (A, B) equals to _______.
(a) (1,0)
(b) (-1, 1)
(c) (0, 1)
(d) (1, 1)
Answer:
(d) (1, 1)

Question 3.
The value of z – \(\bar{Z}\) is ______.
(a) 2 Im (z)
(b) 2 i Im (z)
(c) Im (z)
(d) i Im (z)
Answer:
(b) 2 i Im (z)

Question 4.
If x³ + 12x² + 10ax + 1999 definitely has a positive zero, if and only if ________.
(a) a ≥ 0
(b) a > 0
(c) a < 0
(d) a < 0
Answer:
(c) a < 0

Question 5.
sin(tan^-1 x), |x| < 1 is equal to _______.

Answer:
(d) \(\frac{x}{\sqrt{1+x^{2}}}\)

Question 6.
The centre of the circle inscribed in a square formed by the lines x² – 8x – 12 = 0 and y² – 14y + 45 = 0 is _____.
(a) (4, 7)
(b) (7, 4)
(c) (9, 4)
(d) (4, 9)
Answer:
(a) (4, 7)

Question 7.
The axis of the parabola x² = – 4y is ______.
(a) y= 1
(b) x = 0
(c) y = 0
(d) x = 1
Answer:
(b) x = 0

Question 8.
The coordinates of the point where the line \(\vec{r}=(6 \hat{i}-\hat{j}-3 \hat{k})+t(-\hat{i}+4 \hat{k})\) meets the plane \(\vec{r} \cdot(\hat{i}+\hat{j}-\hat{k})=3\) are _______.
(a) (2, 1, 0)
(b) (7, -1, -7)
(c) (1, 2, -6)
(d) (5, -1, 1)
Answer:
(d) (5, -1, 1)

Question 9.
If the vectors \(\vec{a}=3 \vec{i}+2 \vec{j}+9 \vec{k}\) and \(\vec{b}=\vec{i}+m \vec{j}+3 \vec{k}\) are parallel then m is _________.

Answer:
(b) \(\frac{2}{3}\)

Question 10.
The minimum value of the function |3 – x | + 9 is ________.
(a) 0
(b) 3
(c) 6
(d) 9
Answer:
(d) 9

Question 11.
The curve y² = x² (1 – x²) has ______.
(a) an asymptote x = -1
(b) an asymptote x = 1
(c) two asymptotes x = 1 and x = -1
(d) no asymptote
Answer:
(d) no asymptote

Question 12.
If /(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 13.
A circular template has a radius of 10 cm. The measurement of radius has an approximate error of 0.02 cm. Then the percentage error in calculating area of this template is _______.
(a) 0.2%
(b) 0.4%
(c) 0.04%
(d) 0.08%
Answer:
(b) 0.4%

Question 14.
The value of \(\int_{0}^{\pi} \sin ^{4} x d x\) is _______.

Answer:
(b) \(\frac{3 \pi}{8}\)

Question 15.
\(\int_{a}^{b} f(x) d x\) is _______.

Answer:
(d) \(\int_{a}^{b} f(a+b-x) d x\)

Question 16.
The degree of the differential equation \(y(x)=1+\frac{d y}{d x}+\frac{1}{1.2}\left(\frac{d y}{d x}\right)^{2}+\frac{1}{1.2 .3}\left(\frac{d y}{d x}\right)^{3}+\ldots\) is ________.
(a) 2
(b) 3
(c) 1
(d) 4
Answer:
(c) 1

Question 17.
In finding the differential equation corresponding toy = e^mx where m is the arbitrary constant, then m is ____.
(a) \(\frac{y}{y^{\prime}}\)
(b) \(\frac{y^{\prime}}{y}\)
(c) y’
(d) y
Answer:
(b) \(\frac{y^{\prime}}{y}\)

Question 18.
Let X be random variable with probability density function f(x) = \(\left\{\begin{array}{ll}
2 / x^{3} & x \geq 1 \\
0 & x<1
\end{array}\right.\)
Which of the following statement is correct
(a) both mean and variance exist
(b) mean exists but variance does not exist
(c) both mean and variance do not exist
(d) variance exists but mean does not exist
Answer:
(b) mean exists but variance does not exist

Question 19.
The random variable X has the probability density function f(x) = \(\left\{\begin{array}{cc}
a x+b, & 0<x<1 \\
0, & \text { otherwise }
\end{array}\right.\)
and E(X) = \(\frac{7}{12}\), then a and b are respectively _______.
(a) 1 and \(\frac{1}{2}\)
(b) \(\frac{1}{2}\) and 1
(c) 2 and 1
(d) 1 and 2
Answer:
(a) 1 and \(\frac{1}{2}\)

Question 20.
A binary operation on a set S is a function from ________.
(a) S → S
(b)(S x S) → S
(c) S → (S x S)
(d) (S x S) → (S x S)
Answer:
(b)( S x S) → S

Part – II

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

Question 21.
Solve the following system of homogeneous equations.
3x + 2y + 7z = 0, 4x – 3y – 2z = 0, 5x + 9y + 23z = 0
Answer:
The matrix form of the above equation is

The augmented matrix [A, B] is

The above matrix is in echelon form. Here ρ(A, B) = ρ( A) < number of unknowns.
⇒ The system is consistent with infinite number of solutions. To find the solutions.
Writing the equivalent equations.
We get 3x + 2y + 7z = 0 ……..(1)
-17y – 34z = 0 …….(2)
Taking z = t in (2) we get -17y – 34t = 0
⇒ -17y = 34t
⇒ y= \(\frac{34 t}{-17}\) = -2t
Taking z = t; y = -2t in (1) we get
3x + 2 (-2t) + 7t = 0
3x – 4t + 7t = 0
⇒ 3x = -3t ⇒ x = -t
So the solution is x = -t; y = -2t; and z = t, t∈R

Question 22.
Show that |3z – 5 + i| = 4 represents a circle, and, find its centre and radius.
Answer:
The given equation |3z – 5 + i| = 4 can be written as

It is of the form |z – z| = r and so it represents a circle, whose center and radius are \(\left(\frac{5}{3},-\frac{1}{3}\right)\) and 4/3 respectively.

Question 23.
Find the equation of the circle whose centre is (2, -3) and passing through the intersection of the line 3x – 2y = 1 and 4x + y = 27.
Answer:
Solving 3x – 2y = 1 and 4x + y = 27
Simultaneously, we get x = 5 and y = 7
∴ The point of intersection of the lines is (5, 7)
Now we have to find the equation of a circle whose centre is
(2, -3) and which passes through (5, 7)

∴ Required equation of the circle is
(x – 2)² + (y + 3)² = \((\sqrt{109})^{2}\)
⇒ x² + y² – 4x + 6y – 96 = 0

Question 24.
Find the intercepts cut off by the plane \(\vec{r} \cdot(6 \hat{i}+4 \hat{j}-3 \hat{k})=12\) on the coordinate axes.
Answer:
\(\vec{r} \cdot(6 \vec{i}+4 \vec{j}-3 \vec{k})=12\)
Compare the above equations into \(\vec{r} \cdot \vec{n}=q\) so q = 12
Let a, b, c are intercepts of x-axis, y-axis and z-axis respectively.
Clearly

x – intercept = 2; y – intercept = 3; z – intercept = -4

Question 25.
Find the values in the interval (1, 2) of the mean value theorem satisfied by the function f(x) = x – x² for 1 ≤ x ≤ 2.
Answer:
f(1) = 0 and f(2) = -2. Clearly f(x) is defined and differentiable in 1 < x < 2. Therefore, by the Mean Value Theorem, there exists a c ∈(1, 2) such that
f'(c) = \(\frac{f(2)-f(1)}{2-1}\) = 1 – 2c
That is, 1 – 2c = -2 ⇒ c = \(\frac{3}{2}\)

Question 26.
Show that the percentage error in the n^th root of a number is approximately \(\frac{1}{n}\) times the percentage error in the number.
Answer:

Question 27.
Solve the differential equation: tany \(\frac{d y}{d x}\) = cos (x + y) + cos (x -y)
Answer:
tan y \(\frac{d y}{d x}\) = cos (x + y) + cos(x – y)
tan y \(\frac{d y}{d x}\) = cos x cos y – sin x sin y + cos x cos y + sin x sin y
tan y \(\frac{d y}{d x}\) = 2 cos x cos y
seperating the variables
\(\int \frac{\tan y}{\cos y}\) dy = 2∫cos x dx ⇒ ∫sec y tan y dy = 2∫cos x dx
sec y = 2 sin x + c

Question 28.
The probability density function of X is given by \(f(x)=\left\{\begin{array}{cc}
k e^{-\frac{x}{3}} & \text { for } x>0 \\
0 & \text { for } x \leq 0
\end{array}\right.\)
Find the value of k.
Answer:

Question 29.
Construct the truth table for the following statement. \(\neg(p \wedge \neg q)\).
Answer:

Question 30.
Find an approximate value of \(\int_{1}^{1.5} x^{2} d x\) by applying the right-hand rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}.
Answer:
Here a = 1; b = 1.5; n = 5; f(x) = x²
So, the width of each subinterval is

x₀ = 1; x₁ = 1.1; x₂ = 1.2; x₃ = 1.3; x₄ = 1.4; x₅ = 1.5
The Right hand rule for Riemann sum,
S = [f(x₁) + f(x₂) + f(x₃) + f(x₄) + f(x₅)] Δx
= [f(1.1) + f(1.2) + f(1.3) + f(1.4) + f(1.5)] (0.1)
= [1.21 + 1.44 + 1.69 + 1.96 + 2.25] (0.1)
= [8.55] (0.1)
= 0.855.

Part – III

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

Question 31.
Find a matrix A if adj (A) = \(\left[\begin{array}{ccc}
7 & 7 & -7 \\
-1 & 11 & 7 \\
11 & 5 & 7
\end{array}\right]\)

Question 32.
Obtain the Cartesian form of the locus of z = x + iy in the following case Im[(1 – i)z +1] = 0

Question 33.
If \(\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}, \vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}\), show that \([\vec{a} \vec{b} \vec{c}]\) depends on neither x nor y.

Question 34.
The Taylor’s series expansion of f(x) = sin x about x = \(\frac{\pi}{2}\) is obtained by the following way.

Question 35.
The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error in computing (i) the volume of the cube and (ii) the surface area of cube.

Question 36.
Evaluate \(\int_{0}^{1} \frac{\sin \left(3 \tan ^{-1} x\right) \tan ^{-1} x}{1+x^{2}} d x\)

Question 37.
Find the particular solution of (1 + x³) dy – x² ydx = 0 satisfying the condition y(1) = 2.

Question 38.
If X is the random variable with distribution function F(x) given by,
\(\mathrm{F}(x)=\left\{\begin{array}{ll}
0, & x<0 \\
x, & 0 \leq x<1 \\
1, & 1 \leq x
\end{array}\right.\)
then find (z) the Probability density function f(x)

Question 39.
Show that \(((\neg q) \wedge p) \wedge q\) is a contradiction.

Question 40.
Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis.

Part – IV

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

Question 41.
(a) By using Gaussian elimination method, balance the chemical reaction equation:
C₂H₆ + O₂ → H₂O + CO₂.
[OR]
(b) \(\frac{d y}{d x}+\frac{3 y}{x}=\frac{1}{x^{2}}\), given that y = 2 when x = 1

Question 42.
(a) Find the real values of x and y for the equation \(\frac{(1+i) x-2 i}{3+i}+\frac{(2-3 i) y+i}{3-i}=i\)
[OR]
(b) Find the area between the line y = x + 1 and the curve y = x² – 1.

Question 43.
(a) 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.
[OR]
(b) Evaluate: \(\int_{0}^{\frac{\pi}{2}} \frac{d x}{5+4 \sin ^{2} x}\)

Question 44.
(a) A tunnel through a mountain for a four lane highway is to have a elliptical opening. The total width of the highway (not the opening) is to be 16 m, and the height at the edge of the road must be sufficient for a truck 4 m high to clear if the highest point of the opening is to be 5 m approximately. How wide must the opening be?
[OR]
(b) Using truth table check whether the statements \(\neg(p \vee q) \vee(\neg p \wedge q)\) and \(\neg p\) are logically equivalent.

Question 45.
(a) Find the value of cot^-1 x – cot^-1 (x + 2) = \(\frac{\pi}{12}\), x > 0
[OR]
(b) Verify Euler’s theorem for f(x, y) = \(\frac{1}{\sqrt{x^{2}+y^{2}}}\)

Question 46.
(a) Find the points where the straight line passes through (6, 7, 4) and (8, 4, 9) cuts the xz and yz planes.
[OR]
(b) If X is the random variable with probability density function f(x) given by,
\(f(x)=\left\{\begin{array}{rc}
x+1, & -1 \leq x<0 \\
-x+1, & 0 \leq x<1 \\
0, & \text { otherwise }
\end{array}\right.\)
then find (z) the distribution function f(x) (ii) P (-0.5 ≤ X ≤ 0.5)

Question 47.
(a) Sketch the graph of the function: y = \(x \sqrt{4-x}\)
(b) The velocity v, of a parachute falling vertically satisfies the equation, \(v \frac{d v}{d x}=g\left(1-\frac{v^{2}}{k^{2}}\right)\)
where g and k are constants. If v and x are both initially zero, find v in terms of x.