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Download Tamil Nadu 12th Standard Chemistry question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Generic 3

13 x 3 = 39

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:
f (x) = (x − 2)(x − 7), x ∈ [3,11]

A thermometer was taken from a freezer and placed in a boiling water. It took 22 seconds for the thermometer to raise from −10°C to 100°C. Show that the rate of change of temperature at some time t is 5°C per second.

Prove, using mean value theorem, that \(|sin \alpha-sin\beta|\le |\alpha-\beta|, \alpha, \beta \in R\)

Prove using the Rolle’s theorem that between any two distinct real zeros of the polynomial \(a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\) there is a zero of the polynomial \(na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\)

Without actually solving show that the equation x⁴+2x³-2 = 0 has only one real root in the interval (0, 1).

A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.

A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.

Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram z, −iz , and z−iz

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of \((\vec { a } +\vec { b } ).(\vec { b } \times \vec { c } )+(\vec { b } +\vec { c } ).(\vec { c } \times \vec { a } )+(\vec { c } +\vec { a } )(\vec { a } \times \vec { b } )\)

10.

Solve tan^-1 2x + tan^-1 3x = \(\frac{\pi}{4}\), if 6x²< 1

11.

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).

12.

For any two complex number z₁ and z₂ such that |z₁| = |z₂| = 1 and z₁z₂ \(\neq \) -1, then show that \(\frac { { z }_{ 1 }+{ z }_{ 2 } }{ 1+{ z }_{ 1 }{ z }_{ 2 } } \) is real number.

13.

If α, β and γ are the roots of the cubic equation x³+2x²+3x+4 = 0, form a cubic equation whose roots are, 2α, 2β, 2γ

Generic 5

2 x 5 = 10

14.

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = \(\frac{1}{3}\) (area of ΔABC)

15.

Find the domain of f(x) = sin^-1 \((\frac{|x|-2}{3})+ \) cos^-1\((\frac{1-|x|}{4})\)

Multiple Choice Question

33 x 1 = 33

16.

The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________

(a)

x = 0

(b)

y = 0

(c)

x +y = 0

(d)

x + y = 7

17.

If a particle moves in a straight line according to s = t³-6t²-15t, the time interval during which the velocity is negative and acceleration is positive is __________

(a)

2 < t < 5

(b)

2 ≤ t ≤ 5

(c)

t ≥ 2

(d)

t ≤ 2

18.

Which one is the inverse of the statement (pVq)➝(pΛq)?

(a)

(p∧q)➝(pVq)

(b)

ᄀ(pvq)➝(p∧q)

(c)

(ᄀpvᄀq)➝(ᄀp∧ᄀq)

(d)

(ᄀp∧ᄀq)➝(ᄀpVᄀq)

19.

If \(f(x)=\left\{\begin{array}{ll} 2 x & 0 \leq x \leq a \\ 0 & \text { otherwise } \end{array}\right.\) is a probability density function of a random variable, then the value of a is

(a)

(b)

(c)

(d)

20.

Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are

(a)

50,40

(b)

40,50

(c)

40.75,40

(d)

41,41

21.

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\)

(a)

(b)

(c)

(d)

∞

22.

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)

\(\\ \\ \\ t=\cfrac { 1 }{ 3 } \)

(c)

t =1

(d)

t = 3

23.

If \(\sqrt { a+ib } \) = x + iy, then possible value of \(\sqrt { a-ib }\) is ___________

(a)

x²+y²

(b)

\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

(c)

x+iy

(d)

x-iy

24.

If \(\left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \), then the angle between the vector \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _____________

(a)

\(\frac { \pi }{ 4 } \)

(b)

\(\frac { \pi }{ 3 } \)

(c)

\(\frac { \pi }{ 6 } \)

(d)

\(\frac { \pi }{ 2 } \)

25.

The angle between the vectors \(\overset { \wedge }{ i } -\overset { \wedge }{ j } \) and \(\overset { \wedge }{ j } -\overset { \wedge }{ k } \) is _______________

(a)

\(\frac { \pi }{ 3 } \)

(b)

\(\frac { -2\pi }{ 3 } \)

(c)

\(\frac { -\pi }{ 3 } \)

(d)

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

26.

Th point of curve y = 2x²- 6x - 4 at which the targent is parallel to x - axis is __________

(a)

\(\left( \frac { 5 }{ 2 } ,\frac { -7 }{ 12 } \right) \)

(b)

\(\left( \frac { -5 }{ 2 } ,\frac { -7 }{ 2 } \right) \)

(c)

\(\left( \frac { -5 }{ 2 } ,\frac { 17 }{ 12 } \right) \)

(d)

\(\left( \frac { 3 }{ 2 } ,\frac { -7 }{ 2 } \right) \)

27.

If \(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)= 3 and \(\left| \overset { \rightarrow }{ c } \right| =1\) then \(\left| \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) \right| \) is _____________

(a)

(b)

(c)

(d)

28.

If θ is the angle between the vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow}{b} \), then sin θ is ___________

(a)

\(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \)

(b)

\(\frac { \left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| }{ \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } \)

(c)

\(\sqrt { 1-{ \left( \frac { \overset { \rightarrow }{ a. } \overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \right) }^{ 2 } } \)

(d)

29.

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

(a)

(b)

(c)

(d)

\(\infty\)

30.

If tan^-1(cot \(\theta\)) = 2\(\theta\), then\(\theta\) = _____________

(a)

\(\pm 3\)

(b)

\(\pm \frac { \pi }{ 4 } \)

(c)

\(\pm \frac { \pi }{ 6 } \)

(d)

none

31.

If a parabolic reflector is 20 cm in diameter and 5 cm in diameter and 5 cm deep, then its focus is ____________

(a)

(0, 5)

(b)

(5, 0)

(c)

(10, 0)

(d)

(0, 10)

32.

If \({ sin }^{ -1 }x-cos^{ -1 }x=\frac { \pi }{ 6 } \) then ___________

(a)

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

(b)

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

(c)

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

(d)

none of these

33.

In the non - homogeneous system of equations with 3 unknowns if \(\rho\) (A) = \(\rho\) ([AIB]) = 2, then the system has _______

(a)

unique solution

(b)

one parameter family of solution

(c)

two parameter family of solutions

(d)

in consistent

34.

The quadratic equation whose roots are ∝ and β is ___________

(a)

(x - ∝)(x -β) = 0

(b)

(x - ∝)(x + β) = 0

(c)

∝ + β = \(\frac{b}{a}\)

(d)

∝ β = \(\frac{-c}{a}\)

35.

If A^T is the transpose of a square matrix A, then ___________

(a)

|A| ≠ |A^T|

(b)

|A| = |A^T|

(c)

|A| + |A^T| =0

(d)

|A| = |A^T| only

36.

The augmented matrix of a system of linear equations is \(\left[\begin{array}{cccc} 1 & 2 & 7 & 3 \\ 0 & 1 & 4 & 6 \\ 0 & 0 & \lambda-7 & \mu+5 \end{array}\right]\). The system has infinitely many solutions if

(a)

\(\lambda=7, \mu \neq-5\)

(b)

\(\lambda=-7, \mu=5\)

(c)

\(\lambda \neq 7, \mu \neq-5\)

(d)

\(\lambda=7, \mu=-5\)

37.

If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =

(a)

-40

(b)

-80

(c)

-60

(d)

-20

38.

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)

\(\pm 3\)

(b)

\(\pm 6\)

(c)

-3, 9

(d)

3, -9

39.

If the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -5 }= \frac {z+2 }{ 2 } \) lies in the plane x + 3y - αz + β = 0, then (α, β) is

(a)

(-5, 5)

(b)

(-6, 7)

(c)

(5, -5)

(d)

(6, -7)

40.

The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

(a)

\(\frac { \pi }{ 2 } \)

(b)

\(\frac { \pi }{ 3} \)

(c)

\( { \pi }\)

(d)

\(\frac { \pi }{ 4 } \)

41.

The circle passing through (1, -2) and touching the axis of x at (3, 0) passing through the point

(a)

(-5, 2)

(b)

(2, -5)

(c)

(5, -2)

(d)

(-2, 5)

42.

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse is

(a)

\(\frac { \sqrt { 2 } }{ 2 } \)

(b)

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

(c)

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

(d)

\(\frac { 3 }{ 4 } \)

43.

The circle x²+ y²= 4x + 8y +5 intersects the line 3x−4y = m at two distinct points if

(a)

15< m < 65

(b)

35< m <85

(c)

−85 < m < −35

(d)

−35 < m < 15

44.

sin^-1(2cos²x-1)+cos^-1(1-2sin²x)=

(a)

\(\frac{\pi}{2}\)

(b)

\(\frac{\pi}{3}\)

(c)

\(\frac{\pi}{4}\)

(d)

\(\frac{\pi}{6}\)

45.

If sin^-1 x+sin^-1 y+sin^-1 \(z = \frac{3\pi}{2}\), the value of x²⁰¹⁷+y²⁰¹⁸+z²⁰¹⁹\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

(a)

(b)

(c)

(d)

46.

If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

(a)

(1, 0)

(b)

(−1, 1)

(c)

(0, 1)

(d)

(1, 1)

47.

The polynomial x³- kx²+ 9x has three real zeros if and only if, k satisfies

(a)

|k| ≤ 6

(b)

k = 0

(c)

|k| > 6

(d)

|k| ≥ 6

48.

The solution of the equation |z| - z = 1 + 2i is

(a)

\(\frac { 3 }{ 2 } -2i\)

(b)

\(-\frac { 3 }{ 2 } +2i\)

(c)

\(2-\frac { 3 }{ 2 } i\)

(d)

\(2+\frac { 3 }{ 2 } i\)

Generic 2

1 x 2 = 2

49.

Find the domain of the following functions
\(\frac{1}{2}tan^{-1}(1-x^2)-\frac{\pi}{4}\)

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Answers Key:

Generic 3

13 x 3 = 39

Given f (x) = (x − 2)(x − 7), x ∈ [3,11]
a) f(x) is continuous in [3, 11]
b) f(x) is differentiable in (3, 11)
c) f(11) = (11-2)(11-7)
= (9) (4) = 36
f(3) = (3 - 2)(3 - 7)
= (1)(-4) = - 4
∴ By Lagrange's mean value theorem, there exists c ∈ [3,11] such that f'(c) = \(\frac { f(b)-f(a) }{ b-a } \)
\(\left[ \begin{matrix} f(x)\begin{matrix} = & (x \end{matrix}- & 2) & \begin{matrix} (x & - \end{matrix}7) \\ \begin{matrix} = & { x }^{ 2 } \end{matrix}- & 7x & -2x\begin{matrix} + & 14 \end{matrix} \\ \begin{matrix} = & { x }^{ 2 } \end{matrix}- & 9x & +\begin{matrix} 14 & \end{matrix} \end{matrix} \right] \)
⇒ 2c - 9 = \(\frac{36+4}{11-3}\)
⇒ 3c - 9 = \(\frac{40}{8}\) = 5
⇒ 2c = 14
⇒ c = 7 ∈ [3 , 11]

Let f (t) be the temperature at time t. By the mean value theorem, we have
\(f'(c)=\frac{f(b)-f(a)}{b-a}\)
= \(\frac{100-(-10)}{22}\)
= \(\frac{110}{22}\)
= 5°C per second.
Hence the instantaneous rate of change of temperature at some time t should be 5°C per second.

Let f (x) = sin x which is a differentiable function in any open interval. Consider an interval \([\alpha, \beta]\). Applying the mean value theorem there exists \(c \in (\alpha, \beta)\) such that,
\(\frac{sin \beta - sin \alpha}{\beta-\alpha}=f'(c)=cos(c)\)
Therefore, \(\frac{sin \beta - sin \alpha}{\beta-\alpha}=|cos(c)|\le1\)
Hence, \(|sin\alpha-sin\beta|\le |\alpha-\beta|\)
Remark
If we take \(\beta=0\) in the above problem, we ge \(|sin \alpha|\le |\alpha|\)

Let P(x) = \(a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\). Let \(\alpha<\beta \) be two real zeros of P(x). Therefore, \(P(\alpha)=P(\beta)=0.\) Since P(x) is continuous in \([\alpha, \beta]\) and differentiable in \((\alpha, \beta)\) by an application of Rolle’s theorem there exists \(\gamma \in (\alpha,\beta)\) such that \(P'(\gamma)=0\). Since,
\(P'(x)=na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\) which completes the proof.

Let f (x) = x⁴ + 2x³-2
Then f (x) is continuous in [0, 1] and differentiable in (0, 1)
Now, f'(x) = 4x³+6x²
If f'(x) = 0, then
2x²(2x+3) = 0
Therefore, \(x=0, -\frac{3}{2}\) but \(0,-\frac{3}{2} \notin (0,1)\).
Thus, \(f'(x)>0, \forall x\in (0,1)\).
Hence by the Rolle’s theorem there do not exist \(a,b \in(0,1)\) such that, f(a) = 0 = f(b). Therefore the equation f(x ) = 0 cannot have two roots in the interval (0, 1) . But, f (0, 2) = −2 < 0 and f (1) = 1 > 0 tells us the curve y f = (x) crosses the x -axis between 0 and 1 only once by the Intermediate value theorem. Therefore the equation x⁴ + 2x³ − 2 = 0 has only one real root in the interval (0, 1) .

Given s = 2t² + 3t
s(3) = 2 \(\times\) 3² + 3 (3)
= 2\(\times\)9+9
= 27 m ....(1)
s(6) = 2\(\times\) 6² + 3 (6)
= 72 + 18 = 90m ... (2)
Average velocity = \(\frac { s(6)-s(3) }{ 6-3 } \)
= \(\frac { 90-27 }{ 3 } \) = 21 m/s
(ii) Instantaneous Velocity V(t) = \(\frac { ds }{ dt } \)
Instantaneous Velocity at t = 3
= V(3) = 15 m/sec
Instantaneous Velocity at t = 6
= V(6) = 27m/sec

z = 2+3t
-iz = -i(2+3i)
= 2i -3i²= -2i+3
= 3 - 2i
z - iz = 2+ 3i-3+2i
= -1+5i

Given \(\vec { a } ,\vec { b } ,\vec { c } \) are concurrent edges of a parallelepiped, and its volume is 4 cubic units.
\(\vec { a } .(\vec { b } \times \vec { c } )\) = \(\pm 4\) ..(1)
Consider
\((\vec { a } +\vec { b } ).(\vec { b } \times \vec { c } )+(\vec { b } +\vec { c } ).(\vec { c } \times \vec { a } )+(\vec { c } +\vec { a } )(\vec { a } \times \vec { b } )\)
= \(\vec { a } .(\vec { b } \times \vec { c } )+\vec { b } .(\vec { b } \times \vec { c } )+\vec { b } .(\vec { c } \times \vec { a } )+\vec { c } .(\vec { c } \times \vec { a } )+\vec { c } .(\vec { a } \times \vec { b } )+\vec { a } .(\vec { a } \times \vec { b } )\)
= \(\vec { a } .(\vec { b } \times \vec { c } )+0+\vec { b } .(\vec { b } \times \vec { c } )+0+\vec { b } .(\vec { c } \times \vec { a } )+0\)
\([\because \vec { a } .(\vec { a } \times \vec { b } )=0]\)
= \(\vec { a } .(\vec { b } \times \vec { c } )+\vec { a } .(\vec { b } \times \vec { c } )+\vec { a } .(\vec { b } \times \vec { c } )\)
\(\left[ \because [\vec { a } \vec { b } \vec { c } ]=[\vec { a } \vec { b } \vec { c } ]=[\vec { a } \vec { b } \vec { c } ] \right] \)
= \(3[\vec { a } .(\vec { b } \times \vec { c } )]=3(\pm 4)\) using (1)
= ±12

10.

Now, tan^-1 2x + tan^-1 3x = \({ tan }^{ -1 }\left( \frac { 2x+3x }{ 1-6x^{ 2 } } \right) \), since 6x²< 1.
So, \(tan^{ -1 }\left( \frac { 5x }{ 1-6x^{ 2 } } \right) =\frac { \pi }{ 4 } \), Which implies \(\frac { 5x }{ 1-6x^{ 2 } } =tan\frac { \pi }{ 4 } \) = 1
Thus, 1-6x² = 5x , which gives 6x²+5x−1 = 0
Hence, x = \(\frac{1}{6},-1\). But x = -1 does not satisfy 6x²< 1.
Observe that x = −1 makes the left side of the equation negative whereas the right side is a positive number.
Thus, x = −1 is not a solution.
Hence, x = \(\frac{1}{6}\) is the only solution of the equation.

11.

Vector area of quadrilateral ABCD
= vector area of ΔABC + vector area of ΔACD
\(\frac { 1 }{ 2 } (\vec { AB } \times \vec { AC } )+\frac { 1 }{ 2 } (\vec { AC } \times \vec { AD } )\)
= \(-\frac { 1 }{ 2 } (\vec { AC } \times \vec { AB } )+\frac { 1 }{ 2 } (\vec { AC } \times \vec { AD } )\)
\(\left[ \because \ \vec { b } \times \vec { a } =-(\vec { a } \times \vec { b } ) \right] \)
= \(\frac { 1 }{ 2 } \vec { AC } \times (-\vec { AB } +\vec { AD } )\)
= \(\frac { 1 }{ 2 } \vec { AC } \times (\vec { BA } +\vec { AD } )\) \([\because \vec { AB } =-\vec { BA } ]\)
= \(\frac { 1 }{ 2 } \vec { AC } \times \vec { BD } \) [By Δ law of addition]
∴ Area of the quadrilateral ABCD = \(\frac { 1 }{ 2 } \vec { AC } \times \vec { BD } \)

12.

Given |z₁| = |z₂| = 1
⇒ z₁ \(\bar { z } \) = 1
⇒ z₁= \(\frac { 1 }{ \bar { { z }_{ 1 } } } \)
and z₁ z₂ ≠ -1
Also z₂\(\bar { { z }_{ 2 } } \) = 1
⇒ z₂= \(\frac { 1 }{ \bar { { z }_{ 2 } } } \)
Consider \(\frac { z_{ 1 }+{ z }_{ 2 } }{ 1+{ z }_{ 1 }{ z }_{ 2 } } \)
∴ \(\frac { z_{ 1 }+{ z }_{ 2 } }{ 1+{ z }_{ 1 }{ z }_{ 2 } } =\frac { \frac { 1 }{ \bar { { z }_{ 1 } } } +\frac { 1 }{ \bar { { z }_{ 2 } } } }{ 1+\frac { 1 }{ \bar { { z }1 } } .\frac { 1 }{ \bar { { z }_{ 2 } } } } =\frac { \frac { \bar { { z }_{ 2 } } +\bar { { z }_{ 1 } } }{ \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } } }{ \frac { \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } +1 }{ \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } } } \)
= \(\frac { \bar { { z }_{ 2 } } +\bar { { z }_{ 1 } } }{ \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } +1 } \)
= \(\left( \frac { \overline { { z }_{ 1 }{ +z }_{ 2 } } }{ 1+{ z }_{ 1 }{ z }_{ 2 } } \right) \)
∴ \(\frac { z_{ 1 }+{ z }_{ 2 } }{ 1+{ z }_{ 1 }{ z }_{ 2 } } =\frac { \frac { 1 }{ \bar { { z }_{ 1 } } } +\frac { 1 }{ \bar { { z }_{ 2 } } } }{ 1+\frac { 1 }{ \bar { { z }1 } } .\frac { 1 }{ \bar { { z }_{ 2 } } } } =\frac { \frac { \bar { { z }_{ 2 } } +\bar { { z }_{ 1 } } }{ \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } } }{ \frac { \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } +1 }{ \bar { { z }_{ 1 } } \bar { { z }_{ 2 } } } } \) is real [∵ z = \(\bar { z } \) ⇒ is real]

13.

The roots of x³+2x2+3x+4 = 0 are ∝, β, ૪
∴ ∝+β+૪ = -co-efficient of x²= -2 ...(1)
∝β + β૪ + ૪∝ = co-effficient of x = 3 ....(2)
-∝β૪ = +4 ⇒ ∝β૪ = -4 ...(3)
Form a cubic equation whose roots are 2∝, 2β, 2૪
2∝+2β+2૪ = 2(∝+β+૪) = 2(-2) = -4 [from (1)]
4∝β+4β૪+4૪∝ = 4(∝β+β૪+୪∝) = 4(3) = 12 [from (2)]
(2∝)(2β)(2૪) = 8(∝β૪) = 8(-4) = -32 [from (3)]
∴ The required cubic equation is
x³-(2∝+2β+2૪)x²+ (2∝β+2β૪+2୪∝)x - (2∝)(2β)(2૪) = 0
⇒ x³+(-4)x²+12x+32 = 0
⇒ x³+4x²+12x+32 = 0

Generic 5

2 x 5 = 10

14.

Let the position vector of the vertices of ΔABC be \(\vec { a } \), \(\vec { b } \) and \(\vec { c } \) respectively.
Since G is the centroid of ΔABC, \(\vec { OG } =\frac { \vec { a } +\vec { b } +\vec { c } }{ 3 } \)
Are of ΔGAB, = \(|\vec { AB } \times \vec { AG } |=|(\vec { OB } -\vec { OA } )\times (\vec { OG } -\vec { OA } )|\)
= \(\left| (\vec { b } -\vec { a } )\times \left( \frac { \vec { a } +\vec { b } +\vec { c } }{ 3 } -\vec { a } \right) \right| \)
= \(\left| (\vec { b } -\vec { a } )\times \left( \frac { \vec { a } +\vec { b } +\vec { c } -3\vec { a } }{ 3 } \right) \right| \)
= \(\left| (\vec { b } -\vec { a } )\times \left( \frac { \vec { b } +\vec { c } +2\vec { a } }{ 3 } \right) \right| \)
= \(\frac { 1 }{ 3 } |(\vec { b } -\vec { a } )\times (\vec { a } +\vec { 0 } -2\vec { a } )|\)
= \(\frac { 1 }{ 3 } |\vec { b } \times \vec { b } +\vec { b } \times \vec { c } -2\vec { b } \times \vec { a } -\vec { a } \times \vec { b } -\vec { a } \times \vec { c } +2\vec { a } \times \vec { a } |\)
[∵ cross product is distributive]
= \(\frac { 1 }{ 3 } |\vec { b } \times \vec { c } +2\vec { a } \times \vec { b } -\vec { a } \times \vec { b } +\vec { c } \times \vec { a } |\)
= \(\frac { 1 }{ 3 } |\vec { a } \times \vec { b } +\vec { b } \times \vec { c } +\vec { c } \times \vec { a } |\) \( [\because \vec { b } \times \vec { b } =\vec { 0 } ,\vec { a } \times \vec { a } =\vec { 0 } ,.(1)\vec { a } \times \vec { b } =-\vec { b } \times \vec { a } ]\)
Area of ΔGAC = \(|\vec { CA } \times \vec { AG } |\)
= \(|(\vec { OA } -\vec { OC } )\times (\vec { OG } -\vec { OA } )|\)
\(\left| (\vec { a } -\vec { c } )\times \left( \frac { \vec { a } +\vec { b } +\vec { c } }{ 3 } -\vec { a } \right) \right| \)
= \(\left| (\vec { c } -\vec { a } )\times \left( \frac { \vec { a } +\vec { b } +\vec { c } -3\vec { a } }{ 3 } \right) \right| \)
= \(\frac { 1 }{ 3 } |(\vec { a } -\vec { c } )\times (\vec { b } +\vec { c } -2\vec { a } )|\)
= \(\frac { 1 }{ 3 } |\vec { a } \times \vec { b } +\vec { a } \times \vec { c } -2\vec { a } \times \vec { a } -\vec { c } \times \vec { b } -\vec { c } \times \vec { c } +2\vec { c } \times \vec { a } |\)
= \(\frac { 1 }{ 3 } |\vec { a } \times \vec { b } -\vec { c } \times \vec { a } +\vec { b } \times \vec { c } +2\vec { c } \times \vec { a } |\)
= \(\frac { 1 }{ 3 } |\vec { a } \times \vec { b } +\vec { b } \times \vec { c } +\vec { c } \times \vec { a } |\)
Also area of ΔGBC = \(|\vec { BC } \times \vec { BG } |\)
= \(|\vec { OC } -\vec { OB } )\times (\vec { OG } -\vec { OB } )|\)
= \(\left| (\vec { c } -\vec { a } )\times \left( \frac { \vec { a } +\vec { b } +\vec { c } -\vec { b } }{ 3 } \right) \right| \)
= \(\left| (\vec { c } -\vec { a } )\times \left( \frac { \vec { a } +\vec { b } -2\vec { a } }{ 3 } \right) \right| \)
\(\frac { 1 }{ 3 } |(\vec { c } -\vec { b } )\times (\vec { a } +\vec { c } -2\vec { b } )|\)
= \(\frac { 1 }{ 3 } |\vec { c } \times \vec { a } +\vec { c } \times \vec { c } -2\vec { c } \times \vec { b } -\vec { b } \times \vec { a } -\vec { b } \times \vec { c } +2\vec { b } \times \vec { b } |\)
= \(\frac { 1 }{ 3 } |\vec { c } \times \vec { a } +2\vec { b } \times \vec { c } +\vec { a } \times \vec { b } -\vec { b } \times \vec { c } |\)
\(\frac { 1 }{ 3 } |\vec { c } \times \vec { a } +\vec { b } \times \vec { c } +\vec { a } \times \vec { b } |\)
From (1), (2) and (3),
Area of ΔGAB = Area of ΔGAC = Area of ΔGBC
= \(\frac { 1 }{ 3 } |\vec { a } \times \vec { b } +\vec { b } \times \vec { c } +\vec { c } \times \vec { a } |\)
= \(\frac { 1 }{ 3 } \) Area of ΔABC.

15.

f(x) = sin^-1 \((\frac{|x|-2}{3})+cos^-1(\frac{1-|x|}{4})\)
From the definition of sin^-1
\(-1\le \frac { \left| x \right| -2 }{ 3 } \le 1\)
\(\Rightarrow -3\le \left| x \right| -2\le 3\)
\(\Rightarrow -3+2\le \left| x \right| \le \left| x \right| \le 3+2\)
\(\Rightarrow -1\le \left| x \right| \le 5\)
It reduces to
\(0\le \left| x \right| \le 5\)
\(\Rightarrow 0\le \left| x \right| and\left| x \right| \le 5\)
\(\Rightarrow \left| x \right| \ge 0and-5\le x\le 5\)
From the definition of cos^-1x.
\(-1\le \frac { 1-\left| x \right| }{ 4 } \le 1\)
\(\Rightarrow -4\le 1-\left| x \right| \le 4\)
\(\Rightarrow -4-1\le \left| x \right| \le 4-1\)
\(\Rightarrow -5\le -\left| x \right| \le 3\)
\(\Rightarrow -3\le \left| x \right| >5\)
It reduces to
\(0\le \left| x \right| \le 5\)
\(-5\le |x|\le 5\)
From (1) & (2),
Domain is [-5, 5]

Multiple Choice Question

33 x 1 = 33

16.

(b)
y = 0

17.

(a)
2 < t < 5

18.

19.

(a)
1

20.

21.

(a)
0

22.

(b)
\(\\ \\ \\ t=\cfrac { 1 }{ 3 } \)

23.

(d)
x-iy

24.

(a)
\(\frac { \pi }{ 4 } \)

25.

(d)
\(\frac { 2\pi }{ 3 } \)

26.

(d)
\(\left( \frac { 3 }{ 2 } ,\frac { -7 }{ 2 } \right) \)

27.

(d)
9

28.

(c)
\(\sqrt { 1-{ \left( \frac { \overset { \rightarrow }{ a. } \overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \right) }^{ 2 } } \)

29.

(b)
2

30.

31.

(b)
(5, 0)

32.

(b)
\(\frac { \sqrt { 3 } }{ 2 } \)

33.

(b)
one parameter family of solution

34.

(a)
(x - ∝)(x -β) = 0

35.

(b)
|A| = |A^T|

36.

(d)
\(\lambda=7, \mu=-5\)

37.

(b)
-80

38.

(d)
3, -9

39.

(b)
(-6, 7)

40.

41.

42.

43.

(d)
−35 < m < 15

44.

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

45.

(a)
0

46.

(b)
(−1, 1)

47.

(d)
|k| ≥ 6

48.

(b)
\(-\frac { 3 }{ 2 } +2i\)

Generic 2

1 x 2 = 2

49.

Let \(g(x)=\frac { 1 }{ 2 } { tan }^{ -1 }\left( 1-{ x }^{ 2 } \right) -\frac { \pi }{ 4 } \)
By the definition of tan^-1 x, it is a function with
the entire real line \(\left( -\infty ,\infty \right) \) as its domain.
\(\therefore \) Domain of \(g(x)=\frac { 1 }{ 2 } { tan }^{ -1 }\left( 1-{ x }^{ 2 } \right) -\frac { \pi }{ 4 } \)is R
\(\therefore \) Domain of g(x) is R.