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1

MCQ (Single Correct Answer)

If the roots of the quadratic equation $${x^2} + px + q = 0$$ are $$\tan {30^ \circ }$$ and $$\tan {15^ \circ }$$, respectively, then the value of $$2 + q - p$$ is

A

2

B

3

C

0

D

1

$${x^2} + px + q = 0$$

Sum of roots $$ = \tan {30^ \circ } + \tan {15^ \circ } = - p$$

Products of roots $$ = \tan {30^ \circ }.\tan {15^ \circ } = q$$

$$\tan {45^ \circ } = {{\tan {{30}^ \circ } + \tan {{15}^ \circ }} \over {1 - \tan {{30}^ \circ }.\tan {{15}^ \circ }}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {{ - p} \over {1 - q}} = 1$$

$$ \Rightarrow - p = 1 - q \Rightarrow q - p = 1$$

$$\therefore$$ $$2 + q - p = 3$$

Sum of roots $$ = \tan {30^ \circ } + \tan {15^ \circ } = - p$$

Products of roots $$ = \tan {30^ \circ }.\tan {15^ \circ } = q$$

$$\tan {45^ \circ } = {{\tan {{30}^ \circ } + \tan {{15}^ \circ }} \over {1 - \tan {{30}^ \circ }.\tan {{15}^ \circ }}}$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {{ - p} \over {1 - q}} = 1$$

$$ \Rightarrow - p = 1 - q \Rightarrow q - p = 1$$

$$\therefore$$ $$2 + q - p = 3$$

2

MCQ (Single Correct Answer)

If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals

A

$$-2$$

B

$$3$$

C

$$2$$

D

$$1$$

Let n and (n + 1) be the roots of x^{2} $$-$$ bx + c = 0.

Then, n + (n + 1) = b and n(n + 1) = c

$$\therefore$$ b^{2} $$-$$ 4c = (2n + 1)^{2} $$-$$ 4n(n + 1)

= 4n^{2} + 4n + 1 $$-$$ 4n^{2} $$-$$ 4n = 1

3

MCQ (Single Correct Answer)

The value of $$a$$ for which the sum of the squares of the roots of the equation

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is

A

$$1$$

B

$$0$$

C

$$3$$

D

$$2$$

Given quadratic equation,

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$

Let $$\alpha $$ and $$\beta $$ are the roots of the equation.

$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $$a - 2$$

and $$\alpha $$$$\beta $$ = $$ - a - 1$$

Now $${\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta $$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 2} \right)^2} + 2\left( {a + 1} \right)$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {a^2} - 2a + 6$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 1} \right)^2} + 5$$

$$ \Rightarrow $$ The value of $${\alpha ^2} + {\beta ^2}$$ will be minimum, when $${a - 1}$$ = 0

$$ \Rightarrow $$ $${a = 1}$$

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$

Let $$\alpha $$ and $$\beta $$ are the roots of the equation.

$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $$a - 2$$

and $$\alpha $$$$\beta $$ = $$ - a - 1$$

Now $${\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta $$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 2} \right)^2} + 2\left( {a + 1} \right)$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {a^2} - 2a + 6$$

$$ \Rightarrow $$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 1} \right)^2} + 5$$

$$ \Rightarrow $$ The value of $${\alpha ^2} + {\beta ^2}$$ will be minimum, when $${a - 1}$$ = 0

$$ \Rightarrow $$ $${a = 1}$$

4

MCQ (Single Correct Answer)

If both the roots of the quadratic equation $${x^2} - 2kx + {k^2} + k - 5 = 0$$ are less than 5, then $$k$$ lies in the interval

A

$$\left( {5,6} \right]$$

B

$$\left( {6,\,\infty } \right)$$

C

$$\left( { - \infty ,\,4} \right)$$

D

$$\left[ {4,\,5} \right]$$

both roots are less than $$5,$$

then $$(i)$$ Discriminant $$ \ge 0$$

$$\left( {ii} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p\left( 5 \right) > 0$$

$$\left( {iii} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{Sum\,\,of\,\,roots} \over 2} < 5$$

Hence $$\left( i \right)\,\,\,\,\,\,4{k^2} - 4\left( {{k^2} + k - 5} \right) \ge 0$$

$$4{k^2} - 4{k^2} - 4k + 20 \ge 0$$

$$4k \le 20 \Rightarrow k \le 5$$

$$\left( {ii} \right)\,\,\,\,\,f\left( 5 \right) > 0;25 - 10k + {k^2} + k - 5 > 0$$

or $${k^2} - 9k + 20 > 0$$

or $$k\left( {k - 4} \right) - 5\left( {k - 4} \right) > 0$$

or $$\left( {k - 5} \right)\left( {k - 4} \right) > 0$$

$$ \Rightarrow k \in \left( { - \infty ,4} \right) \cup \left( { - \infty ,5} \right)$$

$$\left( {iii} \right)\,\,\,\,\,\,{{Sum\,\,of\,\,roots} \over 2}$$

$$ = - {b \over {2a}} = {{2k} \over 2} < 5$$

The intersection of $$(i)$$, $$(ii)$$ & $$(iii)$$ gives

$$k \in \left( { - \infty ,4} \right).$$

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Complex Numbers

Quadratic Equation and Inequalities

Permutations and Combinations

Mathematical Induction and Binomial Theorem

Sequences and Series

Matrices and Determinants

Vector Algebra and 3D Geometry

Probability

Statistics

Mathematical Reasoning

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Properties of Triangle

Inverse Trigonometric Functions

Straight Lines and Pair of Straight Lines

Circle

Conic Sections

Functions

Limits, Continuity and Differentiability

Differentiation

Application of Derivatives

Indefinite Integrals

Definite Integrals and Applications of Integrals

Differential Equations