Mathematics

Q.

Find the zeroes of the polynomial $x^2+\frac{1}{6}x-2$ and verify the relation between the coefficient and zeroes of the polynomial.

Q. The number of terms common to the two arithmetic progressions
$3,7,11,\dots ,407$ and $2,9,16,\dots ,709$ is

Q.

Find the zeros of the following quadratic polynomials $x^2-2x-8$ and verify the relationship between the zeros and the coefficients.

Q. The value of ${81}^{\left(1/{\log }_{5}3\right)}+{27}^{{\log }_{9}36}+3^{\left(4/{\log }_{7}9\right)}$ is

Q.

What is the formula of ${\left(A+B\right)}^3$?

Q. If $\alpha ,\beta ,\gamma$ are the roots of $x^3+3x^2+4x+5=0$ , then which of the following is/ are true.

1. $\sum \alpha =3$
2. $\sum \frac{1}{\alpha }=-\frac{4}{5}$
3. $\sum {\alpha }^2=1$
4. $\sum {\beta }^2{\gamma }^2=-14$

Q.

What is the Derivative of $y={\sec }^2\left(x\right)$?

Q. If $y=si{n}^{-1}\frac{\sqrt{(1+x)}+\sqrt{(1-x)}}{2}$, then $\frac{dy}{dx}=$

1. $\frac{1}{\sqrt{(1-x^2)}}$
2. $-\frac{1}{\sqrt{(1-x^2)}}$
3. $-\frac{1}{2\sqrt{(1-x^2)}}$
4. None of these

Q.

The difference between two numbers is $26$ and one number is three times the other. Find them?

Q.

A coin is tossed $10$ times.

The probability of getting exactly six heads is

1. $\frac{512}{513}$

2. $\frac{105}{512}$

3. $\frac{100}{153}$

4. ${}^{10}C_6$

Q.

Find the degree measure of $\frac{1}{4}\mathrm{radian}$.

Q.

Find the zeroes of the quadratic polynomial $x^2+7x+12$ and verify the relationship between the zeroes and the coefficients of the polynomial.

Q. If $\log \frac{a}{b}+\log \frac{b}{a}=\log (a+b),$ then

1. $a+b=0$
2. $a+b=1$
3. $a-b=0$
4. $a-b=1$

Q. Let $S=\{1,2,3,4,5,6,9\}$. Then the number of elements in the set $T=\{A\subseteq S:A\ne \varphi$ and the sum of all the elements of $A$ is not a multiple of $3\}$ is

Q. Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation, $x^2-x-1=0,$ where $\alpha \ne \beta .$ For $n=0,1,2,....,$ let $a_n=p{\alpha }^n+q{\beta }^n.$

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0.$ then $a=0=b.$

If $a_4=28,$ then $p+2q=$

1. $21$
2. $14$
3. $7$
4. $12$

Q. Let $R_1$ and $R_2$ be two relations defined as follows:
$R_1=\left\{\right(a,b)\in R^2:a^2+b^2\in Q\}$ and $R_2=\left\{\right(a,b)\in R^2:a^2+b^2\notin Q\}.$ Then

1. Neither $R_1$ nor $R_2$ is a transitive relation
2. $R_1$ and $R_2$ both are transitive relations
3. $R_1$ is transitive but $R_2$ is not a transitive relation
4. $R_2$ is transitive but $R_1$ is not a transitive relation

Q.

The two events $A$ and $B$ have probabilities $0.25$ and $0.50$ respectively.

The probability that both $A$ and $B$ occur simultaneously is $0.14$.

Then the probability that neither $A$ nor $B$ occurs is

1. $0.39$

2. $0.25$

3. $0.904$

4. None of these

Q.

What angle is a $1$ in $10$ slope?

Q.

A function $f\left(x\right)$ is given by $f\left(x\right)=\frac{5^x}{5^x+5}$, then the sum of the series: $f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+......+f\left(\frac{39}{20}\right)$ is equal to:

1. $\frac{19}{2}$

2. $\frac{49}{2}$

3. $\frac{39}{2}$

4. $\frac{29}{2}$

Q.

Solve $8x+5y=9,3x+2y=4$by any method.

Q. Let $A=\{x\in R:x\text{is not a positive integer}\}$.
Define a function $f:A\to R$ as $f\left(x\right)=\frac{2x}{x-1}$,
then $f$ is :

1. not injective
2. injective but not surjective
3. neither injective nor subjective
4. surjective but not injective

Q. If $f\left(x\right)$ is a polynomial function satisfying
$f\left(x\right)f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$ and $f\left(1\right)=2$ then the number of such functions possible is/are

Q.

Find the value of $x$ if $x=\sqrt{6+\sqrt{6+\sqrt{+6+.....}}}$. Where $x$ is a natural number.

Q.

A train covered a certain distance at a uniform speed. If the speed of train would have been $10km/h$ faster, it would have taken $2hours$ less than the scheduled time. And, if the train were slower by $10km/h$; it would have taken $3hours$ more than the scheduled time. Find the distance covered by the train.

Q. In a class of $5$ students, average weight of the $4$ lightest students is $40$ kgs, average weight of the $4$ heaviest students is $45$ kgs. Then the difference between the maximum and minimum possible average weight is

1. $1.5$
2. $2$
3. $3$
4. $2.5$

Q. Let $z_1,z_2,z_3$ be three complex numbers such that $|z_1|=|z_2|=|z_3|=1$ and $\frac{z_1^2}{z_2{z}_3}+\frac{z_2^2}{z_1{z}_3}+\frac{z_3^2}{z_1{z}_2}=-1.$ Then the possible value(s) of $|z_1+z_2+z_3|$ is/are

1. $0$
2. $1$
3. $2$
4. $3$

Q. For $x\in (0,\frac{3}{2}),$ let $f\left(x\right)=\sqrt{x},g\left(x\right)=\tan x$ and $h\left(x\right)=\frac{1-x^2}{1+x^2}.$ If $\varphi \left(x\right)=\left(\right(hof\left)og\right)\left(x\right),$ then $\varphi \left(\frac{\pi }{3}\right)$ is equal to

1. $\tan \left(\frac{\pi }{12}\right)$
2. $\tan \left(\frac{5\pi }{12}\right)$
3. $\tan \left(\frac{7\pi }{12}\right)$
4. $\tan \left(\frac{11\pi }{12}\right)$

Q.

Let $A$ be a$3\times 3$matrix with $\text{det}\left(A\right)=4$. Let $R_i$ denote the $i^{th}$ row of $A$.

If a matrix $B$ is obtained by performing the operation $R_2\to 2R_2+5R_3$ on $2A$, then $\text{det}\left(B\right)$ is equal to:

1. $64$

2. $16$

3. $80$

4. $128$

Q.

The sum of all three-digit natural numbers which are divisible by $7$ is

1. $25501$
2. $52005$
3. $70336$
4. $84321$

Q. The number of integral values of $m$ for which the equation $(1+m^2)x^2-2(1+3m)x+(1+8m)=0$ has no real root is :

1. $2$
2. $3$
3. infinitely many
4. $1$