Algebra PYQ — Page 82

The sum of integers from 1 to 100 that are divisible by 2 or 5 is

$l, m, n$ are the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ term of a G.P. all positive, then $\left|\begin{array}{ccc}\log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1\end{array}\right|$ equals

If $a, b, c$ are distinct $+v e$ real numbers and $a^2+b^2+c^2=1$ then $a b+b c+c a$ is

Which one is not periodic

If $2 a+3 b+6 c=0(a, b, c \in R)$ then the quadratic equation $a x^2+b x+c=0$ has

If $a>0$ discriminant of $a x^2+2 b x+c$ is -ve, then $\left|\begin{array}{ccc}a & b & a x+b \\ b & c & b x+c \\ a x+b & b x+c & 0\end{array}\right|$ is

If $\alpha \neq \beta$ but $\alpha^2=5 \alpha-3$ and $\beta^2=5 \beta-3$ then the equation having $\alpha / \beta$ and $\beta / \alpha$ as its roots is

Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed) is

If $|z-4| < |z-2|$, its solution is given by

Product of real roots of the equation $t^2 x^2+|x|+9=0$

Fifth term of a GP is 2, then the product of its 9 terms is

The locus of the centre of a circle which touches the circle $\left|z-z_1\right|=a$ and $\left|z-z_2\right|=b$ externally ( $z, z_1$ and $z_2$ are complex numbers) will be

If $p$ and $q$ are the roots of the equation $x^2+p x+q=0$, then

$z$ and $w$ are two non zero complex no.s such that $|z|=|w|$ and $\operatorname{Arg} z+\operatorname{Arg} w=\pi$ then $z$ equals

If the sum of the coefficients in the expansion of $(a+b)^n$ is 4096 , then the greatest coefficient in the expansion is

$1^3-2^3+3^3-4^3+\ldots .+9^3=$

Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are

The coefficients of $x^p$ and $x^q$ in the expansion of $(1+x)^{p+q}$ are

The value of $2^{1 / 4}, 4^{1 / 8}, 8^{1 / 6}+\ldots \ldots \infty$ is

If $f(x+y)=f(x) \cdot f(y) \forall x \cdot y$ and $f(5)=2, f^{\prime}(0)=3$ then $f^{\prime}(5)$ is

The positive integer just greater than $(1+0.0001)^{10000}$ is