IIT-JEE Super Course in Mathematics

Trishna Knowledge Systems

Chapter 5

Sequences And Series - all with Video Answers

Educators

Chapter Questions

Problem 1

Find
(i) the 22 nd term of the progression $4,9,14,19 \ldots \ldots \ldots$
(ii) the first negative term of the progression $19,18 \frac{1}{5}, 17 \frac{2}{5} \ldots \ldots$
(iii) the 11 th term of the series $5+10+20+\ldots \ldots . .$
(iv) the GP if 4 th and 9 th terms are 54 and 13122 respectively.

Sreeraj P

Sreeraj P

Numerade Educator

Problem 2

Find the sum of the
(i) first 61 terms of the series $2+5+8+\ldots \ldots$
(ii) first 25 terms of the AP $\frac{1}{9}, \frac{2}{9}, \frac{3}{9} \ldots .$
(iii) the series $101+99+97+\ldots .+47$
(iv) the series $2+6+18+\ldots \ldots+4374$
(v) product $6^{\frac{1}{2}} \cdot 6^{\frac{1}{4}} \cdot 6^{\frac{1}{8}} \ldots \ldots \ldots \infty$
(vi) series $\frac{1}{7}+\frac{2}{7^{2}}+\frac{1}{7^{3}}+\frac{2}{7^{4}}+\ldots \ldots \ldots \infty$

Gaurav Kalra

Numerade Educator

Problem 3

The ratio of the sum of the first n terms of two APs is $2 \mathrm{n}-15: 2 \mathrm{n}-1$. Find the ratio of the 13 th terms of the two APs.

Sreeraj P

Numerade Educator

Problem 4

For what values of the parameter $\mathrm{k}$ are the three values of $\mathrm{x}$ such that $5^{1+\mathrm{x}}+5^{1-\mathrm{x}}, \frac{\mathrm{k}}{2}, 25^{\mathrm{x}}+25^{-\mathrm{x}}$ are three successive terms of an AP

Gaurav Kalra

Numerade Educator

Problem 5

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{GP}$ and the equations $\mathrm{px}^{2}+2 \mathrm{qx}+\mathrm{r}=0$ and $\mathrm{d} \mathrm{x}^{2}+2 \mathrm{ex}+\mathrm{f}=0$ have a common root, then show that
$\frac{\mathrm{d}}{\mathrm{p}}, \frac{\mathrm{e}}{\mathrm{q}}, \frac{\mathrm{f}}{\mathrm{r}}$ are in $\mathrm{AP}$

Gaurav Kalra

Numerade Educator

Problem 6

The sum of an infinitely decreasing $\mathrm{GP}$ is $\frac{10}{3}$ The sum of the cubes of its terms is $\frac{1000}{63}$. Find the sum of the 4 th powers of the terms of the GP.

Gaurav Kalra

Numerade Educator

Problem 7

Prove that the three successive terms of a GP will form the sides of a triangle if the common ratio $\mathrm{r}$ satisfies the inequality $\frac{\sqrt{5}-1}{2}<\mathrm{r}<1$

Gaurav Kalra

Numerade Educator

Problem 8

The sum of three numbers in a GP is 42 . If the first two numbers are increased by 2 and the third is decreased by 4 then the resulting numbers form an AP Find the numbers.

Urvashi Arora

Numerade Educator

Problem 9

If the equations $x^{2}-p x+q=0, x^{2}-r x+s=0$ have a common root which is the harmonic mean between their other two roots, prove that
(i) $(\mathrm{q}-\mathrm{s})^{2}=(\mathrm{p}-\mathrm{r})(\mathrm{qr}-\mathrm{ps})$
(ii) $p s(3 q+s)=q r(3 s+q)$

Gaurav Kalra

Numerade Educator

Problem 10

Prove that $$
\underbrace{(33 \ldots 3)^{2}}_{n \text { digin }}+\underbrace{22 \ldots 2}_{=\text {figta }}=\underbrace{111 \ldots .1}_{2 n} \text { . }
$$ Hence find the value of $9(11111)^{2}+2(11111)$.

Gaurav Kalra

Numerade Educator

Problem 11

The three angles of a triangle are in AP If the largest angle is twice the smallest angle, then the largest angle is
(a) $40^{\circ}$
(b) $60^{\circ}$
(c) $100^{\circ}$
(d) $80^{\circ}$

Narayan Hari

Numerade Educator

Problem 12

If $\frac{3}{14}, \mathrm{a}, \frac{2}{21}$ are in GP, then $\mathrm{a}=$
(a) $\frac{1}{6}$
(b) $\frac{1}{8}$
(c) $\frac{2}{3}$
(d) $\frac{1}{7}$

Narayan Hari

Numerade Educator

Problem 13

if $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$, then $a, b, c$ are in
(a) AP
(b) $\mathrm{GP}$
(c) HP
(d) AGP

Narayan Hari

Numerade Educator

Problem 14

Sum to $\mathrm{n}$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\ldots \ldots$ is equal to
(a) $2^{\mathrm{n}}-1$
(b) $1-2^{-n}$
(c) $\mathrm{n}+2^{-\mathrm{n}}-1$
(d) $2^{n}-1$

Narayan Hari

Numerade Educator

Problem 15

If $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ be the $\mathrm{AM}, \mathrm{GM}, \mathrm{HM}$ respectively between numbers a and $\mathrm{b}$, then $\mathrm{P}-\mathrm{Q}$ is
(a) $\frac{\mathrm{a}-\mathrm{b}}{\mathrm{a}}$
(b) $\frac{\mathrm{a}+\mathrm{b}}{2}$

Narayan Hari

Numerade Educator

Problem 16

Statement 1 If $a_{1}, a_{2} a_{y^{a}} a_{4}$ are four numbers such that $a_{2} a_{3}-a_{1} a_{4}$ is positive then $a_{1}, a_{2}, a_{y}, a_{4}$ are in AP and
Statement 2
If $a_{1}, a_{2}, a_{y}, a_{4}$ are in AP then $a_{2} a_{3}-a_{1} a_{4}$ is positive.

Narayan Hari

Numerade Educator

Problem 17

Statement 1 The sum of any number of terms from the beginning of the series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots \ldots \ldots$, cannot exceed $2 .$
and
Statement 2
The sum to infinity of a GP whose first term is a and common ratio is $r$ where, $|r|<1$ is finite and is equal to $\frac{a}{1-r}$.

Narayan Hari

Numerade Educator

Problem 18

Statement 1 If $b+c, c+a$ and $a+b$ are in HP then $\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}$ are in HP and
Statement 2 If each term of a sequence in a GP is squared, the resulting series is a GP

Gaurav Kalra

Numerade Educator

Problem 19

Statement 1
If $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in $\mathrm{GP}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in GP
and
Statement 2
If $y^{2}=x z$, then $x, y, z$ are in $G P$

Urvashi Arora

Numerade Educator

Problem 20

Statement 1 The sum of the first 10 terms of the series $\frac{1}{2}-\frac{3}{2}+\frac{9}{2}-\frac{27}{2}+\ldots .$ is $\frac{-1}{8} \times\left(3^{10}-1\right)$
and
Statement 2 In a GP a $+a r+a r^{2}+\ldots$, if $r$ is negative, then $a_{r+1}<a_{s} r=1,2, \ldots .$

Gaurav Kalra

Numerade Educator

Problem 21

The areas $\mathrm{X}_{1}, \mathrm{X}_{2} \ldots \ldots .$ of the squares form a
(a) decreasing GP
(b) increasing GP
(c) AP
(d) HP

Narayan Hari

Numerade Educator

Problem 22

The diagonals $\mathrm{d}_{1}, \mathrm{~d}_{2} \ldots$ of the squares form a
(a) AP
(b) $\mathrm{GP}$
(c) HP
(d) AGP

Narayan Hari

Numerade Educator

Problem 23

The ratio of the areas of the circum circle and the area of the square is
(a) depends on the radius of the circle.
(b) is independent of the radius of circle
(c) vary inversely with respect to one another
(d) cannot say anything definitely

Narayan Hari

Numerade Educator

Problem 24

Cabling to connect the lamp posts is to be done to supply power from the stage to all the posts. The cabling is to be done along the circumference of each circular arrangement. The length of the cable required to connect all the lamp posts is
(a) $26960 \mathrm{~m}$
(b) $26400 \mathrm{~m}$
(c) $53920 \mathrm{~m}$
(d) $52800 \mathrm{~m}$

Narayan Hari

Numerade Educator

Problem 25

If the cost of cable is Rs 40 per metre, the amount to be spent on the purchase of cables, if $5 \%$ wastage is incurred
(a) Rs $11,32,320$
(b) $\mathrm{Rs} 11,08,800$
(c) $\mathrm{Rs} 22,64,640$
(d) Rs $22,17,600$

Narayan Hari

Numerade Educator

Problem 26

The distances between 2 consecutive lampposts on each of these circles along the circumference form a.....when we proceed from the stage to the boundary of the ground
(a) an increasing $\mathrm{AP}$
(b) decreasing AP
(c) $\overrightarrow{A G P}$
(d) HP

Narayan Hari

Numerade Educator

Problem 27

If $\cos (\theta-\alpha), \cos \theta, \cos (\theta+\alpha)$ are in HP, then $\cos \theta \sec \frac{\alpha}{2}$ is equal to
(a) $-1$
(b) $-\sqrt{2}$
(c) $\sqrt{2}$
(d) 2

Narayan Hari

Numerade Educator

Problem 28

If $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are in GP then
(a) $\mathrm{a}+\mathrm{b}, \mathrm{b}+c_{2} \mathrm{c}+\mathrm{d}$ are in $\mathrm{GP}$
(b) $a x^{2}+c$ is a factor of $a x^{3}+b x^{2}+c x+d$
(c) $a^{2}+b^{2}+c^{2}, a b+b c+c d, b^{2}+c^{2}+d^{2}$ are in GP
(d) $a x+c$ is a factor of $a x^{3}+b x^{2}+c x+d$

Gaurav Kalra

Numerade Educator

Problem 29

Three positive numbers $x, y, z$ are in AP then, $x^{2}, y^{2}, z^{2}$ are in HP if $x, y, z$ satisfies
(a) $x^{2}=y^{2}=\frac{z^{2}}{2}$
(b) $2 y^{2}+x z=0$
(c) $x=y=z$
(d) $y^{2}=x z$

Narayan Hari

Numerade Educator

Problem 30

Column I $\quad$ Column II
(a) If $\mathrm{a}^{x}=\mathrm{b}^{\gamma}=\mathrm{c}^{x}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then $\mathrm{x}, y, \mathrm{z}$ are in
(p) AP
(b) Three distinct numbers $a, b, c$ satisfying $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$ are in
(q) $\mathrm{GP}$
(c) If $\mathrm{x}, \mathrm{y}, \mathrm{z}(\mathrm{all}>1)$ are in GP, then $\frac{1}{1+\log \mathrm{x}}, \frac{1}{1+\log \mathrm{y}}, \frac{1}{1+\log \mathrm{z}}$ are in
(r) HP
(d) Three consecutive terms $\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}}$ of a sequence are in
(s) AGP

Gaurav Kalra

Numerade Educator

Problem 31

Number of natural numbers between 250 and 800 which are divisible by 7 is
(a) 80
(b) 79
(c) 63
(d) 70

Narayan Hari

Numerade Educator

Problem 32

If seven times the seventh term of an AP is equal to eleven times its eleventh term, its 18 th term is
(a) $-23$
(b) $-15$
(c) 16
(d) 0

Narayan Hari

Numerade Educator

Problem 33

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, then $\mathrm{b}+\mathrm{c}, \mathrm{c}+\mathrm{a}, \mathrm{a}+\mathrm{b}$ are in
(a) AP
(b) HP
(c) GP
(d) AGP

Narayan Hari

Numerade Educator

Problem 34

The 9th and 7 th terms of a GP are respectively 256 and 64 . Then, the common ratio of the GP is
(a) 6
(b) 4
(c) $-3$
(d) 2

Narayan Hari

Numerade Educator

Problem 35

If the fith term of a GP is 2 , then the product of its first 9 terms is
(a) 64
(b) 512
(c) 128
(d) 256

Narayan Hari

Numerade Educator

Problem 36

If four numbers are in geometric progression, then their logarithms will be in
(a) GP
(b) AP
(c) HP
(d) AGP

Narayan Hari

Numerade Educator

Problem 37

If $\mathrm{a}=1+\mathrm{r}+\mathrm{r}^{2}+\ldots \ldots \ldots \ldots \infty$ and $|\mathrm{r}|<1$, then $\mathrm{r}$ is
(a) $\frac{a}{a-1}$
(b) $\frac{\mathrm{a}-1}{\mathrm{a}}$
(c) $\frac{\mathrm{a}+1}{\mathrm{a}}$
(d) $\frac{2 \mathrm{a}}{\mathrm{a}-1}$

Narayan Hari

Numerade Educator

Problem 38

$16 \times 16^{1 / 2} \times 16^{1 / 4} \times 16^{1 / 8} \times \ldots \ldots \ldots \ldots \ldots \ldots \infty$ equals
(a) 256
(b) 16
(c) 4
(d) 527

Narayan Hari

Numerade Educator

Problem 39

Each term of an infinite geometric progression is twice the sum of all the terms which follows it. The common ratio of this GP is
(a) $\frac{1}{3}$
(b) $\frac{1}{2}$
(c) $\frac{1}{4}$
(d) $\frac{2}{3}$

Narayan Hari

Numerade Educator

Problem 40

If $\mathrm{A}, \mathrm{G}, \mathrm{H}$ are the AM, GM and HM between any two distinct positive real numbers, then out of the following 4 Statements
(i) $\mathrm{G}^{2}=\mathrm{AH}$
(ii) $\mathrm{A}>\mathrm{G}>\mathrm{H}$
(iii) $\mathrm{A}=\mathrm{G}=\mathrm{H}$
(iv) $\mathrm{G}=\mathrm{A}^{2} \mathrm{H}$
(a) Only (i) and (iii) are correct
(b) Only (ii) is correct
(c) Only (i) and (ii) are correct
(d) All are correct

Narayan Hari

Numerade Educator

Problem 41

The AM and GM of 2 numbers are 20 and 4 respectively. Then, their $\mathrm{HM} \div \mathrm{GM}$ is equal to
(a) $\frac{1}{5}$
(b) 5
(c) 20
(d) 80

Narayan Hari

Numerade Educator

Problem 42

The sum to infinity of the arithmeticogeometric series $1+3 \times \frac{1}{2}+5 \times \frac{1}{4}+7 \times \frac{1}{8}+9 \times \frac{1}{16}+\ldots \ldots \ldots .$ to $\infty$ is
(a) $\frac{1}{4}$
(b) $\frac{5}{2}$
(c) 6
(d) 7

Narayan Hari

Numerade Educator

Problem 43

The 20 th term of the series $2 \times 4+4 \times 6+6 \times 8+\ldots \ldots \ldots$ is
(a) 1600
(b) 1680
(c) 420
(d) 840

Narayan Hari

Numerade Educator

Problem 44

If the sum of the first n terms of a series is $5 \mathrm{n}^{2}+2 \mathrm{n}$, then the second term is
(a) 7
(b) 17
(c) 24
(d) 42

Narayan Hari

Numerade Educator

Problem 45

If the roots of the equation $x^{3}-12 x^{2}+39 x-28=0$ are in AP, then the common difference of this AP is
(a) $\pm 1$
(b) $\pm 2$
(c) $\pm 3$
(d) $\pm 4$

Narayan Hari

Numerade Educator

Problem 46

If third term of a GP is 8 , then the product of first five terms of the GP is
(a) $8^{5}$
(b) $8^{10}$
(c) $8^{2}$
(d) $8^{3}$

Narayan Hari

Numerade Educator

Problem 47

If the second number of three numbers in an increasing $\mathrm{GP}$ is doubled, we get an $\mathrm{AP}$, then
(a) common ratio of the GP is 1 or $-1$
(b) common ratio of the GP and common difference of AP are equal.
(c) common difference of AP is equal to both first term and last term.
(d) common ratio of the GP is $2+\sqrt{3}$.

Narayan Hari

Numerade Educator

Problem 48

If $\frac{\mathrm{a}^{\mathrm{n}+1}+\mathrm{b}^{\mathrm{n}+1}}{\mathrm{a}^{\mathrm{m}}+\mathrm{b}^{\mathrm{n}}}$ is the geometric mean between a and $\mathrm{b}$, then the value of $\mathrm{n}$ is
(a) $\frac{1}{2}$
(b) $\frac{-1}{2}$
(c) $\frac{1}{3}$
(d) $\frac{-1}{3}$

Narayan Hari

Numerade Educator

Problem 49

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{GP}$ and $\mathrm{p}, \mathrm{r}, \mathrm{q}$ are in AP then $\mathrm{p}^{2}, \mathrm{q}^{2}, \mathrm{pq}$ are in
(a) AP
(b) $\mathrm{GP}$
(c) AGP
(d) HP

Narayan Hari

Numerade Educator

Problem 50

The recurring decimal $0.555$ is equal to the rational number
(a) $\frac{9}{5}$
(b) $\frac{5}{55}$
(c) $\frac{5}{9}$
(d) $\frac{8}{11}$

Narayan Hari

Numerade Educator

Problem 51

Two AM's $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$, two GM's $\mathrm{G}_{1}$ and $\mathrm{G}_{2}$ and two $\mathrm{HM}$ 's $\mathrm{H}_{1}$ and $\mathrm{H}_{2}$ are inserted between two positive numbers. Then, $\mathrm{H}_{1}^{-1}+\mathrm{H}_{2}^{-1}=$
(a) $\mathrm{A}_{1}^{-1}+\mathrm{A}_{2}^{-1}$
(b) $\mathrm{G}_{1}^{-1}+\mathrm{G}_{2}^{-1}$
(c) $\mathrm{A}_{1} \mathrm{H}_{1}+\mathrm{A}_{2} \mathrm{H}_{2}$
(d) $\frac{A_{1}+A_{2}}{G_{2} G_{2}}$

Narayan Hari

Numerade Educator

Problem 52

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ and $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in $\mathrm{HP}$, then
(a) $a=b=c$
(b) $b=3 a+c$
(c) $b^{2}=\sqrt{\frac{a c}{8}}$
(d) $b^{2}=\sqrt{\frac{a c}{6}}$

Narayan Hari

Numerade Educator

Problem 53

If the ratio of the sum of the first $\mathrm{m}$ terms and the first $\mathrm{n}$ terms of an $\mathrm{AP}$ is $\mathrm{m}^{2}: \mathrm{n}^{2}$, the ratio of its $\mathrm{mth}$ and $\mathrm{nth}$ term will be
(a) $2 \mathrm{~m}-1: 2 \mathrm{n}-1$
(b) $\mathrm{m}: \mathrm{n}$
(c) $2 \mathrm{~m}+1: 2 \mathrm{n}+1$
(d) None of these

Narayan Hari

Numerade Educator

Problem 54

If $\frac{1}{\log _{x} x}+\frac{1}{\log _{\varepsilon} x}=\frac{2}{\log _{b} x}$, then $a, b, c$ are in
(a) AP
(b) $\mathrm{GP}$
(c) HP
(d) None of these

Narayan Hari

Numerade Educator

Problem 55

$a_{1}, a_{2}, \ldots, a_{n}$ are in AP such that $a_{1}+a_{4}+a_{y}+a_{10}+a_{13}+a_{16}=300$. Then, the sum of the first 16 terms of this AP is
(a) 600
(b) 900
(c) 800
(d) 1000

Narayan Hari

Numerade Educator

Problem 56

The ratio of the sum of the first three terms of a GP to the sum of its first six terms is $125: 152$. The common ratio of the GP is
(a) $\frac{1}{5}$
(b) $\frac{2}{5}$
(c) $\frac{3}{5}$
(d) $\frac{4}{5}$

Narayan Hari

Numerade Educator

Problem 57

If $\sqrt{\mathrm{b}}+\sqrt{c}, \sqrt{\mathrm{c}}+\sqrt{\mathrm{a}}, \sqrt{\mathrm{a}}+\sqrt{\mathrm{b}}$ are in $\mathrm{HP}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in
(a) $\mathrm{AP}$
(b) HP
(c) $\mathrm{GP}$
(d) AGP

Narayan Hari

Numerade Educator

Problem 58

If the harmonic mean of two numbers is 4 and their arithmetic mean $\mathrm{A}$ and geometric mean $\mathrm{G}$ satisfy the relation $2 \mathrm{~A}+\mathrm{G}^{2}=27$, the two numbers are
(a) 6,3
(b) 5,4
(c) $5,-2.5$
(d) $-3,1$

Narayan Hari

Numerade Educator

Problem 59

Suppose $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ and $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in $\mathrm{GP}$ If $\mathrm{a}<\mathrm{b}<\mathrm{c}$ and $\mathrm{a}+\mathrm{b}+\mathrm{c}=\frac{3}{2}$, then the value of $\mathrm{c}^{\prime}$ is
(a) $\frac{1}{2 \sqrt{2}}$
(b) $\frac{1}{2 \sqrt{3}}$
(c) $\frac{1}{2}+\frac{1}{\sqrt{3}}$
(d) $\frac{1}{2}+\frac{1}{\sqrt{2}}$

Narayan Hari

Numerade Educator

Problem 60

$\mathrm{x}_{1} \mathrm{x}_{2} ; \mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{z}_{1}, \mathrm{z}_{2}$ are the two AM's, two GM's, two HM's respectively between a and b. $\mathrm{G}_{1}$ is the GM between $\mathrm{y}_{1}$ and $y_{2^{*}} G_{2}$ is the GM between $z_{1}$ and $z_{2} ; A_{1}$ is the AM between $x_{1}$ and $x_{2}$ and $A_{2}$ is the AM between $z_{1}$ and $z_{2}$. Then $A_{1}: A_{2}=$
(a) $\mathrm{G}_{1}: \mathrm{G}_{2}$
(b) $\mathrm{G}_{2}: \mathrm{G}_{1}$
(c) $\mathrm{G}_{1}^{2}: \mathrm{G}_{2}^{2}$
(d) $\mathrm{G}_{2}{ }^{2}: \mathrm{G}_{1}{ }^{2}$

Gaurav Kalra

Numerade Educator

Problem 61

Sum of the series $1+2.2+3.2^{2}+4.2^{3}+\ldots .+100.2^{9 y}$ is
(a) $100.2^{100}+1$
(b) $99.2^{\text {to }}+1$
(c) $99.2^{99}-1$
(d) $100.2^{\text {too }}-1$

Narayan Hari

Numerade Educator

Problem 62

If the harmonic mean between two positive numbers is to their GM as $12: 13$, the numbers are in the ratio
(a) $12: 13$
(b) $\frac{1}{12}: \frac{1}{13}$
(c) $4: 9$
(d) $2: 3$

Narayan Hari

Numerade Educator

Problem 63

If $a, b, c$ are in AP; $p, q, r$ are in $H P$ and ap, bq, cr are in GP, then $\frac{p}{r}+\frac{r}{p}$ is equal to
(a) $\frac{\mathrm{a}}{\mathrm{c}}-\frac{\mathrm{c}}{\mathrm{a}}$
(b) $\frac{\mathrm{a}}{\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}}$
(c) $\frac{b}{q}-\frac{q}{b}$
(d) $\frac{b}{q}+\frac{q}{p}$

Narayan Hari

Numerade Educator

Problem 64

If $\mathrm{a}, \mathrm{b}, \mathrm{x}, \mathrm{y}$ are positive numbers such that $\mathrm{a}+\mathrm{b}+\mathrm{x}+\mathrm{y}=5$, then the maximum possible value of $(\mathrm{a}+\mathrm{b})(\mathrm{x}+\mathrm{y})$ is
(a) $\frac{5}{2}$
(b) 5
(c) $\frac{25}{4}$
(d) $\frac{25}{2}$

Narayan Hari

Numerade Educator

Problem 65

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in AP as well as in GP, then
(a) $\mathrm{a}=\mathrm{b} \neq \mathrm{c}$
(b) $\mathrm{a} \neq \mathrm{b}=\mathrm{c}$
(c) $\mathrm{a} \neq \mathrm{b} \neq \mathrm{c}$
(d) $\mathrm{a}=\mathrm{b}=\mathrm{c}$

Narayan Hari

Numerade Educator

Problem 66

If $\mathrm{a}^{2}+16 \mathrm{~b}^{2}+49 \mathrm{c}^{2}-4 \mathrm{ab}-7 \mathrm{ac}-28 \mathrm{bc}=0$ then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in
(a) AP
(b) $\mathrm{GP}$
(c) HP
(d) None of these

Gaurav Kalra

Numerade Educator

Problem 67

The sum of the numbers less than 450 which are divisible by 2,3 and 5 is
(a) 3210
(b) 1200
(c) 3150
(d) 2350

Narayan Hari

Numerade Educator

Problem 68

The sum $\underbrace{0.555 \ldots 5}_{\text {ndetimus }}+\underbrace{0.535353 \ldots 53}_{=\text {decimal }}$ where, $\mathrm{n}$ is even, is equal to
(a) $12\left(1-\frac{1}{10^{=}}\right)$
(b) $\frac{12}{11}\left(1-\frac{1}{10^{-}}\right)$
(c) $\frac{12}{11}\left(1-\frac{1}{10^{2 n}}\right)$
(d) $12\left(1-\left(\frac{1}{10}\right)^{=/ 2}\right)$

Gaurav Kalra

Numerade Educator

Problem 69

In a GP of positive terms, any term other than the first two terms is equal to the sum of the two preceeding terms. Then, the common ratio of the GP is
(a) $\frac{\sqrt{5}+1}{2}$
(b) $\frac{-1+\sqrt{5}}{2}$
(c) $\frac{-1-\sqrt{5}}{2}$
(d) $\frac{1}{2}$

Narayan Hari

Numerade Educator

Problem 70

If the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ of $\triangle \mathrm{ABC}$ are in GP where $\log \mathrm{a}-\log 2 \mathrm{~b}, \log 2 \mathrm{~b}-\log 3 \mathrm{c}, \log 3 \mathrm{c}-\log \mathrm{a}$ are in $\mathrm{AP}$, then the sides $\mathrm{a}$,
b, c are in the ratio
(a) $4: 6: 9$
(b) $9: 4: 6$
(c) $9: 6: 4$
(d) $3: 6: 8$

Narayan Hari

Numerade Educator

Problem 71

If $\mathrm{x}=\bar{\sum}_{\mathrm{a}=0}^{-} 3^{\frac{-\mathrm{m}}{2}}$ and $\mathrm{y}=\overline{\sum_{\mathrm{n}}=0}^{-} 2^{\frac{-\mathrm{n}}{2}}$, then
(a) $x^{2}+y^{2}+2 x y+4 x+2=0$
(b) $4 x^{2}-y^{2}-12 x+4 y+4=0$
(c) $(x+y)^{2}+2 x y=0$
(d) $x^{2}+y^{2}+6 x+4 x+4=0$

Narayan Hari

Numerade Educator

Problem 72

Let $\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^{4}=\mathrm{f}(\mathrm{n})$. Then $\sum_{\mathrm{r}=1}^{\mathrm{n}}(2 \mathrm{r}-1)^{4}$ is equal to
(a) $\mathrm{f}(2 \mathrm{n})-16 \mathrm{f}(\mathrm{n})$ for all $\mathrm{n} \in \mathrm{N}$
(b) $\mathrm{f}(2 \mathrm{n})-16 \mathrm{f}\left(\frac{\mathrm{n}-1}{2}\right)$ when $\mathrm{n}$ is odd
(c) $\mathrm{f}(\mathrm{n})-16 \mathrm{f}\left(\frac{\mathrm{n}}{2}\right)$ when $\mathrm{n}$ is even
(d) None of these

Narayan Hari

Numerade Educator

Problem 73

If $<a_{2}>$ and $<b_{n}>$ are two sequences given by $a_{1}=2^{1 / 2}+3^{1 / 2}$
$a_{2}=2^{14}+3^{1 / 4}$
$a_{3}=2^{1 / 8}+3^{1 / 8}$
and $b_{1}=2^{1 / 2}-3^{1 / 2}$
$b_{2}=2^{1 / 4}-3^{1 / 4}$
$b_{3}=2^{1 / 8}-3^{1 / 8}$
$\ldots \ldots \ldots \ldots$
Then $\mathrm{a}_{1} \mathrm{a}_{2} \mathrm{a}_{3} \ldots \mathrm{a}_{\mathrm{n}}$ equals
(a) $\mathrm{b}_{1} \mathrm{~b}_{2} \ldots \mathrm{b}_{=}$
(b) $\frac{1}{b_{1} b_{2} \ldots b_{n}}$
(c) $\frac{1}{b_{\mathrm{n}}}$
(d) $\frac{-1}{\mathrm{~b}_{\mathrm{n}}}$

Narayan Hari

Numerade Educator

Problem 74

If $\mathrm{a}, \mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{10^{\prime}} \mathrm{b}$ are in $\mathrm{AP}$ and $\mathrm{a}, \mathrm{h}_{1}, \mathrm{~h}_{2}, \ldots \ldots, \mathrm{h}_{10^{\prime}} \mathrm{b}$ are in $\mathrm{HP}$ such that $\mathrm{a}_{1}+\mathrm{a}_{2}+\ldots+\mathrm{a}_{10}=25$ and $\frac{1}{\mathrm{~h}_{1}}+\frac{1}{\mathrm{~h}_{2}}+\ldots+\frac{1}{\mathrm{~h}_{\mathrm{to}}}=\frac{25}{6}$,
then a and $\mathrm{b}$ are
(a) 1,2
(b) 2,3
(c) $2.5,3.5$
(d) 3,4

Narayan Hari

Numerade Educator

Problem 75

$\frac{4}{19}+\frac{44}{19^{2}}+\frac{444}{19^{3}}+\ldots \ldots . \infty=$
(a) $\frac{38}{81}$
(b) $\frac{4}{19}$
(c) $\frac{36}{171}$
(d) None of these

Narayan Hari

Numerade Educator

Problem 76

If $\frac{x^{2}+3}{(x-3)\left(x^{2}+x-2\right)}=\frac{k_{1}}{x-3}+\frac{k_{2}}{x+2}+\frac{k_{3}}{x-1}$, then $k_{1}, k_{2}, k_{3}$ are respectively
(a) $\frac{2}{3}, \frac{11}{10}, \frac{6}{5}$
(b) $\frac{-2}{3}, \frac{7}{15}, \frac{6}{5}$
(c) $\frac{3}{2}, \frac{15}{7}, \frac{-10}{11}$
(d) $\frac{2}{3}, \frac{5}{7}, \frac{6}{5}$

Narayan Hari

Numerade Educator

Problem 77

If $x, y, a, b, c$ are $>0$, the maximum value of $x y$ when $a^{2} x^{4}+b^{2} y^{4}=c^{6}$ is
(a) $\frac{c^{3}}{2 a b}$
(b) $\sqrt{\frac{c^{3}}{2 a b}}$
(c) $\frac{c^{3}}{\sqrt{2 a b}}$
(d) $\frac{c}{2 \sqrt{a b}}$

Narayan Hari

Numerade Educator

Problem 78

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are distinct and positive, and $(2 \mathrm{a}+3 \mathrm{~b}+5 \mathrm{c})^{10}>\lambda \mathrm{a}^{2} \mathrm{~b}^{3} c^{5}$, then $\lambda=$
(a) 10
(b) $10^{10}$
(c) $10^{\prime / 2}$
(d) $\frac{1}{10}$

Narayan Hari

Numerade Educator

Problem 79

The sum of the first 50 terms of the series: $6+9+16+27+\ldots$ is
(a) 82375
(b) 82039
(c) 39450
(d) 83250

Narayan Hari

Numerade Educator

Problem 80

The sum to infinity of the series $\frac{1}{5.9}+\frac{1}{9.13}+\frac{1}{13.17}+\ldots$
(a) $\frac{1}{5}$
(b) $\frac{1}{20}$
(c) $\frac{1}{10}$
(d) 1

Gaurav Kalra

Numerade Educator

Problem 81

The numbers $36,32,28,24,20$ is a
(a) AGP
(b) $\mathrm{GP}$
(c) HP
(d) $\mathrm{AP}$

Narayan Hari

Numerade Educator

Problem 82

The sum of all 2 digit numbers greater than 19 is
(a) 4760
(b) 4880
(c) 3980
(d) 4580

Narayan Hari

Numerade Educator

Problem 83

If three consecutive terms in an AP are $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$, then $\frac{b-c}{a-b}=$
(a) $\frac{\mathrm{a}}{\mathrm{c}}$
(b) $\frac{\mathrm{b}}{\mathrm{a}}$
(c) a
(d) $\frac{c}{a}$

Narayan Hari

Numerade Educator

Problem 84

The second term of an infinite GP is $\frac{3}{4}$ and the sum to infinity of the GP is 4 . Then, its first term and common ratio are
(a) $\frac{3}{4}, 1$
(b) $3, \frac{1}{4}$
(c) $3, \frac{3}{4}$
(d) $\frac{9}{4}, \frac{3}{4}$

Narayan Hari

Numerade Educator

Problem 85

$1+4+9+16+25+\ldots \ldots \ldots+400=$
(a) 2780
(b) 2870
(c) 4280
(d) 2650

Narayan Hari

Numerade Educator

Problem 86

If $a$ and $b$ two $A M s$ between $c$ and $d$, then $a-c=$
(a) $\frac{\mathrm{b}-\mathrm{c}}{2}$
(b) $\mathrm{b}-\mathrm{c}$
(c) $\frac{\mathrm{d}-\mathrm{c}}{2}$
(d) $\frac{\mathrm{b}-\mathrm{c}}{3}$

Narayan Hari

Numerade Educator

Problem 87

The 20th term of the sequence $\sqrt{5}, \sqrt{20}, \sqrt{45}, \ldots \ldots$ is
(a) $20 \sqrt{3}$
(b) $20 \sqrt{5}$
(c) $10 \sqrt{5}$
(d) $10 \sqrt{3}$

Narayan Hari

Numerade Educator

Problem 88

If $\mathrm{m}, \mathrm{n}, \mathrm{s}, \mathrm{t}$ are in GP, then $\frac{1}{\mathrm{~m}}, \frac{1}{\mathrm{n}}, \frac{1}{\mathrm{~s}}, \frac{1}{\mathrm{t}}$ are in
(a) HP
(b) AGP
(c) AP
(d) $\mathrm{GP}$

Narayan Hari

Numerade Educator

Problem 89

Product 3 numbers in GP is 216 and the sum of their squares is $189 .$ Then, one of the numbers of this set of numbers is
(a) 16
(b) 12
(c) 20
(d) 27

Narayan Hari

Numerade Educator

Problem 90

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A. P, then $7^{\mathrm{a}}, 7^{\mathrm{b},}, 7^{\mathrm{c}}$ are in
(a) HP
(b) AP
(c) AGP
(d) GP

Narayan Hari

Numerade Educator

Problem 91

If the value of $1+2+3+\ldots \ldots+n$ is 55 , then the value of $1^{3}+2^{3}+3^{3}+\ldots+n^{3}$ is
(a) 165
(b) 385
(c) 3025
(d) 555

Narayan Hari

Numerade Educator

Problem 92

The sum of the first n terms of a series is $\frac{n(n+1)(n+2)}{3}$. The 12 th term of the series is
(a) 182
(b) 122
(c) 109
(d) 156

Narayan Hari

Numerade Educator

Problem 93

If $1+6+11+\cdots+\mathrm{x}=148$, then $\mathrm{x}=$
(a) 30
(b) 32
(c) 34
(d) 36

Narayan Hari

Numerade Educator

Problem 94

If $a x^{2}+2 b x+c=0$ and $a_{1} x^{2}+2 b_{1} x+c_{1}=0$ have a common root, and if $\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}$ are in AP then $a_{1}, b_{1}, c_{1}$ are in
(a) GP
(b) AP
(c) HP
(d) AGP

Gaurav Kalra

Numerade Educator

Problem 95

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in GP then $\log _{\mathrm{a}} \mathrm{x}, \log _{\mathrm{b}} \mathrm{x}, \log _{\mathrm{x}} \mathrm{x}$ are in
(a) AP
(b) GP
(c) HP
(d) $\mathrm{AGP}$

Narayan Hari

Numerade Educator

Problem 96

The sum of an infinite GP with common ratio $r(r<1)$ is 4 . The sum of the infinite GP obtained by squaring the terms of this GP is $\frac{16}{3}$. Then the first term and common ratio of the given GP is
(a) $-2, \frac{1}{2}$
(b) $-2, \frac{1}{3}$
(c) $2, \frac{1}{2}$
(d) $3, \frac{-1}{2}$

Gaurav Kalra

Numerade Educator

Problem 97

If $\mathrm{A}, \mathrm{G}, \mathrm{H}$ respectively represent AM, $\mathrm{GM}$ and $\mathrm{HM}$ of $\mathrm{n}$ positive numbers, then
(a) $\mathrm{A}<\mathrm{G}>\mathrm{H}$ always.
(b) $\mathrm{A}=\mathrm{G}=\mathrm{H}$, if the numbers are equal
(c) $\mathrm{A}>\mathrm{H}$ always and $\mathrm{A}=\mathrm{G}$ for equal numbers
(d) $\mathrm{A}>\mathrm{G}$ always, and $\mathrm{G}=\mathrm{H}$ for equal numbers

Gaurav Kalra

Numerade Educator

Problem 98

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ and $\mathrm{a}, \mathrm{mb}, \mathrm{c}$ are in GP, then $\mathrm{a}, \mathrm{m}^{2} \mathrm{~b}, \mathrm{c}$ are in
(a) AP
(b) $\mathrm{GP}$
(c) HP
(d) AGP

Narayan Hari

Numerade Educator

Problem 99

If $t_{1}=2, t_{2}=2$ and $t_{n-1}=t_{n}+1$ for $n \geq 3$ then $t_{5}=$
(a) 1
(b) 0
(c) $-1$
(d) 5

Narayan Hari

Numerade Educator

Problem 100

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, then $2^{\mathrm{ax}+1}, 2^{\mathrm{hx}+1}, 2^{\alpha+1}$, where $\mathrm{x} \neq 0$ are in
(a) AP
(b) GP only if $x>0$
(c) GP only if $x<0$
(d) GP for all $x \neq 0$

Narayan Hari

Numerade Educator

Problem 101

If the $(a+1) t h, 7 t h$ and $(b+1)$ th terms of an $A P$ are in $G P$ with $a, 6, b$ being in $H P$, then 4 th term of this $A P$ is
(a) $-\frac{7}{2}$
(b) $\frac{7}{2}$
(c) 0
(d) 3

Gaurav Kalra

Numerade Educator

Problem 102

$1+\frac{4}{5}+\frac{7}{5^{2}}+\frac{10}{5^{3}}+\ldots \ldots . \infty$
(a) $\frac{16}{35}$
(b) $\frac{11}{8}$
(c) $\frac{35}{16}$
(d) $\frac{7}{16}$

Narayan Hari

Numerade Educator

Problem 103

If $\mathrm{a}_{\mathrm{r}}>0, \mathrm{r} \in \mathrm{N}$ and $\mathrm{a}_{1} \mathrm{a}_{2} \ldots \ldots \mathrm{a}_{2 \mathrm{~m}}$ are in $\mathrm{AP}$, then,
(a) $\mathrm{n}-1$
(b) $\frac{n\left(a_{1}+a_{2 n}\right)}{\sqrt{a_{1}}+\sqrt{a_{n+1}}}$
(c) $\frac{n-1}{\sqrt{a n}+\sqrt{a n+1}}$
(d) None of these

Gaurav Kalra

Numerade Educator

Problem 104

If $a_{1}, a_{2}, a_{3} \ldots$ are in HP and $f(k)=\sum_{r=1}^{n} a_{r}-a_{k}$, then $\frac{a_{1}}{f(1)}, \frac{a_{2}}{f(2)}, \frac{a_{3}}{f(3)}, \ldots \frac{a=}{f(n)}$ are in
(a) AP
(b) GP
(c) HP
(d) AGP

Gaurav Kalra

Numerade Educator

Problem 105

$\mathrm{a}, \mathrm{b}, \mathrm{c}$ are 3 consecutive terms of an $\mathrm{AP}$ If $\tan \mathrm{a}, \tan \mathrm{b}, \tan c\left(\mathrm{~b} \neq \mathrm{multiple}\right.$ of $\left.\frac{\pi}{2}\right)$ are also in $\mathrm{AP}$, then
(a) $\tan \mathrm{b}=2 \tan \mathrm{a}$
(b) $\tan a \times \tan c=\tan b$
(c) $\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}=0$
(d) $\tan \mathrm{a}=\tan \mathrm{b}=\tan \mathrm{c}$

Gaurav Kalra

Numerade Educator

Problem 106

If $\alpha$ and $\beta$ are the roots of $x^{2}-3 x+a=0$ and $\gamma, \delta$ that of $x^{2}-12 x+b=0$ and $\alpha, \beta, \gamma, \delta$ (in that order) form an increasing GP, then
(a) $a=2, b=32$
(b) $a=3, b=12$
(c) $a=4, b=16$
(d) $\mathrm{a}=12, \mathrm{~b}=3$

Gaurav Kalra

Numerade Educator

Problem 107

If $|\mathrm{x}|<1$, then the sum of the infinite series $\left[\mathrm{x}+\frac{1}{2}\right]+\left[\mathrm{x}^{2}+\mathrm{x} \frac{1}{2}+\left(\frac{1}{2}\right)^{2}\right]+\left[\mathrm{x}^{3}+\mathrm{x}^{2} \frac{1}{2}+\mathrm{x}\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3}\right]+\ldots . \infty$
(a) $\frac{1}{2}+x$
(b) $\frac{2-x}{1-x}$
(c) $\frac{1+x}{1-x}$
(d) $\frac{2+x}{2-x}$

Gaurav Kalra

Numerade Educator

Problem 108

The arithmetic mean between two positive numbers a and b where (a > b) is twice their geometric mean. Then $\frac{\text { a }}{\mathrm{b}}$
(a) $2+\sqrt{3}$
(b) $7+4 \sqrt{3}$
(c) $2-\sqrt{3}$
(d) $7-4 \sqrt{3}$

Gaurav Kalra

Numerade Educator

Problem 109

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three distinct numbers in $\mathrm{AP}$ and $\mathrm{b}-\mathrm{a}, \mathrm{c}-\mathrm{b}$, a are in $\mathrm{GP}$, then a: b: $\mathrm{c}=$
(a) $2: 3: 1$
(b) $\frac{1}{2}: \frac{1}{3}: 1$
(c) $1: 2: 3$
(d) $2: 1: 6$

Urvashi Arora

Numerade Educator

Problem 110

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real and $(\mathrm{bc}+\mathrm{ca}+\mathrm{ab})^{2}>\mathrm{k}(\mathrm{a}+\mathrm{b}+\mathrm{c})$, then $\mathrm{k}$ equals
(a) $\mathrm{abc}$
(b) 3
(c) 1
(d) $3 \mathrm{abc}$

Gaurav Kalra

Numerade Educator

Problem 111

Statement 1 There is no AP with non-zero common difference whose terms are all prime numbers and
Statement 2 Any natural number $(>1)$ can be uniquely factored as product of prime numbers.

Gaurav Kalra

Numerade Educator

Problem 112

Statement 1 $\mathrm{S}_{r}$ is the sum of an infinite $\mathrm{G} . \mathrm{P}$ with 1st term $\mathrm{r}$ and common ratio $\frac{1}{\mathrm{r}+1}$. Then $\mathrm{S}_{\mathrm{r}}-\mathrm{r}$ depends on $\mathrm{r}$. and
Statement 2 Sum of an infinite geometric series the common ratio is numerically less than 1 exists only if $\div \mathrm{r} \div<1$

Gaurav Kalra

Numerade Educator

Problem 113

Statement 1 If $\mathrm{a}^{2}+2 \mathrm{bc}, \mathrm{b}^{2}+2 \mathrm{ac}, \mathrm{c}^{2}+2 \mathrm{ab}$ are in AP then, $\mathrm{b}-c_{2} \mathrm{c}-\mathrm{a}, \mathrm{a}-\mathrm{b}$ are in $\mathrm{HP}$
and
Statement 2 If $a_{1}, a_{2}, \ldots . a_{n}$ are in AP then $\frac{a_{1}+k}{h}, \frac{a_{2}+k}{h}, \ldots \ldots .$ in AP

Gaurav Kalra

Numerade Educator

Problem 114

Suppose we start with $\mathrm{x}$ grams of a sample of radioactive material. $\mathrm{y}$ grams of the sample remains after a period of $\mathrm{n}$ life cycle. The relationship between $\mathrm{x}$ and $\mathrm{y}$ is given by
(a) $y=n x+a$
(b) $y=\mathrm{kx} 0<\mathrm{k}<1$
(c) $y=\frac{k}{2}$
(d) $y=\frac{x}{2^{x}}$ where $n=0,1,2 \ldots \ldots$

Gaurav Kalra

Numerade Educator

Problem 115

The half-life of a certain isotope is 2 years. A sample of $3200 \mathrm{~kg}$ of the isotope is considered. If $\mathrm{x} \mathrm{kg}$ of the sample remains after 12 years $\mathrm{x}$ equals
(a) $275 \mathrm{~kg}$
(b) $100 \mathrm{~kg}$
(c) $50 \mathrm{~kg}$
(d) 0

Narayan Hari

Numerade Educator

Problem 116

$12.5 \%$ of an isotope remains after 8 years from the initial stage. After 8 more years, the quantity present will be
(a) $\frac{25}{16} \%$
(b) $6.25 \%$
(c) $0 \%$
(d) $3.125 \%$

Narayan Hari

Numerade Educator

Problem 117

In the nth row of the triangle
(a) last term $=$ sum of the cubes of the first $\mathrm{n}$ natural numbers
(b) first term $=\frac{\left(\mathrm{n}^{2}-\mathrm{n}+2\right)^{2}}{4}$
(c) positive square root of the first term is one more than the sum of first $\mathrm{n}-1$ natural numbers
(d) sum of the elements in the nth row $=\frac{n\left(n^{2}+2\right)\left(3 n^{2}+1\right)}{12}$

Gaurav Kalra

Numerade Educator

Problem 118

If $\sum_{k=1}^{n}\left(\sum_{m=1}^{k} m^{2}\right)=a n^{4}+b n^{3}+c n^{2}+d n+e$ then
(a) $\mathrm{a}=\frac{1}{12}$
(b) $e=0$
(c) $c=\frac{5}{12}$
(d) $\mathrm{d}=\frac{1}{6}$

Gaurav Kalra

Numerade Educator

Problem 119

The number of ordered triplets $(p, q, r)$ where $1 \leq p, q, r \leq 10$ and $p, q, r$ are natural numbers such that $2^{p}+3^{q}+5^{r}$ is a multiple of 4 is
(a) 50 if $\mathrm{p}=1$
(b) 450 if $\mathrm{p} \neq 1$
(c) $500 \mathrm{in}$ all
(d) 75 if $\mathrm{p}=1$

Gaurav Kalra

Numerade Educator

Problem 120

Consider $\mathrm{Q}=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real and distinct. Also, $\mathrm{Q}=0$ has real roots Column 1 $\quad$ Column II
(a) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, then the least value of $\left|\frac{\mathrm{d}}{\mathrm{b}}\right|$ (where $\mathrm{d}$ is the common
(p) $-\sec ^{2} \frac{\pi}{4}$
difference), is
(b) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then $\mathrm{b}$ cannot be equal to
(q) $\cos \frac{\pi}{3}$
(c) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in AP with common difference 'd', and if zero is a root of
(r) $\sin \frac{\pi}{3}$
$\mathrm{Q}=0$, then $\left|\frac{\mathrm{d}}{\mathrm{a}}\right|$ equals
(d) If $a, b, c$ are in $A P$, and $b, c$, a are in GP, then the common ratio $r$ equals
(s) $\sin \frac{\pi}{2}$

Gaurav Kalra

Numerade Educator

Problem 121

If the first three of four given numbers are in AP and the last three are in HP, prove that the four numbers are proportional.

Gaurav Kalra

Numerade Educator

Problem 122

If $\mathrm{S}_{1}, \mathrm{~S}_{2}, \ldots, \mathrm{S}_{2 k}$ are the sums of the first $\mathrm{n}$ terms of $2 \mathrm{k}$ Arithmetic Progressions whose first terms are $1,2,3, \ldots, 2 \mathrm{k}$ and whose common differences are $1,3,5,7, \ldots,(4 \mathrm{k}-1)$, show that
(i) $S_{1}+S_{2}+S_{3}+\ldots+S_{2 k}=\operatorname{kn}(1+2 n k)$
(ii) $\mathrm{S}_{1}-\mathrm{S}_{2}+\mathrm{S}_{3}-\ldots-=-\mathrm{n}^{2} \mathrm{k}$

Gaurav Kalra

Numerade Educator

Problem 123

Find the sum of the first $\mathrm{n}$ terms of the series: $3+7+23+87+\ldots$

Gaurav Kalra

Numerade Educator

Problem 124

If $a_{1}, a, a_{3}, \ldots, a_{n}$ are in AP where $a_{k} \neq \frac{(2 k-1) \pi}{2}$ and $d$ is the common difference, find
(i) $\sec \mathrm{a}_{1} \sec \mathrm{a}_{2}+\mathrm{sec} \mathrm{a}_{2} \sec \mathrm{a}_{3}++\sec \mathrm{a}_{\mathrm{n}-1} \mathrm{sec} \mathrm{a}=$
(ii) $\tan ^{-1}\left(\frac{\mathrm{d}}{1+\mathrm{a}_{1} \mathrm{a}_{2}}\right)+\tan ^{-1}\left(\frac{\mathrm{d}}{1+\mathrm{a}_{2} \mathrm{a}_{3}}\right)+\ldots+\tan ^{-1}\left(\frac{\mathrm{d}}{1+\mathrm{a}_{\mathrm{n}-\mathrm{a}} \mathrm{a}_{\mathrm{n}}}\right)$
Also show that $\frac{1}{a_{1} a_{n}}+\frac{1}{a_{2} a_{n-1}}+\ldots+\frac{1}{a_{n} a_{1}}=\frac{2}{\left(a_{1}+a_{n}\right)}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}\right)$

Gaurav Kalra

Numerade Educator

Problem 125

Obtain the sum of the first 50 terms of the series $\frac{5}{7}+\frac{19}{7^{2}}+\frac{69}{7^{3}}+\frac{263}{7^{4}}+\ldots$

Gaurav Kalra

Numerade Educator

Problem 126

Show that the sum of the cubes of the first $n$ terms of an AP is exactly divisible by the sum of its terms.

Urvashi Arora

Numerade Educator

Problem 127

Prove that the sum to n terms of a geometric series with positive terms is greater than $\mathrm{n}$ times the geometric mean of the first and the nth terms of the series.

Gaurav Kalra

Numerade Educator

Problem 128

The natural numbers are divided into groups in the following way: $(2,3,4) ;(5,6,7,8,9,10,11) ;(12,13,14,15, \ldots$
22). Obtain a formula for the sum of the numbers in the nth group.

Gaurav Kalra

Numerade Educator

Problem 129

If $\mathrm{n}$ is a root of the equation $\mathrm{x}^{2}(1-\mathrm{ac})-\mathrm{x}\left(\mathrm{a}^{2}+\mathrm{c}^{2}\right)-(1+\mathrm{ac})=0$ and if $\mathrm{n}$ harmonic means are inserted between a and c show that the difference between the first and the last mean is equal to $\mathrm{ac}(\mathrm{a}-\mathrm{c})$.

Gaurav Kalra

Numerade Educator

Problem 130

If $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are natural numbers, then prove that
(i) $\left(\frac{a^{3}+b^{3}+c^{3}}{a^{2}+b^{2}+c^{2}}\right)^{a^{2}+b^{2}+c^{2}} \geq a^{a^{2}} b^{b^{2}} c^{c^{2}} \geq\left(\frac{a^{2}+b^{2}+c^{2}}{a+b+c}\right)^{a^{x}+b^{2}+c^{2}}$
(ii) $\frac{(a+b+c+d)^{2} \text { abcd }}{a b+b c+c d+d a} \leq a^{2} e d+b^{2} a d+c^{2} b d+d^{2} b c$

Gaurav Kalra

Numerade Educator

Problem 131

Sum to $\mathrm{n}$ terms the series: $3.8+7.15+11.22+\ldots$ is
(a) $36 \mathrm{n}^{2}+15 \mathrm{n}+3$
(b) $\frac{\mathrm{n}\left(56 \mathrm{n}^{2}+75 \mathrm{n}+13\right)}{6}$
(c) $56 \mathrm{n}^{2}+75 \mathrm{n}+13$
(d) $\frac{16 n^{2}+75 n+1}{6}$

Gaurav Kalra

Numerade Educator

Problem 132

$\mathrm{p}$ and $\mathrm{q}$ are positive integers. $\mathrm{A}_{1}, \mathrm{~A}_{2}, \mathrm{~A}_{3}$ are the three arithmetic means inserted between $\mathrm{p}$ and q. $\mathrm{H}_{1}, \mathrm{H}_{2}, \mathrm{H}_{3}$ are the three harmonic means inserted between $\mathrm{p}$ and $\mathrm{q}$. If $\mathrm{A}_{1} \mathrm{~A}_{2} \mathrm{~A}_{3}=\frac{15}{2}$ and $\mathrm{H}_{1} \mathrm{H}_{2} \mathrm{H}_{3}=\frac{18}{5}$, the values of $\mathrm{p}$ and $\mathrm{q}$ are respectively
(a) 1,2
(b) 3,4
(c) 1,3
(d) 2,4

Gaurav Kalra

Numerade Educator

Problem 133

If $\mathrm{h}_{1}, \mathrm{~h}_{2^{\prime}} \mathrm{h}_{3}, \ldots, \mathrm{h}_{2 \mathrm{n}}$ be the $2 \mathrm{n}$ harmonic means between a and $\mathrm{b}$, then $\frac{\mathrm{h}_{1}+\mathrm{a}}{\mathrm{h}_{1}-\mathrm{a}}+\frac{\mathrm{h}_{2 \mathrm{n}}+\mathrm{b}}{\mathrm{h}_{2 \mathrm{n}}-\mathrm{b}}$
(a) $\mathrm{n}$
(b) $2 \mathrm{n}$
(c) $4 n$
(d) $\underline{1}$

Gaurav Kalra

Numerade Educator

Problem 134

If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are in $\mathrm{AP} ; \ell, \mathrm{m}, \mathrm{n}$ are in $\mathrm{HP} ; \mathrm{p} \ell, \mathrm{qm}, \mathrm{r} \mathrm{n}$ are in $\mathrm{GP}(\mathrm{p} \ell \neq \mathrm{q} \mathrm{m} \neq \mathrm{r} \mathrm{n})$, then $\mathrm{p}: \mathrm{q}: \mathrm{r}=$
(a) $\mathrm{m}: \mathrm{n}: \ell$
(b) $\frac{1}{n}: \frac{1}{m}: \frac{1}{\ell}$
(c) $1: 2: 3$
(d) $\mathrm{n}: \mathrm{m}: \ell$

Gaurav Kalra

Numerade Educator

Problem 135

If $p x^{2}+2 q x+r=0$ and $p_{1} x^{2}+2 q_{1} x+r_{1}=0$ have a common root and $\frac{p_{1}}{p}, \frac{q_{1}}{q}, \frac{r_{1}}{r}$ are in HP, then $p_{1}, q_{1}, r_{1}$ are
(a) in AP
(b) in GP
(c) in HP
(d) not in any progression

Gaurav Kalra

Numerade Educator

Problem 136

An AP and a HP have each the first term $p$ and the last term $q$ and the same number of terms $n$. Then the product of the $(\mathrm{r}+1)$ th term of the first series and the $(\mathrm{n}-\mathrm{r})$ th term of the second series
(a) dependent of $\mathrm{r}$
(b) independent of $\mathrm{r}$
(c) dependent of $\mathrm{n}$
(d) dependent of $\mathrm{r}$ and $\mathrm{n}$

Gaurav Kalra

Numerade Educator

Problem 137

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are the sides of a triangle, and $(\mathrm{a}+\mathrm{b}+\mathrm{c})^{3} \geq \lambda(\mathrm{a}+\mathrm{b}-\mathrm{c})(\mathrm{b}+\mathrm{c}-\mathrm{a})(\mathrm{c}+\mathrm{a}-\mathrm{b})$, then $\lambda$ equals
(a) 9
(b) 3
(c) 8
(d) 27

Gaurav Kalra

Numerade Educator

Problem 138

Consider a GP with $\mathrm{n}$ terms. The product of the AM and HM of all the terms of the GP is equal to
(a) 1
(b) $\mathrm{n}$ (GM)
(c) $\mathrm{GM}$
(d) $(\mathrm{GM})^{2}$

Gaurav Kalra

Numerade Educator

Problem 139

Let $a, b, c$ be the sums of the first $n$ terms, next $n$ terms and next n terms of a GP respectively. Then $a, b, c$ are in
(a) $\mathrm{AP}$
(b) $\mathrm{GP}$
(c) $\mathrm{HP}$
(d) AGP

Gaurav Kalra

Numerade Educator

Problem 140

The sum of first $n$ terms of the series $6+66+666+6666+\ldots \ldots$ is
(a) $\frac{27}{20}\left(10^{-}-1\right)-\frac{2}{3} \mathrm{n}$
(b) $\frac{20}{27}\left(10^{n}-1\right)-n$
(c) $\frac{20}{27}\left(10^{n}-1\right)-\frac{2}{3} \mathrm{n}$
(d) $\frac{1}{27}\left(10^{\mathrm{n}}-1\right)$

Narayan Hari

Numerade Educator

Problem 141

In an AP whose first term is a and the sum of the first p terms is zero, if the sum of the next ' $q$ terms is $\frac{-a(p+q)}{(p-1)} \lambda$, then the value of $\lambda$ is
(a) $\mathrm{p}$
(b) $p+1$
(c) $\mathrm{q}$
(d) $q+1$

Gaurav Kalra

Numerade Educator

Problem 142

If $\mathrm{a}, \mathrm{b}, c, \mathrm{~d}$, are distinct integers in $\mathrm{AP}$ such that $\mathrm{d}=\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}$, then the sum $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}$ equals
(a) 0
(b) 1
(c) 2
(d) $-2$

Gaurav Kalra

Numerade Educator

Problem 143

If two arithmetic means $\mathrm{A}_{1}, \mathrm{~A}_{2}$, two geometric means $\mathrm{G}_{1}, \mathrm{G}_{2}$ and two harmonic means $\mathrm{H}_{1}, \mathrm{H}_{2}$ are inserted between two numbers then $\frac{A_{2}-A_{1}}{H_{2}-H_{1}}$.
(a) $\frac{\mathrm{A}_{1} \mathrm{~A}_{2}}{\mathrm{H}_{1} \mathrm{H}_{2}}$
(b) $\frac{A_{1}+A_{2}}{H_{1}+H_{2}}$
(c) $\frac{A_{1}+A_{2}}{H_{1} H_{2}}$
(d) $\frac{\mathrm{H}_{1}+\mathrm{H}_{2}}{\mathrm{~A}_{1} \mathrm{~A}_{2}}$

Gaurav Kalra

Numerade Educator

Problem 144

If $\mathrm{x}$ is the first of the $\mathrm{n}$ arithmetic means between two numbers a and $\mathrm{b}$ and $\mathrm{y}$ is the first of the $\mathrm{n}$ harmonic means between a and $b$, then their produet $x y$ is equal to
(a) $\left(\frac{\mathrm{na}+\mathrm{b}}{\mathrm{nb}+\mathrm{a}}\right) \mathrm{ab}$
(b) $\frac{\mathrm{a}+\mathrm{nb}}{\mathrm{ab}}$
(c) $\left(\frac{a+b}{a-b}\right) a b$
(d) $\left(\frac{\mathrm{nb}+\mathrm{a}}{\mathrm{na}+\mathrm{b}}\right) \mathrm{ab}$

Gaurav Kalra

Numerade Educator

Problem 145

If $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}$ are in $\mathrm{GP}$ as as $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}$ with the same common ratio, then points $\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right), \mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right), \mathrm{C}\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)$
(a) satisfies $\mathrm{x}_{1}+\mathrm{x}_{2}+\mathrm{x}_{3}=\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}$
(b) form an equilateral triangle
(c) are collinear
(d) form an isosceles triangle

Narayan Hari

Numerade Educator

Problem 146

How many geometric progressions are there containing the terms 48,27 and $64 ?$
(a) 1
(b) 3
(c) 9
(d) infinite

Gaurav Kalra

Numerade Educator

Problem 147

The sum of the series $1+3+5+7+9+\ldots .+101$ is
(a) 2931
(b) 5091
(c) 2601
(d) 2501

Narayan Hari

Numerade Educator

Problem 148

The nth term of an AP is 164 and the sum of its first $n$ terms is $3 n^{2}+5 n$. Then, $n$ equals
(a) 53
(b) 36
(c) 27
(d) 28

Narayan Hari

Numerade Educator

Problem 149

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, then $4\left(\mathrm{~b}^{2}-\mathrm{ac}\right)=$
(a) $a^{2} c+3$
(b) $\frac{\mathrm{a}+\mathrm{c}}{2}$
(c) $\frac{(a+c) a}{3}$
(d) $(\mathrm{a}-\mathrm{c})^{2}$

Narayan Hari

Numerade Educator

Problem 150

Which term of the sequence $4,4 \sqrt{3}, 12,12 \sqrt{3}, \ldots \ldots \ldots \ldots .$ is $36 \times 3^{4}$ ?
(a) 13
(b) 12
(c) 11
(d) 14

Narayan Hari

Numerade Educator

Problem 151

If there are four geometric means between 12 and 384 , then the common ratio is
(a) 3
(b) $1 / 2$
(c) 2
(d) 4

Narayan Hari

Numerade Educator

Problem 152

Sum of the series $1^{3}+2^{3}+3^{3}+4^{3}+\ldots . .+19^{3}$ is
(a) 29130
(b) 19400
(c) 36100
(d) 43250

Narayan Hari

Numerade Educator

Problem 153

If the 4 th term of an AP is 4 , then the sum of its first 7 terms is
(a) 4
(b) 28
(c) 16
(d) 40

Narayan Hari

Numerade Educator

Problem 154

Sum of the series $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}+\ldots \ldots . .$ to 9 terms, is
(a) $\frac{3}{2}$
(b) $\frac{2}{3}$
(c) $\frac{-3}{2}$
(d) $\frac{-2}{3}$

Narayan Hari

Numerade Educator

Problem 155

If the sum of the first n natural numbers is one seventh of the sum of their squares, $\mathrm{n}$ equals
(a) 6
(b) 7
(c) 8
(d) 10

Narayan Hari

Numerade Educator

Problem 156

If the 10 th term of a GP is 9 and its 4 th term is 4 , then its 7 th term is
(a) $4 / 9$
(b) 36
(c) 6
(d) $9 / 4$

Narayan Hari

Numerade Educator

Problem 157

The product of $\mathrm{n}$ geometric means between two given positive numbers a and $\mathrm{b}$ is
(a) $(\mathrm{ab})^{4}$
(b) $(\mathrm{ab})^{7 / 2}$
(c) $(\mathrm{ab})^{2 \mathrm{n}}$
(d) $(\mathrm{ab})^{-\mathrm{n}}$

Gaurav Kalra

Numerade Educator

Problem 158

If $\mathrm{x}, 2 \mathrm{x}+2,3 \mathrm{x}+3$ are in $\mathrm{GP}$, then the 4 th term is
(a) 27
(b) $-27$
(c) $\frac{27}{2}$
(d) $-\frac{27}{2}$

Narayan Hari

Numerade Educator

Problem 159

If $\mathrm{p}$ is the $\mathrm{AM}$ of $\mathrm{q}$ and $\mathrm{r}$ and $\mathrm{q}$ is the $\mathrm{GM}$ of $\mathrm{r}$ and $\mathrm{p}$, then $\mathrm{HM}$ between $\mathrm{p}$ and $\mathrm{q}$ is
(a) $\mathrm{p}^{2}$
(b) $\mathrm{pq}-\mathrm{r}$
(c) $\mathrm{r}$
(d) $\mathrm{pq}+\mathrm{r}$

Narayan Hari

Numerade Educator

Problem 160

If $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in $\mathrm{HP}$, then $\mathrm{a}^{2} \mathrm{~b}^{2}, \mathrm{a}^{2} \mathrm{c}^{2}, \mathrm{~b}^{2} \mathrm{c}^{2}$ are in
(a) GP
(b) AGP
(c) $\mathrm{HP}$
(d) AP

Narayan Hari

Numerade Educator

Problem 161

If $a^{2}, b^{2}, c^{2}$ are in AP, then $\frac{a}{b+c}, \frac{b}{c+a}, \frac{c}{b+a}$ are in
(a) AP
(b) $\mathrm{GP}$
(c) HP
(d) None of these

Gaurav Kalra

Numerade Educator

Problem 162

If three distinct numbers $a, b, c$ are in $H P$ and $a^{2}, b^{2}, c^{2}$ are in AP then
(a) $a+c=b$
(b) $a+b+c=0$
(c) $a+b=c$
(d) $b+c=a$

Urvashi Arora

Numerade Educator

Problem 163

If the three distinct numbers $a, b, c$ are in GP and $a+x, b+x, c+x$ are in $H P$, then the value of $x$ is
(a) $c$
(b) $\mathrm{b}$
(c) a
(d) None of these

Urvashi Arora

Numerade Educator

Problem 164

$\log _{18} 3, \log _{163} 3, \log _{1458} 3$ are in
(a) $\mathrm{AP}$
(b) $\mathrm{GP}$
(c) HP
(d) None of these

Gaurav Kalra

Numerade Educator

Problem 165

If $a_{1}, a_{2}, a_{y}, \ldots$ are in AP such that $a_{i} \neq 0$ and $S_{=}=\frac{1}{a_{1} a_{3}}+\frac{1}{a_{2} a_{4}}+\frac{1}{a_{3} a_{5}}+\ldots+\frac{1}{a_{m} a_{n+2}}$ then $S_{2 n}-S_{=}$ is equal to
(a) $\frac{1}{a_{n}}+\frac{1}{a_{2 m}}$
(b) $\frac{1}{2 \mathrm{~d}}\left[\frac{1}{\mathrm{a}_{1}}+\frac{1}{\mathrm{a}_{2}}+\frac{1}{\mathrm{a}_{2 \mathrm{n}+1}}+\frac{1}{\mathrm{a}_{2 \mathrm{~m}+2}}\right]$
(c) $\frac{\mathrm{nd}}{\left[\frac{1}{\mathrm{a}_{1}}+\frac{1}{\mathrm{a}_{2}}-\frac{1}{\mathrm{a}_{\mathrm{n}+1}}-\frac{1}{\mathrm{a}_{\mathrm{n}+2}}\right]}$
(d) $\frac{\mathrm{n}}{2}\left[\frac{1}{\mathrm{a}_{\mathrm{n}+1} \mathrm{a}_{2 \mathrm{n}+1}}+\frac{1}{\mathrm{a}_{\mathrm{n}+2} \mathrm{a}_{2 \mathrm{~m}+2}}\right]$

Gaurav Kalra

Numerade Educator

Problem 166

The sum of the first $\mathrm{n}$ terms of the series $1^{2}-2^{2}+3^{2}-4^{2}+5^{2}-6^{2}+\ldots$ is
(a) $\frac{\mathrm{n}(\mathrm{n}+1)}{2}$
(b) $\frac{(-1)^{m-1} \mathrm{n}(\mathrm{n}+1)}{2}$
(c) $\frac{-n(n+1)}{2}$
(d) $\mathrm{n}(\mathrm{n}+1)$

Gaurav Kalra

Numerade Educator

Problem 167

greastest volue of $\lambda$ is
(a) $\mathrm{n}$
(b) $2 \mathrm{n}$
(c) $\mathrm{n}^{2}$
(d) 0

Gaurav Kalra

Numerade Educator

Problem 168

The sum of the products taken two at a time of the n numbers $2,4,6,8, \ldots, 2 n$ is
(a) $\left[\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]^{2}$
(b) $\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}$
(c) $\frac{(\mathrm{n}-1) \mathrm{n}(\mathrm{n}+1)}{6}$
(d) $\frac{(\mathrm{n}-1) \mathrm{n}(\mathrm{n}+1)(3 \mathrm{n}+2)}{6}$

Gaurav Kalra

Numerade Educator

Problem 169

When $n$ is even, the sum of the first $n$ terms of the series $1^{2}+2.2^{2}+3^{2}+2.4^{2}+5^{2}+2.6^{2}+\ldots \ldots$ is $\frac{n(n+1)^{2}}{2}$. When $n$
is odd, the sum of the series is
(a) $\frac{n(n+1)^{2}}{4}$
(b) $\frac{n^{2}(n+1)}{2}$
(c) $\frac{3 n(n+1)}{2}$
(d) $\left[\frac{n(n+1)}{4}\right]^{2}$

Gaurav Kalra

Numerade Educator

Problem 170

The infinite geometric series $\sum_{\mathrm{r}=0}^{-} \frac{\mathrm{a}}{\left(\mathrm{y}^{2}-4 \mathrm{y}+5\right)^{\mathrm{r}-\mathrm{l}}}$, where $\mathrm{y}=\mathrm{x}^{2}-6 \mathrm{x}+11$ has a finite sum when $\mathrm{x}$ takes a value
other than
(a) 3
(b) $-1$
(c) 4
(d) $-3$

Narayan Hari

Numerade Educator

Problem 171

Statement 1 If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ then $(\mathrm{b}-\mathrm{a})^{2}=(\mathrm{c}-\mathrm{a})^{2}$
and
Statement 2 If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ then $4\left(\mathrm{~b}^{2}-\mathrm{ac}\right)=(\mathrm{c}-\mathrm{a})^{2}$

Gaurav Kalra

Numerade Educator

Problem 172

Statement 1 If $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}$ are in $\mathrm{AP}$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}$ are in $\mathrm{AP}$
then $\left(a_{1}+2 b_{1}\right),\left(a_{2}+2 b_{2}\right),\left(a_{3}+2 b_{3}\right)$ are in AP
and
Statement 2 If $a_{1}+a_{2}+a_{3}+\ldots$ are in AP then $\left(a_{1}+k\right)+\left(a_{2}+k\right)+\left(a_{3}+k\right)+\ldots$ are also in $A P$

Gaurav Kalra

Numerade Educator

Problem 173

Statement 1 If the sum of the first $\mathrm{n}$ terms of an $\mathrm{AP}$ is zero, then a must be $<0$ and $\mathrm{d}$ must be positive. and
Statement 2 Sum of the first $n$ terms of the $A P a+(a+d)+(a+2 d)+\ldots+(a+(n-1) d)$ is given by $\frac{n}{2}(2 a+(n-1) d)$.

Gaurav Kalra

Numerade Educator

Problem 174

Statement 1
If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in AP, $(b+c-a),(c+a-b),(a+b-c)$ are in AP
and
Statement 2 If $\mathrm{a}_{1}, \mathrm{a}_{2} \mathrm{a}_{3}$ are in $\mathrm{AP}$,
$a_{1}+k, a_{2}+k, a_{3}+k$ where $k \neq 0$ are also in AP

Gaurav Kalra

Numerade Educator

Problem 175

Statement 1 If $3 x+7 y+5 z=45$, maximum value of $x^{3} y^{7} z^{5}$ where, $x, y, z$ are $>0$ is and
Statement 2 If $\mathrm{A}, \mathrm{G}, \mathrm{H}$ denote the $\mathrm{AM}, \mathrm{GM}$ and $\mathrm{HM}$ of two positive numbers, $\mathrm{G}^{2}=\mathrm{AH}$

Gaurav Kalra

Numerade Educator

Problem 176

Statement 1 For $0<\theta<\frac{\pi}{2}, \cos \theta+\sec \theta>2$
and
Statement 2 If $\mathrm{x}$ is any number not equal to zero, $\mathrm{x}+\frac{1}{\mathrm{x}}$ is always greater than or equal to 2 .

Gaurav Kalra

Numerade Educator

Problem 177

Statement 1 If $a_{1}, a_{2}, a_{3}$ denote the 3 arithmetic means inserted between two numbers $x$ and $y$ then, $\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}$ are the 3 harmonic means between the numbers $\mathrm{x}$ and $\mathrm{y}$. and
Statement 2 If $x_{1}, x_{2}, x_{3}$ are in AP, $\frac{1}{x_{1}}, \frac{1}{x_{2}}, \frac{1}{x_{3}}$ are in HP

Gaurav Kalra

Numerade Educator

Problem 178

Statement 1 There cannot be an infinite GP with second term equals $-1$ and whose sum to infinity equals ' $-1$. and
Statement 2 $\mathrm{a}+\mathrm{ar}+\mathrm{ar}^{2}+\mathrm{ar}^{3}+\ldots \infty=\frac{\mathrm{a}}{(1-\mathrm{r})}$, provided $-1<\mathrm{r}<1$

Gaurav Kalra

Numerade Educator

Problem 179

Statement 1 If the terms of a GP are alternately positive and negative, then the common ratio of such a GP has to be a negative number and
Statement 2 nth term of the GP a $+a r+a r^{2}+\ldots$ is ar $^{n-1}$

Gaurav Kalra

Numerade Educator

Problem 180

Statement 1 If the sum of $\mathrm{n}$ consecutive terms of a series is of the form $\left(\mathrm{an}^{2}+\mathrm{bn}+\mathrm{c}\right)$, where $\mathrm{c} \neq 0$ then the series is an AP and
Statement 2 The sum of the first $\mathrm{n}$ terms of an $\mathrm{AP}$ is of the form $\mathrm{An}^{2}+\mathrm{Bn} .$

Gaurav Kalra

Numerade Educator

Problem 181

A rehearsal for a parade on an important occasion is arranged as follows:
Each group is occupied by army men of the same rank only. The groups consist of $1,8,27,64 \ldots$ number of persons. There are 3025 army men of different ranks to be arranged starting from the highest rank. The number of groups necessary to accommodate the above, assuming that enough number of persons of the same rank are available?
(a) 8
(b) 9
(c) 10
(d) 12

Gaurav Kalra

Numerade Educator

Problem 182

In a parade of 4356 army men, the number of persons having the lowest rank is
(a) 1331
(b) 1000
(c) 729
(d) 512

Gaurav Kalra

Numerade Educator

Problem 183

The number of persons in the groups are chosen as $7,11,18,29,47,76,123,199 \ldots .$ the number of persons in the $(\mathrm{n}-2)$ th, nth and $(\mathrm{n}+1)$ th groups $(\mathrm{n} \geq 3)$, form
(a) a GP
(b) a HP
(c) an AP
(d) an AGP

Gaurav Kalra

Numerade Educator

Problem 184

First number in the 15 th group is
(a) 781
(b) 785
(c) 717
(d) 793

Gaurav Kalra

Numerade Educator

Problem 185

Last number in the 15 th group is
(a) 897
(b) 893
(c) 829
(d) 905

Gaurav Kalra

Numerade Educator

Problem 186

Sum of the numbers in the 15 th group is
(a) 24273
(b) 23519
(c) 23448
(d) 24389

Gaurav Kalra

Numerade Educator

Problem 187

$\mathrm{S}_{20}=$
(a) $\frac{1050}{841}$
(b) $\frac{894}{841}$
(c) $\frac{1266}{841}$
(d) $\frac{1066}{841}$

Gaurav Kalra

Numerade Educator

Problem 188

$\lim _{n \rightarrow-} S_{n}=$
(a) $\frac{5}{6}$
(b) $\frac{5}{4}$
(c) $\frac{5}{2}$
(d) $\frac{1}{4}$

Gaurav Kalra

Numerade Educator

Problem 189

$\lim _{n \rightarrow-} S_{n}-S_{25}$ is approximately equal to
(a) $7 \times 10^{-3}$
(b) $9 \times 10^{-3}$
(c) $9 \times 10^{-4}$
(d) $7 \times 10^{-4}$

Gaurav Kalra

Numerade Educator

Problem 190

If $\mathrm{f}(\mathrm{n})=\left[\frac{\mathrm{n}+50}{100}\right]$, where [] stands for greatest integer function then
(a) $\sum_{n=1}^{100} f(n)=51=\sum_{n=50}^{100} f(n)$
(b) $\sum_{m=1}^{1000} \mathrm{f}(\mathrm{n})=5010$
(c) $\sum_{x=350}^{369} f(n)=900$
(d) $\sum_{=1}^{49} f(n)=0$

Gaurav Kalra

Numerade Educator

Problem 191

Let $\alpha$ and $\beta$ denote the roots of the quadratic equation $a x^{2}+2 b x+c=0 .$ Then,
(a) if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$, one of the roots of the equation is $-1$.
(b) if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ one of the roots of the equation is $\frac{\mathrm{c}}{\mathrm{a}}$.
(c) if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$ the roots of the equation are equal.
(d) if $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in HP, $\beta(1+\alpha)^{2}+\alpha\left(1+\beta^{2}\right)=0$

Gaurav Kalra

Numerade Educator

Problem 192

If the series $\mathrm{g}_{1}+\mathrm{g}_{2}+\mathrm{g}_{3}+\ldots+\mathrm{g}_{\mathrm{n}}$ is in GP with common ratio $\mathrm{r}$, then, for $\mathrm{n}>1$
(a) $\frac{1}{\mathrm{~g}_{1}^{2}-\mathrm{g}_{2}^{2}}+\frac{1}{\mathrm{~g}_{2}^{2}-\mathrm{g}_{3}^{2}}+\frac{1}{\mathrm{~g}_{3}^{2}-\mathrm{g}_{4}^{2}}+\ldots+\frac{1}{\mathrm{~g}_{\mathrm{n}-1}^{2}-\mathrm{g}_{\mathrm{n}}^{2}}=\frac{\mathrm{r}^{2}}{\left(1-\mathrm{r}^{2}\right)^{2}}\left\{\frac{1}{\mathrm{~g}_{=}^{2}}-\frac{1}{\mathrm{~g}_{1}^{2}}\right\}$
(b) $\frac{1}{g_{1} g_{2}}+\frac{1}{g_{2} g_{3}}+\frac{1}{g_{3} g_{4}}+\ldots+\frac{1}{g_{z-1} g_{=}}=\frac{r}{\left(1-r^{2}\right)}\left\{\frac{1}{g_{n}^{2}}-\frac{1}{g_{1}^{2}}\right\}$
(c) $\frac{1}{g_{1}^{2}+g_{2}^{2}}+\frac{1}{g_{2}^{2}+g_{3}^{2}}+\frac{1}{g_{3}^{2}+g_{4}^{2}}+\ldots+\frac{1}{g_{n-1}^{2}+g_{n}^{2}}=\frac{r^{2}}{\left(1-r^{4}\right)}\left\{\frac{1}{g_{n}^{2}}-\frac{1}{g_{1}^{2}}\right\}$
(d) $\frac{g_{1}}{g_{1}+g_{2}}+\frac{g_{2}}{g_{2}+g_{3}}+\frac{g_{3}}{g_{3}+g_{4}}+\ldots+\frac{g_{z-1}}{g_{n-1}+g_{n}}=\frac{(n-1)}{(1+r)}$

Gaurav Kalra

Numerade Educator

Problem 193

Let $0<\theta<\frac{\pi}{4}$.
Then,
(a) $\sum_{k=0}^{-} \sin ^{2 k} \theta=\sec ^{2} \theta$
(b) $\sum_{k=0}^{-} \cos ^{2 k} \theta=\operatorname{cosec}^{2} \theta$
(c) $\sum_{k=0}^{-} \tan ^{2 \mathrm{k}} \theta=\frac{\cos ^{2} \theta}{\cos 2 \theta}$
(d) $\sum_{k=0}^{-} \tan ^{2 k} \theta+\sum_{k=0}^{\overline{2}} \cot ^{2 k} \theta=0$

Gaurav Kalra

Numerade Educator

Problem 194

Consider $f(x)=\frac{p}{x+p}+\frac{q}{x+q}-\frac{r}{x+r}-\frac{s}{x+s}$ with $p, q, r, s \neq 0$. If the equation $f(x)=0$ has a pair of equal roots then
(a) p equals ror s
(b) q equals ror s
(c) the HM (p, q) equals $\mathrm{HM}(\mathrm{r}, \mathrm{s})$
(d) the distinct root is the $\mathrm{HM}(\mathrm{p}, \mathrm{q})$ if 0 is the multiple root

Shamshad Waris

Numerade Educator

Problem 195

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive and not all equal, the expression $b c(a-b)(a-c)+c a(b-c)(b-a)+a b(c-a)(c-b)$ is always
(a) positive
(b) negative
(c) $\sum c^{2}(a-b)^{2}$
(d) $\Sigma b^{2} c^{2}-a b c(a+b+c)$

Gaurav Kalra

Numerade Educator

Problem 196

The system of equations $a x+b y+(a \alpha+b) z=0$
$b x+c y+(b \alpha+c) z=0$
$(a \alpha+b) x+(b \alpha+c) y=0$ has trivial solutions only. Then
(a) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in GP
(b) $\alpha$ is a root of the equation $a x^{2}+b x+c=0$
(c) $\mathrm{x}-\alpha$ is a factor of $\mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0$
(d) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{HP}$

Gaurav Kalra

Numerade Educator

Problem 197

The expression $\mathrm{p}(\mathrm{q}-\mathrm{r}) \mathrm{x}^{2}+\mathrm{q}(\mathrm{r}-\mathrm{p}) \mathrm{xy}+\mathrm{r}(\mathrm{p}-\mathrm{q}) \mathrm{y}^{2}$ is a perfect square, where $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are non zero reals
(a) Minimum value of $\mathrm{p}+\mathrm{r}$ is $2 \mathrm{q}$
(b) AM of p, r is q
(c) If $\mathrm{p}, \mathrm{q}, \mathrm{r}$ are the exradii of $\triangle \mathrm{ABC}$ whose smallest side is 4 and largest side is 8 then the third side is 6
(d) The area of $\triangle \mathrm{ABC}$ described in (c) is $3 \sqrt{15}$

Gaurav Kalra

Numerade Educator

Problem 198

(a) $\frac{1}{1.3 .5}+\frac{1}{3.5 .7}+\ldots+\frac{1}{41.43 .45}$ is $/$ are equal to
(p) 68276
(b) $1^{3}-2^{3}+3^{3}-5^{3}+\ldots+51^{3} \mathrm{is} /$ are equal to
(q) $\frac{253}{400}$
(c) $\left[\frac{1+2+\ldots+10}{10^{2}}+\frac{\left(1^{2}+2^{2}+. .+10^{2}\right)}{10^{3}}-\frac{\left(1^{3}+2^{3}+\ldots+10^{3}\right)}{10^{4}}\right]=$
(r) 48048
(d) $1.2 .3 .4+2.3 .4 .5+\ldots . .+10.11 .12 .13 \mathrm{is} /$ are equal to
(s) $\frac{161}{1935}$

Gaurav Kalra

Numerade Educator

Problem 199

(a) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive real numbers different from 1. If $\log _{2}{ }^{100}, 2 \log _{\mathrm{b}}{ }^{10}, 2 \log _{\mathrm{c}}{ }^{5}+\log _{\mathrm{c}}{ }^{4}$
(p) AP are in $\mathrm{HP}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in
(b) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are different positive real numbers such that $\mathrm{a}>\mathrm{b}>\mathrm{c}$. If $2 \log (\mathrm{a}-\mathrm{c}), \log$
(q) GP $\left(a^{2}-c^{2}\right), \log \left(a^{2}+2 b^{2}+c^{2}\right)$ are in $A P$, then $a, b, c$ are in
(c) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ and $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in HP then, $\mathrm{a}^{3}, \mathrm{~b}^{3}, \mathrm{c}^{3}$, are in
(r) HP
(d) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}, \mathrm{b}, c, \mathrm{~d}$ are in $\mathrm{GP}, \mathrm{c}, \mathrm{d}, \mathrm{e}$ are in $\mathrm{HP}$ then $\mathrm{a}, \mathrm{c}, \mathrm{e}$ are in
(s) AGP

Gaurav Kalra

Numerade Educator

Problem 200

(a) If $\mathrm{x}, \mathrm{y}, \mathrm{z}$ be respectively $\mathrm{AM}, \mathrm{GM}, \mathrm{HM}$ between two rational numbers
(p) $\frac{a+b}{a b}$
a and $\mathrm{b}$ then $\mathrm{x}-\mathrm{y}$ is equal to
(b) If $\mathrm{A}_{1}, \mathrm{~A}_{2}$ be two AMs and $\mathrm{G}_{1}, \mathrm{G}_{2}$ be two GMs; between a and $\mathrm{b}$, then
(q) $\left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}}\right)^{2}$
$\frac{\mathrm{A}_{1}+\mathrm{A}_{2}}{\mathrm{G}_{1} \mathrm{G}_{2}}$ is equal to
(c) If $\mathrm{A}_{1}, \mathrm{~A}_{2}$ be two $\mathrm{AMs} ; \mathrm{G}_{1}, \mathrm{G}_{2}$ be two $\mathrm{GMs} ; \mathrm{H}_{1}, \mathrm{H}_{2}$ be two $\mathrm{HMs}$ between
(r) $\frac{a b}{a+b}$
two positive numbers a and b, then $\frac{\mathrm{G}_{1} \mathrm{G}_{2}}{\mathrm{H}_{1} \mathrm{H}_{2}} \frac{\mathrm{H}_{1}+\mathrm{H}_{2}}{\mathrm{~A}_{1}+\mathrm{A}_{2}}$
(d) If $\mathrm{A}_{1}, \mathrm{~A}_{2}$ be two $\mathrm{AMs} \mathrm{G}_{1}, \mathrm{G}_{2}$ be two $\mathrm{GMs} \mathrm{H}_{1}, \mathrm{H}_{2}$ be two $\mathrm{HMs}$ between
(s) 1 two numbers a and b, then $\frac{1}{\mathrm{H}_{1}}+\frac{1}{\mathrm{H}_{2}}$ is equal to

Gaurav Kalra

Numerade Educator