Question 1: Palette 1/32

AP-EAPCET 2025 MCQ

If $S_n=1^3+2^3+\ldots+n^3$ and $T_n=1+2+\ldots+n$, then

A

$S_n=T_{n^3}$

B

$S_n=T_n^3$

C

$S_n=T_{n^2}$

D

$S_n=T_n^2$

Question 2: Palette 2/32

AP-EAPCET 2025 MCQ

$\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\frac{1}{7 \cdot 9}+\ldots$ to 24 terms $=$

A

$\frac{23}{147}$

B

$\frac{6}{35}$

C

$\frac{6}{37}$

D

$\frac{8}{51}$

Question 3: Palette 3/32

AP-EAPCET 2025 MCQ

$ 1+\frac{4}{15}+\frac{4 \cdot 10}{15 \cdot 30}+\frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45}+\ldots . .+\infty= $

A

$\left(\frac{3}{5}\right)^{2 / 3}$

B

$\left(\frac{5}{3}\right)^{2 / 3}$

C

$\left(\frac{3}{5}\right)^{3 / 2}$

D

$\left(\frac{5}{3}\right)^{3 / 2}$

Question 4: Palette 4/32

AP-EAPCET 2025 MCQ

If $t_n=\frac{1}{4}(n+2)(n+3), n \in N$, then which one of the following is true?

Assertion (A) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_{2003}}=\frac{2003}{3009}$

Reason (R) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_n}=\frac{4 n}{(2 n+3)}$

A

(A) and (R) are true and (R) is a correct explanation of (A)

B

(A) and (R) are true, but (R) is not the correct explanation of (A)

C

(A) is true, (R) is false

D

(A) is false, (R) is false

Question 5: Palette 5/32

AP-EAPCET 2025 MCQ

The sum of all integers between 1 and 100 (both inclusive) which are divisible by 5 or 13 is

A

1349

B

1536

C

1237

D

1479

Question 6: Palette 6/32

AP-EAPCET 2025 MCQ

If $x>\sqrt{3}$ and $\frac{x^2+1}{\left(x^2+2\right)\left(x^2+3\right)}$ is expanded in terms of powers of $x$, then the coefficient of $x^{-8}$ is

A

0

B

-81

C

46

D

-46

Question 7: Palette 7/32

AP-EAPCET 2025 MCQ

$ \sum\limits_{k=1}^n k(k+1)(k+2) \ldots(k+r-1)= $

A

$\frac{n(n+1)(n+2) \ldots(n+r)}{r+1}$

B

$\frac{n(n+1)(n+2) \ldots(n+r-1)}{r}$

C

$\frac{n(n+1)(n+2) \ldots(n+r+1)}{r+1}$

D

$\frac{n(n+1)(n+2) \cdot \cdot 2 n}{2 n+1}$

Question 8: Palette 8/32

AP-EAPCET 2025 MCQ

For all $n \in N, \frac{3^n-1}{2} \geq$

A

$n^2\left(2^{\frac{n}{2}}\right)$

B

$n^2\left(3^{\frac{n-1}{2}}\right)$

C

$n^3\left(3^{\frac{n-1}{2}}\right)$

D

$n\left(3^{\frac{n-1}{2}}\right)$

Question 9: Palette 9/32

AP-EAPCET 2025 MCQ

If $2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots$ to $n$ terms $=a n^3+b n^2+c n+d$, then $a-b+c-d=$

A

7

B

5

C

-3

D

-1

Question 10: Palette 10/32

AP-EAPCET 2025 MCQ

For all $n \in N$, if $1^3+2^3+3^3+\ldots n^3>x$, then a value of $x$ among the following is

A

$\frac{n^2}{4}$

B

$n^2$

C

$n^4$

D

$\frac{n^2(n+1)^2}{4}$

Question 11: Palette 11/32

AP-EAPCET 2024 MCQ

The $n$th term of the series $1+(3+5+7)+(9+11+13+15+17)+\ldots$ is

A

$(2 n+1)\left[n^2-(n-1)^2\right]$

B

$(2 n-1)\left[(n-1)^2-n^2\right]$

C

$(2 n+1)\left[(n-1)^2-n^2\right]$

D

$(2 n-1)\left[(n-1)^2+n^2\right]$

Question 12: Palette 12/32

AP-EAPCET 2024 MCQ

The number of ways of selecting- 3 numbers that are in GP from the set $\{1,2,3$, $100\}$ is

A

18

B

52

C

14

D

53

Question 13: Palette 13/32

AP-EAPCET 2024 MCQ

$ 2+3+5+6+8+9+\ldots .2 n \text { terms }= $

A

$3 n^2+2 n$

B

$4 n^2+2 n$

C

$4 n^2$

D

$5 n^2+2 n$

Question 14: Palette 14/32

AP-EAPCET 2024 MCQ

If $\alpha, \beta$ are the roots of the equation $x^2-6 x-2=0$, $\alpha>\beta$ and $a_n=\alpha^n-\beta^n, n \geq 1$, then the value of $\frac{a_{10}-2 a_8}{2 a_9}$ is equal to

A

6

B

4

C

3

D

2

Question 15: Palette 15/32

AP-EAPCET 2024 MCQ

$|x|<1$, The coefficient of $x^2$ in the power series expansion of $\frac{x^4}{(x+1)(x-2)}$ is

A

3

B

0

C

-1

D

-3

Question 16: Palette 16/32

AP-EAPCET 2024 MCQ

If $1 \cdot 3 \cdot 5+3 \cdot 5 \cdot 7+5 \cdot 7 \cdot 9+\ldots n$ terms $=n(n+1) f(n)-3 n$, then $f(l)=$

A

9

B

11

C

12

D

8

Question 17: Palette 17/32

AP-EAPCET 2024 MCQ

The condition that the roots of $x^3-b x^2+c x-d=0$ are in arithmetic progression is

A

$9 c b=2 b^3+27 d$

B

$9 c b=2 d^3+27 b$

C

$9 c b=2 d^3+27 b$

D

$9 c d=2 b^3+27 d$

Question 18: Palette 18/32

AP-EAPCET 2024 MCQ

In the expansion of $\frac{2 x+1}{(1+x)(1-2 x)}$ the sum of the coefficients of the first 5 odd powers of $x$ is

A

$\frac{5}{3}+\frac{8}{9}\left(4^5-1\right)$

B

$\frac{5}{3}+\frac{8}{3}\left(4^5-1\right)$

C

$-\frac{5}{3}+\frac{8}{9}\left(4^5-1\right)$

D

$\frac{5}{3}+\frac{8}{12}\left(4^5+1\right)$

Question 19: Palette 19/32

AP-EAPCET 2024 MCQ

$\frac{1}{1 \cdot 5}+\frac{1}{5 \cdot 9}+\frac{1}{9 \cdot 13}+\ldots$. upto $n$ terms $=$

A

$\frac{1}{4 n+1}$

B

$\frac{4}{4 n+1}$

C

$\frac{n}{4 n+1}$

D

$\frac{4 n+1}{5(4 n+1)}$

Question 20: Palette 20/32

AP-EAPCET 2024 MCQ

If the roots of the equation $4 x^3-12 x^2+11 x+m=0$ are in arithmetic progression, then $m=$

A

-3

B

1

C

2

D

3

Question 21: Palette 21/32

AP-EAPCET 2024 MCQ

If $2 \cdot 4^{2 n+1}+3^{3 n+1}$ is divisible by $k$ for all $n \in N$, then $k=$

A

209

B

11

C

8

D

3

Question 22: Palette 22/32

AP-EAPCET 2024 MCQ

If the roots of the equation $x^3+a x^2+b x+c=0$ are in arithmetic progression. Then,

A

$a^3-3 a b+c=0$

B

$9 a b=2 a^3+27 c$

C

$a^2-2 b c+c=0$

D

$3 a b-3 c-a^3=0$

Question 23: Palette 23/32

AP-EAPCET 2024 MCQ

$ \frac{1}{3 \cdot 7}+\frac{1}{7 \cdot 11}+\frac{1}{11 \cdot 15}+\ldots$ to 50 terms $=$

A

$\frac{50}{203}$

B

$\frac{50}{609}$

C

$\frac{150}{203}$

D

$\frac{25}{609}$

Question 24: Palette 24/32

AP-EAPCET 2024 MCQ

$1+\frac{1}{3}+\frac{1 \cdot 3}{3 \cdot 6}+\frac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9}+\ldots \text { to } \infty= $

A

$\sqrt{5}$

B

$\sqrt{6}$

C

$\sqrt{15}$

D

$\sqrt{3}$

Question 25: Palette 25/32

AP-EAPCET 2024 MCQ

$ 2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots \text { to } 10 \text { terms }= $

A

3355

B

4555

C

1375

D

1380

Question 26: Palette 26/32

AP-EAPCET 2024 MCQ

If the roots of equation $x^3-13 x^2+K x-27=0$ are in geometric progression, then $K=$

A

-30

B

30

C

39

D

-39

Question 27: Palette 27/32

AP-EAPCET 2024 MCQ

$ 1-\frac{2}{3}+\frac{2 \cdot 4}{3 \cdot 6}-\frac{2 \cdot 4 \cdot 6}{3 \cdot 6 \cdot 9}+\ldots \infty= $

A

$\frac{3}{5}$

B

$\left(\frac{2}{5}\right)^{\frac{2}{3}}$

C

$\frac{2}{5}$

D

$\left(\frac{3}{5}\right)^{\frac{2}{3}}$

Question 28: Palette 28/32

AP-EAPCET 2022 MCQ

Suppose that the three points $A, B$ and $C$ in the plane are such that their $x$-coordinates as well as $y$-coordinates are in GP with the same common ratio. Then, the points $A, B$ and $C$

A

constitute a right angled triangle

B

form an isosceles triangle

C

lie on a straight line

D

form an equilateral triangle

Question 29: Palette 29/32

AP-EAPCET 2021 MCQ

Using mathematical induction, the numbers $a_n^{\prime}$ s are defined by $a_0=1, a_{n+1}=3 n^2+n+a_n (n \geq 0)$, then $a_n$ is equal to

A

$n^3+n^2+1$

B

$n^3-n^2+1$

C

$n^3-n^2$

D

$n^3+n^2$

Question 30: Palette 30/32

AP-EAPCET 2021 MCQ

If $1+x^2=\sqrt{3} x$, then $\sum_{n=1}^{24}\left(x^n-\frac{1}{x^n}\right)^2$ is equal to

A

48

B

$-$48

C

$-$24

D

24

Question 31: Palette 31/32

AP-EAPCET 2021 MCQ

Let $p$ and $q$ be the roots of the equation $x^2-2 x+A=0$ and let $r$ and $s$ be the roots of the equation $x^2-18 x+B=0$. If $p < q < r < s$ are in AP then the values of $A$ and $B$ are

A

$-3,77$

B

$3,-77$

C

$3,77$

D

$-3,-77$

Question 32: Palette 32/32

AP-EAPCET 2021 MCQ

Let $f(x)=x^3+a x^2+b x+c$ be polynomial with integer coefficients. If the roots of $f(x)$ are integer and are in Arithmetic Progression, then $a$ cannot take the value

A

$-642$

B

1214

C

1323

D

1626