iCON Education - Binomial Theorem AP-EAPCET Test

Candidate Name

Student

Exam Name

AP-EAPCET

Test Name

Binomial Theorem

Subject

Mathematics

← Back to Binomial Theorem

Remaining Time

00:42:00

Question 1: Palette 1/42

AP-EAPCET 2025 MCQ

In the binomial expansion of $(p-q)^{14}$, if the sum of 7th term and 8 th term is zero, then $\frac{p+q}{p-q}=$

Question 2: Palette 2/42

AP-EAPCET 2025 MCQ

The numerically greatest term in the expansion of $(x+3 y)^{13}$, when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

${ }^{13} C_9\left(\frac{1}{3}\right)^4$

${ }^{13} C_4\left(\frac{1}{2}\right)^9$

${ }^{13} C_9\left(\frac{1}{2}\right)^4$

${ }^{13} C_{10} \frac{1}{2^4}$

Question 3: Palette 3/42

AP-EAPCET 2025 MCQ

The remainder obtained when $(2 m+1)^{2 n}(m, n \in N)$ is divided by 8 is

Question 5: Palette 5/42

AP-EAPCET 2025 MCQ

$ \frac{1}{81^n}-{ }^{2 n} C_1 \frac{10}{81^n}+{ }^{2 n} C_2 \frac{10^2}{81^n}-\ldots+\frac{10^{2 n}}{81^n}= $

$(-1)^n$

Question 6: Palette 6/42

AP-EAPCET 2025 MCQ

If $x$ is positive real number and the first negative term in the expansion of $(1+x)^{\frac{27}{5}}$ is $t_k$, then $k=$

Question 7: Palette 7/42

AP-EAPCET 2025 MCQ

The coefficient of $x^{10}$ in the expansion of $\left(x+\frac{2}{x}-5\right)^{12}$ is

1674

2132

1892

862

Question 8: Palette 8/42

AP-EAPCET 2025 MCQ

Let $S_1=\sum\limits_{j=1}^{10} j(j-1) \cdot{ }^{10} C_j, S_2=\sum\limits_{j=1}^{10} j \cdot{ }^{10} C_j$ and

$ S_3=\sum\limits_{j=1}^{10} j^2 \cdot{ }^{10} C_j $

Assertion (A) $S_3=55 \times 2^9$

Reason (R) $S_1=90 \times 2^8$ and $S_2=10 \times 2^8$

Both $(A)$ and $(R)$ are true and $R$ is the correct explanation of (A)

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

(A) is true, but (R) is false

(A) is false, but (R) is true

Question 9: Palette 9/42

AP-EAPCET 2025 MCQ

If $y=\frac{3}{4}+\frac{3 \cdot 5}{4 \cdot 8}+\frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12}+\ldots+\infty$, then

$y^2-2 y+5=0$

$y^2+2 y-7=0$

$y^2-3 y+4=0$

$y^2+4 y-6=0$

Question 10: Palette 10/42

AP-EAPCET 2025 MCQ

Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $\left(1+x-x^2\right)^6$ is

121

-91

Question 11: Palette 11/42

AP-EAPCET 2025 MCQ

If $11^{12}-11^2=k\left(5 \times 10^9+6 \times 10^9+33 \times 10^8\right. \left.+110 \times 10^7+\ldots+33\right)$, then $k=$

100

200

Question 12: Palette 12/42

AP-EAPCET 2025 MCQ

If $C_0, C_2, \ldots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$, then

$ \left(C_0+C_1\right)-\left(C_2+C_3\right)+\left(C_4+C_5\right)-\left(C_6+C_7\right)+\ldots= $

$2^{n / 2}\left(\cos \frac{n \pi}{4}+i \sin \frac{n \pi}{4}\right)$

$2^{n / 2}\left(\cos \frac{n \pi}{3}+i \sin \frac{n \pi}{3}\right)$

$2^{n / 2}\left(\cos \frac{n \pi}{4}+\sin \frac{n \pi}{4}\right)$

Question 13: Palette 13/42

AP-EAPCET 2025 MCQ

The mean and variance of a binomial distribution are $x$ and 5 respectively. If $x$ is an integer, then the possible values for $x$ are

$6,10,30$

$8,12,28$

$10,15,25$

$9,18,24$

Question 14: Palette 14/42

AP-EAPCET 2025 MCQ

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $\left(1+\alpha x+\beta x^2\right)(1+x)^{11}$ are 396 and 144 respectively, then $\alpha^2+\beta^2=$

Question 15: Palette 15/42

AP-EAPCET 2025 MCQ

If $-\frac{2}{3} < x < \frac{2}{3}$, then the value of the 5 th term in the expansion of $\frac{1}{\sqrt[3]{2-3 x}}$ when $x=\frac{1}{2}$ is

$\frac{35}{256(\sqrt[3]{2})}$

$\frac{35}{768(\sqrt[3]{2})}$

$\frac{7}{768(\sqrt[3]{2})}$

$\frac{105}{256(\sqrt[3]{2})}$

Question 16: Palette 16/42

AP-EAPCET 2025 MCQ

The terms containing $x^r y^s$ (for certain $r$ and $s$ ) are present in both the expansions of $\left(x+y^2\right)^{13}$ and $\left(x^2+y\right)^{14}$. If $\alpha$ is the number of such terms, then the $\operatorname{sum} \alpha \sum_{r, s}(r+s)=$

Question 17: Palette 17/42

AP-EAPCET 2025 MCQ

The coefficient of $x^3$ in the power series expansion of $\frac{1+4 x-3 x^2}{(1+3 x)^3}$ is

-27

153

-153

Question 18: Palette 18/42

AP-EAPCET 2025 MCQ

If $k$ is a positive integer and $10^k$ is a divisor of the number $9^{11}+11^9$, then the greatest value of $k$ is

Question 19: Palette 19/42

AP-EAPCET 2025 MCQ

The number of all possible values of $k$ for which the expansion $(\sqrt{x}+\sqrt[k]{y})^{10}$ will have exactly nine irrational terms is

Question 20: Palette 20/42

AP-EAPCET 2025 MCQ

Coefficient of $x^2$ in the expansion of $\left(x^2+x-2\right)^5$ is

800

756

512

Question 21: Palette 21/42

AP-EAPCET 2025 MCQ

If $P_n$ denotes the product of the binomial coefficients in the expansion of $(1+x)^n$, then $\frac{P_{n+1}}{P_n}=$

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

$\frac{n^n}{n!}$

$\frac{(n+1)^n}{(n+1)!}$

$\frac{(n+1)^{n+1}}{(n+1)!}$

Question 22: Palette 22/42

AP-EAPCET 2025 MCQ

The coefficient of $x^3$ in the expansion of $\frac{x^4+1}{\left(x^2+1\right)(x-1)}$ when it is expressed in terms of positive integral powers of $x$, is

Question 23: Palette 23/42

AP-EAPCET 2025 MCQ

If $(1+x)^n=\sum_{r=0}^n C, x^r$, then the value of $C_0+\left(C_0+C_1\right)+\left(C_0+C_1+C_2\right)+\ldots+ \left(C_0+C_1+C_2+\ldots+C_n\right)$ is

$n R^{n-1}$

$2^n+n$

$(n+2) 2^n$

$(n+2) 2^{n-1}$

Question 24: Palette 24/42

AP-EAPCET 2025 MCQ

If $x$ is so large that terms containing $x^{-3}, x^{-4}, x^{-5}, \ldots$ can be neglected, then the approximate value of $\left(\frac{3 x-5}{4 x^2+3}\right)^{-1 / 5}$ is

$\left(\frac{3}{4 x}\right)^{4 / 5}\left(1-\frac{4}{3 x}-\frac{7}{5 x^2}\right)$

$\left(\frac{4 x}{3}\right)^{4 / 5}\left(1+\frac{4}{3 x}+\frac{13}{5 x^2}\right)$

$\left(\frac{4 x}{3}\right)^{4 / 5}\left(1+\frac{4}{3 x}-\frac{13}{5 x^2}\right)$

$\left(\frac{3}{4 x}\right)^{4 / 5}\left(1-\frac{4}{3 x}+\frac{7}{5 x^2}\right)$

Question 25: Palette 25/42

AP-EAPCET 2024 MCQ

The independent term in the expansion of $\left(1+x+2 x^2\right)\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^9$ is

$\frac{18}{7}$

$\frac{7}{18}$

$-\frac{7}{18}$

$-\frac{18}{7}$

Question 26: Palette 26/42

AP-EAPCET 2024 MCQ

For $|x|<\frac{1}{\sqrt{2}}$, the coefficient of $x$ in the expansion of $\frac{(1-4 x)^2\left(1-2 x^2\right)^{1 / 2}}{(4-x)^{3 / 2}}$ is

$\frac{61}{64}$

$-\frac{61}{64}$

$\frac{69}{64}$

$-\frac{69}{64}$

Question 27: Palette 27/42

AP-EAPCET 2024 MCQ

If $P$ is the greatest divisor of $49^n+16 n-1$ for all $n \in N$, then the number of factors of $P$ is

Question 28: Palette 28/42

AP-EAPCET 2024 MCQ

If the coefficients of $r$ th, $(r+1)$ th and $(r+2)$ th terms in the expansion of $(1+x)^n$ are in the ratio of $4: 15: 42$, then $n-r$ is equal to

Question 29: Palette 29/42

AP-EAPCET 2024 MCQ

If the coefficients of $(2 r+6)$ th and $(r-1)$ th terms in the expansion of $(1+x)^{21}$ are equal, then the value of $r$ is equal to

Question 30: Palette 30/42

AP-EAPCET 2024 MCQ

If the $2 \mathrm{nd}, 3 \mathrm{rd}$ and 4 th terms in the expansion of $(x+a)^n$ are $96,216,216$ respectively and $n$ is a positive integer, then $a+x=$

$n+1$

$n$

$n-1$

$\frac{n}{2}$

Question 31: Palette 31/42

AP-EAPCET 2024 MCQ

If $|x|<1$, then the number of terms in the expansion of $\left[\frac{1}{2}\left(1 \cdot 2+2 \cdot 3 x+3 \cdot 4 x^2+\ldots . \infty\right)\right]^{-25}$

Infinite

101

Question 32: Palette 32/42

AP-EAPCET 2024 MCQ

If the ratio of the terms equidistant from the middle term in the expansion of $(l+x)^{12}$ is $\frac{1}{256}(x \in N)$, then sum of all the terms of the expansion $(1+x)^{12}$ is

$4^{12}$ or $6^{12}$

$3^{12}$ or $5^{12}$

$6^{12}$ or $7^{12}$

$12^{12}$

Question 33: Palette 33/42

AP-EAPCET 2024 MCQ

If the eleventh term in the binomial expansion of $(x+a)^{15}$ is the geometric mean of the eighth and twelfth terms, then the greatest term in the expansion is

7 th term

8 th term

9 th term

10 th term

Question 34: Palette 34/42

AP-EAPCET 2024 MCQ

The sum of the rational terms in the binomial expansion of $\left(\sqrt{2}+3^{1 / 5}\right)^{10}$ is

Question 35: Palette 35/42

AP-EAPCET 2024 MCQ

If the coefficients of $x^5$ and $x^6$ are equal in the expansion of $\left(a+\frac{x}{5}\right)^{65}$, then the coefficient of $x^2$ in the expansion of $\left(a+\frac{x}{5}\right)^4$ is.

$\frac{32}{25}$

$\frac{24}{25}$

Question 36: Palette 36/42

AP-EAPCET 2024 MCQ

If $|x|<\frac{2}{3}$, then the 4th term in the expansion of $(3 x-2)^{\frac{2}{3}}$ is :

$\frac{\sqrt[3]{4}}{6} x^3$

$-\frac{\sqrt[3]{4}}{6} x^3$

$\frac{\sqrt[3]{4}}{8} x^3$

$-\frac{\sqrt[3]{4}}{8} x^3$

Question 37: Palette 37/42

AP-EAPCET 2024 MCQ

The coefficient of $x^5$ in the expansion of $\left(2 x^3-\frac{1}{3 x^2}\right)^5$ is

$\frac{80}{9}$

$\frac{29}{3}$

Question 38: Palette 38/42

AP-EAPCET 2024 MCQ

Numerically greatest term in the expansion of $(5+3 x)^6$ When, $x=1$, is

$3^5 \times 5^3$

$3^3 \times 5^5$

$3^2 \times 5^5$

$3^4 \times 5^4$

Question 39: Palette 39/42

AP-EAPCET 2024 MCQ

The square root of independent term in the expansion of $ \left( 2x^2 + \frac{5}{x} \right)^5 $ is

$\frac{15}{\sqrt{10}}$

$\frac{10}{\sqrt{15}}$

$\frac{30}{\sqrt{5}}$

$\frac{20}{\sqrt{5}}$

Question 40: Palette 40/42

AP-EAPCET 2024 MCQ

The coefficient of $x^5$ in $\left(3+x+x^2\right)^6$ is

540

2178

Question 41: Palette 41/42

AP-EAPCET 2024 MCQ

The absolute value of the difference of the coefficients of $x^4$ and $x^6$ in the expansion of $x^2 - 2x^2 + (x + 1)^4(x^2 - 1)^2$, is

Question 42: Palette 42/42

AP-EAPCET 2022 MCQ

The least value of $n$ so that ${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$

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