1
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

For $$0 < c < b < a$$, let $$(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0$$ and $$\alpha \neq 1$$ be one of its root. Then, among the two statements

(I) If $$\alpha \in(-1,0)$$, then $$b$$ cannot be the geometric mean of $a$ and $$c$$

(II) If $$\alpha \in(0,1)$$, then $$b$$ may be the geometric mean of $$a$$ and $$c$$

A
only (II) is true
B
Both (I) and (II) are true
C
only (I) is true
D
Neither (I) nor (II) is true
2
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

The sum of the series $$\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots$$ up to 10 -terms is

A
$$\frac{45}{109}$$
B
$$-\frac{55}{109}$$
C
$$\frac{55}{109}$$
D
$$-\frac{45}{109}$$
3
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$a$$ and $$b$$ be be two distinct positive real numbers. Let $$11^{\text {th }}$$ term of a GP, whose first term is $$a$$ and third term is $$b$$, is equal to $$p^{\text {th }}$$ term of another GP, whose first term is $$a$$ and fifth term is $$b$$. Then $$p$$ is equal to

A
20
B
24
C
21
D
25
4
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

Let $$S_n$$ denote the sum of first $$n$$ terms of an arithmetic progression. If $$S_{20}=790$$ and $$S_{10}=145$$, then $$\mathrm{S}_{15}-\mathrm{S}_5$$ is :

A
405
B
390
C
410
D
395
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