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JEE Advanced

STATEMENT-1: In water, orthoboric acid behaves as a weak monobasic acid because STATEMENT-2: In water, orthoboric acid acts as a proton donor.

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How many $3×3$ matrices $M$ with entries from ${0,1,2}$ are there, for which the sum of the diagonal entries of $M_{T}M$ is 5? (A) 126 (B)198 (C) 162 (D) 135

A farmer $F_{1}$has a land in the shape of a triangle with vertices at $P(0,0),Q(1,1)$and $R(2,0)$. From this land, a neighbouring farmer $F_{2}$takes away the region which lies between the side $PQ$and a curve of the form $y=x_{n}(n>1)$. If the area of the region taken away by the farmer $F_{2}$is exactly 30% of the area of $PQR$, then the value of $n$is _______.

Let P and Q be distinct points on the parabola $y_{2}=2x$ such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle $ΔOPQ$ is $32$ , then which of the following is (are) the coordinates of $P?$

A pack contains $n$cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of het numbers on the removed cards is $k,$then $k−20=$____________.

Let E1 and E2, be two ellipses whose centers are at the origin.The major axes of E1 and E2, lie along the x-axis and the y-axis, respectively. Let S be the circle $x_{2}+(y−1)_{2}=2$. The straight line x+ y =3 touches the curves S, E1 and E2 at P,Q and R, respectively. Suppose that $PQ=PR=322 $.If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are):

The number of points in $(−∞,∞),$for which $x_{2}−xsinx−cosx=0,$is6 (b) 4 (c) 2 (d) 0

If $y=y(x)$ satisfies the differential equation $8x (9+x )dy=(4+9+x )_{−1}dx,x>0$ and $y(0)=7, $ then $y(256)=$ (A) 16 (B) 80 (C) 3 (D) 9

Let $S$be the set of all column matrices $[b_{1}b_{2}b_{3}]$such that $b_{1},b_{2},b_{3}∈R$and the system of equations (in real variable)$−x+2y+5z=b_{1}$$2x−4y+3z=b_{2}$$x−2y+2z=b_{3}$has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $[b_{1}b_{2}b_{3}]∈S$?(a) $x+2y+3z=b_{1},4y+5z=b_{2}$and $x+2y+6z=b_{3}$(b) $x+y+3z=b_{1},5x+2y+6z=b_{2}$and $−2x−y−3z=b_{3}$(c) $−x+2y−5z=b_{1},2x−4y+10z=b_{2}$and $x−2y+5z=b_{3}$(d) $x+2y+5z=b_{1},2x+3z=b_{2}$and $x+4y−5z=b_{3}$