SECTION - B [MATHS - NUMERIC] - JEE STD 11 Science Questions

Question 14 Marks

Let $S=\left\{p_{1}, p_{2} \ldots \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $\mathrm{A}=\mathrm{S} \cup \mathrm{P}$, where P is the set of all possible products of distinct element of S . Then the number of all ordered pairs ( $x, y$ ), $x \in S$, $y \in A$, such that $x$ divides $y$, is ______________ .

Answer

(5120)
Sol. Let $\frac{y}{x}=\lambda$
$y=\lambda x$ $=10 \times\left({ }^{9} \mathrm{C}_{0}+{ }^{9} \mathrm{C}_{1}+{ }^{9} \mathrm{C}_{2}+{ }^{9} \mathrm{C}_{3}+\ldots .+{ }^{9} \mathrm{C}_{9}\right)$
$=10 \times\left(2^{9}\right)$
$10 \times 512$
5120

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Question 24 Marks

Let be a $3 \times 3$ matrix such that $\mathrm{X}^{\mathrm{T}} \mathrm{AX}=\mathrm{O}$ for all nonzero $3 \times 1$ matrices $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$. If $\mathrm{A}\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\left[\begin{array}{l}1 \\ 4 \\ -5\end{array}\right], \mathrm{A}\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]=\left[\begin{array}{l}0 \\ 4 \\ -8\end{array}\right]$, and
$\operatorname{det}(\operatorname{adj}(2(\mathrm{~A}+\mathrm{I})))=2^{\alpha} 3^{\beta} 5^{\gamma}, \alpha, \beta, \gamma, \in \mathrm{N}$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is

Answer

(44)
Sol. $X^{T} A X=0$
(xyz) $\left(\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3}\end{array}\right)\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=0$
$(x y z)\left(\begin{array}{l}a_{1} x+a_{2} y+a_{3} z \\ b_{1} x+b_{2} y+b_{3} z \\ c_{1} x+c_{2} y+c_{3} z\end{array}\right)=0$
$x\left(a_{1} x+a_{2} y+a_{3} z\right)+y\left(b_{1} x+b_{2} y+b_{3} z\right)$
$+z\left(c_{1} x+c_{2} y+c_{3} z\right)=0$
$a_{1}=0, b_{2}=0 c_{3}=0$
$\mathrm{a}_{2}+\mathrm{b}_{1}=0, \mathrm{a}_{3}+\mathrm{c}_{1}=0, \mathrm{~b}_{3}=\mathrm{c}_{2}=0$
$\mathrm{A}=$ skew symm matrix
$\mathrm{A}=\left(\begin{array}{ccc}0 & \mathrm{x} & \mathrm{y} \\ -\mathrm{x} & 0 & \mathrm{z} \\ -\mathrm{y} & -\mathrm{z} & 0\end{array}\right) ; \quad \mathrm{A}=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 4 \\ -5\end{array}\right)$
$\Rightarrow A=\left(\begin{array}{ccc}0 & x & y \\ -x & 0 & z \\ -y & -z & 0\end{array}\right)\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 4 \\ -5\end{array}\right)$
$x+y=1$
$-x+z=4$
$y+z=5$
$\left(\begin{array}{ccc}0 & x & y \\ -x & 0 & z \\ -y & -z & 0\end{array}\right)\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right)=\left(\begin{array}{l}1 \\ 4 \\ -8\end{array}\right)$
$2 x+y=0 \quad x=-1$
$-x+z=4 \quad y=2$
$-\mathrm{y}-2 \mathrm{z}=-8 \quad \mathrm{z}=3$
$\mathrm{A}=\left(\begin{array}{ccc}0 & -1 & 2 \\ 1 & 0 & 3 \\ -2 & -3 & 0\end{array}\right)$
$2(\mathrm{~A}+\mathrm{I})=\left(\begin{array}{ccc}2 & -2 & 4 \\ 2 & 2 & 6 \\ -2 & -6 & 2\end{array}\right)$
$2(\mathrm{~A}+\mathrm{I})=120 \Rightarrow \operatorname{det}|\operatorname{adi}(2(\mathrm{~A}+\mathrm{I}))|$
$=120^{2}=2^{6} \cdot 3^{2} \cdot 5^{2}$
$\alpha=6, \beta=2, \gamma=2$

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Question 34 Marks

The number of 3-digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is

Answer

(125)
Sol. No, of 3 digits $=999-99=900$
No. of 3 digit numbers divisible by 2 & 3 i.e. by 6
$\frac{900}{6}=150$
No. of 3 digit numbers divisible by $4 \& 9$ i.e. by 36 $\frac{900}{36}=25$
$\therefore$ No of 3 digit numbers divisible by $2 \& 3$ but not by $4 \& 9$
$150-25=125$

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Question 44 Marks

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta=8$ and $\sec ^{2}\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} \beta\right)=36$, then $\alpha^{2}+\beta$ is ______________ .

Answer

(14)
Sol. If $\left(\tan \left(\tan ^{-1}(\alpha)\right)+1\left(\cot \left(\cot ^{-1} \beta\right)\right)^{2}=36\right.$
$\alpha^{2}+\beta^{2}=34$
$\alpha \beta=15$
$\alpha=3, \beta=5$
$\therefore \alpha^{2}+\beta=9+5=14$

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Question 54 Marks

Let f be a differentiable function such that $2(x+2)^{2} f(x)-3(x+2)^{2}=10 \int_{0}^{x}(t+2) f(t) d t$, $x \geq 0$. Then $f(2)$ is equal to ______________ .

Answer

(19)
Sol. Differentiate both sides
$4(x+2) f(x)+2(x+2)^{2} f^{\prime}(x)-6(x+2)=10(x+2) f(x)$
$2(x+2)^{2} f^{\prime}(x)-6(x+2) f(x)=6(x+2)$
$(x+2) \frac{d y}{d x}-3 y=3$
$\int \frac{d y}{d x}=3 \int \frac{d x}{x+2}$
$\ln (y+1)=3 \ln (x+2)+C$
$(y+1)=C(x+2)^{3}$
$f(0)=\frac{3}{2}$
$\mathrm{f}(2)=19$

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JEE SECTION - B [MATHS - NUMERIC]

SECTION - B [MATHS - NUMERIC]

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