Mat Jee Mains
Mat Jee Mains
Mat Jee Mains
Solution:
[latex]=2\left[ {{x}^{6}}+15{{x}^{7}}-15{{x}^{4}}+15{{x}^{8}}-
30{{x}^{5}}+15{{x}^{2}}+{{x}^{9}}-3{{x}^{6}}+3\left( {{x}^{3}}-1 \right) \right][/latex]
2. Let [latex]\sin \left( \alpha -\beta \right)=\frac{5}{13}[/latex]and [latex]\cos \left( \alpha +\beta
\right)=\frac{3}{5}[/latex]where [latex]\alpha ,\beta \in \left( 0,\frac{\pi }{4} \right)[/latex] then
[latex]Tan2\alpha =[/latex]
(A) [latex]\frac{63}{16}[/latex] (B) [latex]\frac{61}{16}[/latex] (C)
[latex]\frac{65}{16}[/latex] (D) [latex]\frac{32}{9}[/latex]
Solution:
[latex]Tan2\alpha =Tan\left[ \left( \alpha +\beta \right)+\left( \alpha -\beta \right) \right][/latex]
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Solution:
[latex]{{x}^{2}}+{{y}^{2}}=16[/latex]
Centre = (0, 0) OA = P
Radius = 4
P [latex]=\frac{n}{\sqrt{2}}[/latex]
[latex]\frac{n}{\sqrt{2}}<4\Rightarrow
n<4\sqrt{2}[/latex]
[latex]=80-\frac{55}{2}=\frac{105}{2}[/latex]
Solution:
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5. The area bounded by the curve [latex]y\le {{x}^{2}}+3x,0\le y\le 4,0\le x\le 3[/latex] is
(A) [latex]\frac{59}{6}[/latex] (B) [latex]\frac{57}{4}[/latex] (C)
[latex]\frac{59}{3}[/latex] (D) [latex]\frac{57}{6}[/latex]
Solution:
[latex]y={{x}^{2}}+3x[/latex]
[latex]y=4[/latex]
[latex]\Rightarrow {{x}^{2}}+3x=4[/latex]
[latex]\Rightarrow {{x}^{2}}+3x-4=0\Rightarrow
x=1\,\,or\,\,x=-4[/latex]
[latex]=\left[
\frac{{{x}^{3}}}{3}+\frac{3{{x}^{2}}}{2} \right]_{0}^{1}+2\left( 4 \right)[/latex]
[latex]=\frac{1}{3}+\frac{3}{2}+8=\frac{59}{6}[/latex]
6. “If you are born in India then you are citizen of India” contrapositive of this statement is
(A) If you are born in India then you are not citizen of India
(B) If you are not citizen of India then you are not born in India
(C) If you are citizen of India then you are not born in India
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(D) If you are citizen of India then you are born in India
Solution:
[latex]\therefore[/latex] Answer is if you are not citizen of India then you are not born in India.
7. [latex]A=\left[ \begin{matrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end{matrix}
\right][/latex] and [latex]{{A}^{32}}=\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \\ \end{matrix} \right][/latex]. then
α may be
(A) 0 (B) [latex]\frac{\pi }{32}[/latex] (C) [latex]\frac{\pi
}{64}[/latex] (D) [latex]\frac{\pi }{16}[/latex]
Solution:
[latex]A=\left[ \begin{matrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end{matrix}
\right]\Rightarrow {{A}^{2}}\left[ \begin{matrix} {{\cos }^{2}}\alpha -{{\sin }^{2}}\alpha & -\cos \alpha \sin \alpha
-\cos \alpha \sin \alpha \\ \sin \alpha \cos \alpha +\sin \alpha \cos \alpha & -{{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha \\
\end{matrix} \right][/latex]
Solution:
[latex]{{y}^{2}}=x-2[/latex]
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[latex]2y.\frac{dy}{dx}=1[/latex]
[latex]\frac{dy}{dx}=\frac{1}{2y}[/latex]
[latex]\frac{1}{2y}=1\Rightarrow
y=\frac{1}{2}\Rightarrow K=\frac{1}{2}[/latex]
[latex]\therefore \,\,N=\left(
\frac{9}{4},\frac{1}{2} \right)[/latex]
Solution:
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10. How many 9 digit numbers can be formed by using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 such that odd numbers
occur at even places.
(A) 160 (B) 175 (C) 180 (D) 220
Solution:
Even places = 4
Odd numbers = 3
Even numbers = 6
11. Let [latex]g\left( x \right)=\ell n\left( x \right)[/latex] and [latex]f\left( x \right)=\left( \frac{1-x\cos x}{1+x\cos
x} \right)[/latex] then [latex]\int\limits_{-\frac{\pi }{4}}^{\frac{\pi }{4}}{g\left[ f\left( x \right)
\right]}.\,dx=[/latex]
(A) [latex]\ell n\,1[/latex] (B) [latex]\ell n\,2[/latex] (C) [latex]\ell
n\,e[/latex] (D) [latex]\ell n\,4[/latex]
Solution:
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Solution:
[latex]2.20{{C}_{0}}+5.20{{C}_{1}}+8.20{{C}_{2}}+...+62.20{{C}_{20}}[/latex]
[latex]=3\sum\limits_{r=0}^{20}{r.\,20{{C}_{r}}}+2.\sum\limits_{r=0}^{20}{20{{C}_{r}}}.[/latex]
[latex]=3.20\sum\limits_{r=1}^{20}{19{{C}_{r-1}}+2\left(
20{{C}_{0}}+20{{C}_{1}}+...+20{{C}_{20}} \right)}[/latex]
13. Sum of natural numbers between 100 and 200 whose HCF with 91 should be more than 1.
(A) 1121 (B) 3210 (C) 3121 (D) 1520
Solution:
[latex]91=13\times 7[/latex]
HCF of 91 and a number is more than 1 means the number should be either multiple of 7 or 13.
[latex]=3121[/latex]
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14. If mean and variance of 7 variates are 8 and 16 respectively and five of them are 2, 4, 10, 12, 14 then the
product of remaining two variates is
(A) 49 (B) 48 (C) 45 (D) 40
Solution:
[latex]\Rightarrow {{x}^{2}}+{{y}^{2}}+2xy=196[/latex]
15. If α and β are the roots of [latex]{{x}^{2}}-2x+2=0[/latex] then the minimum value of n such that
[latex]{{\left( \frac{\alpha }{\beta } \right)}^{n}}=1[/latex]
(A) 4 (B) 3 (C) 2 (D) 5
Solution:
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Solution:
[latex]\frac{dy}{dx}+\frac{2x}{{{x}^{2}}+1}.\,y=\frac{1}{{{\left( {{x}^{2}}+1
\right)}^{2}}}[/latex]
[latex]y.\,IF=\int{Q.\,IF.\,dx}[/latex]
Solution:
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[latex]=2.\,f\left( x \right)[/latex]
18. Given that [latex]A\subset B[/latex] then identify the correct statement.
(A) [latex]P\left( \frac{A}{B} \right)=P\left( A \right)[/latex] (B) [latex]P\left( \frac{A}{B} \right)\le P\left(
A \right)[/latex] (C) [latex]P\left( \frac{A}{B} \right)\ge P\left( A \right)[/latex] (D) [latex]P\left( \frac{A}{B}
\right)=P\left( A \right)-P\left( B \right)[/latex]
Solution:
19. Find the value of ‘c’ for which the following equations have non-trivial solutions.
[latex]cx-y-z=0,-cx+y-cz=0,x+y-cz=0[/latex]
(A) [latex]\frac{1}{2}[/latex] (B) [latex]-1[/latex] (C) 2
(D) 0
Solution:
[latex]\left| \begin{matrix} c & -1 & -1 \\ -c & 1 & -c \\ 1 & 1 & -c \\ \end{matrix} \right|=0[/latex]
20. Let [latex]2y={{\left[ {{\cot }^{-1}}\left( \frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x} \right)
\right]}^{2}}[/latex] then [latex]\frac{dy}{dx}=[/latex]
(A) [latex]x-\frac{\pi }{6}[/latex] (B) [latex]x+\frac{\pi }{6}[/latex] (C) [latex]2x-
\frac{\pi }{6}[/latex] (D) [latex]2x-\frac{\pi }{3}[/latex]
Solution:
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[latex]2.\,\frac{dy}{dx}=2x-\frac{\pi }{3}[/latex]
[latex]\frac{dy}{dx}=x-\frac{\pi }{6}[/latex]
21. Let [latex]{{S}_{1}}[/latex]is set of minima and [latex]{{S}_{2}}[/latex]is set of maxima for the curve
[latex]y=9{{x}^{4}}+12{{x}^{3}}-36{{x}^{2}}-25[/latex] then
(A) [latex]{{S}_{1}}=\left\{ -2,-1 \right\},{{S}_{2}}=\left\{ 0 \right\}[/latex] (B)
[latex]{{S}_{1}}=\left\{ -2,1 \right\},{{S}_{2}}=\left\{ 0 \right\}[/latex]
(C) [latex]{{S}_{1}}=\left\{ -2,1 \right\},{{S}_{2}}=\left\{ -1 \right\}[/latex] (D)
[latex]{{S}_{1}}=\left\{ -2,2 \right\},{{S}_{2}}=\left\{ 0 \right\}[/latex]
Solution:
[latex]y=9{{x}^{4}}+12{{x}^{3}}-36{{x}^{2}}-25[/latex]
[latex]\frac{dy}{dx}=36{{x}^{3}}+36{{x}^{2}}-72x[/latex]
22. Let [latex]{{f}^{11}}\left( x \right)>0[/latex] and [latex]\phi \left( x \right)=f\left( x \right)+f\left( 2-x
\right),x\in \left( 0,2 \right)[/latex] be a function then the function [latex]\phi \left( x \right)[/latex]is
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(A) Increasing in (0, 1) and decreasing in (1, 2) (B) Decreasing in (0, 1) and increasing in (1, 2)
(C) Increasing in (0, 2) (D) Decreasing in (0, 2)
Solution:
[latex]\Rightarrow x>1[/latex]
23. Let vertices of the triangle ABC is A(0, 0), B(0, 1) and C(x, y) and perimeter is 4 then locus of ‘C’ is
(A) [latex]9{{x}^{2}}+8{{y}^{2}}+8y=16[/latex] (B)
[latex]8{{x}^{2}}+9{{y}^{2}}+9y=16[/latex]
(C) [latex]9{{x}^{2}}+8{{y}^{2}}-8y=16[/latex] (D) [latex]8{{x}^{2}}+9{{y}^{2}}-
9x=16[/latex]
Solution:
Perimeter = 4
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[latex]\sqrt{{{x}^{2}}+{{y}^{2}}-2y+1}=3-
\sqrt{{{x}^{2}}+{{y}^{2}}}[/latex]
[latex]\cancel{{{x}^{2}}}+\cancel{{{y}^{2}}}-
2y+1=9+\cancel{{{x}^{2}}}+\cancel{{{y}^{2}}}-
6\sqrt{{{x}^{2}}+{{y}^{2}}}[/latex]
[latex]6\sqrt{{{x}^{2}}+{{y}^{2}}}=8+2y[/latex]
[latex]3\sqrt{{{x}^{2}}+{{y}^{2}}}=4+y[/latex]
[latex]9{{x}^{2}}+8{{y}^{2}}-8y=16[/latex]
24. Let the equation of a line is [latex]3x+5y=15[/latex] and a point P on this line is equidistant from x and y axis.
In which quadrant that P lies.
(A) 1st (B) 3rd (C) 4th (D) none of these
Solution:
[latex]3x+5y=15[/latex]
[latex]\frac{x}{5}+\frac{y}{3}=1[/latex]
If [latex]y=x\Rightarrow 8x=15[/latex]
[latex]\Rightarrow
x=\frac{15}{8},y=\frac{15}{8}[/latex]
If [latex]y=-x\Rightarrow -2x=15[/latex]
[latex]\Rightarrow x=\frac{-15}{2},y=\frac{15}{2}[/latex]
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25. The perpendicular distance of point [latex]\left( 2,-1,4 \right)[/latex] from the line
[latex]\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}[/latex] lies between
(A) (2, 3) (B) (3, 4) (C) (4, 5) (D) (1, 2)
Solution:
[latex]{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}=0[/latex]
26. If a plane passes through intersection of planes [latex]2x-y-4=0[/latex] and [latex]y+2z-4=0[/latex] and also
passes through the point (1, 1, 0). Then the equation of the plane is
(A) [latex]x-y-z=0[/latex] (B) [latex]2x-z=0[/latex] (C) [latex]x+2z-1=0[/latex] (D)
[latex]x-z-1=0[/latex]
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Solution:
[latex]{{P}_{1}}+\lambda {{P}_{2}}=0[/latex]
(1) passing through (1, 1, 0) [latex]\Rightarrow \left( 2-1-4 \right)+\lambda \left( 1-4 \right)=0[/latex]
[latex]\Rightarrow 2x+2y-2z=0[/latex]
[latex]\Rightarrow x+y-z=0[/latex]
27. If [latex]\left| \sqrt{x}-2 \right|+\sqrt{x}\left( \sqrt{x}-4 \right)=2[/latex] then sum of roots of equation is
(A) 12 (B) 8 (C) 4 (D) 16
Solution:
[latex]{{P}^{2}}+P-6=0\,\,\Rightarrow {{P}^{2}}+3P-2P-6=0[/latex]
[latex]\sqrt{x}-2=\pm 2[/latex]
[latex]\sqrt{x}=2+2[/latex] or [latex]-2+2[/latex]
[latex]\sqrt{x}=4[/latex] or 0
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[latex]x=16[/latex] or 0
28. [latex]4{{x}^{2}}+{{y}^{2}}=8,[/latex] Tangent at (1, 2) and another tangent at (a, b) are perpendicular then
[latex]{{a}^{2}}=[/latex]
(A) [latex]\frac{2}{17}[/latex] (B) [latex]\frac{1}{17}[/latex] (C)
[latex]\frac{8}{17}[/latex] (D) [latex]\frac{4}{17}[/latex]
Solution:
[latex]4{{x}^{2}}+{{y}^{2}}=8[/latex]
Tangent at (1, 2) is
[latex]{{m}_{1}}{{m}_{2}}=-1[/latex]
[latex]a=\sqrt{2}\cos \theta[/latex]
29. Find the magnitude of projection of vector [latex]2i+3j+k[/latex] on a vector which is perpendicular to the plane
containing vectors [latex]i+j+k[/latex] and [latex]i+2j+3k[/latex]
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Solution:
[latex]\overrightarrow{n\,}=i-2j+k[/latex]
Solution:
[latex]\alpha =Ta{{n}^{-1}}\frac{4}{3}[/latex]
[latex]\beta =Ta{{n}^{-1}}\frac{5}{12}[/latex]
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