TU IOE | Mathematics
If a set $A$ has 5 elements, the number of proper subsets of $A$ is
Answer: 31
Model set 1, Q1: explanation →The negation of the statement $p \wedge q$ is
Answer: $\neg p \vee \neg q$
Model set 1, Q2: explanation →If $f(x)=2x+3$, then $f^{-1}(7)$ equals
Answer: 2
Model set 1, Q3: explanation →The value of the determinant $\begin{vmatrix}2&3\4&5\end{vmatrix}$ is
Answer:
Model set 1, Q4: explanation →If $A$ is a $3\times3$ matrix with $|A|=4$, then $|2A|$ equals
Answer: 32
Model set 1, Q5: explanation →The value of $i^{47}$ is
Answer:
Model set 1, Q6: explanation →The sum of the first 10 terms of the AP with first term 2 and common difference 3 is
Answer: 155
Model set 1, Q8: explanation →If ${}^{n}C_2=45$, then $n$ equals
Answer: 10
Model set 1, Q9: explanation →The coefficient of $x^2$ in the expansion of $(1+x)^6$ is
Answer: 15
Model set 1, Q10: explanation →In a group, $n(A)=30$, $n(B)=25$ and $n(A\cup B)=45$. The number of elements belonging to exactly one of $A$ and $B$ is
Answer: 35
Model set 1, Q21: explanation →If $A=\begin{pmatrix}1&2\3&4\end{pmatrix}$, then $A^2-5A$ equals
Answer:
Model set 1, Q22: explanation →If $z_1,z_2$ are the roots of $z^2-2z+5=0$, then $z_1^2+z_2^2$ equals
Answer:
Model set 1, Q23: explanation →The sum of the infinite series $1+\dfrac23+\dfrac39+\dfrac4{27}+\cdots$ is
Answer: $\dfrac94$
Model set 1, Q24: explanation →The number of distinct arrangements of the letters of the word LEVEL is
Answer: 30
Model set 1, Q25: explanation →The term independent of $x$ in the expansion of $\left(x^2+\dfrac1x\right)^9$ is
Answer: 84
Model set 1, Q26: explanation →A set $P$ has 3 elements and a set $Q$ has 4 elements. The number of relations from $P$ to $Q$ is
Answer: 4096
Model set 2, Q1: explanation →The negation of the statement $p \wedge \neg q$ is
Answer: $\neg p \vee q$
Model set 2, Q2: explanation →The domain of $f(x)=\sqrt{x^2-5x+6}$ is
Answer: $(-\infty,2]\cup[3,\infty)$
Model set 2, Q3: explanation →The sum of the first 20 terms of the AP $3,7,11,\dots$ is
Answer: 820
Model set 2, Q6: explanation →The number of distinct arrangements of the letters of the word BANANA is
Answer: 60
Model set 2, Q7: explanation →The sum of the coefficients in the expansion of $(2x-1)^5$ is
Answer: 1
Model set 2, Q8: explanation →The term independent of $x$ in the expansion of $\left(x^2-\dfrac1x\right)^9$ is
Answer: 84
Model set 2, Q24: explanation →The sum to infinity of the series $1+\dfrac23+\dfrac39+\dfrac4{27}+\cdots$ is
Answer: $\frac94$
Model set 2, Q25: explanation →A committee of 4 is chosen from 6 men and 4 women. The number of committees with at least 2 women is
Answer: 115
Model set 2, Q26: explanation →The value of $\displaystyle\int_0^1(3x^2+2x),dx$ is
Answer: 2
Model set 1, Q18: explanation →The local minimum value of $f(x)=x^3-12x+5$ is
Answer:
Model set 1, Q31: explanation →The area enclosed between the curves $y=x^2$ and $y=x$ is
Answer: $\dfrac16$
Model set 1, Q32: explanation →The solution of $\dfrac{dy}{dx}=2xy$ with $y(0)=3$ gives $y(1)$ equal to
Answer:
Model set 1, Q33: explanation →$\int_0^{\pi/2}\sin^2x,dx$ equals
Answer: $\frac{\pi}{4}$
Model set 2, Q19: explanation →The general solution of $\dfrac{dy}{dx}+\dfrac{2y}{x}=x^2$ is
Answer: $y=\frac{x^3}{5}+\frac{C}{x^2}$
Model set 2, Q30: explanation →$\int_1^e x\ln x,dx$ equals
Answer: $\frac{e^2+1}{4}$
Model set 2, Q31: explanation →The local maximum value of $f(x)=x^3-12x+5$ is
Answer: 21
Model set 2, Q32: explanation →The distance from the point $(3,4)$ to the line $3x+4y-5=0$ is
Answer: 4
Model set 1, Q7: explanation →The centre of the circle $x^2+y^2-6x+4y-12=0$ is
Answer: $(3,-2)$
Model set 1, Q14: explanation →The length of the tangent drawn from the point $(5,4)$ to the circle $x^2+y^2-2x-4y-4=0$ is
Answer: $\sqrt{11}$
Model set 1, Q28: explanation →The pair of lines $x^2-5xy+6y^2=0$ makes an angle $\theta$ with each other. Then $\tan\theta$ equals
Answer: $\dfrac17$
Model set 1, Q29: explanation →The distance between the parallel lines $3x+4y=5$ and $3x+4y=15$ is
Answer: 2
Model set 2, Q12: explanation →The acute angle between the lines given by $x^2-5xy+6y^2=0$ is
Answer: $\tan^{-1}\frac17$
Model set 2, Q13: explanation →The radius of the circle $x^2+y^2-6x+8y-11=0$ is
Answer: 6
Model set 2, Q14: explanation →The distance of the plane $2x-2y+z=9$ from the origin is
Answer: 3
Model set 1, Q13: explanation →The eccentricity of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ is
Answer: $\dfrac35$
Model set 1, Q15: explanation →The eccentricity of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is
Answer: $\frac35$
Model set 2, Q15: explanation →The distance between the points $(1,2,3)$ and $(4,6,15)$ is
Answer: 13
Model set 2, Q16: explanation →The length of the latus rectum of the hyperbola $\dfrac{x^2}{9}-\dfrac{y^2}{16}=1$ is
Answer: $\frac{32}{3}$
Model set 2, Q28: explanation →The cosine of the angle between two lines with direction ratios $(1,2,2)$ and $(2,3,6)$ is
Answer: $\frac{20}{21}$
Model set 2, Q29: explanation →$\displaystyle\lim_{x\to0}\frac{\sin 3x}{x}$ equals
Answer: 3
Model set 1, Q16: explanation →The derivative of $xe^{x}$ with respect to $x$ is
Answer: $e^{x}(1+x)$
Model set 1, Q17: explanation →$\displaystyle\lim_{x\to\infty}\left(\sqrt{x^2+3x}-x\right)$ equals
Answer: $\dfrac32$
Model set 1, Q30: explanation →$\lim_{x\to0}\dfrac{\sin 5x}{\tan 2x}$ equals
Answer: $\frac52$
Model set 2, Q17: explanation →The derivative of $x\ln x$ at $x=e$ is
Answer: 2
Model set 2, Q18: explanation →The variance of the data 2, 4, 6, 8 is
Answer: 5
Model set 1, Q20: explanation →A bag has 5 red and 3 blue balls. Two balls are drawn at random without replacement. The probability that both are of the same colour is
Answer: $\dfrac{13}{28}$
Model set 1, Q35: explanation →Two fair dice are thrown. The probability that the sum is 9 is
Answer: $\frac19$
Model set 2, Q20: explanation →The mean of 5 observations is 8 and their variance is 4. If a sixth observation equal to 8 is added, the new variance is
Answer: $\frac{10}{3}$
Model set 2, Q34: explanation →Box I has 3 white and 2 black balls, Box II has 1 white and 4 black balls. A box is chosen at random and a ball drawn is white. The probability that it came from Box I is
Answer: $\frac34$
Model set 2, Q35: explanation →If $A$ is a $3\times 3$ matrix with $|A|=4$, then $|2A|$ equals
Answer: 32
Model set 2, Q4: explanation →The value of $(1+i)^8$ is
Answer: 16
Model set 2, Q5: explanation →If $A=\begin{pmatrix}2&1\1&1\end{pmatrix}$, then $A^2-3A+I$ equals
Answer: the zero matrix
Model set 2, Q21: explanation →If $\omega$ is a complex cube root of unity, the value of $\begin{vmatrix}1&1&1\1&\omega&\omega^2\1&\omega^2&\omega\end{vmatrix}$ is
Answer: $-3\sqrt3,i$
Model set 2, Q22: explanation →The principal argument of $\dfrac{1+i}{\sqrt3-i}$ is
Answer: $\frac{5\pi}{12}$
Model set 2, Q23: explanation →The value of $\sin^2 30^\circ+\cos^2 60^\circ$ is
Answer: $\dfrac12$
Model set 1, Q11: explanation →In a triangle $ABC$ with $a=7$, $b=8$, $c=9$, the value of $\cos A$ is
Answer: $\dfrac23$
Model set 1, Q27: explanation →The value of $\tan 15^\circ$ is
Answer: $2-\sqrt3$
Model set 2, Q9: explanation →In triangle $ABC$, $a=7,\ b=8,\ c=9$. Then $\cos A$ equals
Answer: $\frac23$
Model set 2, Q11: explanation →The value of $\sin^{-1}\tfrac12+\cos^{-1}\tfrac12$ is
Answer: $\dfrac{\pi}{2}$
Model set 1, Q12: explanation →$\tan^{-1}1+\tan^{-1}2+\tan^{-1}3$ equals
Answer:
Model set 2, Q10: explanation →The value of $\sin!\left(2\cos^{-1}\frac35\right)$ is
Answer: $\frac{24}{25}$
Model set 2, Q27: explanation →The angle between the vectors $\vec a=(1,2,3)$ and $\vec b=(2,-1,0)$ is
Answer: $90^\circ$
Model set 1, Q19: explanation →If $|\vec a|=3$, $|\vec b|=4$ and $\vec a\cdot\vec b=6$, then $|\vec a\times\vec b|$ equals
Answer: $6\sqrt3$
Model set 1, Q34: explanation →The area of a triangle whose adjacent sides are $\vec a=\hat i+2\hat j+3\hat k$ and $\vec b=2\hat i-\hat j+\hat k$ is
Answer: $\frac{5\sqrt3}{2}$
Model set 2, Q33: explanation →