eSaral - JEE | NEET | Class 8-10 Exam Prep App

For any two sets A and B, prove that :

Question: For any two sets $A$ and $B$, prove that : $A^{\prime}-B^{\prime}=B-A$ Solution: $\mathrm{LHS}=A^{\prime}-B^{\prime}$ $=A^{\prime} \cap\left(B^{\prime}\right)^{\prime} \quad\left[\because C-D=C \cap D^{\prime}\right]$ $=A^{\prime} \cap B$ $=B \cap A^{\prime}$ $=B-A \quad\left[\because C \cap D^{\prime}=C-D\right]$ $\mathrm{RHS}=B-A$So, LHS = RHS...

Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division

Question: Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division $f(x)=4 x^{4}-3 x^{3}-2 x^{2}+x-7, g(x)=x-1$ Solution: Here, $f(x)=4 x^{4}-3 x^{3}-2 x^{2}+x-7$ g(x) = x - 1 from, the remainder theorem when f(x) is divided by g(x) = x - (-1) the remainder will be equal to f(1) Let, g(x) = 0 ⟹ x - 1 = 0 ⟹ x = 1 Substitute the value of x in f(x) $f(1)=4(1)^{4}-3(1)^{3}-2(1)^{2}+1-7$ = 4 - 3 - 2 + 1 - 7 = 5 - 12 = -7 Therefore, the remainder i...

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour.

Question: A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of (A) $1 \mathrm{~m} / \mathrm{h}$ (B) $0.1 \mathrm{~m} / \mathrm{h}$ (C) $1.1 \mathrm{~m} / \mathrm{h}$ (D) $0.5 \mathrm{~m} / \mathrm{h}$ Solution: Letrbe the radius of the cylinder. Then, volume (V)of the cylinder is given by, $\begin{aligned} V =\pi(\text { radius })^{2} \times \text { height } \\ =\pi(10)^{2} h \quad(\text { ra...

Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division

Question: Using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division $f(x)=x^{3}+4 x^{2}-3 x+10, g(x)=x+4$ Solution: Here, $f(x)=x^{3}+4 x^{2}-3 x+10$ g(x) = x + 4 from, the remainder theorem when f(x) is divided by g(x) = x - (- 4) the remainder will be equal to f(- 4) Let, g(x) = 0 ⟹ x + 4 = 0 ⟹ x = - 4 Substitute the value of x in f(x) $f(-4)=(-4)^{3}+4(-4)^{2}-3(-4)+10$ = - 64 + (4*16) + 12 + 10 = - 64 + 64 + 12 + 10 = 12 + 10 = 22 Therefor...

Determine, graphically whether the system of equations x − 2y = 2, 4x − 2y = 5 is consistent or in-consistent.

Question: Determine, graphically whether the system of equationsx 2y= 2, 4x 2y= 5 is consistent or in-consistent. Solution: The given equations are $x-2 y=2$$\ldots(i)$ $4 x-2 y=5$....(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 0-2 y=2$ $\Rightarrow y=-1$ $\Rightarrow x=0, \quad y=-1$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow x-2 \times 0=2$ $\Rightarrow x=2$ $\Rightarrow x=2, \quad y=0$ Use the following table to draw the graph. Draw the graph by plotting the two point...

Find the rational roots of the polynomial f

Question: Find the rational roots of the polynomial $f(x)=2 x^{3}+x^{2}-7 x-6$ Solution: Given that $f(x)=2 x^{3}+x^{2}-7 x-6$ f(x) is a cubic polynomial with an integer coefficient . If the rational root in the form ofp/q, the values of p are limited to factors of 6 which are 1, 2, 3, 6 and the values of q are limited to the highest degree coefficient i.e 2 which are 1, 2 here, the possible rational roots are 1, 2, 3, 6, 1/2, 3/2 Let, x = -1 $f(-1)=2(-1)^{3}+(-1)^{2}-7(-1)-6$ = - 2 + 1 + 7 - 6 ...

Each set X, contains 5 elements and each set Y,

Question: Each set $X$, contains 5 elements and each set $Y$, contains 2 elements and $\bigcup_{r=1}^{20} X_{r}=S=\bigcup_{r=1}^{n} Y_{r}$. If each element of $S$ belong to exactly 10 of the $X_{r}^{\prime} s$ and to eactly 4 of $Y_{r}^{\prime} s$, then find the value of $n$. Solution: It is given that each set $X$ contains 5 elements and $\bigcup_{r=1}^{20} X_{r}=S$. $\therefore n(S)=20 \times 5=100$ But, it is given that each element of $S$ belong to exactly 10 of the $X_{r}$ 's. $\therefore$ ...

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical

Question: Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi vertical angle $\alpha$ is one-third that of the cone and the greatest volume of cylinder is $\frac{4}{27} \pi h^{3} \tan ^{2} \alpha$. Solution: The given right circular cone of fixed height (h)and semi-vertical angle ()can be drawn as: Here, a cylinder of radiusRand heightHis inscribed in the cone. Then, $\angle \mathrm{GAO}=\alpha, \mathrm{OG}=r, \mathrm{OA}=h, ...

Find the integral roots of the polynomial

Question: Find the integral roots of the polynomial $f(x)=x^{3}+6 x^{2}+11 x+6$ Solution: Given, that $f(x)=x^{3}+6 x^{2}+11 x+6$ Clearly we can say that, the polynomial f(x) with an integer coefficient and the highest degree term coefficient which is known as leading factor is 1. So, the roots of f(x) are limited to integer factor of 6, they are 1, 2, 3, 6 Let x = -1 $f(-1)=(-1)^{3}+6(-1)^{2}+11(-1)+6$ = -1 + 6 -11 + 6 = 0 Let x = 2 $f(-2)=(-2)^{3}+6(-2)^{2}+11(-2)+6$ = 8 (6 * 4) - 22 + 6 = 8 +...

Determine graphically the vertices of the triangle, the equations of whose sides are given below :

Question: Determine graphically the vertices of the triangle, the equations of whose sides are given below : (i) $2 y-x=8,5 y-x=14$ and $y-2 x=1$ (ii) $y=x, y=0$ and $3 x+3 y=10$ Solution: (i) Draw the 3 lines as given by equations By takingx=1 = 1 cm onxaxis Andy=1=1cm onyaxis $\frac{y}{4}-\frac{x}{8}=1$ $\frac{y}{2.8}-\frac{x}{14}=1$ $\frac{y}{1}-\frac{x}{0.5}=1$ Clearly from graph points of intersection three lines are (4,2) , (1,3), (2,5) (ii) Draw the 3 lines as given by equations By taking...

Is it true that for any sets A and B,

Question: Is it true that for any sets $A$ and $B, P(A) \cup P(B)=P(A \cup B)$ ? Justify your answer. Solution: False. Let $X \in P(A) \cup P(B)$ $\Rightarrow X \in P(A)$ or $X \in P(B)$ $\Rightarrow X \subset A$ or $X \subset B$ $\Rightarrow X \subset(A \cup B)$ $\Rightarrow X \in P(A \cap B)$ $\therefore P(A) \cup P(B) \subset P(A \cup B)$ ...(1) Again, let $X \in P(A \cup B)$ But $X \notin P(A)$ or $x \notin P(B)$ [For example let $A=\{2,5\}$ and $B=\{1,3,4\}$ and take $X=\{1,2,3,4\}$ ] So, $...

Is it true that for any sets A and B,

Question: Is it true that for any sets $A$ and $B, P(A) \cup P(B)=P(A \cup B)$ ? Justify your answer. Solution: False. Let $X \in P(A) \cup P(B)$ $\Rightarrow X \in P(A)$ or $X \in P(B)$ $\Rightarrow X \subset A$ or $X \subset B$ $\Rightarrow X \subset(A \cup B)$ $\Rightarrow X \in P(A \cap B)$ $\therefore P(A) \cup P(B) \subset P(A \cup B)$ ...(1) Again, let $X \in P(A \cup B)$ But $X \notin P(A)$ or $x \notin P(B)$ [For example let $A=\{2,5\}$ and $B=\{1,3,4\}$ and take $X=\{1,2,3,4\}$ ] So, $...

If x = 0 and x = -1 are the roots of the polynomial f(x)

Question: If $x=0$ and $x=-1$ are the roots of the polynomial $f(x)=2 x^{3}-3 x^{2}+a x+b$, Find the of $a$ and $b$. Solution: We know that, $f(x)=2 x^{3}-3 x^{2}+a x+b$ Given, the values of x are 0 and -1 Substitute x = 0 in f(x) $f(0)=2(0)^{3}-3(0)^{2}+a(0)+b$ = 0 - 0 + 0 + b = b .... 1 Substitute x = (-1) in f(x) $f(-1)=2(-1)^{3}-3(-1)^{2}+a(-1)+b$ = -2 - 3 - a + b = 5 - a + b ..... 2 We need to equate equations 1 and 2 to zero b = 0 and 5 a + b = 0 since, the value of b is zero substituteb =...

Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) :

Question: Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) : $3 x-4 y-1=0$ $2 x-\frac{8}{3} y+5=0$ Solution: The given equations are $3 x-4 y \quad-1=0 \quad \ldots \ldots \ldots .(i)$ $2 x-\frac{8}{3} y+5=0$ $6 x-8 y+15=0 \quad \ldots \ldots \ldots .(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 3 \times 0-4 y=1$ $\Rightarrow y=-1 / 4$ $\Rightarrow x=0, \quad y=-1 / 4$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow ...

If x = −1/2 is zero of the polynomial p(x)

Question: If $x=-1 / 2$ is zero of the polynomial $p(x)=8 x^{3}-a x^{2}-x+2$, Find the value of a Solution: We know that, $p(x)=8 x^{3}-a x^{2}-x+2$ Given that the value of x =-1/2 Substitute the value of x in f(x) $p(-1 / 2)=8(-1 / 2)^{3}-a(-1 / 2)^{2}-(-1 / 2)+2$ =- 8(1/8) - a(1/4) + 1/2 + 2 = -1 -(a/4 + 1/2+ 2) = 1 -(a/4 + 1/2) =3/2 a/4 To, find the value of a, equate p(-1/2) to zero p(-1/2) = 0 3/2 - a/4= 0 On taking L.C.M $\frac{6-a}{4}=0$ $\Rightarrow 6-a=0$ $\Rightarrow a=6$...

For any two sets of A and B, prove that:

Question: For any two sets of A and B, prove that: (i) $A^{\prime} \cup B=U \Rightarrow A \subset B$ (ii) $B^{\prime} \subset A^{\prime} \Rightarrow A \subset B$ Solution: (i) Let $a \in A$. $\Rightarrow a \in \mathrm{U}$ $\Rightarrow a \in A^{\prime} \cup B \quad\left(\because U=A^{\prime} \cup B\right)$ $\Rightarrow a \in B \quad\left(\because a \notin A^{\prime}\right)$ Hence, $A \subset B$ (ii) Let $a \in A$. $\Rightarrow a \notin A^{\prime}$ $\Rightarrow a \notin B^{\prime} \quad\left(\beca...

For any two sets of A and B, prove that:

Question: For any two sets of A and B, prove that: (i) $A^{\prime} \cup B=U \Rightarrow A \subset B$ (ii) $B^{\prime} \subset A^{\prime} \Rightarrow A \subset B$ Solution: (i) Let $a \in A$. $\Rightarrow a \in \mathrm{U}$ $\Rightarrow a \in A^{\prime} \cup B \quad\left(\because U=A^{\prime} \cup B\right)$ $\Rightarrow a \in B \quad\left(\because a \notin A^{\prime}\right)$ Hence, $A \subset B$ (ii) Let $a \in A$. $\Rightarrow a \notin A^{\prime}$ $\Rightarrow a \notin B^{\prime} \quad\left(\beca...

Show that the height of the cylinder of maximum volume

Question: Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2 R}{\sqrt{3}}$. Also find the maximum volume. Solution: A sphere of fixed radius (R)is given. Letrandhbe the radius and the height of the cylinder respectively. From the given figure, we have $h=2 \sqrt{R^{2}-r^{2}}$. The volume $(V)$ of the cylinder is given by, $V=\pi r^{2} h=2 \pi r^{2} \sqrt{R^{2}-r^{2}}$ $\therefore \frac{d V}{d r}=4 \pi r \sqrt{R^{2}-r^{2}}+\frac{2 \p...

If x = 2 is a root of the polynomial f

Question: If $x=2$ is a root of the polynomial $f(x)=2 x^{2}-3 x+7 a$, Find the value of a Solution: We know that, $f(x)=2 x^{2}-3 x+7 a$ Given that x = 2 is the root of f(x) Substitute the value of x in f(x) $f(2)=2(2)^{2}-3(2)+7 a$ = (2 * 4) - 6 + 7a = 8 - 6 + 7a = 7a + 2 Now, equate 7a + 2 to zero ⟹ 7a + 2 = 0 ⟹ 7a = - 2 ⟹ a =- 27 The value of a = - 2/7...

Using properties of sets, show that for any two sets A and B,

Question: Using properties of sets, show that for any two setsAandB, $(A \cup B) \cap\left(A \cap B^{\prime}\right)=A$ Solution: LHS $=(A \cup B) \cup\left(A \cap B^{\prime}\right)$ $\Rightarrow \mathrm{LHS}=\{(A \cup B) \cap A\} \cup\left\{(A \cup B) \cap B^{\prime}\right\}$ $\Rightarrow \mathrm{LHS}=\{(A \cup B) \cap A\} \cup\left\{(A \cup B) \cap B^{\prime}\right\}$ $\Rightarrow \mathrm{LHS}=A \cup\left\{(A \cup B) \cap B^{\prime}\right\}$ $\Rightarrow \mathrm{LHS}=A \cup\left\{\left(A \cap B...

Solve this

Question: 1. $f(x)=3 x+1, x=-1 / 3$ 2. $f(x)=x^{2}-1, x=(1,-1)$ 3. $g(x)=3 x^{2}-2, x=\left(\frac{2}{\sqrt{3}}, \frac{-2}{\sqrt{3}}\right)$ 4. $p(x)=x^{3}-6 x^{2}+11 x-6, x=1,2,3$ 5. $f(x)=5 x-\pi, x=4 / 5$ 6. $f(x)=x^{2}, x=0$ 7. $f(x)=1 x+m, x=-m / 1$ 8. $f(x)=2 x+1, x=1 / 2$ Solution: 1. f(x) = 3x + 1, x = 1/3 we know that , f(x) = 3x + 1 substitute x =1/3in f(x) f(1/3) = 3(1/3) + 1 = -1 + 1 = 0 Since, the result is $0 x=-1 / 3$ is the root of $3 x+1$ 2. $f(x)=x^{2}-1, x=(1,-1)$ we know that,...

Let f be a function defined on [a, b] such that

Question: Letfbe a function defined on [a,b] such thatf'(x) 0, for allx (a,b). Then prove thatfis an increasing function on (a,b). Solution: Let $x_{1}, x_{2} \in(a, b)$ such that $x_{1}x_{2}$. Consider the sub-interval $\left[x_{1}, x_{2}\right] .$ Since $f(x)$ is differentiable on $(a, b)$ and $\left[x_{1}, x_{2}\right] \subset(a, b)$. Therefore, $\mathrm{f}(\mathrm{x})$ is continous on $\left[x_{1}, x_{2}\right]$ and differentiable on $\left(x_{1}, x_{2}\right)$. By the Lagrange's mean value ...

If A and B are sets, then prove that A−B,

Question: If $A$ and $B$ are sets, then prove that $A-B, \mid A \cap B$ and $B-A$ are pair wise disjoint. Solution: (i) $(A-B)$ and $(A \cap B)$ Let $a \in A-B$ $\Rightarrow a \in \mathrm{A}$ and $a \notin B$ $\Rightarrow a \notin A \cap B$ Hence, $(A-B)$ and $A \cap B$ are disjoint sets. (ii) $(B-A)$ and $(A \cap B)$ Let $a \in B-A$ $\Rightarrow a \in B$ and $a \notin A$ $\Rightarrow a \notin \mathrm{A} \cap \mathrm{B}$ Hence, $(B-A)$ and $A \cap B$ a re disjoint sets. (iii) $(A-B)$ and $(B-A)$...

If A and B are sets, then prove that A−B,

Question: If $A$ and $B$ are sets, then prove that $A-B, \mid A \cap B$ and $B-A$ are pair wise disjoint. Solution: (i) $(A-B)$ and $(A \cap B)$ Let $a \in A-B$ $\Rightarrow a \in \mathrm{A}$ and $a \notin B$ $\Rightarrow a \notin A \cap B$ Hence, $(A-B)$ and $A \cap B$ are disjoint sets. (ii) $(B-A)$ and $(A \cap B)$ Let $a \in B-A$ $\Rightarrow a \in B$ and $a \notin A$ $\Rightarrow a \notin \mathrm{A} \cap \mathrm{B}$ Hence, $(B-A)$ and $A \cap B$ a re disjoint sets. (iii) $(A-B)$ and $(B-A)$...

Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) :

Question: Show graphically that each one of the following systems of equations is in-consistent (i.e. has no solution) : $2 y-x=9$ $6 y-3 x=21$ Solution: The given equations are $2 y-x=9$$.(i)$ $6 y-3 x=21$(ii) Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 y-0=9$ $\Rightarrow y=9 / 2$ $\Rightarrow x=0, \quad y=9 / 2$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow 2 y-0=9$ $\Rightarrow y=9 / 2$ $\Rightarrow x=0, \quad y=9 / 2$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow 2...

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