In a school there are 20 teachers who teach mathematics or physics.
Question: In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics? Solution: LetAbe the number of teachers who teach mathematics Bbe the number of teachers who teach physics. Given : $n(A)=12$ $n(A \cup B)=20$ $n(A \cap B)=4$ To find: $n(B)$ We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 20=12+n(B)-4$ $\Rightarrow n(B)=20-8=12$ Therefore, 12 teachers teach physics....
Read More →Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are :
Question: Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are : (i) $y=x, y=2 x$ and $y+x=6$ (ii) $y=x, 3 y=x, x+y=8$ Solution: (i)The given equations are $y=x$....(i) $y=2 x$...(ii) $y+x=6$...(iii) It is seen that the coordinates of the vertices of the obtained triangle are $A(0,0), \mathrm{B}(2,4), \mathrm{C}(3,3)$ (ii)The given equations are $y=x$....(1) $3 y=x$...(11) $x+y=8$....(111) The two points satisfying (i) can be listed in a table as,...
Read More →The points on the curve
Question: The points on the curve $9 y^{2}=x^{3}$, where the normal to the curve makes equal intercepts with the axes are (A) $\left(4, \pm \frac{8}{3}\right)$ (B) $\left(4, \frac{-8}{3}\right)$ (C) $\left(4, \pm \frac{3}{8}\right)$ (D) $\left(\pm 4, \frac{3}{8}\right)$ Solution: The equation of the given curve is $9 y^{2}=x^{3}$. Differentiating with respect tox, we have: $9(2 y) \frac{d y}{d x}=3 x^{2}$ $\Rightarrow \frac{d y}{d x}=\frac{x^{2}}{6 y}$ The slope of the normal to the given curve ...
Read More →If polynomials ax3 + 3x2 − 3
Question: If polynomials ax $^{3}+3 x^{2}-3$ and $2 x^{3}-5 x+a$ when divided by $(x-4)$ leave the remainders as $R_{1}$ and $R_{2}$ respectively. Find the values of a in each of the following cases, if 1. $R_{1}=R_{2}$ 2. $R_{1}+R_{2}=0$ 3. $2 R_{1}-R_{2}=0$ Solution: Here, the polynomials are $f(x)=a x_{3}+3 x_{2}-3$ $p(x)=2 x^{3}-5 x+a$ Let, R1 is the remainder when f(x) is divided by x - 4 $\Rightarrow R_{1}=f(4)$ $\Rightarrow \mathrm{R}_{1}=\mathrm{a}(4)^{3}+3(4)^{2}-3$ $=64 \mathrm{a}+48-3...
Read More →If P and Q are two sets such that P has 40 elements,
Question: If $P$ and $Q$ are two sets such that $P$ has 40 elements, $P \cup Q$ has 60 elements and $P \cap Q$ has 10 elements, how many elements does $Q$ have? Solution: Given: $n(P)=40$ $n(P \cup Q)=60$ $n(P \cap Q)=10$ To find : $n(Q)$ We know : $n(P \cup Q)=n(P)+n(Q)-n(P \cap Q)$ $\Rightarrow 60=40+n(Q)-10$ $\Rightarrow n(Q)=30$...
Read More →Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.
Question: Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis ofy. (i) $2 x-5 y+4=0$ $2 x+y-8=0$ (ii) $3 x+2 y=12$ $5 x-2 y=4$ (iii) $2 x+y-11=0$, $x-y-1=0$ (iv) $x+2 y-7=0$ $2 x-y-4=0$ (v) $3 x+y-5=0$ $2 x-y-5=0$ (vi) $2 x-y-5=0$, $x-y-3=0$ Solution: (i) The given equations are $2 x-5 y+4=0$...(i) $2 x+y-8=0$ ..(ii) The two points satisfying (i) can be listed in a table as, The two points satisfying (ii) can be l...
Read More →If A and B are two sets such that
Question: If $A$ and $B$ are two sets such that $n(A \cup B)=50, n(A)=28$ and $n(B)=32$, find $n(A \cap B)$. Solution: We know: $n(A \cup B)=n(A)+n(B)-n(A \cap B)$ $\Rightarrow 50=28+32-n(A \cap B)$ $\Rightarrow n(A \cap B)=60-50=10$...
Read More →The normal to the curve
Question: The normal to the curve $x^{2}=4 y$ passing $(1,2)$ is (A)x+y= 3 (B)xy= 3 (C)x+y= 1 (D)xy= 1 Solution: The equation of the given curve is $x^{2}=4 y$. Differentiating with respect tox, we have: $2 x=4 \cdot \frac{d y}{d x}$ $\Rightarrow \frac{d y}{d x}=\frac{x}{2}$ The slope of the normal to the given curve at point (h,k) is given by, $\frac{-1}{\left.\frac{d y}{d x}\right]_{(h, k)}}=-\frac{2}{h}$ Equation of the normal at point (h,k) is given as: $y-k=\frac{-2}{h}(x-h)$ Now, it is giv...
Read More →For any two sets A and B, prove that
Question: For any two setsAandB, prove that (i) $(A \cup B)-B=A-B$ (ii) $A-(A \cap B)=A-B$ (iii) $A-(A-B)=A \cap B$ (iv) $A \cup(B-A)=A \cup B$ [NCERT EXEMPLAR] (v) $(A-B) \cup(A \cap B)=A$ [NCERT EXEMPLAR] Solution: (i) $(A \cup B)-B=(A \cup B) \cap B^{\prime} \quad\left(X-Y=X \cap Y^{\prime}\right)$ $=\left(A \cap B^{\prime}\right) \cup\left(B \cap B^{\prime}\right) \quad$ (Distributive law) $=\left(A \cap B^{\prime}\right) \cup \phi$ $=A \cap B$ $=A-B$ (ii) $A-(A \cap B)=A \cap(A \cap B)^{\pr...
Read More →If the polynomial
Question: If the polynomial $2 x^{3}+a x^{2}+3 x-5$ and $x^{3}+x^{2}-4 x+a$ leave the same remainder when divided by $x-2$, Find the value of a Solution: Given, the polymials are $f(x)=2 x^{3}+a x^{2}+3 x-5$ $p(x)=x^{3}+x^{2}-4 x+a$ The remainders are f(2) and p(2) when f(x) and p(x) are divided by x - 2 We know that, f(2) = p(2) (given in problem) we need tocalculate f(2) and p(2) for, f(2) substitute (x = 2) in f(x) $f(2)=2(2)^{3}+a(2)^{2}+3(2)-5$ = (2 * 8) + 4a + 6 - 5 = 16 + 4a + 1 = 4a + 17...
Read More →For any two sets A and B, prove that
Question: For any two setsAandB, prove that (i) $(A \cup B)-B=A-B$ (ii) $A-(A \cap B)=A-B$ (iii) $A-(A-B)=A \cap B$ (iv) $A \cup(B-A)=A \cup B$ [NCERT EXEMPLAR] (v) $(A-B) \cup(A \cap B)=A$ [NCERT EXEMPLAR] Solution: (i) $(A \cup B)-B=(A \cup B) \cap B^{\prime} \quad\left(X-Y=X \cap Y^{\prime}\right)$ $=\left(A \cap B^{\prime}\right) \cup\left(B \cap B^{\prime}\right) \quad$ (Distributive law) $=\left(A \cap B^{\prime}\right) \cup \phi$ $=A \cap B$ $=A-B$ (ii) $A-(A \cap B)=A \cap(A \cap B)^{\pr...
Read More →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 $f(x)=3 x^{4}+2 x^{3}-x^{3} / 3-x / 9+2 / 27, g(x)=x+2 / 3$ Solution: Here, $f(x)=3 x^{4}+2 x^{3}-x^{3} / 3-x / 9+2 / 27$ g(x) =x + 2/3 from remainder theorem when f(x) is divided by g(x) =x - (-2/3), the remainder is equal to f(- 2/3) substitute the value of x in f(x) $f\left(-\frac{2}{3}\right)=3\left(-\frac{2}{3}\right)^{4}+2\left(-\frac{2}{3}\right)^{3}-\frac{\left(-\frac{...
Read More →The normal at the point
Question: The normal at the point $(1,1)$ on the curve $2 y+x^{2}=3$ is (A) $x+y=0$ (B) $x-y=0$ (C) $x+y+1=0$ (D) $x-y=1$ Solution: The equation of the given curve is $2 y+x^{2}=3$. Differentiating with respect tox, we have: $\frac{2 d y}{d x}+2 x=0$ $\Rightarrow \frac{d y}{d x}=-x$ $\left.\therefore \frac{d y}{d x}\right]_{(1,1)}=-1$ The slope of the normal to the given curve at point (1, 1) is $\frac{-1}{\left.\frac{d y}{d x}\right]_{(1,1)}}=1$ Hence, the equation of the normal to the given cu...
Read More →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 $f(x)=9 x^{3}-3 x^{2}+x-5, g(x)=x-2 / 3$ Solution: Here, $f(x)=9 x^{3}-3 x^{2}+x-5$ g(x) =x 2/3 from, the remainder theorem when f(x) is divided by g(x) = x -2/3the remainder will be equal to f(2/3) substitute the value of x in f(x) $f(2 / 3)=9(2 / 3)-3(2 / 3)^{2}+(2 / 3)-5$ =9(8/27) 3(4/9) + 2/3 5 = (8/3) (4/3) + 2/3 5 $=\frac{8-4+2-15}{3}$ $=\frac{10-19}{3}$ =- 9/3 = - 3 The...
Read More →If A, B, C are three sets such that
Question: If $A, B, C$ are three sets such that $A \subset B$, then prove that $C-B \subset C-A$. Solution: Let $a \in C-B$ $\Rightarrow a \in C$ and $a \notin B$ $\Rightarrow a \in C$ and $a \notin A \quad[\because A \subset B]$ $\Rightarrow a \in C-A$ Hence, $C-B \subset C-A$...
Read More →If A, B, C are three sets such that
Question: IfA,B,Care three sets such thatABAB, then prove that$C-B \subset C-A$ Solution: Let $a \in C-B$ $\Rightarrow a \in C$ and $a \notin B$ $\Rightarrow a \in C$ and $a \notin A \quad[\because A \subset B]$ $\Rightarrow a \in C-A$ Hence, $C-B \subset C-A$...
Read More →The line $y=m x+1$ is a tangent to the curve
Question: The line $y=m x+1$ is a tangent to the curve $y^{2}=4 x$ if the value of $m$ is (A) 1 (B) 2 (C) 3 (D) $\frac{1}{2}$ Solution: The equation of the tangent to the given curve is $y=m x+1$. Now, substituting $y=m x+1$ in $y^{2}=4 x$, we get: $\Rightarrow(m x+1)^{2}=4 x$ $\Rightarrow m^{2} x^{2}+1+2 m x-4 x=0$ $\Rightarrow m^{2} x^{2}+x(2 m-4)+1=0$ ...(1) Since a tangent touches the curve at one point, the roots of equation (i) must be equal. Therefore, we have: Discriminant $=0$ $(2 m-4)^...
Read More →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 $f(x)=x^{4}-3 x^{2}+4, g(x)=x-2$ Solution: Here, $f(x)=x^{4}-3 x^{2}+4$ g(x) = x - 2 from, the remainder theorem when f(x) is divided by g(x) = x - 2 the remainder will be equal to f(2) Let, g(x) = 0 ⟹ x - 2 = 0 ⟹ x = 2 Substitute the value of x in f(x) $f(2)=2^{4}-3(2)^{2}+4$ = 16 - (3* 4) + 4 = 16 - 12 + 4 = 20 - 12 = 8 Therefore, the remainder is 8...
Read More →For any two sets A and B, prove the following:
Question: For any two setsAandB, prove the following: (i) $A \cap\left(A^{\prime} \cup B\right)=A \cap B$ (ii) $A-(A-B)=A \cap B$ (iii) $A \cap(A \cup B)^{\prime}=\phi$ (iv) $A-B=A \Delta(A \cap B)$. Solution: (i) $\mathrm{LHS}=A \cap\left(A^{\prime} \cup B\right)$ $=\left(A \cap A^{\prime}\right) \cup(A \cap B)$ $=(\phi) \cup(A \cap B)$ $=A \cap B=\mathrm{RHS}$ Hence proved.(ii) LHS $=A-(A-B)$ $=A-\left(A \cap B^{\prime}\right)$ $=A \cap\left(A \cap B^{\prime}\right)$ $=A \cap\left(A \cap B^{\p...
Read More →Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not :
Question: Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not : (i) $2 x-3 y=6, x+y=1$ (ii) $2 y=4 x-6,2 x=y+3$ Solution: (i) The given equations are $2 x-3 y=6$$.(i)$ $x+y=1$$. .(i i)$ Putting $x=0$ in equation $(i)$, we get: $\Rightarrow 2 \times 0-3 y=6$ $\Rightarrow y=-2$ $x=0, \quad y=-2$ Putting $y=0$ in equation $(i)$ we get: $\Rightarrow 2 x-3 \times 0=6$ $\Rightarrow x=3$ $x=3, \quad y=0$ Use the following table to draw the graph. ...
Read More →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 $f(x)=x^{3}-6 x^{2}+2 x-4, g(x)=1-2 x$ Solution: Here, $f(x)=x^{3}-6 x^{2}+2 x-4$ g(x) = 1 - 2x from, the remainder theorem when f(x) is divided by g(x) = -2(x -1/2), the remainder is equal to f(1/2) Let, g(x) = 0 ⟹ 1 - 2x= 0 ⟹ -2x = -1 ⟹ 2x = 1 ⟹ x =1/2 Substitute the value of x in f(x) $f(1 / 2)=(1 / 2)^{3}-6(1 / 2)^{2}+2(1 / 2)-4$ =1/8 - 8(1/4) + 2(1/2) - 4 =1/8 - (1/2) + 1...
Read More →For any two sets A and B, prove the following:
Question: For any two setsAandB, prove the following: (i) $A \cap\left(A^{\prime} \cup B\right)=A \cap B$ (ii) $A-(A-B)=A \cap B$ (iii) $A \cap(A \cup B)^{\prime}=\phi$ (iv) $A-B=A \Delta(A \cap B)$. Solution: (i) $\mathrm{LHS}=A \cap\left(A^{\prime} \cup B\right)$ $=\left(A \cap A^{\prime}\right) \cup(A \cap B)$ $=(\phi) \cup(A \cap B)$ $=A \cap B=\mathrm{RHS}$ Hence proved.(ii) LHS $=A-(A-B)$ $=A-\left(A \cap B^{\prime}\right)$ $=A \cap\left(A \cap B^{\prime}\right)$ $=A \cap\left(A \cap B^{\p...
Read More →The slope of the tangent to the curve
Question: The slope of the tangent to the curve $x=t^{2}+3 t-8, y=2 t^{2}-2 t-5$ at the point $(2,-1)$ is (A) $\frac{22}{7}$ (B) $\frac{6}{7}$ (C) $\frac{7}{6}$ (D) $\frac{-6}{7}$ Solution: The given curve is $x=t^{2}+3 t-8$ and $y=2 t^{2}-2 t-5$. $\therefore \frac{d x}{d t}=2 t+3$ and $\frac{d y}{d t}=4 t-2$ $\therefore \frac{d y}{d x}=\frac{d y}{d t} \cdot \frac{d t}{d x}=\frac{4 t-2}{2 t+3}$ The given point is (2, 1). Atx= 2, we have: $t^{2}+3 t-8=2$ $\Rightarrow t^{2}+3 t-10=0$ $\Rightarrow(...
Read More →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 $f(x)=4 x^{3}-12 x^{2}+14 x-3, g(x)=2 x-1$ Solution: Here, $f(x)=4 x^{3}-12 x^{2}+14 x-3$ g(x) = 2x - 1 from, the remainder theorem when f(x) is divided by g(x) = 2(x -1/2), the remainder is equal to f(1/2) Let, g(x) = 0 ⟹2x - 1 = 0 ⟹ 2x = 1 ⟹ x =1/2 Substitute the value of x in f(x) $f(1 / 2)=4(1 / 2)^{3}-12(1 / 2)^{2}+14(1 / 2)-3$ =4(1/8) - 12(1/4) + 4(1/2) - 3 =(1/2)- 3 + 7...
Read More →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 $f(x)=2 x^{4}-6 x^{3}+2 x^{2}-x+2, g(x)=x+2$ Solution: Here, $f(x)=2 x^{4}-6 x^{3}+2 x^{2}-x+2$ g(x) = x + 2 from, the remainder theorem when f(x) is divided by g(x) = x - (-2) the remainder will be equal to f(-2) Let, g(x) = 0 ⟹ x + 2 = 0 ⟹ x = - 2 Substitute the value of x in f(x) $f(-2)=2(-2)^{4}-6(-2)^{3}+2(-2)^{2}-(-2)+2$ = (2 * 16) - (6 * (-8)) + (2 * 4) + 2 + 2 = 32 + 4...
Read More →