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

Draw the graph of the equation 2x + y = 6. Find the coordinates of the point,

Question: Draw the graph of the equation 2x+y= 6. Find the coordinates of the point, where the graph cuts thex-axis. Solution: Given equation: $2 x+y=6$ $\Rightarrow y=6-2 x$ When, $x=0, y=6-0=6$. When, $x=1, y=6-2=4$ When, $x=2, y=6-4=2$. Thus, we have the following table Plot the points $(0,6),(1,4)$ and $(2,2)$ on the graph paper. Join these points and extend the line. Clearly, the graph cuts the $x-a x i s$ at $P(3,0)$....

If x = 1 is a common root of ax2 + ax + 2 = 0 and x2 + x + b = 0, then, ab =

Question: If $x=1$ is a common root of $a x^{2}+a x+2=0$ and $x^{2}+x+b=0$, then, $a b=$ (a) 1(b) 2(c) 4(d) 3 Solution: $x=1$ is the common roots given quadric equation are $a x^{2}+a x+2=0$, and $x^{2}+x+b=0$ Then find the value of $a b$. Here, $a x^{2}+a x+2=0$.........(1) $x^{2}+x+b=0$......(2) Putting the value of $x=1$ in equation (2) we get $1^{2}+1+b=0$ $2+b=0$ $b=-2$ Now, putting the value of $x=1$ in equation (1) we get $a+a+2=0$ $2 a+2=0$ $a=\frac{-2}{2}$ $=-1$ $a b=(-1) \times(-2)$ Th...

Find the total number of permutations of the letters of the word 'INSTITUTE'.

Question: Find the total number of permutations of the letters of the word 'INSTITUTE'. Solution: The word 'INSTITUTE' consists of 9 letters including two Is and three Ts. Total number of words that can be formed of the word INSTITUTE = Number of arrangements of 9 things of which 2 are similar to the first kind and 3 are similar to the second kind $=\frac{9 !}{2 ! 3 !}$...

Draw the graph of the equation 2x − 3y = 5. From the graph

Question: Draw the graph of the equation 2x 3y= 5. From the graph, find (i) the value ofywhenx= 4 and (ii) the value ofxwheny= 3. Solution: Given equation : $2 x-3 y=5$ $\Rightarrow 2 x=3 y+5$ $\Rightarrow x=\frac{3 y+5}{2}$ When, $y=-1, x=\frac{-3+5}{2}=\frac{2}{2}=1$ When, $y=-3, x=\frac{-9+5}{2}=\frac{-4}{2}=-2$ Thus, we have the following table: Plot the points $(-2,-3),(1,-1)$ on the graph paper and extend the line in both directions. (i) When x = 4: $4=\frac{3 y+5}{2} \Rightarrow 8=3 y+5$ ...

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together?

Question: In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's are together? Solution: The word ASSASSINATION consists of 13 letters including three As, four Ss, two Ns and two Is. Considering all the Ss are together or as a single letter, we are left with 10 letters. Out of these, there are three As, two Ns and two Is. Number of words in which all the Ss are together = Permutations of 10 letters of which three are similar to the first kind, two are similar ...

How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?

Question: How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4? Solution: Numbers greater than a million can be formed when the first digit can be any one out of the given digits 1, 2, 0, 2, 4, 2, 4, except 0. Number of arrangements of the given digits $1,2,0,2,4,2,4=$ Arrangements of 7 things of which 3 are similar to the first kind, and 2 are similar to the second kind $=\frac{7 !}{2 ! 3 !}$ But, these arrangements also include the numbers in which the fi...

If the sum of the roots of the equation x2 − x = λ(2x − 1) is zero, then λ =

Question: If the sum of the roots of the equation $x^{2}-x=\lambda(2 x-1)$ is zero, then $\lambda=$ (a) $-2$ (b) 2 (c) $-\frac{1}{2}$ (d) $\frac{1}{2}$ Solution: The given quadric equation is $x^{2}-x=\lambda(2 x-1)$, and roots are zero. Then find the value of $\lambda$. Here, $x^{2}-x=\lambda(2 x-1)$ $x^{2}-x-2 \lambda x+\lambda=0$ $x^{2}-(1+2 \lambda) x+\lambda=0$ $a=1, b=-(1+2 \lambda)$ and, $c=\lambda$ As we know that $D=b^{2}-4 a c$ Putting the value of $a=1, b=-(1+2 \lambda)$ and, $c=\lamb...

Draw the graph of the equation x + 2y – 3 = 0.

Question: Draw the graph of the equationx+ 2y 3 = 0.From your graph, find the value ofywhen(i)x= 5(ii)x= 5. Solution: Given equation: x + 2y - 3 = 0 Or, x + 2y = 3When y= 0, x + 0 = 3⇒ x = 3When y = 1, x + 2 = 3⇒ x = 3-2 = 1When y = 2, x + 4 = 3⇒x = 3 - 4 = -1Thus, we have the following table: Now plot the points (3,0) ,(1,1) and (-1,2) on the graph paper.Join the points and extend the line in both the directions.The line segment is the required graph of the equation. When x = 5, $y=\frac{3-x}{2...

In how many ways can 4 red,

Question: In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable? Solution: Number of red discs = 4 Number of yellow discs = 3 Number of green discs = 2 Total number of discs = 9 Total number of arrangements = Number of arrangements of 9 things of which 4 are similar to the first kind, 3 are similar to the second kind and 2 are similar to the third kind $=\frac{9 !}{4 ! 3 ! 2 !}=1260$...

If (a2+b2)x2+2(ab+bd)x+c2+d2=0 has no real roots, then

Question: If $\left(a^{2}+b^{2}\right) x^{2}+2(a b+b d) x+c^{2}+d^{2}=0$ has no real roots, then (a)ab=bc(b)ab=cd(c)ac=bd(d)adbc Solution: The given quadric equation is $\left(a^{2}+b^{2}\right) x^{2}+2(a b+b d) x+c^{2}+d^{2}=0$, and roots are equal. Here, $a=\left(a^{2}+b^{2}\right), b=2(a b+b d)$ and, $c=c^{2}+d^{2}$ As we know that $D=b^{2}-4 a c$ Putting the value of $a=\left(a^{2}+b^{2}\right), b=2(a b+b d)$ and, $c=c^{2}+d^{2}$ $=\{2(a b+b d)\}^{2}-4 \times\left(a^{2}+b^{2}\right) \times\l...

Draw the graph of the equation y = 3x.

Question: Draw the graph of the equationy= 3x.From your graph, find the value ofywhen(i)x= 2(ii)x= 2. Solution: Given equation: y = 3x When x = -2, y = -6.When x = -1, y = -3.Thus, we have the following table: Now plot the points (-2,-6), (-1, -3) on a graph paper.Join the points and extend the line in both the directions.The linesegment is the required graph of the equation.From the graph we can see that whenx= 2,y= 6 Also, when x = -2, y = -6....

A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain.

Question: A biologist studying the genetic code is interested to know the number of possible arrangements of 12 molecules in a chain. The chain contains 4 different molecules represented by the initialsA(for Adenine),C(for Cytosine),G(for Guanine) andT(for Thymine) and 3 molecules of each kind. How many different such arrangements are possible? Solution: Number of molecules in a chain = 12 Number of molecules with initials A = 3 Number of molecules with initials C = 3 Number of molecules with in...

Draw the graph of each of the following equations.

Question: Draw the graph of each of the following equations.(i)x= 4(ii)x+ 4 = 0(iii)y= 3(iv)y= 3(v)x= 2(vi)x= 5(vii)y+ 5 = 0(viii)y= 4 Solution: (i) The equation of given line isx= 4. This equation does not contain the term ofy. So, the graph of this line is parallel toy-axis passing through the point (4, 0). (ii)The equation of given line isx+ 4 = 0 orx=4. This equation does not contain the term ofy.So, the graph of this line is parallel toy-axis passing through the point (4, 0). (iii)The equat...

The number of quadratic equations having real roots and which do not change by squaring their roots is

Question: The number of quadratic equations having real roots and which do not change by squaring their roots is (a) 4(b) 3(c) 2(d) 1 Solution: As we know that the number of quadratic equations having real roots and which do not change by squaring their roots is 2 . Thus, the correct answer is $(c)$...

How many different arrangements can be made by using all the letters in the word 'MATHEMATICS'.

Question: How many different arrangements can be made by using all the letters in the word 'MATHEMATICS'. How many of them begin with C? How many of them begin with T? Solution: The word MATHEMATICS consists of 11 letters that include two Ms, two As, and two Ts. Total number of arrangements of the letters of the word MATHEMATICS $=\frac{11 !}{2 ! 2 ! 2 !}$ Number of words in which the first word is fixed as $C=$ Number of arrangements of the remaining 10 letters, of which there are two As, two M...

If a and b can take values 1, 2, 3, 4. Then the number of the equations

Question: If $a$ and $b$ can take values $1,2,3,4$. Then the number of the equations of the form $a x^{2}+b x+1=0$ having real roots is (a) 10(b) 7(c) 6(d) 12 Solution: Given that the equation $a x^{2}+b x+1=0$. For given equation to have real roots, $\operatorname{discriminant}(D) \geq 0$ $\Rightarrow b^{2}-4 a \geq 0$ $\Rightarrow b^{2} \geq 4 a$ $\Rightarrow b \geq 2 \sqrt{a}$ Now, it is given thataandbcan take the values of 1, 2, 3 and 4. The above condition $b \geq 2 \sqrt{a}$ can be satisf...

There are three copies each of 4 different books.

Question: There are three copies each of 4 different books. In how many ways can they be arranged in a shelf? Solution: Total number of books = 12 $\therefore$ Required number of arrangements $=$ Arrangements of 12 things of which each of the 4 different books has three copies $=\frac{12 !}{3 ! 3 ! 3 ! 3 !}=\frac{12 !}{(3 !)^{4}}$...

If p and q are the roots of the equation x2 − px + q = 0, then

Question: If $p$ and $q$ are the roots of the equation $x^{2}-p x+q=0$, then (a) $p=1, q=-2$ (b) $b=0, q=1$ (c) $p=-2, q=0$ (d) $p=-2, q=1$ Solution: Given that $\rho$ and $q$ be the roots of the equation $x^{2}-p x+q=0$ Then find the value of $p$ and $q$. Here, $a=1, b=-p$ and, $c=q$ pandqbe the roots of the given equation Therefore, sum of the roots $p+q=\frac{-b}{a}$ $=\frac{-p}{1}$ $=-p$ $q=-p-p$......(1) $=-2 p$ Product of the roots $p \times q=\frac{q}{1}$ As we know that $p=\frac{q}{q}$ $...

Find the number of numbers, greater than a million,

Question: Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3. Solution: One million (1,000,000) consists of 7 digits. We have digits 2, 3, 0, 3, 4, 2 and 3. Numbers formed by arranging all these seven digits $=\frac{7 !}{2 ! 3 !}$ But, these numbers also include the numbers whose first digit is 0. This is invalid as in that case the number would be less than a million. Total numbers in which the first digit is fixed as $0=$ Permutations of ...

How many permutations of the letters of the word 'MADHUBANI'

Question: How many permutations of the letters of the word 'MADHUBANI' do not begin with M but end with I? Solution: Number of words that only end with $I=$ Number of permutations of the remaining 8 letters, taken all at a time $=\frac{8 !}{2 !}$ Number of words that start with $M$ and end with $I=$ Permutations of the remaining 7 letters, taken all at a time $=\frac{7 !}{2 !}$ Number of words that do not begin with M but end with I = Number of words that only end with I - Number of words that s...

If x = 1 is a common roots of the equations ax2 + ax + 3 = 0 and x2 + x + b = 0, then ab =

Question: If $x=1$ is a common roots of the equations $a x^{2}+a x+3=0$ and $x^{2}+x+b=0$, then $a b=$ (a) 3(b) 3.5(c) 6(d) 3 Solution: $x=1$ is the common roots given quadric equation are $a x^{2}+a x+3=0$, and $x^{2}+x+b=0$ Then find the value of $q$. Here, $a x^{2}+a x+3=0$........(1) $x^{2}+x+b=0 \ldots$ (2) Putting the value of $x=1$ in equation (1) we get $a \times 1^{2}+a \times 1+3=0$ $a+a+3=0$ $2 a=-3$ $a=-\frac{3}{2}$ Now, putting the value of $x=1$ in equation (2) we get $1^{2}+1+b=0$...

How many words can be formed from the letters of the word 'SERIES' which start with S and end with S?

Question: How many words can be formed from the letters of the word 'SERIES' which start with S and end with S? Solution: The word SERIES consists of 6 letters including two Ss and two Es. The first and the last letters are fixed as S. Now, the remaining four letters can be arranged in $\frac{4 !}{2 !}$ ways $=12$...

The cost of 5 pencils is equal to the cost of 2 ballpoints.

Question: The cost of 5 pencils is equal to the cost of 2 ballpoints. Write a linear equation in two variables to represent this statement. (Take the cost of a pencil to be ₹xand that of a ballpoint to be ₹y). Solution: Letcost of a pencil to be ₹xand that of a ballpoint to be ₹y.Cost of 5 pencils = 5xCost of 2 ballpoints = 2yCost of 5 pencils = cost of 2 ballpoints $\Rightarrow 5 x=2 y$ $\Rightarrow 5 x-2 y=0$...

How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9.

Question: How many different numbers, greater than 50000 can be formed with the digits 0, 1, 1, 5, 9. Solution: Numbers greater than 50000 can either have 5 or 9 in the first place and will consist of 5 digits. Number of arrangements having 5 as the first digit $=\frac{4 !}{2 !}$ Number of arrangement having 9 as the first digit $=\frac{4 !}{2 !}$ $\therefore$ Required arrangements $=\frac{4 !}{2 !}+\frac{4 !}{2 !}=24$...

If x = 3k + 2 and y = 2k – 1 is a solution of the equation

Question: Ifx= 3k+ 2 andy= 2k 1 is a solution of the equation 4x 3y+ 1 = 0, find the value ofk. Solution: Given: 4x 3y+ 1 = 0 .....(1)x= 3k+ 2 andy= 2k 1Putting these values in the equation (1) we get $4(3 k+2)-3(2 k-1)+1=0$ $\Rightarrow 12 k+8-6 k+3+1=0$ $\Rightarrow 6 k+12=0$ $\Rightarrow k+2=0$ $\Rightarrow k=-2$...

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