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

A quadratic equation whose one root is 2 and

Question: A quadratic equation whose one root is 2 and the sum of whose roots is zero, is (a) $x^{2}+4=0$ (b) $x^{2}-4=0$ (c) $4 x^{2}-1=0$ (d) $x^{2}-2=0$ Solution: Let $\alpha$ and $\beta$ be the roots of quadratic equation in such a way that $\alpha=2$ Then, according to question sum of the roots $\alpha+\beta=0$ $2+\beta=0$ $\beta=-2$ And the product of the roots $\alpha \cdot \beta=2 \times(-2)$ $=-4$ As we know that the quadratic equation $x^{2}-(\alpha+\beta) x+\alpha \beta=0$ Putting the...

How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3 ?

Question: How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3 ? (a) 420 (b) 360 (c) 400 (d) 300 Solution: (b) 360 10 lakhs consists of seven digits. Number of arrangements of seven numbers of which 2 are similar of first kind, 3 are similar of second kind $=\frac{7 !}{2 ! 3 !}$ But, these numbers also include the numbers in which the first digit has been considered as 0. This will result in a number less than 10 lakhs. Thus, we need to subtract all those numbers. Numbers in ...

How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3 ?

Question: How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3 ? (a) 420 (b) 360 (c) 400 (d) 300 Solution: (b) 360 10 lakhs consists of seven digits. Number of arrangements of seven numbers of which 2 are similar of first kind, 3 are similar of second kind $=\frac{7 !}{2 ! 3 !}$ But, these numbers also include the numbers in which the first digit has been considered as 0. This will result in a number less than 10 lakhs. Thus, we need to subtract all those numbers. Numbers in ...

If the sum of the roots of the equation x2−(k+6)x+2(2k−1)=0

Question: If the sum of the roots of the equation $x^{2}-(k+6) x+2(2 k-1)=0$ is equal to half of their product, then $k=$ (a) 6(b) 7(c) 1(d) 5 Solution: The given quadric equation is $x^{2}-(k+6) x+2(2 k-1)=0$, and roots are equal Then find the value ofk. Let $\alpha$ and $\beta$ be two roots of given equation And, $a=1, b=-(k+6)$ and,$c=2(2 k-1)$ Then, as we know that sum of the roots $\alpha+\beta=\frac{-b}{a}$ $\alpha+\beta=\frac{-\{-(k+6)\}}{1}$ $=(k+6)$ And the product of the roots $\alpha ...

Draw the graphs of the lines x – y = 1 and 2x + y = 8. Shade the area formed by these two lines and the y-axis.

Question: Draw the graphs of the linesx y= 1 and 2x + y= 8. Shade the area formed by these two lines and they-axis. Also, find this area. Solution: $x-y=1$ $\Rightarrow y=x-1$ When $x=0, y=0-1=-1$ When $x=1, y=1-1=0$ When $x=2, y=2-1=1$ Thus, the points on the linex y= 1 are as given in the following table: Plotting the points (0,1),(1,0) and(2,1) and drawing a line passing through these points, we obtain the graph of of the linex y= 1. $2 x+y=8$ $\Rightarrow y=-2 x+8$ When $x=1, y=-2 \times 1+8...

The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is

Question: The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is (a) 574 (b) 36 (c) 754 (d) 144 Solution: (d) 144 The word ARTICLE consists of 3 vowels that have to be arranged in the three even places. This can be done in 3! ways. And, the remaining 4 consonants can be arranged among themselves in 4! ways. Total number of ways = 3!4! = 144...

The number of five-digit telephone numbers having at least one of their digits repeated is

Question: The number of five-digit telephone numbers having at least one of their digits repeated is (a) 90000 (b) 100000 (c) 30240 (d) 69760 Solution: (d) 69760 Total number of five digit numbers (since there is no restriction of the number $0 X X X X)=10 \times 10 \times 10 \times 10 \times 10=100000$ These numbers also include the numbers where the digits are not being repeated. So, we need to subtract all such numbers. Number of 5 digit numbers that can be formed without any repetition of di...

If 2 is a root of the equation x2 + ax + 12 = 0 and the quadratic

Question: If 2 is a root of the equation $x^{2}+a x+12=0$ and the quadratic equation $x^{2}+a x+q=0$ has equal roots, then $q=$ (a) 12(b) 8(c) 20(d) 16 Solution: $x=2$ is the common roots given quadric equation are $x^{2}+a x+12=0$, and $x^{2}+a x+q=0$ Then find the value ofq. Here, $x^{2}+a x+12=0$......(1) $x^{2}+a x+q=0 \ldots .(2)$ Putting the value of $x=2$ in equation (1) we get $2^{2}+a \times 2+12=0$ $4+2 a+12=0$ $2 a=-16$ $a=-8$ Now, putting the value of $a=-8$ in equation ( 2 ) we get ...

The number of permutations of n different things taking r at a time when 3 particular things are to be included is

Question: The number of permutations ofndifferent things takingrat a time when 3 particular things are to be included is (a) $^{n-3} P_{r-3}$ (b) $^{n-3} P_{r}$ (c) ${ }^{n} P_{r-3}$ (d) $r !^{n-3} C_{r-3}$ Solution: (d) $r !^{n-3} C_{r-3}$ Here, we have to permutenthings of which 3 things are to be included. So, only the remaining $(n-3)$ things are left for permutation, taking $(r-3)$ things at a time. This is because 3 things have already been included. But, these $r$ things can be arranged i...

The number of permutations of n different things taking r at a time when 3 particular things are to be included is

Question: The number of permutations ofndifferent things takingrat a time when 3 particular things are to be included is (a) $^{n-3} P_{r-3}$ (b) $^{n-3} P_{r}$ (c) ${ }^{n} P_{r-3}$ (d) $r !^{n-3} C_{r-3}$ Solution: (d) $r !^{n-3} C_{r-3}$ Here, we have to permutenthings of which 3 things are to be included. So, only the remaining $(n-3)$ things are left for permutation, taking $(r-3)$ things at a time. This is because 3 things have already been included. But, these $r$ things can be arranged i...

Draw the graphs of the lines 2x + y = 6 and 2x – y + 2 = 0. Shade the region bounded by these lines and the x-axis.

Question: Draw the graphs of the lines 2x + y= 6 and 2x y+ 2 = 0. Shade the region bounded by these lines and thex-axis. Find the area of the shaded region. Solution: $2 x+y=6$ $\Rightarrow y=-2 x+6$ When $x=0, y=-2 \times 0+6=0+6=6$ When $x=1, y=-2 \times 1+6=-2+6=4$ When $x=2, y=-2 \times 2+6=-4+6=2$ Thus, the points on the line2x + y = 6are as given in the following table: Plotting the points (0,6),(1,4) and(2,2) and drawing a line passing through these points, we obtain the graph of of the l...

The letters of the word 'ZENITH' are written in all possible orders.

Question: The letters of the word 'ZENITH' are written in all possible orders. How many words are possible if all these words are written out as in a dictionary? What is the rank of the word 'ZENITH'? Solution: In a dictionary, the words are arranged in alphabetical order. Therefore, in the given problem, we must consider the words beginning with E, H, I, N, T and Z. Number of words starting with E = 5! = 120 Number of words starting with H = 5! = 120 Number of words starting with I = 5! = 120 N...

Draw the graph of the line 4x + 3y = 24.

Question: Draw the graph of the line 4x +3y= 24.(i) Write the coordinates of the points where this line intersects thex-axis and they-axis.(ii) Use this graph to find the area of the triangle formed by the graph line and the coordinate axes. Solution: $4 x+3 y=24$ $\Rightarrow 3 y=-4 x+24$ $\Rightarrow y=\frac{-4 x+24}{3}$ When $x=0, y=\frac{-4 \times 0+24}{3}=\frac{0+24}{3}=\frac{24}{3}=8$ When $x=3, y=\frac{-4 \times 3+24}{3}=\frac{-12+24}{3}=\frac{12}{3}=4$ When $x=6, y=\frac{-4 \times 6+24}{...

If sin α and cos α are the roots of the equations ax2 + bx + c = 0, then b2 =

Question: If $\sin \alpha$ and $\cos \alpha$ are the roots of the equations $a x^{2}+b x+c=0$, then $b^{2}=$ (a) $a^{2}-2 a c$ (b) $a^{2}+2 a c$ (c) $a^{2}-a c$ (d) $a^{2}+a c$ Solution: The given quadric equation is $a x^{2}+b x+c=0$, and $\sin \alpha$ and $\cos \beta$ are roots of given equation. And, $a=a, b=b$ and, $c=c$ Then, as we know that sum of the roots $\sin \alpha+\cos \beta=\frac{-b}{a} \ldots$(1) And the product of the roots $\sin \alpha \cdot \cos \beta=\frac{c}{a} \ldots$(2) Squa...

In how many ways can the letters of the word "INTERMEDIATE" be arranged so that:

Question: In how many ways can the letters of the word "INTERMEDIATE" be arranged so that: (i) the vowels always occupy even places? (ii) the relative order of vowels and consonants do not alter? Solution: The word INTERMEDIATE consists of 12 letters that include two Is, two Ts and three Es. (i) There are 6 vowels $(I, I, E, E, E$ and $A)$ that are to be arranged in six even places $=\frac{6 !}{2 ! 3 !}=60$ The remaining 6 consonants can be arranged amongst themselves in $\frac{6 !}{2 !}$ ways, ...

If the sum and product of the roots of the equation kx2 + 6x + 4k = 0 are real, then k =

Question: If the sum and product of the roots of the equation $k x^{2}+6 x+4 k=0$ are real, then $k=$ (a) $-\frac{3}{2}$ (b) $\frac{3}{2}$ (c) $\frac{2}{3}$ (d) $-\frac{2}{3}$ Solution: The given quadric equation is $k x^{2}+6 x+4 k=0$, and roots are equal Then find the value of $c$ Let $\alpha$ and $\beta$ be two roots of given equation And, $a=k, b=6$ and,$c=4 k$ Then, as we know that sum of the roots $\alpha+\beta=\frac{-b}{a}$ $\alpha+\beta=\frac{-6}{k}$ And the product of the roots $\alpha ...

Find the total number of ways in which six '+' and four '−' signs can be arranged in a line such that no two '−' signs occur together.

Question: Find the total number of ways in which six '+' and four '' signs can be arranged in a line such that no two '' signs occur together. Solution: Six 't' signs can be arranged in a row in $\frac{6 !}{6 !}=1$ way Now, we are left with seven places in which four different things can be arranged in ${ }^{7} \mathrm{P}_{4}$ ways. Since all the four '-' signs are identical, four '-' signs can be arranged in $\frac{{ }^{7} P_{4}}{4 !}$ ways, i.e. 35 ways. Number of ways $=1 \times 35=35$...

Draw the graphs of the equations 3x – 2y = 4 and x + y – 3 = 0. On the same graph paper,

Question: Draw the graphs of the equations 3x 2y= 4 andx + y 3 = 0. On the same graph paper, find the coordinates of the point were the two graph lines intersect. Solution: $3 x-2 y=4$ $\Rightarrow 2 y=3 x-4$ $\Rightarrow y=\frac{3 x-4}{2}$ When $x=0, y=\frac{3 \times 0-4}{2}=\frac{0-4}{2}=\frac{-4}{2}=-2$ When $x=2, y=\frac{3 \times 2-4}{2}=\frac{6-4}{2}=\frac{2}{2}=1$ When $x=-2, y=\frac{3 \times(-2)-4}{2}=\frac{-6-4}{2}=\frac{-10}{2}=-5$ Thus, the points on the line3x 2y = 4 are as given in t...

If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered,

Question: If the permutations of a, b, c, d, e taken all together be written down in alphabetical order as in dictionary and numbered, find the rank of the permutation debac. Solution: In a dictionary, the words are listed and ranked in alphabetical order. In the given problem, we need to find the rank of the word 'debac'. For finding the number of words starting with a, we have to find the number of arrangements of the remaining 4 letters. Number of such arrangements = 4! For finding the number...

If x2+k(4x+k−1)+2=0 has equal roots, then k =

Question: If $x^{2}+k(4 x+k-1)+2=0$ has equal roots, then $\mathrm{k}=$ (a) $-\frac{2}{3}, 1$ (b) $\frac{2}{3},-1$ (c) $\frac{3}{2}, \frac{1}{3}$ (d) $-\frac{3}{2},-\frac{1}{3}$ Solution: The given quadric equation is $x^{2}+k(4 x+k-1)+2=0$, and roots are equal Then find the value ofk. $x^{2}+k(4 x+k-1)+2=0$ $x^{2}+4 k x+\left(k^{2}-k+2\right)=0$ Here, $a=1, b=4 k$ and, $c=k^{2}-k+2$ As we know that $D=b^{2}-4 a c$ Putting the value of $a=1, b=4 k$ and, $c=k^{2}-k+2$ $=(4 k)^{2}-4 \times 1 \time...

If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary,

Question: If the letters of the word 'MOTHER' are written in all possible orders and these words are written out as in a dictionary, find the rank of the word 'MOTHER'. Solution: In a dictionary, the words are listed and ranked in alphabetical order. In the given problem, we need to find the rank of the word MOTHER. For finding the number of words starting with E, we have to find the number of arrangements of the remaining 5 letters. Number of such arrangements = 5! For finding the number of wor...

Draw the graph of the equation 3x + 2y = 6. Find the coordinates of the point,

Question: Draw the graph of the equation 3x+ 2y= 6. Find the coordinates of the point, where the graph cuts they-axis. Solution: Given equation: $3 x+2 y=6$. Then, $2 y=6-3 x \Rightarrow y=\frac{6-3 x}{2}$ When $x=2, y=\frac{6-6}{2}=0$ When $x=4, y=\frac{6-12}{2}=-3$ Thus, we get the following table: Plot the points $(2,0),(4,-3)$ on the graph paper. Join the points and extend the graph in both the directions. Clearly, the graph cuts the $y$-axis at $\mathrm{P}(0,3)$....

If the letters of the word 'LATE' be permuted and the words so formed be arranged as in a dictionary,

Question: If the letters of the word 'LATE' be permuted and the words so formed be arranged as in a dictionary, find the rank of the word LATE. Solution: In a dictionary, the words are listed and ranked in alphabetical order. In the given problem, we need to find the rank of the word LATE. For finding the number of words starting with A, we have to find the number of arrangements of the remaining 3 letters. Number of such arrangements = 3! For finding the number of words starting with E, we have...

The value of c for which the equation ax2 + 2bx + c = 0 has equal roots is

Question: The value of $c$ for which the equation $a x^{2}+2 b x+c=0$ has equal roots is (a) $\frac{b^{2}}{a}$ (b) $\frac{b^{2}}{4 a}$ (c) $\frac{a^{2}}{b}$ (d) $\frac{a^{2}}{4 b}$ Solution: The given quadric equation is $a x^{2}+2 b x+c=0$, and roots are equal Then find the value ofc. Let $\alpha$ and $\beta$ be two roots of given equation $\alpha=\beta$ Then, as we know that sum of the roots $\alpha+\beta=\frac{-2 b}{a}$ $\alpha+\alpha=\frac{-2 b}{a}$ $2 a=\frac{-2 b}{a}$ $\alpha=\frac{-b}{a}$...

The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary.

Question: The letters of the word 'SURITI' are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word 'SURITI'. Solution: In a dictionary, the words are arranged in the alphabetical order. Thus, in the given problem, we must consider the words beginning with I, I, R, S, T and U. I will occur at the first place as often as the ways of arranging the remaining 5 letters, when taken all at a time. Thus, I will occur 5! times. Similarly, R will oc...

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