Recent questions tagged quadratic-equations

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CAT 2021 Set-2 | Quantitative Aptitude | Question: 16
Suppose one of the roots of the equation $ax^{2} – bx + c = 0$ is $2 + \sqrt{3},$ where $a, b$ and $c$ are rational numbers and $a \neq 0.$ If $b = c^{3}$ then $|a|$ equals $2$ $4$ $1$ $3$

Suppose one of the roots of the equation $ax^{2} – bx + c = 0$ is $2 + \sqrt{3},$ where $a, b$ and $c$ are rational numbers and $a \neq 0.$ If $b = c^{3}$ then $|a|$ equals $2$ $4$ $1$ $3$

asked Jan 20 in Quantitative Aptitude soujanyareddy13 2.7k points 52 views

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CAT 2021 Set-2 | Quantitative Aptitude | Question: 18
A person buys tea of three different qualities at $₹ \; 800, ₹ \; 500,$ and $₹ \; 300 \; \text{per kg},$ respectively, and the amounts bought are in the proportion $2:3:5.$ She mixes all the tea and sells one-sixth of the mixture at ... $50 \%,$ is $675$ $653$ $692$ $688$

A person buys tea of three different qualities at $₹ \; 800, ₹ \; 500,$ and $₹ \; 300 \; \text{per kg},$ respectively, and the amounts bought are in the proportion $2:3:5.$ She mixes all the tea and sells one-sixth of the mixture at $₹ \; 700 \; \text{per kg}.$ The ... $\text{INR per kg},$ at which she should sell the remaining tea, to make an overall profit of $50 \%,$ is $675$ $653$ $692$ $688$

asked Jan 20 in Quantitative Aptitude soujanyareddy13 2.7k points 102 views

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CAT 2020 Set-3 | Question: 61
Let $\text{m}$ and $\text{n}$ be positive integers, If $x^{2} + mx + 2n = 0$ and $x^{2} + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is $7$ $8$ $5$ $6$

Let $\text{m}$ and $\text{n}$ be positive integers, If $x^{2} + mx + 2n = 0$ and $x^{2} + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is $7$ $8$ $5$ $6$

asked Sep 17, 2021 in Quantitative Aptitude soujanyareddy13 2.7k points 102 views

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CAT 2020 Set-2 | Question: 58
In how many ways can a pair of integers $\textsf{(x , a)}$ be chosen such that $x^{2} – 2 |x| + |a-2| = 0 ?$ $4$ $5$ $6$ $7$

In how many ways can a pair of integers $\textsf{(x , a)}$ be chosen such that $x^{2} – 2 |x| + |a-2| = 0 ?$ $4$ $5$ $6$ $7$

asked Sep 17, 2021 in Quantitative Aptitude soujanyareddy13 2.7k points 94 views

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NIELIT 2019 Feb Scientist D - Section D: 19
If $x= \frac{\sqrt{p^{2}+q^{2}}+\sqrt{p^{2}-q^{2}}}{{\sqrt{p^{2}+q^{2}}-\sqrt{p^{2}-q^{2}}}}$ then $q^{2}x^{2}-2p^{2}x+q^{2}$ equals to : $3$ $-1$ $-2$ $0$

If $x= \frac{\sqrt{p^{2}+q^{2}}+\sqrt{p^{2}-q^{2}}}{{\sqrt{p^{2}+q^{2}}-\sqrt{p^{2}-q^{2}}}}$ then $q^{2}x^{2}-2p^{2}x+q^{2}$ equals to : $3$ $-1$ $-2$ $0$

asked Apr 3, 2020 in Quantitative Aptitude Lakshman Patel RJIT 10.7k points 251 views

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NIELIT 2016 MAR Scientist C - Section A: 12
The expression $(11.98\times 11.98 + 11.98 \times x +0.02 \times 0.02)$ will be a perfect square for $x$ equal to: $2.02$ $0.17$ $0.04$ $1.4$

The expression $(11.98\times 11.98 + 11.98 \times x +0.02 \times 0.02)$ will be a perfect square for $x$ equal to: $2.02$ $0.17$ $0.04$ $1.4$

asked Apr 2, 2020 in Quantitative Aptitude Lakshman Patel RJIT 10.7k points 112 views

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NIELIT 2019 Feb Scientist C - Section D: 9
sum of roots of the equation $\dfrac{3x^{3}-x^{2}+x-1}{3x^{3}-x^{2}-x+1}=\dfrac{4x^{3}-7x^{2}+x+1}{4x^{3}+7x^{2}-x-1}$ is : $0$ $1$ $-1$ $2$

sum of roots of the equation $\dfrac{3x^{3}-x^{2}+x-1}{3x^{3}-x^{2}-x+1}=\dfrac{4x^{3}-7x^{2}+x+1}{4x^{3}+7x^{2}-x-1}$ is : $0$ $1$ $-1$ $2$

asked Apr 1, 2020 in Quantitative Aptitude Lakshman Patel RJIT 10.7k points 210 views

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NIELIT 2019 Feb Scientist C - Section D: 17
If ${m_1}$ and ${m_2}$ are the roots of equation $x^{2}+(\sqrt{3}+2)x+\sqrt{3}-1=0$ then area of the triangle formed by the lines $y={m_1}x, \: \: y={m_2}x, \: \: y=c$ is: $\bigg(\dfrac{\sqrt{33}+\sqrt{11}}{4}\bigg) c^{2} $ ... $\bigg (\dfrac{\sqrt{33}+\sqrt{10}}{4} \bigg ) c^{2}$ $\bigg( \dfrac{\sqrt{33}+\sqrt{21}}{4} \bigg) c^{3}$

If ${m_1}$ and ${m_2}$ are the roots of equation $x^{2}+(\sqrt{3}+2)x+\sqrt{3}-1=0$ then area of the triangle formed by the lines $y={m_1}x, \: \: y={m_2}x, \: \: y=c$ is: $\bigg(\dfrac{\sqrt{33}+\sqrt{11}}{4}\bigg) c^{2} $ $\bigg( \dfrac{\sqrt{32}+\sqrt{11}}{16}\bigg ) c $ $\bigg (\dfrac{\sqrt{33}+\sqrt{10}}{4} \bigg ) c^{2}$ $\bigg( \dfrac{\sqrt{33}+\sqrt{21}}{4} \bigg) c^{3}$

asked Apr 1, 2020 in Quantitative Aptitude Lakshman Patel RJIT 10.7k points 171 views

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NIELIT 2019 Feb Scientist C - Section C: 23
Vidya and Vandana solved a quadratic equation. In solving it, Vidya made a mistake in the constant term and got the roots as $6$ and $2$, while Vandana made a mistake in the coefficient of $x$ only and obtained the root as $-7$ and $-1$. The correct roots of the equation are: $6,-1$ $-7,2$ $-6,-2$ $7,1$

Vidya and Vandana solved a quadratic equation. In solving it, Vidya made a mistake in the constant term and got the roots as $6$ and $2$, while Vandana made a mistake in the coefficient of $x$ only and obtained the root as $-7$ and $-1$. The correct roots of the equation are: $6,-1$ $-7,2$ $-6,-2$ $7,1$

asked Apr 1, 2020 in Quantitative Aptitude Lakshman Patel RJIT 10.7k points 991 views

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CAT 2019 Set-2 | Question: 78
The quadratic equation $x^{2}+bx+c=0$ has two roots $4a$ and $3a$, where a is an integer. Which of the following is a possible value of $b^{2}+c$? $3721$ $549$ $427$ $361$

The quadratic equation $x^{2}+bx+c=0$ has two roots $4a$ and $3a$, where a is an integer. Which of the following is a possible value of $b^{2}+c$? $3721$ $549$ $427$ $361$

asked Mar 20, 2020 in Quantitative Aptitude go_editor 13.3k points 269 views

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11

CAT 2019 Set-2 | Question: 88
Let $A$ be a real number. Then the roots of the equation $x^{2}-4x-\log _{2}A=0$ are real and distinct if and only if $A> \frac{1}{16}$ $A> \frac{1}{8}$ $A< \frac{1}{16}$ $A< \frac{1}{8}$

Let $A$ be a real number. Then the roots of the equation $x^{2}-4x-\log _{2}A=0$ are real and distinct if and only if $A> \frac{1}{16}$ $A> \frac{1}{8}$ $A< \frac{1}{16}$ $A< \frac{1}{8}$

asked Mar 20, 2020 in Quantitative Aptitude go_editor 13.3k points 155 views

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CAT 2018 Set-2 | Question: 77
If $a$ and $b$ are integers such that $2x^{2}- ax + 2 > 0$ and $x^{2}-bx+8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a-6b$ is _________

If $a$ and $b$ are integers such that $2x^{2}- ax + 2 > 0$ and $x^{2}-bx+8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a-6b$ is _________

asked Mar 20, 2020 in Quantitative Aptitude go_editor 13.3k points 192 views

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13

CAT 2017 Set-2 | Question: 90
The minimum possible value of the sum of the squares of the roots of the equation $x^{2}+\left ( a+3 \right )x-\left ( a+5 \right )=0$ is $1$ $2$ $3$ $4$

The minimum possible value of the sum of the squares of the roots of the equation $x^{2}+\left ( a+3 \right )x-\left ( a+5 \right )=0$ is $1$ $2$ $3$ $4$

asked Mar 16, 2020 in Quantitative Aptitude go_editor 13.3k points 240 views

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CAT 2017 Set-1 | Question: 93
If $f_{1}\left ( x \right )=x^{2}+11x+n$ and $f_{2}\left ( x \right )=x$, then the largest positive integer $n$ for which the equation $f_{1}\left ( x \right )=f_{2}\left ( x \right )$ has two distinct real roots, is $24$ $23$ $19$ $10$

If $f_{1}\left ( x \right )=x^{2}+11x+n$ and $f_{2}\left ( x \right )=x$, then the largest positive integer $n$ for which the equation $f_{1}\left ( x \right )=f_{2}\left ( x \right )$ has two distinct real roots, is $24$ $23$ $19$ $10$

asked Mar 13, 2020 in Quantitative Aptitude go_editor 13.3k points 147 views

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CAT 2016 | Question: 82
Given the quadratic equation $x^2 – (\text{A} – 3)x – (\text{A} – 2)$, for what value of $\text{A}$ will the sum of the squares of the roots be zero? $-2$ $3$ $6$ $\text{None of these}$

Given the quadratic equation $x^2 – (\text{A} – 3)x – (\text{A} – 2)$, for what value of $\text{A}$ will the sum of the squares of the roots be zero? $-2$ $3$ $6$ $\text{None of these}$

asked Mar 11, 2020 in Quantitative Aptitude go_editor 13.3k points 259 views

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CAT 2016 | Question: 99
If both $a$ and $b$ belong to the set $\{1,2,3,4\}$, then the number of equations of the form $ax^2+bx+1=0$ having real roots is _______

If both $a$ and $b$ belong to the set $\{1,2,3,4\}$, then the number of equations of the form $ax^2+bx+1=0$ having real roots is _______

asked Mar 11, 2020 in Quantitative Aptitude go_editor 13.3k points 108 views

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CAT 2012 | Question: 23
If $ax^{2}+bx+c= 0$ and $2a,b$ and $2c$ are in arithmetic progression, which of the following are the roots of the equation? $a,c \\$ $-a,-c \\$ $-\dfrac{a}{2},-\dfrac{c}{2} \\$ $-\dfrac{c}{a},-1$

If $ax^{2}+bx+c= 0$ and $2a,b$ and $2c$ are in arithmetic progression, which of the following are the roots of the equation? $a,c \\$ $-a,-c \\$ $-\dfrac{a}{2},-\dfrac{c}{2} \\$ $-\dfrac{c}{a},-1$

asked Mar 5, 2020 in Quantitative Aptitude Chandanachandu 308 points 153 views

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CAT 2012 | Question: 14
If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2}-10x+15= 0$, then find the quadratic equation whose roots are $\bigg(\alpha+\dfrac{\alpha }{\beta }\bigg)$ and $\bigg(\beta +\dfrac{\beta}{\alpha}\bigg)$ $15x^{2}+71x+210= 0$ $5x^{2}-22x+56= 0$ $3x^{2}-44x+78= 0$ Cannot be determined

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^{2}-10x+15= 0$, then find the quadratic equation whose roots are $\bigg(\alpha+\dfrac{\alpha }{\beta }\bigg)$ and $\bigg(\beta +\dfrac{\beta}{\alpha}\bigg)$ $15x^{2}+71x+210= 0$ $5x^{2}-22x+56= 0$ $3x^{2}-44x+78= 0$ Cannot be determined

asked Mar 5, 2020 in Quantitative Aptitude Chandanachandu 308 points 157 views

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CAT 2012 | Question: 5
If the roots of the equation $(a^{2}+b^{2})x^{2}+2(b^{2}+c^{2})x+(b^{2}+c^{2})= 0$ are real, which of the following must hold true? $c^{2}\geq a^{2}$ $c^{4}\geq a^{2}(b^{2}+c^{2})$ $b^{2}\geq a^{2}$ $a^{4}\leq b^{2}(a^{2}+c^{2})$

If the roots of the equation $(a^{2}+b^{2})x^{2}+2(b^{2}+c^{2})x+(b^{2}+c^{2})= 0$ are real, which of the following must hold true? $c^{2}\geq a^{2}$ $c^{4}\geq a^{2}(b^{2}+c^{2})$ $b^{2}\geq a^{2}$ $a^{4}\leq b^{2}(a^{2}+c^{2})$

asked Mar 5, 2020 in Quantitative Aptitude Chandanachandu 308 points 148 views

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20

CAT 2010 | Question: 12
For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive? $x^{2}-y^{2}=0$ $(x-k)^{2}+y^{2}=1$ $2$ $0$ $\sqrt{2}$ $\sqrt{-2}$

For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive? $x^{2}-y^{2}=0$ $(x-k)^{2}+y^{2}=1$ $2$ $0$ $\sqrt{2}$ $\sqrt{-2}$

asked Mar 1, 2020 in Quantitative Aptitude Arjun 8.3k points 879 views

4 votes

3 answers

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CAT 1994 | Question: 62
If one root of $x^{2} + px + 12 = 0$ is $4$, while the equation $x^{2} + px + q = 0$ has equal roots, then the value of $q$ is: $49/4$ $4/49$ $4$ $\frac{1}{4}$

If one root of $x^{2} + px + 12 = 0$ is $4$, while the equation $x^{2} + px + q = 0$ has equal roots, then the value of $q$ is: $49/4$ $4/49$ $4$ $\frac{1}{4}$

asked Oct 28, 2017 in Quantitative Aptitude makhdoom ghaya 7.9k points 471 views

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1 answer

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CAT 1995 | Question: 82
One root of $x^{2} + kx – 8 = 0$ is square of the other. Then, the value of k is: $2$ $8$ $-8$ $-2$

One root of $x^{2} + kx – 8 = 0$ is square of the other. Then, the value of k is: $2$ $8$ $-8$ $-2$

asked Aug 30, 2017 in Quantitative Aptitude makhdoom ghaya 7.9k points 248 views

–1 vote

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CAT 1996 | Question: 112
Given the quadratic equation $x^{2}-(A-3) x- (A-2) = 0$, for what value of $A$ will the sum of the squares of the roots be zero? $-2$ $3$ $6$ None of these

Given the quadratic equation $x^{2}-(A-3) x- (A-2) = 0$, for what value of $A$ will the sum of the squares of the roots be zero? $-2$ $3$ $6$ None of these

asked Jul 16, 2017 in Quantitative Aptitude makhdoom ghaya 7.9k points 255 views

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CAT 2003 | Question: 2-88
If both $a$ and $b$ belong to the set $\{1, 2, 3, 4\}$, then the number of equations of the form $ax^2 + bx + 1 = 0$ having real roots is $10$ $7$ $6$ $12$

If both $a$ and $b$ belong to the set $\{1, 2, 3, 4\}$, then the number of equations of the form $ax^2 + bx + 1 = 0$ having real roots is $10$ $7$ $6$ $12$

asked May 5, 2016 in Quantitative Aptitude go_editor 13.3k points 255 views

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25

CAT 2003 | Question: 1-124
Choose $1$ if the question can be answered by one of the statements alone but not by the other. Choose $2$ if the question can be answered by using either statement alone. Choose $3$ if the question can be answered by using both the statements together, but cannot be answered by ... $-3 \leq a \leq 3$ One of the roots of the equation $4x^2 - 4x +1 =0$ is $a.$

Choose $1$ if the question can be answered by one of the statements alone but not by the other. Choose $2$ if the question can be answered by using either statement alone. Choose $3$ if the question can be answered by using both the statements together, but cannot be answered by using either statement ... $-3 \leq a \leq 3$ One of the roots of the equation $4x^2 - 4x +1 =0$ is $a.$

asked May 3, 2016 in Quantitative Aptitude go_editor 13.3k points 143 views

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26

CAT 2003 | Question: 1-122
Choose $1$ if the question can be answered by one of the statements alone but not by the other. Choose $2$ if the question can be answered by using either statement alone. Choose $3$ if the question can be answered by using both the statements together, but cannot be answered by ... $\frac{1}{2}$ the ratio of $c$ and $b$ is $1.$

Choose $1$ if the question can be answered by one of the statements alone but not by the other. Choose $2$ if the question can be answered by using either statement alone. Choose $3$ if the question can be answered by using both the statements together, but cannot be answered by using either statement ... $\frac{1}{2}$ the ratio of $c$ and $b$ is $1.$

asked May 3, 2016 in Quantitative Aptitude go_editor 13.3k points 249 views

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CAT 2008 | Question: 11
Let $f(x) = ax^2 + bx +c$, where $a, b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2)$ and that $3$ is a root of $f(x)=0$. What is the value of $a+b+c?$ $9$ $14$ $13$ $37$ cannot be determined

Let $f(x) = ax^2 + bx +c$, where $a, b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2)$ and that $3$ is a root of $f(x)=0$. What is the value of $a+b+c?$ $9$ $14$ $13$ $37$ cannot be determined

asked Apr 30, 2016 in Quantitative Aptitude go_editor 13.3k points 365 views

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28

CAT 2001 | Question: 45
Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots $(4, 3).$ Keshab made a mistake in writing down the coefficient of $x.$ He got the root as $(3, 2).$ What will be the exact roots of the original quadratic equation? $(6, 1)$ $(–3, –4)$ $(4, 3)$ $(–4, –3)$

Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots $(4, 3).$ Keshab made a mistake in writing down the coefficient of $x.$ He got the root as $(3, 2).$ What will be the exact roots of the original quadratic equation? $(6, 1)$ $(–3, –4)$ $(4, 3)$ $(–4, –3)$

asked Apr 1, 2016 in Quantitative Aptitude go_editor 13.3k points 249 views

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29

CAT 2002 | Question: 65
The number of roots of $\frac{A^2}{x} + \frac{b^2}{x-1} =1$ is $1$ $2$ $3$ None of these

The number of roots of $\frac{A^2}{x} + \frac{b^2}{x-1} =1$ is $1$ $2$ $3$ None of these

asked Mar 2, 2016 in Quantitative Aptitude go_editor 13.3k points 127 views

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CAT 2003 | Question: 1-129
Let $p$ and $q$ be the roots of the quadratic equation $x^2 - (a-2) x-a -1 =0.$ What is the minimum possible value of $p^2 + q^2?$ $0$ $3$ $4$ $5$

Let $p$ and $q$ be the roots of the quadratic equation $x^2 - (a-2) x-a -1 =0.$ What is the minimum possible value of $p^2 + q^2?$ $0$ $3$ $4$ $5$

asked Feb 8, 2016 in Quantitative Aptitude go_editor 13.3k points 180 views

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31

CAT 2004 | Question: 51
Let $f(x) = ax^2 - b |x|$, where $a$ and $b$ are constants. Then at $x=0, f(x)$ is, maximized whenever $a>0, b>0$ maximized whenever $a>0, b<0$ minimized whenever $a>0, b>0$ minimized whenever $a>0, b<0$

Let $f(x) = ax^2 - b |x|$, where $a$ and $b$ are constants. Then at $x=0, f(x)$ is, maximized whenever $a>0, b>0$ maximized whenever $a>0, b<0$ minimized whenever $a>0, b>0$ minimized whenever $a>0, b<0$

asked Jan 12, 2016 in Quantitative Aptitude go_editor 13.3k points 92 views

0 votes

1 answer

32

CAT 2005 | Question: 10
For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive? $x^2 - y^2 =0$ $(x-k)^2 + y^2 =1$ $2$ $0$ $\sqrt{2}$ -$\sqrt{2}$

For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive? $x^2 - y^2 =0$ $(x-k)^2 + y^2 =1$ $2$ $0$ $\sqrt{2}$ -$\sqrt{2}$

asked Dec 29, 2015 in Quantitative Aptitude go_editor 13.3k points 149 views

2 votes

1 answer

33

CAT 2007 | Question: 02
A quadratic function f(x) attains a maximum of $3$ at $x=1.$ The value of the function at $x=0$ is $1.$ What is the value of $f(x)$ at $x=10?$ $-119$ $-159$ $-110$ $-180$ $-105$

A quadratic function f(x) attains a maximum of $3$ at $x=1.$ The value of the function at $x=0$ is $1.$ What is the value of $f(x)$ at $x=10?$ $-119$ $-159$ $-110$ $-180$ $-105$

asked Dec 6, 2015 in Quantitative Aptitude go_editor 13.3k points 667 views

2 votes

1 answer

34

CAT 2008 | Question: 10
Let $f(x) = ax^2 + bx +c$, where $a$, $b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2) $ and that 3 is a root of $f(x)=0$. What is the other root of $f(x)=0?$ $-7$ $-4$ $2$ $6$ cannot be determined

Let $f(x) = ax^2 + bx +c$, where $a$, $b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2) $ and that 3 is a root of $f(x)=0$. What is the other root of $f(x)=0?$ $-7$ $-4$ $2$ $6$ cannot be determined

asked Nov 28, 2015 in Quantitative Aptitude go_editor 13.3k points 359 views

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