{"id":212652,"date":"2022-07-07T17:35:04","date_gmt":"2022-07-07T12:05:04","guid":{"rendered":"https:\/\/cracku.in\/blog\/?p=212652"},"modified":"2022-07-07T17:35:04","modified_gmt":"2022-07-07T12:05:04","slug":"cat-quadratic-equation-questions-pdf","status":"publish","type":"post","link":"https:\/\/cracku.in\/blog\/cat-quadratic-equation-questions-pdf\/","title":{"rendered":"CAT Quadratic Equation Questions PDF [Most Important]"},"content":{"rendered":"

Quadratic Equations<\/b> is an important topic in the CAT<\/strong> Quant<\/strong> (Algebra) section. <\/span>If you find these questions a bit tough, make sure you solve more CAT Quadratic Equation questions. Learn all the important formulas<\/strong><\/span> and tricks on how to answer questions<\/strong><\/span><\/a><\/span> on<\/strong><\/span> Quadratic<\/strong><\/span> Equations<\/strong><\/span><\/a>. You can check out these Quadratic Equation<\/span> questions from the CAT Previous year papers<\/strong><\/a>.<\/span> Practice a good number of sums in the CAT Quadratic Equation<\/strong> so that you can answer these questions with ease in the exam. In this post, we will look into some important Quadratic Equation Questions in the CAT quants. These are a good source of practice for CAT 2022 preparation; If you want to practice these questions, you can download this Important CAT Quadratic Equation Questions<\/strong> and answers PDF<\/strong> along with the video solutions below, which are completely Free.<\/p>\n

Download Quadratic Equation Questions for CAT<\/a><\/p>\n

Enroll for CAT 2022 Online Course<\/a><\/p>\n

Question 1:\u00a0<\/b>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?<\/p>\n

a)\u00a0-119<\/p>\n

b)\u00a0-159<\/p>\n

c)\u00a0-110<\/p>\n

d)\u00a0-180<\/p>\n

e)\u00a0-105<\/p>\n

1)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

Let the function be $ax^2 + bx + c$.<\/p>\n

We know that x=0 value is 1 so c=1.<\/p>\n

So equation is $ax^2 + bx + 1$.<\/p>\n

Now max value is 3 at x = 1.<\/p>\n

So after substituting we get a + b = 2.<\/p>\n

If f(x) attains a maximum at ‘a’ then the differential of f(x) at x=a, that is, f'(a)=0.<\/p>\n

So in this question f'(1)=0<\/p>\n

=> 2*(1)*a+b = 0<\/p>\n

=> 2a+b = 0.<\/p>\n

Solving the equations we get a=-2 and b=4.<\/p>\n

$ -2x^2 + 4x + 1$ is the equation and on substituting x=10, we get -159.<\/p>\n

Question 2:\u00a0<\/b>If the roots of the equation $x^3 – ax^2 + bx – c = 0$ are three consecutive integers, then what is the smallest possible value of b?
\n[CAT 2008]<\/p>\n

a)\u00a0$\\frac{-1}{\\sqrt 3}$<\/p>\n

b)\u00a0$-1$<\/p>\n

c)\u00a0$0$<\/p>\n

d)\u00a0$1$<\/p>\n

e)\u00a0$\\frac{1}{\\sqrt 3}$<\/p>\n

2)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

b = sum of the roots taken 2 at a time.
\nLet the roots be n-1, n and n+1.
\nTherefore, $b = (n-1)n + n(n+1) + (n+1)(n-1) = n^2 – n + n^2 + n + n^2 – 1$
\n$b = 3n^2 – 1$. The smallest value is -1.<\/p>\n

Question 3:\u00a0<\/b>Let p and q be the roots of the quadratic equation $x^2 – (\\alpha – 2) x – \\alpha -1= 0$ . What is the minimum possible value of $p^2 + q^2$?<\/p>\n

a)\u00a00<\/p>\n

b)\u00a03<\/p>\n

c)\u00a04<\/p>\n

d)\u00a05<\/p>\n

3)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

Let $\\alpha $ be equal to k.<\/p>\n

=> f(x) = $x^2-(k-2)x-(k+1) = 0$<\/p>\n

p and q are the roots<\/p>\n

=> p+q = k-2 and pq = -1-k<\/p>\n

We know that $(p+q)^2 = p^2 + q^2 + 2pq$<\/p>\n

=> $ (k-2)^2 = p^2 + q^2 + 2(-1-k)$<\/p>\n

=> $p^2 + q^2 = k^2 + 4 – 4k + 2 + 2k$<\/p>\n

=> $p^2 + q^2 = k^2 – 2k + 6$<\/p>\n

This is in the form of a quadratic equation.<\/p>\n

The coefficient of $k^2$ is positive. Therefore this equation has a minimum value.<\/p>\n

We know that the minimum value occurs at x = $\\frac{-b}{2a}$<\/p>\n

Here a = 1, b = -2 and c = 6<\/p>\n

=> Minimum value occurs at k = $\\frac{2}{2}$ = 1<\/p>\n

If we substitute k = 1 in $k^2-2k+6$, we get 1 -2 + 6 = 5.<\/p>\n

Hence 5 is the minimum value that $p^2+q^2$ can attain.<\/p>\n

Question 4:\u00a0<\/b>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 roots as (3, 2). What will be the exact roots of the original quadratic equation?<\/p>\n

a)\u00a0(6, 1)<\/p>\n

b)\u00a0(-3, -4)<\/p>\n

c)\u00a0(4, 3)<\/p>\n

d)\u00a0(-4, -3)<\/p>\n

4)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

We know that quadratic equation can be written as $x^2$-(sum of roots)*x+(product of the roots)=0.
\nUjakar ended up with the roots (4, 3) so the equation is $x^2$-(7)*x+(12)=0 where the constant term is wrong.<\/p>\n

Keshab got the roots as (3, 2) so the equation is $x^2$-(5)*x+(6)=0 where the coefficient of x is wrong .<\/p>\n

So the correct equation is $x^2$-(7)*x+(6)=0. The roots of above equations are (6,1).<\/p>\n

Question 5:\u00a0<\/b>If the roots $x_1$ and $x_2$ are the roots of the quadratic equation $x^2 -2x+c=0$ also satisfy the equation $7x_2 – 4x_1 = 47$, then which of the following is true?<\/p>\n

a)\u00a0c = -15<\/p>\n

b)\u00a0$x_1 = -5$ and $x_2=3$<\/p>\n

c)\u00a0$x_1 = 4.5$ and $x_2 = -2.5$<\/p>\n

d)\u00a0None of these<\/p>\n

5)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

$x_1 + x_2 = 2$
\nand $7x_2 – 4x_1 = 47$
\nSo $x_1 = -3$ and $x_2 = 5$
\nAnd $c = x_1 \\times x_2 = -15$<\/p>\n

Checkout: CAT Free Practice Questions and Videos<\/strong><\/a><\/em><\/p>\n

Question 6:\u00a0<\/b>If $x+1=x^{2}$ and $x>0$, then $2x^{4}$\u00a0 is<\/p>\n

a)\u00a0$6+4\\sqrt{5}$<\/p>\n

b)\u00a0$3+3\\sqrt{5}$<\/p>\n

c)\u00a0$5+3\\sqrt{5}$<\/p>\n

d)\u00a0$7+3\\sqrt{5}$<\/p>\n

6)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

We know that $x^2 – x – 1=0$
\nTherefore $x^4 = (x+1)^2 = x^2+2x+1 = x+1 + 2x+1 = 3x+2$
\nTherefore, $2x^4 = 6x+4$<\/p>\n

We know that $x>0$ therefore, we can calculate the value of $x$ to be $\\frac{1+\\sqrt{5}}{2}$
\nHence, $2x^4 = 6x+4 = 3+3\\sqrt{5}+4 = 3\\sqrt{5}+7$<\/p>\n

Question 7:\u00a0<\/b>If $U^{2}+(U-2V-1)^{2}$= \u2212$4V(U+V)$ , then what is the value of $U+3V$ ?<\/p>\n

a)\u00a0$0$<\/p>\n

b)\u00a0$\\dfrac{1}{2}$<\/p>\n

c)\u00a0$\\dfrac{-1}{4}$<\/p>\n

d)\u00a0$\\dfrac{1}{4}$<\/p>\n

7)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

Given that $U^{2}+(U-2V-1)^{2}$= \u2212$4V(U+V)$<\/p>\n

$\\Rightarrow$ $U^{2}+(U-2V-1)(U-2V-1)$= \u2212$4V(U+V)$<\/p>\n

$\\Rightarrow$ $U^{2}+(U^2-2UV-U-2UV+4V^2+2V-U+2V+1)$ =\u00a0\u2212$4V(U+V)$<\/p>\n

$\\Rightarrow$ $U^{2}+(U^2-4UV-2U+4V^2+4V+1)$ =\u00a0\u2212$4V(U+V)$<\/p>\n

$\\Rightarrow$ $2U^2-4UV-2U+4V^2+4V+1=\u22124UV-4V^2$<\/p>\n

$\\Rightarrow$ $2U^2-2U+8V^2+4V+1=0$<\/p>\n

$\\Rightarrow$ $2[U^2-U+\\dfrac{1}{4}]+8[V^2+\\dfrac{V}{2}+\\dfrac{1}{16}]=0$<\/p>\n

$\\Rightarrow$ $2(U-\\dfrac{1}{2})^2+8(V+\\dfrac{1}{4})^2=0$<\/p>\n

Sum of two square terms is zero i.e. individual square term is equal to zero.<\/p>\n

$U-\\dfrac{1}{2}$ = 0 and $V+\\dfrac{1}{4}$ = 0<\/p>\n

U = $\\dfrac{1}{2}$ and\u00a0V = $-\\dfrac{1}{4}$<\/p>\n

Therefore,\u00a0$U+3V$ =\u00a0$\\dfrac{1}{2}$+$\\dfrac{-1*3}{4}$ =\u00a0$\\dfrac{-1}{4}$. Hence, option C is the correct answer.<\/p>\n

Question 8:\u00a0<\/b>If a and b are integers such that $2x^2\u2212ax+2>0$ and $x^2\u2212bx+8\u22650$ for all real numbers $x$, then the largest possible value of $2a\u22126b$ is<\/p>\n

8)\u00a0Answer:\u00a036<\/b><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

Let f(x) = $2x^2\u2212ax+2$. We can see that f(x) is a quadratic function.<\/p>\n

For, f(x) > 0, Discriminant (D) < 0<\/p>\n

$\\Rightarrow$ $(-a)^2-4*2*2<0$<\/p>\n

$\\Rightarrow$ (a-4)(a+4)<0<\/p>\n

$\\Rightarrow$ a $\\epsilon$ (-4, 4)<\/p>\n

Therefore, integer values that ‘a’ can take = {-3, -2, -1, 0, 1, 2, 3}<\/p>\n

Let g(x) = $x^2\u2212bx+8$. We can see that g(x) is also a quadratic function.<\/p>\n

For, g(x)\u22650, Discriminant (D) $\\leq$ 0<\/p>\n

$\\Rightarrow$ $(-b)^2-4*8*1<0$<\/p>\n

$\\Rightarrow$ $(b-\\sqrt{32})(b+\\sqrt{32})<0$<\/p>\n

$\\Rightarrow$ b\u00a0$\\epsilon$ (-$\\sqrt{32}$, $\\sqrt{32}$)<\/p>\n

Therefore, integer values that ‘b’ can take = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}<\/p>\n

We have to find out the\u00a0largest possible value of $2a\u22126b$. The largest possible value will occur when ‘a’ is maximum and ‘b’ is minimum.<\/p>\n

a$_{max}$ = 3,\u00a0b$_{min}$ = -5<\/p>\n

Therefore, the\u00a0largest possible value of $2a\u22126b$ = 2*3 – 6*(-5) = 36.<\/p>\n

Question 9:\u00a0<\/b>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<\/p>\n

a)\u00a0$A > \\frac{1}{16}$<\/p>\n

b)\u00a0$A < \\frac{1}{16}$<\/p>\n

c)\u00a0$A < \\frac{1}{8}$<\/p>\n

d)\u00a0$A > \\frac{1}{8}$<\/p>\n

9)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

The roots of\u00a0$x^2 – 4x – log_{2}{A} = 0$ will be real and distinct if and only if the discriminant is greater than zero<\/p>\n

16+4*$log_{2}{A}$ > 0<\/p>\n

$log_{2}{A}$ > -4<\/p>\n

A> 1\/16<\/p>\n

Question 10:\u00a0<\/b>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 $b^2 + c$?<\/p>\n

a)\u00a03721<\/p>\n

b)\u00a0361<\/p>\n

c)\u00a0427<\/p>\n

d)\u00a0549<\/p>\n

10)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

Given,<\/p>\n

The quadratic equation $x^2 + bx + c = 0$ has two roots 4a and 3a<\/p>\n

7a=-b<\/p>\n

12$a^2$ = c<\/p>\n

We have to find the value of\u00a0 $b^2 + c$ = 49$a^2$+ 12$a^2$=61$a^2$<\/p>\n

Now lets verify the options<\/p>\n

61$a^2$ = 3721 ==> a= 7.8 which is not an integer<\/p>\n

61$a^2$ = 361 ==> a= 2.42 which is not an integer<\/p>\n

61$a^2$ = 427 ==> a= 2.64 which is not an integer<\/p>\n

61$a^2$ = 3721 ==> a= 3 which is an integer<\/p>\n

Question 11:\u00a0<\/b>The product of the distinct roots of $\\mid x^2 – x – 6 \\mid = x + 2$ is<\/p>\n

a)\u00a0\u221216<\/p>\n

b)\u00a0-4<\/p>\n

c)\u00a0-24<\/p>\n

d)\u00a0-8<\/p>\n

11)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

We have,\u00a0$\\mid x^2 – x – 6 \\mid = x + 2$<\/p>\n

=> |(x-3)(x+2)|=x+2<\/p>\n

For x<-2, (3-x)(-x-2)=x+2<\/p>\n

=> x-3=1\u00a0 \u00a0=>x=4 (Rejected as x<-2)<\/p>\n

For -2$\\le\\ $x<3,\u00a0(3-x)(x+2)=x+2\u00a0 \u00a0 =>x=2,-2<\/p>\n

For x$\\ge\\ $3,\u00a0(x-3)(x+2)=x+2\u00a0 \u00a0=>x=4<\/p>\n

Hence the product =4*-2*2=-16<\/p>\n

Question 12:\u00a0<\/b>The number of solutions to the equation $\\mid x \\mid (6x^2 + 1) = 5x^2$ is<\/p>\n

12)\u00a0Answer:\u00a05<\/b><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

For x <0, -x($6x^2+1$) =\u00a0$5x^2$<\/p>\n

=>\u00a0($6x^2+1$) = -5x<\/p>\n

=>\u00a0($6x^2 + 5x+ 1$) = 0<\/p>\n

=>($6x^2 + 3x+2x+ 1$) = 0<\/p>\n

=> (3x+1)(2x+1)=0\u00a0 \u00a0 =>x=$\\ -\\frac{\\ 1}{3}$\u00a0 or x=$\\ -\\frac{\\ 1}{2}$<\/p>\n

For x=0, LHS=RHS=0\u00a0 \u00a0 (Hence, 1 solution)<\/p>\n

For x >0, x($6x^2+1$) = $5x^2$<\/p>\n

=> ($6x^2 – 5x+ 1$) = 0<\/p>\n

=>(3x-1)(2x-1)=0\u00a0 \u00a0 =>x=$\\ \\frac{\\ 1}{3}$\u00a0 \u00a0or\u00a0 \u00a0x=$\\ \\frac{\\ 1}{2}$<\/p>\n

Hence, the total number of solutions = 5<\/p>\n

Question 13:\u00a0<\/b>How many disticnt positive integer-valued solutions exist to the equation $(x^{2}-7x+11)^{(x^{2}-13x+42)}=1$ ?<\/p>\n

a)\u00a08<\/p>\n

b)\u00a04<\/p>\n

c)\u00a02<\/p>\n

d)\u00a06<\/p>\n

13)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

$(x^{2}-7x+11)^{(x^{2}-13x+42)}=1$<\/p>\n

if\u00a0$(x^{2}-13x+42)$=0 or\u00a0$(x^{2}-7x+11)$=1 or\u00a0$(x^{2}-7x+11)$=-1 and\u00a0$(x^{2}-13x+42)$ is even number<\/p>\n

For x=6,7 the value $(x^{2}-13x+42)$=0<\/p>\n

$(x^{2}-7x+11)$=1 for x=5,2.<\/p>\n

$(x^{2}-7x+11)$=-1 for x=3,4 and for X=3 or 4,\u00a0$(x^{2}-13x+42)$ is even number.<\/p>\n

.’. {2,3,4,5,6,7} is the solution set of x.<\/p>\n

.’. x can take six values.<\/p>\n

Question 14:\u00a0<\/b>The number of distinct real roots of the equation $(x+\\frac{1}{x})^{2}-3(x+\\frac{1}{x})+2=0$ equals<\/p>\n

14)\u00a0Answer:\u00a01<\/b><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

Let $a=x+\\frac{1}{x}$
\nSo, the given equation is $a^2-3a+2=0$
\nSo, $a$ can be either 2 or 1.<\/p>\n

If $a=1$, $x+\\frac{1}{x}=1$ and it has no real roots.
\nIf $a=2$, $x+\\frac{1}{x}=2$ and it has exactly one real root which is $x=1$<\/p>\n

So, the total number of distinct real roots of the given equation is 1<\/p>\n

Question 15:\u00a0<\/b>Let m and n be positive integers, If $x^{2}+mx+2n=0$ and $x^{2}+2nx+m=0$ have real\u00a0roots, then the smallest possible value of $m+n$ is<\/p>\n

a)\u00a07<\/p>\n

b)\u00a06<\/p>\n

c)\u00a08<\/p>\n

d)\u00a05<\/p>\n

15)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

View Video Solution<\/a><\/p>\n

Solution:<\/b><\/p>\n

To have real roots the discriminant should be greater than or equal to 0.<\/p>\n

So, $m^2-8n\\ge0\\ \\&\\ 4n^2-4m\\ge0$<\/p>\n

=> $m^2\\ge8n\\ \\&\\ n^2\\ge m$<\/p>\n

Since m,n are positive integers the value of m+n will be minimum when m=4 and n=2.<\/p>\n

.’. m+n=6.<\/p>\n

Enroll for CAT 2022 Complete Course <\/a><\/p><\/div>\n

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Download CAT Quant Formulas PDF<\/a><\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Quadratic Equations is an important topic in the CAT Quant (Algebra) section. If you find these questions a bit tough, make sure you solve more CAT Quadratic Equation questions. Learn all the important formulas and tricks on how to answer questions on Quadratic Equations. You can check out these Quadratic Equation questions from the CAT […]<\/p>\n","protected":false},"author":32,"featured_media":212667,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_mi_skip_tracking":false,"footnotes":""},"categories":[3],"tags":[5119,4147],"class_list":{"0":"post-212652","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-cat","8":"tag-cat-2022","9":"tag-quadratic-equations"},"better_featured_image":{"id":212667,"alt_text":"CAT Quadratic Equations Questions PDF","caption":"CAT Quadratic Equations Questions PDF","description":"CAT Quadratic Equations Questions 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