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Linear and Quadratic Equations questions for IPMAT, with solutions

10 free Linear and Quadratic Equations practice questions for IPMAT Indore, Rohtak and JIPMAT, 2 from past papers, each with the answer and a full worked solution. No sign-in. Try each one first, then open the solution. The lessons for this chapter are in Linear and Quadratic Equations on Learn.

1. Question 1Type the answerWritten by IPMATwayExponential Equations

   If $3^{x-1}+3^{x+1}=90$, then $x$ equals

   Show the answer and solution

   Answer: 3

   Factor $3^{x-1}$: $3^{x-1}(1+3^{2})=90$, so $3^{x-1}=9=3^{2}$ and $x=3$.

2. Question 2Type the answerWritten by IPMATwayExponential Equations

   If $5^{x}+5^{x+1}=750$, then $x$ equals

   Show the answer and solution

   Answer: 3

   Factor $5^{x}$: $5^{x}(1+5)=750$, so $5^{x}=125=5^{3}$ and $x=3$.

3. Question 3Type the answerWritten by IPMATwayHigher-Degree Equations

   The product of all real roots of $x^{4}-7x^{2}+12=0$ is

   Show the answer and solution

   Answer: 12

   $(x^{2}-3)(x^{2}-4)=0$ gives $x=\pm\sqrt{3}$ and $x=\pm2$; the product is $(-3)(-4)=12$.

4. Question 4Multiple choiceWritten by IPMATwayLinear Equations

   The system $x+2y=3$ and $2x+4y=7$ has

   1. A.exactly two solutions
   2. B.a unique solution
   3. C.no solution
   4. D.infinitely many solutions
   Show the answer and solution

   Answer: C. no solution

   $2x+4y=2(x+2y)=6$ for every solution of the first equation, which contradicts $2x+4y=7$. No solution.

5. Question 5Type the answerWritten by IPMATwayLinear Equations

   A two-digit number is such that six times the tens digit added to ten times the units digit equals the number itself. The number is

   Show the answer and solution

   Answer: 94

   $4t=9u$ with digits forces $u=4$, $t=9$ (the next multiple would need $t=18$). The number is $94$.

6. Question 6Multiple choiceAsked in JIPMAT 2023Quadratic Equations

   Statement (I): $(x^{2}+3x+1)=(x-2)^{2}$ is not a quadratic equation

   Statement (II): The nature of roots of quadric equations $x^{2}+2x\sqrt{3}+3=0$ are real and equal.

   In light of the above statements, choose the most appropriate answer from the options given below.

   1. A.Both Statement (I) and Statement (II) are true.
   2. B.Both Statement (I) and Statement (II) are false.
   3. C.Statement (I) is true but Statement (II) is false.
   4. D.Statement (I) is false but Statement (II) is true.
   Show the answer and solution

   Answer: A. Both Statement (I) and Statement (II) are true.

   Expanding, $x^{2}+3x+1=x^{2}-4x+4$ reduces to $7x=3$, which is linear, so Statement I is true. For Statement II the discriminant is $12-12=0$: real and equal roots. Both true.

7. Question 7Multiple choiceAsked in JIPMAT 2024Quadratic Equations

   If $\alpha$ and $\beta$ are the roots of the equation $ax^{2}+bx+c=0$, then value of $\frac{1}{a\alpha+b}+\frac{1}{a\beta+b}$ is:

   1. A.$\frac{a}{bc}$
   2. B.$\frac{b}{ac}$
   3. C.$\frac{c}{ab}$
   4. D.$\frac{1}{abc}$
   Show the answer and solution

   Answer: B. $\frac{b}{ac}$

   From $\alpha+\beta=-\frac{b}{a}$, $a\alpha+b=-a\beta$ and $a\beta+b=-a\alpha$. The expression is $-\frac{1}{a}\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)=-\frac{1}{a}\cdot\frac{\alpha+\beta}{\alpha\beta}=-\frac{1}{a}\cdot\frac{-b/a}{c/a}=\frac{b}{ac}$. Check with $x^{2}-5x+6$: $\frac{1}{2-5}+\frac{1}{3-5}=-\frac{5}{6}=\frac{-5}{6}$.

8. Question 8Type the answerWritten by IPMATwayQuadratic Equations

   The value of $p$ for which the sum of the squares of the roots of $x^{2}-(p-3)x+(p-4)=0$ is least is

   Show the answer and solution

   Answer: 4

   $\alpha^{2}+\beta^{2}=S^{2}-2P=(p-3)^{2}-2(p-4)=(p-4)^{2}+1$, least at $p=4$. The roots are real for every $p$ since $D=(p-5)^{2}\ge0$.

9. Question 9Multiple choiceWritten by IPMATwayLinear Equations

   For which value of $k$ does the system $(k+1)x+8y=4k$, $\;kx+(k+3)y=3k-1$ have no solution?

   1. A.$k=1$
   2. B.$k=1$ or $k=3$
   3. C.$k=-3$
   4. D.$k=3$
   Show the answer and solution

   Answer: D. $k=3$

   The system has no solution when $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$. From the first equality, $(k+1)(k+3)=8k\Rightarrow k^2-4k+3=0\Rightarrow k=1$ or $3$. For $k=1$: $2x+8y=4$ and $x+4y=2$ are identical (ratio $2$ for all three), so infinitely many solutions. For $k=3$: $4x+8y=12$ and $3x+6y=8$; ratios $\frac43,\frac43,\frac{12}{8}=\frac32$, so the constants break the pattern: parallel lines, no solution. Answer $k=3$.

10. Question 10Multiple choiceWritten by IPMATwayLinear Equations

    A new temperature scale $X$ marks $20^\circ X$ at the freezing point of water and $140^\circ X$ at its boiling point. At what Celsius temperature does the $X$ scale show the same number as the Fahrenheit scale?

    1. A.$-40^\circ$C
    2. B.$-20^\circ$C
    3. C.$20^\circ$C
    4. D.$-4^\circ$C
    Show the answer and solution

    Answer: B. $-20^\circ$C

    The $X$ scale spans $120$ units over $100^\circ$C, so $X=20+1.2C$. Fahrenheit is $F=1.8C+32$. Setting $X=F$: $20+1.2C=1.8C+32\Rightarrow-12=0.6C\Rightarrow C=-20$. Check: $X=20-24=-4$ and $F=-36+32=-4$. The question asks for the Celsius temperature, $-20^\circ$C, not the shared reading $-4$.