System Of Linear Equations

01

PYQ 2024

easy

mathematics ID: ts-eamce

The system of simultaneous linear equations :

1

Infinitely many solutions

2

No solution

3

Unique solution having z = 2

4

Unique solution having z =1⁄2

02

PYQ 2024

hard

mathematics ID: ts-eamce

If represents the system of linear equations then AX = D ?

1

2

3

4

03

PYQ 2024

medium

mathematics ID: ts-eamce

The system of equations,, and has:

1

unique solution

2

infinitely many solutions

3

no solution

4

finite number of solutions \bigskip

04

PYQ 2024

medium

mathematics ID: ts-eamce

The system of equations has:

1

only zero solution for any values of .

2

non-zero solution for any values of .

3

non-zero solution whenever .

4

non-zero solution whenever .

05

PYQ 2024

medium

mathematics ID: ts-eamce

If the homogeneous system of linear equations,, has a non-trivial solution, then

1

2

2

6

3

-6

4

-2

06

PYQ 2024

hard

mathematics ID: ts-eamce

Solving the System of Linear Equations If (x,y,z) = (α,β,γ) is the unique solution of the system of simultaneous linear equations: 3x - 4y + 2z + 7 = 0, 2x + 3y - z = 10, x - 2y - 3z = 3, then α = ?

1

2

3

4

07

PYQ 2025

easy

mathematics ID: ts-eamce

If are the roots of the equation and,, are the roots of the equation, then

1

2

3

4

08

PYQ 2025

medium

mathematics ID: ts-eamce

$If local maximum of exists at, then$

1

$0$

2

$-1$

3

$1$

4

$2$

Official Solution

Correct Option: (3)

$Step 1: Use the point condition. For a local maximum at : Step 2: Use derivative condition. The first derivative: Set numerator zero at : Step 3: Find . From, . Thus,$

09

PYQ 2025

medium

mathematics ID: ts-eamce

$If is a root of the equation, then sum of the squares of the other roots is$

1

$0$

2

$2+i$

3

$-4-4i$

4

$8/3$

Official Solution

Correct Option: (3)

$Step 1: Complex conjugate root theorem. Since the polynomial has real coefficients, if is a root, is also a root. Step 2: Form quadratic factor. Step 3: Divide original polynomial. Step 4: Find other roots. Solve to get and . Step 5: Sum of squares of other roots. Step 6: Note on answer key. The calculation gives, but the answer key lists . Likely a typographical error in the original question.$

10

PYQ 2025

medium

mathematics ID: ts-eamce

$If and for, and if, then$

1

$g(K) attains its maximum at the midpoint of (a,b)$

2

$g(K) attains its minimum at two points in (a,b)$

3

$g(K) attains its both maximum and minimum in (a,b)$

4

$g(K) attains no maximum and no minimum in (a,b)$

Official Solution

Correct Option: (4)

$Step 1: Condition for quadratic . For, we require and . Step 2: Apply discriminant condition. Here,, so Step 3: Substitute . Step 4: Solve inequality. Step 5: Analyze on . is an upward parabola. Vertex at, so is strictly increasing on . Step 6: Conclude. A strictly increasing function on an open interval does not attain maximum or minimum inside the interval. Thus, has no maximum or minimum in .$

11

PYQ 2025

medium

mathematics ID: ts-eamce

$If the quotient and remainder obtained when the expression is divided by the expression are and respectively, then$

1

2

3

4

Official Solution

Correct Option: (2)

$We perform polynomial long division to find the quotient and remainder. Divide by . Step 1: Divide by to get . . Subtract this from : . Bring down the remaining terms: . Step 2: Divide by to get . . Subtract this: . Bring down the remaining term: . Step 3: Divide by to get . . Subtract this: . The division is complete. The quotient is . Comparing with, we get . The remainder is . Comparing with, we get . We need to calculate the value of . . Now we check the options using and . Option (A): . Option (B): . Option (C): . Option (D): . The value of is 70, which matches option (B).$

12

PYQ 2025

medium

mathematics ID: ts-eamce

$If are the roots of the equation such that and, then$

1

2

3

4

Official Solution

Correct Option: (4)

$The given equation is a reciprocal equation of type I, since the coefficients are symmetric. . Divide the equation by (since is not a root): . Group the terms: . Let . Then, which means . Substitute this into the equation: . . . We solve this quadratic equation for : . The two possible values for are: . . The roots of the original equation come in reciprocal pairs. Let the pairs be and . Given and . So and . The sums are and . These sums are the values of we found. So, . We are given the condition, which implies . Let's compare the two values: . Since, we must have and . The required ratio is .$

13

PYQ 2025

medium

mathematics ID: ts-eamce

$Sum of all the roots of the equation is$

1

$8$

2

$0$

3

$6$

4

$9$

Official Solution

Correct Option: (3)

$We are given the equation . To solve this, we remove the outermost absolute value. This gives two possibilities. Case 1: . . This leads to two sub-cases: Sub-case 1a: . Sub-case 1b: . Case 2: . . This leads to two more sub-cases: Sub-case 2a: . Sub-case 2b: . The roots of the equation are . The sum of all the roots is . Sum = .$

14

PYQ 2025

medium

mathematics ID: ts-eamce

$If and are two distinct roots of the equation, then$

1

2

3

4

Official Solution

Correct Option: (2)

$Given that and are the roots of the quadratic equation . From the properties of quadratic equations, we can write the sum and product of the roots. Sum of the roots: . Product of the roots: . Since, we have, which implies . Now, let's work with the sum of the roots. . Using the identity, we get: . Rearranging gives . We know the double angle identity . Multiplying both sides by 2: . Therefore, . Since we established that, we can substitute for . .$

15

PYQ 2025

medium

mathematics ID: ts-eamce

$The equation having the multiple root of the equation as its root is$

1

2

3

4

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept If a polynomial equation has a multiple root (a root with multiplicity greater than 1), then that root is also a root of the derivative equation, . Therefore, the multiple root must be a common root of and . Step 2: Key Formula or Approach 1. Let . 2. Find the derivative, . 3. Solve the equation to find the potential multiple roots. 4. Substitute these potential roots back into the original equation to identify the actual multiple root. 5. Check which of the given quadratic equations has this multiple root as one of its roots. Step 3: Detailed Explanation 1. Given . 2. Find the derivative: 3. Solve : Divide by 4 to simplify: Let's find the roots of this cubic equation by testing integer divisors of -4 (i.e.,). Let . . So, is a root. . So, is a root. Since it's a cubic, there's one more root. As the sum of coefficients is 0, one root is 1. We found another root is -2. The sum of roots is -3. So . So the roots of are . 4. Identify the multiple root of : The possible multiple roots of are the roots of, which are and . We test them in . Test : . So, is not the multiple root of . Test : . Since and, is a multiple root of . 5. Find the correct equation: We need to find which of the given options has as a root. (A) . (B) . (C) . This is correct. (D) . Step 4: Final Answer The multiple root of the given equation is . The equation from the options that has as a root is .$

16

PYQ 2025

medium

mathematics ID: ts-eamce

$If are the roots of the equation such that and are integers and are irrational numbers, then$

1

$5$

2

$7$

3

$11$

4

$13$

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept For a polynomial with integer coefficients, any rational roots must be of the form , where is a divisor of the constant term and is a divisor of the leading coefficient (Rational Root Theorem). Also, irrational roots of polynomials with rational coefficients occur in conjugate pairs. Step 2: Key Formula or Approach Let . 1. Use the Rational Root Theorem to find the possible integer roots, and . 2. Once the integer roots are found, use polynomial division to find the remaining quadratic factor whose roots are and . 3. Use Vieta's formulas on the quadratic factor to find the value of . 4. Calculate the final expression. Step 3: Detailed Explanation 1. Find integer roots: According to the Rational Root Theorem, any integer root must be a divisor of the constant term, -2. The possible integer roots are . Let's test these values in : . So, is a root. Since there might be another integer root, let's use synthetic division to find the remaining polynomial. \begin{center} \begin{tabular}{c|ccccc} 1 & 1 & -4 & 3 & 2 & -2 & & 1 & -3 & 0 & 2 \hline & 1 & -3 & 0 & 2 & 0 \end{tabular} \end{center} The reduced polynomial is . Let's test the possible integer roots in : . So, is a repeated root. The two integer roots are and . 2. Find the quadratic factor: We divide by . \begin{center} \begin{tabular}{c|cccc} 1 & 1 & -3 & 0 & 2 & & 1 & -2 & -2 \hline & 1 & -2 & -2 & 0 \end{tabular} \end{center} The remaining quadratic factor is . The roots of are the irrational roots and . 3. Find : For the quadratic equation , by Vieta's formulas: Sum of roots: . Product of roots: . We need to find . We use the identity: 4. Calculate the final expression: We need to calculate . Substituting the values we found: . Step 4: Final Answer The value of the expression is 11.$

17

PYQ 2025

medium

mathematics ID: ts-eamce

$If both roots of the equation exceed 1, then the range of 'a' is$

1

2

3

4

Official Solution

Correct Option: (2)

$Step 1: Understanding the Concept For a quadratic equation with real roots, the location of the roots relative to a certain number can be determined by a set of conditions. Here, we need both roots to be greater than 1. Step 2: Key Formula or Approach Let . For both roots of to be greater than a number (here), the following three conditions must be satisfied: 1. The discriminant must be non-negative: . 2. The value of the function at must have the same sign as the leading coefficient : . 3. The x-coordinate of the vertex must be greater than : . Step 3: Detailed Explanation Here,,,, and . Condition 1: This inequality holds when or . So, . Condition 2: Since, we need . So, . Condition 3: So, . Now, we need to find the intersection of the solution sets from all three conditions. From (1): From (2): From (3): Let's find the intersection. Intersection of (2) and (3): . Now, intersect this with (1): We compare the values: and . The interval does not overlap with . We need to find the intersection of and . Since, the intersection is . Step 4: Final Answer The range of values for 'a' that satisfies all three conditions is .$

18

PYQ 2025

medium

mathematics ID: ts-eamce

$If the equations and have a common root then the sum of the roots of the equation is$

1

$-3$

2

$3$

3

$6$

4

$-6$

Official Solution

Correct Option: (3)

$Step 1: Understanding the Concept If two quadratic equations have a common root, that root must satisfy both equations. We can solve the two equations to find a condition on the coefficients, and then find the value of the common root. Step 2: Key Formula or Approach Let be the common root of the equations and . Then must satisfy both: 1) 2) Subtracting equation (2) from equation (1) will eliminate the term, allowing us to solve for . Step 3: Detailed Explanation Subtracting the two equations: This gives two possibilities: or . Case 1: . If, both equations become . The discriminant of this equation is . The roots are imaginary, but they are both common. So is a valid possibility. Case 2: . If the common root is, it must satisfy either equation. Let's substitute it into the first equation: Let's check if also satisfies the second equation with : It does. So is also a valid possibility. The question doesn't specify any conditions on or the roots. Let's evaluate the required sum for both values of . The final equation is . The sum of the roots of a quadratic equation is . Sum of roots = . If, the sum of the roots is . This is not among the options. If, the sum of the roots is . This is option (C). Step 4: Final Answer We found two possible values for : and . For, the sum of roots of is . For, the sum of roots of is . Since is one of the options, we choose this as the intended answer.$

High-Yield Trend

Chapter Questions
18 MCQs

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System Of Linear Equations

High-Yield Trend

Chapter Questions 18 MCQs

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Chapter Questions
18 MCQs