LFM Pure

jee-main 2021 Q72 Linear System and Inverse Existence View

The system of equations $kx + y + z = 1$, $x + ky + z = k$ and $x + y + zk = k^2$ has no solution if $k$ is equal to:
(1) 0
(2) 1
(3) $-1$
(4) $-2$

jee-main 2021 Q72 View

If the following system of linear equations $2 x + y + z = 5$ $x - y + z = 3$ $x + y + a z = b$ has no solution, then :
(1) $a = - \frac { 1 } { 3 } , b \neq \frac { 7 } { 3 }$
(2) $a \neq \frac { 1 } { 3 } , b = \frac { 7 } { 3 }$
(3) $a \neq - \frac { 1 } { 3 } , b = \frac { 7 } { 3 }$
(4) $a = \frac { 1 } { 3 } , b \neq \frac { 7 } { 3 }$

jee-main 2021 Q86 View

If the system of equations $$\begin{aligned} & k x + y + 2 z = 1 \\ & 3 x - y - 2 z = 2 \\ & - 2 x - 2 y - 4 z = 3 \end{aligned}$$ has infinitely many solutions, then $k$ is equal to

jee-main 2022 Q68 Linear System and Inverse Existence View

Let the system of linear equations $x + y + a z = 2$ $3 x + y + z = 4$ $x + 2 z = 1$ have a unique solution $\left( x ^ { * } , y ^ { * } , z ^ { * } \right)$. If $\left( \left( a , x ^ { * } \right) , \left( y ^ { * } , \alpha \right) \right.$ and $\left( x ^ { * } , - y ^ { * } \right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is:
(1) 4
(2) 3
(3) 2
(4) 1

jee-main 2022 Q68 View

If the system of linear equations $8 x + y + 4 z = - 2$ $x + y + z = 0$ $\lambda x - 3 y = \mu$ has infinitely many solutions, then the distance of the point $\left( \lambda , \mu , - \frac { 1 } { 2 } \right)$ from the plane $8 x + y + 4 z + 2 = 0$ is:
(1) $3 \sqrt { 5 }$
(2) 4
(3) $\frac { 26 } { 9 }$
(4) $\frac { 10 } { 3 }$

jee-main 2022 Q69 Linear System and Inverse Existence View

If the system of equations $\alpha x + y + z = 5 , x + 2 y + 3 z = 4 , x + 3 y + 5 z = \beta$ has infinitely many solutions, then the ordered pair $( \alpha , \beta )$ is equal to
(1) $( 1 , - 3 )$
(2) $( - 1,3 )$
(3) $( 1,3 )$
(4) $( - 1 , - 3 )$

jee-main 2022 Q69 Linear System and Inverse Existence View

If the system of linear equations $2x + 3y - z = -2$ $x + y + z = 4$ $x - y + |\lambda|z = 4\lambda - 4$ where $\lambda \in \mathbb{R}$, has no solution, then
(1) $\lambda = 7$
(2) $\lambda = -7$
(3) $\lambda = 8$
(4) $\lambda^2 = 1$

jee-main 2022 Q70 Linear System and Inverse Existence View

If the system of equations $x + y + z = 6$ $2x + 5y + \alpha z = \beta$ $x + 2y + 3z = 14$ has infinitely many solutions, then $\alpha + \beta$ is equal to
(1) 8
(2) 36
(3) 44
(4) 48

jee-main 2022 Q70 View

The ordered pair $( a , b )$, for which the system of linear equations $3 x - 2 y + z = b$ $5 x - 8 y + 9 z = 3$ $2 x + y + a z = - 1$ has no solution, is
(1) $\left( 3 , \frac { 1 } { 3 } \right)$
(2) $\left( - 3 , \frac { 1 } { 3 } \right)$
(3) $\left( - 3 , - \frac { 1 } { 3 } \right)$
(4) $\left( 3 , - \frac { 1 } { 3 } \right)$

jee-main 2022 Q71 Linear System and Inverse Existence View

The system of equations $-kx + 3y - 14z = 25$ $-15x + 4y - kz = 3$ $-4x + y + 3z = 4$ is consistent for all $k$ in the set
(1) $R$
(2) $R - \{-11, 13\}$
(3) $R - \{-13\}$
(4) $R - \{-11, 11\}$

jee-main 2022 Q72 Linear System and Inverse Existence View

If the system of linear equations $2 x + y - z = 7$ $x - 3 y + 2 z = 1$ $x + 4 y + \delta z = k$, where $\delta , k \in R$ has infinitely many solutions, then $\delta + k$ is equal to
(1) $- 3$
(2) 3
(3) 6
(4) 9

jee-main 2023 Q69 Linear System and Inverse Existence View

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x + y + z = 1$, $2 x + N y + 2 z = 2$, $3 x + 3 y + N z = 3$ has unique solution is $\frac { k } { 6 }$, then the sum of value of $k$ and all possible values of $N$ is
(1) 18
(2) 19
(3) 20
(4) 21

jee-main 2023 Q69 View

For the system of linear equations $x + y + z = 6$ $\alpha x + \beta y + 7z = 3$ $x + 2y + 3z = 14$ which of the following is NOT true?
(1) If $\alpha = \beta = 7$, then the system has no solution
(2) If $\alpha = \beta$ and $\alpha \neq 7$ then the system has a unique solution.
(3) There is a unique point $(\alpha, \beta)$ on the line $x + 2y + 18 = 0$ for which the system has infinitely many solutions
(4) For every point $(\alpha, \beta) \neq (7,7)$ on the line $x - 2y + 7 = 0$, the system has infinitely many solutions.

jee-main 2023 Q69 View

For the system of linear equations $$2x - y + 3z = 5$$ $$3x + 2y - z = 7$$ $$4x + 5y + \alpha z = \beta$$ which of the following is NOT correct?
(1) The system has infinitely many solutions for $\alpha = -5$ and $\beta = 9$
(2) The system has infinitely many solutions for $\alpha = -6$ and $\beta = 9$
(3) The system is inconsistent for $\alpha = -5$ and $\beta = 8$
(4) The system has a unique solution for $\alpha \neq -5$ and $\beta = 8$

jee-main 2023 Q71 View

If the system of equations $$\begin{aligned} & 2 x + y - z = 5 \\ & 2 x - 5 y + \lambda z = \mu \\ & x + 2 y - 5 z = 7 \end{aligned}$$ has infinitely many solutions, then $( \lambda + \mu ) ^ { 2 } + ( \lambda - \mu ) ^ { 2 }$ is equal to
(1) 904
(2) 916
(3) 912
(4) 920

jee-main 2023 Q75 View

Let $S _ { 1 }$ and $S _ { 2 }$ be respectively the sets of all $a \in R - \{ 0 \}$ for which the system of linear equations $a x + 2 a y - 3 a z = 1$ $( 2 a + 1 ) x + ( 2 a + 3 ) y + ( a + 1 ) z = 2$ $( 3 a + 5 ) x + ( a + 5 ) y + ( a + 2 ) z = 3$ has unique solution and infinitely many solutions. Then
(1) $\mathrm { n } \left( S _ { 1 } \right) = 2$ and $S _ { 2 }$ is an infinite set
(2) $S _ { 1 }$ is an infinite set and $n \left( S _ { 2 } \right) = 2$
(3) $S _ { 1 } = \phi$ and $S _ { 2 } = \mathbb { R } - \{ 0 \}$
(4) $S _ { 1 } = \mathbb { R } - \{ 0 \}$ and $S _ { 2 } = \phi$

jee-main 2023 Q75 View

Consider the following system of questions
$$\begin{aligned} & \alpha x + 2 y + z = 1 \\ & 2 \alpha x + 3 y + z = 1 \\ & 3 x + \alpha y + 2 z = \beta \end{aligned}$$
For some $\alpha , \beta \in \mathbb { R }$. Then which of the following is NOT correct.
(1) It has no solution if $\alpha = - 1$ and $\beta \neq 2$
(2) It has no solution for $\alpha = - 1$ and for all $\beta \in \mathbb { R }$
(3) It has no solution for $\alpha = 3$ and for all $\beta \neq 2$
(4) It has a solution for all $\alpha \neq - 1$ and $\beta = 2$

jee-main 2023 Q75 View

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations $x - y + z = 5$ $2x + 2y + \alpha z = 8$ $3x - y + 4z = \beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of
(1) $x^{2} - 10x + 16 = 0$
(2) $x^{2} + 18x + 56 = 0$
(3) $x^{2} - 18x + 56 = 0$
(4) $x^{2} + 14x + 24 = 0$

jee-main 2023 Q77 Linear System and Inverse Existence View

For the system of equations $x + y + z = 6$ $x + 2y + \alpha z = 10$ $x + 3y + 5z = \beta$, which one of the following is NOT true?
(1) System has no solution for $\alpha = 3, \beta = 24$
(2) System has a unique solution for $\alpha = -3, \beta = 14$
(3) System has infinitely many solutions for $\alpha = 3, \beta = 14$
(4) System has a unique solution for $\alpha = 3, \beta = 14$

jee-main 2024 Q69 View

Consider the system of linear equations $x + y + z = 5$, $x + 2y + \lambda^2 z = 9$ and $x + 3y + \lambda z = \mu$, where $\lambda, \mu \in R$. Then, which of the following statement is NOT correct?
(1) System has infinite number of solution if $\lambda = 1$
(2) System is inconsistent if $\lambda = 1$ and $\mu \neq 13$ and $\mu = 13$
(3) System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$
(4) System is consistent if $\lambda \neq 1$ and $\mu = 13$

jee-main 2024 Q70 View

The values of $m , n$, for which the system of equations \begin{align*} x + y + z &= 4, 2x + 5y + 5z &= 17, x + 2y + \mathrm{m}z &= \mathrm{n} \end{align*} has infinitely many solutions, satisfy the equation:
(1) $m ^ { 2 } + n ^ { 2 } - m n = 39$
(2) $m ^ { 2 } + n ^ { 2 } - m - n = 46$
(3) $m ^ { 2 } + n ^ { 2 } + m + n = 64$
(4) $m ^ { 2 } + n ^ { 2 } + m n = 68$

jee-main 2024 Q70 View

If the system of equations $x + 4 y - z = \lambda , 7 x + 9 y + \mu z = - 3,5 x + y + 2 z = - 1$ has infinitely many solutions, then $( 2 \mu + 3 \lambda )$ is equal to : (1) 3 (2) - 3 (3) - 2 (4) 2

jee-main 2024 Q71 Linear System with Parameter — Infinite Solutions View

Let the system of equations $x + 2y + 3z = 5$, $2x + 3y + z = 9$, $4x + 3y + \lambda z = \mu$ have infinite number of solutions. Then $\lambda + 2\mu$ is equal to:
(1) 28
(2) 17
(3) 22
(4) 15

jee-main 2024 Q71 Linear System with Parameter — Infinite Solutions View

Consider the system of linear equation $x + y + z = 4 \mu , x + 2 y + 2 \lambda z = 10 \mu , x + 3 y + 4 \lambda ^ { 2 } z = \mu ^ { 2 } + 15$, where $\lambda , \mu \in \mathrm { R }$. Which one of the following statements is NOT correct?
(1) The system has unique solution if $\lambda \neq \frac { 1 } { 2 }$ and $\mu \neq 1$
(2) The system is inconsistent if $\lambda = \frac { 1 } { 2 }$ and $\mu \neq 1, 15$
(3) The system has infinite number of solutions if $\lambda = \frac { 1 } { 2 }$ and $\mu = 15$
(4) The system is consistent if $\lambda \neq \frac { 1 } { 2 }$

jee-main 2024 Q76 Substitution Combined with Symmetry or Companion Integral View

If $( a , b )$ be the orthocentre of the triangle whose vertices are $( 1,2 ) , ( 2,3 )$ and $( 3,1 )$, and $I _ { 1 } = \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { x } \sin \left( 4 \mathrm { x } - \mathrm { x } ^ { 2 } \right) \mathrm { dx } , \mathrm { I } _ { 2 } = \int _ { \mathrm { a } } ^ { \mathrm { b } } \sin \left( 4 \mathrm { x } - \mathrm { x } ^ { 2 } \right) \mathrm { dx }$, then $36 \frac { I _ { 1 } } { I _ { 2 } }$ is equal to:
(1) 72
(2) 88
(3) 80
(4) 66

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164