Class 12 Maths Sample paper 2021-22: best sample paper for practice.

A) Let [latex]\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}[/latex] be defined by [latex]f(x)=\left\{\begin{array}{l}2 x: x>3 \\ x^{2}: 1 Then [latex]f(-1)+f(2)+f(4)[/latex] is

B) Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F = 4x
+ 6y be the objective function. Maximum of F – Minimum of F =

C) 3. Find the value of b for which the function [latex]f(x)=\left\{\begin{array}{ll}5 x-4 & , 0

D) Let A be a non-singular square matrix of order 3 × 3. Then |adj A| is equal to

E) A fruit grower can use two types of fertilizer in his garden, brand P and brand Q. The amounts
(in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in
the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg
of potash, and at most 310 kg of chlorine.

If the grower wants to maximize the amount of nitrogen added to the garden, how many bags
of each brand should be added? What is the maximum amount of nitrogen added?

F) The equation of normal to the curve
[latex]3 x^{2}-y^{2}=8[/latex]
= 8 which is parallel to the line x + 3y = 8 is

G) If [latex]\omega[/latex] is a complex cube root of unity then the value of [latex]\left|\begin{array}{ccc}1 & \omega & 1+\omega \\ 1+\omega & 1 & \omega \\ \omega & 1+\omega & 1\end{array}\right|[/latex]

H) If [latex]\mathrm{y}=\mathrm{x}^{2} \sin \frac{1}{x}[/latex] then [latex]\frac{d y}{d x}=[/latex] ?

I) Determine the maximum value of [latex]\mathrm{Z}=11 \mathrm{x}+7 \mathrm{y}[/latex] subject to the constraints : [latex] 2 \mathrm{x}+\mathrm{y} \leq 6, \mathrm{x} \leq 2, \mathrm{x} \geq 0[/latex], [latex]y \geq 0[/latex].

J) If [latex]A=\left[\begin{array}{ccc}0 & 2 & -3 \\ -2 & 0 & -1 \\ 3 & 1 & 0\end{array}\right][/latex] then [latex]\mathrm{A}[/latex] is a

K) If [latex]f(x)=\left\{\begin{array}{ll}m x+1, & \text { if } x \leq \frac{\pi}{2} \\ \sin x+n, & \text { if } x>\frac{\pi}{2}\end{array}\right.[/latex] is continuous at [latex]x=\frac{\pi}{2}[/latex] then

L) The feasible region for an LPP is shown in the Figure. Let F = 3x - 4y be the objective function.
Maximum value of F is.

M) The value of [latex]\mathrm{k}[/latex] for which [latex]\mathrm{f}(\mathrm{x})=\left\{\begin{aligned} \frac{\sin 5 x}{3 x}, & \text { if } x \neq 0 \\ k, & \text { if } x=0 \end{aligned}\right.[/latex] is continuous at x=0 is

N) The function [latex]f(x)=\frac{4-x^{2}}{4 x-x^{3}}[/latex] is

O) If [latex]y=x^{\mathrm{n}-1} \log x[/latex] then [latex]x^{2} y_{2}+(3-2 n) x y_{1}[/latex] is equal to

P) The function f(x) = tanx - x

Q) The point on the curve y²
= 4x which is nearest to the point (2,1) is

R) [latex]\sin ^{-1}\left(\frac{-1}{2}\right)+2 \cos ^{-1}\left(\frac{-\sqrt{3}}{2}\right)=[/latex] ?

S) If [latex]\mathrm{x}^{\mathrm{y}}=\mathrm{e}^{\mathrm{x}-\mathrm{y}}[/latex], then [latex]\frac{d y}{d x}[/latex] is

T) The curves x= y²
and xy = k cut orthogonally when

U) Attempt any 16 questions
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to [latex]\mathrm{b} \mathrm{a}, \mathrm{b} \in \mathrm{T}[/latex]. Then [latex]\mathrm{R}[/latex] is

V) f(x) = sin x [latex]\sqrt{3}[/latex] cos x is maximum when x =

W) The feasible region for an LPP is shown in Figure. Evaluate Z = 4x + y at each of the corner
points of this region. Find the minimum value of Z, if it exists

X) 24. The slope of the tangent to the curve [latex]x=3 t^{2}+1, y=t^{3}-1[/latex] at x=1 is

Y) If [latex]y=\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}+\frac{1}{1+x^{b-a}+x^{c-a}} [/latex] , then [latex] \frac{d y}{d x}[/latex] is equal to

Z) [latex]\cot \left(\tan ^{-1} \mathrm{x}+\cot ^{-1} \mathrm{x}\right) [/latex]

AA) If the set A contains 5 elements and the set B contains 6 elements, then the number of one one and onto mappings from A to B is

AB) [latex] \tan \left[2 \tan ^{-1} \frac{1}{5}-\frac{\pi}{4}\right]=?[/latex]

AC) If [latex]\mathrm{y}=\tan ^{-1}\left(\frac{\sqrt{a}+\sqrt{x}}{1-\sqrt{a x}}\right) [/latex] then [latex]\frac{d y}{d x}= [/latex]?

AD) If A’ is the transpose of a square matrix A, then

AE) If [latex]\sqrt{1-x^{6}}+\sqrt{1-y^{6}}=\mathrm{a}^{3}\left(\mathrm{x}^{3}-\mathrm{y}^{3}\right) [/latex] then [latex]\frac{d y}{d x} [/latex] is equal to

AF) If [latex] y=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}[/latex] , then [latex] \frac{d y}{d x}[/latex] is equal to

AG) If the function [latex]f(x)=2 x^{2}-k x+5 [/latex] is increasing on (1,2), then  k lies in the interval

AH) [latex]\operatorname{Sin}\left(\tan ^{-1} x\right),|x|

AI) [latex]\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|=? [/latex]

AJ) The feasible region (shaded) for an LPP is shown in the Figure. Minimum of Z = 4x + 3y occurs at
the point

AK) If A is an invertible matrix of order 2 , then [latex]\operatorname{det}\left(\mathrm{A}^{-1}\right)  [/latex] is equal to

AL) Let [latex]\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}[/latex], then [latex]\mathrm{f}(\mathrm{x}) [/latex] has a

AM) If [latex] f(x)=\sqrt{1-\sqrt{1-x^{2}}}[/latex], then [latex]\mathrm{f}(\mathrm{x})[/latex] is

AN) Which of the following functions from Z into Z are bijections?

AO) If [latex]\tan ^{-1}\left\{\frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}\right\}=\alpha [/latex] , then [latex]\mathrm{x}^{2}= [/latex]

AP) Maximize [latex]\mathrm{Z}=-\mathrm{x}+2 \mathrm{y} [/latex] , subject to the constraints : [latex]\mathrm{x} \geq 3, \mathrm{x}+\mathrm{y} \geq 5, \mathrm{x}+2 \mathrm{y} \geq 6, \mathrm{y} \geq 0 [/latex]

AQ) If f is derivable at x = a , then [latex]\operatorname{Lt}_{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a} [/latex] is equal to

AR) If A, B are two n × n non - singular matrices, then what can you infer about AB?

AS) Let S be the set of all real numbers and let R be a relation on S, defined by [latex]\mathrm{a} \mathrm{Rb} \Leftrightarrow(1+\mathrm{ab})>0 [/latex] Then , R is

AT) Question No. 46 to 50 are based on the given text. Read the text carefully and answer the
questions:
Consider 2 families A and B. Suppose there are 4 men, 4 women and 4 children in family A and 2 men, 2
women and 2 children in family B. The recommended daily amount of calories is 2400 for a man, 1900
for a woman, 1800 for children and 45 grams of proteins for a man, 55 grams for a woman and 33
grams for children.

The requirement of calories and proteins for each person in matrix form can be represented
as

AU) The requirement of calories of family A is

AV) The requirement of proteins for family B is

AW) If A and B are two matrices such that [latex]\mathrm{AB}=\mathrm{B} [/latex] and [latex] \mathrm{BA}=\mathrm{A}[/latex] , then [latex] \mathrm{A}^{2}+\mathrm{B}^{2}[/latex] equals

AX) If [latex] \mathrm{A}=\left(\mathrm{a}_{\mathrm{ij}}\right)_{\mathrm{m}} \times \mathrm{n}, \mathrm{B}=\left(\mathrm{b}_{\mathrm{ij}}\right)_{\mathrm{n}} \times \mathrm{p}[/latex] and [latex]\mathrm{C}=\left(\mathrm{c}_{\mathrm{ij}}\right) \mathrm{p} \times \mathrm{q} [/latex] , then the product  (BC)A is possible only when