Question 1

Q1. Let A={1,2,3,4}. The total number of bijective functions from set A to itself is:

Accepted Answer

Option C: 24. Bijective functions matlab function strictly one-one aur onto dono hona chahiye. A function from a set of n elements to itself is bijective if and only if it is a permutation of the set. So, total bijective functions =n!=4!=24.

Question 2

Q2. The principal value of tan-1(tan67 π ) is:

Accepted Answer

Option B: 6π. Yeh ek classic trap hai! tan-1x ki principal value branch (-2π ,2π ) hoti hai, aur 67π is range ke bahar hai. Hum isko likhenge tan( π+6π )= tan(6 π ). So, correct principal value is 6π .

Question 3

Q3. If A and B are symmetric matrices of the same order, then AB-BA is always a:

Accepted Answer

Option B: Skew-symmetric matrix. Let's take the transpose: (AB-BA)T=(AB)T-(BA)T=BTAT-ATBT. Kyunki dono symmetric hain, BT=B aur AT=A. Toh yeh ban gaya BA-AB=-(AB-BA). Jab XT=-X hota hai, toh matrix skew-symmetric hoti hai.

Question 4

Q4. If a is a non-zero vector of magnitude a, and λ is a non-zero scalar, then λ a represents a unit vector if:

Accepted Answer

Option D: a= ∣ λ ∣ 1. Unit vector ka magnitude 1 hota hai. Isliye, ∣ λ a ∣ =1 ⟹∣ λ ∣∣ a ∣ =1. Hamein pata hai ∣ a ∣ =a, toh equation bani ∣ λ ∣ a=1 ⟹ a= ∣ λ ∣ 1 .

Question 5

Q5. If A is a square matrix such that A2=A, then (I+A)3-7A is equal to:

Accepted Answer

Option C: I. Binomial expansion use karenge: (I+A)3=I3+3I2A+3IA2+A3. Since A2=A, we also get A3=A. Yeh expand hoke banega I+3A+3A+A=I+7A. Ab expression mein daalo: (I+7A)-7A=I.

Question 6

Q6. The value of k for which the function f(x)=8x21-cos4x for x=0, and f(0)=k, is continuous at x=0 is:

Accepted Answer

Option B: 1. Continuous hone ke liye limx→0 f(x)=f(0) hona chahiye. 1-cos4x=2sin2(2x). Limit banegi 8x22sin2(2x) =4x2sin2(2x) =(2xsin2x )2. Standard limit lim θ→0 θ sin θ =1 se, answer 12=1 aayega. So, k=1.

Question 7

Q7. The Integrating Factor (I.F.) for the differential equation xdxdy -y=2x2 is:

Accepted Answer

Option C: x1. Pehle standard form dxdy +Py=Q mein laane ke liye x se divide karo: dxdy -x1 y=2x. Yahan P=-x1 . I.F. ka formula hai e∫Pdx=e∫-1/xdx=e-logx=elog(x-1)=x-1=x1 .

Question 8

Q8. The value of the integral ∫ex(x1 -x21 )dx is:

Accepted Answer

Option B: xex +C. CBSE directly tests the competency form ∫ex[f(x)+f′(x)]dx=exf(x)+C. Yahan observe karo, agar f(x)=x1 hai, toh uska exact derivative f′(x)=-x21 hota hai. So direct answer is ex(x1 )+C.

Question 9

Q9. In a Linear Programming Problem, the objective function is Z=4x+3y. The corner points of the bounded feasible region are (0,0),(0,40),(20,40),(60,20),(60,0). The maximum value of Z occurs at:

Accepted Answer

Option C: (60,20). Har corner point par Z ki value check karenge. Z(20,40)=80+120=200. Z(60,0)=240+0=240. Z(60,20)=240+60=300. Maximum value 300 hai, jo (60,20) par aati hai.

Question 10

Q10. If A and B are two independent events such that P(A)=0.3 and P(B)=0.6, then P(A and not B) is:

Accepted Answer

Option A: 0.12. P(A and not B) ko P(A∩B′) likhte hain. Kyunki events independent hain, P(A∩B′)=P(A)×P(B′). Hamein pata hai P(B′)=1-P(B)=1-0.6=0.4. Ab multiply kar do: 0.3×0.4=0.12.

Question 11

Q11. A spherical balloon is being inflated such that its volume is increasing at a constant rate of 8 cm3/s. The rate of increase of its radius when the radius is 2 cm is:

Accepted Answer

Option B: 2π1 cm/s. Volume V=34 π r3. Differentiate with respect to time t: dtdV =4 π r2dtdr . Given dtdV =8 and r=2. Values put karne par: 8=4π(4) dtdr ⟹ 8=16 π dtdr ⟹ dtdr =16 π8 =2π1 .

Question 12

Q12. The area of the region bounded by the curve x=y2 and the y-axis, between y=1 and y=2 is:

Accepted Answer

Option A: 37 sq units. Real-life geometry application. Jab y-axis ke respect mein area nikalna ho, toh hum ∫xdy use karte hain. Limits 1 se 2 hain. Area =∫12 y2dy=[3y3 ]12 =38 -31 =37 .

Question 13

Q13. Two aircraft flight paths are represented by the straight lines r =(i^-j^ )+ λ(2 i^+k^) and r =(2i^-j^ )+ μ( i^+j^ -k^). To guarantee that these aircraft will never collide in mid-air, their shortest distance must be:

Accepted Answer

Option C: Non-zero. 3D Geometry application! Do lines (flight paths) space mein sirf tabhi intersect karti hain (collide hoti hain) jab unke beech ki shortest distance perfectly zero ho. Collision avoid karne ke liye, shortest distance strictly non-zero honi chahiye.

Question 14

Q14. Assertion (A): The degree of the differential equation x(dxdy )3+y(dx2d2y )=ex is 1. Reason (R): The highest power of the highest order derivative is called the degree of the differential equation, provided it is a polynomial equation in derivatives.

Accepted Answer

Option A: Both A and R are true and R is the correct explanation of A. Is equation mein highest order derivative dx2d2y hai, aur uski total power 1 hai. Assertion bilkul sahi hai. Reason bhi perfectly degree ki definition de raha hai aur explain kar raha hai ki humne power 3 wali term ko kyu ignore kiya.

Question 15

Q15. Assertion (A): If A is a square matrix of order 3 and ∣ A ∣ =4, then ∣ 2A ∣ =8. Reason (R): For a square matrix A of order n, and any scalar k, ∣ kA ∣ =kn ∣ A ∣ .

Accepted Answer

Option D: A is false but R is true. Reason ka rule perfectly correct hai ( ∣ kA ∣ =kn ∣ A ∣ ). Agar yeh rule Assertion pe lagayein: order n=3, k=2, aur ∣ A ∣ =4. Toh ∣ 2A ∣ =23×4=8×4=32 hona chahiye. Assertion keh raha hai value 8 hai, jo ki galat hai. Passage for Q16 & Q17: A marketing agency is designing a promotional rectangular poster. The central printed area must exactly be 50 cm2. The design requires blank margins at the top and bottom of 2 cm each, and at the left and right sides of 1 cm each. Let the width of the inner printed area be x cm.

Question 16

Q16. The total area of the poster paper A(x), expressed as a function of x, is:

Accepted Answer

Option A: (x+2)(x50 +4). Printed area dimensions: width = x, height = y. We know xy=50 ⟹ y=x50 . Total poster ki width (margins add karke) =x+1+1=x+2. Total poster ki height =y+2+2=y+4=x50 +4. Total Area =width×height.

Question 17

Q17. To minimize the cost of paper, the total area A(x) must be minimized. What width x of the printed area minimizes the total poster area?

Accepted Answer

Option B: 5 cm. A(x)=(x+2)(x50 +4)=50+4x+x100 +8. Differentiating w.r.t x: A′(x)=4-x2100 . For maxima/minima, A′(x)=0 ⟹ 4=x2100 ⟹ x2=25 ⟹ x=5 cm.

Question 18

Q18. For any 3×3 non-singular matrix A, if ∣ A ∣ =3, what is the exact value of ∣ adj(adjA) ∣ ?

Accepted Answer

Option C: 81. A brilliant HOTS question based on advanced determinant properties. The direct formula is ∣ adj(adjA) ∣ = ∣ A ∣ (n-1)2. Yahan order n=3 hai. So power will be (3-1)2=22=4. Evaluating it: 34=81.

Question 19

Q19. The value of the definite integral ∫03 ∣ x-2 ∣ dx is:

Accepted Answer

Option B: 25. Modulus function hai, toh interval break karna padega jahan x-2=0 (i.e., at x=2). ∫02 (-(x-2))dx+∫23 (x-2)dx. Isko geometrically do triangles ke area ki tarah calculate karo: 21 (2)(2)+21 (1)(1)=2+0.5=2.5 or 25 .

Question 20

Q20. The general solution of the differential equation dxdy =ex+y is:

Accepted Answer

Option A: ex+e-y=C. Highly repeated exam pattern! Expand the exponent: dxdy =ex ⋅ ey. Separation of variables karo: eydy =exdx ⟹ e-ydy=exdx. Dono side integrate karne par: -e-y=ex+C1 , which rearranges beautifully to ex+e-y=C.

Mock Test 06 Performance Solutions

Detailed Question Analysis

Q1. Let A={1,2,3,4}. The total number of bijective functions from set A to itself is:

Q2. The principal value of tan-1(tan67 π ) is:

Q3. If A and B are symmetric matrices of the same order, then AB-BA is always a:

Q4. If a is a non-zero vector of magnitude a, and λ is a non-zero scalar, then λ a represents a unit vector if:

Q5. If A is a square matrix such that A2=A, then (I+A)3-7A is equal to:

Q6. The value of k for which the function f(x)=8x21-cos4x for x=0, and f(0)=k, is continuous at x=0 is:

Q7. The Integrating Factor (I.F.) for the differential equation xdxdy -y=2x2 is:

Q8. The value of the integral ∫ex(x1 -x21 )dx is:

Q9. In a Linear Programming Problem, the objective function is Z=4x+3y. The corner points of the bounded feasible region are (0,0),(0,40),(20,40),(60,20),(60,0). The maximum value of Z occurs at:

Q10. If A and B are two independent events such that P(A)=0.3 and P(B)=0.6, then P(A and not B) is:

Q11. A spherical balloon is being inflated such that its volume is increasing at a constant rate of 8 cm3/s. The rate of increase of its radius when the radius is 2 cm is:

Q12. The area of the region bounded by the curve x=y2 and the y-axis, between y=1 and y=2 is:

Q13. Two aircraft flight paths are represented by the straight lines r =(i^-j^ )+ λ(2 i^+k^) and r =(2i^-j^ )+ μ( i^+j^ -k^). To guarantee that these aircraft will never collide in mid-air, their shortest distance must be:

Q14. Assertion (A): The degree of the differential equation x(dxdy )3+y(dx2d2y )=ex is 1. Reason (R): The highest power of the highest order derivative is called the degree of the differential equation, provided it is a polynomial equation in derivatives.

Q15. Assertion (A): If A is a square matrix of order 3 and ∣ A ∣ =4, then ∣ 2A ∣ =8. Reason (R): For a square matrix A of order n, and any scalar k, ∣ kA ∣ =kn ∣ A ∣ .

Q16. The total area of the poster paper A(x), expressed as a function of x, is:

Q17. To minimize the cost of paper, the total area A(x) must be minimized. What width x of the printed area minimizes the total poster area?

Q18. For any 3×3 non-singular matrix A, if ∣ A ∣ =3, what is the exact value of ∣ adj(adjA) ∣ ?

Q19. The value of the definite integral ∫03 ∣ x-2 ∣ dx is:

Q20. The general solution of the differential equation dxdy =ex+y is:

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Mock Test 06 Performance Solutions

Detailed Question Analysis

Q1. Let A={1,2,3,4}. The total number of bijective functions from set A to itself is:

Q2. The principal value of tan-1(tan67 π ) is:

Q3. If A and B are symmetric matrices of the same order, then AB-BA is always a:

Q4. If a is a non-zero vector of magnitude a, and λ is a non-zero scalar, then λ a represents a unit vector if:

Q5. If A is a square matrix such that A2=A, then (I+A)3-7A is equal to:

Q6. The value of k for which the function f(x)=8x21-cos4x for x=0, and f(0)=k, is continuous at x=0 is:

Q7. The Integrating Factor (I.F.) for the differential equation xdxdy -y=2x2 is:

Q8. The value of the integral ∫ex(x1 -x21 )dx is:

Q9. In a Linear Programming Problem, the objective function is Z=4x+3y. The corner points of the bounded feasible region are (0,0),(0,40),(20,40),(60,20),(60,0). The maximum value of Z occurs at:

Q10. If A and B are two independent events such that P(A)=0.3 and P(B)=0.6, then P(A and not B) is:

Q11. A spherical balloon is being inflated such that its volume is increasing at a constant rate of 8 cm3/s. The rate of increase of its radius when the radius is 2 cm is:

Q12. The area of the region bounded by the curve x=y2 and the y-axis, between y=1 and y=2 is:

Q13. Two aircraft flight paths are represented by the straight lines r =(i^-j^ )+ λ(2 i^+k^) and r =(2i^-j^ )+ μ( i^+j^ -k^). To guarantee that these aircraft will never collide in mid-air, their shortest distance must be:

Q14. Assertion (A): The degree of the differential equation x(dxdy )3+y(dx2d2y )=ex is 1. Reason (R): The highest power of the highest order derivative is called the degree of the differential equation, provided it is a polynomial equation in derivatives.

Q15. Assertion (A): If A is a square matrix of order 3 and ∣ A ∣ =4, then ∣ 2A ∣ =8. Reason (R): For a square matrix A of order n, and any scalar k, ∣ kA ∣ =kn ∣ A ∣ .

Q16. The total area of the poster paper A(x), expressed as a function of x, is:

Q17. To minimize the cost of paper, the total area A(x) must be minimized. What width x of the printed area minimizes the total poster area?

Q18. For any 3×3 non-singular matrix A, if ∣ A ∣ =3, what is the exact value of ∣ adj(adjA) ∣ ?

Q19. The value of the definite integral ∫03 ∣ x-2 ∣ dx is:

Q20. The general solution of the differential equation dxdy =ex+y is:

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