TEST PAPER -1 CLASS- XII (2021-22)

1) Fill the gap: The value of the determinant  3    1975     1978
                        4    1982     1986
                        5    1995     2000  is_____
OR
State whether the following statement is true or false :
" the product of two non zero matrices must be a non-zero metrix."

2) If y= log log x, x> 1 which one of the following answer is true?
A) x dy/dx= 1  B)(x logx)dy/dx= 1
C) (logx)dy/dx= 1
D) (logx)dy/dx= x
OR
* If x= a(t - sint) and y= a(1+cos t) then which one of the following is the value of dy/dx.
A) - cot(t/2)        B) cot t
C) - tan(t/2)        D) cot(t/2)

3) If y= cos²x, then Which one of the following is the value of d²y/dx² ?
A) - 2 cos 2x    B) 2 cos 2x
C) -2 sin 2x      D) cos 2x

4) State whether the following statement is true or false:
the gradient of the tangent at the point (8,-4) to the parabola y²= 8(x-6) is - 1.
OR
* State whether the following is true or false:
f(x)= (x-1)(3-x) had an extreme value at the point x= 2.

5) The equation of the normal to the ellipse x²+ 4y²= 4 which is parallel to the line 8x+ 3y= 0 be..
A) 8x+ 3y= ±12   B) 16x+ 6y= ±15
C) 40x+ 15y= ±36  D) 24x+ 9y= ±15

6) The difference between the maximum and minimum values of the function f(x)= x³/3 - 2x² + 3x +1 is...
A) 4    B) 2     C) 1    D) 4/3.

7) The minimum value of 1/2 (7 - cos 2x) is...
A) 7/2     B) 4     C) 5/2      D) 3.

8) The length of the rectangle of maximum area that can be inscribed in a semicircle of radius 1 unit, so that two vertices lie on the diameter, is..
A) √2. B) 2  C) √2/3 D) √3 unit

9) If the tangent at the point P on the circle x²+ y²+ 6x + 6y= 2 meets the straight line 5x - 2y+6= 0 at the point Q on the y-axis, then the length of PQ is...
A) 4      B) 2√5   C) 5.     D) 3√5

10) Equations of the tangent and normal drawn at the point (6.0) on the ellipse x²/36+ y²/9= 0 respectively are...
A) x= 6, y= 0.  B) x+y= 6, y-x+6= 0
C) x= 0, y= 3  D) x= -6, y= 0

11) If the function f(x)= x²(x-2)² is an increasing function of 3, then
A) 1< x<2      B) x<0 
C)0< x<1 or x>2.  D)1< x<2 or 0< x

12) If f(x)=kx³ - 9x²+ 9x+3 is an increasing function then...
A) k< 3 B) k≤3 C) k>3 D) k is indeterminate.

13) The function f(x)= 1- x³ - x⁵ is decreasing for.....
A) 1≤ x≤5 B) all real values of x
C) x≤ 3.   D) x≥ 5

14) If v= 4πr³/3, then the rate (in cubic unit) at which v is increasing when r= 10 and dr/dt= 0.01, is ..
A) 4π.   B) π  C) 40π.   D) 4π/3

15) If the time rate of change of the radius of a sphere is 1/2π, then the rate of change of its surface area (in sq cm), when the radius is 5cm is..
A) 20      B) 10.    C) 4     D) 5

16) If f(x)= x      when 0≤ x ≤ 1
               2x -1 when x> 1 then
A) f(x) is discontinues at x= 1
B) f(x) is discontinues but not differentiable at x= 1
C) f(x) is differentiable at x= 1
D) none of these

17) Which of the following statements is not true?
A) a polynomial function is always continuous
B) a differentiable function is always continuous
C) a continuous function is always differentiable
D) log x is continuous for all x> 0

18) If the function
f(x)= (x²-9)/(x-3) when x≠ 3
          2x+a when x= 3 is continuous at x=3, then the value of a is...
A) 3      B) 6      C) 0        D) 4

19) If y= √(x+1) - √(x-1), then the value of (x²-1)d²y/dx² + x dy/dx is
A) 2y      B) -2y    C) y/4    D) y/2

20) In the mean value theorem f(b) - f(a) = (b - a) f'(c), (a<c<b), if f(x)= x³ - 3x -1, a= -11/7, b= 13/7 then the value of c is...
A) 0      B) 1      C) -1        D) ±1

21) The derivative of the function
tan⁻¹[{2x√(1-x²)}/(1-2x²)] w.r.t. the function tan⁻¹[{√(1+x²) - 1}/x] at x= 0 is...
A) 1     B) 2      C) 4           D) 8

22) If f(x)g(x)= k(a constant) and g"(x)/g'(x)= f"(x)/f'(x) + a.f'(x)/f(x) , then the value of a is...
A) 4      B) -4   C) 2    D) -2

23) If 2x= y¹⁾⁵ + y⁻¹⁾⁵ and (x²-1)y₂ + xy₁= ky, then the value of k is..
A) 5      B) -5       C) 25     D) -25

24) lim ₓ→∞{1 - 4/(x-1)}³ˣ⁻¹=
A) e⁴   B) e³    C) e¹²    D)1/e¹²

25) If the function f(x)= 4x³ + ax² + bx -1 satisfies all the conditions of Rolle's theorem in -1/4 ≤x ≤ 1 and f'(1/2)=0, then the values of a and bare
A) a=2, b =-3       B) a=1, b =-4
C) a=-1, b =4       D)  a=-4, b =-1

26) lim ₓ→∞ {(n-3)/(n+2)ⁿ is..
A) 1/e⁵ B) 1/e⁴ C) 1/e² D) 1/e

27) The value of c in Rolle's theorem f(x)= 2x³ - 5x² - 4x +3, x belongs [1/2, 3] is...
A) -1/3  B) 2/3    C) 2       D) -2

28) If sin⁻¹(x/5) + cosec⁻¹(5/4)= π/2, then the value of x is..
A) 1     B) 2      C) 3          D) 4

29) The equation sin⁻¹x - cos⁻¹x = cos⁻¹(√3/2) has
A) unique solution
B) two solution
C) no solution
D) infinite number of solutions

30) If x+y+z= xyz, then the value of (tan⁻¹x + tan⁻¹y + tan⁻¹z) is equal to
A) 3π/2   B) π    C) π/2     D) 2π

31) The value of cos⁻¹{(3+ 5 cosx)/(5+ 3cosx)} is...
A) 1/2 tan⁻¹(2 tan(x/2))
B) 2 tan⁻¹(1/2 tan(x/2))
C) tan⁻¹(1/2 tan x)
D) 2 tan⁻¹(2 tan(x/2))

32) If the determinant of the matrix a₁       b₁        c₁
            a₂        b₂        c₂
            a₃        b₃        c₃ is denoted by D, then the determinant of the matrix  a₁+3b₁- 4c₁    b₁       4c₁
              a₂ +3₂- 4c₂     b₂       4c₂
              a₃ +3b₃-4c₃    b₃       4c₃ will be
A) D B) 2D C) 3D D) 4D

33) If A= 3    - 5
               -4      2 , then the value of A² - 5A is equal to
A) I     B) 14I     C) O     D) none

34) If A= 1     2    and B= 1      2
                 2     3                 2      1
                 3     4
Then,
A) both AB and BA exist
B) neither AB nor BA exist
C) AB exists but BA does not exist
D) AB does not exist but BA exist

35) The value of the determinant b²c²         bc         b+ c
c²a²         ca         c+ a
b²a²         ab         a+ b
A) abc(a²+b²+c²)         B) 0
C) abc(bc+ca+ab)
D) (a+b+c)(a²+b²+c²) (ab+bc+ca)

36) The maximum value of the function 3cosx - 4 sinx -2 is..
A) 0   B)1   C) 4       D) 3

37) If y= cot⁻¹{(b- ax)/(a+ bx)}, then the value of dy/dx is...
A) 1  B) 1/(1+x²) C) -1 D)-1/(1+x²)

38) If y= cot⁻¹{8x⁴ - 8x²+1}, then the value of dy/dx.
A) 4/√(1-x²)       B) -4/√(1-x²)
C) 4/(1+x²)        D) - 4/(1+x²)

39) State whether sin(sin⁻¹(√2))= √2 is true or false
OR
Find y in terms of x where tan⁻¹{x/√(1- x²)}= sin⁻¹y.

40) tan⁻¹x + tan⁻¹y=π/4, then...
A) x+y+z+1=0     B) x+y+xy-1=0
C) x+y- xy+1=0   D) x+y- xy-1=0

REVISED CMA

16/8/21

1) If the 3rd and 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference.                              3 and 2

2) find the sum of all natural numbers between 100 and 1000 which are multiple of 5.      98450

3) how many terms of the AP -6, -11/2, -5,.... are needed to give the sum -25 ?                            5 or 20.

4) Determine the sum of the first 35 terms of an AP if a₂ = 2 and a₇ = 22.                        2310

5) If the first term of an AP is 2 and the sum of first five terms is equal to one fourth of the sum of the next five terms, show that the 20th term is --112

6) Insert 3 arithmetic mean between 2 and 10.                  4,6,8

7) The sum of three decreasing numbers in AP is 27. If -1, -1, 3 are added to them respectively, the resulting series is in GP. The numbers are 

A) 5,8,13 B)15,9,3 C)13,9,5 D) 17,9,1

8) The sum of all odd numbers between 1 and 100 which are divisible by 3, is..

A) 83667 B) 90000 C) 83660 D) n 

9) If 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is.

A) 87   B) 88   C) 89   D) 90 

10) If the sum of p terms of an AP is q and the sum of q terms is q, then the sum of the p + q terms will be..

A) 0    B) p-q   C) p+q   D) -(p-q)

11) If the sum of n terms of AP be n² - n and its common difference is 6, then its first term is..

A) 2    B) 3      C) 1     D) 4 

12) Sum of all two digit numbers which when divided by 4 yield Unity as reminder is..

A) 1200   B)1210.  C)1250.  D) n

13) In n AM's introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3:1, then the value of n is..

A) 6      B) 8      C) 4     D) n 

14) The 1st and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be.

A) 5     B) 6      C) 7         D) 8 

15) Find the sum of all odd integers from 1 to 1001.                      251001

16) If the ratio between the sums of n terms of two AP is (7n+1):(4n+27) find the ratio of their 11th term.   148: 111

17) If the sum of m terms of an AP be n and the sum of n terms be m, show that the sum of m+n terms is -(m+n).

18) If the sum of n terms of an AP is (pn+ qn²), where p and q are constants, find the common difference.                                   2q

19) In an AP, the first term is 2 and the sum of first five terms is one-fourth of the sum of next terms. Show that the 20th term is - 112 and the sum of first 20 term is -1100.

21) If the sum of n terms of an AP is given by (3n²+ 4n), find its rth term.                                        6r +1

22) The digits of a three-digit numbers are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.                                    852

23) Between 1 and 31, m numbers have been inserted in such a way that the ratio of 7th and (m-1)th numbers is 5:9. Find the value of m.                                 14

24) In the arithmetic progression whose common difference is non zero, the sum of the first 3n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2n terms the next to 2n terms is 

A) 1/5. B) 2/3  C) 3/4 D) none

25) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are:

A) 5,10,15,20     B) 4,10,16,22

C) 3,7,11,15       D) none

26) The first and the last term of an AP are a and l respectively. if S is the sum of all the terms of the AP. and the common difference is given by (l²-a²)/{k -(l+a)}, then k is

A) S   B) 2S      C) 3S    D) none

27) If the sum of the first n even natural number is equal to K times the sum of the first n odd natural numbers, then k is..

A) 1/n B) (n-1)/n  C)(n+1)/2n D)(n+1)/n  

28) If the first, second and last term of an AP are a,b and 2a respectively, then its sum is 

A) ab/{2(b-a)}       B) ab/(b-a)

C) 3ab/{2(b-a)}     D) none

29) If x is the sum of an arithmetic progression of n odd number of terms and y the sum of the terms of the series in odd places, then x/y is

A) 2n/(n+1)               B) n/(n+1)

C) (n+1)/2n               D) (n+1)/n 

30) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is

A) 3200 B) 1600 C) 200 D) 2800

31) The number of terms of the AP 3, 7, 11, 15, ... to be so that the sum is 406 is...

A) 5 B) 10 C) 12  D) 14   E) 20

32) If a(1/b+ 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP , then

A) a, b, c are in AP

B) 1/a, 1/b, 1/c are in AP

C) a, b, c are in HP

D) 1/a, 1/b, 1/c are in GP. 

33) If the sum of the three numbers in AP be 18 then what is the middle term ?                                       6

34) The fifth term and the 11th term of an AP are 41 and 20 respectively. Find the first term. What will be the sum of first 11 terms of the AP. ?          425/2

35) The n-th term of an AP is p. Show that sum of first (2n-1) terms is (2n-1)p.

36) The middle term, of an AP having 11th term is 12. Find the sum of the 11 terms of that progression.                               132

37) There are n arithmetic means between 4 and 31. If the second mean : last mean=5: 14 then find the value of n.                         8

38) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then show that the sum of the first P +Q terms is zero.

) Find the sum upto n terms of the series 1²- 2²+ 3²- 4²+ 5²- 6²+.. ..          -n/2 (n+1) (n= 2r)

39) if the sum of p terms of an AP is to the sum of q terms as p²:q², show that (pth term)/(qth term)= (2p-1)/(2q-1).

40) The first term of an AP is a, the second term is b and the last term is c. Show that the sum is {(a+c)(b+c-2a)}/{2(b-a)}.

41) The sides of a right angled triangle are in AP. if the smallest side is 5cm then find the largest side.                                  25/3

42) find the sum of natural numbers from 1 to 200 excluding those divisible by 5.                           16000 

43) Show that the sum of all odd numbers between 2 and 1000 which are divisible by 3 is 83667 and of those not divisible by 3 is 166332.

44) Find the 14 A. M which can be inserted between 5 and 8 and show that their sum is 14 times the Arithmetic mean between 5 and 8.

45) Divide 25/2 into five parts in AP, such that the first and the last parts are in the ratio 2: 3.     2,9/4,5/2, 11/3, 3.

46) For what value of m, the sequence 2(4m+7), 6m + 1/2, 12m-7 forms an AP.               -3/4 

47) Find the 20th term of the AP 80, 75, 70,... Calculate the number of terms required to make the sum equal to zero.                         35 

48) Prove that if unity is added to the sum of any number of terms of the AP 3, 5,7,9...the resulting sum is a perfect square.

49) The sum of n terms of the series 25, 22, 19, 16,.. is 116. Find the number of terms and the last term. The given series is AP.  18405

50) Find the sum of all natural numbers from 100 to 300:

a) which is divisible by 4.       10200

b) excluding those which are divisible by 4.                        30000

c) which are exactly divisible by 5. 

d) which are exactly divisible by 4 and 5.               8200, 2200

e) which are exactly divisible by 4 or 5.                                        16200

Profit and Loss(Basic)

A) Find Selling Price:

1) C. P= ₹78, Profit= ₹30.

2) C. P= ₹121, Profit= ₹13.

3) C. P= ₹1028, Profit=₹ 329

4) C. P=₹ 998.50, Profit= ₹37.25.

5) C. P= ₹1937, Profit=₹ 789.

6) C. P= ₹12000, Profit= ₹3000.

7) C. P= ₹178.30, Profit=₹ 52.13.

8) C. P= ₹2.35, Profit= ₹5.75.

9) C. P=₹ 1999.99, Profit= ₹22.34

10) C. P=₹ 841, Profit= ₹329.

11) C. P= ₹178, loss= ₹49.

12) C. P= ₹782, loss=₹ 93.

13) C. P= ₹849.60, loss= ₹90.10.

14) C. P= ₹1009.11, loss= ₹39.21.

15) C. P= ₹896.79, loss= ₹321.

16) C. P= ₹8920, loss= ₹651.

B) Find the Cost Price:

1) S.P= ₹79, loss= ₹9.

2) S.P= ₹892, loss= ₹19.

3) S.P= ₹892, loss= ₹19.

4) S.P= ₹9927, Profit= ₹342.

5) S.P= ₹21.32, loss= ₹3.96.

6) S.P= ₹873, loss= ₹415.

7) S.P= ₹3975, Profit= ₹653.

8) S.P= ₹4321, Profit= ₹19.

9) S.P= ₹653, loss= ₹213.50.

10) S.P= ₹5641, loss= ₹327.

C) Find the Profit or Loss, when

1) S.P= ₹55, C. P= ₹72.60

2) C.P= ₹55, S. P= 72.60

3) C.P= ₹490, S. P= 416.50

4) C.P= ₹4000, S. P= 4160.

5) C.P= ₹567.77 S. P= 526.50

6) C.P= ₹8769, S. P= 9887.

7) C.P= ₹490, S. P= 416.50

REVISED QUESTIONS PAPER (8)

11/8/21

1) The compound interest on a certain sum of money at 5% per annum for 2 years is ₹246. calculate the simple interest on the same sum for 3 years at 6% per annum.                                     432

2) what sum of money amount to ₹3630 in two years at 10% p.a compound interest.                3000

3) on a certain sum of money, the difference between the compound interest for a year, payable half-yearly, and the simple interest for a year is ₹180. Find the sum left out, if the rate of interest in both the cases is 10% p.a.                   72000

4) A man borrows ₹ 5000 at 12% compound interest p.a, interest payable every six months. He pays back ₹1800 at the end of every six months. calculate the third payment he had to make at the end of eight months in order to clear the entire loan.                      2024.60

5) Calculate the compound interest for the second year on ₹800 invested for 3 years at 10% p.a.          880

6) A man invests ₹5000 for 3 years at a certain rate of interest compounded annually. At the end of one year amounts ₹5600, calculate,
a)  the rate of interest per annum.
b) the interest accrued in the second year.
c) the amount at the end of the third year.             12%, 672, 7024.64

7) A man invests ₹46875 4% per annum compound interest for 3 years. Calculate
a)the interest for the first year.
b) The amount standing to his credit at the end of second year.
c) the interest for the third year.      1875, 50700, 2028

8) A person invests ₹5600 at 14% p.a. compound interest for two years. calculate:
a) the interest for the first year.
b) the amount at the end of 1st year.
c) The interest for the second year, correct to nearest rupees.        784, 6384, 894

9) the compound interest, calculated yearly, on a certain sum of money for the second year is 880 and for the third year ₹968. calculate the rate of interest and the sum of money.     10%, 8000

10) A certain sum of money amounts ₹5292 in two years and to ₹5556.60 in three years, interest being compounded annually. find the rate%.             5%

11) At what rate percent, per annum compound interest, would 80000 amounts to ₹ 88200 in 2 years; interest being compounded half yearly ?            5%

12) A sum of money is lent out at compound interest for 2 years at 20% p.a, C. I being reckoned  yearly. If the same sum of money was lent out at compound interest at the same rate per annum, C. I being reckoned half yearly. It would have fetched ₹482 more by the way of interest. calculate the sum of money lent out.             200000

1) Student A scores 20 marks in an examination out of 30 while another student B scores 40 marks out of 70. who has performed better ?     A

 

2) company A increases distance by 1 crore rupees while a company B increased its sales by 10 crore rupees. which company has grown more?                                        B

3) The population of a city grew from 20 lakh 22 lakh. Find the

a) percentage change.

b) percentage change based on the final value of the population.      10%, 9.09%

4) what is the percentage value of the ratio 53/81 ?        

5) what is the percentage value of the ratio 223/72.

6) A's salary increases by 20% and then decreases by 20%. what is the net percentage change in A's salary?                        4%

7) B's salary 25% more than A's salary. By what percentage is A's salary less B's salary?         20%

8) A sells his 30% cheaper than B and 30% dearer than C, By what percentage is the cost of C's goods cheaper than B's goods.    46.15%

9) The length and the breadth of a rectangular changed by +20% and by -10% respectively. What is the percentage change in the area of the rectangle.                          8%

10) Due to a 25% price hike in the price of rice, a person is able to purchase 20 kg less of rice for ₹400. Find the initial price.       ₹5

11) A's salary is 20% lower than B's salary, which is 15% lower than C's salary. By how much percent is C's salary more than A's salary?           

12) which of the following is the largest number ?

a) 20% 200 b) 7% of 500

c) 1300% of 3. d) 600% of 7

13) If 25% of a number is 75, then 45% of that number is...               135

14) what is 20% of 50% of 75% of 70 ?                                         5.25

15) If we express 41(3/17)% as a fraction, then it is equal to.      7/17 

16) Mr. Abhimanyu Banerjee is worried about the balance of his monthly budget. The price of petrol has increased by 40%. By what percent should he reduce the consumption of petrol so that he is able to balance his budget?    28.56

17) in above question Banerjee wanted to limit the increase in his expenditure to 5% on his basic expenditure on petrol then what should be the corresponding decrease in consumtion so that expenditure exceeds only by 5%?    25

18) Ram sells his goods 25% cheaper than Shyam and 25% dearer than bram. how much percentage in Bram's good cheaper than Shyam's?                          40%

19) In an election between two candidates Bhiku gets 65% of the total valid votes. If the total votes was 6000, what is the number of valid votes that the other candidate Mhatre gets if 25% of the total votes were declared invalid ?   1575

20) In a medical certificate, by mistake a candidate gave his height as 25% more than normal. In the interview panel, he clarified that his weight was 5 feet 5 inches. find the percentage correction made by the candidate from his stated height to his actual height to his actual height.                                          20

21) Arijit Sharma generally wears his is father father's coat. unfortunately, His cousin Sourya poked him one day that he was wearing a coat of length more than his height by 15%. If the length of Arijit's father's coat is 120cm then find the actual length of his coat. 104.34

22) A number is mistakenly divided by 5 instead of being multiplied by 5. find the percentage change in the result due to this mistake.     2400%

23) Harsh wanted to subtract 5 from the number. Unfortunately. He added 5 instead of subtracting. find the percentage change in the result. 33.33%

24) If 65% of x= 13% of y, then find the value of x if y= 2000.          400

25) 50 % of a% of b is 75% of b% of c. which of the following is ?

A) 1.5a B) 0.667a. C) 0.5a D) 1.25a E) 1.66a

26) A landowner increased the length and the breadth of a rectangular plot by 10% and 20% respectively. find the percentage change in the cost of the plot assuming land prices are uniform throughout his plot.            

27) The height of a triangle is increased by 40%. what can be the maximum percentage increase in length of the base so that increase in area is restricted to a maximum of 60%.                     14.28%

28) the salary of it is 30% more than that of Varun. find by what percentage is the salary of Varun less than that of an Amit.      23.07%