In this question paper, it covers the important one question from the chapter

Matrices and Determinants
Algebra
Analytical Geometry
Trigonometry
Differential Calculus

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Questions:

Answer ALL Questions

The value of x if \(\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0\) is_________.

(a)

0, - 1

(b)

0, 1

(c)

- 1, 1

(d)

- 1, - 1

adj (AB) is equal to ________.

(a)

adj A adj B

(b)

adj A^T adj B^T

(c)

adj B adj A

(d)

adj B^T adj A^T

If A = \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)such that ad - bc \(\neq\) 0 then A^-1 is ________.

(a)

\({{1}\over{ad-bc}}\begin{pmatrix} d & b \\-c & a\end{pmatrix}\)

(b)

\({{1}\over{ad-bc}}\begin{pmatrix} d & b \\c & a\end{pmatrix}\)

(c)

\({{1}\over{ad-bc}}\begin{pmatrix} d & -b \\-c & a\end{pmatrix}\)

(d)

\({{1}\over{ad-bc}}\begin{pmatrix} d & -b \\c & a\end{pmatrix}\)

The inverse matrix of \(\begin{pmatrix} 3 & 1 \\ 5 & 2\end{pmatrix}\) is ________.

(a)

\(\begin{pmatrix} 2 & -1 \\-5 & 3 \end{pmatrix}\)

(b)

\(\begin{pmatrix} -2 & 5 \\1 & -3 \end{pmatrix}\)

(c)

\(\begin{pmatrix} 3 & -1 \\-5 & -3 \end{pmatrix}\)

(d)

\(\begin{pmatrix} -3 & 5 \\1 & -2 \end{pmatrix}\)

If A and B are non-singular matrix then, which of the following is incorrect?

(a)

A² = I implies A^-1= A

(b)

I^-1 = I

(c)

If AX = B, then X = B^-1 A

(d)

If A is square matrix of order 3 then |adj A|= |A|²

The value of \(\begin{vmatrix} x & x^2 & -yz & 1 \\ y & y^2 & -zx & 1 \\ z & z^2 & -xy &1 \end{vmatrix}\) is ________.

(a)

(b)

(c)

-1

(d)

-xyz

If A = \(\begin{vmatrix}cos \theta & sin \theta \\ -sin \theta&cons\theta \end{vmatrix},\) then |2A| is equal to ________.

(a)

4 cos 2 \(\theta\)

(b)

(c)

(d)

If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and A_ij is cofactor of a_ij, then value of \(\triangle\) is given by ________.

(a)

a₁₁ A₃₁ + a₁₂ A₃₂ + a₁₃ A₃₃

(b)

a₁₁ A₁₁ + a₁₂A₂₁ + a₁₃ A₃₁

(c)

a₂₁ A₁₁ + a₂₂ A₁₂ + a₂₃ A₁₃

(d)

a₁₁ A₁₁ + a₂₁ A₂₁ + a₃₁ A₃₁

If \(\begin{vmatrix} 4 & 3 \\ 3 & 1 \end{vmatrix}=-5\) then value of \(\begin{vmatrix} 20 & 15 \\ 15 & 5 \end{vmatrix}\) is ________.

(a)

-5

(b)

-125

(c)

-25

(d)

10.

If nC₃ = nC₂, then the value of nC₄ is _______.

(a)

(b)

(c)

(d)

11.

The number of ways selecting 4 players out of 5 is _______.

(a)

(b)

(c)

(d)

12.

If nP_r = 720 (nC_r), then r is equal to ______.

(a)

(b)

(c)

(d)

13.

The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for n \(\in\) N is ________.

(a)

(b)

(c)

(d)

14.

If n is a positive integer, then the number of terms in the expansion (x + a)ⁿ is _______.

(a)

(b)

n + 1

(c)

n-1

(d)

15.

The middle term in the expansion of \({ \left( x+\frac { 1 }{ x } \right) }^{ 10 }\) is _______.

(a)

10C₄\(\left( \frac { 1 }{ x } \right) \)

(b)

10C₅

(c)

10C₆

(d)

10C₇x⁴

16.

If \(\frac { kx }{ (x+4)(2x-1) } =\frac { 4 }{ x+4 } +\frac { 1 }{ 2x-1 } \) then k is equal to _______.

(a)

(b)

(c)

(d)

17.

The number of 3 letter words that can be formed from the letters of the word number when the repetition is allowed are ________.

(a)

206

(b)

133

(c)

216

(d)

300

18.

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is _________.

(a)

(b)

(c)

(d)

19.

The value of (5C₀ + 5C₁) + (5C₁ + 5C₂) + (5C₂ + 5C₃) + (5C₃ + 5C₄) + (5C₄ + 5C₅ ) is ________.

(a)

2⁶-2

(b)

2⁵-1

(c)

2⁸

(d)

2⁷

20.

Sum of Binomial co-efficient in a particular expansion is 256, then number of terms in the expansion is ________.

(a)

(b)

(c)

(d)

21.

Sum of the binomial coefficients is ________.

(a)

2ⁿ

(b)

n²

(c)

(d)

n + 17

22.

If the lines 2x - 3y - 5 = 0 and 3x - 4y - 7 = 0 are the diameters of a circle, then its centre is _______.

(a)

(-1, 1)

(b)

(1,1)

(c)

(1, -1 )

(d)

(-1, -1)

23.

The locus of the point P which moves such that P is at equidistance from their coordinate axes is _______.

(a)

\(y={1\over x}\)

(b)

y = -x

(c)

y = x

(d)

\(y=-{1\over x}\)

24.

The locus of the point P which moves such that P is always at equidistance from the line x + 2y+ 7 = 0 is _______.

(a)

x+2y+2 = 0

(b)

x - 2y + 1 = 0

(c)

2x - y + 2 = 0

(d)

3x + y + 1 = 0

25.

Length of the latus rectum of the parabola y² = - 25x is _______.

(a)

(b)

-5

(c)

(d)

-25

26.

If the centre of the circle is (-a, -b) and radius is \(\sqrt{a^2-b^2}\), then the equation of circle is _______.

(a)

x²+y²+2ax+2by+2b²= 0

(b)

x²+y²+2ax +2by-2b²= 0

(c)

x² +y² - 2ax - 2by - 2b² = 0

(d)

x² + y - 2ax - 2by + 2b² = 0

27.

Combined equation of co-ordinate axes is _______.

(a)

x²-y²= 0

(b)

x²+y²= 0

(c)

xy = c

(d)

xy = 0

28.

If the circle touches x axis, y axis and the line x = 6 then the length of the diameter of the circle is _______.

(a)

(b)

(c)

(d)

29.

The distance between directrix and focus of a parabola y² = 4ax is _______.

(a)

(b)

(c)

(d)

30.

The radian measure of 37^o30^' is ______.

(a)

\(\frac{5\pi}{24}\)

(b)

\(\frac{3\pi}{24}\)

(c)

\(\frac{7\pi}{24}\)

(d)

\(\frac{9\pi}{24}\)

31.

If \(\tan\theta=\frac{1}{\sqrt5}\) and \(\theta\) lies in the first quadrant, then \(\cos\theta\) is _______.

(a)

\(\frac{1}{\sqrt6}\)

(b)

\(\frac{-1}{\sqrt6}\)

(c)

\(\frac{\sqrt5}{\sqrt6}\)

(d)

\(\frac{-\sqrt5}{\sqrt6}\)

32.

The value of \(\sin(-420^o)\) is _______.

(a)

\(\frac{\sqrt3}{2}\)

(b)

\(-\frac{\sqrt3}{2}\)

(c)

\(\frac{1}{2}\)

(d)

\(\frac{-1}{2}\)

33.

The value of \(\sin 28^o\cos 17^o+\cos 28^o\sin 17^o\) is _______.

(a)

\(\frac{1}{\sqrt2}\)

(b)

(c)

\(\frac{-1}{\sqrt2}\)

(d)

34.

The value of sec A sin(270^o + A) is ______.

(a)

-1

(b)

cos² A

(c)

sec² A

(d)

35.

The value of 1 - 2sin²45^o is _______.

(a)

(b)

\(\frac{1}{2}\)

(c)

\(\frac14\)

(d)

36.

The value of \(\frac{2\tan30^o}{1+tan^230}\) is _____.

(a)

\(\frac12\)

(b)

\(\frac{1}{\sqrt3}\)

(c)

\(\frac{\sqrt{3}}{2}\)

(d)

\(\sqrt3\)

37.

If \(\sin A=\frac{1}{2}\) then \(4\cos^3A-3\cos A\) is ________.

(a)

(b)

(c)

\(\frac{\sqrt3}{2}\)

(d)

\(\frac{1}{\sqrt{2}}\)

38.

The value of \(\frac{3 \tan 10^{\circ}-\tan ^3 10^{\circ}}{1-3 \tan ^2 10^{\circ}}\) is _______,

(a)

\(\frac{1}{\sqrt3}\)

(b)

\(\frac{1}{2}\)

(c)

\(\frac{\sqrt3}2\)

(d)

\(\frac{1}{\sqrt2}\)

39.

If \(\alpha\) and \(\beta\) be between 0 and \(\frac{\pi}{2}\) and if \(\cos(\alpha+\beta)=\frac{12}{13}\) and \(\sin(\alpha-\beta)=\frac{3}{5}\) then \(\sin2\alpha\) is _____.

(a)

\(\frac{16}{15}\)

(b)

(c)

\(\frac{56}{65}\)

(d)

\(\frac{64}{65}\)

40.

The value of \(\frac{1}{cosec(-45^o)}\) is _______.

(a)

\(\frac{-1}{\sqrt2}\)

(b)

\(\frac{1}{\sqrt2}\)

(c)

\(\sqrt2\)

(d)

\(-\sqrt2\)

41.

If f(x) = x² - x + 1, then f (x + 1) is _______.

(a)

x²

(b)

(c)

(d)

x² + x + 1

42.

If \(f\left( x \right) =\begin{cases} { x }^{ 2 }-4x\quad ifx\ge 2 \\ x+2\quad ifx<2 \end{cases}\) then f(0) is _______.

(a)

(b)

(c)

-1

(d)

43.

The graph of the line y = 3 is _______.

(a)

Parallel to x-axis

(b)

Parallel to y-axis

(c)

Passing through the origin

(d)

Perpendicular to x-axis

44.

The graph of y = 2x² is passing through _______.

(a)

(0,0)

(b)

(2,1)

(c)

(2,0)

(d)

(0,2)

45.

Which one of the following functions has the property f (x) = \(f\left( \frac { 1 }{ x } \right) \), provided \(x \neq 0\)

(a)

\(f\left( x \right) =\frac { { x }^{ 2 }-1 }{ x } \)

(b)

\(f\left( x \right) =\frac { 1-{ x }^{ 2 } }{ x } \)

(c)

f(x) = x

(d)

\(f\left( x \right) =\frac { { x }^{ 2 }+1 }{ x } \)

46.

The range of f(x) = |x|, for all \(x\in R\), is ________.

(a)

(0, \(\infty \))

(b)

(0, \(\infty \))

(c)

(-\(\infty \), \(\infty \))

(d)

(1, \(\infty \))

47.

The graph of f(x) = e^x is identical to that of ________.

(a)

f(x) = a^x, a > 1

(b)

f(x) = a^x, a < 1

(c)

f(x) = a^x, 0 < a < 1

(d)

y = ax +b, a \(\ne\) 0

48.

For what value of x, f(x) = \(\frac{x+2}{x-1}\) is not continuous?

(a)

-2

(b)

(c)

(d)

-1

49.

\(\frac{d}{dx}(\frac{1}{x})\) is equal to ________.

(a)

\(-\frac{1}{x^2}\)

(b)

\(-\frac{1}{x}\)

(c)

log x

(d)

\(\frac{1}{x^2}\)

50.

If y = log x then y₂ = ________.

(a)

\(\frac{1}{x}\)

(b)

\(-\frac{1}{x^2}\)

(c)

\(-\frac{2}{x^2}\)

(d)

e²

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Answers Key:

Answer ALL Questions

\(\left|\begin{array}{lll} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{array}\right|=0 \Rightarrow-1\left[x^2-x\right]=0\)
\(\Rightarrow x(x-1)=0 \Rightarrow x=0,1\)

\(|A|=\frac{1}{a d-b c} ; \quad A^{-1}=\frac{1}{a d-b c}\left(\begin{array}{cc} d & -b \\ -c & a \end{array}\right)\)

\(|A|=6-5=1\)
\(A^{-1}=\left(\begin{array}{cc} 2 & -1 \\ -5 & 3 \end{array}\right)\)

(Since \(\mathrm{X}=\mathrm{A}^{-1} \mathrm{~B}\) is the correct answer)

\(\left|\begin{array}{ccc} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{array}\right|+\left|\begin{array}{ccc} x & -y z & 1 \\ y & -z x & 1 \\ z & -x y & 1 \end{array}\right|\)
Multiply R₁, R₂, R₃, of second determinant by x, y, z respectively
\(\left|\begin{array}{ccc} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{array}\right|-\frac{1}{x y z}\left|\begin{array}{ccc} x^2 & x y z & x \\ y^2 & x y z & y \\ z^2 & x y z & z \end{array}\right|\)
\(=\left|\begin{array}{lll} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{array}\right|-\frac{x y z}{x y z}\left|\begin{array}{ccc} x^2 & 1 & x \\ y^2 & 1 & y \\ z^2 & 1 & z \end{array}\right|\)
\(=\left|\begin{array}{lll} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{array}\right|-\left|\begin{array}{lll} x & x^2 & 1 \\ y & y^2 & 1 \\ z & z^2 & 1 \end{array}\right|=0\)

\(|A|=\cos ^2 \theta+\sin ^2 \theta=1\)
\(|2 A|=2^2(1)=4\)

(Corresponding co-factor)

\(\left|\begin{array}{cc} 20 & 15 \\ 15 & 5 \end{array}\right|=5 \times 5\left|\begin{array}{ll} 4 & 3 \\ 3 & 1 \end{array}\right|\)
\(=5 \times 5 \times-5\)
\(=-125\)

10.

x + y = n
3 + 2 = 5 = n
nC₄ = 5C₄ = 5C₁ = 5

11.

No. of ways = 5C₄ = 5C₁ = 5

12.

\(n P_r =720(\mathrm{nCr}) \)
\(\frac{n !}{(n-r) !} =720\left(\frac{n !}{r !(n-r) !}\right) \)
\(r ! =720 \)
\(r ! =6 ! \)
r = 6

13.

Since if n = 1 then (1) (2) (3) (4) = 24 is divisible by = 24

14.

(b)
n + 1

15.

\(\text { M.T } =t_{\frac{n}{2}}+1=t_{5+1}=t_6 \quad r=5 \)
\(t_6 =10 C_5\left(x^5\right)\left(\frac{1}{x^5}\right)=10 C_5 \)

16.

Equality coefficient of x in the numerator
kx = 4 (2n) + 1 (x)
kx = 9x

17.

The number of ways = 6 x 6 x 6 = 216

18.

The number of parallelograms = 4C₂ x 3C₂

19.

Since sum of all binomial coefficients is 2n
5C₀ + 5C₁ + 5C₂ + 5C₃ + 5C₄ + 5C₅+ (5C₁ + 5C₂ + 5C₃ + 5C₄)
= 2⁵ + (2⁵ - (5C₀ + 5C₅)
= 32 + 32 - (1 + 1)
= 64 - 2 = 2⁶ - 2

20.

2ⁿ = 256 = 2⁸
n = 8
Number of terms = n + 1 = 9

21.

(a)
2ⁿ

22.

23.

24.

(parallel line)

25.

4a = 5

26.

\((x+a)^2+(y+b)^2=a^2-b^2\)
\(x^2+a^2+2 a x+y^2+b^2+2 b y=a^2-b^2\)

27.

Equation of x axis y = 0, equation of y axis x = 0
\(\therefore\) combined equation xy = 0

28.

29.

(b)
2a

30.

\(37^{\circ} 30^{\prime}=37 \frac{1}{2}^{\circ}=\frac{75}{2} \times \frac{\pi}{180}=\frac{5 \pi}{24}\)

31.

\(\sec \theta =\sqrt{1+\tan ^2 \theta}=\sqrt{1+\frac{1}{5}}=\sqrt{\frac{6}{5}} \)

32.

\(\sin \left(-420^{\circ}\right) =-\sin 420^{\circ}=-\sin (360+60) =-\sin 60^{\circ}=\frac{-\sqrt{3}}{2} \)

33.

\(\sin (28+17)=\sin 45^{\circ}=\frac{1}{\sqrt{2}}\)

34.

\(\sec A(-\cos A)=\frac{1}{\cos A}(-\cos A)=-1\)

35.

1 - 2sin²45^o= cos 2(45^o)
= cos 90^o = 0

36.

\(\frac{2 \tan 30^{\circ}}{1+\tan ^2 30^{\circ}} =\sin 2\left(30^{\circ}\right)=\sin 60^{\circ}=\frac{\sqrt{3}}{2} \)

37.

\(\sin A=\frac{1}{2} \Rightarrow A=30^{\circ} \)
\(4 \cos ^3 A-3 \cos \mathrm{A} =\cos 3 \mathrm{~A} \)
\(=\cos 3(30) \)
\(=\cos 90^{\circ} =0\)

38.

\(\frac{3 \tan 10^{\circ}-\tan ^3 10^{\circ}}{1-3 \tan ^2 10^{\circ}} =\tan 3(10) =\tan 30^{\circ}=\frac{1}{\sqrt{3}} \)

39.

\(\cos (\alpha+\beta)=\frac{12}{13} \Rightarrow \sin (\alpha+\beta)=\frac{5}{13} \)
\(\sin (\alpha-\beta)=3 / 5 \Rightarrow \cos (\alpha-\beta)=4 / 5 \)
\(\sin 2 \alpha=\sin ((\alpha+\beta)+(\alpha-\beta)) \)
\(= \sin (\alpha+\beta) \cos (\alpha-\beta) +\cos (\alpha+\beta) \sin (\alpha-\beta) \)
\(= \frac{5}{13} \cdot \frac{4}{5}+\frac{3}{5} \cdot \frac{12}{13}=\frac{56}{65}\)

40.

\(\frac{-1}{\operatorname{cosec} 45^{\circ}}=\frac{-1}{\sqrt{2}}\)

41.

\(f(x+1) =(x+1)^2-(x+1)+1 \)
\(=x^2+2 x+1-x-1+1 \)
\(=x^2+x+1 \)

42.

f(0) = 0 + 2 = 2

43.

(a)
Parallel to x-axis

44.

(a)
(0,0)

45.

\(f(x)=\frac{x^2+1}{x} \)
\(f\left(\frac{1}{x}\right)=\frac{\frac{1}{x^2}+1}{\frac{1}{x}}=\frac{1+x^2}{x^2} \times \frac{x}{1}=\frac{1+x^2}{x}=f(x) \)

46.