Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Prove that:
$open vertical bar table row cell straight a squared end cell bc cell ac plus straight c squared end cell row cell straight a squared plus ab end cell cell straight b squared end cell ac row ab cell straight b squared plus bc end cell cell straight c squared end cell end table close vertical bar space equals space 4 straight a squared straight b squared straight c squared.$

72 Views

92.

Prove that:
$open vertical bar table row straight a cell space space space straight b minus straight c end cell cell space space space straight c minus straight b end cell row cell straight a minus straight c end cell straight b cell space space space space straight c minus straight a end cell row cell straight a minus straight b end cell cell space space space straight b minus straight a end cell cell space space space straight c end cell end table close vertical bar space equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis$

73 Views

93.

Using properties of determinats, prove the following:
$open vertical bar table row straight a cell space straight b end cell cell space space straight c end cell row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals space straight a cubed plus straight b cubed plus straight c cubed minus 3 abc.$

74 Views

94.

Prove that
$open vertical bar table row cell 1 plus straight a end cell 1 1 row 1 cell 1 plus straight b end cell 1 row 1 1 cell 1 plus straight c end cell end table close vertical bar space equals space straight a space straight b space straight c space open parentheses 1 plus 1 over straight a plus 1 over straight b plus 1 over straight c close parentheses space space or space space straight a space straight b plus straight b space straight c plus straight c space straight a plus straight a space straight b space straight c$

106 Views

95.

Show that:
$increment space equals space open vertical bar table row cell left parenthesis straight y plus straight z right parenthesis squared end cell cell straight x space straight y end cell cell straight z space straight x end cell row cell straight x space straight y end cell cell left parenthesis straight x plus straight z right parenthesis squared end cell cell straight y space straight z end cell row cell straight x space straight z end cell cell straight y space straight z end cell cell left parenthesis straight x plus straight y right parenthesis squared end cell end table close vertical bar space equals space 2 xyz space left parenthesis straight x plus straight y plus straight z right parenthesis squared$

78 Views

96.
If a, b and c are real numbers, and
$increment space equals space open vertical bar table row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight c plus straight a end cell cell space straight a plus straight b end cell cell space straight b plus straight c end cell row cell straight a plus straight b end cell cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 0 comma$ show that either $straight a plus straight b plus straight c space equals space 0 space or space space straight a space equals space straight b space equals space straight c.$

increment space equals space open vertical bar table row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight c plus straight a end cell cell straight a plus straight b end cell cell straight b plus straight c end cell row cell straight a plus straight b end cell cell straight b plus straight c end cell cell straight c plus straight a end cell end table close vertical bar space equals space open vertical bar table row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight a plus straight b end cell cell straight b plus straight c end cell row cell 2 left parenthesis straight a plus straight b plus straight c right parenthesis end cell cell straight b plus straight c end cell cell straight c plus straight a end cell end table close vertical bar comma space by space straight C subscript 1 space rightwards arrow space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3

equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell space straight c plus straight a end cell cell space space straight a plus straight b end cell row 1 cell space straight a plus straight b end cell cell space space straight b plus straight c end cell row 1 cell space straight b plus straight c end cell cell space space straight c plus straight a end cell end table close vertical bar equals 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell space straight c plus straight a end cell cell space straight a plus straight b end cell row 1 cell space straight a plus straight b end cell cell space straight b plus straight c end cell row 1 cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar comma space by space straight R subscript 2 space rightwards arrow space straight R subscript 2 space minus straight R subscript 1 comma space straight R subscript 3 space rightwards arrow space straight R subscript 3 space minus straight R subscript 1 equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space open vertical bar table row cell straight b minus straight c end cell cell space space straight c minus straight a end cell row cell straight b minus straight a end cell cell space space straight c minus straight b end cell end table close vertical bar equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space open square brackets left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight b right parenthesis space minus space left parenthesis straight b minus straight a right parenthesis space left parenthesis straight c minus straight a right parenthesis close square brackets equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis space open square brackets negative straight b squared minus straight c squared plus 2 bc minus bc plus ab plus ca minus straight a squared close square brackets equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis ab plus bc plus ca minus straight a squared minus straight b squared minus straight c squared right parenthesis equals space minus left parenthesis straight a plus straight b plus straight c right parenthesis thin space open square brackets 2 straight a squared plus 2 straight b squared plus 2 straight c squared minus 2 ab minus 2 bc minus 2 ca close square brackets equals negative left parenthesis straight a plus straight b plus straight c right parenthesis space left square bracket left parenthesis straight a squared plus straight b squared minus 2 ab right parenthesis plus left parenthesis straight b squared plus straight c squared minus 2 bc right parenthesis plus left parenthesis straight c squared plus straight a squared minus 2 ca right parenthesis right square bracket equals negative left parenthesis straight a plus straight b plus straight c right parenthesis space left square bracket left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared right square bracket Now space increment space equals space 0 space give space us space space space space space space minus left parenthesis straight a plus straight b plus straight c right parenthesis thin space left square bracket left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared space equals space 0 rightwards double arrow space space space space space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left square bracket left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared right square bracket space equals space 0 therefore space space space space space space space space space space space space space either space space straight a plus straight b plus straight c space equals space 0 or space space space space space space space space left parenthesis straight a minus straight b right parenthesis squared plus left parenthesis straight b minus straight c right parenthesis squared plus left parenthesis straight c minus straight a right parenthesis squared rightwards double arrow space space space space space space space space space space space space space left parenthesis straight a minus straight b right parenthesis squared space equals space 0 comma space space space space space left parenthesis straight b minus straight c right parenthesis squared space equals space 0 comma space space space left parenthesis straight c minus straight a right parenthesis squared space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space straight a equals straight b comma space space straight b equals space straight c comma space space straight c equals straight a rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space straight a equals straight b equals straight c therefore space space space space space increment space equals space 0 space space space space rightwards double arrow space space either space straight a plus straight b plus straight c space equals space 0 space or space space space straight a equals straight b equals straight c. space space space space space space space space space space

71 Views

Short Answer Type

97.

Using properties of determinants, prove that
$open vertical bar table row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight c plus straight a end cell cell space straight a plus straight b end cell cell space straight b plus straight c end cell row cell straight a plus straight b end cell cell space straight b plus straight c end cell cell space straight c plus straight a end cell end table close vertical bar space equals space 2 left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis ab plus bc plus ca minus straight a squared minus straight b squared minus straight c squared right parenthesis$

72 Views

Long Answer Type

98.

Using properties of determinants, show that:
$open vertical bar table row cell straight b squared straight c end cell bc cell straight b plus straight c end cell row cell straight c squared straight a squared end cell ca cell straight c plus straight a end cell row cell straight a squared straight b squared end cell ab cell straight a plus straight b end cell end table close vertical bar space equals space 0$

77 Views

99.

Prove that:
$open vertical bar table row cell straight a plus straight b end cell cell space space straight b plus straight c end cell cell space space space straight c plus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell row cell straight c plus straight a end cell cell space space straight a plus straight b end cell cell space space space straight b plus straight c end cell end table close vertical bar space equals space 2 space open vertical bar table row straight a cell space space straight b end cell cell space straight c end cell row straight b cell space space straight c end cell cell space straight a end cell row straight c cell space straight a end cell cell space straight b end cell end table close vertical bar.$